WORST_CASE(Omega(n^1), O(n^3)) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^3). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 209 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), 1340 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (14) CpxRNTS (15) InliningProof [UPPER BOUND(ID), 381 ms] (16) CpxRNTS (17) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 1 ms] (20) CpxRNTS (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 51 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 3 ms] (26) CpxRNTS (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 332 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 166 ms] (32) CpxRNTS (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 67 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 41 ms] (38) CpxRNTS (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 125 ms] (42) CpxRNTS (43) IntTrsBoundProof [UPPER BOUND(ID), 2 ms] (44) CpxRNTS (45) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (46) CpxRNTS (47) IntTrsBoundProof [UPPER BOUND(ID), 69 ms] (48) CpxRNTS (49) IntTrsBoundProof [UPPER BOUND(ID), 32 ms] (50) CpxRNTS (51) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (52) CpxRNTS (53) IntTrsBoundProof [UPPER BOUND(ID), 1530 ms] (54) CpxRNTS (55) IntTrsBoundProof [UPPER BOUND(ID), 734 ms] (56) CpxRNTS (57) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (58) CpxRNTS (59) IntTrsBoundProof [UPPER BOUND(ID), 903 ms] (60) CpxRNTS (61) IntTrsBoundProof [UPPER BOUND(ID), 560 ms] (62) CpxRNTS (63) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (64) CpxRNTS (65) IntTrsBoundProof [UPPER BOUND(ID), 539 ms] (66) CpxRNTS (67) IntTrsBoundProof [UPPER BOUND(ID), 129 ms] (68) CpxRNTS (69) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (70) CpxRNTS (71) IntTrsBoundProof [UPPER BOUND(ID), 228 ms] (72) CpxRNTS (73) IntTrsBoundProof [UPPER BOUND(ID), 2 ms] (74) CpxRNTS (75) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (76) CpxRNTS (77) IntTrsBoundProof [UPPER BOUND(ID), 96 ms] (78) CpxRNTS (79) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] (80) CpxRNTS (81) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (82) CpxRNTS (83) IntTrsBoundProof [UPPER BOUND(ID), 157 ms] (84) CpxRNTS (85) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] (86) CpxRNTS (87) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (88) CpxRNTS (89) IntTrsBoundProof [UPPER BOUND(ID), 187 ms] (90) CpxRNTS (91) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] (92) CpxRNTS (93) FinalProof [FINISHED, 0 ms] (94) BOUNDS(1, n^3) (95) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CpxRelTRS (97) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (98) typed CpxTrs (99) OrderProof [LOWER BOUND(ID), 0 ms] (100) typed CpxTrs (101) RewriteLemmaProof [LOWER BOUND(ID), 232 ms] (102) BEST (103) proven lower bound (104) LowerBoundPropagationProof [FINISHED, 0 ms] (105) BOUNDS(n^1, INF) (106) typed CpxTrs (107) RewriteLemmaProof [LOWER BOUND(ID), 142 ms] (108) 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: p(0) -> 0 p(s(x)) -> x le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) minus(x, y) -> if(le(x, y), x, y) if(true, x, y) -> 0 if(false, x, y) -> s(minus(p(x), 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(s(x_1)) -> s(encArg(x_1)) encArg(true) -> true encArg(false) -> false encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_p(x_1) -> p(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) 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: p(0) -> 0 p(s(x)) -> x le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) minus(x, y) -> if(le(x, y), x, y) if(true, x, y) -> 0 if(false, x, y) -> s(minus(p(x), y)) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(true) -> true encArg(false) -> false encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_p(x_1) -> p(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) 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: p(0) -> 0 p(s(x)) -> x le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) minus(x, y) -> if(le(x, y), x, y) if(true, x, y) -> 0 if(false, x, y) -> s(minus(p(x), y)) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(true) -> true encArg(false) -> false encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_p(x_1) -> p(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) 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: p(0) -> 0 [1] p(s(x)) -> x [1] le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [1] minus(x, y) -> if(le(x, y), x, y) [1] if(true, x, y) -> 0 [1] if(false, x, y) -> s(minus(p(x), y)) [1] encArg(0) -> 0 [0] encArg(s(x_1)) -> s(encArg(x_1)) [0] encArg(true) -> true [0] encArg(false) -> false [0] encArg(cons_p(x_1)) -> p(encArg(x_1)) [0] encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) [0] encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) [0] encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encode_p(x_1) -> p(encArg(x_1)) [0] encode_0 -> 0 [0] encode_s(x_1) -> s(encArg(x_1)) [0] encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) [0] encode_true -> true [0] encode_false -> false [0] encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) [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: p(0) -> 0 [1] p(s(x)) -> x [1] le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [1] minus(x, y) -> if(le(x, y), x, y) [1] if(true, x, y) -> 0 [1] if(false, x, y) -> s(minus(p(x), y)) [1] encArg(0) -> 0 [0] encArg(s(x_1)) -> s(encArg(x_1)) [0] encArg(true) -> true [0] encArg(false) -> false [0] encArg(cons_p(x_1)) -> p(encArg(x_1)) [0] encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) [0] encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) [0] encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encode_p(x_1) -> p(encArg(x_1)) [0] encode_0 -> 0 [0] encode_s(x_1) -> s(encArg(x_1)) [0] encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) [0] encode_true -> true [0] encode_false -> false [0] encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) [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: p :: 0:s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0:s:true:false:cons_p:cons_le:cons_minus:cons_if 0 :: 0:s:true:false:cons_p:cons_le:cons_minus:cons_if s :: 0:s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0:s:true:false:cons_p:cons_le:cons_minus:cons_if le :: 0:s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0:s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0:s:true:false:cons_p:cons_le:cons_minus:cons_if true :: 0:s:true:false:cons_p:cons_le:cons_minus:cons_if false :: 0:s:true:false:cons_p:cons_le:cons_minus:cons_if minus :: 0:s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0:s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0:s:true:false:cons_p:cons_le:cons_minus:cons_if if :: 0:s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0:s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0:s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0:s:true:false:cons_p:cons_le:cons_minus:cons_if encArg :: 0:s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0:s:true:false:cons_p:cons_le:cons_minus:cons_if cons_p :: 0:s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0:s:true:false:cons_p:cons_le:cons_minus:cons_if cons_le :: 0:s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0:s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0:s:true:false:cons_p:cons_le:cons_minus:cons_if cons_minus :: 0:s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0:s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0:s:true:false:cons_p:cons_le:cons_minus:cons_if cons_if :: 0:s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0:s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0:s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0:s:true:false:cons_p:cons_le:cons_minus:cons_if encode_p :: 0:s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0:s:true:false:cons_p:cons_le:cons_minus:cons_if encode_0 :: 0:s:true:false:cons_p:cons_le:cons_minus:cons_if encode_s :: 0:s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0:s:true:false:cons_p:cons_le:cons_minus:cons_if encode_le :: 0:s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0:s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0:s:true:false:cons_p:cons_le:cons_minus:cons_if encode_true :: 0:s:true:false:cons_p:cons_le:cons_minus:cons_if encode_false :: 0:s:true:false:cons_p:cons_le:cons_minus:cons_if encode_minus :: 0:s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0:s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0:s:true:false:cons_p:cons_le:cons_minus:cons_if encode_if :: 0:s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0:s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0:s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0:s:true:false:cons_p:cons_le:cons_minus: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: p_1 le_2 if_3 minus_2 encArg_1 encode_p_1 encode_0 encode_s_1 encode_le_2 encode_true encode_false encode_minus_2 encode_if_3 Due to the following rules being added: encArg(v0) -> null_encArg [0] encode_p(v0) -> null_encode_p [0] encode_0 -> null_encode_0 [0] encode_s(v0) -> null_encode_s [0] encode_le(v0, v1) -> null_encode_le [0] encode_true -> null_encode_true [0] encode_false -> null_encode_false [0] encode_minus(v0, v1) -> null_encode_minus [0] encode_if(v0, v1, v2) -> null_encode_if [0] p(v0) -> null_p [0] le(v0, v1) -> null_le [0] if(v0, v1, v2) -> null_if [0] And the following fresh constants: null_encArg, null_encode_p, null_encode_0, null_encode_s, null_encode_le, null_encode_true, null_encode_false, null_encode_minus, null_encode_if, null_p, null_le, 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: p(0) -> 0 [1] p(s(x)) -> x [1] le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [1] minus(x, y) -> if(le(x, y), x, y) [1] if(true, x, y) -> 0 [1] if(false, x, y) -> s(minus(p(x), y)) [1] encArg(0) -> 0 [0] encArg(s(x_1)) -> s(encArg(x_1)) [0] encArg(true) -> true [0] encArg(false) -> false [0] encArg(cons_p(x_1)) -> p(encArg(x_1)) [0] encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) [0] encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) [0] encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encode_p(x_1) -> p(encArg(x_1)) [0] encode_0 -> 0 [0] encode_s(x_1) -> s(encArg(x_1)) [0] encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) [0] encode_true -> true [0] encode_false -> false [0] encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) [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_p(v0) -> null_encode_p [0] encode_0 -> null_encode_0 [0] encode_s(v0) -> null_encode_s [0] encode_le(v0, v1) -> null_encode_le [0] encode_true -> null_encode_true [0] encode_false -> null_encode_false [0] encode_minus(v0, v1) -> null_encode_minus [0] encode_if(v0, v1, v2) -> null_encode_if [0] p(v0) -> null_p [0] le(v0, v1) -> null_le [0] if(v0, v1, v2) -> null_if [0] The TRS has the following type information: p :: 0:s:true:false:cons_p:cons_le:cons_minus:cons_if:null_encArg:null_encode_p:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_minus:null_encode_if:null_p:null_le:null_if -> 0:s:true:false:cons_p:cons_le:cons_minus:cons_if:null_encArg:null_encode_p:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_minus:null_encode_if:null_p:null_le:null_if 0 :: 0:s:true:false:cons_p:cons_le:cons_minus:cons_if:null_encArg:null_encode_p:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_minus:null_encode_if:null_p:null_le:null_if s :: 0:s:true:false:cons_p:cons_le:cons_minus:cons_if:null_encArg:null_encode_p:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_minus:null_encode_if:null_p:null_le:null_if -> 0:s:true:false:cons_p:cons_le:cons_minus:cons_if:null_encArg:null_encode_p:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_minus:null_encode_if:null_p:null_le:null_if le :: 0:s:true:false:cons_p:cons_le:cons_minus:cons_if:null_encArg:null_encode_p:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_minus:null_encode_if:null_p:null_le:null_if -> 0:s:true:false:cons_p:cons_le:cons_minus:cons_if:null_encArg:null_encode_p:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_minus:null_encode_if:null_p:null_le:null_if -> 0:s:true:false:cons_p:cons_le:cons_minus:cons_if:null_encArg:null_encode_p:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_minus:null_encode_if:null_p:null_le:null_if true :: 0:s:true:false:cons_p:cons_le:cons_minus:cons_if:null_encArg:null_encode_p:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_minus:null_encode_if:null_p:null_le:null_if false :: 0:s:true:false:cons_p:cons_le:cons_minus:cons_if:null_encArg:null_encode_p:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_minus:null_encode_if:null_p:null_le:null_if minus :: 0:s:true:false:cons_p:cons_le:cons_minus:cons_if:null_encArg:null_encode_p:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_minus:null_encode_if:null_p:null_le:null_if -> 0:s:true:false:cons_p:cons_le:cons_minus:cons_if:null_encArg:null_encode_p:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_minus:null_encode_if:null_p:null_le:null_if -> 0:s:true:false:cons_p:cons_le:cons_minus:cons_if:null_encArg:null_encode_p:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_minus:null_encode_if:null_p:null_le:null_if if :: 0:s:true:false:cons_p:cons_le:cons_minus:cons_if:null_encArg:null_encode_p:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_minus:null_encode_if:null_p:null_le:null_if -> 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0:s:true:false:cons_p:cons_le:cons_minus:cons_if:null_encArg:null_encode_p:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_minus:null_encode_if:null_p:null_le:null_if null_if :: 0:s:true:false:cons_p:cons_le:cons_minus:cons_if:null_encArg:null_encode_p:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_minus:null_encode_if:null_p:null_le: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: p(0) -> 0 [1] p(s(x)) -> x [1] le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [1] minus(0, y) -> if(true, 0, y) [2] minus(s(x'), 0) -> if(false, s(x'), 0) [2] minus(s(x''), s(y')) -> if(le(x'', y'), s(x''), s(y')) [2] minus(x, y) -> if(null_le, x, y) [1] if(true, x, y) -> 0 [1] if(false, 0, y) -> s(minus(0, y)) [2] if(false, s(x1), y) -> s(minus(x1, y)) [2] if(false, x, y) -> s(minus(null_p, y)) [1] encArg(0) -> 0 [0] encArg(s(x_1)) -> s(encArg(x_1)) [0] encArg(true) -> true [0] encArg(false) -> false [0] encArg(cons_p(0)) -> p(0) [0] encArg(cons_p(s(x_1'))) -> p(s(encArg(x_1'))) [0] encArg(cons_p(true)) -> p(true) [0] encArg(cons_p(false)) -> p(false) [0] encArg(cons_p(cons_p(x_1''))) -> p(p(encArg(x_1''))) [0] encArg(cons_p(cons_le(x_11, x_2'))) -> p(le(encArg(x_11), encArg(x_2'))) [0] encArg(cons_p(cons_minus(x_12, x_2''))) -> p(minus(encArg(x_12), encArg(x_2''))) [0] encArg(cons_p(cons_if(x_13, x_21, x_3'))) -> p(if(encArg(x_13), encArg(x_21), encArg(x_3'))) [0] encArg(cons_p(x_1)) -> p(null_encArg) [0] encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) [0] encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) [0] encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encode_p(0) -> p(0) [0] encode_p(s(x_1559)) -> p(s(encArg(x_1559))) [0] encode_p(true) -> p(true) [0] encode_p(false) -> p(false) [0] encode_p(cons_p(x_1560)) -> p(p(encArg(x_1560))) [0] encode_p(cons_le(x_1561, x_2335)) -> p(le(encArg(x_1561), encArg(x_2335))) [0] encode_p(cons_minus(x_1562, x_2336)) -> p(minus(encArg(x_1562), encArg(x_2336))) [0] encode_p(cons_if(x_1563, x_2337, x_3111)) -> p(if(encArg(x_1563), encArg(x_2337), encArg(x_3111))) [0] encode_p(x_1) -> p(null_encArg) [0] encode_0 -> 0 [0] encode_s(x_1) -> s(encArg(x_1)) [0] encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) [0] encode_true -> true [0] encode_false -> false [0] encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) [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_p(v0) -> null_encode_p [0] encode_0 -> null_encode_0 [0] encode_s(v0) -> null_encode_s [0] encode_le(v0, v1) -> null_encode_le [0] encode_true -> null_encode_true [0] encode_false -> null_encode_false [0] encode_minus(v0, v1) -> null_encode_minus [0] encode_if(v0, v1, v2) -> null_encode_if [0] p(v0) -> null_p [0] le(v0, v1) -> null_le [0] if(v0, v1, v2) -> null_if [0] The TRS has the following type information: p :: 0:s:true:false:cons_p:cons_le:cons_minus:cons_if:null_encArg:null_encode_p:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_minus:null_encode_if:null_p:null_le:null_if -> 0:s:true:false:cons_p:cons_le:cons_minus:cons_if:null_encArg:null_encode_p:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_minus:null_encode_if:null_p:null_le:null_if 0 :: 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0:s:true:false:cons_p:cons_le:cons_minus:cons_if:null_encArg:null_encode_p:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_minus:null_encode_if:null_p:null_le:null_if -> 0:s:true:false:cons_p:cons_le:cons_minus:cons_if:null_encArg:null_encode_p:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_minus:null_encode_if:null_p:null_le:null_if true :: 0:s:true:false:cons_p:cons_le:cons_minus:cons_if:null_encArg:null_encode_p:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_minus:null_encode_if:null_p:null_le:null_if false :: 0:s:true:false:cons_p:cons_le:cons_minus:cons_if:null_encArg:null_encode_p:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_minus:null_encode_if:null_p:null_le:null_if minus :: 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0:s:true:false:cons_p:cons_le:cons_minus:cons_if:null_encArg:null_encode_p:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_minus:null_encode_if:null_p:null_le:null_if encode_true :: 0:s:true:false:cons_p:cons_le:cons_minus:cons_if:null_encArg:null_encode_p:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_minus:null_encode_if:null_p:null_le:null_if encode_false :: 0:s:true:false:cons_p:cons_le:cons_minus:cons_if:null_encArg:null_encode_p:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_minus:null_encode_if:null_p:null_le:null_if encode_minus :: 0:s:true:false:cons_p:cons_le:cons_minus:cons_if:null_encArg:null_encode_p:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_minus:null_encode_if:null_p:null_le:null_if -> 0:s:true:false:cons_p:cons_le:cons_minus:cons_if:null_encArg:null_encode_p:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_minus:null_encode_if:null_p:null_le:null_if -> 0:s:true:false:cons_p:cons_le:cons_minus:cons_if:null_encArg:null_encode_p:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_minus:null_encode_if:null_p:null_le:null_if encode_if :: 0:s:true:false:cons_p:cons_le:cons_minus:cons_if:null_encArg:null_encode_p:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_minus:null_encode_if:null_p:null_le:null_if -> 0:s:true:false:cons_p:cons_le:cons_minus:cons_if:null_encArg:null_encode_p:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_minus:null_encode_if:null_p:null_le:null_if -> 0:s:true:false:cons_p:cons_le:cons_minus:cons_if:null_encArg:null_encode_p:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_minus:null_encode_if:null_p:null_le:null_if -> 0:s:true:false:cons_p:cons_le:cons_minus:cons_if:null_encArg:null_encode_p:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_minus:null_encode_if:null_p:null_le:null_if null_encArg :: 0:s:true:false:cons_p:cons_le:cons_minus:cons_if:null_encArg:null_encode_p:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_minus:null_encode_if:null_p:null_le:null_if null_encode_p :: 0:s:true:false:cons_p:cons_le:cons_minus:cons_if:null_encArg:null_encode_p:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_minus:null_encode_if:null_p:null_le:null_if null_encode_0 :: 0:s:true:false:cons_p:cons_le:cons_minus:cons_if:null_encArg:null_encode_p:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_minus:null_encode_if:null_p:null_le:null_if null_encode_s :: 0:s:true:false:cons_p:cons_le:cons_minus:cons_if:null_encArg:null_encode_p:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_minus:null_encode_if:null_p:null_le:null_if null_encode_le :: 0:s:true:false:cons_p:cons_le:cons_minus:cons_if:null_encArg:null_encode_p:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_minus:null_encode_if:null_p:null_le:null_if null_encode_true :: 0:s:true:false:cons_p:cons_le:cons_minus:cons_if:null_encArg:null_encode_p:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_minus:null_encode_if:null_p:null_le:null_if null_encode_false :: 0:s:true:false:cons_p:cons_le:cons_minus:cons_if:null_encArg:null_encode_p:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_minus:null_encode_if:null_p:null_le:null_if null_encode_minus :: 0:s:true:false:cons_p:cons_le:cons_minus:cons_if:null_encArg:null_encode_p:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_minus:null_encode_if:null_p:null_le:null_if null_encode_if :: 0:s:true:false:cons_p:cons_le:cons_minus:cons_if:null_encArg:null_encode_p:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_minus:null_encode_if:null_p:null_le:null_if null_p :: 0:s:true:false:cons_p:cons_le:cons_minus:cons_if:null_encArg:null_encode_p:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_minus:null_encode_if:null_p:null_le:null_if null_le :: 0:s:true:false:cons_p:cons_le:cons_minus:cons_if:null_encArg:null_encode_p:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_minus:null_encode_if:null_p:null_le:null_if null_if :: 0:s:true:false:cons_p:cons_le:cons_minus:cons_if:null_encArg:null_encode_p:null_encode_0:null_encode_s:null_encode_le:null_encode_true:null_encode_false:null_encode_minus:null_encode_if:null_p:null_le: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 => 2 false => 1 null_encArg => 0 null_encode_p => 0 null_encode_0 => 0 null_encode_s => 0 null_encode_le => 0 null_encode_true => 0 null_encode_false => 0 null_encode_minus => 0 null_encode_if => 0 null_p => 0 null_le => 0 null_if => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 0 }-> p(p(encArg(x_1''))) :|: z = 1 + (1 + x_1''), x_1'' >= 0 encArg(z) -{ 0 }-> p(minus(encArg(x_12), encArg(x_2''))) :|: z = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0 encArg(z) -{ 0 }-> p(le(encArg(x_11), encArg(x_2'))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') encArg(z) -{ 0 }-> p(if(encArg(x_13), encArg(x_21), encArg(x_3'))) :|: z = 1 + (1 + x_13 + x_21 + x_3'), x_3' >= 0, x_13 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> p(2) :|: z = 1 + 2 encArg(z) -{ 0 }-> p(1) :|: z = 1 + 1 encArg(z) -{ 0 }-> p(0) :|: z = 1 + 0 encArg(z) -{ 0 }-> p(0) :|: z = 1 + x_1, x_1 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(x_1')) :|: z = 1 + (1 + x_1'), x_1' >= 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 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, 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 }-> 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_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: 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_le(z, z') -{ 0 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2 encode_le(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 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_p(z) -{ 0 }-> p(p(encArg(x_1560))) :|: x_1560 >= 0, z = 1 + x_1560 encode_p(z) -{ 0 }-> p(minus(encArg(x_1562), encArg(x_2336))) :|: x_2336 >= 0, x_1562 >= 0, z = 1 + x_1562 + x_2336 encode_p(z) -{ 0 }-> p(le(encArg(x_1561), encArg(x_2335))) :|: x_1561 >= 0, x_2335 >= 0, z = 1 + x_1561 + x_2335 encode_p(z) -{ 0 }-> p(if(encArg(x_1563), encArg(x_2337), encArg(x_3111))) :|: z = 1 + x_1563 + x_2337 + x_3111, x_2337 >= 0, x_1563 >= 0, x_3111 >= 0 encode_p(z) -{ 0 }-> p(2) :|: z = 2 encode_p(z) -{ 0 }-> p(1) :|: z = 1 encode_p(z) -{ 0 }-> p(0) :|: z = 0 encode_p(z) -{ 0 }-> p(0) :|: x_1 >= 0, z = x_1 encode_p(z) -{ 0 }-> p(1 + encArg(x_1559)) :|: x_1559 >= 0, z = 1 + x_1559 encode_p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_s(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_s(z) -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z = x_1 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: 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 + minus(x1, y) :|: x1 >= 0, z'' = y, z = 1, y >= 0, z' = 1 + x1 if(z, z', z'') -{ 2 }-> 1 + minus(0, y) :|: z'' = y, z = 1, y >= 0, z' = 0 if(z, z', z'') -{ 1 }-> 1 + minus(0, y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0 le(z, z') -{ 1 }-> le(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x le(z, z') -{ 1 }-> 2 :|: y >= 0, z = 0, z' = y le(z, z') -{ 1 }-> 1 :|: x >= 0, z = 1 + x, z' = 0 le(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 minus(z, z') -{ 2 }-> if(le(x'', y'), 1 + x'', 1 + y') :|: z = 1 + x'', y' >= 0, z' = 1 + y', x'' >= 0 minus(z, z') -{ 2 }-> if(2, 0, y) :|: y >= 0, z = 0, z' = y minus(z, z') -{ 2 }-> if(1, 1 + x', 0) :|: z = 1 + x', x' >= 0, z' = 0 minus(z, z') -{ 1 }-> if(0, x, y) :|: x >= 0, y >= 0, z = x, z' = y p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 ---------------------------------------- (15) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> p(p(encArg(x_1''))) :|: z = 1 + (1 + x_1''), x_1'' >= 0 encArg(z) -{ 0 }-> p(minus(encArg(x_12), encArg(x_2''))) :|: z = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0 encArg(z) -{ 0 }-> p(le(encArg(x_11), encArg(x_2'))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') encArg(z) -{ 0 }-> p(if(encArg(x_13), encArg(x_21), encArg(x_3'))) :|: z = 1 + (1 + x_13 + x_21 + x_3'), x_3' >= 0, x_13 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(x_1')) :|: z = 1 + (1 + x_1'), x_1' >= 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 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, 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 }-> 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) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z = 1 + x_1, x_1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + x_1, x_1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 1 + encArg(x_1) :|: z = 1 + x_1, x_1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: 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_le(z, z') -{ 0 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2 encode_le(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 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_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> p(p(encArg(x_1560))) :|: x_1560 >= 0, z = 1 + x_1560 encode_p(z) -{ 0 }-> p(minus(encArg(x_1562), encArg(x_2336))) :|: x_2336 >= 0, x_1562 >= 0, z = 1 + x_1562 + x_2336 encode_p(z) -{ 0 }-> p(le(encArg(x_1561), encArg(x_2335))) :|: x_1561 >= 0, x_2335 >= 0, z = 1 + x_1561 + x_2335 encode_p(z) -{ 0 }-> p(if(encArg(x_1563), encArg(x_2337), encArg(x_3111))) :|: z = 1 + x_1563 + x_2337 + x_3111, x_2337 >= 0, x_1563 >= 0, x_3111 >= 0 encode_p(z) -{ 0 }-> p(1 + encArg(x_1559)) :|: x_1559 >= 0, z = 1 + x_1559 encode_p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: x_1 >= 0, z = x_1, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: x_1 >= 0, z = x_1, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encode_s(z) -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z = x_1 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: 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 + minus(x1, y) :|: x1 >= 0, z'' = y, z = 1, y >= 0, z' = 1 + x1 if(z, z', z'') -{ 2 }-> 1 + minus(0, y) :|: z'' = y, z = 1, y >= 0, z' = 0 if(z, z', z'') -{ 1 }-> 1 + minus(0, y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0 le(z, z') -{ 1 }-> le(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x le(z, z') -{ 1 }-> 2 :|: y >= 0, z = 0, z' = y le(z, z') -{ 1 }-> 1 :|: x >= 0, z = 1 + x, z' = 0 le(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 minus(z, z') -{ 2 }-> if(le(x'', y'), 1 + x'', 1 + y') :|: z = 1 + x'', y' >= 0, z' = 1 + y', x'' >= 0 minus(z, z') -{ 2 }-> if(2, 0, y) :|: y >= 0, z = 0, z' = y minus(z, z') -{ 2 }-> if(1, 1 + x', 0) :|: z = 1 + x', x' >= 0, z' = 0 minus(z, z') -{ 1 }-> if(0, x, y) :|: x >= 0, y >= 0, z = x, z' = y p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 ---------------------------------------- (17) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(minus(encArg(x_12), encArg(x_2''))) :|: z = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0 encArg(z) -{ 0 }-> p(le(encArg(x_11), encArg(x_2'))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') encArg(z) -{ 0 }-> p(if(encArg(x_13), encArg(x_21), encArg(x_3'))) :|: z = 1 + (1 + x_13 + x_21 + x_3'), x_3' >= 0, x_13 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 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 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, 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 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 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_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 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_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(minus(encArg(x_1562), encArg(x_2336))) :|: x_2336 >= 0, x_1562 >= 0, z = 1 + x_1562 + x_2336 encode_p(z) -{ 0 }-> p(le(encArg(x_1561), encArg(x_2335))) :|: x_1561 >= 0, x_2335 >= 0, z = 1 + x_1561 + x_2335 encode_p(z) -{ 0 }-> p(if(encArg(x_1563), encArg(x_2337), encArg(x_3111))) :|: z = 1 + x_1563 + x_2337 + x_3111, x_2337 >= 0, x_1563 >= 0, x_3111 >= 0 encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: 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 + minus(0, z'') :|: z = 1, z'' >= 0, z' = 0 if(z, z', z'') -{ 1 }-> 1 + minus(0, z'') :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 2 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z = 1, z'' >= 0 le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 2 }-> if(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 2 }-> if(2, 0, z') :|: z' >= 0, z = 0 minus(z, z') -{ 2 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 minus(z, z') -{ 1 }-> if(0, z, z') :|: z >= 0, z' >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 ---------------------------------------- (19) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { encode_0 } { le } { encode_false } { encode_true } { p } { if, minus } { encArg } { encode_p } { encode_if } { encode_minus } { encode_le } { encode_s } ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(minus(encArg(x_12), encArg(x_2''))) :|: z = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0 encArg(z) -{ 0 }-> p(le(encArg(x_11), encArg(x_2'))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') encArg(z) -{ 0 }-> p(if(encArg(x_13), encArg(x_21), encArg(x_3'))) :|: z = 1 + (1 + x_13 + x_21 + x_3'), x_3' >= 0, x_13 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 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 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, 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 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 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_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 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_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(minus(encArg(x_1562), encArg(x_2336))) :|: x_2336 >= 0, x_1562 >= 0, z = 1 + x_1562 + x_2336 encode_p(z) -{ 0 }-> p(le(encArg(x_1561), encArg(x_2335))) :|: x_1561 >= 0, x_2335 >= 0, z = 1 + x_1561 + x_2335 encode_p(z) -{ 0 }-> p(if(encArg(x_1563), encArg(x_2337), encArg(x_3111))) :|: z = 1 + x_1563 + x_2337 + x_3111, x_2337 >= 0, x_1563 >= 0, x_3111 >= 0 encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: 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 + minus(0, z'') :|: z = 1, z'' >= 0, z' = 0 if(z, z', z'') -{ 1 }-> 1 + minus(0, z'') :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 2 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z = 1, z'' >= 0 le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 2 }-> if(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 2 }-> if(2, 0, z') :|: z' >= 0, z = 0 minus(z, z') -{ 2 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 minus(z, z') -{ 1 }-> if(0, z, z') :|: z >= 0, z' >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encode_0}, {le}, {encode_false}, {encode_true}, {p}, {if,minus}, {encArg}, {encode_p}, {encode_if}, {encode_minus}, {encode_le}, {encode_s} ---------------------------------------- (21) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(minus(encArg(x_12), encArg(x_2''))) :|: z = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0 encArg(z) -{ 0 }-> p(le(encArg(x_11), encArg(x_2'))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') encArg(z) -{ 0 }-> p(if(encArg(x_13), encArg(x_21), encArg(x_3'))) :|: z = 1 + (1 + x_13 + x_21 + x_3'), x_3' >= 0, x_13 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 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 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, 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 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 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_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 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_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(minus(encArg(x_1562), encArg(x_2336))) :|: x_2336 >= 0, x_1562 >= 0, z = 1 + x_1562 + x_2336 encode_p(z) -{ 0 }-> p(le(encArg(x_1561), encArg(x_2335))) :|: x_1561 >= 0, x_2335 >= 0, z = 1 + x_1561 + x_2335 encode_p(z) -{ 0 }-> p(if(encArg(x_1563), encArg(x_2337), encArg(x_3111))) :|: z = 1 + x_1563 + x_2337 + x_3111, x_2337 >= 0, x_1563 >= 0, x_3111 >= 0 encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: 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 + minus(0, z'') :|: z = 1, z'' >= 0, z' = 0 if(z, z', z'') -{ 1 }-> 1 + minus(0, z'') :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 2 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z = 1, z'' >= 0 le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 2 }-> if(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 2 }-> if(2, 0, z') :|: z' >= 0, z = 0 minus(z, z') -{ 2 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 minus(z, z') -{ 1 }-> if(0, z, z') :|: z >= 0, z' >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encode_0}, {le}, {encode_false}, {encode_true}, {p}, {if,minus}, {encArg}, {encode_p}, {encode_if}, {encode_minus}, {encode_le}, {encode_s} ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: encode_0 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(minus(encArg(x_12), encArg(x_2''))) :|: z = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0 encArg(z) -{ 0 }-> p(le(encArg(x_11), encArg(x_2'))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') encArg(z) -{ 0 }-> p(if(encArg(x_13), encArg(x_21), encArg(x_3'))) :|: z = 1 + (1 + x_13 + x_21 + x_3'), x_3' >= 0, x_13 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 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 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, 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 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 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_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 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_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(minus(encArg(x_1562), encArg(x_2336))) :|: x_2336 >= 0, x_1562 >= 0, z = 1 + x_1562 + x_2336 encode_p(z) -{ 0 }-> p(le(encArg(x_1561), encArg(x_2335))) :|: x_1561 >= 0, x_2335 >= 0, z = 1 + x_1561 + x_2335 encode_p(z) -{ 0 }-> p(if(encArg(x_1563), encArg(x_2337), encArg(x_3111))) :|: z = 1 + x_1563 + x_2337 + x_3111, x_2337 >= 0, x_1563 >= 0, x_3111 >= 0 encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: 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 + minus(0, z'') :|: z = 1, z'' >= 0, z' = 0 if(z, z', z'') -{ 1 }-> 1 + minus(0, z'') :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 2 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z = 1, z'' >= 0 le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 2 }-> if(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 2 }-> if(2, 0, z') :|: z' >= 0, z = 0 minus(z, z') -{ 2 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 minus(z, z') -{ 1 }-> if(0, z, z') :|: z >= 0, z' >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encode_0}, {le}, {encode_false}, {encode_true}, {p}, {if,minus}, {encArg}, {encode_p}, {encode_if}, {encode_minus}, {encode_le}, {encode_s} Previous analysis results are: encode_0: runtime: ?, size: O(1) [0] ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: encode_0 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(minus(encArg(x_12), encArg(x_2''))) :|: z = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0 encArg(z) -{ 0 }-> p(le(encArg(x_11), encArg(x_2'))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') encArg(z) -{ 0 }-> p(if(encArg(x_13), encArg(x_21), encArg(x_3'))) :|: z = 1 + (1 + x_13 + x_21 + x_3'), x_3' >= 0, x_13 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 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 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, 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 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 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_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 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_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(minus(encArg(x_1562), encArg(x_2336))) :|: x_2336 >= 0, x_1562 >= 0, z = 1 + x_1562 + x_2336 encode_p(z) -{ 0 }-> p(le(encArg(x_1561), encArg(x_2335))) :|: x_1561 >= 0, x_2335 >= 0, z = 1 + x_1561 + x_2335 encode_p(z) -{ 0 }-> p(if(encArg(x_1563), encArg(x_2337), encArg(x_3111))) :|: z = 1 + x_1563 + x_2337 + x_3111, x_2337 >= 0, x_1563 >= 0, x_3111 >= 0 encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: 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 + minus(0, z'') :|: z = 1, z'' >= 0, z' = 0 if(z, z', z'') -{ 1 }-> 1 + minus(0, z'') :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 2 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z = 1, z'' >= 0 le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 2 }-> if(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 2 }-> if(2, 0, z') :|: z' >= 0, z = 0 minus(z, z') -{ 2 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 minus(z, z') -{ 1 }-> if(0, z, z') :|: z >= 0, z' >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {le}, {encode_false}, {encode_true}, {p}, {if,minus}, {encArg}, {encode_p}, {encode_if}, {encode_minus}, {encode_le}, {encode_s} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] ---------------------------------------- (27) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(minus(encArg(x_12), encArg(x_2''))) :|: z = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0 encArg(z) -{ 0 }-> p(le(encArg(x_11), encArg(x_2'))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') encArg(z) -{ 0 }-> p(if(encArg(x_13), encArg(x_21), encArg(x_3'))) :|: z = 1 + (1 + x_13 + x_21 + x_3'), x_3' >= 0, x_13 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 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 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, 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 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 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_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 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_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(minus(encArg(x_1562), encArg(x_2336))) :|: x_2336 >= 0, x_1562 >= 0, z = 1 + x_1562 + x_2336 encode_p(z) -{ 0 }-> p(le(encArg(x_1561), encArg(x_2335))) :|: x_1561 >= 0, x_2335 >= 0, z = 1 + x_1561 + x_2335 encode_p(z) -{ 0 }-> p(if(encArg(x_1563), encArg(x_2337), encArg(x_3111))) :|: z = 1 + x_1563 + x_2337 + x_3111, x_2337 >= 0, x_1563 >= 0, x_3111 >= 0 encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: 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 + minus(0, z'') :|: z = 1, z'' >= 0, z' = 0 if(z, z', z'') -{ 1 }-> 1 + minus(0, z'') :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 2 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z = 1, z'' >= 0 le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 2 }-> if(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 2 }-> if(2, 0, z') :|: z' >= 0, z = 0 minus(z, z') -{ 2 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 minus(z, z') -{ 1 }-> if(0, z, z') :|: z >= 0, z' >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {le}, {encode_false}, {encode_true}, {p}, {if,minus}, {encArg}, {encode_p}, {encode_if}, {encode_minus}, {encode_le}, {encode_s} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: le after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(minus(encArg(x_12), encArg(x_2''))) :|: z = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0 encArg(z) -{ 0 }-> p(le(encArg(x_11), encArg(x_2'))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') encArg(z) -{ 0 }-> p(if(encArg(x_13), encArg(x_21), encArg(x_3'))) :|: z = 1 + (1 + x_13 + x_21 + x_3'), x_3' >= 0, x_13 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 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 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, 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 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 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_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 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_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(minus(encArg(x_1562), encArg(x_2336))) :|: x_2336 >= 0, x_1562 >= 0, z = 1 + x_1562 + x_2336 encode_p(z) -{ 0 }-> p(le(encArg(x_1561), encArg(x_2335))) :|: x_1561 >= 0, x_2335 >= 0, z = 1 + x_1561 + x_2335 encode_p(z) -{ 0 }-> p(if(encArg(x_1563), encArg(x_2337), encArg(x_3111))) :|: z = 1 + x_1563 + x_2337 + x_3111, x_2337 >= 0, x_1563 >= 0, x_3111 >= 0 encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: 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 + minus(0, z'') :|: z = 1, z'' >= 0, z' = 0 if(z, z', z'') -{ 1 }-> 1 + minus(0, z'') :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 2 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z = 1, z'' >= 0 le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 2 }-> if(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 2 }-> if(2, 0, z') :|: z' >= 0, z = 0 minus(z, z') -{ 2 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 minus(z, z') -{ 1 }-> if(0, z, z') :|: z >= 0, z' >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {le}, {encode_false}, {encode_true}, {p}, {if,minus}, {encArg}, {encode_p}, {encode_if}, {encode_minus}, {encode_le}, {encode_s} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] le: runtime: ?, size: O(1) [2] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: le after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z' ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(minus(encArg(x_12), encArg(x_2''))) :|: z = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0 encArg(z) -{ 0 }-> p(le(encArg(x_11), encArg(x_2'))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') encArg(z) -{ 0 }-> p(if(encArg(x_13), encArg(x_21), encArg(x_3'))) :|: z = 1 + (1 + x_13 + x_21 + x_3'), x_3' >= 0, x_13 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 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 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, 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 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 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_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 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_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(minus(encArg(x_1562), encArg(x_2336))) :|: x_2336 >= 0, x_1562 >= 0, z = 1 + x_1562 + x_2336 encode_p(z) -{ 0 }-> p(le(encArg(x_1561), encArg(x_2335))) :|: x_1561 >= 0, x_2335 >= 0, z = 1 + x_1561 + x_2335 encode_p(z) -{ 0 }-> p(if(encArg(x_1563), encArg(x_2337), encArg(x_3111))) :|: z = 1 + x_1563 + x_2337 + x_3111, x_2337 >= 0, x_1563 >= 0, x_3111 >= 0 encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: 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 + minus(0, z'') :|: z = 1, z'' >= 0, z' = 0 if(z, z', z'') -{ 1 }-> 1 + minus(0, z'') :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 2 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z = 1, z'' >= 0 le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 2 }-> if(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 2 }-> if(2, 0, z') :|: z' >= 0, z = 0 minus(z, z') -{ 2 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 minus(z, z') -{ 1 }-> if(0, z, z') :|: z >= 0, z' >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encode_false}, {encode_true}, {p}, {if,minus}, {encArg}, {encode_p}, {encode_if}, {encode_minus}, {encode_le}, {encode_s} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] le: runtime: O(n^1) [2 + z'], size: O(1) [2] ---------------------------------------- (33) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(minus(encArg(x_12), encArg(x_2''))) :|: z = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0 encArg(z) -{ 0 }-> p(le(encArg(x_11), encArg(x_2'))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') encArg(z) -{ 0 }-> p(if(encArg(x_13), encArg(x_21), encArg(x_3'))) :|: z = 1 + (1 + x_13 + x_21 + x_3'), x_3' >= 0, x_13 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 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 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, 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 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 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_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 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_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(minus(encArg(x_1562), encArg(x_2336))) :|: x_2336 >= 0, x_1562 >= 0, z = 1 + x_1562 + x_2336 encode_p(z) -{ 0 }-> p(le(encArg(x_1561), encArg(x_2335))) :|: x_1561 >= 0, x_2335 >= 0, z = 1 + x_1561 + x_2335 encode_p(z) -{ 0 }-> p(if(encArg(x_1563), encArg(x_2337), encArg(x_3111))) :|: z = 1 + x_1563 + x_2337 + x_3111, x_2337 >= 0, x_1563 >= 0, x_3111 >= 0 encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: 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 + minus(0, z'') :|: z = 1, z'' >= 0, z' = 0 if(z, z', z'') -{ 1 }-> 1 + minus(0, z'') :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 2 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z = 1, z'' >= 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 3 + z' }-> if(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 2, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 2 }-> if(2, 0, z') :|: z' >= 0, z = 0 minus(z, z') -{ 2 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 minus(z, z') -{ 1 }-> if(0, z, z') :|: z >= 0, z' >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encode_false}, {encode_true}, {p}, {if,minus}, {encArg}, {encode_p}, {encode_if}, {encode_minus}, {encode_le}, {encode_s} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] le: runtime: O(n^1) [2 + z'], size: O(1) [2] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: encode_false after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(minus(encArg(x_12), encArg(x_2''))) :|: z = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0 encArg(z) -{ 0 }-> p(le(encArg(x_11), encArg(x_2'))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') encArg(z) -{ 0 }-> p(if(encArg(x_13), encArg(x_21), encArg(x_3'))) :|: z = 1 + (1 + x_13 + x_21 + x_3'), x_3' >= 0, x_13 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 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 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, 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 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 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_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 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_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(minus(encArg(x_1562), encArg(x_2336))) :|: x_2336 >= 0, x_1562 >= 0, z = 1 + x_1562 + x_2336 encode_p(z) -{ 0 }-> p(le(encArg(x_1561), encArg(x_2335))) :|: x_1561 >= 0, x_2335 >= 0, z = 1 + x_1561 + x_2335 encode_p(z) -{ 0 }-> p(if(encArg(x_1563), encArg(x_2337), encArg(x_3111))) :|: z = 1 + x_1563 + x_2337 + x_3111, x_2337 >= 0, x_1563 >= 0, x_3111 >= 0 encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: 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 + minus(0, z'') :|: z = 1, z'' >= 0, z' = 0 if(z, z', z'') -{ 1 }-> 1 + minus(0, z'') :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 2 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z = 1, z'' >= 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 3 + z' }-> if(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 2, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 2 }-> if(2, 0, z') :|: z' >= 0, z = 0 minus(z, z') -{ 2 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 minus(z, z') -{ 1 }-> if(0, z, z') :|: z >= 0, z' >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encode_false}, {encode_true}, {p}, {if,minus}, {encArg}, {encode_p}, {encode_if}, {encode_minus}, {encode_le}, {encode_s} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] le: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_false: runtime: ?, size: O(1) [1] ---------------------------------------- (37) 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 ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(minus(encArg(x_12), encArg(x_2''))) :|: z = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0 encArg(z) -{ 0 }-> p(le(encArg(x_11), encArg(x_2'))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') encArg(z) -{ 0 }-> p(if(encArg(x_13), encArg(x_21), encArg(x_3'))) :|: z = 1 + (1 + x_13 + x_21 + x_3'), x_3' >= 0, x_13 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 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 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, 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 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 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_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 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_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(minus(encArg(x_1562), encArg(x_2336))) :|: x_2336 >= 0, x_1562 >= 0, z = 1 + x_1562 + x_2336 encode_p(z) -{ 0 }-> p(le(encArg(x_1561), encArg(x_2335))) :|: x_1561 >= 0, x_2335 >= 0, z = 1 + x_1561 + x_2335 encode_p(z) -{ 0 }-> p(if(encArg(x_1563), encArg(x_2337), encArg(x_3111))) :|: z = 1 + x_1563 + x_2337 + x_3111, x_2337 >= 0, x_1563 >= 0, x_3111 >= 0 encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: 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 + minus(0, z'') :|: z = 1, z'' >= 0, z' = 0 if(z, z', z'') -{ 1 }-> 1 + minus(0, z'') :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 2 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z = 1, z'' >= 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 3 + z' }-> if(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 2, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 2 }-> if(2, 0, z') :|: z' >= 0, z = 0 minus(z, z') -{ 2 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 minus(z, z') -{ 1 }-> if(0, z, z') :|: z >= 0, z' >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encode_true}, {p}, {if,minus}, {encArg}, {encode_p}, {encode_if}, {encode_minus}, {encode_le}, {encode_s} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] le: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_false: runtime: O(1) [0], size: O(1) [1] ---------------------------------------- (39) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(minus(encArg(x_12), encArg(x_2''))) :|: z = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0 encArg(z) -{ 0 }-> p(le(encArg(x_11), encArg(x_2'))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') encArg(z) -{ 0 }-> p(if(encArg(x_13), encArg(x_21), encArg(x_3'))) :|: z = 1 + (1 + x_13 + x_21 + x_3'), x_3' >= 0, x_13 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 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 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, 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 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 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_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 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_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(minus(encArg(x_1562), encArg(x_2336))) :|: x_2336 >= 0, x_1562 >= 0, z = 1 + x_1562 + x_2336 encode_p(z) -{ 0 }-> p(le(encArg(x_1561), encArg(x_2335))) :|: x_1561 >= 0, x_2335 >= 0, z = 1 + x_1561 + x_2335 encode_p(z) -{ 0 }-> p(if(encArg(x_1563), encArg(x_2337), encArg(x_3111))) :|: z = 1 + x_1563 + x_2337 + x_3111, x_2337 >= 0, x_1563 >= 0, x_3111 >= 0 encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: 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 + minus(0, z'') :|: z = 1, z'' >= 0, z' = 0 if(z, z', z'') -{ 1 }-> 1 + minus(0, z'') :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 2 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z = 1, z'' >= 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 3 + z' }-> if(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 2, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 2 }-> if(2, 0, z') :|: z' >= 0, z = 0 minus(z, z') -{ 2 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 minus(z, z') -{ 1 }-> if(0, z, z') :|: z >= 0, z' >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encode_true}, {p}, {if,minus}, {encArg}, {encode_p}, {encode_if}, {encode_minus}, {encode_le}, {encode_s} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] le: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_false: runtime: O(1) [0], size: O(1) [1] ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: encode_true after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(minus(encArg(x_12), encArg(x_2''))) :|: z = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0 encArg(z) -{ 0 }-> p(le(encArg(x_11), encArg(x_2'))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') encArg(z) -{ 0 }-> p(if(encArg(x_13), encArg(x_21), encArg(x_3'))) :|: z = 1 + (1 + x_13 + x_21 + x_3'), x_3' >= 0, x_13 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 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 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, 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 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 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_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 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_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(minus(encArg(x_1562), encArg(x_2336))) :|: x_2336 >= 0, x_1562 >= 0, z = 1 + x_1562 + x_2336 encode_p(z) -{ 0 }-> p(le(encArg(x_1561), encArg(x_2335))) :|: x_1561 >= 0, x_2335 >= 0, z = 1 + x_1561 + x_2335 encode_p(z) -{ 0 }-> p(if(encArg(x_1563), encArg(x_2337), encArg(x_3111))) :|: z = 1 + x_1563 + x_2337 + x_3111, x_2337 >= 0, x_1563 >= 0, x_3111 >= 0 encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: 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 + minus(0, z'') :|: z = 1, z'' >= 0, z' = 0 if(z, z', z'') -{ 1 }-> 1 + minus(0, z'') :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 2 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z = 1, z'' >= 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 3 + z' }-> if(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 2, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 2 }-> if(2, 0, z') :|: z' >= 0, z = 0 minus(z, z') -{ 2 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 minus(z, z') -{ 1 }-> if(0, z, z') :|: z >= 0, z' >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encode_true}, {p}, {if,minus}, {encArg}, {encode_p}, {encode_if}, {encode_minus}, {encode_le}, {encode_s} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] le: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: ?, size: O(1) [2] ---------------------------------------- (43) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: encode_true after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(minus(encArg(x_12), encArg(x_2''))) :|: z = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0 encArg(z) -{ 0 }-> p(le(encArg(x_11), encArg(x_2'))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') encArg(z) -{ 0 }-> p(if(encArg(x_13), encArg(x_21), encArg(x_3'))) :|: z = 1 + (1 + x_13 + x_21 + x_3'), x_3' >= 0, x_13 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 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 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, 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 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 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_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 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_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(minus(encArg(x_1562), encArg(x_2336))) :|: x_2336 >= 0, x_1562 >= 0, z = 1 + x_1562 + x_2336 encode_p(z) -{ 0 }-> p(le(encArg(x_1561), encArg(x_2335))) :|: x_1561 >= 0, x_2335 >= 0, z = 1 + x_1561 + x_2335 encode_p(z) -{ 0 }-> p(if(encArg(x_1563), encArg(x_2337), encArg(x_3111))) :|: z = 1 + x_1563 + x_2337 + x_3111, x_2337 >= 0, x_1563 >= 0, x_3111 >= 0 encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: 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 + minus(0, z'') :|: z = 1, z'' >= 0, z' = 0 if(z, z', z'') -{ 1 }-> 1 + minus(0, z'') :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 2 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z = 1, z'' >= 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 3 + z' }-> if(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 2, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 2 }-> if(2, 0, z') :|: z' >= 0, z = 0 minus(z, z') -{ 2 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 minus(z, z') -{ 1 }-> if(0, z, z') :|: z >= 0, z' >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {p}, {if,minus}, {encArg}, {encode_p}, {encode_if}, {encode_minus}, {encode_le}, {encode_s} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] le: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (45) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(minus(encArg(x_12), encArg(x_2''))) :|: z = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0 encArg(z) -{ 0 }-> p(le(encArg(x_11), encArg(x_2'))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') encArg(z) -{ 0 }-> p(if(encArg(x_13), encArg(x_21), encArg(x_3'))) :|: z = 1 + (1 + x_13 + x_21 + x_3'), x_3' >= 0, x_13 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 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 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, 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 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 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_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 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_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(minus(encArg(x_1562), encArg(x_2336))) :|: x_2336 >= 0, x_1562 >= 0, z = 1 + x_1562 + x_2336 encode_p(z) -{ 0 }-> p(le(encArg(x_1561), encArg(x_2335))) :|: x_1561 >= 0, x_2335 >= 0, z = 1 + x_1561 + x_2335 encode_p(z) -{ 0 }-> p(if(encArg(x_1563), encArg(x_2337), encArg(x_3111))) :|: z = 1 + x_1563 + x_2337 + x_3111, x_2337 >= 0, x_1563 >= 0, x_3111 >= 0 encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: 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 + minus(0, z'') :|: z = 1, z'' >= 0, z' = 0 if(z, z', z'') -{ 1 }-> 1 + minus(0, z'') :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 2 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z = 1, z'' >= 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 3 + z' }-> if(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 2, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 2 }-> if(2, 0, z') :|: z' >= 0, z = 0 minus(z, z') -{ 2 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 minus(z, z') -{ 1 }-> if(0, z, z') :|: z >= 0, z' >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {p}, {if,minus}, {encArg}, {encode_p}, {encode_if}, {encode_minus}, {encode_le}, {encode_s} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] le: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (47) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: p after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(minus(encArg(x_12), encArg(x_2''))) :|: z = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0 encArg(z) -{ 0 }-> p(le(encArg(x_11), encArg(x_2'))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') encArg(z) -{ 0 }-> p(if(encArg(x_13), encArg(x_21), encArg(x_3'))) :|: z = 1 + (1 + x_13 + x_21 + x_3'), x_3' >= 0, x_13 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 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 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, 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 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 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_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 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_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(minus(encArg(x_1562), encArg(x_2336))) :|: x_2336 >= 0, x_1562 >= 0, z = 1 + x_1562 + x_2336 encode_p(z) -{ 0 }-> p(le(encArg(x_1561), encArg(x_2335))) :|: x_1561 >= 0, x_2335 >= 0, z = 1 + x_1561 + x_2335 encode_p(z) -{ 0 }-> p(if(encArg(x_1563), encArg(x_2337), encArg(x_3111))) :|: z = 1 + x_1563 + x_2337 + x_3111, x_2337 >= 0, x_1563 >= 0, x_3111 >= 0 encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: 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 + minus(0, z'') :|: z = 1, z'' >= 0, z' = 0 if(z, z', z'') -{ 1 }-> 1 + minus(0, z'') :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 2 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z = 1, z'' >= 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 3 + z' }-> if(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 2, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 2 }-> if(2, 0, z') :|: z' >= 0, z = 0 minus(z, z') -{ 2 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 minus(z, z') -{ 1 }-> if(0, z, z') :|: z >= 0, z' >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {p}, {if,minus}, {encArg}, {encode_p}, {encode_if}, {encode_minus}, {encode_le}, {encode_s} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] le: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: ?, size: O(n^1) [z] ---------------------------------------- (49) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: p after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(minus(encArg(x_12), encArg(x_2''))) :|: z = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0 encArg(z) -{ 0 }-> p(le(encArg(x_11), encArg(x_2'))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') encArg(z) -{ 0 }-> p(if(encArg(x_13), encArg(x_21), encArg(x_3'))) :|: z = 1 + (1 + x_13 + x_21 + x_3'), x_3' >= 0, x_13 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 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 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, 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 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 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_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 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_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(minus(encArg(x_1562), encArg(x_2336))) :|: x_2336 >= 0, x_1562 >= 0, z = 1 + x_1562 + x_2336 encode_p(z) -{ 0 }-> p(le(encArg(x_1561), encArg(x_2335))) :|: x_1561 >= 0, x_2335 >= 0, z = 1 + x_1561 + x_2335 encode_p(z) -{ 0 }-> p(if(encArg(x_1563), encArg(x_2337), encArg(x_3111))) :|: z = 1 + x_1563 + x_2337 + x_3111, x_2337 >= 0, x_1563 >= 0, x_3111 >= 0 encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: 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 + minus(0, z'') :|: z = 1, z'' >= 0, z' = 0 if(z, z', z'') -{ 1 }-> 1 + minus(0, z'') :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 2 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z = 1, z'' >= 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 3 + z' }-> if(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 2, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 2 }-> if(2, 0, z') :|: z' >= 0, z = 0 minus(z, z') -{ 2 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 minus(z, z') -{ 1 }-> if(0, z, z') :|: z >= 0, z' >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {if,minus}, {encArg}, {encode_p}, {encode_if}, {encode_minus}, {encode_le}, {encode_s} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] le: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] ---------------------------------------- (51) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(minus(encArg(x_12), encArg(x_2''))) :|: z = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0 encArg(z) -{ 0 }-> p(le(encArg(x_11), encArg(x_2'))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') encArg(z) -{ 0 }-> p(if(encArg(x_13), encArg(x_21), encArg(x_3'))) :|: z = 1 + (1 + x_13 + x_21 + x_3'), x_3' >= 0, x_13 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 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 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, 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 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 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_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 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_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(minus(encArg(x_1562), encArg(x_2336))) :|: x_2336 >= 0, x_1562 >= 0, z = 1 + x_1562 + x_2336 encode_p(z) -{ 0 }-> p(le(encArg(x_1561), encArg(x_2335))) :|: x_1561 >= 0, x_2335 >= 0, z = 1 + x_1561 + x_2335 encode_p(z) -{ 0 }-> p(if(encArg(x_1563), encArg(x_2337), encArg(x_3111))) :|: z = 1 + x_1563 + x_2337 + x_3111, x_2337 >= 0, x_1563 >= 0, x_3111 >= 0 encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: 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 + minus(0, z'') :|: z = 1, z'' >= 0, z' = 0 if(z, z', z'') -{ 1 }-> 1 + minus(0, z'') :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 2 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z = 1, z'' >= 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 3 + z' }-> if(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 2, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 2 }-> if(2, 0, z') :|: z' >= 0, z = 0 minus(z, z') -{ 2 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 minus(z, z') -{ 1 }-> if(0, z, z') :|: z >= 0, z' >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {if,minus}, {encArg}, {encode_p}, {encode_if}, {encode_minus}, {encode_le}, {encode_s} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] le: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] ---------------------------------------- (53) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: if after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z' Computed SIZE bound using CoFloCo for: minus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (54) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(minus(encArg(x_12), encArg(x_2''))) :|: z = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0 encArg(z) -{ 0 }-> p(le(encArg(x_11), encArg(x_2'))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') encArg(z) -{ 0 }-> p(if(encArg(x_13), encArg(x_21), encArg(x_3'))) :|: z = 1 + (1 + x_13 + x_21 + x_3'), x_3' >= 0, x_13 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 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 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, 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 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 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_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 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_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(minus(encArg(x_1562), encArg(x_2336))) :|: x_2336 >= 0, x_1562 >= 0, z = 1 + x_1562 + x_2336 encode_p(z) -{ 0 }-> p(le(encArg(x_1561), encArg(x_2335))) :|: x_1561 >= 0, x_2335 >= 0, z = 1 + x_1561 + x_2335 encode_p(z) -{ 0 }-> p(if(encArg(x_1563), encArg(x_2337), encArg(x_3111))) :|: z = 1 + x_1563 + x_2337 + x_3111, x_2337 >= 0, x_1563 >= 0, x_3111 >= 0 encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: 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 + minus(0, z'') :|: z = 1, z'' >= 0, z' = 0 if(z, z', z'') -{ 1 }-> 1 + minus(0, z'') :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 2 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z = 1, z'' >= 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 3 + z' }-> if(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 2, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 2 }-> if(2, 0, z') :|: z' >= 0, z = 0 minus(z, z') -{ 2 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 minus(z, z') -{ 1 }-> if(0, z, z') :|: z >= 0, z' >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {if,minus}, {encArg}, {encode_p}, {encode_if}, {encode_minus}, {encode_le}, {encode_s} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] le: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] if: runtime: ?, size: O(n^1) [1 + z'] minus: runtime: ?, size: O(n^1) [1 + z] ---------------------------------------- (55) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: if after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 10 + 5*z' + z'*z'' + z'' Computed RUNTIME bound using KoAT for: minus after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 48 + 15*z + 2*z*z' + 4*z' ---------------------------------------- (56) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(minus(encArg(x_12), encArg(x_2''))) :|: z = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0 encArg(z) -{ 0 }-> p(le(encArg(x_11), encArg(x_2'))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') encArg(z) -{ 0 }-> p(if(encArg(x_13), encArg(x_21), encArg(x_3'))) :|: z = 1 + (1 + x_13 + x_21 + x_3'), x_3' >= 0, x_13 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 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 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, 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 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 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_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 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_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(minus(encArg(x_1562), encArg(x_2336))) :|: x_2336 >= 0, x_1562 >= 0, z = 1 + x_1562 + x_2336 encode_p(z) -{ 0 }-> p(le(encArg(x_1561), encArg(x_2335))) :|: x_1561 >= 0, x_2335 >= 0, z = 1 + x_1561 + x_2335 encode_p(z) -{ 0 }-> p(if(encArg(x_1563), encArg(x_2337), encArg(x_3111))) :|: z = 1 + x_1563 + x_2337 + x_3111, x_2337 >= 0, x_1563 >= 0, x_3111 >= 0 encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: 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 + minus(0, z'') :|: z = 1, z'' >= 0, z' = 0 if(z, z', z'') -{ 1 }-> 1 + minus(0, z'') :|: z = 1, z' >= 0, z'' >= 0 if(z, z', z'') -{ 2 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z = 1, z'' >= 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 3 + z' }-> if(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 2, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 2 }-> if(2, 0, z') :|: z' >= 0, z = 0 minus(z, z') -{ 2 }-> if(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 minus(z, z') -{ 1 }-> if(0, z, z') :|: z >= 0, z' >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encArg}, {encode_p}, {encode_if}, {encode_minus}, {encode_le}, {encode_s} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] le: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] if: runtime: O(n^2) [10 + 5*z' + z'*z'' + z''], size: O(n^1) [1 + z'] minus: runtime: O(n^2) [48 + 15*z + 2*z*z' + 4*z'], size: O(n^1) [1 + z] ---------------------------------------- (57) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (58) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(minus(encArg(x_12), encArg(x_2''))) :|: z = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0 encArg(z) -{ 0 }-> p(le(encArg(x_11), encArg(x_2'))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') encArg(z) -{ 0 }-> p(if(encArg(x_13), encArg(x_21), encArg(x_3'))) :|: z = 1 + (1 + x_13 + x_21 + x_3'), x_3' >= 0, x_13 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 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 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, 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 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 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_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 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_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(minus(encArg(x_1562), encArg(x_2336))) :|: x_2336 >= 0, x_1562 >= 0, z = 1 + x_1562 + x_2336 encode_p(z) -{ 0 }-> p(le(encArg(x_1561), encArg(x_2335))) :|: x_1561 >= 0, x_2335 >= 0, z = 1 + x_1561 + x_2335 encode_p(z) -{ 0 }-> p(if(encArg(x_1563), encArg(x_2337), encArg(x_3111))) :|: z = 1 + x_1563 + x_2337 + x_3111, x_2337 >= 0, x_1563 >= 0, x_3111 >= 0 encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: 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'') -{ 50 + 4*z'' }-> 1 + s4 :|: s4 >= 0, s4 <= 0 + 1, z = 1, z'' >= 0, z' = 0 if(z, z', z'') -{ 35 + 15*z' + 2*z'*z'' + 2*z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1 + 1, z' - 1 >= 0, z = 1, z'' >= 0 if(z, z', z'') -{ 49 + 4*z'' }-> 1 + s6 :|: s6 >= 0, s6 <= 0 + 1, z = 1, z' >= 0, z'' >= 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 12 + z' }-> s'' :|: s'' >= 0, s'' <= 0 + 1, z' >= 0, z = 0 minus(z, z') -{ 12 + 5*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1) + 1, z - 1 >= 0, z' = 0 minus(z, z') -{ 13 + 5*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1) + 1, s' >= 0, s' <= 2, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 11 + 5*z + z*z' + z' }-> s3 :|: s3 >= 0, s3 <= z + 1, z >= 0, z' >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encArg}, {encode_p}, {encode_if}, {encode_minus}, {encode_le}, {encode_s} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] le: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] if: runtime: O(n^2) [10 + 5*z' + z'*z'' + z''], size: O(n^1) [1 + z'] minus: runtime: O(n^2) [48 + 15*z + 2*z*z' + 4*z'], size: O(n^1) [1 + z] ---------------------------------------- (59) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: encArg after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (60) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(minus(encArg(x_12), encArg(x_2''))) :|: z = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0 encArg(z) -{ 0 }-> p(le(encArg(x_11), encArg(x_2'))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') encArg(z) -{ 0 }-> p(if(encArg(x_13), encArg(x_21), encArg(x_3'))) :|: z = 1 + (1 + x_13 + x_21 + x_3'), x_3' >= 0, x_13 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 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 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, 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 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 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_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 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_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(minus(encArg(x_1562), encArg(x_2336))) :|: x_2336 >= 0, x_1562 >= 0, z = 1 + x_1562 + x_2336 encode_p(z) -{ 0 }-> p(le(encArg(x_1561), encArg(x_2335))) :|: x_1561 >= 0, x_2335 >= 0, z = 1 + x_1561 + x_2335 encode_p(z) -{ 0 }-> p(if(encArg(x_1563), encArg(x_2337), encArg(x_3111))) :|: z = 1 + x_1563 + x_2337 + x_3111, x_2337 >= 0, x_1563 >= 0, x_3111 >= 0 encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: 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'') -{ 50 + 4*z'' }-> 1 + s4 :|: s4 >= 0, s4 <= 0 + 1, z = 1, z'' >= 0, z' = 0 if(z, z', z'') -{ 35 + 15*z' + 2*z'*z'' + 2*z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1 + 1, z' - 1 >= 0, z = 1, z'' >= 0 if(z, z', z'') -{ 49 + 4*z'' }-> 1 + s6 :|: s6 >= 0, s6 <= 0 + 1, z = 1, z' >= 0, z'' >= 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 12 + z' }-> s'' :|: s'' >= 0, s'' <= 0 + 1, z' >= 0, z = 0 minus(z, z') -{ 12 + 5*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1) + 1, z - 1 >= 0, z' = 0 minus(z, z') -{ 13 + 5*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1) + 1, s' >= 0, s' <= 2, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 11 + 5*z + z*z' + z' }-> s3 :|: s3 >= 0, s3 <= z + 1, z >= 0, z' >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encArg}, {encode_p}, {encode_if}, {encode_minus}, {encode_le}, {encode_s} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] le: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] if: runtime: O(n^2) [10 + 5*z' + z'*z'' + z''], size: O(n^1) [1 + z'] minus: runtime: O(n^2) [48 + 15*z + 2*z*z' + 4*z'], size: O(n^1) [1 + z] encArg: runtime: ?, size: O(n^1) [1 + z] ---------------------------------------- (61) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: encArg after applying outer abstraction to obtain an ITS, resulting in: O(n^3) with polynomial bound: 188 + 628*z + 198*z^2 + 18*z^3 ---------------------------------------- (62) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0 encArg(z) -{ 0 }-> p(minus(encArg(x_12), encArg(x_2''))) :|: z = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0 encArg(z) -{ 0 }-> p(le(encArg(x_11), encArg(x_2'))) :|: x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') encArg(z) -{ 0 }-> p(if(encArg(x_13), encArg(x_21), encArg(x_3'))) :|: z = 1 + (1 + x_13 + x_21 + x_3'), x_3' >= 0, x_13 >= 0, x_21 >= 0 encArg(z) -{ 0 }-> p(1 + encArg(z - 2)) :|: z - 2 >= 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 }-> le(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, 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 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 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_le(z, z') -{ 0 }-> le(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 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_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> p(minus(encArg(x_1562), encArg(x_2336))) :|: x_2336 >= 0, x_1562 >= 0, z = 1 + x_1562 + x_2336 encode_p(z) -{ 0 }-> p(le(encArg(x_1561), encArg(x_2335))) :|: x_1561 >= 0, x_2335 >= 0, z = 1 + x_1561 + x_2335 encode_p(z) -{ 0 }-> p(if(encArg(x_1563), encArg(x_2337), encArg(x_3111))) :|: z = 1 + x_1563 + x_2337 + x_3111, x_2337 >= 0, x_1563 >= 0, x_3111 >= 0 encode_p(z) -{ 0 }-> p(1 + encArg(z - 1)) :|: z - 1 >= 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: 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'') -{ 50 + 4*z'' }-> 1 + s4 :|: s4 >= 0, s4 <= 0 + 1, z = 1, z'' >= 0, z' = 0 if(z, z', z'') -{ 35 + 15*z' + 2*z'*z'' + 2*z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1 + 1, z' - 1 >= 0, z = 1, z'' >= 0 if(z, z', z'') -{ 49 + 4*z'' }-> 1 + s6 :|: s6 >= 0, s6 <= 0 + 1, z = 1, z' >= 0, z'' >= 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 12 + z' }-> s'' :|: s'' >= 0, s'' <= 0 + 1, z' >= 0, z = 0 minus(z, z') -{ 12 + 5*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1) + 1, z - 1 >= 0, z' = 0 minus(z, z') -{ 13 + 5*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1) + 1, s' >= 0, s' <= 2, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 11 + 5*z + z*z' + z' }-> s3 :|: s3 >= 0, s3 <= z + 1, z >= 0, z' >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encode_p}, {encode_if}, {encode_minus}, {encode_le}, {encode_s} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] le: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] if: runtime: O(n^2) [10 + 5*z' + z'*z'' + z''], size: O(n^1) [1 + z'] minus: runtime: O(n^2) [48 + 15*z + 2*z*z' + 4*z'], size: O(n^1) [1 + z] encArg: runtime: O(n^3) [188 + 628*z + 198*z^2 + 18*z^3], size: O(n^1) [1 + z] ---------------------------------------- (63) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (64) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 378 + s9 + 628*x_1 + 198*x_1^2 + 18*x_1^3 + 628*x_2 + 198*x_2^2 + 18*x_2^3 }-> s10 :|: s8 >= 0, s8 <= x_1 + 1, s9 >= 0, s9 <= x_2 + 1, s10 >= 0, s10 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 424 + 15*s11 + 2*s11*s12 + 4*s12 + 628*x_1 + 198*x_1^2 + 18*x_1^3 + 628*x_2 + 198*x_2^2 + 18*x_2^3 }-> s13 :|: s11 >= 0, s11 <= x_1 + 1, s12 >= 0, s12 <= x_2 + 1, s13 >= 0, s13 <= s11 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 574 + 5*s15 + s15*s16 + s16 + 628*x_1 + 198*x_1^2 + 18*x_1^3 + 628*x_2 + 198*x_2^2 + 18*x_2^3 + 628*x_3 + 198*x_3^2 + 18*x_3^3 }-> s17 :|: s14 >= 0, s14 <= x_1 + 1, s15 >= 0, s15 <= x_2 + 1, s16 >= 0, s16 <= x_3 + 1, s17 >= 0, s17 <= s15 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ -419 + 52*z + 90*z^2 + 18*z^3 }-> s30 :|: s29 >= 0, s29 <= z - 2 + 1, s30 >= 0, s30 <= 1 + s29, z - 2 >= 0 encArg(z) -{ -418 + 52*z + 90*z^2 + 18*z^3 }-> s33 :|: s31 >= 0, s31 <= z - 2 + 1, s32 >= 0, s32 <= s31, s33 >= 0, s33 <= s32, z - 2 >= 0 encArg(z) -{ 379 + s35 + 628*x_11 + 198*x_11^2 + 18*x_11^3 + 628*x_2' + 198*x_2'^2 + 18*x_2'^3 }-> s37 :|: s34 >= 0, s34 <= x_11 + 1, s35 >= 0, s35 <= x_2' + 1, s36 >= 0, s36 <= 2, s37 >= 0, s37 <= s36, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') encArg(z) -{ 425 + 15*s38 + 2*s38*s39 + 4*s39 + 628*x_12 + 198*x_12^2 + 18*x_12^3 + 628*x_2'' + 198*x_2''^2 + 18*x_2''^3 }-> s41 :|: s38 >= 0, s38 <= x_12 + 1, s39 >= 0, s39 <= x_2'' + 1, s40 >= 0, s40 <= s38 + 1, s41 >= 0, s41 <= s40, z = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0 encArg(z) -{ 575 + 5*s43 + s43*s44 + s44 + 628*x_13 + 198*x_13^2 + 18*x_13^3 + 628*x_21 + 198*x_21^2 + 18*x_21^3 + 628*x_3' + 198*x_3'^2 + 18*x_3'^3 }-> s46 :|: s42 >= 0, s42 <= x_13 + 1, s43 >= 0, s43 <= x_21 + 1, s44 >= 0, s44 <= x_3' + 1, s45 >= 0, s45 <= s43 + 1, s46 >= 0, s46 <= s45, z = 1 + (1 + x_13 + x_21 + x_3'), x_3' >= 0, x_13 >= 0, x_21 >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ -260 + 286*z + 144*z^2 + 18*z^3 }-> 1 + s7 :|: s7 >= 0, s7 <= z - 1 + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 574 + 5*s26 + s26*s27 + s27 + 628*z + 198*z^2 + 18*z^3 + 628*z' + 198*z'^2 + 18*z'^3 + 628*z'' + 198*z''^2 + 18*z''^3 }-> s28 :|: s25 >= 0, s25 <= z + 1, s26 >= 0, s26 <= z' + 1, s27 >= 0, s27 <= z'' + 1, s28 >= 0, s28 <= s26 + 1, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_le(z, z') -{ 378 + s20 + 628*z + 198*z^2 + 18*z^3 + 628*z' + 198*z'^2 + 18*z'^3 }-> s21 :|: s19 >= 0, s19 <= z + 1, s20 >= 0, s20 <= z' + 1, s21 >= 0, s21 <= 2, z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 424 + 15*s22 + 2*s22*s23 + 4*s23 + 628*z + 198*z^2 + 18*z^3 + 628*z' + 198*z'^2 + 18*z'^3 }-> s24 :|: s22 >= 0, s22 <= z + 1, s23 >= 0, s23 <= z' + 1, s24 >= 0, s24 <= s22 + 1, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_p(z) -{ -259 + 286*z + 144*z^2 + 18*z^3 }-> s48 :|: s47 >= 0, s47 <= z - 1 + 1, s48 >= 0, s48 <= 1 + s47, z - 1 >= 0 encode_p(z) -{ -258 + 286*z + 144*z^2 + 18*z^3 }-> s51 :|: s49 >= 0, s49 <= z - 1 + 1, s50 >= 0, s50 <= s49, s51 >= 0, s51 <= s50, z - 1 >= 0 encode_p(z) -{ 379 + s53 + 628*x_1561 + 198*x_1561^2 + 18*x_1561^3 + 628*x_2335 + 198*x_2335^2 + 18*x_2335^3 }-> s55 :|: s52 >= 0, s52 <= x_1561 + 1, s53 >= 0, s53 <= x_2335 + 1, s54 >= 0, s54 <= 2, s55 >= 0, s55 <= s54, x_1561 >= 0, x_2335 >= 0, z = 1 + x_1561 + x_2335 encode_p(z) -{ 425 + 15*s56 + 2*s56*s57 + 4*s57 + 628*x_1562 + 198*x_1562^2 + 18*x_1562^3 + 628*x_2336 + 198*x_2336^2 + 18*x_2336^3 }-> s59 :|: s56 >= 0, s56 <= x_1562 + 1, s57 >= 0, s57 <= x_2336 + 1, s58 >= 0, s58 <= s56 + 1, s59 >= 0, s59 <= s58, x_2336 >= 0, x_1562 >= 0, z = 1 + x_1562 + x_2336 encode_p(z) -{ 575 + 5*s61 + s61*s62 + s62 + 628*x_1563 + 198*x_1563^2 + 18*x_1563^3 + 628*x_2337 + 198*x_2337^2 + 18*x_2337^3 + 628*x_3111 + 198*x_3111^2 + 18*x_3111^3 }-> s64 :|: s60 >= 0, s60 <= x_1563 + 1, s61 >= 0, s61 <= x_2337 + 1, s62 >= 0, s62 <= x_3111 + 1, s63 >= 0, s63 <= s61 + 1, s64 >= 0, s64 <= s63, z = 1 + x_1563 + x_2337 + x_3111, x_2337 >= 0, x_1563 >= 0, x_3111 >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 188 + 628*z + 198*z^2 + 18*z^3 }-> 1 + s18 :|: s18 >= 0, s18 <= z + 1, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 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'') -{ 50 + 4*z'' }-> 1 + s4 :|: s4 >= 0, s4 <= 0 + 1, z = 1, z'' >= 0, z' = 0 if(z, z', z'') -{ 35 + 15*z' + 2*z'*z'' + 2*z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1 + 1, z' - 1 >= 0, z = 1, z'' >= 0 if(z, z', z'') -{ 49 + 4*z'' }-> 1 + s6 :|: s6 >= 0, s6 <= 0 + 1, z = 1, z' >= 0, z'' >= 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 12 + z' }-> s'' :|: s'' >= 0, s'' <= 0 + 1, z' >= 0, z = 0 minus(z, z') -{ 12 + 5*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1) + 1, z - 1 >= 0, z' = 0 minus(z, z') -{ 13 + 5*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1) + 1, s' >= 0, s' <= 2, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 11 + 5*z + z*z' + z' }-> s3 :|: s3 >= 0, s3 <= z + 1, z >= 0, z' >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encode_p}, {encode_if}, {encode_minus}, {encode_le}, {encode_s} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] le: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] if: runtime: O(n^2) [10 + 5*z' + z'*z'' + z''], size: O(n^1) [1 + z'] minus: runtime: O(n^2) [48 + 15*z + 2*z*z' + 4*z'], size: O(n^1) [1 + z] encArg: runtime: O(n^3) [188 + 628*z + 198*z^2 + 18*z^3], size: O(n^1) [1 + z] ---------------------------------------- (65) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: encode_p after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (66) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 378 + s9 + 628*x_1 + 198*x_1^2 + 18*x_1^3 + 628*x_2 + 198*x_2^2 + 18*x_2^3 }-> s10 :|: s8 >= 0, s8 <= x_1 + 1, s9 >= 0, s9 <= x_2 + 1, s10 >= 0, s10 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 424 + 15*s11 + 2*s11*s12 + 4*s12 + 628*x_1 + 198*x_1^2 + 18*x_1^3 + 628*x_2 + 198*x_2^2 + 18*x_2^3 }-> s13 :|: s11 >= 0, s11 <= x_1 + 1, s12 >= 0, s12 <= x_2 + 1, s13 >= 0, s13 <= s11 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 574 + 5*s15 + s15*s16 + s16 + 628*x_1 + 198*x_1^2 + 18*x_1^3 + 628*x_2 + 198*x_2^2 + 18*x_2^3 + 628*x_3 + 198*x_3^2 + 18*x_3^3 }-> s17 :|: s14 >= 0, s14 <= x_1 + 1, s15 >= 0, s15 <= x_2 + 1, s16 >= 0, s16 <= x_3 + 1, s17 >= 0, s17 <= s15 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ -419 + 52*z + 90*z^2 + 18*z^3 }-> s30 :|: s29 >= 0, s29 <= z - 2 + 1, s30 >= 0, s30 <= 1 + s29, z - 2 >= 0 encArg(z) -{ -418 + 52*z + 90*z^2 + 18*z^3 }-> s33 :|: s31 >= 0, s31 <= z - 2 + 1, s32 >= 0, s32 <= s31, s33 >= 0, s33 <= s32, z - 2 >= 0 encArg(z) -{ 379 + s35 + 628*x_11 + 198*x_11^2 + 18*x_11^3 + 628*x_2' + 198*x_2'^2 + 18*x_2'^3 }-> s37 :|: s34 >= 0, s34 <= x_11 + 1, s35 >= 0, s35 <= x_2' + 1, s36 >= 0, s36 <= 2, s37 >= 0, s37 <= s36, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') encArg(z) -{ 425 + 15*s38 + 2*s38*s39 + 4*s39 + 628*x_12 + 198*x_12^2 + 18*x_12^3 + 628*x_2'' + 198*x_2''^2 + 18*x_2''^3 }-> s41 :|: s38 >= 0, s38 <= x_12 + 1, s39 >= 0, s39 <= x_2'' + 1, s40 >= 0, s40 <= s38 + 1, s41 >= 0, s41 <= s40, z = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0 encArg(z) -{ 575 + 5*s43 + s43*s44 + s44 + 628*x_13 + 198*x_13^2 + 18*x_13^3 + 628*x_21 + 198*x_21^2 + 18*x_21^3 + 628*x_3' + 198*x_3'^2 + 18*x_3'^3 }-> s46 :|: s42 >= 0, s42 <= x_13 + 1, s43 >= 0, s43 <= x_21 + 1, s44 >= 0, s44 <= x_3' + 1, s45 >= 0, s45 <= s43 + 1, s46 >= 0, s46 <= s45, z = 1 + (1 + x_13 + x_21 + x_3'), x_3' >= 0, x_13 >= 0, x_21 >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ -260 + 286*z + 144*z^2 + 18*z^3 }-> 1 + s7 :|: s7 >= 0, s7 <= z - 1 + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 574 + 5*s26 + s26*s27 + s27 + 628*z + 198*z^2 + 18*z^3 + 628*z' + 198*z'^2 + 18*z'^3 + 628*z'' + 198*z''^2 + 18*z''^3 }-> s28 :|: s25 >= 0, s25 <= z + 1, s26 >= 0, s26 <= z' + 1, s27 >= 0, s27 <= z'' + 1, s28 >= 0, s28 <= s26 + 1, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_le(z, z') -{ 378 + s20 + 628*z + 198*z^2 + 18*z^3 + 628*z' + 198*z'^2 + 18*z'^3 }-> s21 :|: s19 >= 0, s19 <= z + 1, s20 >= 0, s20 <= z' + 1, s21 >= 0, s21 <= 2, z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 424 + 15*s22 + 2*s22*s23 + 4*s23 + 628*z + 198*z^2 + 18*z^3 + 628*z' + 198*z'^2 + 18*z'^3 }-> s24 :|: s22 >= 0, s22 <= z + 1, s23 >= 0, s23 <= z' + 1, s24 >= 0, s24 <= s22 + 1, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_p(z) -{ -259 + 286*z + 144*z^2 + 18*z^3 }-> s48 :|: s47 >= 0, s47 <= z - 1 + 1, s48 >= 0, s48 <= 1 + s47, z - 1 >= 0 encode_p(z) -{ -258 + 286*z + 144*z^2 + 18*z^3 }-> s51 :|: s49 >= 0, s49 <= z - 1 + 1, s50 >= 0, s50 <= s49, s51 >= 0, s51 <= s50, z - 1 >= 0 encode_p(z) -{ 379 + s53 + 628*x_1561 + 198*x_1561^2 + 18*x_1561^3 + 628*x_2335 + 198*x_2335^2 + 18*x_2335^3 }-> s55 :|: s52 >= 0, s52 <= x_1561 + 1, s53 >= 0, s53 <= x_2335 + 1, s54 >= 0, s54 <= 2, s55 >= 0, s55 <= s54, x_1561 >= 0, x_2335 >= 0, z = 1 + x_1561 + x_2335 encode_p(z) -{ 425 + 15*s56 + 2*s56*s57 + 4*s57 + 628*x_1562 + 198*x_1562^2 + 18*x_1562^3 + 628*x_2336 + 198*x_2336^2 + 18*x_2336^3 }-> s59 :|: s56 >= 0, s56 <= x_1562 + 1, s57 >= 0, s57 <= x_2336 + 1, s58 >= 0, s58 <= s56 + 1, s59 >= 0, s59 <= s58, x_2336 >= 0, x_1562 >= 0, z = 1 + x_1562 + x_2336 encode_p(z) -{ 575 + 5*s61 + s61*s62 + s62 + 628*x_1563 + 198*x_1563^2 + 18*x_1563^3 + 628*x_2337 + 198*x_2337^2 + 18*x_2337^3 + 628*x_3111 + 198*x_3111^2 + 18*x_3111^3 }-> s64 :|: s60 >= 0, s60 <= x_1563 + 1, s61 >= 0, s61 <= x_2337 + 1, s62 >= 0, s62 <= x_3111 + 1, s63 >= 0, s63 <= s61 + 1, s64 >= 0, s64 <= s63, z = 1 + x_1563 + x_2337 + x_3111, x_2337 >= 0, x_1563 >= 0, x_3111 >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 188 + 628*z + 198*z^2 + 18*z^3 }-> 1 + s18 :|: s18 >= 0, s18 <= z + 1, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 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'') -{ 50 + 4*z'' }-> 1 + s4 :|: s4 >= 0, s4 <= 0 + 1, z = 1, z'' >= 0, z' = 0 if(z, z', z'') -{ 35 + 15*z' + 2*z'*z'' + 2*z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1 + 1, z' - 1 >= 0, z = 1, z'' >= 0 if(z, z', z'') -{ 49 + 4*z'' }-> 1 + s6 :|: s6 >= 0, s6 <= 0 + 1, z = 1, z' >= 0, z'' >= 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 12 + z' }-> s'' :|: s'' >= 0, s'' <= 0 + 1, z' >= 0, z = 0 minus(z, z') -{ 12 + 5*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1) + 1, z - 1 >= 0, z' = 0 minus(z, z') -{ 13 + 5*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1) + 1, s' >= 0, s' <= 2, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 11 + 5*z + z*z' + z' }-> s3 :|: s3 >= 0, s3 <= z + 1, z >= 0, z' >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encode_p}, {encode_if}, {encode_minus}, {encode_le}, {encode_s} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] le: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] if: runtime: O(n^2) [10 + 5*z' + z'*z'' + z''], size: O(n^1) [1 + z'] minus: runtime: O(n^2) [48 + 15*z + 2*z*z' + 4*z'], size: O(n^1) [1 + z] encArg: runtime: O(n^3) [188 + 628*z + 198*z^2 + 18*z^3], size: O(n^1) [1 + z] encode_p: runtime: ?, size: O(n^1) [1 + z] ---------------------------------------- (67) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: encode_p after applying outer abstraction to obtain an ITS, resulting in: O(n^3) with polynomial bound: 1412 + 5000*z + 1677*z^2 + 162*z^3 ---------------------------------------- (68) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 378 + s9 + 628*x_1 + 198*x_1^2 + 18*x_1^3 + 628*x_2 + 198*x_2^2 + 18*x_2^3 }-> s10 :|: s8 >= 0, s8 <= x_1 + 1, s9 >= 0, s9 <= x_2 + 1, s10 >= 0, s10 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 424 + 15*s11 + 2*s11*s12 + 4*s12 + 628*x_1 + 198*x_1^2 + 18*x_1^3 + 628*x_2 + 198*x_2^2 + 18*x_2^3 }-> s13 :|: s11 >= 0, s11 <= x_1 + 1, s12 >= 0, s12 <= x_2 + 1, s13 >= 0, s13 <= s11 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 574 + 5*s15 + s15*s16 + s16 + 628*x_1 + 198*x_1^2 + 18*x_1^3 + 628*x_2 + 198*x_2^2 + 18*x_2^3 + 628*x_3 + 198*x_3^2 + 18*x_3^3 }-> s17 :|: s14 >= 0, s14 <= x_1 + 1, s15 >= 0, s15 <= x_2 + 1, s16 >= 0, s16 <= x_3 + 1, s17 >= 0, s17 <= s15 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ -419 + 52*z + 90*z^2 + 18*z^3 }-> s30 :|: s29 >= 0, s29 <= z - 2 + 1, s30 >= 0, s30 <= 1 + s29, z - 2 >= 0 encArg(z) -{ -418 + 52*z + 90*z^2 + 18*z^3 }-> s33 :|: s31 >= 0, s31 <= z - 2 + 1, s32 >= 0, s32 <= s31, s33 >= 0, s33 <= s32, z - 2 >= 0 encArg(z) -{ 379 + s35 + 628*x_11 + 198*x_11^2 + 18*x_11^3 + 628*x_2' + 198*x_2'^2 + 18*x_2'^3 }-> s37 :|: s34 >= 0, s34 <= x_11 + 1, s35 >= 0, s35 <= x_2' + 1, s36 >= 0, s36 <= 2, s37 >= 0, s37 <= s36, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') encArg(z) -{ 425 + 15*s38 + 2*s38*s39 + 4*s39 + 628*x_12 + 198*x_12^2 + 18*x_12^3 + 628*x_2'' + 198*x_2''^2 + 18*x_2''^3 }-> s41 :|: s38 >= 0, s38 <= x_12 + 1, s39 >= 0, s39 <= x_2'' + 1, s40 >= 0, s40 <= s38 + 1, s41 >= 0, s41 <= s40, z = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0 encArg(z) -{ 575 + 5*s43 + s43*s44 + s44 + 628*x_13 + 198*x_13^2 + 18*x_13^3 + 628*x_21 + 198*x_21^2 + 18*x_21^3 + 628*x_3' + 198*x_3'^2 + 18*x_3'^3 }-> s46 :|: s42 >= 0, s42 <= x_13 + 1, s43 >= 0, s43 <= x_21 + 1, s44 >= 0, s44 <= x_3' + 1, s45 >= 0, s45 <= s43 + 1, s46 >= 0, s46 <= s45, z = 1 + (1 + x_13 + x_21 + x_3'), x_3' >= 0, x_13 >= 0, x_21 >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ -260 + 286*z + 144*z^2 + 18*z^3 }-> 1 + s7 :|: s7 >= 0, s7 <= z - 1 + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 574 + 5*s26 + s26*s27 + s27 + 628*z + 198*z^2 + 18*z^3 + 628*z' + 198*z'^2 + 18*z'^3 + 628*z'' + 198*z''^2 + 18*z''^3 }-> s28 :|: s25 >= 0, s25 <= z + 1, s26 >= 0, s26 <= z' + 1, s27 >= 0, s27 <= z'' + 1, s28 >= 0, s28 <= s26 + 1, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_le(z, z') -{ 378 + s20 + 628*z + 198*z^2 + 18*z^3 + 628*z' + 198*z'^2 + 18*z'^3 }-> s21 :|: s19 >= 0, s19 <= z + 1, s20 >= 0, s20 <= z' + 1, s21 >= 0, s21 <= 2, z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 424 + 15*s22 + 2*s22*s23 + 4*s23 + 628*z + 198*z^2 + 18*z^3 + 628*z' + 198*z'^2 + 18*z'^3 }-> s24 :|: s22 >= 0, s22 <= z + 1, s23 >= 0, s23 <= z' + 1, s24 >= 0, s24 <= s22 + 1, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_p(z) -{ -259 + 286*z + 144*z^2 + 18*z^3 }-> s48 :|: s47 >= 0, s47 <= z - 1 + 1, s48 >= 0, s48 <= 1 + s47, z - 1 >= 0 encode_p(z) -{ -258 + 286*z + 144*z^2 + 18*z^3 }-> s51 :|: s49 >= 0, s49 <= z - 1 + 1, s50 >= 0, s50 <= s49, s51 >= 0, s51 <= s50, z - 1 >= 0 encode_p(z) -{ 379 + s53 + 628*x_1561 + 198*x_1561^2 + 18*x_1561^3 + 628*x_2335 + 198*x_2335^2 + 18*x_2335^3 }-> s55 :|: s52 >= 0, s52 <= x_1561 + 1, s53 >= 0, s53 <= x_2335 + 1, s54 >= 0, s54 <= 2, s55 >= 0, s55 <= s54, x_1561 >= 0, x_2335 >= 0, z = 1 + x_1561 + x_2335 encode_p(z) -{ 425 + 15*s56 + 2*s56*s57 + 4*s57 + 628*x_1562 + 198*x_1562^2 + 18*x_1562^3 + 628*x_2336 + 198*x_2336^2 + 18*x_2336^3 }-> s59 :|: s56 >= 0, s56 <= x_1562 + 1, s57 >= 0, s57 <= x_2336 + 1, s58 >= 0, s58 <= s56 + 1, s59 >= 0, s59 <= s58, x_2336 >= 0, x_1562 >= 0, z = 1 + x_1562 + x_2336 encode_p(z) -{ 575 + 5*s61 + s61*s62 + s62 + 628*x_1563 + 198*x_1563^2 + 18*x_1563^3 + 628*x_2337 + 198*x_2337^2 + 18*x_2337^3 + 628*x_3111 + 198*x_3111^2 + 18*x_3111^3 }-> s64 :|: s60 >= 0, s60 <= x_1563 + 1, s61 >= 0, s61 <= x_2337 + 1, s62 >= 0, s62 <= x_3111 + 1, s63 >= 0, s63 <= s61 + 1, s64 >= 0, s64 <= s63, z = 1 + x_1563 + x_2337 + x_3111, x_2337 >= 0, x_1563 >= 0, x_3111 >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 188 + 628*z + 198*z^2 + 18*z^3 }-> 1 + s18 :|: s18 >= 0, s18 <= z + 1, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 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'') -{ 50 + 4*z'' }-> 1 + s4 :|: s4 >= 0, s4 <= 0 + 1, z = 1, z'' >= 0, z' = 0 if(z, z', z'') -{ 35 + 15*z' + 2*z'*z'' + 2*z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1 + 1, z' - 1 >= 0, z = 1, z'' >= 0 if(z, z', z'') -{ 49 + 4*z'' }-> 1 + s6 :|: s6 >= 0, s6 <= 0 + 1, z = 1, z' >= 0, z'' >= 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 12 + z' }-> s'' :|: s'' >= 0, s'' <= 0 + 1, z' >= 0, z = 0 minus(z, z') -{ 12 + 5*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1) + 1, z - 1 >= 0, z' = 0 minus(z, z') -{ 13 + 5*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1) + 1, s' >= 0, s' <= 2, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 11 + 5*z + z*z' + z' }-> s3 :|: s3 >= 0, s3 <= z + 1, z >= 0, z' >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encode_if}, {encode_minus}, {encode_le}, {encode_s} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] le: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] if: runtime: O(n^2) [10 + 5*z' + z'*z'' + z''], size: O(n^1) [1 + z'] minus: runtime: O(n^2) [48 + 15*z + 2*z*z' + 4*z'], size: O(n^1) [1 + z] encArg: runtime: O(n^3) [188 + 628*z + 198*z^2 + 18*z^3], size: O(n^1) [1 + z] encode_p: runtime: O(n^3) [1412 + 5000*z + 1677*z^2 + 162*z^3], size: O(n^1) [1 + z] ---------------------------------------- (69) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (70) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 378 + s9 + 628*x_1 + 198*x_1^2 + 18*x_1^3 + 628*x_2 + 198*x_2^2 + 18*x_2^3 }-> s10 :|: s8 >= 0, s8 <= x_1 + 1, s9 >= 0, s9 <= x_2 + 1, s10 >= 0, s10 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 424 + 15*s11 + 2*s11*s12 + 4*s12 + 628*x_1 + 198*x_1^2 + 18*x_1^3 + 628*x_2 + 198*x_2^2 + 18*x_2^3 }-> s13 :|: s11 >= 0, s11 <= x_1 + 1, s12 >= 0, s12 <= x_2 + 1, s13 >= 0, s13 <= s11 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 574 + 5*s15 + s15*s16 + s16 + 628*x_1 + 198*x_1^2 + 18*x_1^3 + 628*x_2 + 198*x_2^2 + 18*x_2^3 + 628*x_3 + 198*x_3^2 + 18*x_3^3 }-> s17 :|: s14 >= 0, s14 <= x_1 + 1, s15 >= 0, s15 <= x_2 + 1, s16 >= 0, s16 <= x_3 + 1, s17 >= 0, s17 <= s15 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ -419 + 52*z + 90*z^2 + 18*z^3 }-> s30 :|: s29 >= 0, s29 <= z - 2 + 1, s30 >= 0, s30 <= 1 + s29, z - 2 >= 0 encArg(z) -{ -418 + 52*z + 90*z^2 + 18*z^3 }-> s33 :|: s31 >= 0, s31 <= z - 2 + 1, s32 >= 0, s32 <= s31, s33 >= 0, s33 <= s32, z - 2 >= 0 encArg(z) -{ 379 + s35 + 628*x_11 + 198*x_11^2 + 18*x_11^3 + 628*x_2' + 198*x_2'^2 + 18*x_2'^3 }-> s37 :|: s34 >= 0, s34 <= x_11 + 1, s35 >= 0, s35 <= x_2' + 1, s36 >= 0, s36 <= 2, s37 >= 0, s37 <= s36, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') encArg(z) -{ 425 + 15*s38 + 2*s38*s39 + 4*s39 + 628*x_12 + 198*x_12^2 + 18*x_12^3 + 628*x_2'' + 198*x_2''^2 + 18*x_2''^3 }-> s41 :|: s38 >= 0, s38 <= x_12 + 1, s39 >= 0, s39 <= x_2'' + 1, s40 >= 0, s40 <= s38 + 1, s41 >= 0, s41 <= s40, z = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0 encArg(z) -{ 575 + 5*s43 + s43*s44 + s44 + 628*x_13 + 198*x_13^2 + 18*x_13^3 + 628*x_21 + 198*x_21^2 + 18*x_21^3 + 628*x_3' + 198*x_3'^2 + 18*x_3'^3 }-> s46 :|: s42 >= 0, s42 <= x_13 + 1, s43 >= 0, s43 <= x_21 + 1, s44 >= 0, s44 <= x_3' + 1, s45 >= 0, s45 <= s43 + 1, s46 >= 0, s46 <= s45, z = 1 + (1 + x_13 + x_21 + x_3'), x_3' >= 0, x_13 >= 0, x_21 >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ -260 + 286*z + 144*z^2 + 18*z^3 }-> 1 + s7 :|: s7 >= 0, s7 <= z - 1 + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 574 + 5*s26 + s26*s27 + s27 + 628*z + 198*z^2 + 18*z^3 + 628*z' + 198*z'^2 + 18*z'^3 + 628*z'' + 198*z''^2 + 18*z''^3 }-> s28 :|: s25 >= 0, s25 <= z + 1, s26 >= 0, s26 <= z' + 1, s27 >= 0, s27 <= z'' + 1, s28 >= 0, s28 <= s26 + 1, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_le(z, z') -{ 378 + s20 + 628*z + 198*z^2 + 18*z^3 + 628*z' + 198*z'^2 + 18*z'^3 }-> s21 :|: s19 >= 0, s19 <= z + 1, s20 >= 0, s20 <= z' + 1, s21 >= 0, s21 <= 2, z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 424 + 15*s22 + 2*s22*s23 + 4*s23 + 628*z + 198*z^2 + 18*z^3 + 628*z' + 198*z'^2 + 18*z'^3 }-> s24 :|: s22 >= 0, s22 <= z + 1, s23 >= 0, s23 <= z' + 1, s24 >= 0, s24 <= s22 + 1, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_p(z) -{ -259 + 286*z + 144*z^2 + 18*z^3 }-> s48 :|: s47 >= 0, s47 <= z - 1 + 1, s48 >= 0, s48 <= 1 + s47, z - 1 >= 0 encode_p(z) -{ -258 + 286*z + 144*z^2 + 18*z^3 }-> s51 :|: s49 >= 0, s49 <= z - 1 + 1, s50 >= 0, s50 <= s49, s51 >= 0, s51 <= s50, z - 1 >= 0 encode_p(z) -{ 379 + s53 + 628*x_1561 + 198*x_1561^2 + 18*x_1561^3 + 628*x_2335 + 198*x_2335^2 + 18*x_2335^3 }-> s55 :|: s52 >= 0, s52 <= x_1561 + 1, s53 >= 0, s53 <= x_2335 + 1, s54 >= 0, s54 <= 2, s55 >= 0, s55 <= s54, x_1561 >= 0, x_2335 >= 0, z = 1 + x_1561 + x_2335 encode_p(z) -{ 425 + 15*s56 + 2*s56*s57 + 4*s57 + 628*x_1562 + 198*x_1562^2 + 18*x_1562^3 + 628*x_2336 + 198*x_2336^2 + 18*x_2336^3 }-> s59 :|: s56 >= 0, s56 <= x_1562 + 1, s57 >= 0, s57 <= x_2336 + 1, s58 >= 0, s58 <= s56 + 1, s59 >= 0, s59 <= s58, x_2336 >= 0, x_1562 >= 0, z = 1 + x_1562 + x_2336 encode_p(z) -{ 575 + 5*s61 + s61*s62 + s62 + 628*x_1563 + 198*x_1563^2 + 18*x_1563^3 + 628*x_2337 + 198*x_2337^2 + 18*x_2337^3 + 628*x_3111 + 198*x_3111^2 + 18*x_3111^3 }-> s64 :|: s60 >= 0, s60 <= x_1563 + 1, s61 >= 0, s61 <= x_2337 + 1, s62 >= 0, s62 <= x_3111 + 1, s63 >= 0, s63 <= s61 + 1, s64 >= 0, s64 <= s63, z = 1 + x_1563 + x_2337 + x_3111, x_2337 >= 0, x_1563 >= 0, x_3111 >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 188 + 628*z + 198*z^2 + 18*z^3 }-> 1 + s18 :|: s18 >= 0, s18 <= z + 1, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 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'') -{ 50 + 4*z'' }-> 1 + s4 :|: s4 >= 0, s4 <= 0 + 1, z = 1, z'' >= 0, z' = 0 if(z, z', z'') -{ 35 + 15*z' + 2*z'*z'' + 2*z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1 + 1, z' - 1 >= 0, z = 1, z'' >= 0 if(z, z', z'') -{ 49 + 4*z'' }-> 1 + s6 :|: s6 >= 0, s6 <= 0 + 1, z = 1, z' >= 0, z'' >= 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 12 + z' }-> s'' :|: s'' >= 0, s'' <= 0 + 1, z' >= 0, z = 0 minus(z, z') -{ 12 + 5*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1) + 1, z - 1 >= 0, z' = 0 minus(z, z') -{ 13 + 5*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1) + 1, s' >= 0, s' <= 2, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 11 + 5*z + z*z' + z' }-> s3 :|: s3 >= 0, s3 <= z + 1, z >= 0, z' >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encode_if}, {encode_minus}, {encode_le}, {encode_s} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] le: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] if: runtime: O(n^2) [10 + 5*z' + z'*z'' + z''], size: O(n^1) [1 + z'] minus: runtime: O(n^2) [48 + 15*z + 2*z*z' + 4*z'], size: O(n^1) [1 + z] encArg: runtime: O(n^3) [188 + 628*z + 198*z^2 + 18*z^3], size: O(n^1) [1 + z] encode_p: runtime: O(n^3) [1412 + 5000*z + 1677*z^2 + 162*z^3], size: O(n^1) [1 + z] ---------------------------------------- (71) 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: 2 + z' ---------------------------------------- (72) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 378 + s9 + 628*x_1 + 198*x_1^2 + 18*x_1^3 + 628*x_2 + 198*x_2^2 + 18*x_2^3 }-> s10 :|: s8 >= 0, s8 <= x_1 + 1, s9 >= 0, s9 <= x_2 + 1, s10 >= 0, s10 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 424 + 15*s11 + 2*s11*s12 + 4*s12 + 628*x_1 + 198*x_1^2 + 18*x_1^3 + 628*x_2 + 198*x_2^2 + 18*x_2^3 }-> s13 :|: s11 >= 0, s11 <= x_1 + 1, s12 >= 0, s12 <= x_2 + 1, s13 >= 0, s13 <= s11 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 574 + 5*s15 + s15*s16 + s16 + 628*x_1 + 198*x_1^2 + 18*x_1^3 + 628*x_2 + 198*x_2^2 + 18*x_2^3 + 628*x_3 + 198*x_3^2 + 18*x_3^3 }-> s17 :|: s14 >= 0, s14 <= x_1 + 1, s15 >= 0, s15 <= x_2 + 1, s16 >= 0, s16 <= x_3 + 1, s17 >= 0, s17 <= s15 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ -419 + 52*z + 90*z^2 + 18*z^3 }-> s30 :|: s29 >= 0, s29 <= z - 2 + 1, s30 >= 0, s30 <= 1 + s29, z - 2 >= 0 encArg(z) -{ -418 + 52*z + 90*z^2 + 18*z^3 }-> s33 :|: s31 >= 0, s31 <= z - 2 + 1, s32 >= 0, s32 <= s31, s33 >= 0, s33 <= s32, z - 2 >= 0 encArg(z) -{ 379 + s35 + 628*x_11 + 198*x_11^2 + 18*x_11^3 + 628*x_2' + 198*x_2'^2 + 18*x_2'^3 }-> s37 :|: s34 >= 0, s34 <= x_11 + 1, s35 >= 0, s35 <= x_2' + 1, s36 >= 0, s36 <= 2, s37 >= 0, s37 <= s36, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') encArg(z) -{ 425 + 15*s38 + 2*s38*s39 + 4*s39 + 628*x_12 + 198*x_12^2 + 18*x_12^3 + 628*x_2'' + 198*x_2''^2 + 18*x_2''^3 }-> s41 :|: s38 >= 0, s38 <= x_12 + 1, s39 >= 0, s39 <= x_2'' + 1, s40 >= 0, s40 <= s38 + 1, s41 >= 0, s41 <= s40, z = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0 encArg(z) -{ 575 + 5*s43 + s43*s44 + s44 + 628*x_13 + 198*x_13^2 + 18*x_13^3 + 628*x_21 + 198*x_21^2 + 18*x_21^3 + 628*x_3' + 198*x_3'^2 + 18*x_3'^3 }-> s46 :|: s42 >= 0, s42 <= x_13 + 1, s43 >= 0, s43 <= x_21 + 1, s44 >= 0, s44 <= x_3' + 1, s45 >= 0, s45 <= s43 + 1, s46 >= 0, s46 <= s45, z = 1 + (1 + x_13 + x_21 + x_3'), x_3' >= 0, x_13 >= 0, x_21 >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ -260 + 286*z + 144*z^2 + 18*z^3 }-> 1 + s7 :|: s7 >= 0, s7 <= z - 1 + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 574 + 5*s26 + s26*s27 + s27 + 628*z + 198*z^2 + 18*z^3 + 628*z' + 198*z'^2 + 18*z'^3 + 628*z'' + 198*z''^2 + 18*z''^3 }-> s28 :|: s25 >= 0, s25 <= z + 1, s26 >= 0, s26 <= z' + 1, s27 >= 0, s27 <= z'' + 1, s28 >= 0, s28 <= s26 + 1, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_le(z, z') -{ 378 + s20 + 628*z + 198*z^2 + 18*z^3 + 628*z' + 198*z'^2 + 18*z'^3 }-> s21 :|: s19 >= 0, s19 <= z + 1, s20 >= 0, s20 <= z' + 1, s21 >= 0, s21 <= 2, z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 424 + 15*s22 + 2*s22*s23 + 4*s23 + 628*z + 198*z^2 + 18*z^3 + 628*z' + 198*z'^2 + 18*z'^3 }-> s24 :|: s22 >= 0, s22 <= z + 1, s23 >= 0, s23 <= z' + 1, s24 >= 0, s24 <= s22 + 1, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_p(z) -{ -259 + 286*z + 144*z^2 + 18*z^3 }-> s48 :|: s47 >= 0, s47 <= z - 1 + 1, s48 >= 0, s48 <= 1 + s47, z - 1 >= 0 encode_p(z) -{ -258 + 286*z + 144*z^2 + 18*z^3 }-> s51 :|: s49 >= 0, s49 <= z - 1 + 1, s50 >= 0, s50 <= s49, s51 >= 0, s51 <= s50, z - 1 >= 0 encode_p(z) -{ 379 + s53 + 628*x_1561 + 198*x_1561^2 + 18*x_1561^3 + 628*x_2335 + 198*x_2335^2 + 18*x_2335^3 }-> s55 :|: s52 >= 0, s52 <= x_1561 + 1, s53 >= 0, s53 <= x_2335 + 1, s54 >= 0, s54 <= 2, s55 >= 0, s55 <= s54, x_1561 >= 0, x_2335 >= 0, z = 1 + x_1561 + x_2335 encode_p(z) -{ 425 + 15*s56 + 2*s56*s57 + 4*s57 + 628*x_1562 + 198*x_1562^2 + 18*x_1562^3 + 628*x_2336 + 198*x_2336^2 + 18*x_2336^3 }-> s59 :|: s56 >= 0, s56 <= x_1562 + 1, s57 >= 0, s57 <= x_2336 + 1, s58 >= 0, s58 <= s56 + 1, s59 >= 0, s59 <= s58, x_2336 >= 0, x_1562 >= 0, z = 1 + x_1562 + x_2336 encode_p(z) -{ 575 + 5*s61 + s61*s62 + s62 + 628*x_1563 + 198*x_1563^2 + 18*x_1563^3 + 628*x_2337 + 198*x_2337^2 + 18*x_2337^3 + 628*x_3111 + 198*x_3111^2 + 18*x_3111^3 }-> s64 :|: s60 >= 0, s60 <= x_1563 + 1, s61 >= 0, s61 <= x_2337 + 1, s62 >= 0, s62 <= x_3111 + 1, s63 >= 0, s63 <= s61 + 1, s64 >= 0, s64 <= s63, z = 1 + x_1563 + x_2337 + x_3111, x_2337 >= 0, x_1563 >= 0, x_3111 >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 188 + 628*z + 198*z^2 + 18*z^3 }-> 1 + s18 :|: s18 >= 0, s18 <= z + 1, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 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'') -{ 50 + 4*z'' }-> 1 + s4 :|: s4 >= 0, s4 <= 0 + 1, z = 1, z'' >= 0, z' = 0 if(z, z', z'') -{ 35 + 15*z' + 2*z'*z'' + 2*z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1 + 1, z' - 1 >= 0, z = 1, z'' >= 0 if(z, z', z'') -{ 49 + 4*z'' }-> 1 + s6 :|: s6 >= 0, s6 <= 0 + 1, z = 1, z' >= 0, z'' >= 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 12 + z' }-> s'' :|: s'' >= 0, s'' <= 0 + 1, z' >= 0, z = 0 minus(z, z') -{ 12 + 5*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1) + 1, z - 1 >= 0, z' = 0 minus(z, z') -{ 13 + 5*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1) + 1, s' >= 0, s' <= 2, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 11 + 5*z + z*z' + z' }-> s3 :|: s3 >= 0, s3 <= z + 1, z >= 0, z' >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encode_if}, {encode_minus}, {encode_le}, {encode_s} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] le: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] if: runtime: O(n^2) [10 + 5*z' + z'*z'' + z''], size: O(n^1) [1 + z'] minus: runtime: O(n^2) [48 + 15*z + 2*z*z' + 4*z'], size: O(n^1) [1 + z] encArg: runtime: O(n^3) [188 + 628*z + 198*z^2 + 18*z^3], size: O(n^1) [1 + z] encode_p: runtime: O(n^3) [1412 + 5000*z + 1677*z^2 + 162*z^3], size: O(n^1) [1 + z] encode_if: runtime: ?, size: O(n^1) [2 + z'] ---------------------------------------- (73) 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: 581 + 628*z + 198*z^2 + 18*z^3 + 634*z' + z'*z'' + 198*z'^2 + 18*z'^3 + 630*z'' + 198*z''^2 + 18*z''^3 ---------------------------------------- (74) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 378 + s9 + 628*x_1 + 198*x_1^2 + 18*x_1^3 + 628*x_2 + 198*x_2^2 + 18*x_2^3 }-> s10 :|: s8 >= 0, s8 <= x_1 + 1, s9 >= 0, s9 <= x_2 + 1, s10 >= 0, s10 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 424 + 15*s11 + 2*s11*s12 + 4*s12 + 628*x_1 + 198*x_1^2 + 18*x_1^3 + 628*x_2 + 198*x_2^2 + 18*x_2^3 }-> s13 :|: s11 >= 0, s11 <= x_1 + 1, s12 >= 0, s12 <= x_2 + 1, s13 >= 0, s13 <= s11 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 574 + 5*s15 + s15*s16 + s16 + 628*x_1 + 198*x_1^2 + 18*x_1^3 + 628*x_2 + 198*x_2^2 + 18*x_2^3 + 628*x_3 + 198*x_3^2 + 18*x_3^3 }-> s17 :|: s14 >= 0, s14 <= x_1 + 1, s15 >= 0, s15 <= x_2 + 1, s16 >= 0, s16 <= x_3 + 1, s17 >= 0, s17 <= s15 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ -419 + 52*z + 90*z^2 + 18*z^3 }-> s30 :|: s29 >= 0, s29 <= z - 2 + 1, s30 >= 0, s30 <= 1 + s29, z - 2 >= 0 encArg(z) -{ -418 + 52*z + 90*z^2 + 18*z^3 }-> s33 :|: s31 >= 0, s31 <= z - 2 + 1, s32 >= 0, s32 <= s31, s33 >= 0, s33 <= s32, z - 2 >= 0 encArg(z) -{ 379 + s35 + 628*x_11 + 198*x_11^2 + 18*x_11^3 + 628*x_2' + 198*x_2'^2 + 18*x_2'^3 }-> s37 :|: s34 >= 0, s34 <= x_11 + 1, s35 >= 0, s35 <= x_2' + 1, s36 >= 0, s36 <= 2, s37 >= 0, s37 <= s36, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') encArg(z) -{ 425 + 15*s38 + 2*s38*s39 + 4*s39 + 628*x_12 + 198*x_12^2 + 18*x_12^3 + 628*x_2'' + 198*x_2''^2 + 18*x_2''^3 }-> s41 :|: s38 >= 0, s38 <= x_12 + 1, s39 >= 0, s39 <= x_2'' + 1, s40 >= 0, s40 <= s38 + 1, s41 >= 0, s41 <= s40, z = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0 encArg(z) -{ 575 + 5*s43 + s43*s44 + s44 + 628*x_13 + 198*x_13^2 + 18*x_13^3 + 628*x_21 + 198*x_21^2 + 18*x_21^3 + 628*x_3' + 198*x_3'^2 + 18*x_3'^3 }-> s46 :|: s42 >= 0, s42 <= x_13 + 1, s43 >= 0, s43 <= x_21 + 1, s44 >= 0, s44 <= x_3' + 1, s45 >= 0, s45 <= s43 + 1, s46 >= 0, s46 <= s45, z = 1 + (1 + x_13 + x_21 + x_3'), x_3' >= 0, x_13 >= 0, x_21 >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ -260 + 286*z + 144*z^2 + 18*z^3 }-> 1 + s7 :|: s7 >= 0, s7 <= z - 1 + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 574 + 5*s26 + s26*s27 + s27 + 628*z + 198*z^2 + 18*z^3 + 628*z' + 198*z'^2 + 18*z'^3 + 628*z'' + 198*z''^2 + 18*z''^3 }-> s28 :|: s25 >= 0, s25 <= z + 1, s26 >= 0, s26 <= z' + 1, s27 >= 0, s27 <= z'' + 1, s28 >= 0, s28 <= s26 + 1, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_le(z, z') -{ 378 + s20 + 628*z + 198*z^2 + 18*z^3 + 628*z' + 198*z'^2 + 18*z'^3 }-> s21 :|: s19 >= 0, s19 <= z + 1, s20 >= 0, s20 <= z' + 1, s21 >= 0, s21 <= 2, z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 424 + 15*s22 + 2*s22*s23 + 4*s23 + 628*z + 198*z^2 + 18*z^3 + 628*z' + 198*z'^2 + 18*z'^3 }-> s24 :|: s22 >= 0, s22 <= z + 1, s23 >= 0, s23 <= z' + 1, s24 >= 0, s24 <= s22 + 1, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_p(z) -{ -259 + 286*z + 144*z^2 + 18*z^3 }-> s48 :|: s47 >= 0, s47 <= z - 1 + 1, s48 >= 0, s48 <= 1 + s47, z - 1 >= 0 encode_p(z) -{ -258 + 286*z + 144*z^2 + 18*z^3 }-> s51 :|: s49 >= 0, s49 <= z - 1 + 1, s50 >= 0, s50 <= s49, s51 >= 0, s51 <= s50, z - 1 >= 0 encode_p(z) -{ 379 + s53 + 628*x_1561 + 198*x_1561^2 + 18*x_1561^3 + 628*x_2335 + 198*x_2335^2 + 18*x_2335^3 }-> s55 :|: s52 >= 0, s52 <= x_1561 + 1, s53 >= 0, s53 <= x_2335 + 1, s54 >= 0, s54 <= 2, s55 >= 0, s55 <= s54, x_1561 >= 0, x_2335 >= 0, z = 1 + x_1561 + x_2335 encode_p(z) -{ 425 + 15*s56 + 2*s56*s57 + 4*s57 + 628*x_1562 + 198*x_1562^2 + 18*x_1562^3 + 628*x_2336 + 198*x_2336^2 + 18*x_2336^3 }-> s59 :|: s56 >= 0, s56 <= x_1562 + 1, s57 >= 0, s57 <= x_2336 + 1, s58 >= 0, s58 <= s56 + 1, s59 >= 0, s59 <= s58, x_2336 >= 0, x_1562 >= 0, z = 1 + x_1562 + x_2336 encode_p(z) -{ 575 + 5*s61 + s61*s62 + s62 + 628*x_1563 + 198*x_1563^2 + 18*x_1563^3 + 628*x_2337 + 198*x_2337^2 + 18*x_2337^3 + 628*x_3111 + 198*x_3111^2 + 18*x_3111^3 }-> s64 :|: s60 >= 0, s60 <= x_1563 + 1, s61 >= 0, s61 <= x_2337 + 1, s62 >= 0, s62 <= x_3111 + 1, s63 >= 0, s63 <= s61 + 1, s64 >= 0, s64 <= s63, z = 1 + x_1563 + x_2337 + x_3111, x_2337 >= 0, x_1563 >= 0, x_3111 >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 188 + 628*z + 198*z^2 + 18*z^3 }-> 1 + s18 :|: s18 >= 0, s18 <= z + 1, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 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'') -{ 50 + 4*z'' }-> 1 + s4 :|: s4 >= 0, s4 <= 0 + 1, z = 1, z'' >= 0, z' = 0 if(z, z', z'') -{ 35 + 15*z' + 2*z'*z'' + 2*z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1 + 1, z' - 1 >= 0, z = 1, z'' >= 0 if(z, z', z'') -{ 49 + 4*z'' }-> 1 + s6 :|: s6 >= 0, s6 <= 0 + 1, z = 1, z' >= 0, z'' >= 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 12 + z' }-> s'' :|: s'' >= 0, s'' <= 0 + 1, z' >= 0, z = 0 minus(z, z') -{ 12 + 5*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1) + 1, z - 1 >= 0, z' = 0 minus(z, z') -{ 13 + 5*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1) + 1, s' >= 0, s' <= 2, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 11 + 5*z + z*z' + z' }-> s3 :|: s3 >= 0, s3 <= z + 1, z >= 0, z' >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encode_minus}, {encode_le}, {encode_s} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] le: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] if: runtime: O(n^2) [10 + 5*z' + z'*z'' + z''], size: O(n^1) [1 + z'] minus: runtime: O(n^2) [48 + 15*z + 2*z*z' + 4*z'], size: O(n^1) [1 + z] encArg: runtime: O(n^3) [188 + 628*z + 198*z^2 + 18*z^3], size: O(n^1) [1 + z] encode_p: runtime: O(n^3) [1412 + 5000*z + 1677*z^2 + 162*z^3], size: O(n^1) [1 + z] encode_if: runtime: O(n^3) [581 + 628*z + 198*z^2 + 18*z^3 + 634*z' + z'*z'' + 198*z'^2 + 18*z'^3 + 630*z'' + 198*z''^2 + 18*z''^3], size: O(n^1) [2 + z'] ---------------------------------------- (75) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (76) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 378 + s9 + 628*x_1 + 198*x_1^2 + 18*x_1^3 + 628*x_2 + 198*x_2^2 + 18*x_2^3 }-> s10 :|: s8 >= 0, s8 <= x_1 + 1, s9 >= 0, s9 <= x_2 + 1, s10 >= 0, s10 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 424 + 15*s11 + 2*s11*s12 + 4*s12 + 628*x_1 + 198*x_1^2 + 18*x_1^3 + 628*x_2 + 198*x_2^2 + 18*x_2^3 }-> s13 :|: s11 >= 0, s11 <= x_1 + 1, s12 >= 0, s12 <= x_2 + 1, s13 >= 0, s13 <= s11 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 574 + 5*s15 + s15*s16 + s16 + 628*x_1 + 198*x_1^2 + 18*x_1^3 + 628*x_2 + 198*x_2^2 + 18*x_2^3 + 628*x_3 + 198*x_3^2 + 18*x_3^3 }-> s17 :|: s14 >= 0, s14 <= x_1 + 1, s15 >= 0, s15 <= x_2 + 1, s16 >= 0, s16 <= x_3 + 1, s17 >= 0, s17 <= s15 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ -419 + 52*z + 90*z^2 + 18*z^3 }-> s30 :|: s29 >= 0, s29 <= z - 2 + 1, s30 >= 0, s30 <= 1 + s29, z - 2 >= 0 encArg(z) -{ -418 + 52*z + 90*z^2 + 18*z^3 }-> s33 :|: s31 >= 0, s31 <= z - 2 + 1, s32 >= 0, s32 <= s31, s33 >= 0, s33 <= s32, z - 2 >= 0 encArg(z) -{ 379 + s35 + 628*x_11 + 198*x_11^2 + 18*x_11^3 + 628*x_2' + 198*x_2'^2 + 18*x_2'^3 }-> s37 :|: s34 >= 0, s34 <= x_11 + 1, s35 >= 0, s35 <= x_2' + 1, s36 >= 0, s36 <= 2, s37 >= 0, s37 <= s36, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') encArg(z) -{ 425 + 15*s38 + 2*s38*s39 + 4*s39 + 628*x_12 + 198*x_12^2 + 18*x_12^3 + 628*x_2'' + 198*x_2''^2 + 18*x_2''^3 }-> s41 :|: s38 >= 0, s38 <= x_12 + 1, s39 >= 0, s39 <= x_2'' + 1, s40 >= 0, s40 <= s38 + 1, s41 >= 0, s41 <= s40, z = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0 encArg(z) -{ 575 + 5*s43 + s43*s44 + s44 + 628*x_13 + 198*x_13^2 + 18*x_13^3 + 628*x_21 + 198*x_21^2 + 18*x_21^3 + 628*x_3' + 198*x_3'^2 + 18*x_3'^3 }-> s46 :|: s42 >= 0, s42 <= x_13 + 1, s43 >= 0, s43 <= x_21 + 1, s44 >= 0, s44 <= x_3' + 1, s45 >= 0, s45 <= s43 + 1, s46 >= 0, s46 <= s45, z = 1 + (1 + x_13 + x_21 + x_3'), x_3' >= 0, x_13 >= 0, x_21 >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ -260 + 286*z + 144*z^2 + 18*z^3 }-> 1 + s7 :|: s7 >= 0, s7 <= z - 1 + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 574 + 5*s26 + s26*s27 + s27 + 628*z + 198*z^2 + 18*z^3 + 628*z' + 198*z'^2 + 18*z'^3 + 628*z'' + 198*z''^2 + 18*z''^3 }-> s28 :|: s25 >= 0, s25 <= z + 1, s26 >= 0, s26 <= z' + 1, s27 >= 0, s27 <= z'' + 1, s28 >= 0, s28 <= s26 + 1, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_le(z, z') -{ 378 + s20 + 628*z + 198*z^2 + 18*z^3 + 628*z' + 198*z'^2 + 18*z'^3 }-> s21 :|: s19 >= 0, s19 <= z + 1, s20 >= 0, s20 <= z' + 1, s21 >= 0, s21 <= 2, z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 424 + 15*s22 + 2*s22*s23 + 4*s23 + 628*z + 198*z^2 + 18*z^3 + 628*z' + 198*z'^2 + 18*z'^3 }-> s24 :|: s22 >= 0, s22 <= z + 1, s23 >= 0, s23 <= z' + 1, s24 >= 0, s24 <= s22 + 1, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_p(z) -{ -259 + 286*z + 144*z^2 + 18*z^3 }-> s48 :|: s47 >= 0, s47 <= z - 1 + 1, s48 >= 0, s48 <= 1 + s47, z - 1 >= 0 encode_p(z) -{ -258 + 286*z + 144*z^2 + 18*z^3 }-> s51 :|: s49 >= 0, s49 <= z - 1 + 1, s50 >= 0, s50 <= s49, s51 >= 0, s51 <= s50, z - 1 >= 0 encode_p(z) -{ 379 + s53 + 628*x_1561 + 198*x_1561^2 + 18*x_1561^3 + 628*x_2335 + 198*x_2335^2 + 18*x_2335^3 }-> s55 :|: s52 >= 0, s52 <= x_1561 + 1, s53 >= 0, s53 <= x_2335 + 1, s54 >= 0, s54 <= 2, s55 >= 0, s55 <= s54, x_1561 >= 0, x_2335 >= 0, z = 1 + x_1561 + x_2335 encode_p(z) -{ 425 + 15*s56 + 2*s56*s57 + 4*s57 + 628*x_1562 + 198*x_1562^2 + 18*x_1562^3 + 628*x_2336 + 198*x_2336^2 + 18*x_2336^3 }-> s59 :|: s56 >= 0, s56 <= x_1562 + 1, s57 >= 0, s57 <= x_2336 + 1, s58 >= 0, s58 <= s56 + 1, s59 >= 0, s59 <= s58, x_2336 >= 0, x_1562 >= 0, z = 1 + x_1562 + x_2336 encode_p(z) -{ 575 + 5*s61 + s61*s62 + s62 + 628*x_1563 + 198*x_1563^2 + 18*x_1563^3 + 628*x_2337 + 198*x_2337^2 + 18*x_2337^3 + 628*x_3111 + 198*x_3111^2 + 18*x_3111^3 }-> s64 :|: s60 >= 0, s60 <= x_1563 + 1, s61 >= 0, s61 <= x_2337 + 1, s62 >= 0, s62 <= x_3111 + 1, s63 >= 0, s63 <= s61 + 1, s64 >= 0, s64 <= s63, z = 1 + x_1563 + x_2337 + x_3111, x_2337 >= 0, x_1563 >= 0, x_3111 >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 188 + 628*z + 198*z^2 + 18*z^3 }-> 1 + s18 :|: s18 >= 0, s18 <= z + 1, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 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'') -{ 50 + 4*z'' }-> 1 + s4 :|: s4 >= 0, s4 <= 0 + 1, z = 1, z'' >= 0, z' = 0 if(z, z', z'') -{ 35 + 15*z' + 2*z'*z'' + 2*z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1 + 1, z' - 1 >= 0, z = 1, z'' >= 0 if(z, z', z'') -{ 49 + 4*z'' }-> 1 + s6 :|: s6 >= 0, s6 <= 0 + 1, z = 1, z' >= 0, z'' >= 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 12 + z' }-> s'' :|: s'' >= 0, s'' <= 0 + 1, z' >= 0, z = 0 minus(z, z') -{ 12 + 5*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1) + 1, z - 1 >= 0, z' = 0 minus(z, z') -{ 13 + 5*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1) + 1, s' >= 0, s' <= 2, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 11 + 5*z + z*z' + z' }-> s3 :|: s3 >= 0, s3 <= z + 1, z >= 0, z' >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encode_minus}, {encode_le}, {encode_s} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] le: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] if: runtime: O(n^2) [10 + 5*z' + z'*z'' + z''], size: O(n^1) [1 + z'] minus: runtime: O(n^2) [48 + 15*z + 2*z*z' + 4*z'], size: O(n^1) [1 + z] encArg: runtime: O(n^3) [188 + 628*z + 198*z^2 + 18*z^3], size: O(n^1) [1 + z] encode_p: runtime: O(n^3) [1412 + 5000*z + 1677*z^2 + 162*z^3], size: O(n^1) [1 + z] encode_if: runtime: O(n^3) [581 + 628*z + 198*z^2 + 18*z^3 + 634*z' + z'*z'' + 198*z'^2 + 18*z'^3 + 630*z'' + 198*z''^2 + 18*z''^3], size: O(n^1) [2 + z'] ---------------------------------------- (77) 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: 2 + z ---------------------------------------- (78) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 378 + s9 + 628*x_1 + 198*x_1^2 + 18*x_1^3 + 628*x_2 + 198*x_2^2 + 18*x_2^3 }-> s10 :|: s8 >= 0, s8 <= x_1 + 1, s9 >= 0, s9 <= x_2 + 1, s10 >= 0, s10 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 424 + 15*s11 + 2*s11*s12 + 4*s12 + 628*x_1 + 198*x_1^2 + 18*x_1^3 + 628*x_2 + 198*x_2^2 + 18*x_2^3 }-> s13 :|: s11 >= 0, s11 <= x_1 + 1, s12 >= 0, s12 <= x_2 + 1, s13 >= 0, s13 <= s11 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 574 + 5*s15 + s15*s16 + s16 + 628*x_1 + 198*x_1^2 + 18*x_1^3 + 628*x_2 + 198*x_2^2 + 18*x_2^3 + 628*x_3 + 198*x_3^2 + 18*x_3^3 }-> s17 :|: s14 >= 0, s14 <= x_1 + 1, s15 >= 0, s15 <= x_2 + 1, s16 >= 0, s16 <= x_3 + 1, s17 >= 0, s17 <= s15 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ -419 + 52*z + 90*z^2 + 18*z^3 }-> s30 :|: s29 >= 0, s29 <= z - 2 + 1, s30 >= 0, s30 <= 1 + s29, z - 2 >= 0 encArg(z) -{ -418 + 52*z + 90*z^2 + 18*z^3 }-> s33 :|: s31 >= 0, s31 <= z - 2 + 1, s32 >= 0, s32 <= s31, s33 >= 0, s33 <= s32, z - 2 >= 0 encArg(z) -{ 379 + s35 + 628*x_11 + 198*x_11^2 + 18*x_11^3 + 628*x_2' + 198*x_2'^2 + 18*x_2'^3 }-> s37 :|: s34 >= 0, s34 <= x_11 + 1, s35 >= 0, s35 <= x_2' + 1, s36 >= 0, s36 <= 2, s37 >= 0, s37 <= s36, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') encArg(z) -{ 425 + 15*s38 + 2*s38*s39 + 4*s39 + 628*x_12 + 198*x_12^2 + 18*x_12^3 + 628*x_2'' + 198*x_2''^2 + 18*x_2''^3 }-> s41 :|: s38 >= 0, s38 <= x_12 + 1, s39 >= 0, s39 <= x_2'' + 1, s40 >= 0, s40 <= s38 + 1, s41 >= 0, s41 <= s40, z = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0 encArg(z) -{ 575 + 5*s43 + s43*s44 + s44 + 628*x_13 + 198*x_13^2 + 18*x_13^3 + 628*x_21 + 198*x_21^2 + 18*x_21^3 + 628*x_3' + 198*x_3'^2 + 18*x_3'^3 }-> s46 :|: s42 >= 0, s42 <= x_13 + 1, s43 >= 0, s43 <= x_21 + 1, s44 >= 0, s44 <= x_3' + 1, s45 >= 0, s45 <= s43 + 1, s46 >= 0, s46 <= s45, z = 1 + (1 + x_13 + x_21 + x_3'), x_3' >= 0, x_13 >= 0, x_21 >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ -260 + 286*z + 144*z^2 + 18*z^3 }-> 1 + s7 :|: s7 >= 0, s7 <= z - 1 + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 574 + 5*s26 + s26*s27 + s27 + 628*z + 198*z^2 + 18*z^3 + 628*z' + 198*z'^2 + 18*z'^3 + 628*z'' + 198*z''^2 + 18*z''^3 }-> s28 :|: s25 >= 0, s25 <= z + 1, s26 >= 0, s26 <= z' + 1, s27 >= 0, s27 <= z'' + 1, s28 >= 0, s28 <= s26 + 1, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_le(z, z') -{ 378 + s20 + 628*z + 198*z^2 + 18*z^3 + 628*z' + 198*z'^2 + 18*z'^3 }-> s21 :|: s19 >= 0, s19 <= z + 1, s20 >= 0, s20 <= z' + 1, s21 >= 0, s21 <= 2, z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 424 + 15*s22 + 2*s22*s23 + 4*s23 + 628*z + 198*z^2 + 18*z^3 + 628*z' + 198*z'^2 + 18*z'^3 }-> s24 :|: s22 >= 0, s22 <= z + 1, s23 >= 0, s23 <= z' + 1, s24 >= 0, s24 <= s22 + 1, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_p(z) -{ -259 + 286*z + 144*z^2 + 18*z^3 }-> s48 :|: s47 >= 0, s47 <= z - 1 + 1, s48 >= 0, s48 <= 1 + s47, z - 1 >= 0 encode_p(z) -{ -258 + 286*z + 144*z^2 + 18*z^3 }-> s51 :|: s49 >= 0, s49 <= z - 1 + 1, s50 >= 0, s50 <= s49, s51 >= 0, s51 <= s50, z - 1 >= 0 encode_p(z) -{ 379 + s53 + 628*x_1561 + 198*x_1561^2 + 18*x_1561^3 + 628*x_2335 + 198*x_2335^2 + 18*x_2335^3 }-> s55 :|: s52 >= 0, s52 <= x_1561 + 1, s53 >= 0, s53 <= x_2335 + 1, s54 >= 0, s54 <= 2, s55 >= 0, s55 <= s54, x_1561 >= 0, x_2335 >= 0, z = 1 + x_1561 + x_2335 encode_p(z) -{ 425 + 15*s56 + 2*s56*s57 + 4*s57 + 628*x_1562 + 198*x_1562^2 + 18*x_1562^3 + 628*x_2336 + 198*x_2336^2 + 18*x_2336^3 }-> s59 :|: s56 >= 0, s56 <= x_1562 + 1, s57 >= 0, s57 <= x_2336 + 1, s58 >= 0, s58 <= s56 + 1, s59 >= 0, s59 <= s58, x_2336 >= 0, x_1562 >= 0, z = 1 + x_1562 + x_2336 encode_p(z) -{ 575 + 5*s61 + s61*s62 + s62 + 628*x_1563 + 198*x_1563^2 + 18*x_1563^3 + 628*x_2337 + 198*x_2337^2 + 18*x_2337^3 + 628*x_3111 + 198*x_3111^2 + 18*x_3111^3 }-> s64 :|: s60 >= 0, s60 <= x_1563 + 1, s61 >= 0, s61 <= x_2337 + 1, s62 >= 0, s62 <= x_3111 + 1, s63 >= 0, s63 <= s61 + 1, s64 >= 0, s64 <= s63, z = 1 + x_1563 + x_2337 + x_3111, x_2337 >= 0, x_1563 >= 0, x_3111 >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 188 + 628*z + 198*z^2 + 18*z^3 }-> 1 + s18 :|: s18 >= 0, s18 <= z + 1, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 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'') -{ 50 + 4*z'' }-> 1 + s4 :|: s4 >= 0, s4 <= 0 + 1, z = 1, z'' >= 0, z' = 0 if(z, z', z'') -{ 35 + 15*z' + 2*z'*z'' + 2*z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1 + 1, z' - 1 >= 0, z = 1, z'' >= 0 if(z, z', z'') -{ 49 + 4*z'' }-> 1 + s6 :|: s6 >= 0, s6 <= 0 + 1, z = 1, z' >= 0, z'' >= 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 12 + z' }-> s'' :|: s'' >= 0, s'' <= 0 + 1, z' >= 0, z = 0 minus(z, z') -{ 12 + 5*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1) + 1, z - 1 >= 0, z' = 0 minus(z, z') -{ 13 + 5*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1) + 1, s' >= 0, s' <= 2, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 11 + 5*z + z*z' + z' }-> s3 :|: s3 >= 0, s3 <= z + 1, z >= 0, z' >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encode_minus}, {encode_le}, {encode_s} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] le: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] if: runtime: O(n^2) [10 + 5*z' + z'*z'' + z''], size: O(n^1) [1 + z'] minus: runtime: O(n^2) [48 + 15*z + 2*z*z' + 4*z'], size: O(n^1) [1 + z] encArg: runtime: O(n^3) [188 + 628*z + 198*z^2 + 18*z^3], size: O(n^1) [1 + z] encode_p: runtime: O(n^3) [1412 + 5000*z + 1677*z^2 + 162*z^3], size: O(n^1) [1 + z] encode_if: runtime: O(n^3) [581 + 628*z + 198*z^2 + 18*z^3 + 634*z' + z'*z'' + 198*z'^2 + 18*z'^3 + 630*z'' + 198*z''^2 + 18*z''^3], size: O(n^1) [2 + z'] encode_minus: runtime: ?, size: O(n^1) [2 + z] ---------------------------------------- (79) 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: 445 + 645*z + 2*z*z' + 198*z^2 + 18*z^3 + 634*z' + 198*z'^2 + 18*z'^3 ---------------------------------------- (80) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 378 + s9 + 628*x_1 + 198*x_1^2 + 18*x_1^3 + 628*x_2 + 198*x_2^2 + 18*x_2^3 }-> s10 :|: s8 >= 0, s8 <= x_1 + 1, s9 >= 0, s9 <= x_2 + 1, s10 >= 0, s10 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 424 + 15*s11 + 2*s11*s12 + 4*s12 + 628*x_1 + 198*x_1^2 + 18*x_1^3 + 628*x_2 + 198*x_2^2 + 18*x_2^3 }-> s13 :|: s11 >= 0, s11 <= x_1 + 1, s12 >= 0, s12 <= x_2 + 1, s13 >= 0, s13 <= s11 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 574 + 5*s15 + s15*s16 + s16 + 628*x_1 + 198*x_1^2 + 18*x_1^3 + 628*x_2 + 198*x_2^2 + 18*x_2^3 + 628*x_3 + 198*x_3^2 + 18*x_3^3 }-> s17 :|: s14 >= 0, s14 <= x_1 + 1, s15 >= 0, s15 <= x_2 + 1, s16 >= 0, s16 <= x_3 + 1, s17 >= 0, s17 <= s15 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ -419 + 52*z + 90*z^2 + 18*z^3 }-> s30 :|: s29 >= 0, s29 <= z - 2 + 1, s30 >= 0, s30 <= 1 + s29, z - 2 >= 0 encArg(z) -{ -418 + 52*z + 90*z^2 + 18*z^3 }-> s33 :|: s31 >= 0, s31 <= z - 2 + 1, s32 >= 0, s32 <= s31, s33 >= 0, s33 <= s32, z - 2 >= 0 encArg(z) -{ 379 + s35 + 628*x_11 + 198*x_11^2 + 18*x_11^3 + 628*x_2' + 198*x_2'^2 + 18*x_2'^3 }-> s37 :|: s34 >= 0, s34 <= x_11 + 1, s35 >= 0, s35 <= x_2' + 1, s36 >= 0, s36 <= 2, s37 >= 0, s37 <= s36, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') encArg(z) -{ 425 + 15*s38 + 2*s38*s39 + 4*s39 + 628*x_12 + 198*x_12^2 + 18*x_12^3 + 628*x_2'' + 198*x_2''^2 + 18*x_2''^3 }-> s41 :|: s38 >= 0, s38 <= x_12 + 1, s39 >= 0, s39 <= x_2'' + 1, s40 >= 0, s40 <= s38 + 1, s41 >= 0, s41 <= s40, z = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0 encArg(z) -{ 575 + 5*s43 + s43*s44 + s44 + 628*x_13 + 198*x_13^2 + 18*x_13^3 + 628*x_21 + 198*x_21^2 + 18*x_21^3 + 628*x_3' + 198*x_3'^2 + 18*x_3'^3 }-> s46 :|: s42 >= 0, s42 <= x_13 + 1, s43 >= 0, s43 <= x_21 + 1, s44 >= 0, s44 <= x_3' + 1, s45 >= 0, s45 <= s43 + 1, s46 >= 0, s46 <= s45, z = 1 + (1 + x_13 + x_21 + x_3'), x_3' >= 0, x_13 >= 0, x_21 >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ -260 + 286*z + 144*z^2 + 18*z^3 }-> 1 + s7 :|: s7 >= 0, s7 <= z - 1 + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 574 + 5*s26 + s26*s27 + s27 + 628*z + 198*z^2 + 18*z^3 + 628*z' + 198*z'^2 + 18*z'^3 + 628*z'' + 198*z''^2 + 18*z''^3 }-> s28 :|: s25 >= 0, s25 <= z + 1, s26 >= 0, s26 <= z' + 1, s27 >= 0, s27 <= z'' + 1, s28 >= 0, s28 <= s26 + 1, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_le(z, z') -{ 378 + s20 + 628*z + 198*z^2 + 18*z^3 + 628*z' + 198*z'^2 + 18*z'^3 }-> s21 :|: s19 >= 0, s19 <= z + 1, s20 >= 0, s20 <= z' + 1, s21 >= 0, s21 <= 2, z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 424 + 15*s22 + 2*s22*s23 + 4*s23 + 628*z + 198*z^2 + 18*z^3 + 628*z' + 198*z'^2 + 18*z'^3 }-> s24 :|: s22 >= 0, s22 <= z + 1, s23 >= 0, s23 <= z' + 1, s24 >= 0, s24 <= s22 + 1, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_p(z) -{ -259 + 286*z + 144*z^2 + 18*z^3 }-> s48 :|: s47 >= 0, s47 <= z - 1 + 1, s48 >= 0, s48 <= 1 + s47, z - 1 >= 0 encode_p(z) -{ -258 + 286*z + 144*z^2 + 18*z^3 }-> s51 :|: s49 >= 0, s49 <= z - 1 + 1, s50 >= 0, s50 <= s49, s51 >= 0, s51 <= s50, z - 1 >= 0 encode_p(z) -{ 379 + s53 + 628*x_1561 + 198*x_1561^2 + 18*x_1561^3 + 628*x_2335 + 198*x_2335^2 + 18*x_2335^3 }-> s55 :|: s52 >= 0, s52 <= x_1561 + 1, s53 >= 0, s53 <= x_2335 + 1, s54 >= 0, s54 <= 2, s55 >= 0, s55 <= s54, x_1561 >= 0, x_2335 >= 0, z = 1 + x_1561 + x_2335 encode_p(z) -{ 425 + 15*s56 + 2*s56*s57 + 4*s57 + 628*x_1562 + 198*x_1562^2 + 18*x_1562^3 + 628*x_2336 + 198*x_2336^2 + 18*x_2336^3 }-> s59 :|: s56 >= 0, s56 <= x_1562 + 1, s57 >= 0, s57 <= x_2336 + 1, s58 >= 0, s58 <= s56 + 1, s59 >= 0, s59 <= s58, x_2336 >= 0, x_1562 >= 0, z = 1 + x_1562 + x_2336 encode_p(z) -{ 575 + 5*s61 + s61*s62 + s62 + 628*x_1563 + 198*x_1563^2 + 18*x_1563^3 + 628*x_2337 + 198*x_2337^2 + 18*x_2337^3 + 628*x_3111 + 198*x_3111^2 + 18*x_3111^3 }-> s64 :|: s60 >= 0, s60 <= x_1563 + 1, s61 >= 0, s61 <= x_2337 + 1, s62 >= 0, s62 <= x_3111 + 1, s63 >= 0, s63 <= s61 + 1, s64 >= 0, s64 <= s63, z = 1 + x_1563 + x_2337 + x_3111, x_2337 >= 0, x_1563 >= 0, x_3111 >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 188 + 628*z + 198*z^2 + 18*z^3 }-> 1 + s18 :|: s18 >= 0, s18 <= z + 1, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 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'') -{ 50 + 4*z'' }-> 1 + s4 :|: s4 >= 0, s4 <= 0 + 1, z = 1, z'' >= 0, z' = 0 if(z, z', z'') -{ 35 + 15*z' + 2*z'*z'' + 2*z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1 + 1, z' - 1 >= 0, z = 1, z'' >= 0 if(z, z', z'') -{ 49 + 4*z'' }-> 1 + s6 :|: s6 >= 0, s6 <= 0 + 1, z = 1, z' >= 0, z'' >= 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 12 + z' }-> s'' :|: s'' >= 0, s'' <= 0 + 1, z' >= 0, z = 0 minus(z, z') -{ 12 + 5*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1) + 1, z - 1 >= 0, z' = 0 minus(z, z') -{ 13 + 5*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1) + 1, s' >= 0, s' <= 2, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 11 + 5*z + z*z' + z' }-> s3 :|: s3 >= 0, s3 <= z + 1, z >= 0, z' >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encode_le}, {encode_s} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] le: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] if: runtime: O(n^2) [10 + 5*z' + z'*z'' + z''], size: O(n^1) [1 + z'] minus: runtime: O(n^2) [48 + 15*z + 2*z*z' + 4*z'], size: O(n^1) [1 + z] encArg: runtime: O(n^3) [188 + 628*z + 198*z^2 + 18*z^3], size: O(n^1) [1 + z] encode_p: runtime: O(n^3) [1412 + 5000*z + 1677*z^2 + 162*z^3], size: O(n^1) [1 + z] encode_if: runtime: O(n^3) [581 + 628*z + 198*z^2 + 18*z^3 + 634*z' + z'*z'' + 198*z'^2 + 18*z'^3 + 630*z'' + 198*z''^2 + 18*z''^3], size: O(n^1) [2 + z'] encode_minus: runtime: O(n^3) [445 + 645*z + 2*z*z' + 198*z^2 + 18*z^3 + 634*z' + 198*z'^2 + 18*z'^3], size: O(n^1) [2 + z] ---------------------------------------- (81) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (82) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 378 + s9 + 628*x_1 + 198*x_1^2 + 18*x_1^3 + 628*x_2 + 198*x_2^2 + 18*x_2^3 }-> s10 :|: s8 >= 0, s8 <= x_1 + 1, s9 >= 0, s9 <= x_2 + 1, s10 >= 0, s10 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 424 + 15*s11 + 2*s11*s12 + 4*s12 + 628*x_1 + 198*x_1^2 + 18*x_1^3 + 628*x_2 + 198*x_2^2 + 18*x_2^3 }-> s13 :|: s11 >= 0, s11 <= x_1 + 1, s12 >= 0, s12 <= x_2 + 1, s13 >= 0, s13 <= s11 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 574 + 5*s15 + s15*s16 + s16 + 628*x_1 + 198*x_1^2 + 18*x_1^3 + 628*x_2 + 198*x_2^2 + 18*x_2^3 + 628*x_3 + 198*x_3^2 + 18*x_3^3 }-> s17 :|: s14 >= 0, s14 <= x_1 + 1, s15 >= 0, s15 <= x_2 + 1, s16 >= 0, s16 <= x_3 + 1, s17 >= 0, s17 <= s15 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ -419 + 52*z + 90*z^2 + 18*z^3 }-> s30 :|: s29 >= 0, s29 <= z - 2 + 1, s30 >= 0, s30 <= 1 + s29, z - 2 >= 0 encArg(z) -{ -418 + 52*z + 90*z^2 + 18*z^3 }-> s33 :|: s31 >= 0, s31 <= z - 2 + 1, s32 >= 0, s32 <= s31, s33 >= 0, s33 <= s32, z - 2 >= 0 encArg(z) -{ 379 + s35 + 628*x_11 + 198*x_11^2 + 18*x_11^3 + 628*x_2' + 198*x_2'^2 + 18*x_2'^3 }-> s37 :|: s34 >= 0, s34 <= x_11 + 1, s35 >= 0, s35 <= x_2' + 1, s36 >= 0, s36 <= 2, s37 >= 0, s37 <= s36, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') encArg(z) -{ 425 + 15*s38 + 2*s38*s39 + 4*s39 + 628*x_12 + 198*x_12^2 + 18*x_12^3 + 628*x_2'' + 198*x_2''^2 + 18*x_2''^3 }-> s41 :|: s38 >= 0, s38 <= x_12 + 1, s39 >= 0, s39 <= x_2'' + 1, s40 >= 0, s40 <= s38 + 1, s41 >= 0, s41 <= s40, z = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0 encArg(z) -{ 575 + 5*s43 + s43*s44 + s44 + 628*x_13 + 198*x_13^2 + 18*x_13^3 + 628*x_21 + 198*x_21^2 + 18*x_21^3 + 628*x_3' + 198*x_3'^2 + 18*x_3'^3 }-> s46 :|: s42 >= 0, s42 <= x_13 + 1, s43 >= 0, s43 <= x_21 + 1, s44 >= 0, s44 <= x_3' + 1, s45 >= 0, s45 <= s43 + 1, s46 >= 0, s46 <= s45, z = 1 + (1 + x_13 + x_21 + x_3'), x_3' >= 0, x_13 >= 0, x_21 >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ -260 + 286*z + 144*z^2 + 18*z^3 }-> 1 + s7 :|: s7 >= 0, s7 <= z - 1 + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 574 + 5*s26 + s26*s27 + s27 + 628*z + 198*z^2 + 18*z^3 + 628*z' + 198*z'^2 + 18*z'^3 + 628*z'' + 198*z''^2 + 18*z''^3 }-> s28 :|: s25 >= 0, s25 <= z + 1, s26 >= 0, s26 <= z' + 1, s27 >= 0, s27 <= z'' + 1, s28 >= 0, s28 <= s26 + 1, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_le(z, z') -{ 378 + s20 + 628*z + 198*z^2 + 18*z^3 + 628*z' + 198*z'^2 + 18*z'^3 }-> s21 :|: s19 >= 0, s19 <= z + 1, s20 >= 0, s20 <= z' + 1, s21 >= 0, s21 <= 2, z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 424 + 15*s22 + 2*s22*s23 + 4*s23 + 628*z + 198*z^2 + 18*z^3 + 628*z' + 198*z'^2 + 18*z'^3 }-> s24 :|: s22 >= 0, s22 <= z + 1, s23 >= 0, s23 <= z' + 1, s24 >= 0, s24 <= s22 + 1, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_p(z) -{ -259 + 286*z + 144*z^2 + 18*z^3 }-> s48 :|: s47 >= 0, s47 <= z - 1 + 1, s48 >= 0, s48 <= 1 + s47, z - 1 >= 0 encode_p(z) -{ -258 + 286*z + 144*z^2 + 18*z^3 }-> s51 :|: s49 >= 0, s49 <= z - 1 + 1, s50 >= 0, s50 <= s49, s51 >= 0, s51 <= s50, z - 1 >= 0 encode_p(z) -{ 379 + s53 + 628*x_1561 + 198*x_1561^2 + 18*x_1561^3 + 628*x_2335 + 198*x_2335^2 + 18*x_2335^3 }-> s55 :|: s52 >= 0, s52 <= x_1561 + 1, s53 >= 0, s53 <= x_2335 + 1, s54 >= 0, s54 <= 2, s55 >= 0, s55 <= s54, x_1561 >= 0, x_2335 >= 0, z = 1 + x_1561 + x_2335 encode_p(z) -{ 425 + 15*s56 + 2*s56*s57 + 4*s57 + 628*x_1562 + 198*x_1562^2 + 18*x_1562^3 + 628*x_2336 + 198*x_2336^2 + 18*x_2336^3 }-> s59 :|: s56 >= 0, s56 <= x_1562 + 1, s57 >= 0, s57 <= x_2336 + 1, s58 >= 0, s58 <= s56 + 1, s59 >= 0, s59 <= s58, x_2336 >= 0, x_1562 >= 0, z = 1 + x_1562 + x_2336 encode_p(z) -{ 575 + 5*s61 + s61*s62 + s62 + 628*x_1563 + 198*x_1563^2 + 18*x_1563^3 + 628*x_2337 + 198*x_2337^2 + 18*x_2337^3 + 628*x_3111 + 198*x_3111^2 + 18*x_3111^3 }-> s64 :|: s60 >= 0, s60 <= x_1563 + 1, s61 >= 0, s61 <= x_2337 + 1, s62 >= 0, s62 <= x_3111 + 1, s63 >= 0, s63 <= s61 + 1, s64 >= 0, s64 <= s63, z = 1 + x_1563 + x_2337 + x_3111, x_2337 >= 0, x_1563 >= 0, x_3111 >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 188 + 628*z + 198*z^2 + 18*z^3 }-> 1 + s18 :|: s18 >= 0, s18 <= z + 1, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 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'') -{ 50 + 4*z'' }-> 1 + s4 :|: s4 >= 0, s4 <= 0 + 1, z = 1, z'' >= 0, z' = 0 if(z, z', z'') -{ 35 + 15*z' + 2*z'*z'' + 2*z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1 + 1, z' - 1 >= 0, z = 1, z'' >= 0 if(z, z', z'') -{ 49 + 4*z'' }-> 1 + s6 :|: s6 >= 0, s6 <= 0 + 1, z = 1, z' >= 0, z'' >= 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 12 + z' }-> s'' :|: s'' >= 0, s'' <= 0 + 1, z' >= 0, z = 0 minus(z, z') -{ 12 + 5*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1) + 1, z - 1 >= 0, z' = 0 minus(z, z') -{ 13 + 5*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1) + 1, s' >= 0, s' <= 2, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 11 + 5*z + z*z' + z' }-> s3 :|: s3 >= 0, s3 <= z + 1, z >= 0, z' >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encode_le}, {encode_s} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] le: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] if: runtime: O(n^2) [10 + 5*z' + z'*z'' + z''], size: O(n^1) [1 + z'] minus: runtime: O(n^2) [48 + 15*z + 2*z*z' + 4*z'], size: O(n^1) [1 + z] encArg: runtime: O(n^3) [188 + 628*z + 198*z^2 + 18*z^3], size: O(n^1) [1 + z] encode_p: runtime: O(n^3) [1412 + 5000*z + 1677*z^2 + 162*z^3], size: O(n^1) [1 + z] encode_if: runtime: O(n^3) [581 + 628*z + 198*z^2 + 18*z^3 + 634*z' + z'*z'' + 198*z'^2 + 18*z'^3 + 630*z'' + 198*z''^2 + 18*z''^3], size: O(n^1) [2 + z'] encode_minus: runtime: O(n^3) [445 + 645*z + 2*z*z' + 198*z^2 + 18*z^3 + 634*z' + 198*z'^2 + 18*z'^3], size: O(n^1) [2 + z] ---------------------------------------- (83) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: encode_le after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (84) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 378 + s9 + 628*x_1 + 198*x_1^2 + 18*x_1^3 + 628*x_2 + 198*x_2^2 + 18*x_2^3 }-> s10 :|: s8 >= 0, s8 <= x_1 + 1, s9 >= 0, s9 <= x_2 + 1, s10 >= 0, s10 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 424 + 15*s11 + 2*s11*s12 + 4*s12 + 628*x_1 + 198*x_1^2 + 18*x_1^3 + 628*x_2 + 198*x_2^2 + 18*x_2^3 }-> s13 :|: s11 >= 0, s11 <= x_1 + 1, s12 >= 0, s12 <= x_2 + 1, s13 >= 0, s13 <= s11 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 574 + 5*s15 + s15*s16 + s16 + 628*x_1 + 198*x_1^2 + 18*x_1^3 + 628*x_2 + 198*x_2^2 + 18*x_2^3 + 628*x_3 + 198*x_3^2 + 18*x_3^3 }-> s17 :|: s14 >= 0, s14 <= x_1 + 1, s15 >= 0, s15 <= x_2 + 1, s16 >= 0, s16 <= x_3 + 1, s17 >= 0, s17 <= s15 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ -419 + 52*z + 90*z^2 + 18*z^3 }-> s30 :|: s29 >= 0, s29 <= z - 2 + 1, s30 >= 0, s30 <= 1 + s29, z - 2 >= 0 encArg(z) -{ -418 + 52*z + 90*z^2 + 18*z^3 }-> s33 :|: s31 >= 0, s31 <= z - 2 + 1, s32 >= 0, s32 <= s31, s33 >= 0, s33 <= s32, z - 2 >= 0 encArg(z) -{ 379 + s35 + 628*x_11 + 198*x_11^2 + 18*x_11^3 + 628*x_2' + 198*x_2'^2 + 18*x_2'^3 }-> s37 :|: s34 >= 0, s34 <= x_11 + 1, s35 >= 0, s35 <= x_2' + 1, s36 >= 0, s36 <= 2, s37 >= 0, s37 <= s36, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') encArg(z) -{ 425 + 15*s38 + 2*s38*s39 + 4*s39 + 628*x_12 + 198*x_12^2 + 18*x_12^3 + 628*x_2'' + 198*x_2''^2 + 18*x_2''^3 }-> s41 :|: s38 >= 0, s38 <= x_12 + 1, s39 >= 0, s39 <= x_2'' + 1, s40 >= 0, s40 <= s38 + 1, s41 >= 0, s41 <= s40, z = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0 encArg(z) -{ 575 + 5*s43 + s43*s44 + s44 + 628*x_13 + 198*x_13^2 + 18*x_13^3 + 628*x_21 + 198*x_21^2 + 18*x_21^3 + 628*x_3' + 198*x_3'^2 + 18*x_3'^3 }-> s46 :|: s42 >= 0, s42 <= x_13 + 1, s43 >= 0, s43 <= x_21 + 1, s44 >= 0, s44 <= x_3' + 1, s45 >= 0, s45 <= s43 + 1, s46 >= 0, s46 <= s45, z = 1 + (1 + x_13 + x_21 + x_3'), x_3' >= 0, x_13 >= 0, x_21 >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ -260 + 286*z + 144*z^2 + 18*z^3 }-> 1 + s7 :|: s7 >= 0, s7 <= z - 1 + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 574 + 5*s26 + s26*s27 + s27 + 628*z + 198*z^2 + 18*z^3 + 628*z' + 198*z'^2 + 18*z'^3 + 628*z'' + 198*z''^2 + 18*z''^3 }-> s28 :|: s25 >= 0, s25 <= z + 1, s26 >= 0, s26 <= z' + 1, s27 >= 0, s27 <= z'' + 1, s28 >= 0, s28 <= s26 + 1, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_le(z, z') -{ 378 + s20 + 628*z + 198*z^2 + 18*z^3 + 628*z' + 198*z'^2 + 18*z'^3 }-> s21 :|: s19 >= 0, s19 <= z + 1, s20 >= 0, s20 <= z' + 1, s21 >= 0, s21 <= 2, z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 424 + 15*s22 + 2*s22*s23 + 4*s23 + 628*z + 198*z^2 + 18*z^3 + 628*z' + 198*z'^2 + 18*z'^3 }-> s24 :|: s22 >= 0, s22 <= z + 1, s23 >= 0, s23 <= z' + 1, s24 >= 0, s24 <= s22 + 1, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_p(z) -{ -259 + 286*z + 144*z^2 + 18*z^3 }-> s48 :|: s47 >= 0, s47 <= z - 1 + 1, s48 >= 0, s48 <= 1 + s47, z - 1 >= 0 encode_p(z) -{ -258 + 286*z + 144*z^2 + 18*z^3 }-> s51 :|: s49 >= 0, s49 <= z - 1 + 1, s50 >= 0, s50 <= s49, s51 >= 0, s51 <= s50, z - 1 >= 0 encode_p(z) -{ 379 + s53 + 628*x_1561 + 198*x_1561^2 + 18*x_1561^3 + 628*x_2335 + 198*x_2335^2 + 18*x_2335^3 }-> s55 :|: s52 >= 0, s52 <= x_1561 + 1, s53 >= 0, s53 <= x_2335 + 1, s54 >= 0, s54 <= 2, s55 >= 0, s55 <= s54, x_1561 >= 0, x_2335 >= 0, z = 1 + x_1561 + x_2335 encode_p(z) -{ 425 + 15*s56 + 2*s56*s57 + 4*s57 + 628*x_1562 + 198*x_1562^2 + 18*x_1562^3 + 628*x_2336 + 198*x_2336^2 + 18*x_2336^3 }-> s59 :|: s56 >= 0, s56 <= x_1562 + 1, s57 >= 0, s57 <= x_2336 + 1, s58 >= 0, s58 <= s56 + 1, s59 >= 0, s59 <= s58, x_2336 >= 0, x_1562 >= 0, z = 1 + x_1562 + x_2336 encode_p(z) -{ 575 + 5*s61 + s61*s62 + s62 + 628*x_1563 + 198*x_1563^2 + 18*x_1563^3 + 628*x_2337 + 198*x_2337^2 + 18*x_2337^3 + 628*x_3111 + 198*x_3111^2 + 18*x_3111^3 }-> s64 :|: s60 >= 0, s60 <= x_1563 + 1, s61 >= 0, s61 <= x_2337 + 1, s62 >= 0, s62 <= x_3111 + 1, s63 >= 0, s63 <= s61 + 1, s64 >= 0, s64 <= s63, z = 1 + x_1563 + x_2337 + x_3111, x_2337 >= 0, x_1563 >= 0, x_3111 >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 188 + 628*z + 198*z^2 + 18*z^3 }-> 1 + s18 :|: s18 >= 0, s18 <= z + 1, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 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'') -{ 50 + 4*z'' }-> 1 + s4 :|: s4 >= 0, s4 <= 0 + 1, z = 1, z'' >= 0, z' = 0 if(z, z', z'') -{ 35 + 15*z' + 2*z'*z'' + 2*z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1 + 1, z' - 1 >= 0, z = 1, z'' >= 0 if(z, z', z'') -{ 49 + 4*z'' }-> 1 + s6 :|: s6 >= 0, s6 <= 0 + 1, z = 1, z' >= 0, z'' >= 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 12 + z' }-> s'' :|: s'' >= 0, s'' <= 0 + 1, z' >= 0, z = 0 minus(z, z') -{ 12 + 5*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1) + 1, z - 1 >= 0, z' = 0 minus(z, z') -{ 13 + 5*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1) + 1, s' >= 0, s' <= 2, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 11 + 5*z + z*z' + z' }-> s3 :|: s3 >= 0, s3 <= z + 1, z >= 0, z' >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encode_le}, {encode_s} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] le: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] if: runtime: O(n^2) [10 + 5*z' + z'*z'' + z''], size: O(n^1) [1 + z'] minus: runtime: O(n^2) [48 + 15*z + 2*z*z' + 4*z'], size: O(n^1) [1 + z] encArg: runtime: O(n^3) [188 + 628*z + 198*z^2 + 18*z^3], size: O(n^1) [1 + z] encode_p: runtime: O(n^3) [1412 + 5000*z + 1677*z^2 + 162*z^3], size: O(n^1) [1 + z] encode_if: runtime: O(n^3) [581 + 628*z + 198*z^2 + 18*z^3 + 634*z' + z'*z'' + 198*z'^2 + 18*z'^3 + 630*z'' + 198*z''^2 + 18*z''^3], size: O(n^1) [2 + z'] encode_minus: runtime: O(n^3) [445 + 645*z + 2*z*z' + 198*z^2 + 18*z^3 + 634*z' + 198*z'^2 + 18*z'^3], size: O(n^1) [2 + z] encode_le: runtime: ?, size: O(1) [2] ---------------------------------------- (85) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: encode_le after applying outer abstraction to obtain an ITS, resulting in: O(n^3) with polynomial bound: 379 + 628*z + 198*z^2 + 18*z^3 + 629*z' + 198*z'^2 + 18*z'^3 ---------------------------------------- (86) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 378 + s9 + 628*x_1 + 198*x_1^2 + 18*x_1^3 + 628*x_2 + 198*x_2^2 + 18*x_2^3 }-> s10 :|: s8 >= 0, s8 <= x_1 + 1, s9 >= 0, s9 <= x_2 + 1, s10 >= 0, s10 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 424 + 15*s11 + 2*s11*s12 + 4*s12 + 628*x_1 + 198*x_1^2 + 18*x_1^3 + 628*x_2 + 198*x_2^2 + 18*x_2^3 }-> s13 :|: s11 >= 0, s11 <= x_1 + 1, s12 >= 0, s12 <= x_2 + 1, s13 >= 0, s13 <= s11 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 574 + 5*s15 + s15*s16 + s16 + 628*x_1 + 198*x_1^2 + 18*x_1^3 + 628*x_2 + 198*x_2^2 + 18*x_2^3 + 628*x_3 + 198*x_3^2 + 18*x_3^3 }-> s17 :|: s14 >= 0, s14 <= x_1 + 1, s15 >= 0, s15 <= x_2 + 1, s16 >= 0, s16 <= x_3 + 1, s17 >= 0, s17 <= s15 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ -419 + 52*z + 90*z^2 + 18*z^3 }-> s30 :|: s29 >= 0, s29 <= z - 2 + 1, s30 >= 0, s30 <= 1 + s29, z - 2 >= 0 encArg(z) -{ -418 + 52*z + 90*z^2 + 18*z^3 }-> s33 :|: s31 >= 0, s31 <= z - 2 + 1, s32 >= 0, s32 <= s31, s33 >= 0, s33 <= s32, z - 2 >= 0 encArg(z) -{ 379 + s35 + 628*x_11 + 198*x_11^2 + 18*x_11^3 + 628*x_2' + 198*x_2'^2 + 18*x_2'^3 }-> s37 :|: s34 >= 0, s34 <= x_11 + 1, s35 >= 0, s35 <= x_2' + 1, s36 >= 0, s36 <= 2, s37 >= 0, s37 <= s36, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') encArg(z) -{ 425 + 15*s38 + 2*s38*s39 + 4*s39 + 628*x_12 + 198*x_12^2 + 18*x_12^3 + 628*x_2'' + 198*x_2''^2 + 18*x_2''^3 }-> s41 :|: s38 >= 0, s38 <= x_12 + 1, s39 >= 0, s39 <= x_2'' + 1, s40 >= 0, s40 <= s38 + 1, s41 >= 0, s41 <= s40, z = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0 encArg(z) -{ 575 + 5*s43 + s43*s44 + s44 + 628*x_13 + 198*x_13^2 + 18*x_13^3 + 628*x_21 + 198*x_21^2 + 18*x_21^3 + 628*x_3' + 198*x_3'^2 + 18*x_3'^3 }-> s46 :|: s42 >= 0, s42 <= x_13 + 1, s43 >= 0, s43 <= x_21 + 1, s44 >= 0, s44 <= x_3' + 1, s45 >= 0, s45 <= s43 + 1, s46 >= 0, s46 <= s45, z = 1 + (1 + x_13 + x_21 + x_3'), x_3' >= 0, x_13 >= 0, x_21 >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ -260 + 286*z + 144*z^2 + 18*z^3 }-> 1 + s7 :|: s7 >= 0, s7 <= z - 1 + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 574 + 5*s26 + s26*s27 + s27 + 628*z + 198*z^2 + 18*z^3 + 628*z' + 198*z'^2 + 18*z'^3 + 628*z'' + 198*z''^2 + 18*z''^3 }-> s28 :|: s25 >= 0, s25 <= z + 1, s26 >= 0, s26 <= z' + 1, s27 >= 0, s27 <= z'' + 1, s28 >= 0, s28 <= s26 + 1, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_le(z, z') -{ 378 + s20 + 628*z + 198*z^2 + 18*z^3 + 628*z' + 198*z'^2 + 18*z'^3 }-> s21 :|: s19 >= 0, s19 <= z + 1, s20 >= 0, s20 <= z' + 1, s21 >= 0, s21 <= 2, z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 424 + 15*s22 + 2*s22*s23 + 4*s23 + 628*z + 198*z^2 + 18*z^3 + 628*z' + 198*z'^2 + 18*z'^3 }-> s24 :|: s22 >= 0, s22 <= z + 1, s23 >= 0, s23 <= z' + 1, s24 >= 0, s24 <= s22 + 1, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_p(z) -{ -259 + 286*z + 144*z^2 + 18*z^3 }-> s48 :|: s47 >= 0, s47 <= z - 1 + 1, s48 >= 0, s48 <= 1 + s47, z - 1 >= 0 encode_p(z) -{ -258 + 286*z + 144*z^2 + 18*z^3 }-> s51 :|: s49 >= 0, s49 <= z - 1 + 1, s50 >= 0, s50 <= s49, s51 >= 0, s51 <= s50, z - 1 >= 0 encode_p(z) -{ 379 + s53 + 628*x_1561 + 198*x_1561^2 + 18*x_1561^3 + 628*x_2335 + 198*x_2335^2 + 18*x_2335^3 }-> s55 :|: s52 >= 0, s52 <= x_1561 + 1, s53 >= 0, s53 <= x_2335 + 1, s54 >= 0, s54 <= 2, s55 >= 0, s55 <= s54, x_1561 >= 0, x_2335 >= 0, z = 1 + x_1561 + x_2335 encode_p(z) -{ 425 + 15*s56 + 2*s56*s57 + 4*s57 + 628*x_1562 + 198*x_1562^2 + 18*x_1562^3 + 628*x_2336 + 198*x_2336^2 + 18*x_2336^3 }-> s59 :|: s56 >= 0, s56 <= x_1562 + 1, s57 >= 0, s57 <= x_2336 + 1, s58 >= 0, s58 <= s56 + 1, s59 >= 0, s59 <= s58, x_2336 >= 0, x_1562 >= 0, z = 1 + x_1562 + x_2336 encode_p(z) -{ 575 + 5*s61 + s61*s62 + s62 + 628*x_1563 + 198*x_1563^2 + 18*x_1563^3 + 628*x_2337 + 198*x_2337^2 + 18*x_2337^3 + 628*x_3111 + 198*x_3111^2 + 18*x_3111^3 }-> s64 :|: s60 >= 0, s60 <= x_1563 + 1, s61 >= 0, s61 <= x_2337 + 1, s62 >= 0, s62 <= x_3111 + 1, s63 >= 0, s63 <= s61 + 1, s64 >= 0, s64 <= s63, z = 1 + x_1563 + x_2337 + x_3111, x_2337 >= 0, x_1563 >= 0, x_3111 >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 188 + 628*z + 198*z^2 + 18*z^3 }-> 1 + s18 :|: s18 >= 0, s18 <= z + 1, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 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'') -{ 50 + 4*z'' }-> 1 + s4 :|: s4 >= 0, s4 <= 0 + 1, z = 1, z'' >= 0, z' = 0 if(z, z', z'') -{ 35 + 15*z' + 2*z'*z'' + 2*z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1 + 1, z' - 1 >= 0, z = 1, z'' >= 0 if(z, z', z'') -{ 49 + 4*z'' }-> 1 + s6 :|: s6 >= 0, s6 <= 0 + 1, z = 1, z' >= 0, z'' >= 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 12 + z' }-> s'' :|: s'' >= 0, s'' <= 0 + 1, z' >= 0, z = 0 minus(z, z') -{ 12 + 5*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1) + 1, z - 1 >= 0, z' = 0 minus(z, z') -{ 13 + 5*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1) + 1, s' >= 0, s' <= 2, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 11 + 5*z + z*z' + z' }-> s3 :|: s3 >= 0, s3 <= z + 1, z >= 0, z' >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encode_s} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] le: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] if: runtime: O(n^2) [10 + 5*z' + z'*z'' + z''], size: O(n^1) [1 + z'] minus: runtime: O(n^2) [48 + 15*z + 2*z*z' + 4*z'], size: O(n^1) [1 + z] encArg: runtime: O(n^3) [188 + 628*z + 198*z^2 + 18*z^3], size: O(n^1) [1 + z] encode_p: runtime: O(n^3) [1412 + 5000*z + 1677*z^2 + 162*z^3], size: O(n^1) [1 + z] encode_if: runtime: O(n^3) [581 + 628*z + 198*z^2 + 18*z^3 + 634*z' + z'*z'' + 198*z'^2 + 18*z'^3 + 630*z'' + 198*z''^2 + 18*z''^3], size: O(n^1) [2 + z'] encode_minus: runtime: O(n^3) [445 + 645*z + 2*z*z' + 198*z^2 + 18*z^3 + 634*z' + 198*z'^2 + 18*z'^3], size: O(n^1) [2 + z] encode_le: runtime: O(n^3) [379 + 628*z + 198*z^2 + 18*z^3 + 629*z' + 198*z'^2 + 18*z'^3], size: O(1) [2] ---------------------------------------- (87) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (88) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 378 + s9 + 628*x_1 + 198*x_1^2 + 18*x_1^3 + 628*x_2 + 198*x_2^2 + 18*x_2^3 }-> s10 :|: s8 >= 0, s8 <= x_1 + 1, s9 >= 0, s9 <= x_2 + 1, s10 >= 0, s10 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 424 + 15*s11 + 2*s11*s12 + 4*s12 + 628*x_1 + 198*x_1^2 + 18*x_1^3 + 628*x_2 + 198*x_2^2 + 18*x_2^3 }-> s13 :|: s11 >= 0, s11 <= x_1 + 1, s12 >= 0, s12 <= x_2 + 1, s13 >= 0, s13 <= s11 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 574 + 5*s15 + s15*s16 + s16 + 628*x_1 + 198*x_1^2 + 18*x_1^3 + 628*x_2 + 198*x_2^2 + 18*x_2^3 + 628*x_3 + 198*x_3^2 + 18*x_3^3 }-> s17 :|: s14 >= 0, s14 <= x_1 + 1, s15 >= 0, s15 <= x_2 + 1, s16 >= 0, s16 <= x_3 + 1, s17 >= 0, s17 <= s15 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ -419 + 52*z + 90*z^2 + 18*z^3 }-> s30 :|: s29 >= 0, s29 <= z - 2 + 1, s30 >= 0, s30 <= 1 + s29, z - 2 >= 0 encArg(z) -{ -418 + 52*z + 90*z^2 + 18*z^3 }-> s33 :|: s31 >= 0, s31 <= z - 2 + 1, s32 >= 0, s32 <= s31, s33 >= 0, s33 <= s32, z - 2 >= 0 encArg(z) -{ 379 + s35 + 628*x_11 + 198*x_11^2 + 18*x_11^3 + 628*x_2' + 198*x_2'^2 + 18*x_2'^3 }-> s37 :|: s34 >= 0, s34 <= x_11 + 1, s35 >= 0, s35 <= x_2' + 1, s36 >= 0, s36 <= 2, s37 >= 0, s37 <= s36, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') encArg(z) -{ 425 + 15*s38 + 2*s38*s39 + 4*s39 + 628*x_12 + 198*x_12^2 + 18*x_12^3 + 628*x_2'' + 198*x_2''^2 + 18*x_2''^3 }-> s41 :|: s38 >= 0, s38 <= x_12 + 1, s39 >= 0, s39 <= x_2'' + 1, s40 >= 0, s40 <= s38 + 1, s41 >= 0, s41 <= s40, z = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0 encArg(z) -{ 575 + 5*s43 + s43*s44 + s44 + 628*x_13 + 198*x_13^2 + 18*x_13^3 + 628*x_21 + 198*x_21^2 + 18*x_21^3 + 628*x_3' + 198*x_3'^2 + 18*x_3'^3 }-> s46 :|: s42 >= 0, s42 <= x_13 + 1, s43 >= 0, s43 <= x_21 + 1, s44 >= 0, s44 <= x_3' + 1, s45 >= 0, s45 <= s43 + 1, s46 >= 0, s46 <= s45, z = 1 + (1 + x_13 + x_21 + x_3'), x_3' >= 0, x_13 >= 0, x_21 >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ -260 + 286*z + 144*z^2 + 18*z^3 }-> 1 + s7 :|: s7 >= 0, s7 <= z - 1 + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 574 + 5*s26 + s26*s27 + s27 + 628*z + 198*z^2 + 18*z^3 + 628*z' + 198*z'^2 + 18*z'^3 + 628*z'' + 198*z''^2 + 18*z''^3 }-> s28 :|: s25 >= 0, s25 <= z + 1, s26 >= 0, s26 <= z' + 1, s27 >= 0, s27 <= z'' + 1, s28 >= 0, s28 <= s26 + 1, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_le(z, z') -{ 378 + s20 + 628*z + 198*z^2 + 18*z^3 + 628*z' + 198*z'^2 + 18*z'^3 }-> s21 :|: s19 >= 0, s19 <= z + 1, s20 >= 0, s20 <= z' + 1, s21 >= 0, s21 <= 2, z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 424 + 15*s22 + 2*s22*s23 + 4*s23 + 628*z + 198*z^2 + 18*z^3 + 628*z' + 198*z'^2 + 18*z'^3 }-> s24 :|: s22 >= 0, s22 <= z + 1, s23 >= 0, s23 <= z' + 1, s24 >= 0, s24 <= s22 + 1, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_p(z) -{ -259 + 286*z + 144*z^2 + 18*z^3 }-> s48 :|: s47 >= 0, s47 <= z - 1 + 1, s48 >= 0, s48 <= 1 + s47, z - 1 >= 0 encode_p(z) -{ -258 + 286*z + 144*z^2 + 18*z^3 }-> s51 :|: s49 >= 0, s49 <= z - 1 + 1, s50 >= 0, s50 <= s49, s51 >= 0, s51 <= s50, z - 1 >= 0 encode_p(z) -{ 379 + s53 + 628*x_1561 + 198*x_1561^2 + 18*x_1561^3 + 628*x_2335 + 198*x_2335^2 + 18*x_2335^3 }-> s55 :|: s52 >= 0, s52 <= x_1561 + 1, s53 >= 0, s53 <= x_2335 + 1, s54 >= 0, s54 <= 2, s55 >= 0, s55 <= s54, x_1561 >= 0, x_2335 >= 0, z = 1 + x_1561 + x_2335 encode_p(z) -{ 425 + 15*s56 + 2*s56*s57 + 4*s57 + 628*x_1562 + 198*x_1562^2 + 18*x_1562^3 + 628*x_2336 + 198*x_2336^2 + 18*x_2336^3 }-> s59 :|: s56 >= 0, s56 <= x_1562 + 1, s57 >= 0, s57 <= x_2336 + 1, s58 >= 0, s58 <= s56 + 1, s59 >= 0, s59 <= s58, x_2336 >= 0, x_1562 >= 0, z = 1 + x_1562 + x_2336 encode_p(z) -{ 575 + 5*s61 + s61*s62 + s62 + 628*x_1563 + 198*x_1563^2 + 18*x_1563^3 + 628*x_2337 + 198*x_2337^2 + 18*x_2337^3 + 628*x_3111 + 198*x_3111^2 + 18*x_3111^3 }-> s64 :|: s60 >= 0, s60 <= x_1563 + 1, s61 >= 0, s61 <= x_2337 + 1, s62 >= 0, s62 <= x_3111 + 1, s63 >= 0, s63 <= s61 + 1, s64 >= 0, s64 <= s63, z = 1 + x_1563 + x_2337 + x_3111, x_2337 >= 0, x_1563 >= 0, x_3111 >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 188 + 628*z + 198*z^2 + 18*z^3 }-> 1 + s18 :|: s18 >= 0, s18 <= z + 1, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 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'') -{ 50 + 4*z'' }-> 1 + s4 :|: s4 >= 0, s4 <= 0 + 1, z = 1, z'' >= 0, z' = 0 if(z, z', z'') -{ 35 + 15*z' + 2*z'*z'' + 2*z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1 + 1, z' - 1 >= 0, z = 1, z'' >= 0 if(z, z', z'') -{ 49 + 4*z'' }-> 1 + s6 :|: s6 >= 0, s6 <= 0 + 1, z = 1, z' >= 0, z'' >= 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 12 + z' }-> s'' :|: s'' >= 0, s'' <= 0 + 1, z' >= 0, z = 0 minus(z, z') -{ 12 + 5*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1) + 1, z - 1 >= 0, z' = 0 minus(z, z') -{ 13 + 5*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1) + 1, s' >= 0, s' <= 2, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 11 + 5*z + z*z' + z' }-> s3 :|: s3 >= 0, s3 <= z + 1, z >= 0, z' >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encode_s} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] le: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] if: runtime: O(n^2) [10 + 5*z' + z'*z'' + z''], size: O(n^1) [1 + z'] minus: runtime: O(n^2) [48 + 15*z + 2*z*z' + 4*z'], size: O(n^1) [1 + z] encArg: runtime: O(n^3) [188 + 628*z + 198*z^2 + 18*z^3], size: O(n^1) [1 + z] encode_p: runtime: O(n^3) [1412 + 5000*z + 1677*z^2 + 162*z^3], size: O(n^1) [1 + z] encode_if: runtime: O(n^3) [581 + 628*z + 198*z^2 + 18*z^3 + 634*z' + z'*z'' + 198*z'^2 + 18*z'^3 + 630*z'' + 198*z''^2 + 18*z''^3], size: O(n^1) [2 + z'] encode_minus: runtime: O(n^3) [445 + 645*z + 2*z*z' + 198*z^2 + 18*z^3 + 634*z' + 198*z'^2 + 18*z'^3], size: O(n^1) [2 + z] encode_le: runtime: O(n^3) [379 + 628*z + 198*z^2 + 18*z^3 + 629*z' + 198*z'^2 + 18*z'^3], size: O(1) [2] ---------------------------------------- (89) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: encode_s after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z ---------------------------------------- (90) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 378 + s9 + 628*x_1 + 198*x_1^2 + 18*x_1^3 + 628*x_2 + 198*x_2^2 + 18*x_2^3 }-> s10 :|: s8 >= 0, s8 <= x_1 + 1, s9 >= 0, s9 <= x_2 + 1, s10 >= 0, s10 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 424 + 15*s11 + 2*s11*s12 + 4*s12 + 628*x_1 + 198*x_1^2 + 18*x_1^3 + 628*x_2 + 198*x_2^2 + 18*x_2^3 }-> s13 :|: s11 >= 0, s11 <= x_1 + 1, s12 >= 0, s12 <= x_2 + 1, s13 >= 0, s13 <= s11 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 574 + 5*s15 + s15*s16 + s16 + 628*x_1 + 198*x_1^2 + 18*x_1^3 + 628*x_2 + 198*x_2^2 + 18*x_2^3 + 628*x_3 + 198*x_3^2 + 18*x_3^3 }-> s17 :|: s14 >= 0, s14 <= x_1 + 1, s15 >= 0, s15 <= x_2 + 1, s16 >= 0, s16 <= x_3 + 1, s17 >= 0, s17 <= s15 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ -419 + 52*z + 90*z^2 + 18*z^3 }-> s30 :|: s29 >= 0, s29 <= z - 2 + 1, s30 >= 0, s30 <= 1 + s29, z - 2 >= 0 encArg(z) -{ -418 + 52*z + 90*z^2 + 18*z^3 }-> s33 :|: s31 >= 0, s31 <= z - 2 + 1, s32 >= 0, s32 <= s31, s33 >= 0, s33 <= s32, z - 2 >= 0 encArg(z) -{ 379 + s35 + 628*x_11 + 198*x_11^2 + 18*x_11^3 + 628*x_2' + 198*x_2'^2 + 18*x_2'^3 }-> s37 :|: s34 >= 0, s34 <= x_11 + 1, s35 >= 0, s35 <= x_2' + 1, s36 >= 0, s36 <= 2, s37 >= 0, s37 <= s36, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') encArg(z) -{ 425 + 15*s38 + 2*s38*s39 + 4*s39 + 628*x_12 + 198*x_12^2 + 18*x_12^3 + 628*x_2'' + 198*x_2''^2 + 18*x_2''^3 }-> s41 :|: s38 >= 0, s38 <= x_12 + 1, s39 >= 0, s39 <= x_2'' + 1, s40 >= 0, s40 <= s38 + 1, s41 >= 0, s41 <= s40, z = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0 encArg(z) -{ 575 + 5*s43 + s43*s44 + s44 + 628*x_13 + 198*x_13^2 + 18*x_13^3 + 628*x_21 + 198*x_21^2 + 18*x_21^3 + 628*x_3' + 198*x_3'^2 + 18*x_3'^3 }-> s46 :|: s42 >= 0, s42 <= x_13 + 1, s43 >= 0, s43 <= x_21 + 1, s44 >= 0, s44 <= x_3' + 1, s45 >= 0, s45 <= s43 + 1, s46 >= 0, s46 <= s45, z = 1 + (1 + x_13 + x_21 + x_3'), x_3' >= 0, x_13 >= 0, x_21 >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ -260 + 286*z + 144*z^2 + 18*z^3 }-> 1 + s7 :|: s7 >= 0, s7 <= z - 1 + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 574 + 5*s26 + s26*s27 + s27 + 628*z + 198*z^2 + 18*z^3 + 628*z' + 198*z'^2 + 18*z'^3 + 628*z'' + 198*z''^2 + 18*z''^3 }-> s28 :|: s25 >= 0, s25 <= z + 1, s26 >= 0, s26 <= z' + 1, s27 >= 0, s27 <= z'' + 1, s28 >= 0, s28 <= s26 + 1, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_le(z, z') -{ 378 + s20 + 628*z + 198*z^2 + 18*z^3 + 628*z' + 198*z'^2 + 18*z'^3 }-> s21 :|: s19 >= 0, s19 <= z + 1, s20 >= 0, s20 <= z' + 1, s21 >= 0, s21 <= 2, z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 424 + 15*s22 + 2*s22*s23 + 4*s23 + 628*z + 198*z^2 + 18*z^3 + 628*z' + 198*z'^2 + 18*z'^3 }-> s24 :|: s22 >= 0, s22 <= z + 1, s23 >= 0, s23 <= z' + 1, s24 >= 0, s24 <= s22 + 1, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_p(z) -{ -259 + 286*z + 144*z^2 + 18*z^3 }-> s48 :|: s47 >= 0, s47 <= z - 1 + 1, s48 >= 0, s48 <= 1 + s47, z - 1 >= 0 encode_p(z) -{ -258 + 286*z + 144*z^2 + 18*z^3 }-> s51 :|: s49 >= 0, s49 <= z - 1 + 1, s50 >= 0, s50 <= s49, s51 >= 0, s51 <= s50, z - 1 >= 0 encode_p(z) -{ 379 + s53 + 628*x_1561 + 198*x_1561^2 + 18*x_1561^3 + 628*x_2335 + 198*x_2335^2 + 18*x_2335^3 }-> s55 :|: s52 >= 0, s52 <= x_1561 + 1, s53 >= 0, s53 <= x_2335 + 1, s54 >= 0, s54 <= 2, s55 >= 0, s55 <= s54, x_1561 >= 0, x_2335 >= 0, z = 1 + x_1561 + x_2335 encode_p(z) -{ 425 + 15*s56 + 2*s56*s57 + 4*s57 + 628*x_1562 + 198*x_1562^2 + 18*x_1562^3 + 628*x_2336 + 198*x_2336^2 + 18*x_2336^3 }-> s59 :|: s56 >= 0, s56 <= x_1562 + 1, s57 >= 0, s57 <= x_2336 + 1, s58 >= 0, s58 <= s56 + 1, s59 >= 0, s59 <= s58, x_2336 >= 0, x_1562 >= 0, z = 1 + x_1562 + x_2336 encode_p(z) -{ 575 + 5*s61 + s61*s62 + s62 + 628*x_1563 + 198*x_1563^2 + 18*x_1563^3 + 628*x_2337 + 198*x_2337^2 + 18*x_2337^3 + 628*x_3111 + 198*x_3111^2 + 18*x_3111^3 }-> s64 :|: s60 >= 0, s60 <= x_1563 + 1, s61 >= 0, s61 <= x_2337 + 1, s62 >= 0, s62 <= x_3111 + 1, s63 >= 0, s63 <= s61 + 1, s64 >= 0, s64 <= s63, z = 1 + x_1563 + x_2337 + x_3111, x_2337 >= 0, x_1563 >= 0, x_3111 >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 188 + 628*z + 198*z^2 + 18*z^3 }-> 1 + s18 :|: s18 >= 0, s18 <= z + 1, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 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'') -{ 50 + 4*z'' }-> 1 + s4 :|: s4 >= 0, s4 <= 0 + 1, z = 1, z'' >= 0, z' = 0 if(z, z', z'') -{ 35 + 15*z' + 2*z'*z'' + 2*z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1 + 1, z' - 1 >= 0, z = 1, z'' >= 0 if(z, z', z'') -{ 49 + 4*z'' }-> 1 + s6 :|: s6 >= 0, s6 <= 0 + 1, z = 1, z' >= 0, z'' >= 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 12 + z' }-> s'' :|: s'' >= 0, s'' <= 0 + 1, z' >= 0, z = 0 minus(z, z') -{ 12 + 5*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1) + 1, z - 1 >= 0, z' = 0 minus(z, z') -{ 13 + 5*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1) + 1, s' >= 0, s' <= 2, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 11 + 5*z + z*z' + z' }-> s3 :|: s3 >= 0, s3 <= z + 1, z >= 0, z' >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {encode_s} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] le: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] if: runtime: O(n^2) [10 + 5*z' + z'*z'' + z''], size: O(n^1) [1 + z'] minus: runtime: O(n^2) [48 + 15*z + 2*z*z' + 4*z'], size: O(n^1) [1 + z] encArg: runtime: O(n^3) [188 + 628*z + 198*z^2 + 18*z^3], size: O(n^1) [1 + z] encode_p: runtime: O(n^3) [1412 + 5000*z + 1677*z^2 + 162*z^3], size: O(n^1) [1 + z] encode_if: runtime: O(n^3) [581 + 628*z + 198*z^2 + 18*z^3 + 634*z' + z'*z'' + 198*z'^2 + 18*z'^3 + 630*z'' + 198*z''^2 + 18*z''^3], size: O(n^1) [2 + z'] encode_minus: runtime: O(n^3) [445 + 645*z + 2*z*z' + 198*z^2 + 18*z^3 + 634*z' + 198*z'^2 + 18*z'^3], size: O(n^1) [2 + z] encode_le: runtime: O(n^3) [379 + 628*z + 198*z^2 + 18*z^3 + 629*z' + 198*z'^2 + 18*z'^3], size: O(1) [2] encode_s: runtime: ?, size: O(n^1) [2 + z] ---------------------------------------- (91) 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: 188 + 628*z + 198*z^2 + 18*z^3 ---------------------------------------- (92) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 378 + s9 + 628*x_1 + 198*x_1^2 + 18*x_1^3 + 628*x_2 + 198*x_2^2 + 18*x_2^3 }-> s10 :|: s8 >= 0, s8 <= x_1 + 1, s9 >= 0, s9 <= x_2 + 1, s10 >= 0, s10 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 424 + 15*s11 + 2*s11*s12 + 4*s12 + 628*x_1 + 198*x_1^2 + 18*x_1^3 + 628*x_2 + 198*x_2^2 + 18*x_2^3 }-> s13 :|: s11 >= 0, s11 <= x_1 + 1, s12 >= 0, s12 <= x_2 + 1, s13 >= 0, s13 <= s11 + 1, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 574 + 5*s15 + s15*s16 + s16 + 628*x_1 + 198*x_1^2 + 18*x_1^3 + 628*x_2 + 198*x_2^2 + 18*x_2^3 + 628*x_3 + 198*x_3^2 + 18*x_3^3 }-> s17 :|: s14 >= 0, s14 <= x_1 + 1, s15 >= 0, s15 <= x_2 + 1, s16 >= 0, s16 <= x_3 + 1, s17 >= 0, s17 <= s15 + 1, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ -419 + 52*z + 90*z^2 + 18*z^3 }-> s30 :|: s29 >= 0, s29 <= z - 2 + 1, s30 >= 0, s30 <= 1 + s29, z - 2 >= 0 encArg(z) -{ -418 + 52*z + 90*z^2 + 18*z^3 }-> s33 :|: s31 >= 0, s31 <= z - 2 + 1, s32 >= 0, s32 <= s31, s33 >= 0, s33 <= s32, z - 2 >= 0 encArg(z) -{ 379 + s35 + 628*x_11 + 198*x_11^2 + 18*x_11^3 + 628*x_2' + 198*x_2'^2 + 18*x_2'^3 }-> s37 :|: s34 >= 0, s34 <= x_11 + 1, s35 >= 0, s35 <= x_2' + 1, s36 >= 0, s36 <= 2, s37 >= 0, s37 <= s36, x_11 >= 0, x_2' >= 0, z = 1 + (1 + x_11 + x_2') encArg(z) -{ 425 + 15*s38 + 2*s38*s39 + 4*s39 + 628*x_12 + 198*x_12^2 + 18*x_12^3 + 628*x_2'' + 198*x_2''^2 + 18*x_2''^3 }-> s41 :|: s38 >= 0, s38 <= x_12 + 1, s39 >= 0, s39 <= x_2'' + 1, s40 >= 0, s40 <= s38 + 1, s41 >= 0, s41 <= s40, z = 1 + (1 + x_12 + x_2''), x_2'' >= 0, x_12 >= 0 encArg(z) -{ 575 + 5*s43 + s43*s44 + s44 + 628*x_13 + 198*x_13^2 + 18*x_13^3 + 628*x_21 + 198*x_21^2 + 18*x_21^3 + 628*x_3' + 198*x_3'^2 + 18*x_3'^3 }-> s46 :|: s42 >= 0, s42 <= x_13 + 1, s43 >= 0, s43 <= x_21 + 1, s44 >= 0, s44 <= x_3' + 1, s45 >= 0, s45 <= s43 + 1, s46 >= 0, s46 <= s45, z = 1 + (1 + x_13 + x_21 + x_3'), x_3' >= 0, x_13 >= 0, x_21 >= 0 encArg(z) -{ 1 }-> x :|: z = 1 + 2, x >= 0, 2 = 1 + x encArg(z) -{ 1 }-> x :|: z = 1 + 1, x >= 0, 1 = 1 + x encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 1 }-> 0 :|: z = 1 + 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 0, v0 >= 0, 0 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 2, v0 >= 0, 2 = v0 encArg(z) -{ 0 }-> 0 :|: z = 1 + 1, v0 >= 0, 1 = v0 encArg(z) -{ 1 }-> 0 :|: z - 1 >= 0, 0 = 0 encArg(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, 0 = v0 encArg(z) -{ -260 + 286*z + 144*z^2 + 18*z^3 }-> 1 + s7 :|: s7 >= 0, s7 <= z - 1 + 1, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_if(z, z', z'') -{ 574 + 5*s26 + s26*s27 + s27 + 628*z + 198*z^2 + 18*z^3 + 628*z' + 198*z'^2 + 18*z'^3 + 628*z'' + 198*z''^2 + 18*z''^3 }-> s28 :|: s25 >= 0, s25 <= z + 1, s26 >= 0, s26 <= z' + 1, s27 >= 0, s27 <= z'' + 1, s28 >= 0, s28 <= s26 + 1, z >= 0, z'' >= 0, z' >= 0 encode_if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_le(z, z') -{ 378 + s20 + 628*z + 198*z^2 + 18*z^3 + 628*z' + 198*z'^2 + 18*z'^3 }-> s21 :|: s19 >= 0, s19 <= z + 1, s20 >= 0, s20 <= z' + 1, s21 >= 0, s21 <= 2, z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 424 + 15*s22 + 2*s22*s23 + 4*s23 + 628*z + 198*z^2 + 18*z^3 + 628*z' + 198*z'^2 + 18*z'^3 }-> s24 :|: s22 >= 0, s22 <= z + 1, s23 >= 0, s23 <= z' + 1, s24 >= 0, s24 <= s22 + 1, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_p(z) -{ -259 + 286*z + 144*z^2 + 18*z^3 }-> s48 :|: s47 >= 0, s47 <= z - 1 + 1, s48 >= 0, s48 <= 1 + s47, z - 1 >= 0 encode_p(z) -{ -258 + 286*z + 144*z^2 + 18*z^3 }-> s51 :|: s49 >= 0, s49 <= z - 1 + 1, s50 >= 0, s50 <= s49, s51 >= 0, s51 <= s50, z - 1 >= 0 encode_p(z) -{ 379 + s53 + 628*x_1561 + 198*x_1561^2 + 18*x_1561^3 + 628*x_2335 + 198*x_2335^2 + 18*x_2335^3 }-> s55 :|: s52 >= 0, s52 <= x_1561 + 1, s53 >= 0, s53 <= x_2335 + 1, s54 >= 0, s54 <= 2, s55 >= 0, s55 <= s54, x_1561 >= 0, x_2335 >= 0, z = 1 + x_1561 + x_2335 encode_p(z) -{ 425 + 15*s56 + 2*s56*s57 + 4*s57 + 628*x_1562 + 198*x_1562^2 + 18*x_1562^3 + 628*x_2336 + 198*x_2336^2 + 18*x_2336^3 }-> s59 :|: s56 >= 0, s56 <= x_1562 + 1, s57 >= 0, s57 <= x_2336 + 1, s58 >= 0, s58 <= s56 + 1, s59 >= 0, s59 <= s58, x_2336 >= 0, x_1562 >= 0, z = 1 + x_1562 + x_2336 encode_p(z) -{ 575 + 5*s61 + s61*s62 + s62 + 628*x_1563 + 198*x_1563^2 + 18*x_1563^3 + 628*x_2337 + 198*x_2337^2 + 18*x_2337^3 + 628*x_3111 + 198*x_3111^2 + 18*x_3111^3 }-> s64 :|: s60 >= 0, s60 <= x_1563 + 1, s61 >= 0, s61 <= x_2337 + 1, s62 >= 0, s62 <= x_3111 + 1, s63 >= 0, s63 <= s61 + 1, s64 >= 0, s64 <= s63, z = 1 + x_1563 + x_2337 + x_3111, x_2337 >= 0, x_1563 >= 0, x_3111 >= 0 encode_p(z) -{ 1 }-> x :|: z = 2, x >= 0, 2 = 1 + x encode_p(z) -{ 1 }-> x :|: z = 1, x >= 0, 1 = 1 + x encode_p(z) -{ 0 }-> 0 :|: z >= 0 encode_p(z) -{ 1 }-> 0 :|: z = 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z = 0, v0 >= 0, 0 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 2, v0 >= 0, 2 = v0 encode_p(z) -{ 0 }-> 0 :|: z = 1, v0 >= 0, 1 = v0 encode_p(z) -{ 1 }-> 0 :|: z >= 0, 0 = 0 encode_p(z) -{ 0 }-> 0 :|: z >= 0, v0 >= 0, 0 = v0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 188 + 628*z + 198*z^2 + 18*z^3 }-> 1 + s18 :|: s18 >= 0, s18 <= z + 1, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 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'') -{ 50 + 4*z'' }-> 1 + s4 :|: s4 >= 0, s4 <= 0 + 1, z = 1, z'' >= 0, z' = 0 if(z, z', z'') -{ 35 + 15*z' + 2*z'*z'' + 2*z'' }-> 1 + s5 :|: s5 >= 0, s5 <= z' - 1 + 1, z' - 1 >= 0, z = 1, z'' >= 0 if(z, z', z'') -{ 49 + 4*z'' }-> 1 + s6 :|: s6 >= 0, s6 <= 0 + 1, z = 1, z' >= 0, z'' >= 0 le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 2, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 2 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 12 + z' }-> s'' :|: s'' >= 0, s'' <= 0 + 1, z' >= 0, z = 0 minus(z, z') -{ 12 + 5*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1) + 1, z - 1 >= 0, z' = 0 minus(z, z') -{ 13 + 5*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1) + 1, s' >= 0, s' <= 2, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 11 + 5*z + z*z' + z' }-> s3 :|: s3 >= 0, s3 <= z + 1, z >= 0, z' >= 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] le: runtime: O(n^1) [2 + z'], size: O(1) [2] encode_false: runtime: O(1) [0], size: O(1) [1] encode_true: runtime: O(1) [0], size: O(1) [2] p: runtime: O(1) [1], size: O(n^1) [z] if: runtime: O(n^2) [10 + 5*z' + z'*z'' + z''], size: O(n^1) [1 + z'] minus: runtime: O(n^2) [48 + 15*z + 2*z*z' + 4*z'], size: O(n^1) [1 + z] encArg: runtime: O(n^3) [188 + 628*z + 198*z^2 + 18*z^3], size: O(n^1) [1 + z] encode_p: runtime: O(n^3) [1412 + 5000*z + 1677*z^2 + 162*z^3], size: O(n^1) [1 + z] encode_if: runtime: O(n^3) [581 + 628*z + 198*z^2 + 18*z^3 + 634*z' + z'*z'' + 198*z'^2 + 18*z'^3 + 630*z'' + 198*z''^2 + 18*z''^3], size: O(n^1) [2 + z'] encode_minus: runtime: O(n^3) [445 + 645*z + 2*z*z' + 198*z^2 + 18*z^3 + 634*z' + 198*z'^2 + 18*z'^3], size: O(n^1) [2 + z] encode_le: runtime: O(n^3) [379 + 628*z + 198*z^2 + 18*z^3 + 629*z' + 198*z'^2 + 18*z'^3], size: O(1) [2] encode_s: runtime: O(n^3) [188 + 628*z + 198*z^2 + 18*z^3], size: O(n^1) [2 + z] ---------------------------------------- (93) FinalProof (FINISHED) Computed overall runtime complexity ---------------------------------------- (94) BOUNDS(1, n^3) ---------------------------------------- (95) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (96) 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: p(0') -> 0' p(s(x)) -> x le(0', y) -> true le(s(x), 0') -> false le(s(x), s(y)) -> le(x, y) minus(x, y) -> if(le(x, y), x, y) if(true, x, y) -> 0' if(false, x, y) -> s(minus(p(x), y)) The (relative) TRS S consists of the following rules: encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(true) -> true encArg(false) -> false encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_p(x_1) -> p(encArg(x_1)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) Rewrite Strategy: INNERMOST ---------------------------------------- (97) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (98) Obligation: Innermost TRS: Rules: p(0') -> 0' p(s(x)) -> x le(0', y) -> true le(s(x), 0') -> false le(s(x), s(y)) -> le(x, y) minus(x, y) -> if(le(x, y), x, y) if(true, x, y) -> 0' if(false, x, y) -> s(minus(p(x), y)) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(true) -> true encArg(false) -> false encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_p(x_1) -> p(encArg(x_1)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) Types: p :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if 0' :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if s :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if le :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if true :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if false :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if minus :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if if :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if encArg :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if cons_p :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if cons_le :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if cons_minus :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if cons_if :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if encode_p :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if encode_0 :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if encode_s :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if encode_le :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if encode_true :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if encode_false :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if encode_minus :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if encode_if :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if hole_0':s:true:false:cons_p:cons_le:cons_minus:cons_if1_4 :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if gen_0':s:true:false:cons_p:cons_le:cons_minus:cons_if2_4 :: Nat -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if ---------------------------------------- (99) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: le, minus, if, encArg They will be analysed ascendingly in the following order: le < minus le < encArg minus = if minus < encArg if < encArg ---------------------------------------- (100) Obligation: Innermost TRS: Rules: p(0') -> 0' p(s(x)) -> x le(0', y) -> true le(s(x), 0') -> false le(s(x), s(y)) -> le(x, y) minus(x, y) -> if(le(x, y), x, y) if(true, x, y) -> 0' if(false, x, y) -> s(minus(p(x), y)) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(true) -> true encArg(false) -> false encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_p(x_1) -> p(encArg(x_1)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) Types: p :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if 0' :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if s :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if le :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if true :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if false :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if minus :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if if :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if encArg :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if cons_p :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if cons_le :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if cons_minus :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if cons_if :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if encode_p :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if encode_0 :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if encode_s :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if encode_le :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if encode_true :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if encode_false :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if encode_minus :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if encode_if :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if hole_0':s:true:false:cons_p:cons_le:cons_minus:cons_if1_4 :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if gen_0':s:true:false:cons_p:cons_le:cons_minus:cons_if2_4 :: Nat -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if Generator Equations: gen_0':s:true:false:cons_p:cons_le:cons_minus:cons_if2_4(0) <=> 0' gen_0':s:true:false:cons_p:cons_le:cons_minus:cons_if2_4(+(x, 1)) <=> s(gen_0':s:true:false:cons_p:cons_le:cons_minus:cons_if2_4(x)) The following defined symbols remain to be analysed: le, minus, if, encArg They will be analysed ascendingly in the following order: le < minus le < encArg minus = if minus < encArg if < encArg ---------------------------------------- (101) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: le(gen_0':s:true:false:cons_p:cons_le:cons_minus:cons_if2_4(n4_4), gen_0':s:true:false:cons_p:cons_le:cons_minus:cons_if2_4(n4_4)) -> true, rt in Omega(1 + n4_4) Induction Base: le(gen_0':s:true:false:cons_p:cons_le:cons_minus:cons_if2_4(0), gen_0':s:true:false:cons_p:cons_le:cons_minus:cons_if2_4(0)) ->_R^Omega(1) true Induction Step: le(gen_0':s:true:false:cons_p:cons_le:cons_minus:cons_if2_4(+(n4_4, 1)), gen_0':s:true:false:cons_p:cons_le:cons_minus:cons_if2_4(+(n4_4, 1))) ->_R^Omega(1) le(gen_0':s:true:false:cons_p:cons_le:cons_minus:cons_if2_4(n4_4), gen_0':s:true:false:cons_p:cons_le:cons_minus: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). ---------------------------------------- (102) Complex Obligation (BEST) ---------------------------------------- (103) Obligation: Proved the lower bound n^1 for the following obligation: Innermost TRS: Rules: p(0') -> 0' p(s(x)) -> x le(0', y) -> true le(s(x), 0') -> false le(s(x), s(y)) -> le(x, y) minus(x, y) -> if(le(x, y), x, y) if(true, x, y) -> 0' if(false, x, y) -> s(minus(p(x), y)) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(true) -> true encArg(false) -> false encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_p(x_1) -> p(encArg(x_1)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) Types: p :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if 0' :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if s :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if le :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if true :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if false :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if minus :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if if :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if encArg :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if cons_p :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if cons_le :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if cons_minus :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if cons_if :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if encode_p :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if encode_0 :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if encode_s :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if encode_le :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if encode_true :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if encode_false :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if encode_minus :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if encode_if :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if hole_0':s:true:false:cons_p:cons_le:cons_minus:cons_if1_4 :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if gen_0':s:true:false:cons_p:cons_le:cons_minus:cons_if2_4 :: Nat -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if Generator Equations: gen_0':s:true:false:cons_p:cons_le:cons_minus:cons_if2_4(0) <=> 0' gen_0':s:true:false:cons_p:cons_le:cons_minus:cons_if2_4(+(x, 1)) <=> s(gen_0':s:true:false:cons_p:cons_le:cons_minus:cons_if2_4(x)) The following defined symbols remain to be analysed: le, minus, if, encArg They will be analysed ascendingly in the following order: le < minus le < encArg minus = if minus < encArg if < encArg ---------------------------------------- (104) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (105) BOUNDS(n^1, INF) ---------------------------------------- (106) Obligation: Innermost TRS: Rules: p(0') -> 0' p(s(x)) -> x le(0', y) -> true le(s(x), 0') -> false le(s(x), s(y)) -> le(x, y) minus(x, y) -> if(le(x, y), x, y) if(true, x, y) -> 0' if(false, x, y) -> s(minus(p(x), y)) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(true) -> true encArg(false) -> false encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_p(x_1) -> p(encArg(x_1)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) Types: p :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if 0' :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if s :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if le :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if true :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if false :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if minus :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if if :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if encArg :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if cons_p :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if cons_le :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if cons_minus :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if cons_if :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if encode_p :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if encode_0 :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if encode_s :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if encode_le :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if encode_true :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if encode_false :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if encode_minus :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if encode_if :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if hole_0':s:true:false:cons_p:cons_le:cons_minus:cons_if1_4 :: 0':s:true:false:cons_p:cons_le:cons_minus:cons_if gen_0':s:true:false:cons_p:cons_le:cons_minus:cons_if2_4 :: Nat -> 0':s:true:false:cons_p:cons_le:cons_minus:cons_if Lemmas: le(gen_0':s:true:false:cons_p:cons_le:cons_minus:cons_if2_4(n4_4), gen_0':s:true:false:cons_p:cons_le:cons_minus:cons_if2_4(n4_4)) -> true, rt in Omega(1 + n4_4) Generator Equations: gen_0':s:true:false:cons_p:cons_le:cons_minus:cons_if2_4(0) <=> 0' gen_0':s:true:false:cons_p:cons_le:cons_minus:cons_if2_4(+(x, 1)) <=> s(gen_0':s:true:false:cons_p:cons_le:cons_minus:cons_if2_4(x)) The following defined symbols remain to be analysed: if, minus, encArg They will be analysed ascendingly in the following order: minus = if minus < encArg if < encArg ---------------------------------------- (107) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_0':s:true:false:cons_p:cons_le:cons_minus:cons_if2_4(n536_4)) -> gen_0':s:true:false:cons_p:cons_le:cons_minus:cons_if2_4(n536_4), rt in Omega(0) Induction Base: encArg(gen_0':s:true:false:cons_p:cons_le:cons_minus:cons_if2_4(0)) ->_R^Omega(0) 0' Induction Step: encArg(gen_0':s:true:false:cons_p:cons_le:cons_minus:cons_if2_4(+(n536_4, 1))) ->_R^Omega(0) s(encArg(gen_0':s:true:false:cons_p:cons_le:cons_minus:cons_if2_4(n536_4))) ->_IH s(gen_0':s:true:false:cons_p:cons_le:cons_minus:cons_if2_4(c537_4)) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (108) BOUNDS(1, INF)