/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), O(n^4)) 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^4). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 251 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), 10.3 s] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (14) CpxRNTS (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxRNTS (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) ResultPropagationProof [UPPER BOUND(ID), 2 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 85 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 6 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 370 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 166 ms] (30) CpxRNTS (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 96 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 2 ms] (36) CpxRNTS (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 91 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 32 ms] (42) CpxRNTS (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 719 ms] (46) CpxRNTS (47) IntTrsBoundProof [UPPER BOUND(ID), 338 ms] (48) CpxRNTS (49) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (50) CpxRNTS (51) IntTrsBoundProof [UPPER BOUND(ID), 2375 ms] (52) CpxRNTS (53) IntTrsBoundProof [UPPER BOUND(ID), 117 ms] (54) CpxRNTS (55) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (56) CpxRNTS (57) IntTrsBoundProof [UPPER BOUND(ID), 663 ms] (58) CpxRNTS (59) IntTrsBoundProof [UPPER BOUND(ID), 4 ms] (60) CpxRNTS (61) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (62) CpxRNTS (63) IntTrsBoundProof [UPPER BOUND(ID), 160 ms] (64) CpxRNTS (65) IntTrsBoundProof [UPPER BOUND(ID), 3 ms] (66) CpxRNTS (67) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (68) CpxRNTS (69) IntTrsBoundProof [UPPER BOUND(ID), 218 ms] (70) CpxRNTS (71) IntTrsBoundProof [UPPER BOUND(ID), 3 ms] (72) CpxRNTS (73) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (74) CpxRNTS (75) IntTrsBoundProof [UPPER BOUND(ID), 217 ms] (76) CpxRNTS (77) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] (78) CpxRNTS (79) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (80) CpxRNTS (81) IntTrsBoundProof [UPPER BOUND(ID), 218 ms] (82) CpxRNTS (83) IntTrsBoundProof [UPPER BOUND(ID), 3 ms] (84) CpxRNTS (85) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (86) CpxRNTS (87) IntTrsBoundProof [UPPER BOUND(ID), 177 ms] (88) CpxRNTS (89) IntTrsBoundProof [UPPER BOUND(ID), 2 ms] (90) CpxRNTS (91) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (92) CpxRNTS (93) IntTrsBoundProof [UPPER BOUND(ID), 196 ms] (94) CpxRNTS (95) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] (96) CpxRNTS (97) FinalProof [FINISHED, 0 ms] (98) BOUNDS(1, n^4) (99) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CpxRelTRS (101) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (102) typed CpxTrs (103) OrderProof [LOWER BOUND(ID), 0 ms] (104) typed CpxTrs (105) RewriteLemmaProof [LOWER BOUND(ID), 286 ms] (106) BEST (107) proven lower bound (108) LowerBoundPropagationProof [FINISHED, 0 ms] (109) BOUNDS(n^1, INF) (110) typed CpxTrs (111) RewriteLemmaProof [LOWER BOUND(ID), 268 ms] (112) BOUNDS(1, INF) ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^4). The TRS R consists of the following rules: le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) minus(0, y) -> 0 minus(s(x), y) -> if_minus(le(s(x), y), s(x), y) if_minus(true, s(x), y) -> 0 if_minus(false, s(x), y) -> s(minus(x, y)) gcd(0, y) -> y gcd(s(x), 0) -> s(x) gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y)) if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x)) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(0) -> 0 encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false 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_minus(x_1, x_2, x_3)) -> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_gcd(x_1, x_2)) -> gcd(encArg(x_1), encArg(x_2)) encArg(cons_if_gcd(x_1, x_2, x_3)) -> if_gcd(encArg(x_1), encArg(x_2), encArg(x_3)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_if_minus(x_1, x_2, x_3) -> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) encode_gcd(x_1, x_2) -> gcd(encArg(x_1), encArg(x_2)) encode_if_gcd(x_1, x_2, x_3) -> if_gcd(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^4). The TRS R consists of the following rules: le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) minus(0, y) -> 0 minus(s(x), y) -> if_minus(le(s(x), y), s(x), y) if_minus(true, s(x), y) -> 0 if_minus(false, s(x), y) -> s(minus(x, y)) gcd(0, y) -> y gcd(s(x), 0) -> s(x) gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y)) if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x)) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(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_minus(x_1, x_2, x_3)) -> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_gcd(x_1, x_2)) -> gcd(encArg(x_1), encArg(x_2)) encArg(cons_if_gcd(x_1, x_2, x_3)) -> if_gcd(encArg(x_1), encArg(x_2), encArg(x_3)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_if_minus(x_1, x_2, x_3) -> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) encode_gcd(x_1, x_2) -> gcd(encArg(x_1), encArg(x_2)) encode_if_gcd(x_1, x_2, x_3) -> if_gcd(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^4). The TRS R consists of the following rules: le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) minus(0, y) -> 0 minus(s(x), y) -> if_minus(le(s(x), y), s(x), y) if_minus(true, s(x), y) -> 0 if_minus(false, s(x), y) -> s(minus(x, y)) gcd(0, y) -> y gcd(s(x), 0) -> s(x) gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y)) if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x)) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(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_minus(x_1, x_2, x_3)) -> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_gcd(x_1, x_2)) -> gcd(encArg(x_1), encArg(x_2)) encArg(cons_if_gcd(x_1, x_2, x_3)) -> if_gcd(encArg(x_1), encArg(x_2), encArg(x_3)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_if_minus(x_1, x_2, x_3) -> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) encode_gcd(x_1, x_2) -> gcd(encArg(x_1), encArg(x_2)) encode_if_gcd(x_1, x_2, x_3) -> if_gcd(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^4). The TRS R consists of the following rules: le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [1] minus(0, y) -> 0 [1] minus(s(x), y) -> if_minus(le(s(x), y), s(x), y) [1] if_minus(true, s(x), y) -> 0 [1] if_minus(false, s(x), y) -> s(minus(x, y)) [1] gcd(0, y) -> y [1] gcd(s(x), 0) -> s(x) [1] gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) [1] if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y)) [1] if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x)) [1] encArg(0) -> 0 [0] encArg(true) -> true [0] encArg(s(x_1)) -> s(encArg(x_1)) [0] encArg(false) -> false [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_minus(x_1, x_2, x_3)) -> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encArg(cons_gcd(x_1, x_2)) -> gcd(encArg(x_1), encArg(x_2)) [0] encArg(cons_if_gcd(x_1, x_2, x_3)) -> if_gcd(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) [0] encode_0 -> 0 [0] encode_true -> true [0] encode_s(x_1) -> s(encArg(x_1)) [0] encode_false -> false [0] encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) [0] encode_if_minus(x_1, x_2, x_3) -> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encode_gcd(x_1, x_2) -> gcd(encArg(x_1), encArg(x_2)) [0] encode_if_gcd(x_1, x_2, x_3) -> if_gcd(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: le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [1] minus(0, y) -> 0 [1] minus(s(x), y) -> if_minus(le(s(x), y), s(x), y) [1] if_minus(true, s(x), y) -> 0 [1] if_minus(false, s(x), y) -> s(minus(x, y)) [1] gcd(0, y) -> y [1] gcd(s(x), 0) -> s(x) [1] gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) [1] if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y)) [1] if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x)) [1] encArg(0) -> 0 [0] encArg(true) -> true [0] encArg(s(x_1)) -> s(encArg(x_1)) [0] encArg(false) -> false [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_minus(x_1, x_2, x_3)) -> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encArg(cons_gcd(x_1, x_2)) -> gcd(encArg(x_1), encArg(x_2)) [0] encArg(cons_if_gcd(x_1, x_2, x_3)) -> if_gcd(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) [0] encode_0 -> 0 [0] encode_true -> true [0] encode_s(x_1) -> s(encArg(x_1)) [0] encode_false -> false [0] encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) [0] encode_if_minus(x_1, x_2, x_3) -> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encode_gcd(x_1, x_2) -> gcd(encArg(x_1), encArg(x_2)) [0] encode_if_gcd(x_1, x_2, x_3) -> if_gcd(encArg(x_1), encArg(x_2), encArg(x_3)) [0] The TRS has the following type information: le :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd 0 :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd true :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd s :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd false :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd if_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd gcd :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd if_gcd :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encArg :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd cons_le :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd cons_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd cons_if_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd cons_gcd :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd cons_if_gcd :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encode_le :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encode_0 :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encode_true :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encode_s :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encode_false :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encode_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encode_if_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encode_gcd :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encode_if_gcd :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd 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: le_2 minus_2 if_minus_3 gcd_2 if_gcd_3 encArg_1 encode_le_2 encode_0 encode_true encode_s_1 encode_false encode_minus_2 encode_if_minus_3 encode_gcd_2 encode_if_gcd_3 Due to the following rules being added: encArg(v0) -> null_encArg [0] encode_le(v0, v1) -> null_encode_le [0] encode_0 -> null_encode_0 [0] encode_true -> null_encode_true [0] encode_s(v0) -> null_encode_s [0] encode_false -> null_encode_false [0] encode_minus(v0, v1) -> null_encode_minus [0] encode_if_minus(v0, v1, v2) -> null_encode_if_minus [0] encode_gcd(v0, v1) -> null_encode_gcd [0] encode_if_gcd(v0, v1, v2) -> null_encode_if_gcd [0] le(v0, v1) -> null_le [0] minus(v0, v1) -> null_minus [0] if_minus(v0, v1, v2) -> null_if_minus [0] gcd(v0, v1) -> null_gcd [0] if_gcd(v0, v1, v2) -> null_if_gcd [0] And the following fresh constants: null_encArg, null_encode_le, null_encode_0, null_encode_true, null_encode_s, null_encode_false, null_encode_minus, null_encode_if_minus, null_encode_gcd, null_encode_if_gcd, null_le, null_minus, null_if_minus, null_gcd, null_if_gcd ---------------------------------------- (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: le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [1] minus(0, y) -> 0 [1] minus(s(x), y) -> if_minus(le(s(x), y), s(x), y) [1] if_minus(true, s(x), y) -> 0 [1] if_minus(false, s(x), y) -> s(minus(x, y)) [1] gcd(0, y) -> y [1] gcd(s(x), 0) -> s(x) [1] gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) [1] if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y)) [1] if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x)) [1] encArg(0) -> 0 [0] encArg(true) -> true [0] encArg(s(x_1)) -> s(encArg(x_1)) [0] encArg(false) -> false [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_minus(x_1, x_2, x_3)) -> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encArg(cons_gcd(x_1, x_2)) -> gcd(encArg(x_1), encArg(x_2)) [0] encArg(cons_if_gcd(x_1, x_2, x_3)) -> if_gcd(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) [0] encode_0 -> 0 [0] encode_true -> true [0] encode_s(x_1) -> s(encArg(x_1)) [0] encode_false -> false [0] encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) [0] encode_if_minus(x_1, x_2, x_3) -> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encode_gcd(x_1, x_2) -> gcd(encArg(x_1), encArg(x_2)) [0] encode_if_gcd(x_1, x_2, x_3) -> if_gcd(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encArg(v0) -> null_encArg [0] encode_le(v0, v1) -> null_encode_le [0] encode_0 -> null_encode_0 [0] encode_true -> null_encode_true [0] encode_s(v0) -> null_encode_s [0] encode_false -> null_encode_false [0] encode_minus(v0, v1) -> null_encode_minus [0] encode_if_minus(v0, v1, v2) -> null_encode_if_minus [0] encode_gcd(v0, v1) -> null_encode_gcd [0] encode_if_gcd(v0, v1, v2) -> null_encode_if_gcd [0] le(v0, v1) -> null_le [0] minus(v0, v1) -> null_minus [0] if_minus(v0, v1, v2) -> null_if_minus [0] gcd(v0, v1) -> null_gcd [0] if_gcd(v0, v1, v2) -> null_if_gcd [0] The TRS has the following type information: le :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd 0 :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd true :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd s :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd false :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd if_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd gcd :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd if_gcd :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd encArg :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd cons_le :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd cons_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd cons_if_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd cons_gcd :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd cons_if_gcd :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd encode_le :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd encode_0 :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd encode_true :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd encode_s :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd encode_false :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd encode_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd encode_if_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd encode_gcd :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd encode_if_gcd :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd null_encArg :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd null_encode_le :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd null_encode_0 :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd null_encode_true :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd null_encode_s :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd null_encode_false :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd null_encode_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd null_encode_if_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd null_encode_gcd :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd null_encode_if_gcd :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd null_le :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd null_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd null_if_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd null_gcd :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd null_if_gcd :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd 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: le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [1] minus(0, y) -> 0 [1] minus(s(x), 0) -> if_minus(false, s(x), 0) [2] minus(s(x), s(y')) -> if_minus(le(x, y'), s(x), s(y')) [2] minus(s(x), y) -> if_minus(null_le, s(x), y) [1] if_minus(true, s(x), y) -> 0 [1] if_minus(false, s(x), y) -> s(minus(x, y)) [1] gcd(0, y) -> y [1] gcd(s(x), 0) -> s(x) [1] gcd(s(x), s(0)) -> if_gcd(true, s(x), s(0)) [2] gcd(s(0), s(s(x'))) -> if_gcd(false, s(0), s(s(x'))) [2] gcd(s(s(y'')), s(s(x''))) -> if_gcd(le(x'', y''), s(s(y'')), s(s(x''))) [2] gcd(s(x), s(y)) -> if_gcd(null_le, s(x), s(y)) [1] if_gcd(true, s(0), s(y)) -> gcd(0, s(y)) [2] if_gcd(true, s(s(x1)), s(y)) -> gcd(if_minus(le(s(x1), y), s(x1), y), s(y)) [2] if_gcd(true, s(x), s(y)) -> gcd(null_minus, s(y)) [1] if_gcd(false, s(x), s(0)) -> gcd(0, s(x)) [2] if_gcd(false, s(x), s(s(x2))) -> gcd(if_minus(le(s(x2), x), s(x2), x), s(x)) [2] if_gcd(false, s(x), s(y)) -> gcd(null_minus, s(x)) [1] encArg(0) -> 0 [0] encArg(true) -> true [0] encArg(s(x_1)) -> s(encArg(x_1)) [0] encArg(false) -> false [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_minus(x_1, x_2, x_3)) -> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encArg(cons_gcd(x_1, x_2)) -> gcd(encArg(x_1), encArg(x_2)) [0] encArg(cons_if_gcd(x_1, x_2, x_3)) -> if_gcd(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) [0] encode_0 -> 0 [0] encode_true -> true [0] encode_s(x_1) -> s(encArg(x_1)) [0] encode_false -> false [0] encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) [0] encode_if_minus(x_1, x_2, x_3) -> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encode_gcd(x_1, x_2) -> gcd(encArg(x_1), encArg(x_2)) [0] encode_if_gcd(x_1, x_2, x_3) -> if_gcd(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encArg(v0) -> null_encArg [0] encode_le(v0, v1) -> null_encode_le [0] encode_0 -> null_encode_0 [0] encode_true -> null_encode_true [0] encode_s(v0) -> null_encode_s [0] encode_false -> null_encode_false [0] encode_minus(v0, v1) -> null_encode_minus [0] encode_if_minus(v0, v1, v2) -> null_encode_if_minus [0] encode_gcd(v0, v1) -> null_encode_gcd [0] encode_if_gcd(v0, v1, v2) -> null_encode_if_gcd [0] le(v0, v1) -> null_le [0] minus(v0, v1) -> null_minus [0] if_minus(v0, v1, v2) -> null_if_minus [0] gcd(v0, v1) -> null_gcd [0] if_gcd(v0, v1, v2) -> null_if_gcd [0] The TRS has the following type information: le :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd 0 :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd true :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd s :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd false :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd if_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd gcd :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd if_gcd :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd encArg :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd cons_le :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd cons_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd cons_if_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd cons_gcd :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd cons_if_gcd :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd encode_le :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd encode_0 :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd encode_true :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd encode_s :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd encode_false :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd encode_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd encode_if_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd encode_gcd :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 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0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd -> 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd null_encArg :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd null_encode_le :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd null_encode_0 :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd null_encode_true :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd null_encode_s :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd null_encode_false :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd null_encode_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd null_encode_if_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd null_encode_gcd :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd null_encode_if_gcd :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd null_le :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd null_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd null_if_minus :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd null_gcd :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd null_if_gcd :: 0:true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd:null_encArg:null_encode_le:null_encode_0:null_encode_true:null_encode_s:null_encode_false:null_encode_minus:null_encode_if_minus:null_encode_gcd:null_encode_if_gcd:null_le:null_minus:null_if_minus:null_gcd:null_if_gcd 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_le => 0 null_encode_0 => 0 null_encode_true => 0 null_encode_s => 0 null_encode_false => 0 null_encode_minus => 0 null_encode_if_minus => 0 null_encode_gcd => 0 null_encode_if_gcd => 0 null_le => 0 null_minus => 0 null_if_minus => 0 null_gcd => 0 null_if_gcd => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: 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_minus(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> if_gcd(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 }-> gcd(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 encArg(z) -{ 0 }-> 1 + encArg(x_1) :|: z = 1 + x_1, x_1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gcd(z, z') -{ 0 }-> gcd(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2 encode_gcd(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 encode_if_gcd(z, z', z'') -{ 0 }-> if_gcd(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_gcd(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 encode_if_minus(z, z', z'') -{ 0 }-> if_minus(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_minus(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_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 :|: gcd(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y gcd(z, z') -{ 2 }-> if_gcd(le(x'', y''), 1 + (1 + y''), 1 + (1 + x'')) :|: z' = 1 + (1 + x''), z = 1 + (1 + y''), y'' >= 0, x'' >= 0 gcd(z, z') -{ 2 }-> if_gcd(2, 1 + x, 1 + 0) :|: x >= 0, z' = 1 + 0, z = 1 + x gcd(z, z') -{ 2 }-> if_gcd(1, 1 + 0, 1 + (1 + x')) :|: z' = 1 + (1 + x'), z = 1 + 0, x' >= 0 gcd(z, z') -{ 1 }-> if_gcd(0, 1 + x, 1 + y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x gcd(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 gcd(z, z') -{ 1 }-> 1 + x :|: x >= 0, z = 1 + x, z' = 0 if_gcd(z, z', z'') -{ 2 }-> gcd(if_minus(le(1 + x1, y), 1 + x1, y), 1 + y) :|: z = 2, x1 >= 0, z' = 1 + (1 + x1), y >= 0, z'' = 1 + y if_gcd(z, z', z'') -{ 2 }-> gcd(if_minus(le(1 + x2, x), 1 + x2, x), 1 + x) :|: z' = 1 + x, z = 1, x >= 0, z'' = 1 + (1 + x2), x2 >= 0 if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + x) :|: z' = 1 + x, z = 1, x >= 0, z'' = 1 + 0 if_gcd(z, z', z'') -{ 1 }-> gcd(0, 1 + x) :|: z' = 1 + x, z = 1, x >= 0, y >= 0, z'' = 1 + y if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + y) :|: z = 2, y >= 0, z'' = 1 + y, z' = 1 + 0 if_gcd(z, z', z'') -{ 1 }-> gcd(0, 1 + y) :|: z = 2, z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y if_gcd(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' = 1 + x, z'' = y, x >= 0, y >= 0 if_minus(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 if_minus(z, z', z'') -{ 1 }-> 1 + minus(x, y) :|: z' = 1 + 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_minus(le(x, y'), 1 + x, 1 + y') :|: x >= 0, y' >= 0, z = 1 + x, z' = 1 + y' minus(z, z') -{ 2 }-> if_minus(1, 1 + x, 0) :|: x >= 0, z = 1 + x, z' = 0 minus(z, z') -{ 1 }-> if_minus(0, 1 + x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y minus(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 ---------------------------------------- (15) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: 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_minus(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> if_gcd(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 }-> gcd(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gcd(z, z') -{ 0 }-> gcd(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> if_gcd(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_if_minus(z, z', z'') -{ 0 }-> if_minus(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if_minus(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_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 gcd(z, z') -{ 2 }-> if_gcd(le(z' - 2, z - 2), 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0 gcd(z, z') -{ 2 }-> if_gcd(2, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 2 }-> if_gcd(1, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0 gcd(z, z') -{ 1 }-> if_gcd(0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0 if_gcd(z, z', z'') -{ 2 }-> gcd(if_minus(le(1 + (z' - 2), z'' - 1), 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: z = 2, z' - 2 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 2 }-> gcd(if_minus(le(1 + (z'' - 2), z' - 1), 1 + (z'' - 2), z' - 1), 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' - 2 >= 0 if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' = 1 + 0 if_gcd(z, z', z'') -{ 1 }-> gcd(0, 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z'' - 1)) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0 if_gcd(z, z', z'') -{ 1 }-> gcd(0, 1 + (z'' - 1)) :|: z = 2, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0 if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z = 1, z' - 1 >= 0, 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_minus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 2 }-> if_minus(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 minus(z, z') -{ 1 }-> if_minus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { encode_0 } { le } { encode_false } { encode_true } { minus, if_minus } { gcd, if_gcd } { encArg } { encode_if_gcd } { encode_gcd } { encode_if_minus } { encode_minus } { encode_le } { encode_s } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: 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_minus(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> if_gcd(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 }-> gcd(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gcd(z, z') -{ 0 }-> gcd(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> if_gcd(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_if_minus(z, z', z'') -{ 0 }-> if_minus(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if_minus(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_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 gcd(z, z') -{ 2 }-> if_gcd(le(z' - 2, z - 2), 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0 gcd(z, z') -{ 2 }-> if_gcd(2, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 2 }-> if_gcd(1, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0 gcd(z, z') -{ 1 }-> if_gcd(0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0 if_gcd(z, z', z'') -{ 2 }-> gcd(if_minus(le(1 + (z' - 2), z'' - 1), 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: z = 2, z' - 2 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 2 }-> gcd(if_minus(le(1 + (z'' - 2), z' - 1), 1 + (z'' - 2), z' - 1), 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' - 2 >= 0 if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' = 1 + 0 if_gcd(z, z', z'') -{ 1 }-> gcd(0, 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z'' - 1)) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0 if_gcd(z, z', z'') -{ 1 }-> gcd(0, 1 + (z'' - 1)) :|: z = 2, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0 if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z = 1, z' - 1 >= 0, 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_minus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 2 }-> if_minus(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 minus(z, z') -{ 1 }-> if_minus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encode_0}, {le}, {encode_false}, {encode_true}, {minus,if_minus}, {gcd,if_gcd}, {encArg}, {encode_if_gcd}, {encode_gcd}, {encode_if_minus}, {encode_minus}, {encode_le}, {encode_s} ---------------------------------------- (19) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: 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_minus(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> if_gcd(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 }-> gcd(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gcd(z, z') -{ 0 }-> gcd(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> if_gcd(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_if_minus(z, z', z'') -{ 0 }-> if_minus(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if_minus(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_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 gcd(z, z') -{ 2 }-> if_gcd(le(z' - 2, z - 2), 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0 gcd(z, z') -{ 2 }-> if_gcd(2, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 2 }-> if_gcd(1, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0 gcd(z, z') -{ 1 }-> if_gcd(0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0 if_gcd(z, z', z'') -{ 2 }-> gcd(if_minus(le(1 + (z' - 2), z'' - 1), 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: z = 2, z' - 2 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 2 }-> gcd(if_minus(le(1 + (z'' - 2), z' - 1), 1 + (z'' - 2), z' - 1), 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' - 2 >= 0 if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' = 1 + 0 if_gcd(z, z', z'') -{ 1 }-> gcd(0, 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z'' - 1)) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0 if_gcd(z, z', z'') -{ 1 }-> gcd(0, 1 + (z'' - 1)) :|: z = 2, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0 if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z = 1, z' - 1 >= 0, 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_minus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 2 }-> if_minus(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 minus(z, z') -{ 1 }-> if_minus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encode_0}, {le}, {encode_false}, {encode_true}, {minus,if_minus}, {gcd,if_gcd}, {encArg}, {encode_if_gcd}, {encode_gcd}, {encode_if_minus}, {encode_minus}, {encode_le}, {encode_s} ---------------------------------------- (21) 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 ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: 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_minus(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> if_gcd(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 }-> gcd(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gcd(z, z') -{ 0 }-> gcd(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> if_gcd(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_if_minus(z, z', z'') -{ 0 }-> if_minus(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if_minus(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_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 gcd(z, z') -{ 2 }-> if_gcd(le(z' - 2, z - 2), 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0 gcd(z, z') -{ 2 }-> if_gcd(2, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 2 }-> if_gcd(1, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0 gcd(z, z') -{ 1 }-> if_gcd(0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0 if_gcd(z, z', z'') -{ 2 }-> gcd(if_minus(le(1 + (z' - 2), z'' - 1), 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: z = 2, z' - 2 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 2 }-> gcd(if_minus(le(1 + (z'' - 2), z' - 1), 1 + (z'' - 2), z' - 1), 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' - 2 >= 0 if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' = 1 + 0 if_gcd(z, z', z'') -{ 1 }-> gcd(0, 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z'' - 1)) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0 if_gcd(z, z', z'') -{ 1 }-> gcd(0, 1 + (z'' - 1)) :|: z = 2, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0 if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z = 1, z' - 1 >= 0, 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_minus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 2 }-> if_minus(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 minus(z, z') -{ 1 }-> if_minus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encode_0}, {le}, {encode_false}, {encode_true}, {minus,if_minus}, {gcd,if_gcd}, {encArg}, {encode_if_gcd}, {encode_gcd}, {encode_if_minus}, {encode_minus}, {encode_le}, {encode_s} Previous analysis results are: encode_0: runtime: ?, size: O(1) [0] ---------------------------------------- (23) 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 ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: 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_minus(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> if_gcd(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 }-> gcd(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gcd(z, z') -{ 0 }-> gcd(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> if_gcd(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_if_minus(z, z', z'') -{ 0 }-> if_minus(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if_minus(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_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 gcd(z, z') -{ 2 }-> if_gcd(le(z' - 2, z - 2), 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0 gcd(z, z') -{ 2 }-> if_gcd(2, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 2 }-> if_gcd(1, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0 gcd(z, z') -{ 1 }-> if_gcd(0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0 if_gcd(z, z', z'') -{ 2 }-> gcd(if_minus(le(1 + (z' - 2), z'' - 1), 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: z = 2, z' - 2 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 2 }-> gcd(if_minus(le(1 + (z'' - 2), z' - 1), 1 + (z'' - 2), z' - 1), 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' - 2 >= 0 if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' = 1 + 0 if_gcd(z, z', z'') -{ 1 }-> gcd(0, 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z'' - 1)) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0 if_gcd(z, z', z'') -{ 1 }-> gcd(0, 1 + (z'' - 1)) :|: z = 2, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0 if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z = 1, z' - 1 >= 0, 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_minus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 2 }-> if_minus(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 minus(z, z') -{ 1 }-> if_minus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {le}, {encode_false}, {encode_true}, {minus,if_minus}, {gcd,if_gcd}, {encArg}, {encode_if_gcd}, {encode_gcd}, {encode_if_minus}, {encode_minus}, {encode_le}, {encode_s} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] ---------------------------------------- (25) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: 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_minus(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> if_gcd(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 }-> gcd(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gcd(z, z') -{ 0 }-> gcd(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> if_gcd(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_if_minus(z, z', z'') -{ 0 }-> if_minus(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if_minus(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_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 gcd(z, z') -{ 2 }-> if_gcd(le(z' - 2, z - 2), 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0 gcd(z, z') -{ 2 }-> if_gcd(2, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 2 }-> if_gcd(1, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0 gcd(z, z') -{ 1 }-> if_gcd(0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0 if_gcd(z, z', z'') -{ 2 }-> gcd(if_minus(le(1 + (z' - 2), z'' - 1), 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: z = 2, z' - 2 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 2 }-> gcd(if_minus(le(1 + (z'' - 2), z' - 1), 1 + (z'' - 2), z' - 1), 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' - 2 >= 0 if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' = 1 + 0 if_gcd(z, z', z'') -{ 1 }-> gcd(0, 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z'' - 1)) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0 if_gcd(z, z', z'') -{ 1 }-> gcd(0, 1 + (z'' - 1)) :|: z = 2, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0 if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z = 1, z' - 1 >= 0, 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_minus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 2 }-> if_minus(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 minus(z, z') -{ 1 }-> if_minus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {le}, {encode_false}, {encode_true}, {minus,if_minus}, {gcd,if_gcd}, {encArg}, {encode_if_gcd}, {encode_gcd}, {encode_if_minus}, {encode_minus}, {encode_le}, {encode_s} Previous analysis results are: encode_0: runtime: O(1) [0], size: O(1) [0] ---------------------------------------- (27) 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 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: 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_minus(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> if_gcd(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 }-> gcd(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gcd(z, z') -{ 0 }-> gcd(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> if_gcd(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_if_minus(z, z', z'') -{ 0 }-> if_minus(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if_minus(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_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 gcd(z, z') -{ 2 }-> if_gcd(le(z' - 2, z - 2), 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0 gcd(z, z') -{ 2 }-> if_gcd(2, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 2 }-> if_gcd(1, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0 gcd(z, z') -{ 1 }-> if_gcd(0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0 if_gcd(z, z', z'') -{ 2 }-> gcd(if_minus(le(1 + (z' - 2), z'' - 1), 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: z = 2, z' - 2 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 2 }-> gcd(if_minus(le(1 + (z'' - 2), z' - 1), 1 + (z'' - 2), z' - 1), 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' - 2 >= 0 if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' = 1 + 0 if_gcd(z, z', z'') -{ 1 }-> gcd(0, 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z'' - 1)) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0 if_gcd(z, z', z'') -{ 1 }-> gcd(0, 1 + (z'' - 1)) :|: z = 2, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0 if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z = 1, z' - 1 >= 0, 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_minus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 2 }-> if_minus(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 minus(z, z') -{ 1 }-> if_minus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {le}, {encode_false}, {encode_true}, {minus,if_minus}, {gcd,if_gcd}, {encArg}, {encode_if_gcd}, {encode_gcd}, {encode_if_minus}, {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] ---------------------------------------- (29) 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' ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: 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_minus(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> if_gcd(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 }-> gcd(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gcd(z, z') -{ 0 }-> gcd(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> if_gcd(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_if_minus(z, z', z'') -{ 0 }-> if_minus(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if_minus(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_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 gcd(z, z') -{ 2 }-> if_gcd(le(z' - 2, z - 2), 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0 gcd(z, z') -{ 2 }-> if_gcd(2, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 2 }-> if_gcd(1, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0 gcd(z, z') -{ 1 }-> if_gcd(0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0 if_gcd(z, z', z'') -{ 2 }-> gcd(if_minus(le(1 + (z' - 2), z'' - 1), 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: z = 2, z' - 2 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 2 }-> gcd(if_minus(le(1 + (z'' - 2), z' - 1), 1 + (z'' - 2), z' - 1), 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' - 2 >= 0 if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' = 1 + 0 if_gcd(z, z', z'') -{ 1 }-> gcd(0, 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z'' - 1)) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0 if_gcd(z, z', z'') -{ 1 }-> gcd(0, 1 + (z'' - 1)) :|: z = 2, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0 if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z = 1, z' - 1 >= 0, 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_minus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 2 }-> if_minus(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 minus(z, z') -{ 1 }-> if_minus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encode_false}, {encode_true}, {minus,if_minus}, {gcd,if_gcd}, {encArg}, {encode_if_gcd}, {encode_gcd}, {encode_if_minus}, {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] ---------------------------------------- (31) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: 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_minus(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> if_gcd(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 }-> gcd(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gcd(z, z') -{ 0 }-> gcd(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> if_gcd(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_if_minus(z, z', z'') -{ 0 }-> if_minus(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if_minus(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_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 gcd(z, z') -{ 2 + z }-> if_gcd(s'', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0 gcd(z, z') -{ 2 }-> if_gcd(2, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 2 }-> if_gcd(1, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0 gcd(z, z') -{ 1 }-> if_gcd(0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0 if_gcd(z, z', z'') -{ 3 + z'' }-> gcd(if_minus(s1, 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 3 + z' }-> gcd(if_minus(s2, 1 + (z'' - 2), z' - 1), 1 + (z' - 1)) :|: s2 >= 0, s2 <= 2, z = 1, z' - 1 >= 0, z'' - 2 >= 0 if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' = 1 + 0 if_gcd(z, z', z'') -{ 1 }-> gcd(0, 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z'' - 1)) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0 if_gcd(z, z', z'') -{ 1 }-> gcd(0, 1 + (z'' - 1)) :|: z = 2, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0 if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z = 1, z' - 1 >= 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') -{ 3 + z' }-> if_minus(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 2 }-> if_minus(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 minus(z, z') -{ 1 }-> if_minus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encode_false}, {encode_true}, {minus,if_minus}, {gcd,if_gcd}, {encArg}, {encode_if_gcd}, {encode_gcd}, {encode_if_minus}, {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) 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 ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: 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_minus(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> if_gcd(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 }-> gcd(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gcd(z, z') -{ 0 }-> gcd(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> if_gcd(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_if_minus(z, z', z'') -{ 0 }-> if_minus(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if_minus(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_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 gcd(z, z') -{ 2 + z }-> if_gcd(s'', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0 gcd(z, z') -{ 2 }-> if_gcd(2, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 2 }-> if_gcd(1, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0 gcd(z, z') -{ 1 }-> if_gcd(0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0 if_gcd(z, z', z'') -{ 3 + z'' }-> gcd(if_minus(s1, 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 3 + z' }-> gcd(if_minus(s2, 1 + (z'' - 2), z' - 1), 1 + (z' - 1)) :|: s2 >= 0, s2 <= 2, z = 1, z' - 1 >= 0, z'' - 2 >= 0 if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' = 1 + 0 if_gcd(z, z', z'') -{ 1 }-> gcd(0, 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z'' - 1)) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0 if_gcd(z, z', z'') -{ 1 }-> gcd(0, 1 + (z'' - 1)) :|: z = 2, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0 if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z = 1, z' - 1 >= 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') -{ 3 + z' }-> if_minus(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 2 }-> if_minus(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 minus(z, z') -{ 1 }-> if_minus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encode_false}, {encode_true}, {minus,if_minus}, {gcd,if_gcd}, {encArg}, {encode_if_gcd}, {encode_gcd}, {encode_if_minus}, {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] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: encode_false after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: 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_minus(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> if_gcd(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 }-> gcd(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gcd(z, z') -{ 0 }-> gcd(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> if_gcd(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_if_minus(z, z', z'') -{ 0 }-> if_minus(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if_minus(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_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 gcd(z, z') -{ 2 + z }-> if_gcd(s'', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0 gcd(z, z') -{ 2 }-> if_gcd(2, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 2 }-> if_gcd(1, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0 gcd(z, z') -{ 1 }-> if_gcd(0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0 if_gcd(z, z', z'') -{ 3 + z'' }-> gcd(if_minus(s1, 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 3 + z' }-> gcd(if_minus(s2, 1 + (z'' - 2), z' - 1), 1 + (z' - 1)) :|: s2 >= 0, s2 <= 2, z = 1, z' - 1 >= 0, z'' - 2 >= 0 if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' = 1 + 0 if_gcd(z, z', z'') -{ 1 }-> gcd(0, 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z'' - 1)) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0 if_gcd(z, z', z'') -{ 1 }-> gcd(0, 1 + (z'' - 1)) :|: z = 2, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0 if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z = 1, z' - 1 >= 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') -{ 3 + z' }-> if_minus(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 2 }-> if_minus(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 minus(z, z') -{ 1 }-> if_minus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encode_true}, {minus,if_minus}, {gcd,if_gcd}, {encArg}, {encode_if_gcd}, {encode_gcd}, {encode_if_minus}, {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] ---------------------------------------- (37) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: 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_minus(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> if_gcd(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 }-> gcd(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gcd(z, z') -{ 0 }-> gcd(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> if_gcd(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_if_minus(z, z', z'') -{ 0 }-> if_minus(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if_minus(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_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 gcd(z, z') -{ 2 + z }-> if_gcd(s'', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0 gcd(z, z') -{ 2 }-> if_gcd(2, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 2 }-> if_gcd(1, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0 gcd(z, z') -{ 1 }-> if_gcd(0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0 if_gcd(z, z', z'') -{ 3 + z'' }-> gcd(if_minus(s1, 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 3 + z' }-> gcd(if_minus(s2, 1 + (z'' - 2), z' - 1), 1 + (z' - 1)) :|: s2 >= 0, s2 <= 2, z = 1, z' - 1 >= 0, z'' - 2 >= 0 if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' = 1 + 0 if_gcd(z, z', z'') -{ 1 }-> gcd(0, 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z'' - 1)) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0 if_gcd(z, z', z'') -{ 1 }-> gcd(0, 1 + (z'' - 1)) :|: z = 2, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0 if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z = 1, z' - 1 >= 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') -{ 3 + z' }-> if_minus(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 2 }-> if_minus(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 minus(z, z') -{ 1 }-> if_minus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encode_true}, {minus,if_minus}, {gcd,if_gcd}, {encArg}, {encode_if_gcd}, {encode_gcd}, {encode_if_minus}, {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) 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 ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: 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_minus(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> if_gcd(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 }-> gcd(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gcd(z, z') -{ 0 }-> gcd(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> if_gcd(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_if_minus(z, z', z'') -{ 0 }-> if_minus(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if_minus(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_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 gcd(z, z') -{ 2 + z }-> if_gcd(s'', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0 gcd(z, z') -{ 2 }-> if_gcd(2, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 2 }-> if_gcd(1, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0 gcd(z, z') -{ 1 }-> if_gcd(0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0 if_gcd(z, z', z'') -{ 3 + z'' }-> gcd(if_minus(s1, 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 3 + z' }-> gcd(if_minus(s2, 1 + (z'' - 2), z' - 1), 1 + (z' - 1)) :|: s2 >= 0, s2 <= 2, z = 1, z' - 1 >= 0, z'' - 2 >= 0 if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' = 1 + 0 if_gcd(z, z', z'') -{ 1 }-> gcd(0, 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z'' - 1)) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0 if_gcd(z, z', z'') -{ 1 }-> gcd(0, 1 + (z'' - 1)) :|: z = 2, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0 if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z = 1, z' - 1 >= 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') -{ 3 + z' }-> if_minus(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 2 }-> if_minus(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 minus(z, z') -{ 1 }-> if_minus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encode_true}, {minus,if_minus}, {gcd,if_gcd}, {encArg}, {encode_if_gcd}, {encode_gcd}, {encode_if_minus}, {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] ---------------------------------------- (41) 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 ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: 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_minus(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> if_gcd(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 }-> gcd(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gcd(z, z') -{ 0 }-> gcd(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> if_gcd(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_if_minus(z, z', z'') -{ 0 }-> if_minus(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if_minus(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_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 gcd(z, z') -{ 2 + z }-> if_gcd(s'', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0 gcd(z, z') -{ 2 }-> if_gcd(2, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 2 }-> if_gcd(1, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0 gcd(z, z') -{ 1 }-> if_gcd(0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0 if_gcd(z, z', z'') -{ 3 + z'' }-> gcd(if_minus(s1, 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 3 + z' }-> gcd(if_minus(s2, 1 + (z'' - 2), z' - 1), 1 + (z' - 1)) :|: s2 >= 0, s2 <= 2, z = 1, z' - 1 >= 0, z'' - 2 >= 0 if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' = 1 + 0 if_gcd(z, z', z'') -{ 1 }-> gcd(0, 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z'' - 1)) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0 if_gcd(z, z', z'') -{ 1 }-> gcd(0, 1 + (z'' - 1)) :|: z = 2, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0 if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z = 1, z' - 1 >= 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') -{ 3 + z' }-> if_minus(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 2 }-> if_minus(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 minus(z, z') -{ 1 }-> if_minus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {minus,if_minus}, {gcd,if_gcd}, {encArg}, {encode_if_gcd}, {encode_gcd}, {encode_if_minus}, {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] ---------------------------------------- (43) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: 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_minus(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> if_gcd(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 }-> gcd(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gcd(z, z') -{ 0 }-> gcd(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> if_gcd(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_if_minus(z, z', z'') -{ 0 }-> if_minus(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if_minus(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_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 gcd(z, z') -{ 2 + z }-> if_gcd(s'', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0 gcd(z, z') -{ 2 }-> if_gcd(2, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 2 }-> if_gcd(1, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0 gcd(z, z') -{ 1 }-> if_gcd(0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0 if_gcd(z, z', z'') -{ 3 + z'' }-> gcd(if_minus(s1, 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 3 + z' }-> gcd(if_minus(s2, 1 + (z'' - 2), z' - 1), 1 + (z' - 1)) :|: s2 >= 0, s2 <= 2, z = 1, z' - 1 >= 0, z'' - 2 >= 0 if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' = 1 + 0 if_gcd(z, z', z'') -{ 1 }-> gcd(0, 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z'' - 1)) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0 if_gcd(z, z', z'') -{ 1 }-> gcd(0, 1 + (z'' - 1)) :|: z = 2, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0 if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z = 1, z' - 1 >= 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') -{ 3 + z' }-> if_minus(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 2 }-> if_minus(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 minus(z, z') -{ 1 }-> if_minus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {minus,if_minus}, {gcd,if_gcd}, {encArg}, {encode_if_gcd}, {encode_gcd}, {encode_if_minus}, {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) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: minus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z Computed SIZE bound using CoFloCo for: if_minus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: 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_minus(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> if_gcd(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 }-> gcd(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gcd(z, z') -{ 0 }-> gcd(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> if_gcd(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_if_minus(z, z', z'') -{ 0 }-> if_minus(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if_minus(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_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 gcd(z, z') -{ 2 + z }-> if_gcd(s'', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0 gcd(z, z') -{ 2 }-> if_gcd(2, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 2 }-> if_gcd(1, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0 gcd(z, z') -{ 1 }-> if_gcd(0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0 if_gcd(z, z', z'') -{ 3 + z'' }-> gcd(if_minus(s1, 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 3 + z' }-> gcd(if_minus(s2, 1 + (z'' - 2), z' - 1), 1 + (z' - 1)) :|: s2 >= 0, s2 <= 2, z = 1, z' - 1 >= 0, z'' - 2 >= 0 if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' = 1 + 0 if_gcd(z, z', z'') -{ 1 }-> gcd(0, 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z'' - 1)) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0 if_gcd(z, z', z'') -{ 1 }-> gcd(0, 1 + (z'' - 1)) :|: z = 2, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0 if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z = 1, z' - 1 >= 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') -{ 3 + z' }-> if_minus(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 2 }-> if_minus(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 minus(z, z') -{ 1 }-> if_minus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {minus,if_minus}, {gcd,if_gcd}, {encArg}, {encode_if_gcd}, {encode_gcd}, {encode_if_minus}, {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] minus: runtime: ?, size: O(n^1) [z] if_minus: runtime: ?, size: O(n^1) [z'] ---------------------------------------- (47) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: minus after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 8 + 4*z + z*z' + 2*z' Computed RUNTIME bound using KoAT for: if_minus after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 6 + 4*z' + z'*z'' + z'' ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: 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_minus(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> if_gcd(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 }-> gcd(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gcd(z, z') -{ 0 }-> gcd(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> if_gcd(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_if_minus(z, z', z'') -{ 0 }-> if_minus(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if_minus(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_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 gcd(z, z') -{ 2 + z }-> if_gcd(s'', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0 gcd(z, z') -{ 2 }-> if_gcd(2, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 2 }-> if_gcd(1, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0 gcd(z, z') -{ 1 }-> if_gcd(0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0 if_gcd(z, z', z'') -{ 3 + z'' }-> gcd(if_minus(s1, 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 3 + z' }-> gcd(if_minus(s2, 1 + (z'' - 2), z' - 1), 1 + (z' - 1)) :|: s2 >= 0, s2 <= 2, z = 1, z' - 1 >= 0, z'' - 2 >= 0 if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' = 1 + 0 if_gcd(z, z', z'') -{ 1 }-> gcd(0, 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z'' - 1)) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0 if_gcd(z, z', z'') -{ 1 }-> gcd(0, 1 + (z'' - 1)) :|: z = 2, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0 if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z = 1, z' - 1 >= 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') -{ 3 + z' }-> if_minus(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 2 }-> if_minus(1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 minus(z, z') -{ 1 }-> if_minus(0, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {gcd,if_gcd}, {encArg}, {encode_if_gcd}, {encode_gcd}, {encode_if_minus}, {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] minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z] if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z'] ---------------------------------------- (49) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: 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_minus(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> if_gcd(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 }-> gcd(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gcd(z, z') -{ 0 }-> gcd(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> if_gcd(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_if_minus(z, z', z'') -{ 0 }-> if_minus(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if_minus(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_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 gcd(z, z') -{ 2 + z }-> if_gcd(s'', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0 gcd(z, z') -{ 2 }-> if_gcd(2, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 2 }-> if_gcd(1, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0 gcd(z, z') -{ 1 }-> if_gcd(0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0 if_gcd(z, z', z'') -{ 5 + 3*z' + z'*z'' + z'' }-> gcd(s7, 1 + (z'' - 1)) :|: s7 >= 0, s7 <= 1 + (z' - 2), s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 5 + z' + z'*z'' + 3*z'' }-> gcd(s8, 1 + (z' - 1)) :|: s8 >= 0, s8 <= 1 + (z'' - 2), s2 >= 0, s2 <= 2, z = 1, z' - 1 >= 0, z'' - 2 >= 0 if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' = 1 + 0 if_gcd(z, z', z'') -{ 1 }-> gcd(0, 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z'' - 1)) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0 if_gcd(z, z', z'') -{ 1 }-> gcd(0, 1 + (z'' - 1)) :|: z = 2, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0 if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s6 :|: s6 >= 0, s6 <= z' - 1, z = 1, z' - 1 >= 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') -{ 8 + 4*z }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), z - 1 >= 0, z' = 0 minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s5 :|: s5 >= 0, s5 <= 1 + (z - 1), z - 1 >= 0, z' >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {gcd,if_gcd}, {encArg}, {encode_if_gcd}, {encode_gcd}, {encode_if_minus}, {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] minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z] if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z'] ---------------------------------------- (51) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: gcd after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z + z' Computed SIZE bound using KoAT for: if_gcd after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' + z'' ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: 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_minus(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> if_gcd(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 }-> gcd(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gcd(z, z') -{ 0 }-> gcd(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> if_gcd(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_if_minus(z, z', z'') -{ 0 }-> if_minus(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if_minus(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_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 gcd(z, z') -{ 2 + z }-> if_gcd(s'', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0 gcd(z, z') -{ 2 }-> if_gcd(2, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 2 }-> if_gcd(1, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0 gcd(z, z') -{ 1 }-> if_gcd(0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0 if_gcd(z, z', z'') -{ 5 + 3*z' + z'*z'' + z'' }-> gcd(s7, 1 + (z'' - 1)) :|: s7 >= 0, s7 <= 1 + (z' - 2), s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 5 + z' + z'*z'' + 3*z'' }-> gcd(s8, 1 + (z' - 1)) :|: s8 >= 0, s8 <= 1 + (z'' - 2), s2 >= 0, s2 <= 2, z = 1, z' - 1 >= 0, z'' - 2 >= 0 if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' = 1 + 0 if_gcd(z, z', z'') -{ 1 }-> gcd(0, 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z'' - 1)) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0 if_gcd(z, z', z'') -{ 1 }-> gcd(0, 1 + (z'' - 1)) :|: z = 2, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0 if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s6 :|: s6 >= 0, s6 <= z' - 1, z = 1, z' - 1 >= 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') -{ 8 + 4*z }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), z - 1 >= 0, z' = 0 minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s5 :|: s5 >= 0, s5 <= 1 + (z - 1), z - 1 >= 0, z' >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {gcd,if_gcd}, {encArg}, {encode_if_gcd}, {encode_gcd}, {encode_if_minus}, {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] minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z] if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z'] gcd: runtime: ?, size: O(n^1) [z + z'] if_gcd: runtime: ?, size: O(n^1) [z' + z''] ---------------------------------------- (53) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: gcd after applying outer abstraction to obtain an ITS, resulting in: O(n^3) with polynomial bound: 2 + 64*z + 52*z*z' + 12*z*z'^2 + 26*z^2 + 12*z^2*z' + 4*z^3 + 64*z' + 26*z'^2 + 4*z'^3 Computed RUNTIME bound using KoAT for: if_gcd after applying outer abstraction to obtain an ITS, resulting in: O(n^3) with polynomial bound: 14 + 180*z' + 58*z'*z'' + 24*z'*z''^2 + 80*z'^2 + 24*z'^2*z'' + 16*z'^3 + 180*z'' + 80*z''^2 + 16*z''^3 ---------------------------------------- (54) Obligation: Complexity RNTS consisting of the following rules: 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_minus(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> if_gcd(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 }-> gcd(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gcd(z, z') -{ 0 }-> gcd(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> if_gcd(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_if_minus(z, z', z'') -{ 0 }-> if_minus(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if_minus(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_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 gcd(z, z') -{ 2 + z }-> if_gcd(s'', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0 gcd(z, z') -{ 2 }-> if_gcd(2, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 2 }-> if_gcd(1, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0 gcd(z, z') -{ 1 }-> if_gcd(0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0 if_gcd(z, z', z'') -{ 5 + 3*z' + z'*z'' + z'' }-> gcd(s7, 1 + (z'' - 1)) :|: s7 >= 0, s7 <= 1 + (z' - 2), s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 5 + z' + z'*z'' + 3*z'' }-> gcd(s8, 1 + (z' - 1)) :|: s8 >= 0, s8 <= 1 + (z'' - 2), s2 >= 0, s2 <= 2, z = 1, z' - 1 >= 0, z'' - 2 >= 0 if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' = 1 + 0 if_gcd(z, z', z'') -{ 1 }-> gcd(0, 1 + (z' - 1)) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 2 }-> gcd(0, 1 + (z'' - 1)) :|: z = 2, z'' - 1 >= 0, z' = 1 + 0 if_gcd(z, z', z'') -{ 1 }-> gcd(0, 1 + (z'' - 1)) :|: z = 2, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0 if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s6 :|: s6 >= 0, s6 <= z' - 1, z = 1, z' - 1 >= 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') -{ 8 + 4*z }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), z - 1 >= 0, z' = 0 minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s5 :|: s5 >= 0, s5 <= 1 + (z - 1), z - 1 >= 0, z' >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encArg}, {encode_if_gcd}, {encode_gcd}, {encode_if_minus}, {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] minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z] if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z'] gcd: runtime: O(n^3) [2 + 64*z + 52*z*z' + 12*z*z'^2 + 26*z^2 + 12*z^2*z' + 4*z^3 + 64*z' + 26*z'^2 + 4*z'^3], size: O(n^1) [z + z'] if_gcd: runtime: O(n^3) [14 + 180*z' + 58*z'*z'' + 24*z'*z''^2 + 80*z'^2 + 24*z'^2*z'' + 16*z'^3 + 180*z'' + 80*z''^2 + 16*z''^3], size: O(n^1) [z' + z''] ---------------------------------------- (55) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (56) Obligation: Complexity RNTS consisting of the following rules: 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_minus(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> if_gcd(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 }-> gcd(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gcd(z, z') -{ 0 }-> gcd(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> if_gcd(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_if_minus(z, z', z'') -{ 0 }-> if_minus(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if_minus(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_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gcd(z, z') -{ 292 + 262*z' + 104*z'^2 + 16*z'^3 }-> s10 :|: s10 >= 0, s10 <= 1 + 0 + (1 + (1 + (z' - 2))), z = 1 + 0, z' - 2 >= 0 gcd(z, z') -{ 16 + 181*z + 58*z*z' + 24*z*z'^2 + 80*z^2 + 24*z^2*z' + 16*z^3 + 180*z' + 80*z'^2 + 16*z'^3 }-> s11 :|: s11 >= 0, s11 <= 1 + (1 + (z - 2)) + (1 + (1 + (z' - 2))), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0 gcd(z, z') -{ 15 + 180*z + 58*z*z' + 24*z*z'^2 + 80*z^2 + 24*z^2*z' + 16*z^3 + 180*z' + 80*z'^2 + 16*z'^3 }-> s12 :|: s12 >= 0, s12 <= 1 + (z - 1) + (1 + (z' - 1)), z - 1 >= 0, z' - 1 >= 0 gcd(z, z') -{ 292 + 262*z + 104*z^2 + 16*z^3 }-> s9 :|: s9 >= 0, s9 <= 1 + (z - 1) + (1 + 0), z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0 if_gcd(z, z', z'') -{ 4 + 64*z'' + 26*z''^2 + 4*z''^3 }-> s13 :|: s13 >= 0, s13 <= 0 + (1 + (z'' - 1)), z = 2, z'' - 1 >= 0, z' = 1 + 0 if_gcd(z, z', z'') -{ 7 + 64*s7 + 52*s7*z'' + 12*s7*z''^2 + 26*s7^2 + 12*s7^2*z'' + 4*s7^3 + 3*z' + z'*z'' + 65*z'' + 26*z''^2 + 4*z''^3 }-> s14 :|: s14 >= 0, s14 <= s7 + (1 + (z'' - 1)), s7 >= 0, s7 <= 1 + (z' - 2), s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 3 + 64*z'' + 26*z''^2 + 4*z''^3 }-> s15 :|: s15 >= 0, s15 <= 0 + (1 + (z'' - 1)), z = 2, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 4 + 64*z' + 26*z'^2 + 4*z'^3 }-> s16 :|: s16 >= 0, s16 <= 0 + (1 + (z' - 1)), z = 1, z' - 1 >= 0, z'' = 1 + 0 if_gcd(z, z', z'') -{ 7 + 64*s8 + 52*s8*z' + 12*s8*z'^2 + 26*s8^2 + 12*s8^2*z' + 4*s8^3 + 65*z' + z'*z'' + 26*z'^2 + 4*z'^3 + 3*z'' }-> s17 :|: s17 >= 0, s17 <= s8 + (1 + (z' - 1)), s8 >= 0, s8 <= 1 + (z'' - 2), s2 >= 0, s2 <= 2, z = 1, z' - 1 >= 0, z'' - 2 >= 0 if_gcd(z, z', z'') -{ 3 + 64*z' + 26*z'^2 + 4*z'^3 }-> s18 :|: s18 >= 0, s18 <= 0 + (1 + (z' - 1)), z = 1, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0 if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s6 :|: s6 >= 0, s6 <= z' - 1, z = 1, z' - 1 >= 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') -{ 8 + 4*z }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), z - 1 >= 0, z' = 0 minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s5 :|: s5 >= 0, s5 <= 1 + (z - 1), z - 1 >= 0, z' >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encArg}, {encode_if_gcd}, {encode_gcd}, {encode_if_minus}, {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] minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z] if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z'] gcd: runtime: O(n^3) [2 + 64*z + 52*z*z' + 12*z*z'^2 + 26*z^2 + 12*z^2*z' + 4*z^3 + 64*z' + 26*z'^2 + 4*z'^3], size: O(n^1) [z + z'] if_gcd: runtime: O(n^3) [14 + 180*z' + 58*z'*z'' + 24*z'*z''^2 + 80*z'^2 + 24*z'^2*z'' + 16*z'^3 + 180*z'' + 80*z''^2 + 16*z''^3], size: O(n^1) [z' + z''] ---------------------------------------- (57) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: encArg after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + 2*z ---------------------------------------- (58) Obligation: Complexity RNTS consisting of the following rules: 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_minus(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> if_gcd(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 }-> gcd(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gcd(z, z') -{ 0 }-> gcd(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> if_gcd(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_if_minus(z, z', z'') -{ 0 }-> if_minus(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if_minus(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_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gcd(z, z') -{ 292 + 262*z' + 104*z'^2 + 16*z'^3 }-> s10 :|: s10 >= 0, s10 <= 1 + 0 + (1 + (1 + (z' - 2))), z = 1 + 0, z' - 2 >= 0 gcd(z, z') -{ 16 + 181*z + 58*z*z' + 24*z*z'^2 + 80*z^2 + 24*z^2*z' + 16*z^3 + 180*z' + 80*z'^2 + 16*z'^3 }-> s11 :|: s11 >= 0, s11 <= 1 + (1 + (z - 2)) + (1 + (1 + (z' - 2))), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0 gcd(z, z') -{ 15 + 180*z + 58*z*z' + 24*z*z'^2 + 80*z^2 + 24*z^2*z' + 16*z^3 + 180*z' + 80*z'^2 + 16*z'^3 }-> s12 :|: s12 >= 0, s12 <= 1 + (z - 1) + (1 + (z' - 1)), z - 1 >= 0, z' - 1 >= 0 gcd(z, z') -{ 292 + 262*z + 104*z^2 + 16*z^3 }-> s9 :|: s9 >= 0, s9 <= 1 + (z - 1) + (1 + 0), z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0 if_gcd(z, z', z'') -{ 4 + 64*z'' + 26*z''^2 + 4*z''^3 }-> s13 :|: s13 >= 0, s13 <= 0 + (1 + (z'' - 1)), z = 2, z'' - 1 >= 0, z' = 1 + 0 if_gcd(z, z', z'') -{ 7 + 64*s7 + 52*s7*z'' + 12*s7*z''^2 + 26*s7^2 + 12*s7^2*z'' + 4*s7^3 + 3*z' + z'*z'' + 65*z'' + 26*z''^2 + 4*z''^3 }-> s14 :|: s14 >= 0, s14 <= s7 + (1 + (z'' - 1)), s7 >= 0, s7 <= 1 + (z' - 2), s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 3 + 64*z'' + 26*z''^2 + 4*z''^3 }-> s15 :|: s15 >= 0, s15 <= 0 + (1 + (z'' - 1)), z = 2, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 4 + 64*z' + 26*z'^2 + 4*z'^3 }-> s16 :|: s16 >= 0, s16 <= 0 + (1 + (z' - 1)), z = 1, z' - 1 >= 0, z'' = 1 + 0 if_gcd(z, z', z'') -{ 7 + 64*s8 + 52*s8*z' + 12*s8*z'^2 + 26*s8^2 + 12*s8^2*z' + 4*s8^3 + 65*z' + z'*z'' + 26*z'^2 + 4*z'^3 + 3*z'' }-> s17 :|: s17 >= 0, s17 <= s8 + (1 + (z' - 1)), s8 >= 0, s8 <= 1 + (z'' - 2), s2 >= 0, s2 <= 2, z = 1, z' - 1 >= 0, z'' - 2 >= 0 if_gcd(z, z', z'') -{ 3 + 64*z' + 26*z'^2 + 4*z'^3 }-> s18 :|: s18 >= 0, s18 <= 0 + (1 + (z' - 1)), z = 1, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0 if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s6 :|: s6 >= 0, s6 <= z' - 1, z = 1, z' - 1 >= 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') -{ 8 + 4*z }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), z - 1 >= 0, z' = 0 minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s5 :|: s5 >= 0, s5 <= 1 + (z - 1), z - 1 >= 0, z' >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encArg}, {encode_if_gcd}, {encode_gcd}, {encode_if_minus}, {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] minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z] if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z'] gcd: runtime: O(n^3) [2 + 64*z + 52*z*z' + 12*z*z'^2 + 26*z^2 + 12*z^2*z' + 4*z^3 + 64*z' + 26*z'^2 + 4*z'^3], size: O(n^1) [z + z'] if_gcd: runtime: O(n^3) [14 + 180*z' + 58*z'*z'' + 24*z'*z''^2 + 80*z'^2 + 24*z'^2*z'' + 16*z'^3 + 180*z'' + 80*z''^2 + 16*z''^3], size: O(n^1) [z' + z''] encArg: runtime: ?, size: O(n^1) [2 + 2*z] ---------------------------------------- (59) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: encArg after applying outer abstraction to obtain an ITS, resulting in: O(n^4) with polynomial bound: 3224 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 ---------------------------------------- (60) Obligation: Complexity RNTS consisting of the following rules: 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_minus(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 0 }-> if_gcd(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 }-> gcd(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 0 }-> 2 :|: z = 2 encArg(z) -{ 0 }-> 1 :|: z = 1 encArg(z) -{ 0 }-> 0 :|: z = 0 encArg(z) -{ 0 }-> 0 :|: z >= 0 encArg(z) -{ 0 }-> 1 + encArg(z - 1) :|: z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gcd(z, z') -{ 0 }-> gcd(encArg(z), encArg(z')) :|: z >= 0, z' >= 0 encode_gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> if_gcd(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_if_minus(z, z', z'') -{ 0 }-> if_minus(encArg(z), encArg(z'), encArg(z'')) :|: z >= 0, z'' >= 0, z' >= 0 encode_if_minus(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_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 0 }-> 1 + encArg(z) :|: z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gcd(z, z') -{ 292 + 262*z' + 104*z'^2 + 16*z'^3 }-> s10 :|: s10 >= 0, s10 <= 1 + 0 + (1 + (1 + (z' - 2))), z = 1 + 0, z' - 2 >= 0 gcd(z, z') -{ 16 + 181*z + 58*z*z' + 24*z*z'^2 + 80*z^2 + 24*z^2*z' + 16*z^3 + 180*z' + 80*z'^2 + 16*z'^3 }-> s11 :|: s11 >= 0, s11 <= 1 + (1 + (z - 2)) + (1 + (1 + (z' - 2))), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0 gcd(z, z') -{ 15 + 180*z + 58*z*z' + 24*z*z'^2 + 80*z^2 + 24*z^2*z' + 16*z^3 + 180*z' + 80*z'^2 + 16*z'^3 }-> s12 :|: s12 >= 0, s12 <= 1 + (z - 1) + (1 + (z' - 1)), z - 1 >= 0, z' - 1 >= 0 gcd(z, z') -{ 292 + 262*z + 104*z^2 + 16*z^3 }-> s9 :|: s9 >= 0, s9 <= 1 + (z - 1) + (1 + 0), z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0 if_gcd(z, z', z'') -{ 4 + 64*z'' + 26*z''^2 + 4*z''^3 }-> s13 :|: s13 >= 0, s13 <= 0 + (1 + (z'' - 1)), z = 2, z'' - 1 >= 0, z' = 1 + 0 if_gcd(z, z', z'') -{ 7 + 64*s7 + 52*s7*z'' + 12*s7*z''^2 + 26*s7^2 + 12*s7^2*z'' + 4*s7^3 + 3*z' + z'*z'' + 65*z'' + 26*z''^2 + 4*z''^3 }-> s14 :|: s14 >= 0, s14 <= s7 + (1 + (z'' - 1)), s7 >= 0, s7 <= 1 + (z' - 2), s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 3 + 64*z'' + 26*z''^2 + 4*z''^3 }-> s15 :|: s15 >= 0, s15 <= 0 + (1 + (z'' - 1)), z = 2, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 4 + 64*z' + 26*z'^2 + 4*z'^3 }-> s16 :|: s16 >= 0, s16 <= 0 + (1 + (z' - 1)), z = 1, z' - 1 >= 0, z'' = 1 + 0 if_gcd(z, z', z'') -{ 7 + 64*s8 + 52*s8*z' + 12*s8*z'^2 + 26*s8^2 + 12*s8^2*z' + 4*s8^3 + 65*z' + z'*z'' + 26*z'^2 + 4*z'^3 + 3*z'' }-> s17 :|: s17 >= 0, s17 <= s8 + (1 + (z' - 1)), s8 >= 0, s8 <= 1 + (z'' - 2), s2 >= 0, s2 <= 2, z = 1, z' - 1 >= 0, z'' - 2 >= 0 if_gcd(z, z', z'') -{ 3 + 64*z' + 26*z'^2 + 4*z'^3 }-> s18 :|: s18 >= 0, s18 <= 0 + (1 + (z' - 1)), z = 1, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0 if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s6 :|: s6 >= 0, s6 <= z' - 1, z = 1, z' - 1 >= 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') -{ 8 + 4*z }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), z - 1 >= 0, z' = 0 minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s5 :|: s5 >= 0, s5 <= 1 + (z - 1), z - 1 >= 0, z' >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encode_if_gcd}, {encode_gcd}, {encode_if_minus}, {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] minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z] if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z'] gcd: runtime: O(n^3) [2 + 64*z + 52*z*z' + 12*z*z'^2 + 26*z^2 + 12*z^2*z' + 4*z^3 + 64*z' + 26*z'^2 + 4*z'^3], size: O(n^1) [z + z'] if_gcd: runtime: O(n^3) [14 + 180*z' + 58*z'*z'' + 24*z'*z''^2 + 80*z'^2 + 24*z'^2*z'' + 16*z'^3 + 180*z'' + 80*z''^2 + 16*z''^3], size: O(n^1) [z' + z''] encArg: runtime: O(n^4) [3224 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4], size: O(n^1) [2 + 2*z] ---------------------------------------- (61) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (62) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 6450 + s21 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s22 :|: s20 >= 0, s20 <= 2 * x_1 + 2, s21 >= 0, s21 <= 2 * x_2 + 2, s22 >= 0, s22 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 6456 + 4*s23 + s23*s24 + 2*s24 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s25 :|: s23 >= 0, s23 <= 2 * x_1 + 2, s24 >= 0, s24 <= 2 * x_2 + 2, s25 >= 0, s25 <= s23, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 9678 + 4*s27 + s27*s28 + s28 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 + 15952*x_3 + 22824*x_3^2 + 12848*x_3^3 + 2688*x_3^4 }-> s29 :|: s26 >= 0, s26 <= 2 * x_1 + 2, s27 >= 0, s27 <= 2 * x_2 + 2, s28 >= 0, s28 <= 2 * x_3 + 2, s29 >= 0, s29 <= s27, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 6450 + 64*s30 + 52*s30*s31 + 12*s30*s31^2 + 26*s30^2 + 12*s30^2*s31 + 4*s30^3 + 64*s31 + 26*s31^2 + 4*s31^3 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s32 :|: s30 >= 0, s30 <= 2 * x_1 + 2, s31 >= 0, s31 <= 2 * x_2 + 2, s32 >= 0, s32 <= s30 + s31, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 9686 + 180*s34 + 58*s34*s35 + 24*s34*s35^2 + 80*s34^2 + 24*s34^2*s35 + 16*s34^3 + 180*s35 + 80*s35^2 + 16*s35^3 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 + 15952*x_3 + 22824*x_3^2 + 12848*x_3^3 + 2688*x_3^4 }-> s36 :|: s33 >= 0, s33 <= 2 * x_1 + 2, s34 >= 0, s34 <= 2 * x_2 + 2, s35 >= 0, s35 <= 2 * x_3 + 2, s36 >= 0, s36 <= s34 + s35, 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) -{ -64 + -1904*z + 408*z^2 + 2096*z^3 + 2688*z^4 }-> 1 + s19 :|: s19 >= 0, s19 <= 2 * (z - 1) + 2, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gcd(z, z') -{ 6450 + 64*s48 + 52*s48*s49 + 12*s48*s49^2 + 26*s48^2 + 12*s48^2*s49 + 4*s48^3 + 64*s49 + 26*s49^2 + 4*s49^3 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s50 :|: s48 >= 0, s48 <= 2 * z + 2, s49 >= 0, s49 <= 2 * z' + 2, s50 >= 0, s50 <= s48 + s49, z >= 0, z' >= 0 encode_gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 9686 + 180*s52 + 58*s52*s53 + 24*s52*s53^2 + 80*s52^2 + 24*s52^2*s53 + 16*s52^3 + 180*s53 + 80*s53^2 + 16*s53^3 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 + 15952*z'' + 22824*z''^2 + 12848*z''^3 + 2688*z''^4 }-> s54 :|: s51 >= 0, s51 <= 2 * z + 2, s52 >= 0, s52 <= 2 * z' + 2, s53 >= 0, s53 <= 2 * z'' + 2, s54 >= 0, s54 <= s52 + s53, z >= 0, z'' >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_if_minus(z, z', z'') -{ 9678 + 4*s45 + s45*s46 + s46 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 + 15952*z'' + 22824*z''^2 + 12848*z''^3 + 2688*z''^4 }-> s47 :|: s44 >= 0, s44 <= 2 * z + 2, s45 >= 0, s45 <= 2 * z' + 2, s46 >= 0, s46 <= 2 * z'' + 2, s47 >= 0, s47 <= s45, z >= 0, z'' >= 0, z' >= 0 encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_le(z, z') -{ 6450 + s38 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s39 :|: s37 >= 0, s37 <= 2 * z + 2, s38 >= 0, s38 <= 2 * z' + 2, s39 >= 0, s39 <= 2, z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 6456 + 4*s41 + s41*s42 + 2*s42 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s43 :|: s41 >= 0, s41 <= 2 * z + 2, s42 >= 0, s42 <= 2 * z' + 2, s43 >= 0, s43 <= s41, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 3224 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 }-> 1 + s40 :|: s40 >= 0, s40 <= 2 * z + 2, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gcd(z, z') -{ 292 + 262*z' + 104*z'^2 + 16*z'^3 }-> s10 :|: s10 >= 0, s10 <= 1 + 0 + (1 + (1 + (z' - 2))), z = 1 + 0, z' - 2 >= 0 gcd(z, z') -{ 16 + 181*z + 58*z*z' + 24*z*z'^2 + 80*z^2 + 24*z^2*z' + 16*z^3 + 180*z' + 80*z'^2 + 16*z'^3 }-> s11 :|: s11 >= 0, s11 <= 1 + (1 + (z - 2)) + (1 + (1 + (z' - 2))), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0 gcd(z, z') -{ 15 + 180*z + 58*z*z' + 24*z*z'^2 + 80*z^2 + 24*z^2*z' + 16*z^3 + 180*z' + 80*z'^2 + 16*z'^3 }-> s12 :|: s12 >= 0, s12 <= 1 + (z - 1) + (1 + (z' - 1)), z - 1 >= 0, z' - 1 >= 0 gcd(z, z') -{ 292 + 262*z + 104*z^2 + 16*z^3 }-> s9 :|: s9 >= 0, s9 <= 1 + (z - 1) + (1 + 0), z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0 if_gcd(z, z', z'') -{ 4 + 64*z'' + 26*z''^2 + 4*z''^3 }-> s13 :|: s13 >= 0, s13 <= 0 + (1 + (z'' - 1)), z = 2, z'' - 1 >= 0, z' = 1 + 0 if_gcd(z, z', z'') -{ 7 + 64*s7 + 52*s7*z'' + 12*s7*z''^2 + 26*s7^2 + 12*s7^2*z'' + 4*s7^3 + 3*z' + z'*z'' + 65*z'' + 26*z''^2 + 4*z''^3 }-> s14 :|: s14 >= 0, s14 <= s7 + (1 + (z'' - 1)), s7 >= 0, s7 <= 1 + (z' - 2), s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 3 + 64*z'' + 26*z''^2 + 4*z''^3 }-> s15 :|: s15 >= 0, s15 <= 0 + (1 + (z'' - 1)), z = 2, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 4 + 64*z' + 26*z'^2 + 4*z'^3 }-> s16 :|: s16 >= 0, s16 <= 0 + (1 + (z' - 1)), z = 1, z' - 1 >= 0, z'' = 1 + 0 if_gcd(z, z', z'') -{ 7 + 64*s8 + 52*s8*z' + 12*s8*z'^2 + 26*s8^2 + 12*s8^2*z' + 4*s8^3 + 65*z' + z'*z'' + 26*z'^2 + 4*z'^3 + 3*z'' }-> s17 :|: s17 >= 0, s17 <= s8 + (1 + (z' - 1)), s8 >= 0, s8 <= 1 + (z'' - 2), s2 >= 0, s2 <= 2, z = 1, z' - 1 >= 0, z'' - 2 >= 0 if_gcd(z, z', z'') -{ 3 + 64*z' + 26*z'^2 + 4*z'^3 }-> s18 :|: s18 >= 0, s18 <= 0 + (1 + (z' - 1)), z = 1, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0 if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s6 :|: s6 >= 0, s6 <= z' - 1, z = 1, z' - 1 >= 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') -{ 8 + 4*z }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), z - 1 >= 0, z' = 0 minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s5 :|: s5 >= 0, s5 <= 1 + (z - 1), z - 1 >= 0, z' >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encode_if_gcd}, {encode_gcd}, {encode_if_minus}, {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] minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z] if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z'] gcd: runtime: O(n^3) [2 + 64*z + 52*z*z' + 12*z*z'^2 + 26*z^2 + 12*z^2*z' + 4*z^3 + 64*z' + 26*z'^2 + 4*z'^3], size: O(n^1) [z + z'] if_gcd: runtime: O(n^3) [14 + 180*z' + 58*z'*z'' + 24*z'*z''^2 + 80*z'^2 + 24*z'^2*z'' + 16*z'^3 + 180*z'' + 80*z''^2 + 16*z''^3], size: O(n^1) [z' + z''] encArg: runtime: O(n^4) [3224 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4], size: O(n^1) [2 + 2*z] ---------------------------------------- (63) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: encode_if_gcd after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 4 + 2*z' + 2*z'' ---------------------------------------- (64) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 6450 + s21 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s22 :|: s20 >= 0, s20 <= 2 * x_1 + 2, s21 >= 0, s21 <= 2 * x_2 + 2, s22 >= 0, s22 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 6456 + 4*s23 + s23*s24 + 2*s24 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s25 :|: s23 >= 0, s23 <= 2 * x_1 + 2, s24 >= 0, s24 <= 2 * x_2 + 2, s25 >= 0, s25 <= s23, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 9678 + 4*s27 + s27*s28 + s28 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 + 15952*x_3 + 22824*x_3^2 + 12848*x_3^3 + 2688*x_3^4 }-> s29 :|: s26 >= 0, s26 <= 2 * x_1 + 2, s27 >= 0, s27 <= 2 * x_2 + 2, s28 >= 0, s28 <= 2 * x_3 + 2, s29 >= 0, s29 <= s27, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 6450 + 64*s30 + 52*s30*s31 + 12*s30*s31^2 + 26*s30^2 + 12*s30^2*s31 + 4*s30^3 + 64*s31 + 26*s31^2 + 4*s31^3 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s32 :|: s30 >= 0, s30 <= 2 * x_1 + 2, s31 >= 0, s31 <= 2 * x_2 + 2, s32 >= 0, s32 <= s30 + s31, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 9686 + 180*s34 + 58*s34*s35 + 24*s34*s35^2 + 80*s34^2 + 24*s34^2*s35 + 16*s34^3 + 180*s35 + 80*s35^2 + 16*s35^3 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 + 15952*x_3 + 22824*x_3^2 + 12848*x_3^3 + 2688*x_3^4 }-> s36 :|: s33 >= 0, s33 <= 2 * x_1 + 2, s34 >= 0, s34 <= 2 * x_2 + 2, s35 >= 0, s35 <= 2 * x_3 + 2, s36 >= 0, s36 <= s34 + s35, 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) -{ -64 + -1904*z + 408*z^2 + 2096*z^3 + 2688*z^4 }-> 1 + s19 :|: s19 >= 0, s19 <= 2 * (z - 1) + 2, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gcd(z, z') -{ 6450 + 64*s48 + 52*s48*s49 + 12*s48*s49^2 + 26*s48^2 + 12*s48^2*s49 + 4*s48^3 + 64*s49 + 26*s49^2 + 4*s49^3 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s50 :|: s48 >= 0, s48 <= 2 * z + 2, s49 >= 0, s49 <= 2 * z' + 2, s50 >= 0, s50 <= s48 + s49, z >= 0, z' >= 0 encode_gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 9686 + 180*s52 + 58*s52*s53 + 24*s52*s53^2 + 80*s52^2 + 24*s52^2*s53 + 16*s52^3 + 180*s53 + 80*s53^2 + 16*s53^3 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 + 15952*z'' + 22824*z''^2 + 12848*z''^3 + 2688*z''^4 }-> s54 :|: s51 >= 0, s51 <= 2 * z + 2, s52 >= 0, s52 <= 2 * z' + 2, s53 >= 0, s53 <= 2 * z'' + 2, s54 >= 0, s54 <= s52 + s53, z >= 0, z'' >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_if_minus(z, z', z'') -{ 9678 + 4*s45 + s45*s46 + s46 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 + 15952*z'' + 22824*z''^2 + 12848*z''^3 + 2688*z''^4 }-> s47 :|: s44 >= 0, s44 <= 2 * z + 2, s45 >= 0, s45 <= 2 * z' + 2, s46 >= 0, s46 <= 2 * z'' + 2, s47 >= 0, s47 <= s45, z >= 0, z'' >= 0, z' >= 0 encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_le(z, z') -{ 6450 + s38 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s39 :|: s37 >= 0, s37 <= 2 * z + 2, s38 >= 0, s38 <= 2 * z' + 2, s39 >= 0, s39 <= 2, z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 6456 + 4*s41 + s41*s42 + 2*s42 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s43 :|: s41 >= 0, s41 <= 2 * z + 2, s42 >= 0, s42 <= 2 * z' + 2, s43 >= 0, s43 <= s41, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 3224 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 }-> 1 + s40 :|: s40 >= 0, s40 <= 2 * z + 2, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gcd(z, z') -{ 292 + 262*z' + 104*z'^2 + 16*z'^3 }-> s10 :|: s10 >= 0, s10 <= 1 + 0 + (1 + (1 + (z' - 2))), z = 1 + 0, z' - 2 >= 0 gcd(z, z') -{ 16 + 181*z + 58*z*z' + 24*z*z'^2 + 80*z^2 + 24*z^2*z' + 16*z^3 + 180*z' + 80*z'^2 + 16*z'^3 }-> s11 :|: s11 >= 0, s11 <= 1 + (1 + (z - 2)) + (1 + (1 + (z' - 2))), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0 gcd(z, z') -{ 15 + 180*z + 58*z*z' + 24*z*z'^2 + 80*z^2 + 24*z^2*z' + 16*z^3 + 180*z' + 80*z'^2 + 16*z'^3 }-> s12 :|: s12 >= 0, s12 <= 1 + (z - 1) + (1 + (z' - 1)), z - 1 >= 0, z' - 1 >= 0 gcd(z, z') -{ 292 + 262*z + 104*z^2 + 16*z^3 }-> s9 :|: s9 >= 0, s9 <= 1 + (z - 1) + (1 + 0), z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0 if_gcd(z, z', z'') -{ 4 + 64*z'' + 26*z''^2 + 4*z''^3 }-> s13 :|: s13 >= 0, s13 <= 0 + (1 + (z'' - 1)), z = 2, z'' - 1 >= 0, z' = 1 + 0 if_gcd(z, z', z'') -{ 7 + 64*s7 + 52*s7*z'' + 12*s7*z''^2 + 26*s7^2 + 12*s7^2*z'' + 4*s7^3 + 3*z' + z'*z'' + 65*z'' + 26*z''^2 + 4*z''^3 }-> s14 :|: s14 >= 0, s14 <= s7 + (1 + (z'' - 1)), s7 >= 0, s7 <= 1 + (z' - 2), s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 3 + 64*z'' + 26*z''^2 + 4*z''^3 }-> s15 :|: s15 >= 0, s15 <= 0 + (1 + (z'' - 1)), z = 2, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 4 + 64*z' + 26*z'^2 + 4*z'^3 }-> s16 :|: s16 >= 0, s16 <= 0 + (1 + (z' - 1)), z = 1, z' - 1 >= 0, z'' = 1 + 0 if_gcd(z, z', z'') -{ 7 + 64*s8 + 52*s8*z' + 12*s8*z'^2 + 26*s8^2 + 12*s8^2*z' + 4*s8^3 + 65*z' + z'*z'' + 26*z'^2 + 4*z'^3 + 3*z'' }-> s17 :|: s17 >= 0, s17 <= s8 + (1 + (z' - 1)), s8 >= 0, s8 <= 1 + (z'' - 2), s2 >= 0, s2 <= 2, z = 1, z' - 1 >= 0, z'' - 2 >= 0 if_gcd(z, z', z'') -{ 3 + 64*z' + 26*z'^2 + 4*z'^3 }-> s18 :|: s18 >= 0, s18 <= 0 + (1 + (z' - 1)), z = 1, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0 if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s6 :|: s6 >= 0, s6 <= z' - 1, z = 1, z' - 1 >= 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') -{ 8 + 4*z }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), z - 1 >= 0, z' = 0 minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s5 :|: s5 >= 0, s5 <= 1 + (z - 1), z - 1 >= 0, z' >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encode_if_gcd}, {encode_gcd}, {encode_if_minus}, {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] minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z] if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z'] gcd: runtime: O(n^3) [2 + 64*z + 52*z*z' + 12*z*z'^2 + 26*z^2 + 12*z^2*z' + 4*z^3 + 64*z' + 26*z'^2 + 4*z'^3], size: O(n^1) [z + z'] if_gcd: runtime: O(n^3) [14 + 180*z' + 58*z'*z'' + 24*z'*z''^2 + 80*z'^2 + 24*z'^2*z'' + 16*z'^3 + 180*z'' + 80*z''^2 + 16*z''^3], size: O(n^1) [z' + z''] encArg: runtime: O(n^4) [3224 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4], size: O(n^1) [2 + 2*z] encode_if_gcd: runtime: ?, size: O(n^1) [4 + 2*z' + 2*z''] ---------------------------------------- (65) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: encode_if_gcd after applying outer abstraction to obtain an ITS, resulting in: O(n^4) with polynomial bound: 11918 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 18144*z' + 1000*z'*z'' + 192*z'*z''^2 + 23720*z'^2 + 192*z'^2*z'' + 12976*z'^3 + 2688*z'^4 + 18144*z'' + 23720*z''^2 + 12976*z''^3 + 2688*z''^4 ---------------------------------------- (66) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 6450 + s21 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s22 :|: s20 >= 0, s20 <= 2 * x_1 + 2, s21 >= 0, s21 <= 2 * x_2 + 2, s22 >= 0, s22 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 6456 + 4*s23 + s23*s24 + 2*s24 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s25 :|: s23 >= 0, s23 <= 2 * x_1 + 2, s24 >= 0, s24 <= 2 * x_2 + 2, s25 >= 0, s25 <= s23, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 9678 + 4*s27 + s27*s28 + s28 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 + 15952*x_3 + 22824*x_3^2 + 12848*x_3^3 + 2688*x_3^4 }-> s29 :|: s26 >= 0, s26 <= 2 * x_1 + 2, s27 >= 0, s27 <= 2 * x_2 + 2, s28 >= 0, s28 <= 2 * x_3 + 2, s29 >= 0, s29 <= s27, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 6450 + 64*s30 + 52*s30*s31 + 12*s30*s31^2 + 26*s30^2 + 12*s30^2*s31 + 4*s30^3 + 64*s31 + 26*s31^2 + 4*s31^3 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s32 :|: s30 >= 0, s30 <= 2 * x_1 + 2, s31 >= 0, s31 <= 2 * x_2 + 2, s32 >= 0, s32 <= s30 + s31, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 9686 + 180*s34 + 58*s34*s35 + 24*s34*s35^2 + 80*s34^2 + 24*s34^2*s35 + 16*s34^3 + 180*s35 + 80*s35^2 + 16*s35^3 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 + 15952*x_3 + 22824*x_3^2 + 12848*x_3^3 + 2688*x_3^4 }-> s36 :|: s33 >= 0, s33 <= 2 * x_1 + 2, s34 >= 0, s34 <= 2 * x_2 + 2, s35 >= 0, s35 <= 2 * x_3 + 2, s36 >= 0, s36 <= s34 + s35, 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) -{ -64 + -1904*z + 408*z^2 + 2096*z^3 + 2688*z^4 }-> 1 + s19 :|: s19 >= 0, s19 <= 2 * (z - 1) + 2, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gcd(z, z') -{ 6450 + 64*s48 + 52*s48*s49 + 12*s48*s49^2 + 26*s48^2 + 12*s48^2*s49 + 4*s48^3 + 64*s49 + 26*s49^2 + 4*s49^3 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s50 :|: s48 >= 0, s48 <= 2 * z + 2, s49 >= 0, s49 <= 2 * z' + 2, s50 >= 0, s50 <= s48 + s49, z >= 0, z' >= 0 encode_gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 9686 + 180*s52 + 58*s52*s53 + 24*s52*s53^2 + 80*s52^2 + 24*s52^2*s53 + 16*s52^3 + 180*s53 + 80*s53^2 + 16*s53^3 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 + 15952*z'' + 22824*z''^2 + 12848*z''^3 + 2688*z''^4 }-> s54 :|: s51 >= 0, s51 <= 2 * z + 2, s52 >= 0, s52 <= 2 * z' + 2, s53 >= 0, s53 <= 2 * z'' + 2, s54 >= 0, s54 <= s52 + s53, z >= 0, z'' >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_if_minus(z, z', z'') -{ 9678 + 4*s45 + s45*s46 + s46 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 + 15952*z'' + 22824*z''^2 + 12848*z''^3 + 2688*z''^4 }-> s47 :|: s44 >= 0, s44 <= 2 * z + 2, s45 >= 0, s45 <= 2 * z' + 2, s46 >= 0, s46 <= 2 * z'' + 2, s47 >= 0, s47 <= s45, z >= 0, z'' >= 0, z' >= 0 encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_le(z, z') -{ 6450 + s38 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s39 :|: s37 >= 0, s37 <= 2 * z + 2, s38 >= 0, s38 <= 2 * z' + 2, s39 >= 0, s39 <= 2, z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 6456 + 4*s41 + s41*s42 + 2*s42 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s43 :|: s41 >= 0, s41 <= 2 * z + 2, s42 >= 0, s42 <= 2 * z' + 2, s43 >= 0, s43 <= s41, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 3224 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 }-> 1 + s40 :|: s40 >= 0, s40 <= 2 * z + 2, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gcd(z, z') -{ 292 + 262*z' + 104*z'^2 + 16*z'^3 }-> s10 :|: s10 >= 0, s10 <= 1 + 0 + (1 + (1 + (z' - 2))), z = 1 + 0, z' - 2 >= 0 gcd(z, z') -{ 16 + 181*z + 58*z*z' + 24*z*z'^2 + 80*z^2 + 24*z^2*z' + 16*z^3 + 180*z' + 80*z'^2 + 16*z'^3 }-> s11 :|: s11 >= 0, s11 <= 1 + (1 + (z - 2)) + (1 + (1 + (z' - 2))), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0 gcd(z, z') -{ 15 + 180*z + 58*z*z' + 24*z*z'^2 + 80*z^2 + 24*z^2*z' + 16*z^3 + 180*z' + 80*z'^2 + 16*z'^3 }-> s12 :|: s12 >= 0, s12 <= 1 + (z - 1) + (1 + (z' - 1)), z - 1 >= 0, z' - 1 >= 0 gcd(z, z') -{ 292 + 262*z + 104*z^2 + 16*z^3 }-> s9 :|: s9 >= 0, s9 <= 1 + (z - 1) + (1 + 0), z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0 if_gcd(z, z', z'') -{ 4 + 64*z'' + 26*z''^2 + 4*z''^3 }-> s13 :|: s13 >= 0, s13 <= 0 + (1 + (z'' - 1)), z = 2, z'' - 1 >= 0, z' = 1 + 0 if_gcd(z, z', z'') -{ 7 + 64*s7 + 52*s7*z'' + 12*s7*z''^2 + 26*s7^2 + 12*s7^2*z'' + 4*s7^3 + 3*z' + z'*z'' + 65*z'' + 26*z''^2 + 4*z''^3 }-> s14 :|: s14 >= 0, s14 <= s7 + (1 + (z'' - 1)), s7 >= 0, s7 <= 1 + (z' - 2), s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 3 + 64*z'' + 26*z''^2 + 4*z''^3 }-> s15 :|: s15 >= 0, s15 <= 0 + (1 + (z'' - 1)), z = 2, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 4 + 64*z' + 26*z'^2 + 4*z'^3 }-> s16 :|: s16 >= 0, s16 <= 0 + (1 + (z' - 1)), z = 1, z' - 1 >= 0, z'' = 1 + 0 if_gcd(z, z', z'') -{ 7 + 64*s8 + 52*s8*z' + 12*s8*z'^2 + 26*s8^2 + 12*s8^2*z' + 4*s8^3 + 65*z' + z'*z'' + 26*z'^2 + 4*z'^3 + 3*z'' }-> s17 :|: s17 >= 0, s17 <= s8 + (1 + (z' - 1)), s8 >= 0, s8 <= 1 + (z'' - 2), s2 >= 0, s2 <= 2, z = 1, z' - 1 >= 0, z'' - 2 >= 0 if_gcd(z, z', z'') -{ 3 + 64*z' + 26*z'^2 + 4*z'^3 }-> s18 :|: s18 >= 0, s18 <= 0 + (1 + (z' - 1)), z = 1, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0 if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s6 :|: s6 >= 0, s6 <= z' - 1, z = 1, z' - 1 >= 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') -{ 8 + 4*z }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), z - 1 >= 0, z' = 0 minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s5 :|: s5 >= 0, s5 <= 1 + (z - 1), z - 1 >= 0, z' >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encode_gcd}, {encode_if_minus}, {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] minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z] if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z'] gcd: runtime: O(n^3) [2 + 64*z + 52*z*z' + 12*z*z'^2 + 26*z^2 + 12*z^2*z' + 4*z^3 + 64*z' + 26*z'^2 + 4*z'^3], size: O(n^1) [z + z'] if_gcd: runtime: O(n^3) [14 + 180*z' + 58*z'*z'' + 24*z'*z''^2 + 80*z'^2 + 24*z'^2*z'' + 16*z'^3 + 180*z'' + 80*z''^2 + 16*z''^3], size: O(n^1) [z' + z''] encArg: runtime: O(n^4) [3224 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4], size: O(n^1) [2 + 2*z] encode_if_gcd: runtime: O(n^4) [11918 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 18144*z' + 1000*z'*z'' + 192*z'*z''^2 + 23720*z'^2 + 192*z'^2*z'' + 12976*z'^3 + 2688*z'^4 + 18144*z'' + 23720*z''^2 + 12976*z''^3 + 2688*z''^4], size: O(n^1) [4 + 2*z' + 2*z''] ---------------------------------------- (67) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (68) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 6450 + s21 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s22 :|: s20 >= 0, s20 <= 2 * x_1 + 2, s21 >= 0, s21 <= 2 * x_2 + 2, s22 >= 0, s22 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 6456 + 4*s23 + s23*s24 + 2*s24 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s25 :|: s23 >= 0, s23 <= 2 * x_1 + 2, s24 >= 0, s24 <= 2 * x_2 + 2, s25 >= 0, s25 <= s23, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 9678 + 4*s27 + s27*s28 + s28 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 + 15952*x_3 + 22824*x_3^2 + 12848*x_3^3 + 2688*x_3^4 }-> s29 :|: s26 >= 0, s26 <= 2 * x_1 + 2, s27 >= 0, s27 <= 2 * x_2 + 2, s28 >= 0, s28 <= 2 * x_3 + 2, s29 >= 0, s29 <= s27, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 6450 + 64*s30 + 52*s30*s31 + 12*s30*s31^2 + 26*s30^2 + 12*s30^2*s31 + 4*s30^3 + 64*s31 + 26*s31^2 + 4*s31^3 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s32 :|: s30 >= 0, s30 <= 2 * x_1 + 2, s31 >= 0, s31 <= 2 * x_2 + 2, s32 >= 0, s32 <= s30 + s31, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 9686 + 180*s34 + 58*s34*s35 + 24*s34*s35^2 + 80*s34^2 + 24*s34^2*s35 + 16*s34^3 + 180*s35 + 80*s35^2 + 16*s35^3 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 + 15952*x_3 + 22824*x_3^2 + 12848*x_3^3 + 2688*x_3^4 }-> s36 :|: s33 >= 0, s33 <= 2 * x_1 + 2, s34 >= 0, s34 <= 2 * x_2 + 2, s35 >= 0, s35 <= 2 * x_3 + 2, s36 >= 0, s36 <= s34 + s35, 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) -{ -64 + -1904*z + 408*z^2 + 2096*z^3 + 2688*z^4 }-> 1 + s19 :|: s19 >= 0, s19 <= 2 * (z - 1) + 2, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gcd(z, z') -{ 6450 + 64*s48 + 52*s48*s49 + 12*s48*s49^2 + 26*s48^2 + 12*s48^2*s49 + 4*s48^3 + 64*s49 + 26*s49^2 + 4*s49^3 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s50 :|: s48 >= 0, s48 <= 2 * z + 2, s49 >= 0, s49 <= 2 * z' + 2, s50 >= 0, s50 <= s48 + s49, z >= 0, z' >= 0 encode_gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 9686 + 180*s52 + 58*s52*s53 + 24*s52*s53^2 + 80*s52^2 + 24*s52^2*s53 + 16*s52^3 + 180*s53 + 80*s53^2 + 16*s53^3 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 + 15952*z'' + 22824*z''^2 + 12848*z''^3 + 2688*z''^4 }-> s54 :|: s51 >= 0, s51 <= 2 * z + 2, s52 >= 0, s52 <= 2 * z' + 2, s53 >= 0, s53 <= 2 * z'' + 2, s54 >= 0, s54 <= s52 + s53, z >= 0, z'' >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_if_minus(z, z', z'') -{ 9678 + 4*s45 + s45*s46 + s46 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 + 15952*z'' + 22824*z''^2 + 12848*z''^3 + 2688*z''^4 }-> s47 :|: s44 >= 0, s44 <= 2 * z + 2, s45 >= 0, s45 <= 2 * z' + 2, s46 >= 0, s46 <= 2 * z'' + 2, s47 >= 0, s47 <= s45, z >= 0, z'' >= 0, z' >= 0 encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_le(z, z') -{ 6450 + s38 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s39 :|: s37 >= 0, s37 <= 2 * z + 2, s38 >= 0, s38 <= 2 * z' + 2, s39 >= 0, s39 <= 2, z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 6456 + 4*s41 + s41*s42 + 2*s42 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s43 :|: s41 >= 0, s41 <= 2 * z + 2, s42 >= 0, s42 <= 2 * z' + 2, s43 >= 0, s43 <= s41, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 3224 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 }-> 1 + s40 :|: s40 >= 0, s40 <= 2 * z + 2, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gcd(z, z') -{ 292 + 262*z' + 104*z'^2 + 16*z'^3 }-> s10 :|: s10 >= 0, s10 <= 1 + 0 + (1 + (1 + (z' - 2))), z = 1 + 0, z' - 2 >= 0 gcd(z, z') -{ 16 + 181*z + 58*z*z' + 24*z*z'^2 + 80*z^2 + 24*z^2*z' + 16*z^3 + 180*z' + 80*z'^2 + 16*z'^3 }-> s11 :|: s11 >= 0, s11 <= 1 + (1 + (z - 2)) + (1 + (1 + (z' - 2))), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0 gcd(z, z') -{ 15 + 180*z + 58*z*z' + 24*z*z'^2 + 80*z^2 + 24*z^2*z' + 16*z^3 + 180*z' + 80*z'^2 + 16*z'^3 }-> s12 :|: s12 >= 0, s12 <= 1 + (z - 1) + (1 + (z' - 1)), z - 1 >= 0, z' - 1 >= 0 gcd(z, z') -{ 292 + 262*z + 104*z^2 + 16*z^3 }-> s9 :|: s9 >= 0, s9 <= 1 + (z - 1) + (1 + 0), z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0 if_gcd(z, z', z'') -{ 4 + 64*z'' + 26*z''^2 + 4*z''^3 }-> s13 :|: s13 >= 0, s13 <= 0 + (1 + (z'' - 1)), z = 2, z'' - 1 >= 0, z' = 1 + 0 if_gcd(z, z', z'') -{ 7 + 64*s7 + 52*s7*z'' + 12*s7*z''^2 + 26*s7^2 + 12*s7^2*z'' + 4*s7^3 + 3*z' + z'*z'' + 65*z'' + 26*z''^2 + 4*z''^3 }-> s14 :|: s14 >= 0, s14 <= s7 + (1 + (z'' - 1)), s7 >= 0, s7 <= 1 + (z' - 2), s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 3 + 64*z'' + 26*z''^2 + 4*z''^3 }-> s15 :|: s15 >= 0, s15 <= 0 + (1 + (z'' - 1)), z = 2, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 4 + 64*z' + 26*z'^2 + 4*z'^3 }-> s16 :|: s16 >= 0, s16 <= 0 + (1 + (z' - 1)), z = 1, z' - 1 >= 0, z'' = 1 + 0 if_gcd(z, z', z'') -{ 7 + 64*s8 + 52*s8*z' + 12*s8*z'^2 + 26*s8^2 + 12*s8^2*z' + 4*s8^3 + 65*z' + z'*z'' + 26*z'^2 + 4*z'^3 + 3*z'' }-> s17 :|: s17 >= 0, s17 <= s8 + (1 + (z' - 1)), s8 >= 0, s8 <= 1 + (z'' - 2), s2 >= 0, s2 <= 2, z = 1, z' - 1 >= 0, z'' - 2 >= 0 if_gcd(z, z', z'') -{ 3 + 64*z' + 26*z'^2 + 4*z'^3 }-> s18 :|: s18 >= 0, s18 <= 0 + (1 + (z' - 1)), z = 1, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0 if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s6 :|: s6 >= 0, s6 <= z' - 1, z = 1, z' - 1 >= 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') -{ 8 + 4*z }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), z - 1 >= 0, z' = 0 minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s5 :|: s5 >= 0, s5 <= 1 + (z - 1), z - 1 >= 0, z' >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encode_gcd}, {encode_if_minus}, {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] minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z] if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z'] gcd: runtime: O(n^3) [2 + 64*z + 52*z*z' + 12*z*z'^2 + 26*z^2 + 12*z^2*z' + 4*z^3 + 64*z' + 26*z'^2 + 4*z'^3], size: O(n^1) [z + z'] if_gcd: runtime: O(n^3) [14 + 180*z' + 58*z'*z'' + 24*z'*z''^2 + 80*z'^2 + 24*z'^2*z'' + 16*z'^3 + 180*z'' + 80*z''^2 + 16*z''^3], size: O(n^1) [z' + z''] encArg: runtime: O(n^4) [3224 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4], size: O(n^1) [2 + 2*z] encode_if_gcd: runtime: O(n^4) [11918 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 18144*z' + 1000*z'*z'' + 192*z'*z''^2 + 23720*z'^2 + 192*z'^2*z'' + 12976*z'^3 + 2688*z'^4 + 18144*z'' + 23720*z''^2 + 12976*z''^3 + 2688*z''^4], size: O(n^1) [4 + 2*z' + 2*z''] ---------------------------------------- (69) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: encode_gcd after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 4 + 2*z + 2*z' ---------------------------------------- (70) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 6450 + s21 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s22 :|: s20 >= 0, s20 <= 2 * x_1 + 2, s21 >= 0, s21 <= 2 * x_2 + 2, s22 >= 0, s22 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 6456 + 4*s23 + s23*s24 + 2*s24 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s25 :|: s23 >= 0, s23 <= 2 * x_1 + 2, s24 >= 0, s24 <= 2 * x_2 + 2, s25 >= 0, s25 <= s23, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 9678 + 4*s27 + s27*s28 + s28 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 + 15952*x_3 + 22824*x_3^2 + 12848*x_3^3 + 2688*x_3^4 }-> s29 :|: s26 >= 0, s26 <= 2 * x_1 + 2, s27 >= 0, s27 <= 2 * x_2 + 2, s28 >= 0, s28 <= 2 * x_3 + 2, s29 >= 0, s29 <= s27, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 6450 + 64*s30 + 52*s30*s31 + 12*s30*s31^2 + 26*s30^2 + 12*s30^2*s31 + 4*s30^3 + 64*s31 + 26*s31^2 + 4*s31^3 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s32 :|: s30 >= 0, s30 <= 2 * x_1 + 2, s31 >= 0, s31 <= 2 * x_2 + 2, s32 >= 0, s32 <= s30 + s31, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 9686 + 180*s34 + 58*s34*s35 + 24*s34*s35^2 + 80*s34^2 + 24*s34^2*s35 + 16*s34^3 + 180*s35 + 80*s35^2 + 16*s35^3 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 + 15952*x_3 + 22824*x_3^2 + 12848*x_3^3 + 2688*x_3^4 }-> s36 :|: s33 >= 0, s33 <= 2 * x_1 + 2, s34 >= 0, s34 <= 2 * x_2 + 2, s35 >= 0, s35 <= 2 * x_3 + 2, s36 >= 0, s36 <= s34 + s35, 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) -{ -64 + -1904*z + 408*z^2 + 2096*z^3 + 2688*z^4 }-> 1 + s19 :|: s19 >= 0, s19 <= 2 * (z - 1) + 2, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gcd(z, z') -{ 6450 + 64*s48 + 52*s48*s49 + 12*s48*s49^2 + 26*s48^2 + 12*s48^2*s49 + 4*s48^3 + 64*s49 + 26*s49^2 + 4*s49^3 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s50 :|: s48 >= 0, s48 <= 2 * z + 2, s49 >= 0, s49 <= 2 * z' + 2, s50 >= 0, s50 <= s48 + s49, z >= 0, z' >= 0 encode_gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 9686 + 180*s52 + 58*s52*s53 + 24*s52*s53^2 + 80*s52^2 + 24*s52^2*s53 + 16*s52^3 + 180*s53 + 80*s53^2 + 16*s53^3 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 + 15952*z'' + 22824*z''^2 + 12848*z''^3 + 2688*z''^4 }-> s54 :|: s51 >= 0, s51 <= 2 * z + 2, s52 >= 0, s52 <= 2 * z' + 2, s53 >= 0, s53 <= 2 * z'' + 2, s54 >= 0, s54 <= s52 + s53, z >= 0, z'' >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_if_minus(z, z', z'') -{ 9678 + 4*s45 + s45*s46 + s46 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 + 15952*z'' + 22824*z''^2 + 12848*z''^3 + 2688*z''^4 }-> s47 :|: s44 >= 0, s44 <= 2 * z + 2, s45 >= 0, s45 <= 2 * z' + 2, s46 >= 0, s46 <= 2 * z'' + 2, s47 >= 0, s47 <= s45, z >= 0, z'' >= 0, z' >= 0 encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_le(z, z') -{ 6450 + s38 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s39 :|: s37 >= 0, s37 <= 2 * z + 2, s38 >= 0, s38 <= 2 * z' + 2, s39 >= 0, s39 <= 2, z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 6456 + 4*s41 + s41*s42 + 2*s42 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s43 :|: s41 >= 0, s41 <= 2 * z + 2, s42 >= 0, s42 <= 2 * z' + 2, s43 >= 0, s43 <= s41, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 3224 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 }-> 1 + s40 :|: s40 >= 0, s40 <= 2 * z + 2, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gcd(z, z') -{ 292 + 262*z' + 104*z'^2 + 16*z'^3 }-> s10 :|: s10 >= 0, s10 <= 1 + 0 + (1 + (1 + (z' - 2))), z = 1 + 0, z' - 2 >= 0 gcd(z, z') -{ 16 + 181*z + 58*z*z' + 24*z*z'^2 + 80*z^2 + 24*z^2*z' + 16*z^3 + 180*z' + 80*z'^2 + 16*z'^3 }-> s11 :|: s11 >= 0, s11 <= 1 + (1 + (z - 2)) + (1 + (1 + (z' - 2))), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0 gcd(z, z') -{ 15 + 180*z + 58*z*z' + 24*z*z'^2 + 80*z^2 + 24*z^2*z' + 16*z^3 + 180*z' + 80*z'^2 + 16*z'^3 }-> s12 :|: s12 >= 0, s12 <= 1 + (z - 1) + (1 + (z' - 1)), z - 1 >= 0, z' - 1 >= 0 gcd(z, z') -{ 292 + 262*z + 104*z^2 + 16*z^3 }-> s9 :|: s9 >= 0, s9 <= 1 + (z - 1) + (1 + 0), z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0 if_gcd(z, z', z'') -{ 4 + 64*z'' + 26*z''^2 + 4*z''^3 }-> s13 :|: s13 >= 0, s13 <= 0 + (1 + (z'' - 1)), z = 2, z'' - 1 >= 0, z' = 1 + 0 if_gcd(z, z', z'') -{ 7 + 64*s7 + 52*s7*z'' + 12*s7*z''^2 + 26*s7^2 + 12*s7^2*z'' + 4*s7^3 + 3*z' + z'*z'' + 65*z'' + 26*z''^2 + 4*z''^3 }-> s14 :|: s14 >= 0, s14 <= s7 + (1 + (z'' - 1)), s7 >= 0, s7 <= 1 + (z' - 2), s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 3 + 64*z'' + 26*z''^2 + 4*z''^3 }-> s15 :|: s15 >= 0, s15 <= 0 + (1 + (z'' - 1)), z = 2, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 4 + 64*z' + 26*z'^2 + 4*z'^3 }-> s16 :|: s16 >= 0, s16 <= 0 + (1 + (z' - 1)), z = 1, z' - 1 >= 0, z'' = 1 + 0 if_gcd(z, z', z'') -{ 7 + 64*s8 + 52*s8*z' + 12*s8*z'^2 + 26*s8^2 + 12*s8^2*z' + 4*s8^3 + 65*z' + z'*z'' + 26*z'^2 + 4*z'^3 + 3*z'' }-> s17 :|: s17 >= 0, s17 <= s8 + (1 + (z' - 1)), s8 >= 0, s8 <= 1 + (z'' - 2), s2 >= 0, s2 <= 2, z = 1, z' - 1 >= 0, z'' - 2 >= 0 if_gcd(z, z', z'') -{ 3 + 64*z' + 26*z'^2 + 4*z'^3 }-> s18 :|: s18 >= 0, s18 <= 0 + (1 + (z' - 1)), z = 1, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0 if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s6 :|: s6 >= 0, s6 <= z' - 1, z = 1, z' - 1 >= 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') -{ 8 + 4*z }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), z - 1 >= 0, z' = 0 minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s5 :|: s5 >= 0, s5 <= 1 + (z - 1), z - 1 >= 0, z' >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encode_gcd}, {encode_if_minus}, {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] minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z] if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z'] gcd: runtime: O(n^3) [2 + 64*z + 52*z*z' + 12*z*z'^2 + 26*z^2 + 12*z^2*z' + 4*z^3 + 64*z' + 26*z'^2 + 4*z'^3], size: O(n^1) [z + z'] if_gcd: runtime: O(n^3) [14 + 180*z' + 58*z'*z'' + 24*z'*z''^2 + 80*z'^2 + 24*z'^2*z'' + 16*z'^3 + 180*z'' + 80*z''^2 + 16*z''^3], size: O(n^1) [z' + z''] encArg: runtime: O(n^4) [3224 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4], size: O(n^1) [2 + 2*z] encode_if_gcd: runtime: O(n^4) [11918 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 18144*z' + 1000*z'*z'' + 192*z'*z''^2 + 23720*z'^2 + 192*z'^2*z'' + 12976*z'^3 + 2688*z'^4 + 18144*z'' + 23720*z''^2 + 12976*z''^3 + 2688*z''^4], size: O(n^1) [4 + 2*z' + 2*z''] encode_gcd: runtime: ?, size: O(n^1) [4 + 2*z + 2*z'] ---------------------------------------- (71) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: encode_gcd after applying outer abstraction to obtain an ITS, resulting in: O(n^4) with polynomial bound: 7378 + 16880*z + 592*z*z' + 96*z*z'^2 + 23120*z^2 + 96*z^2*z' + 12880*z^3 + 2688*z^4 + 16880*z' + 23120*z'^2 + 12880*z'^3 + 2688*z'^4 ---------------------------------------- (72) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 6450 + s21 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s22 :|: s20 >= 0, s20 <= 2 * x_1 + 2, s21 >= 0, s21 <= 2 * x_2 + 2, s22 >= 0, s22 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 6456 + 4*s23 + s23*s24 + 2*s24 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s25 :|: s23 >= 0, s23 <= 2 * x_1 + 2, s24 >= 0, s24 <= 2 * x_2 + 2, s25 >= 0, s25 <= s23, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 9678 + 4*s27 + s27*s28 + s28 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 + 15952*x_3 + 22824*x_3^2 + 12848*x_3^3 + 2688*x_3^4 }-> s29 :|: s26 >= 0, s26 <= 2 * x_1 + 2, s27 >= 0, s27 <= 2 * x_2 + 2, s28 >= 0, s28 <= 2 * x_3 + 2, s29 >= 0, s29 <= s27, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 6450 + 64*s30 + 52*s30*s31 + 12*s30*s31^2 + 26*s30^2 + 12*s30^2*s31 + 4*s30^3 + 64*s31 + 26*s31^2 + 4*s31^3 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s32 :|: s30 >= 0, s30 <= 2 * x_1 + 2, s31 >= 0, s31 <= 2 * x_2 + 2, s32 >= 0, s32 <= s30 + s31, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 9686 + 180*s34 + 58*s34*s35 + 24*s34*s35^2 + 80*s34^2 + 24*s34^2*s35 + 16*s34^3 + 180*s35 + 80*s35^2 + 16*s35^3 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 + 15952*x_3 + 22824*x_3^2 + 12848*x_3^3 + 2688*x_3^4 }-> s36 :|: s33 >= 0, s33 <= 2 * x_1 + 2, s34 >= 0, s34 <= 2 * x_2 + 2, s35 >= 0, s35 <= 2 * x_3 + 2, s36 >= 0, s36 <= s34 + s35, 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) -{ -64 + -1904*z + 408*z^2 + 2096*z^3 + 2688*z^4 }-> 1 + s19 :|: s19 >= 0, s19 <= 2 * (z - 1) + 2, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gcd(z, z') -{ 6450 + 64*s48 + 52*s48*s49 + 12*s48*s49^2 + 26*s48^2 + 12*s48^2*s49 + 4*s48^3 + 64*s49 + 26*s49^2 + 4*s49^3 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s50 :|: s48 >= 0, s48 <= 2 * z + 2, s49 >= 0, s49 <= 2 * z' + 2, s50 >= 0, s50 <= s48 + s49, z >= 0, z' >= 0 encode_gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 9686 + 180*s52 + 58*s52*s53 + 24*s52*s53^2 + 80*s52^2 + 24*s52^2*s53 + 16*s52^3 + 180*s53 + 80*s53^2 + 16*s53^3 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 + 15952*z'' + 22824*z''^2 + 12848*z''^3 + 2688*z''^4 }-> s54 :|: s51 >= 0, s51 <= 2 * z + 2, s52 >= 0, s52 <= 2 * z' + 2, s53 >= 0, s53 <= 2 * z'' + 2, s54 >= 0, s54 <= s52 + s53, z >= 0, z'' >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_if_minus(z, z', z'') -{ 9678 + 4*s45 + s45*s46 + s46 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 + 15952*z'' + 22824*z''^2 + 12848*z''^3 + 2688*z''^4 }-> s47 :|: s44 >= 0, s44 <= 2 * z + 2, s45 >= 0, s45 <= 2 * z' + 2, s46 >= 0, s46 <= 2 * z'' + 2, s47 >= 0, s47 <= s45, z >= 0, z'' >= 0, z' >= 0 encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_le(z, z') -{ 6450 + s38 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s39 :|: s37 >= 0, s37 <= 2 * z + 2, s38 >= 0, s38 <= 2 * z' + 2, s39 >= 0, s39 <= 2, z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 6456 + 4*s41 + s41*s42 + 2*s42 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s43 :|: s41 >= 0, s41 <= 2 * z + 2, s42 >= 0, s42 <= 2 * z' + 2, s43 >= 0, s43 <= s41, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 3224 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 }-> 1 + s40 :|: s40 >= 0, s40 <= 2 * z + 2, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gcd(z, z') -{ 292 + 262*z' + 104*z'^2 + 16*z'^3 }-> s10 :|: s10 >= 0, s10 <= 1 + 0 + (1 + (1 + (z' - 2))), z = 1 + 0, z' - 2 >= 0 gcd(z, z') -{ 16 + 181*z + 58*z*z' + 24*z*z'^2 + 80*z^2 + 24*z^2*z' + 16*z^3 + 180*z' + 80*z'^2 + 16*z'^3 }-> s11 :|: s11 >= 0, s11 <= 1 + (1 + (z - 2)) + (1 + (1 + (z' - 2))), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0 gcd(z, z') -{ 15 + 180*z + 58*z*z' + 24*z*z'^2 + 80*z^2 + 24*z^2*z' + 16*z^3 + 180*z' + 80*z'^2 + 16*z'^3 }-> s12 :|: s12 >= 0, s12 <= 1 + (z - 1) + (1 + (z' - 1)), z - 1 >= 0, z' - 1 >= 0 gcd(z, z') -{ 292 + 262*z + 104*z^2 + 16*z^3 }-> s9 :|: s9 >= 0, s9 <= 1 + (z - 1) + (1 + 0), z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0 if_gcd(z, z', z'') -{ 4 + 64*z'' + 26*z''^2 + 4*z''^3 }-> s13 :|: s13 >= 0, s13 <= 0 + (1 + (z'' - 1)), z = 2, z'' - 1 >= 0, z' = 1 + 0 if_gcd(z, z', z'') -{ 7 + 64*s7 + 52*s7*z'' + 12*s7*z''^2 + 26*s7^2 + 12*s7^2*z'' + 4*s7^3 + 3*z' + z'*z'' + 65*z'' + 26*z''^2 + 4*z''^3 }-> s14 :|: s14 >= 0, s14 <= s7 + (1 + (z'' - 1)), s7 >= 0, s7 <= 1 + (z' - 2), s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 3 + 64*z'' + 26*z''^2 + 4*z''^3 }-> s15 :|: s15 >= 0, s15 <= 0 + (1 + (z'' - 1)), z = 2, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 4 + 64*z' + 26*z'^2 + 4*z'^3 }-> s16 :|: s16 >= 0, s16 <= 0 + (1 + (z' - 1)), z = 1, z' - 1 >= 0, z'' = 1 + 0 if_gcd(z, z', z'') -{ 7 + 64*s8 + 52*s8*z' + 12*s8*z'^2 + 26*s8^2 + 12*s8^2*z' + 4*s8^3 + 65*z' + z'*z'' + 26*z'^2 + 4*z'^3 + 3*z'' }-> s17 :|: s17 >= 0, s17 <= s8 + (1 + (z' - 1)), s8 >= 0, s8 <= 1 + (z'' - 2), s2 >= 0, s2 <= 2, z = 1, z' - 1 >= 0, z'' - 2 >= 0 if_gcd(z, z', z'') -{ 3 + 64*z' + 26*z'^2 + 4*z'^3 }-> s18 :|: s18 >= 0, s18 <= 0 + (1 + (z' - 1)), z = 1, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0 if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s6 :|: s6 >= 0, s6 <= z' - 1, z = 1, z' - 1 >= 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') -{ 8 + 4*z }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), z - 1 >= 0, z' = 0 minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s5 :|: s5 >= 0, s5 <= 1 + (z - 1), z - 1 >= 0, z' >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encode_if_minus}, {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] minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z] if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z'] gcd: runtime: O(n^3) [2 + 64*z + 52*z*z' + 12*z*z'^2 + 26*z^2 + 12*z^2*z' + 4*z^3 + 64*z' + 26*z'^2 + 4*z'^3], size: O(n^1) [z + z'] if_gcd: runtime: O(n^3) [14 + 180*z' + 58*z'*z'' + 24*z'*z''^2 + 80*z'^2 + 24*z'^2*z'' + 16*z'^3 + 180*z'' + 80*z''^2 + 16*z''^3], size: O(n^1) [z' + z''] encArg: runtime: O(n^4) [3224 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4], size: O(n^1) [2 + 2*z] encode_if_gcd: runtime: O(n^4) [11918 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 18144*z' + 1000*z'*z'' + 192*z'*z''^2 + 23720*z'^2 + 192*z'^2*z'' + 12976*z'^3 + 2688*z'^4 + 18144*z'' + 23720*z''^2 + 12976*z''^3 + 2688*z''^4], size: O(n^1) [4 + 2*z' + 2*z''] encode_gcd: runtime: O(n^4) [7378 + 16880*z + 592*z*z' + 96*z*z'^2 + 23120*z^2 + 96*z^2*z' + 12880*z^3 + 2688*z^4 + 16880*z' + 23120*z'^2 + 12880*z'^3 + 2688*z'^4], size: O(n^1) [4 + 2*z + 2*z'] ---------------------------------------- (73) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (74) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 6450 + s21 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s22 :|: s20 >= 0, s20 <= 2 * x_1 + 2, s21 >= 0, s21 <= 2 * x_2 + 2, s22 >= 0, s22 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 6456 + 4*s23 + s23*s24 + 2*s24 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s25 :|: s23 >= 0, s23 <= 2 * x_1 + 2, s24 >= 0, s24 <= 2 * x_2 + 2, s25 >= 0, s25 <= s23, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 9678 + 4*s27 + s27*s28 + s28 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 + 15952*x_3 + 22824*x_3^2 + 12848*x_3^3 + 2688*x_3^4 }-> s29 :|: s26 >= 0, s26 <= 2 * x_1 + 2, s27 >= 0, s27 <= 2 * x_2 + 2, s28 >= 0, s28 <= 2 * x_3 + 2, s29 >= 0, s29 <= s27, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 6450 + 64*s30 + 52*s30*s31 + 12*s30*s31^2 + 26*s30^2 + 12*s30^2*s31 + 4*s30^3 + 64*s31 + 26*s31^2 + 4*s31^3 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s32 :|: s30 >= 0, s30 <= 2 * x_1 + 2, s31 >= 0, s31 <= 2 * x_2 + 2, s32 >= 0, s32 <= s30 + s31, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 9686 + 180*s34 + 58*s34*s35 + 24*s34*s35^2 + 80*s34^2 + 24*s34^2*s35 + 16*s34^3 + 180*s35 + 80*s35^2 + 16*s35^3 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 + 15952*x_3 + 22824*x_3^2 + 12848*x_3^3 + 2688*x_3^4 }-> s36 :|: s33 >= 0, s33 <= 2 * x_1 + 2, s34 >= 0, s34 <= 2 * x_2 + 2, s35 >= 0, s35 <= 2 * x_3 + 2, s36 >= 0, s36 <= s34 + s35, 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) -{ -64 + -1904*z + 408*z^2 + 2096*z^3 + 2688*z^4 }-> 1 + s19 :|: s19 >= 0, s19 <= 2 * (z - 1) + 2, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gcd(z, z') -{ 6450 + 64*s48 + 52*s48*s49 + 12*s48*s49^2 + 26*s48^2 + 12*s48^2*s49 + 4*s48^3 + 64*s49 + 26*s49^2 + 4*s49^3 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s50 :|: s48 >= 0, s48 <= 2 * z + 2, s49 >= 0, s49 <= 2 * z' + 2, s50 >= 0, s50 <= s48 + s49, z >= 0, z' >= 0 encode_gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 9686 + 180*s52 + 58*s52*s53 + 24*s52*s53^2 + 80*s52^2 + 24*s52^2*s53 + 16*s52^3 + 180*s53 + 80*s53^2 + 16*s53^3 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 + 15952*z'' + 22824*z''^2 + 12848*z''^3 + 2688*z''^4 }-> s54 :|: s51 >= 0, s51 <= 2 * z + 2, s52 >= 0, s52 <= 2 * z' + 2, s53 >= 0, s53 <= 2 * z'' + 2, s54 >= 0, s54 <= s52 + s53, z >= 0, z'' >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_if_minus(z, z', z'') -{ 9678 + 4*s45 + s45*s46 + s46 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 + 15952*z'' + 22824*z''^2 + 12848*z''^3 + 2688*z''^4 }-> s47 :|: s44 >= 0, s44 <= 2 * z + 2, s45 >= 0, s45 <= 2 * z' + 2, s46 >= 0, s46 <= 2 * z'' + 2, s47 >= 0, s47 <= s45, z >= 0, z'' >= 0, z' >= 0 encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_le(z, z') -{ 6450 + s38 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s39 :|: s37 >= 0, s37 <= 2 * z + 2, s38 >= 0, s38 <= 2 * z' + 2, s39 >= 0, s39 <= 2, z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 6456 + 4*s41 + s41*s42 + 2*s42 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s43 :|: s41 >= 0, s41 <= 2 * z + 2, s42 >= 0, s42 <= 2 * z' + 2, s43 >= 0, s43 <= s41, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 3224 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 }-> 1 + s40 :|: s40 >= 0, s40 <= 2 * z + 2, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gcd(z, z') -{ 292 + 262*z' + 104*z'^2 + 16*z'^3 }-> s10 :|: s10 >= 0, s10 <= 1 + 0 + (1 + (1 + (z' - 2))), z = 1 + 0, z' - 2 >= 0 gcd(z, z') -{ 16 + 181*z + 58*z*z' + 24*z*z'^2 + 80*z^2 + 24*z^2*z' + 16*z^3 + 180*z' + 80*z'^2 + 16*z'^3 }-> s11 :|: s11 >= 0, s11 <= 1 + (1 + (z - 2)) + (1 + (1 + (z' - 2))), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0 gcd(z, z') -{ 15 + 180*z + 58*z*z' + 24*z*z'^2 + 80*z^2 + 24*z^2*z' + 16*z^3 + 180*z' + 80*z'^2 + 16*z'^3 }-> s12 :|: s12 >= 0, s12 <= 1 + (z - 1) + (1 + (z' - 1)), z - 1 >= 0, z' - 1 >= 0 gcd(z, z') -{ 292 + 262*z + 104*z^2 + 16*z^3 }-> s9 :|: s9 >= 0, s9 <= 1 + (z - 1) + (1 + 0), z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0 if_gcd(z, z', z'') -{ 4 + 64*z'' + 26*z''^2 + 4*z''^3 }-> s13 :|: s13 >= 0, s13 <= 0 + (1 + (z'' - 1)), z = 2, z'' - 1 >= 0, z' = 1 + 0 if_gcd(z, z', z'') -{ 7 + 64*s7 + 52*s7*z'' + 12*s7*z''^2 + 26*s7^2 + 12*s7^2*z'' + 4*s7^3 + 3*z' + z'*z'' + 65*z'' + 26*z''^2 + 4*z''^3 }-> s14 :|: s14 >= 0, s14 <= s7 + (1 + (z'' - 1)), s7 >= 0, s7 <= 1 + (z' - 2), s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 3 + 64*z'' + 26*z''^2 + 4*z''^3 }-> s15 :|: s15 >= 0, s15 <= 0 + (1 + (z'' - 1)), z = 2, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 4 + 64*z' + 26*z'^2 + 4*z'^3 }-> s16 :|: s16 >= 0, s16 <= 0 + (1 + (z' - 1)), z = 1, z' - 1 >= 0, z'' = 1 + 0 if_gcd(z, z', z'') -{ 7 + 64*s8 + 52*s8*z' + 12*s8*z'^2 + 26*s8^2 + 12*s8^2*z' + 4*s8^3 + 65*z' + z'*z'' + 26*z'^2 + 4*z'^3 + 3*z'' }-> s17 :|: s17 >= 0, s17 <= s8 + (1 + (z' - 1)), s8 >= 0, s8 <= 1 + (z'' - 2), s2 >= 0, s2 <= 2, z = 1, z' - 1 >= 0, z'' - 2 >= 0 if_gcd(z, z', z'') -{ 3 + 64*z' + 26*z'^2 + 4*z'^3 }-> s18 :|: s18 >= 0, s18 <= 0 + (1 + (z' - 1)), z = 1, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0 if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s6 :|: s6 >= 0, s6 <= z' - 1, z = 1, z' - 1 >= 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') -{ 8 + 4*z }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), z - 1 >= 0, z' = 0 minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s5 :|: s5 >= 0, s5 <= 1 + (z - 1), z - 1 >= 0, z' >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encode_if_minus}, {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] minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z] if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z'] gcd: runtime: O(n^3) [2 + 64*z + 52*z*z' + 12*z*z'^2 + 26*z^2 + 12*z^2*z' + 4*z^3 + 64*z' + 26*z'^2 + 4*z'^3], size: O(n^1) [z + z'] if_gcd: runtime: O(n^3) [14 + 180*z' + 58*z'*z'' + 24*z'*z''^2 + 80*z'^2 + 24*z'^2*z'' + 16*z'^3 + 180*z'' + 80*z''^2 + 16*z''^3], size: O(n^1) [z' + z''] encArg: runtime: O(n^4) [3224 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4], size: O(n^1) [2 + 2*z] encode_if_gcd: runtime: O(n^4) [11918 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 18144*z' + 1000*z'*z'' + 192*z'*z''^2 + 23720*z'^2 + 192*z'^2*z'' + 12976*z'^3 + 2688*z'^4 + 18144*z'' + 23720*z''^2 + 12976*z''^3 + 2688*z''^4], size: O(n^1) [4 + 2*z' + 2*z''] encode_gcd: runtime: O(n^4) [7378 + 16880*z + 592*z*z' + 96*z*z'^2 + 23120*z^2 + 96*z^2*z' + 12880*z^3 + 2688*z^4 + 16880*z' + 23120*z'^2 + 12880*z'^3 + 2688*z'^4], size: O(n^1) [4 + 2*z + 2*z'] ---------------------------------------- (75) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: encode_if_minus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + 2*z' ---------------------------------------- (76) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 6450 + s21 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s22 :|: s20 >= 0, s20 <= 2 * x_1 + 2, s21 >= 0, s21 <= 2 * x_2 + 2, s22 >= 0, s22 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 6456 + 4*s23 + s23*s24 + 2*s24 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s25 :|: s23 >= 0, s23 <= 2 * x_1 + 2, s24 >= 0, s24 <= 2 * x_2 + 2, s25 >= 0, s25 <= s23, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 9678 + 4*s27 + s27*s28 + s28 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 + 15952*x_3 + 22824*x_3^2 + 12848*x_3^3 + 2688*x_3^4 }-> s29 :|: s26 >= 0, s26 <= 2 * x_1 + 2, s27 >= 0, s27 <= 2 * x_2 + 2, s28 >= 0, s28 <= 2 * x_3 + 2, s29 >= 0, s29 <= s27, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 6450 + 64*s30 + 52*s30*s31 + 12*s30*s31^2 + 26*s30^2 + 12*s30^2*s31 + 4*s30^3 + 64*s31 + 26*s31^2 + 4*s31^3 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s32 :|: s30 >= 0, s30 <= 2 * x_1 + 2, s31 >= 0, s31 <= 2 * x_2 + 2, s32 >= 0, s32 <= s30 + s31, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 9686 + 180*s34 + 58*s34*s35 + 24*s34*s35^2 + 80*s34^2 + 24*s34^2*s35 + 16*s34^3 + 180*s35 + 80*s35^2 + 16*s35^3 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 + 15952*x_3 + 22824*x_3^2 + 12848*x_3^3 + 2688*x_3^4 }-> s36 :|: s33 >= 0, s33 <= 2 * x_1 + 2, s34 >= 0, s34 <= 2 * x_2 + 2, s35 >= 0, s35 <= 2 * x_3 + 2, s36 >= 0, s36 <= s34 + s35, 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) -{ -64 + -1904*z + 408*z^2 + 2096*z^3 + 2688*z^4 }-> 1 + s19 :|: s19 >= 0, s19 <= 2 * (z - 1) + 2, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gcd(z, z') -{ 6450 + 64*s48 + 52*s48*s49 + 12*s48*s49^2 + 26*s48^2 + 12*s48^2*s49 + 4*s48^3 + 64*s49 + 26*s49^2 + 4*s49^3 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s50 :|: s48 >= 0, s48 <= 2 * z + 2, s49 >= 0, s49 <= 2 * z' + 2, s50 >= 0, s50 <= s48 + s49, z >= 0, z' >= 0 encode_gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 9686 + 180*s52 + 58*s52*s53 + 24*s52*s53^2 + 80*s52^2 + 24*s52^2*s53 + 16*s52^3 + 180*s53 + 80*s53^2 + 16*s53^3 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 + 15952*z'' + 22824*z''^2 + 12848*z''^3 + 2688*z''^4 }-> s54 :|: s51 >= 0, s51 <= 2 * z + 2, s52 >= 0, s52 <= 2 * z' + 2, s53 >= 0, s53 <= 2 * z'' + 2, s54 >= 0, s54 <= s52 + s53, z >= 0, z'' >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_if_minus(z, z', z'') -{ 9678 + 4*s45 + s45*s46 + s46 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 + 15952*z'' + 22824*z''^2 + 12848*z''^3 + 2688*z''^4 }-> s47 :|: s44 >= 0, s44 <= 2 * z + 2, s45 >= 0, s45 <= 2 * z' + 2, s46 >= 0, s46 <= 2 * z'' + 2, s47 >= 0, s47 <= s45, z >= 0, z'' >= 0, z' >= 0 encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_le(z, z') -{ 6450 + s38 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s39 :|: s37 >= 0, s37 <= 2 * z + 2, s38 >= 0, s38 <= 2 * z' + 2, s39 >= 0, s39 <= 2, z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 6456 + 4*s41 + s41*s42 + 2*s42 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s43 :|: s41 >= 0, s41 <= 2 * z + 2, s42 >= 0, s42 <= 2 * z' + 2, s43 >= 0, s43 <= s41, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 3224 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 }-> 1 + s40 :|: s40 >= 0, s40 <= 2 * z + 2, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gcd(z, z') -{ 292 + 262*z' + 104*z'^2 + 16*z'^3 }-> s10 :|: s10 >= 0, s10 <= 1 + 0 + (1 + (1 + (z' - 2))), z = 1 + 0, z' - 2 >= 0 gcd(z, z') -{ 16 + 181*z + 58*z*z' + 24*z*z'^2 + 80*z^2 + 24*z^2*z' + 16*z^3 + 180*z' + 80*z'^2 + 16*z'^3 }-> s11 :|: s11 >= 0, s11 <= 1 + (1 + (z - 2)) + (1 + (1 + (z' - 2))), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0 gcd(z, z') -{ 15 + 180*z + 58*z*z' + 24*z*z'^2 + 80*z^2 + 24*z^2*z' + 16*z^3 + 180*z' + 80*z'^2 + 16*z'^3 }-> s12 :|: s12 >= 0, s12 <= 1 + (z - 1) + (1 + (z' - 1)), z - 1 >= 0, z' - 1 >= 0 gcd(z, z') -{ 292 + 262*z + 104*z^2 + 16*z^3 }-> s9 :|: s9 >= 0, s9 <= 1 + (z - 1) + (1 + 0), z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0 if_gcd(z, z', z'') -{ 4 + 64*z'' + 26*z''^2 + 4*z''^3 }-> s13 :|: s13 >= 0, s13 <= 0 + (1 + (z'' - 1)), z = 2, z'' - 1 >= 0, z' = 1 + 0 if_gcd(z, z', z'') -{ 7 + 64*s7 + 52*s7*z'' + 12*s7*z''^2 + 26*s7^2 + 12*s7^2*z'' + 4*s7^3 + 3*z' + z'*z'' + 65*z'' + 26*z''^2 + 4*z''^3 }-> s14 :|: s14 >= 0, s14 <= s7 + (1 + (z'' - 1)), s7 >= 0, s7 <= 1 + (z' - 2), s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 3 + 64*z'' + 26*z''^2 + 4*z''^3 }-> s15 :|: s15 >= 0, s15 <= 0 + (1 + (z'' - 1)), z = 2, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 4 + 64*z' + 26*z'^2 + 4*z'^3 }-> s16 :|: s16 >= 0, s16 <= 0 + (1 + (z' - 1)), z = 1, z' - 1 >= 0, z'' = 1 + 0 if_gcd(z, z', z'') -{ 7 + 64*s8 + 52*s8*z' + 12*s8*z'^2 + 26*s8^2 + 12*s8^2*z' + 4*s8^3 + 65*z' + z'*z'' + 26*z'^2 + 4*z'^3 + 3*z'' }-> s17 :|: s17 >= 0, s17 <= s8 + (1 + (z' - 1)), s8 >= 0, s8 <= 1 + (z'' - 2), s2 >= 0, s2 <= 2, z = 1, z' - 1 >= 0, z'' - 2 >= 0 if_gcd(z, z', z'') -{ 3 + 64*z' + 26*z'^2 + 4*z'^3 }-> s18 :|: s18 >= 0, s18 <= 0 + (1 + (z' - 1)), z = 1, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0 if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s6 :|: s6 >= 0, s6 <= z' - 1, z = 1, z' - 1 >= 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') -{ 8 + 4*z }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), z - 1 >= 0, z' = 0 minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s5 :|: s5 >= 0, s5 <= 1 + (z - 1), z - 1 >= 0, z' >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {encode_if_minus}, {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] minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z] if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z'] gcd: runtime: O(n^3) [2 + 64*z + 52*z*z' + 12*z*z'^2 + 26*z^2 + 12*z^2*z' + 4*z^3 + 64*z' + 26*z'^2 + 4*z'^3], size: O(n^1) [z + z'] if_gcd: runtime: O(n^3) [14 + 180*z' + 58*z'*z'' + 24*z'*z''^2 + 80*z'^2 + 24*z'^2*z'' + 16*z'^3 + 180*z'' + 80*z''^2 + 16*z''^3], size: O(n^1) [z' + z''] encArg: runtime: O(n^4) [3224 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4], size: O(n^1) [2 + 2*z] encode_if_gcd: runtime: O(n^4) [11918 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 18144*z' + 1000*z'*z'' + 192*z'*z''^2 + 23720*z'^2 + 192*z'^2*z'' + 12976*z'^3 + 2688*z'^4 + 18144*z'' + 23720*z''^2 + 12976*z''^3 + 2688*z''^4], size: O(n^1) [4 + 2*z' + 2*z''] encode_gcd: runtime: O(n^4) [7378 + 16880*z + 592*z*z' + 96*z*z'^2 + 23120*z^2 + 96*z^2*z' + 12880*z^3 + 2688*z^4 + 16880*z' + 23120*z'^2 + 12880*z'^3 + 2688*z'^4], size: O(n^1) [4 + 2*z + 2*z'] encode_if_minus: runtime: ?, size: O(n^1) [2 + 2*z'] ---------------------------------------- (77) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: encode_if_minus after applying outer abstraction to obtain an ITS, resulting in: O(n^4) with polynomial bound: 9692 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15964*z' + 4*z'*z'' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 + 15958*z'' + 22824*z''^2 + 12848*z''^3 + 2688*z''^4 ---------------------------------------- (78) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 6450 + s21 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s22 :|: s20 >= 0, s20 <= 2 * x_1 + 2, s21 >= 0, s21 <= 2 * x_2 + 2, s22 >= 0, s22 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 6456 + 4*s23 + s23*s24 + 2*s24 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s25 :|: s23 >= 0, s23 <= 2 * x_1 + 2, s24 >= 0, s24 <= 2 * x_2 + 2, s25 >= 0, s25 <= s23, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 9678 + 4*s27 + s27*s28 + s28 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 + 15952*x_3 + 22824*x_3^2 + 12848*x_3^3 + 2688*x_3^4 }-> s29 :|: s26 >= 0, s26 <= 2 * x_1 + 2, s27 >= 0, s27 <= 2 * x_2 + 2, s28 >= 0, s28 <= 2 * x_3 + 2, s29 >= 0, s29 <= s27, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 6450 + 64*s30 + 52*s30*s31 + 12*s30*s31^2 + 26*s30^2 + 12*s30^2*s31 + 4*s30^3 + 64*s31 + 26*s31^2 + 4*s31^3 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s32 :|: s30 >= 0, s30 <= 2 * x_1 + 2, s31 >= 0, s31 <= 2 * x_2 + 2, s32 >= 0, s32 <= s30 + s31, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 9686 + 180*s34 + 58*s34*s35 + 24*s34*s35^2 + 80*s34^2 + 24*s34^2*s35 + 16*s34^3 + 180*s35 + 80*s35^2 + 16*s35^3 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 + 15952*x_3 + 22824*x_3^2 + 12848*x_3^3 + 2688*x_3^4 }-> s36 :|: s33 >= 0, s33 <= 2 * x_1 + 2, s34 >= 0, s34 <= 2 * x_2 + 2, s35 >= 0, s35 <= 2 * x_3 + 2, s36 >= 0, s36 <= s34 + s35, 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) -{ -64 + -1904*z + 408*z^2 + 2096*z^3 + 2688*z^4 }-> 1 + s19 :|: s19 >= 0, s19 <= 2 * (z - 1) + 2, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gcd(z, z') -{ 6450 + 64*s48 + 52*s48*s49 + 12*s48*s49^2 + 26*s48^2 + 12*s48^2*s49 + 4*s48^3 + 64*s49 + 26*s49^2 + 4*s49^3 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s50 :|: s48 >= 0, s48 <= 2 * z + 2, s49 >= 0, s49 <= 2 * z' + 2, s50 >= 0, s50 <= s48 + s49, z >= 0, z' >= 0 encode_gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 9686 + 180*s52 + 58*s52*s53 + 24*s52*s53^2 + 80*s52^2 + 24*s52^2*s53 + 16*s52^3 + 180*s53 + 80*s53^2 + 16*s53^3 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 + 15952*z'' + 22824*z''^2 + 12848*z''^3 + 2688*z''^4 }-> s54 :|: s51 >= 0, s51 <= 2 * z + 2, s52 >= 0, s52 <= 2 * z' + 2, s53 >= 0, s53 <= 2 * z'' + 2, s54 >= 0, s54 <= s52 + s53, z >= 0, z'' >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_if_minus(z, z', z'') -{ 9678 + 4*s45 + s45*s46 + s46 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 + 15952*z'' + 22824*z''^2 + 12848*z''^3 + 2688*z''^4 }-> s47 :|: s44 >= 0, s44 <= 2 * z + 2, s45 >= 0, s45 <= 2 * z' + 2, s46 >= 0, s46 <= 2 * z'' + 2, s47 >= 0, s47 <= s45, z >= 0, z'' >= 0, z' >= 0 encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_le(z, z') -{ 6450 + s38 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s39 :|: s37 >= 0, s37 <= 2 * z + 2, s38 >= 0, s38 <= 2 * z' + 2, s39 >= 0, s39 <= 2, z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 6456 + 4*s41 + s41*s42 + 2*s42 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s43 :|: s41 >= 0, s41 <= 2 * z + 2, s42 >= 0, s42 <= 2 * z' + 2, s43 >= 0, s43 <= s41, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 3224 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 }-> 1 + s40 :|: s40 >= 0, s40 <= 2 * z + 2, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gcd(z, z') -{ 292 + 262*z' + 104*z'^2 + 16*z'^3 }-> s10 :|: s10 >= 0, s10 <= 1 + 0 + (1 + (1 + (z' - 2))), z = 1 + 0, z' - 2 >= 0 gcd(z, z') -{ 16 + 181*z + 58*z*z' + 24*z*z'^2 + 80*z^2 + 24*z^2*z' + 16*z^3 + 180*z' + 80*z'^2 + 16*z'^3 }-> s11 :|: s11 >= 0, s11 <= 1 + (1 + (z - 2)) + (1 + (1 + (z' - 2))), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0 gcd(z, z') -{ 15 + 180*z + 58*z*z' + 24*z*z'^2 + 80*z^2 + 24*z^2*z' + 16*z^3 + 180*z' + 80*z'^2 + 16*z'^3 }-> s12 :|: s12 >= 0, s12 <= 1 + (z - 1) + (1 + (z' - 1)), z - 1 >= 0, z' - 1 >= 0 gcd(z, z') -{ 292 + 262*z + 104*z^2 + 16*z^3 }-> s9 :|: s9 >= 0, s9 <= 1 + (z - 1) + (1 + 0), z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0 if_gcd(z, z', z'') -{ 4 + 64*z'' + 26*z''^2 + 4*z''^3 }-> s13 :|: s13 >= 0, s13 <= 0 + (1 + (z'' - 1)), z = 2, z'' - 1 >= 0, z' = 1 + 0 if_gcd(z, z', z'') -{ 7 + 64*s7 + 52*s7*z'' + 12*s7*z''^2 + 26*s7^2 + 12*s7^2*z'' + 4*s7^3 + 3*z' + z'*z'' + 65*z'' + 26*z''^2 + 4*z''^3 }-> s14 :|: s14 >= 0, s14 <= s7 + (1 + (z'' - 1)), s7 >= 0, s7 <= 1 + (z' - 2), s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 3 + 64*z'' + 26*z''^2 + 4*z''^3 }-> s15 :|: s15 >= 0, s15 <= 0 + (1 + (z'' - 1)), z = 2, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 4 + 64*z' + 26*z'^2 + 4*z'^3 }-> s16 :|: s16 >= 0, s16 <= 0 + (1 + (z' - 1)), z = 1, z' - 1 >= 0, z'' = 1 + 0 if_gcd(z, z', z'') -{ 7 + 64*s8 + 52*s8*z' + 12*s8*z'^2 + 26*s8^2 + 12*s8^2*z' + 4*s8^3 + 65*z' + z'*z'' + 26*z'^2 + 4*z'^3 + 3*z'' }-> s17 :|: s17 >= 0, s17 <= s8 + (1 + (z' - 1)), s8 >= 0, s8 <= 1 + (z'' - 2), s2 >= 0, s2 <= 2, z = 1, z' - 1 >= 0, z'' - 2 >= 0 if_gcd(z, z', z'') -{ 3 + 64*z' + 26*z'^2 + 4*z'^3 }-> s18 :|: s18 >= 0, s18 <= 0 + (1 + (z' - 1)), z = 1, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0 if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s6 :|: s6 >= 0, s6 <= z' - 1, z = 1, z' - 1 >= 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') -{ 8 + 4*z }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), z - 1 >= 0, z' = 0 minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s5 :|: s5 >= 0, s5 <= 1 + (z - 1), z - 1 >= 0, z' >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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] minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z] if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z'] gcd: runtime: O(n^3) [2 + 64*z + 52*z*z' + 12*z*z'^2 + 26*z^2 + 12*z^2*z' + 4*z^3 + 64*z' + 26*z'^2 + 4*z'^3], size: O(n^1) [z + z'] if_gcd: runtime: O(n^3) [14 + 180*z' + 58*z'*z'' + 24*z'*z''^2 + 80*z'^2 + 24*z'^2*z'' + 16*z'^3 + 180*z'' + 80*z''^2 + 16*z''^3], size: O(n^1) [z' + z''] encArg: runtime: O(n^4) [3224 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4], size: O(n^1) [2 + 2*z] encode_if_gcd: runtime: O(n^4) [11918 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 18144*z' + 1000*z'*z'' + 192*z'*z''^2 + 23720*z'^2 + 192*z'^2*z'' + 12976*z'^3 + 2688*z'^4 + 18144*z'' + 23720*z''^2 + 12976*z''^3 + 2688*z''^4], size: O(n^1) [4 + 2*z' + 2*z''] encode_gcd: runtime: O(n^4) [7378 + 16880*z + 592*z*z' + 96*z*z'^2 + 23120*z^2 + 96*z^2*z' + 12880*z^3 + 2688*z^4 + 16880*z' + 23120*z'^2 + 12880*z'^3 + 2688*z'^4], size: O(n^1) [4 + 2*z + 2*z'] encode_if_minus: runtime: O(n^4) [9692 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15964*z' + 4*z'*z'' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 + 15958*z'' + 22824*z''^2 + 12848*z''^3 + 2688*z''^4], size: O(n^1) [2 + 2*z'] ---------------------------------------- (79) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (80) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 6450 + s21 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s22 :|: s20 >= 0, s20 <= 2 * x_1 + 2, s21 >= 0, s21 <= 2 * x_2 + 2, s22 >= 0, s22 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 6456 + 4*s23 + s23*s24 + 2*s24 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s25 :|: s23 >= 0, s23 <= 2 * x_1 + 2, s24 >= 0, s24 <= 2 * x_2 + 2, s25 >= 0, s25 <= s23, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 9678 + 4*s27 + s27*s28 + s28 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 + 15952*x_3 + 22824*x_3^2 + 12848*x_3^3 + 2688*x_3^4 }-> s29 :|: s26 >= 0, s26 <= 2 * x_1 + 2, s27 >= 0, s27 <= 2 * x_2 + 2, s28 >= 0, s28 <= 2 * x_3 + 2, s29 >= 0, s29 <= s27, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 6450 + 64*s30 + 52*s30*s31 + 12*s30*s31^2 + 26*s30^2 + 12*s30^2*s31 + 4*s30^3 + 64*s31 + 26*s31^2 + 4*s31^3 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s32 :|: s30 >= 0, s30 <= 2 * x_1 + 2, s31 >= 0, s31 <= 2 * x_2 + 2, s32 >= 0, s32 <= s30 + s31, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 9686 + 180*s34 + 58*s34*s35 + 24*s34*s35^2 + 80*s34^2 + 24*s34^2*s35 + 16*s34^3 + 180*s35 + 80*s35^2 + 16*s35^3 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 + 15952*x_3 + 22824*x_3^2 + 12848*x_3^3 + 2688*x_3^4 }-> s36 :|: s33 >= 0, s33 <= 2 * x_1 + 2, s34 >= 0, s34 <= 2 * x_2 + 2, s35 >= 0, s35 <= 2 * x_3 + 2, s36 >= 0, s36 <= s34 + s35, 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) -{ -64 + -1904*z + 408*z^2 + 2096*z^3 + 2688*z^4 }-> 1 + s19 :|: s19 >= 0, s19 <= 2 * (z - 1) + 2, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gcd(z, z') -{ 6450 + 64*s48 + 52*s48*s49 + 12*s48*s49^2 + 26*s48^2 + 12*s48^2*s49 + 4*s48^3 + 64*s49 + 26*s49^2 + 4*s49^3 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s50 :|: s48 >= 0, s48 <= 2 * z + 2, s49 >= 0, s49 <= 2 * z' + 2, s50 >= 0, s50 <= s48 + s49, z >= 0, z' >= 0 encode_gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 9686 + 180*s52 + 58*s52*s53 + 24*s52*s53^2 + 80*s52^2 + 24*s52^2*s53 + 16*s52^3 + 180*s53 + 80*s53^2 + 16*s53^3 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 + 15952*z'' + 22824*z''^2 + 12848*z''^3 + 2688*z''^4 }-> s54 :|: s51 >= 0, s51 <= 2 * z + 2, s52 >= 0, s52 <= 2 * z' + 2, s53 >= 0, s53 <= 2 * z'' + 2, s54 >= 0, s54 <= s52 + s53, z >= 0, z'' >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_if_minus(z, z', z'') -{ 9678 + 4*s45 + s45*s46 + s46 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 + 15952*z'' + 22824*z''^2 + 12848*z''^3 + 2688*z''^4 }-> s47 :|: s44 >= 0, s44 <= 2 * z + 2, s45 >= 0, s45 <= 2 * z' + 2, s46 >= 0, s46 <= 2 * z'' + 2, s47 >= 0, s47 <= s45, z >= 0, z'' >= 0, z' >= 0 encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_le(z, z') -{ 6450 + s38 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s39 :|: s37 >= 0, s37 <= 2 * z + 2, s38 >= 0, s38 <= 2 * z' + 2, s39 >= 0, s39 <= 2, z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 6456 + 4*s41 + s41*s42 + 2*s42 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s43 :|: s41 >= 0, s41 <= 2 * z + 2, s42 >= 0, s42 <= 2 * z' + 2, s43 >= 0, s43 <= s41, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 3224 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 }-> 1 + s40 :|: s40 >= 0, s40 <= 2 * z + 2, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gcd(z, z') -{ 292 + 262*z' + 104*z'^2 + 16*z'^3 }-> s10 :|: s10 >= 0, s10 <= 1 + 0 + (1 + (1 + (z' - 2))), z = 1 + 0, z' - 2 >= 0 gcd(z, z') -{ 16 + 181*z + 58*z*z' + 24*z*z'^2 + 80*z^2 + 24*z^2*z' + 16*z^3 + 180*z' + 80*z'^2 + 16*z'^3 }-> s11 :|: s11 >= 0, s11 <= 1 + (1 + (z - 2)) + (1 + (1 + (z' - 2))), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0 gcd(z, z') -{ 15 + 180*z + 58*z*z' + 24*z*z'^2 + 80*z^2 + 24*z^2*z' + 16*z^3 + 180*z' + 80*z'^2 + 16*z'^3 }-> s12 :|: s12 >= 0, s12 <= 1 + (z - 1) + (1 + (z' - 1)), z - 1 >= 0, z' - 1 >= 0 gcd(z, z') -{ 292 + 262*z + 104*z^2 + 16*z^3 }-> s9 :|: s9 >= 0, s9 <= 1 + (z - 1) + (1 + 0), z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0 if_gcd(z, z', z'') -{ 4 + 64*z'' + 26*z''^2 + 4*z''^3 }-> s13 :|: s13 >= 0, s13 <= 0 + (1 + (z'' - 1)), z = 2, z'' - 1 >= 0, z' = 1 + 0 if_gcd(z, z', z'') -{ 7 + 64*s7 + 52*s7*z'' + 12*s7*z''^2 + 26*s7^2 + 12*s7^2*z'' + 4*s7^3 + 3*z' + z'*z'' + 65*z'' + 26*z''^2 + 4*z''^3 }-> s14 :|: s14 >= 0, s14 <= s7 + (1 + (z'' - 1)), s7 >= 0, s7 <= 1 + (z' - 2), s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 3 + 64*z'' + 26*z''^2 + 4*z''^3 }-> s15 :|: s15 >= 0, s15 <= 0 + (1 + (z'' - 1)), z = 2, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 4 + 64*z' + 26*z'^2 + 4*z'^3 }-> s16 :|: s16 >= 0, s16 <= 0 + (1 + (z' - 1)), z = 1, z' - 1 >= 0, z'' = 1 + 0 if_gcd(z, z', z'') -{ 7 + 64*s8 + 52*s8*z' + 12*s8*z'^2 + 26*s8^2 + 12*s8^2*z' + 4*s8^3 + 65*z' + z'*z'' + 26*z'^2 + 4*z'^3 + 3*z'' }-> s17 :|: s17 >= 0, s17 <= s8 + (1 + (z' - 1)), s8 >= 0, s8 <= 1 + (z'' - 2), s2 >= 0, s2 <= 2, z = 1, z' - 1 >= 0, z'' - 2 >= 0 if_gcd(z, z', z'') -{ 3 + 64*z' + 26*z'^2 + 4*z'^3 }-> s18 :|: s18 >= 0, s18 <= 0 + (1 + (z' - 1)), z = 1, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0 if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s6 :|: s6 >= 0, s6 <= z' - 1, z = 1, z' - 1 >= 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') -{ 8 + 4*z }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), z - 1 >= 0, z' = 0 minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s5 :|: s5 >= 0, s5 <= 1 + (z - 1), z - 1 >= 0, z' >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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] minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z] if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z'] gcd: runtime: O(n^3) [2 + 64*z + 52*z*z' + 12*z*z'^2 + 26*z^2 + 12*z^2*z' + 4*z^3 + 64*z' + 26*z'^2 + 4*z'^3], size: O(n^1) [z + z'] if_gcd: runtime: O(n^3) [14 + 180*z' + 58*z'*z'' + 24*z'*z''^2 + 80*z'^2 + 24*z'^2*z'' + 16*z'^3 + 180*z'' + 80*z''^2 + 16*z''^3], size: O(n^1) [z' + z''] encArg: runtime: O(n^4) [3224 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4], size: O(n^1) [2 + 2*z] encode_if_gcd: runtime: O(n^4) [11918 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 18144*z' + 1000*z'*z'' + 192*z'*z''^2 + 23720*z'^2 + 192*z'^2*z'' + 12976*z'^3 + 2688*z'^4 + 18144*z'' + 23720*z''^2 + 12976*z''^3 + 2688*z''^4], size: O(n^1) [4 + 2*z' + 2*z''] encode_gcd: runtime: O(n^4) [7378 + 16880*z + 592*z*z' + 96*z*z'^2 + 23120*z^2 + 96*z^2*z' + 12880*z^3 + 2688*z^4 + 16880*z' + 23120*z'^2 + 12880*z'^3 + 2688*z'^4], size: O(n^1) [4 + 2*z + 2*z'] encode_if_minus: runtime: O(n^4) [9692 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15964*z' + 4*z'*z'' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 + 15958*z'' + 22824*z''^2 + 12848*z''^3 + 2688*z''^4], size: O(n^1) [2 + 2*z'] ---------------------------------------- (81) 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 + 2*z ---------------------------------------- (82) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 6450 + s21 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s22 :|: s20 >= 0, s20 <= 2 * x_1 + 2, s21 >= 0, s21 <= 2 * x_2 + 2, s22 >= 0, s22 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 6456 + 4*s23 + s23*s24 + 2*s24 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s25 :|: s23 >= 0, s23 <= 2 * x_1 + 2, s24 >= 0, s24 <= 2 * x_2 + 2, s25 >= 0, s25 <= s23, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 9678 + 4*s27 + s27*s28 + s28 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 + 15952*x_3 + 22824*x_3^2 + 12848*x_3^3 + 2688*x_3^4 }-> s29 :|: s26 >= 0, s26 <= 2 * x_1 + 2, s27 >= 0, s27 <= 2 * x_2 + 2, s28 >= 0, s28 <= 2 * x_3 + 2, s29 >= 0, s29 <= s27, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 6450 + 64*s30 + 52*s30*s31 + 12*s30*s31^2 + 26*s30^2 + 12*s30^2*s31 + 4*s30^3 + 64*s31 + 26*s31^2 + 4*s31^3 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s32 :|: s30 >= 0, s30 <= 2 * x_1 + 2, s31 >= 0, s31 <= 2 * x_2 + 2, s32 >= 0, s32 <= s30 + s31, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 9686 + 180*s34 + 58*s34*s35 + 24*s34*s35^2 + 80*s34^2 + 24*s34^2*s35 + 16*s34^3 + 180*s35 + 80*s35^2 + 16*s35^3 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 + 15952*x_3 + 22824*x_3^2 + 12848*x_3^3 + 2688*x_3^4 }-> s36 :|: s33 >= 0, s33 <= 2 * x_1 + 2, s34 >= 0, s34 <= 2 * x_2 + 2, s35 >= 0, s35 <= 2 * x_3 + 2, s36 >= 0, s36 <= s34 + s35, 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) -{ -64 + -1904*z + 408*z^2 + 2096*z^3 + 2688*z^4 }-> 1 + s19 :|: s19 >= 0, s19 <= 2 * (z - 1) + 2, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gcd(z, z') -{ 6450 + 64*s48 + 52*s48*s49 + 12*s48*s49^2 + 26*s48^2 + 12*s48^2*s49 + 4*s48^3 + 64*s49 + 26*s49^2 + 4*s49^3 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s50 :|: s48 >= 0, s48 <= 2 * z + 2, s49 >= 0, s49 <= 2 * z' + 2, s50 >= 0, s50 <= s48 + s49, z >= 0, z' >= 0 encode_gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 9686 + 180*s52 + 58*s52*s53 + 24*s52*s53^2 + 80*s52^2 + 24*s52^2*s53 + 16*s52^3 + 180*s53 + 80*s53^2 + 16*s53^3 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 + 15952*z'' + 22824*z''^2 + 12848*z''^3 + 2688*z''^4 }-> s54 :|: s51 >= 0, s51 <= 2 * z + 2, s52 >= 0, s52 <= 2 * z' + 2, s53 >= 0, s53 <= 2 * z'' + 2, s54 >= 0, s54 <= s52 + s53, z >= 0, z'' >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_if_minus(z, z', z'') -{ 9678 + 4*s45 + s45*s46 + s46 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 + 15952*z'' + 22824*z''^2 + 12848*z''^3 + 2688*z''^4 }-> s47 :|: s44 >= 0, s44 <= 2 * z + 2, s45 >= 0, s45 <= 2 * z' + 2, s46 >= 0, s46 <= 2 * z'' + 2, s47 >= 0, s47 <= s45, z >= 0, z'' >= 0, z' >= 0 encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_le(z, z') -{ 6450 + s38 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s39 :|: s37 >= 0, s37 <= 2 * z + 2, s38 >= 0, s38 <= 2 * z' + 2, s39 >= 0, s39 <= 2, z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 6456 + 4*s41 + s41*s42 + 2*s42 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s43 :|: s41 >= 0, s41 <= 2 * z + 2, s42 >= 0, s42 <= 2 * z' + 2, s43 >= 0, s43 <= s41, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 3224 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 }-> 1 + s40 :|: s40 >= 0, s40 <= 2 * z + 2, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gcd(z, z') -{ 292 + 262*z' + 104*z'^2 + 16*z'^3 }-> s10 :|: s10 >= 0, s10 <= 1 + 0 + (1 + (1 + (z' - 2))), z = 1 + 0, z' - 2 >= 0 gcd(z, z') -{ 16 + 181*z + 58*z*z' + 24*z*z'^2 + 80*z^2 + 24*z^2*z' + 16*z^3 + 180*z' + 80*z'^2 + 16*z'^3 }-> s11 :|: s11 >= 0, s11 <= 1 + (1 + (z - 2)) + (1 + (1 + (z' - 2))), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0 gcd(z, z') -{ 15 + 180*z + 58*z*z' + 24*z*z'^2 + 80*z^2 + 24*z^2*z' + 16*z^3 + 180*z' + 80*z'^2 + 16*z'^3 }-> s12 :|: s12 >= 0, s12 <= 1 + (z - 1) + (1 + (z' - 1)), z - 1 >= 0, z' - 1 >= 0 gcd(z, z') -{ 292 + 262*z + 104*z^2 + 16*z^3 }-> s9 :|: s9 >= 0, s9 <= 1 + (z - 1) + (1 + 0), z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0 if_gcd(z, z', z'') -{ 4 + 64*z'' + 26*z''^2 + 4*z''^3 }-> s13 :|: s13 >= 0, s13 <= 0 + (1 + (z'' - 1)), z = 2, z'' - 1 >= 0, z' = 1 + 0 if_gcd(z, z', z'') -{ 7 + 64*s7 + 52*s7*z'' + 12*s7*z''^2 + 26*s7^2 + 12*s7^2*z'' + 4*s7^3 + 3*z' + z'*z'' + 65*z'' + 26*z''^2 + 4*z''^3 }-> s14 :|: s14 >= 0, s14 <= s7 + (1 + (z'' - 1)), s7 >= 0, s7 <= 1 + (z' - 2), s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 3 + 64*z'' + 26*z''^2 + 4*z''^3 }-> s15 :|: s15 >= 0, s15 <= 0 + (1 + (z'' - 1)), z = 2, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 4 + 64*z' + 26*z'^2 + 4*z'^3 }-> s16 :|: s16 >= 0, s16 <= 0 + (1 + (z' - 1)), z = 1, z' - 1 >= 0, z'' = 1 + 0 if_gcd(z, z', z'') -{ 7 + 64*s8 + 52*s8*z' + 12*s8*z'^2 + 26*s8^2 + 12*s8^2*z' + 4*s8^3 + 65*z' + z'*z'' + 26*z'^2 + 4*z'^3 + 3*z'' }-> s17 :|: s17 >= 0, s17 <= s8 + (1 + (z' - 1)), s8 >= 0, s8 <= 1 + (z'' - 2), s2 >= 0, s2 <= 2, z = 1, z' - 1 >= 0, z'' - 2 >= 0 if_gcd(z, z', z'') -{ 3 + 64*z' + 26*z'^2 + 4*z'^3 }-> s18 :|: s18 >= 0, s18 <= 0 + (1 + (z' - 1)), z = 1, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0 if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s6 :|: s6 >= 0, s6 <= z' - 1, z = 1, z' - 1 >= 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') -{ 8 + 4*z }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), z - 1 >= 0, z' = 0 minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s5 :|: s5 >= 0, s5 <= 1 + (z - 1), z - 1 >= 0, z' >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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] minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z] if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z'] gcd: runtime: O(n^3) [2 + 64*z + 52*z*z' + 12*z*z'^2 + 26*z^2 + 12*z^2*z' + 4*z^3 + 64*z' + 26*z'^2 + 4*z'^3], size: O(n^1) [z + z'] if_gcd: runtime: O(n^3) [14 + 180*z' + 58*z'*z'' + 24*z'*z''^2 + 80*z'^2 + 24*z'^2*z'' + 16*z'^3 + 180*z'' + 80*z''^2 + 16*z''^3], size: O(n^1) [z' + z''] encArg: runtime: O(n^4) [3224 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4], size: O(n^1) [2 + 2*z] encode_if_gcd: runtime: O(n^4) [11918 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 18144*z' + 1000*z'*z'' + 192*z'*z''^2 + 23720*z'^2 + 192*z'^2*z'' + 12976*z'^3 + 2688*z'^4 + 18144*z'' + 23720*z''^2 + 12976*z''^3 + 2688*z''^4], size: O(n^1) [4 + 2*z' + 2*z''] encode_gcd: runtime: O(n^4) [7378 + 16880*z + 592*z*z' + 96*z*z'^2 + 23120*z^2 + 96*z^2*z' + 12880*z^3 + 2688*z^4 + 16880*z' + 23120*z'^2 + 12880*z'^3 + 2688*z'^4], size: O(n^1) [4 + 2*z + 2*z'] encode_if_minus: runtime: O(n^4) [9692 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15964*z' + 4*z'*z'' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 + 15958*z'' + 22824*z''^2 + 12848*z''^3 + 2688*z''^4], size: O(n^1) [2 + 2*z'] encode_minus: runtime: ?, size: O(n^1) [2 + 2*z] ---------------------------------------- (83) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: encode_minus after applying outer abstraction to obtain an ITS, resulting in: O(n^4) with polynomial bound: 6472 + 15964*z + 4*z*z' + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15960*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 ---------------------------------------- (84) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 6450 + s21 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s22 :|: s20 >= 0, s20 <= 2 * x_1 + 2, s21 >= 0, s21 <= 2 * x_2 + 2, s22 >= 0, s22 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 6456 + 4*s23 + s23*s24 + 2*s24 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s25 :|: s23 >= 0, s23 <= 2 * x_1 + 2, s24 >= 0, s24 <= 2 * x_2 + 2, s25 >= 0, s25 <= s23, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 9678 + 4*s27 + s27*s28 + s28 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 + 15952*x_3 + 22824*x_3^2 + 12848*x_3^3 + 2688*x_3^4 }-> s29 :|: s26 >= 0, s26 <= 2 * x_1 + 2, s27 >= 0, s27 <= 2 * x_2 + 2, s28 >= 0, s28 <= 2 * x_3 + 2, s29 >= 0, s29 <= s27, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 6450 + 64*s30 + 52*s30*s31 + 12*s30*s31^2 + 26*s30^2 + 12*s30^2*s31 + 4*s30^3 + 64*s31 + 26*s31^2 + 4*s31^3 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s32 :|: s30 >= 0, s30 <= 2 * x_1 + 2, s31 >= 0, s31 <= 2 * x_2 + 2, s32 >= 0, s32 <= s30 + s31, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 9686 + 180*s34 + 58*s34*s35 + 24*s34*s35^2 + 80*s34^2 + 24*s34^2*s35 + 16*s34^3 + 180*s35 + 80*s35^2 + 16*s35^3 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 + 15952*x_3 + 22824*x_3^2 + 12848*x_3^3 + 2688*x_3^4 }-> s36 :|: s33 >= 0, s33 <= 2 * x_1 + 2, s34 >= 0, s34 <= 2 * x_2 + 2, s35 >= 0, s35 <= 2 * x_3 + 2, s36 >= 0, s36 <= s34 + s35, 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) -{ -64 + -1904*z + 408*z^2 + 2096*z^3 + 2688*z^4 }-> 1 + s19 :|: s19 >= 0, s19 <= 2 * (z - 1) + 2, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gcd(z, z') -{ 6450 + 64*s48 + 52*s48*s49 + 12*s48*s49^2 + 26*s48^2 + 12*s48^2*s49 + 4*s48^3 + 64*s49 + 26*s49^2 + 4*s49^3 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s50 :|: s48 >= 0, s48 <= 2 * z + 2, s49 >= 0, s49 <= 2 * z' + 2, s50 >= 0, s50 <= s48 + s49, z >= 0, z' >= 0 encode_gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 9686 + 180*s52 + 58*s52*s53 + 24*s52*s53^2 + 80*s52^2 + 24*s52^2*s53 + 16*s52^3 + 180*s53 + 80*s53^2 + 16*s53^3 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 + 15952*z'' + 22824*z''^2 + 12848*z''^3 + 2688*z''^4 }-> s54 :|: s51 >= 0, s51 <= 2 * z + 2, s52 >= 0, s52 <= 2 * z' + 2, s53 >= 0, s53 <= 2 * z'' + 2, s54 >= 0, s54 <= s52 + s53, z >= 0, z'' >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_if_minus(z, z', z'') -{ 9678 + 4*s45 + s45*s46 + s46 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 + 15952*z'' + 22824*z''^2 + 12848*z''^3 + 2688*z''^4 }-> s47 :|: s44 >= 0, s44 <= 2 * z + 2, s45 >= 0, s45 <= 2 * z' + 2, s46 >= 0, s46 <= 2 * z'' + 2, s47 >= 0, s47 <= s45, z >= 0, z'' >= 0, z' >= 0 encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_le(z, z') -{ 6450 + s38 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s39 :|: s37 >= 0, s37 <= 2 * z + 2, s38 >= 0, s38 <= 2 * z' + 2, s39 >= 0, s39 <= 2, z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 6456 + 4*s41 + s41*s42 + 2*s42 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s43 :|: s41 >= 0, s41 <= 2 * z + 2, s42 >= 0, s42 <= 2 * z' + 2, s43 >= 0, s43 <= s41, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 3224 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 }-> 1 + s40 :|: s40 >= 0, s40 <= 2 * z + 2, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gcd(z, z') -{ 292 + 262*z' + 104*z'^2 + 16*z'^3 }-> s10 :|: s10 >= 0, s10 <= 1 + 0 + (1 + (1 + (z' - 2))), z = 1 + 0, z' - 2 >= 0 gcd(z, z') -{ 16 + 181*z + 58*z*z' + 24*z*z'^2 + 80*z^2 + 24*z^2*z' + 16*z^3 + 180*z' + 80*z'^2 + 16*z'^3 }-> s11 :|: s11 >= 0, s11 <= 1 + (1 + (z - 2)) + (1 + (1 + (z' - 2))), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0 gcd(z, z') -{ 15 + 180*z + 58*z*z' + 24*z*z'^2 + 80*z^2 + 24*z^2*z' + 16*z^3 + 180*z' + 80*z'^2 + 16*z'^3 }-> s12 :|: s12 >= 0, s12 <= 1 + (z - 1) + (1 + (z' - 1)), z - 1 >= 0, z' - 1 >= 0 gcd(z, z') -{ 292 + 262*z + 104*z^2 + 16*z^3 }-> s9 :|: s9 >= 0, s9 <= 1 + (z - 1) + (1 + 0), z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0 if_gcd(z, z', z'') -{ 4 + 64*z'' + 26*z''^2 + 4*z''^3 }-> s13 :|: s13 >= 0, s13 <= 0 + (1 + (z'' - 1)), z = 2, z'' - 1 >= 0, z' = 1 + 0 if_gcd(z, z', z'') -{ 7 + 64*s7 + 52*s7*z'' + 12*s7*z''^2 + 26*s7^2 + 12*s7^2*z'' + 4*s7^3 + 3*z' + z'*z'' + 65*z'' + 26*z''^2 + 4*z''^3 }-> s14 :|: s14 >= 0, s14 <= s7 + (1 + (z'' - 1)), s7 >= 0, s7 <= 1 + (z' - 2), s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 3 + 64*z'' + 26*z''^2 + 4*z''^3 }-> s15 :|: s15 >= 0, s15 <= 0 + (1 + (z'' - 1)), z = 2, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 4 + 64*z' + 26*z'^2 + 4*z'^3 }-> s16 :|: s16 >= 0, s16 <= 0 + (1 + (z' - 1)), z = 1, z' - 1 >= 0, z'' = 1 + 0 if_gcd(z, z', z'') -{ 7 + 64*s8 + 52*s8*z' + 12*s8*z'^2 + 26*s8^2 + 12*s8^2*z' + 4*s8^3 + 65*z' + z'*z'' + 26*z'^2 + 4*z'^3 + 3*z'' }-> s17 :|: s17 >= 0, s17 <= s8 + (1 + (z' - 1)), s8 >= 0, s8 <= 1 + (z'' - 2), s2 >= 0, s2 <= 2, z = 1, z' - 1 >= 0, z'' - 2 >= 0 if_gcd(z, z', z'') -{ 3 + 64*z' + 26*z'^2 + 4*z'^3 }-> s18 :|: s18 >= 0, s18 <= 0 + (1 + (z' - 1)), z = 1, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0 if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s6 :|: s6 >= 0, s6 <= z' - 1, z = 1, z' - 1 >= 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') -{ 8 + 4*z }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), z - 1 >= 0, z' = 0 minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s5 :|: s5 >= 0, s5 <= 1 + (z - 1), z - 1 >= 0, z' >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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] minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z] if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z'] gcd: runtime: O(n^3) [2 + 64*z + 52*z*z' + 12*z*z'^2 + 26*z^2 + 12*z^2*z' + 4*z^3 + 64*z' + 26*z'^2 + 4*z'^3], size: O(n^1) [z + z'] if_gcd: runtime: O(n^3) [14 + 180*z' + 58*z'*z'' + 24*z'*z''^2 + 80*z'^2 + 24*z'^2*z'' + 16*z'^3 + 180*z'' + 80*z''^2 + 16*z''^3], size: O(n^1) [z' + z''] encArg: runtime: O(n^4) [3224 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4], size: O(n^1) [2 + 2*z] encode_if_gcd: runtime: O(n^4) [11918 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 18144*z' + 1000*z'*z'' + 192*z'*z''^2 + 23720*z'^2 + 192*z'^2*z'' + 12976*z'^3 + 2688*z'^4 + 18144*z'' + 23720*z''^2 + 12976*z''^3 + 2688*z''^4], size: O(n^1) [4 + 2*z' + 2*z''] encode_gcd: runtime: O(n^4) [7378 + 16880*z + 592*z*z' + 96*z*z'^2 + 23120*z^2 + 96*z^2*z' + 12880*z^3 + 2688*z^4 + 16880*z' + 23120*z'^2 + 12880*z'^3 + 2688*z'^4], size: O(n^1) [4 + 2*z + 2*z'] encode_if_minus: runtime: O(n^4) [9692 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15964*z' + 4*z'*z'' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 + 15958*z'' + 22824*z''^2 + 12848*z''^3 + 2688*z''^4], size: O(n^1) [2 + 2*z'] encode_minus: runtime: O(n^4) [6472 + 15964*z + 4*z*z' + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15960*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4], size: O(n^1) [2 + 2*z] ---------------------------------------- (85) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (86) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 6450 + s21 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s22 :|: s20 >= 0, s20 <= 2 * x_1 + 2, s21 >= 0, s21 <= 2 * x_2 + 2, s22 >= 0, s22 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 6456 + 4*s23 + s23*s24 + 2*s24 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s25 :|: s23 >= 0, s23 <= 2 * x_1 + 2, s24 >= 0, s24 <= 2 * x_2 + 2, s25 >= 0, s25 <= s23, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 9678 + 4*s27 + s27*s28 + s28 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 + 15952*x_3 + 22824*x_3^2 + 12848*x_3^3 + 2688*x_3^4 }-> s29 :|: s26 >= 0, s26 <= 2 * x_1 + 2, s27 >= 0, s27 <= 2 * x_2 + 2, s28 >= 0, s28 <= 2 * x_3 + 2, s29 >= 0, s29 <= s27, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 6450 + 64*s30 + 52*s30*s31 + 12*s30*s31^2 + 26*s30^2 + 12*s30^2*s31 + 4*s30^3 + 64*s31 + 26*s31^2 + 4*s31^3 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s32 :|: s30 >= 0, s30 <= 2 * x_1 + 2, s31 >= 0, s31 <= 2 * x_2 + 2, s32 >= 0, s32 <= s30 + s31, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 9686 + 180*s34 + 58*s34*s35 + 24*s34*s35^2 + 80*s34^2 + 24*s34^2*s35 + 16*s34^3 + 180*s35 + 80*s35^2 + 16*s35^3 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 + 15952*x_3 + 22824*x_3^2 + 12848*x_3^3 + 2688*x_3^4 }-> s36 :|: s33 >= 0, s33 <= 2 * x_1 + 2, s34 >= 0, s34 <= 2 * x_2 + 2, s35 >= 0, s35 <= 2 * x_3 + 2, s36 >= 0, s36 <= s34 + s35, 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) -{ -64 + -1904*z + 408*z^2 + 2096*z^3 + 2688*z^4 }-> 1 + s19 :|: s19 >= 0, s19 <= 2 * (z - 1) + 2, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gcd(z, z') -{ 6450 + 64*s48 + 52*s48*s49 + 12*s48*s49^2 + 26*s48^2 + 12*s48^2*s49 + 4*s48^3 + 64*s49 + 26*s49^2 + 4*s49^3 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s50 :|: s48 >= 0, s48 <= 2 * z + 2, s49 >= 0, s49 <= 2 * z' + 2, s50 >= 0, s50 <= s48 + s49, z >= 0, z' >= 0 encode_gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 9686 + 180*s52 + 58*s52*s53 + 24*s52*s53^2 + 80*s52^2 + 24*s52^2*s53 + 16*s52^3 + 180*s53 + 80*s53^2 + 16*s53^3 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 + 15952*z'' + 22824*z''^2 + 12848*z''^3 + 2688*z''^4 }-> s54 :|: s51 >= 0, s51 <= 2 * z + 2, s52 >= 0, s52 <= 2 * z' + 2, s53 >= 0, s53 <= 2 * z'' + 2, s54 >= 0, s54 <= s52 + s53, z >= 0, z'' >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_if_minus(z, z', z'') -{ 9678 + 4*s45 + s45*s46 + s46 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 + 15952*z'' + 22824*z''^2 + 12848*z''^3 + 2688*z''^4 }-> s47 :|: s44 >= 0, s44 <= 2 * z + 2, s45 >= 0, s45 <= 2 * z' + 2, s46 >= 0, s46 <= 2 * z'' + 2, s47 >= 0, s47 <= s45, z >= 0, z'' >= 0, z' >= 0 encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_le(z, z') -{ 6450 + s38 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s39 :|: s37 >= 0, s37 <= 2 * z + 2, s38 >= 0, s38 <= 2 * z' + 2, s39 >= 0, s39 <= 2, z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 6456 + 4*s41 + s41*s42 + 2*s42 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s43 :|: s41 >= 0, s41 <= 2 * z + 2, s42 >= 0, s42 <= 2 * z' + 2, s43 >= 0, s43 <= s41, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 3224 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 }-> 1 + s40 :|: s40 >= 0, s40 <= 2 * z + 2, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gcd(z, z') -{ 292 + 262*z' + 104*z'^2 + 16*z'^3 }-> s10 :|: s10 >= 0, s10 <= 1 + 0 + (1 + (1 + (z' - 2))), z = 1 + 0, z' - 2 >= 0 gcd(z, z') -{ 16 + 181*z + 58*z*z' + 24*z*z'^2 + 80*z^2 + 24*z^2*z' + 16*z^3 + 180*z' + 80*z'^2 + 16*z'^3 }-> s11 :|: s11 >= 0, s11 <= 1 + (1 + (z - 2)) + (1 + (1 + (z' - 2))), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0 gcd(z, z') -{ 15 + 180*z + 58*z*z' + 24*z*z'^2 + 80*z^2 + 24*z^2*z' + 16*z^3 + 180*z' + 80*z'^2 + 16*z'^3 }-> s12 :|: s12 >= 0, s12 <= 1 + (z - 1) + (1 + (z' - 1)), z - 1 >= 0, z' - 1 >= 0 gcd(z, z') -{ 292 + 262*z + 104*z^2 + 16*z^3 }-> s9 :|: s9 >= 0, s9 <= 1 + (z - 1) + (1 + 0), z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0 if_gcd(z, z', z'') -{ 4 + 64*z'' + 26*z''^2 + 4*z''^3 }-> s13 :|: s13 >= 0, s13 <= 0 + (1 + (z'' - 1)), z = 2, z'' - 1 >= 0, z' = 1 + 0 if_gcd(z, z', z'') -{ 7 + 64*s7 + 52*s7*z'' + 12*s7*z''^2 + 26*s7^2 + 12*s7^2*z'' + 4*s7^3 + 3*z' + z'*z'' + 65*z'' + 26*z''^2 + 4*z''^3 }-> s14 :|: s14 >= 0, s14 <= s7 + (1 + (z'' - 1)), s7 >= 0, s7 <= 1 + (z' - 2), s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 3 + 64*z'' + 26*z''^2 + 4*z''^3 }-> s15 :|: s15 >= 0, s15 <= 0 + (1 + (z'' - 1)), z = 2, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 4 + 64*z' + 26*z'^2 + 4*z'^3 }-> s16 :|: s16 >= 0, s16 <= 0 + (1 + (z' - 1)), z = 1, z' - 1 >= 0, z'' = 1 + 0 if_gcd(z, z', z'') -{ 7 + 64*s8 + 52*s8*z' + 12*s8*z'^2 + 26*s8^2 + 12*s8^2*z' + 4*s8^3 + 65*z' + z'*z'' + 26*z'^2 + 4*z'^3 + 3*z'' }-> s17 :|: s17 >= 0, s17 <= s8 + (1 + (z' - 1)), s8 >= 0, s8 <= 1 + (z'' - 2), s2 >= 0, s2 <= 2, z = 1, z' - 1 >= 0, z'' - 2 >= 0 if_gcd(z, z', z'') -{ 3 + 64*z' + 26*z'^2 + 4*z'^3 }-> s18 :|: s18 >= 0, s18 <= 0 + (1 + (z' - 1)), z = 1, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0 if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s6 :|: s6 >= 0, s6 <= z' - 1, z = 1, z' - 1 >= 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') -{ 8 + 4*z }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), z - 1 >= 0, z' = 0 minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s5 :|: s5 >= 0, s5 <= 1 + (z - 1), z - 1 >= 0, z' >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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] minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z] if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z'] gcd: runtime: O(n^3) [2 + 64*z + 52*z*z' + 12*z*z'^2 + 26*z^2 + 12*z^2*z' + 4*z^3 + 64*z' + 26*z'^2 + 4*z'^3], size: O(n^1) [z + z'] if_gcd: runtime: O(n^3) [14 + 180*z' + 58*z'*z'' + 24*z'*z''^2 + 80*z'^2 + 24*z'^2*z'' + 16*z'^3 + 180*z'' + 80*z''^2 + 16*z''^3], size: O(n^1) [z' + z''] encArg: runtime: O(n^4) [3224 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4], size: O(n^1) [2 + 2*z] encode_if_gcd: runtime: O(n^4) [11918 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 18144*z' + 1000*z'*z'' + 192*z'*z''^2 + 23720*z'^2 + 192*z'^2*z'' + 12976*z'^3 + 2688*z'^4 + 18144*z'' + 23720*z''^2 + 12976*z''^3 + 2688*z''^4], size: O(n^1) [4 + 2*z' + 2*z''] encode_gcd: runtime: O(n^4) [7378 + 16880*z + 592*z*z' + 96*z*z'^2 + 23120*z^2 + 96*z^2*z' + 12880*z^3 + 2688*z^4 + 16880*z' + 23120*z'^2 + 12880*z'^3 + 2688*z'^4], size: O(n^1) [4 + 2*z + 2*z'] encode_if_minus: runtime: O(n^4) [9692 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15964*z' + 4*z'*z'' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 + 15958*z'' + 22824*z''^2 + 12848*z''^3 + 2688*z''^4], size: O(n^1) [2 + 2*z'] encode_minus: runtime: O(n^4) [6472 + 15964*z + 4*z*z' + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15960*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4], size: O(n^1) [2 + 2*z] ---------------------------------------- (87) 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 ---------------------------------------- (88) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 6450 + s21 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s22 :|: s20 >= 0, s20 <= 2 * x_1 + 2, s21 >= 0, s21 <= 2 * x_2 + 2, s22 >= 0, s22 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 6456 + 4*s23 + s23*s24 + 2*s24 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s25 :|: s23 >= 0, s23 <= 2 * x_1 + 2, s24 >= 0, s24 <= 2 * x_2 + 2, s25 >= 0, s25 <= s23, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 9678 + 4*s27 + s27*s28 + s28 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 + 15952*x_3 + 22824*x_3^2 + 12848*x_3^3 + 2688*x_3^4 }-> s29 :|: s26 >= 0, s26 <= 2 * x_1 + 2, s27 >= 0, s27 <= 2 * x_2 + 2, s28 >= 0, s28 <= 2 * x_3 + 2, s29 >= 0, s29 <= s27, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 6450 + 64*s30 + 52*s30*s31 + 12*s30*s31^2 + 26*s30^2 + 12*s30^2*s31 + 4*s30^3 + 64*s31 + 26*s31^2 + 4*s31^3 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s32 :|: s30 >= 0, s30 <= 2 * x_1 + 2, s31 >= 0, s31 <= 2 * x_2 + 2, s32 >= 0, s32 <= s30 + s31, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 9686 + 180*s34 + 58*s34*s35 + 24*s34*s35^2 + 80*s34^2 + 24*s34^2*s35 + 16*s34^3 + 180*s35 + 80*s35^2 + 16*s35^3 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 + 15952*x_3 + 22824*x_3^2 + 12848*x_3^3 + 2688*x_3^4 }-> s36 :|: s33 >= 0, s33 <= 2 * x_1 + 2, s34 >= 0, s34 <= 2 * x_2 + 2, s35 >= 0, s35 <= 2 * x_3 + 2, s36 >= 0, s36 <= s34 + s35, 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) -{ -64 + -1904*z + 408*z^2 + 2096*z^3 + 2688*z^4 }-> 1 + s19 :|: s19 >= 0, s19 <= 2 * (z - 1) + 2, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gcd(z, z') -{ 6450 + 64*s48 + 52*s48*s49 + 12*s48*s49^2 + 26*s48^2 + 12*s48^2*s49 + 4*s48^3 + 64*s49 + 26*s49^2 + 4*s49^3 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s50 :|: s48 >= 0, s48 <= 2 * z + 2, s49 >= 0, s49 <= 2 * z' + 2, s50 >= 0, s50 <= s48 + s49, z >= 0, z' >= 0 encode_gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 9686 + 180*s52 + 58*s52*s53 + 24*s52*s53^2 + 80*s52^2 + 24*s52^2*s53 + 16*s52^3 + 180*s53 + 80*s53^2 + 16*s53^3 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 + 15952*z'' + 22824*z''^2 + 12848*z''^3 + 2688*z''^4 }-> s54 :|: s51 >= 0, s51 <= 2 * z + 2, s52 >= 0, s52 <= 2 * z' + 2, s53 >= 0, s53 <= 2 * z'' + 2, s54 >= 0, s54 <= s52 + s53, z >= 0, z'' >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_if_minus(z, z', z'') -{ 9678 + 4*s45 + s45*s46 + s46 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 + 15952*z'' + 22824*z''^2 + 12848*z''^3 + 2688*z''^4 }-> s47 :|: s44 >= 0, s44 <= 2 * z + 2, s45 >= 0, s45 <= 2 * z' + 2, s46 >= 0, s46 <= 2 * z'' + 2, s47 >= 0, s47 <= s45, z >= 0, z'' >= 0, z' >= 0 encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_le(z, z') -{ 6450 + s38 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s39 :|: s37 >= 0, s37 <= 2 * z + 2, s38 >= 0, s38 <= 2 * z' + 2, s39 >= 0, s39 <= 2, z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 6456 + 4*s41 + s41*s42 + 2*s42 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s43 :|: s41 >= 0, s41 <= 2 * z + 2, s42 >= 0, s42 <= 2 * z' + 2, s43 >= 0, s43 <= s41, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 3224 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 }-> 1 + s40 :|: s40 >= 0, s40 <= 2 * z + 2, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gcd(z, z') -{ 292 + 262*z' + 104*z'^2 + 16*z'^3 }-> s10 :|: s10 >= 0, s10 <= 1 + 0 + (1 + (1 + (z' - 2))), z = 1 + 0, z' - 2 >= 0 gcd(z, z') -{ 16 + 181*z + 58*z*z' + 24*z*z'^2 + 80*z^2 + 24*z^2*z' + 16*z^3 + 180*z' + 80*z'^2 + 16*z'^3 }-> s11 :|: s11 >= 0, s11 <= 1 + (1 + (z - 2)) + (1 + (1 + (z' - 2))), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0 gcd(z, z') -{ 15 + 180*z + 58*z*z' + 24*z*z'^2 + 80*z^2 + 24*z^2*z' + 16*z^3 + 180*z' + 80*z'^2 + 16*z'^3 }-> s12 :|: s12 >= 0, s12 <= 1 + (z - 1) + (1 + (z' - 1)), z - 1 >= 0, z' - 1 >= 0 gcd(z, z') -{ 292 + 262*z + 104*z^2 + 16*z^3 }-> s9 :|: s9 >= 0, s9 <= 1 + (z - 1) + (1 + 0), z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0 if_gcd(z, z', z'') -{ 4 + 64*z'' + 26*z''^2 + 4*z''^3 }-> s13 :|: s13 >= 0, s13 <= 0 + (1 + (z'' - 1)), z = 2, z'' - 1 >= 0, z' = 1 + 0 if_gcd(z, z', z'') -{ 7 + 64*s7 + 52*s7*z'' + 12*s7*z''^2 + 26*s7^2 + 12*s7^2*z'' + 4*s7^3 + 3*z' + z'*z'' + 65*z'' + 26*z''^2 + 4*z''^3 }-> s14 :|: s14 >= 0, s14 <= s7 + (1 + (z'' - 1)), s7 >= 0, s7 <= 1 + (z' - 2), s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 3 + 64*z'' + 26*z''^2 + 4*z''^3 }-> s15 :|: s15 >= 0, s15 <= 0 + (1 + (z'' - 1)), z = 2, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 4 + 64*z' + 26*z'^2 + 4*z'^3 }-> s16 :|: s16 >= 0, s16 <= 0 + (1 + (z' - 1)), z = 1, z' - 1 >= 0, z'' = 1 + 0 if_gcd(z, z', z'') -{ 7 + 64*s8 + 52*s8*z' + 12*s8*z'^2 + 26*s8^2 + 12*s8^2*z' + 4*s8^3 + 65*z' + z'*z'' + 26*z'^2 + 4*z'^3 + 3*z'' }-> s17 :|: s17 >= 0, s17 <= s8 + (1 + (z' - 1)), s8 >= 0, s8 <= 1 + (z'' - 2), s2 >= 0, s2 <= 2, z = 1, z' - 1 >= 0, z'' - 2 >= 0 if_gcd(z, z', z'') -{ 3 + 64*z' + 26*z'^2 + 4*z'^3 }-> s18 :|: s18 >= 0, s18 <= 0 + (1 + (z' - 1)), z = 1, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0 if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s6 :|: s6 >= 0, s6 <= z' - 1, z = 1, z' - 1 >= 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') -{ 8 + 4*z }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), z - 1 >= 0, z' = 0 minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s5 :|: s5 >= 0, s5 <= 1 + (z - 1), z - 1 >= 0, z' >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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] minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z] if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z'] gcd: runtime: O(n^3) [2 + 64*z + 52*z*z' + 12*z*z'^2 + 26*z^2 + 12*z^2*z' + 4*z^3 + 64*z' + 26*z'^2 + 4*z'^3], size: O(n^1) [z + z'] if_gcd: runtime: O(n^3) [14 + 180*z' + 58*z'*z'' + 24*z'*z''^2 + 80*z'^2 + 24*z'^2*z'' + 16*z'^3 + 180*z'' + 80*z''^2 + 16*z''^3], size: O(n^1) [z' + z''] encArg: runtime: O(n^4) [3224 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4], size: O(n^1) [2 + 2*z] encode_if_gcd: runtime: O(n^4) [11918 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 18144*z' + 1000*z'*z'' + 192*z'*z''^2 + 23720*z'^2 + 192*z'^2*z'' + 12976*z'^3 + 2688*z'^4 + 18144*z'' + 23720*z''^2 + 12976*z''^3 + 2688*z''^4], size: O(n^1) [4 + 2*z' + 2*z''] encode_gcd: runtime: O(n^4) [7378 + 16880*z + 592*z*z' + 96*z*z'^2 + 23120*z^2 + 96*z^2*z' + 12880*z^3 + 2688*z^4 + 16880*z' + 23120*z'^2 + 12880*z'^3 + 2688*z'^4], size: O(n^1) [4 + 2*z + 2*z'] encode_if_minus: runtime: O(n^4) [9692 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15964*z' + 4*z'*z'' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 + 15958*z'' + 22824*z''^2 + 12848*z''^3 + 2688*z''^4], size: O(n^1) [2 + 2*z'] encode_minus: runtime: O(n^4) [6472 + 15964*z + 4*z*z' + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15960*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4], size: O(n^1) [2 + 2*z] encode_le: runtime: ?, size: O(1) [2] ---------------------------------------- (89) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: encode_le after applying outer abstraction to obtain an ITS, resulting in: O(n^4) with polynomial bound: 6452 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15954*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 ---------------------------------------- (90) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 6450 + s21 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s22 :|: s20 >= 0, s20 <= 2 * x_1 + 2, s21 >= 0, s21 <= 2 * x_2 + 2, s22 >= 0, s22 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 6456 + 4*s23 + s23*s24 + 2*s24 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s25 :|: s23 >= 0, s23 <= 2 * x_1 + 2, s24 >= 0, s24 <= 2 * x_2 + 2, s25 >= 0, s25 <= s23, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 9678 + 4*s27 + s27*s28 + s28 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 + 15952*x_3 + 22824*x_3^2 + 12848*x_3^3 + 2688*x_3^4 }-> s29 :|: s26 >= 0, s26 <= 2 * x_1 + 2, s27 >= 0, s27 <= 2 * x_2 + 2, s28 >= 0, s28 <= 2 * x_3 + 2, s29 >= 0, s29 <= s27, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 6450 + 64*s30 + 52*s30*s31 + 12*s30*s31^2 + 26*s30^2 + 12*s30^2*s31 + 4*s30^3 + 64*s31 + 26*s31^2 + 4*s31^3 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s32 :|: s30 >= 0, s30 <= 2 * x_1 + 2, s31 >= 0, s31 <= 2 * x_2 + 2, s32 >= 0, s32 <= s30 + s31, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 9686 + 180*s34 + 58*s34*s35 + 24*s34*s35^2 + 80*s34^2 + 24*s34^2*s35 + 16*s34^3 + 180*s35 + 80*s35^2 + 16*s35^3 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 + 15952*x_3 + 22824*x_3^2 + 12848*x_3^3 + 2688*x_3^4 }-> s36 :|: s33 >= 0, s33 <= 2 * x_1 + 2, s34 >= 0, s34 <= 2 * x_2 + 2, s35 >= 0, s35 <= 2 * x_3 + 2, s36 >= 0, s36 <= s34 + s35, 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) -{ -64 + -1904*z + 408*z^2 + 2096*z^3 + 2688*z^4 }-> 1 + s19 :|: s19 >= 0, s19 <= 2 * (z - 1) + 2, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gcd(z, z') -{ 6450 + 64*s48 + 52*s48*s49 + 12*s48*s49^2 + 26*s48^2 + 12*s48^2*s49 + 4*s48^3 + 64*s49 + 26*s49^2 + 4*s49^3 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s50 :|: s48 >= 0, s48 <= 2 * z + 2, s49 >= 0, s49 <= 2 * z' + 2, s50 >= 0, s50 <= s48 + s49, z >= 0, z' >= 0 encode_gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 9686 + 180*s52 + 58*s52*s53 + 24*s52*s53^2 + 80*s52^2 + 24*s52^2*s53 + 16*s52^3 + 180*s53 + 80*s53^2 + 16*s53^3 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 + 15952*z'' + 22824*z''^2 + 12848*z''^3 + 2688*z''^4 }-> s54 :|: s51 >= 0, s51 <= 2 * z + 2, s52 >= 0, s52 <= 2 * z' + 2, s53 >= 0, s53 <= 2 * z'' + 2, s54 >= 0, s54 <= s52 + s53, z >= 0, z'' >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_if_minus(z, z', z'') -{ 9678 + 4*s45 + s45*s46 + s46 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 + 15952*z'' + 22824*z''^2 + 12848*z''^3 + 2688*z''^4 }-> s47 :|: s44 >= 0, s44 <= 2 * z + 2, s45 >= 0, s45 <= 2 * z' + 2, s46 >= 0, s46 <= 2 * z'' + 2, s47 >= 0, s47 <= s45, z >= 0, z'' >= 0, z' >= 0 encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_le(z, z') -{ 6450 + s38 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s39 :|: s37 >= 0, s37 <= 2 * z + 2, s38 >= 0, s38 <= 2 * z' + 2, s39 >= 0, s39 <= 2, z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 6456 + 4*s41 + s41*s42 + 2*s42 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s43 :|: s41 >= 0, s41 <= 2 * z + 2, s42 >= 0, s42 <= 2 * z' + 2, s43 >= 0, s43 <= s41, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 3224 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 }-> 1 + s40 :|: s40 >= 0, s40 <= 2 * z + 2, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gcd(z, z') -{ 292 + 262*z' + 104*z'^2 + 16*z'^3 }-> s10 :|: s10 >= 0, s10 <= 1 + 0 + (1 + (1 + (z' - 2))), z = 1 + 0, z' - 2 >= 0 gcd(z, z') -{ 16 + 181*z + 58*z*z' + 24*z*z'^2 + 80*z^2 + 24*z^2*z' + 16*z^3 + 180*z' + 80*z'^2 + 16*z'^3 }-> s11 :|: s11 >= 0, s11 <= 1 + (1 + (z - 2)) + (1 + (1 + (z' - 2))), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0 gcd(z, z') -{ 15 + 180*z + 58*z*z' + 24*z*z'^2 + 80*z^2 + 24*z^2*z' + 16*z^3 + 180*z' + 80*z'^2 + 16*z'^3 }-> s12 :|: s12 >= 0, s12 <= 1 + (z - 1) + (1 + (z' - 1)), z - 1 >= 0, z' - 1 >= 0 gcd(z, z') -{ 292 + 262*z + 104*z^2 + 16*z^3 }-> s9 :|: s9 >= 0, s9 <= 1 + (z - 1) + (1 + 0), z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0 if_gcd(z, z', z'') -{ 4 + 64*z'' + 26*z''^2 + 4*z''^3 }-> s13 :|: s13 >= 0, s13 <= 0 + (1 + (z'' - 1)), z = 2, z'' - 1 >= 0, z' = 1 + 0 if_gcd(z, z', z'') -{ 7 + 64*s7 + 52*s7*z'' + 12*s7*z''^2 + 26*s7^2 + 12*s7^2*z'' + 4*s7^3 + 3*z' + z'*z'' + 65*z'' + 26*z''^2 + 4*z''^3 }-> s14 :|: s14 >= 0, s14 <= s7 + (1 + (z'' - 1)), s7 >= 0, s7 <= 1 + (z' - 2), s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 3 + 64*z'' + 26*z''^2 + 4*z''^3 }-> s15 :|: s15 >= 0, s15 <= 0 + (1 + (z'' - 1)), z = 2, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 4 + 64*z' + 26*z'^2 + 4*z'^3 }-> s16 :|: s16 >= 0, s16 <= 0 + (1 + (z' - 1)), z = 1, z' - 1 >= 0, z'' = 1 + 0 if_gcd(z, z', z'') -{ 7 + 64*s8 + 52*s8*z' + 12*s8*z'^2 + 26*s8^2 + 12*s8^2*z' + 4*s8^3 + 65*z' + z'*z'' + 26*z'^2 + 4*z'^3 + 3*z'' }-> s17 :|: s17 >= 0, s17 <= s8 + (1 + (z' - 1)), s8 >= 0, s8 <= 1 + (z'' - 2), s2 >= 0, s2 <= 2, z = 1, z' - 1 >= 0, z'' - 2 >= 0 if_gcd(z, z', z'') -{ 3 + 64*z' + 26*z'^2 + 4*z'^3 }-> s18 :|: s18 >= 0, s18 <= 0 + (1 + (z' - 1)), z = 1, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0 if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s6 :|: s6 >= 0, s6 <= z' - 1, z = 1, z' - 1 >= 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') -{ 8 + 4*z }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), z - 1 >= 0, z' = 0 minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s5 :|: s5 >= 0, s5 <= 1 + (z - 1), z - 1 >= 0, z' >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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] minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z] if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z'] gcd: runtime: O(n^3) [2 + 64*z + 52*z*z' + 12*z*z'^2 + 26*z^2 + 12*z^2*z' + 4*z^3 + 64*z' + 26*z'^2 + 4*z'^3], size: O(n^1) [z + z'] if_gcd: runtime: O(n^3) [14 + 180*z' + 58*z'*z'' + 24*z'*z''^2 + 80*z'^2 + 24*z'^2*z'' + 16*z'^3 + 180*z'' + 80*z''^2 + 16*z''^3], size: O(n^1) [z' + z''] encArg: runtime: O(n^4) [3224 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4], size: O(n^1) [2 + 2*z] encode_if_gcd: runtime: O(n^4) [11918 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 18144*z' + 1000*z'*z'' + 192*z'*z''^2 + 23720*z'^2 + 192*z'^2*z'' + 12976*z'^3 + 2688*z'^4 + 18144*z'' + 23720*z''^2 + 12976*z''^3 + 2688*z''^4], size: O(n^1) [4 + 2*z' + 2*z''] encode_gcd: runtime: O(n^4) [7378 + 16880*z + 592*z*z' + 96*z*z'^2 + 23120*z^2 + 96*z^2*z' + 12880*z^3 + 2688*z^4 + 16880*z' + 23120*z'^2 + 12880*z'^3 + 2688*z'^4], size: O(n^1) [4 + 2*z + 2*z'] encode_if_minus: runtime: O(n^4) [9692 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15964*z' + 4*z'*z'' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 + 15958*z'' + 22824*z''^2 + 12848*z''^3 + 2688*z''^4], size: O(n^1) [2 + 2*z'] encode_minus: runtime: O(n^4) [6472 + 15964*z + 4*z*z' + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15960*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4], size: O(n^1) [2 + 2*z] encode_le: runtime: O(n^4) [6452 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15954*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4], size: O(1) [2] ---------------------------------------- (91) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (92) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 6450 + s21 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s22 :|: s20 >= 0, s20 <= 2 * x_1 + 2, s21 >= 0, s21 <= 2 * x_2 + 2, s22 >= 0, s22 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 6456 + 4*s23 + s23*s24 + 2*s24 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s25 :|: s23 >= 0, s23 <= 2 * x_1 + 2, s24 >= 0, s24 <= 2 * x_2 + 2, s25 >= 0, s25 <= s23, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 9678 + 4*s27 + s27*s28 + s28 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 + 15952*x_3 + 22824*x_3^2 + 12848*x_3^3 + 2688*x_3^4 }-> s29 :|: s26 >= 0, s26 <= 2 * x_1 + 2, s27 >= 0, s27 <= 2 * x_2 + 2, s28 >= 0, s28 <= 2 * x_3 + 2, s29 >= 0, s29 <= s27, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 6450 + 64*s30 + 52*s30*s31 + 12*s30*s31^2 + 26*s30^2 + 12*s30^2*s31 + 4*s30^3 + 64*s31 + 26*s31^2 + 4*s31^3 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s32 :|: s30 >= 0, s30 <= 2 * x_1 + 2, s31 >= 0, s31 <= 2 * x_2 + 2, s32 >= 0, s32 <= s30 + s31, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 9686 + 180*s34 + 58*s34*s35 + 24*s34*s35^2 + 80*s34^2 + 24*s34^2*s35 + 16*s34^3 + 180*s35 + 80*s35^2 + 16*s35^3 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 + 15952*x_3 + 22824*x_3^2 + 12848*x_3^3 + 2688*x_3^4 }-> s36 :|: s33 >= 0, s33 <= 2 * x_1 + 2, s34 >= 0, s34 <= 2 * x_2 + 2, s35 >= 0, s35 <= 2 * x_3 + 2, s36 >= 0, s36 <= s34 + s35, 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) -{ -64 + -1904*z + 408*z^2 + 2096*z^3 + 2688*z^4 }-> 1 + s19 :|: s19 >= 0, s19 <= 2 * (z - 1) + 2, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gcd(z, z') -{ 6450 + 64*s48 + 52*s48*s49 + 12*s48*s49^2 + 26*s48^2 + 12*s48^2*s49 + 4*s48^3 + 64*s49 + 26*s49^2 + 4*s49^3 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s50 :|: s48 >= 0, s48 <= 2 * z + 2, s49 >= 0, s49 <= 2 * z' + 2, s50 >= 0, s50 <= s48 + s49, z >= 0, z' >= 0 encode_gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 9686 + 180*s52 + 58*s52*s53 + 24*s52*s53^2 + 80*s52^2 + 24*s52^2*s53 + 16*s52^3 + 180*s53 + 80*s53^2 + 16*s53^3 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 + 15952*z'' + 22824*z''^2 + 12848*z''^3 + 2688*z''^4 }-> s54 :|: s51 >= 0, s51 <= 2 * z + 2, s52 >= 0, s52 <= 2 * z' + 2, s53 >= 0, s53 <= 2 * z'' + 2, s54 >= 0, s54 <= s52 + s53, z >= 0, z'' >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_if_minus(z, z', z'') -{ 9678 + 4*s45 + s45*s46 + s46 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 + 15952*z'' + 22824*z''^2 + 12848*z''^3 + 2688*z''^4 }-> s47 :|: s44 >= 0, s44 <= 2 * z + 2, s45 >= 0, s45 <= 2 * z' + 2, s46 >= 0, s46 <= 2 * z'' + 2, s47 >= 0, s47 <= s45, z >= 0, z'' >= 0, z' >= 0 encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_le(z, z') -{ 6450 + s38 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s39 :|: s37 >= 0, s37 <= 2 * z + 2, s38 >= 0, s38 <= 2 * z' + 2, s39 >= 0, s39 <= 2, z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 6456 + 4*s41 + s41*s42 + 2*s42 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s43 :|: s41 >= 0, s41 <= 2 * z + 2, s42 >= 0, s42 <= 2 * z' + 2, s43 >= 0, s43 <= s41, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 3224 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 }-> 1 + s40 :|: s40 >= 0, s40 <= 2 * z + 2, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gcd(z, z') -{ 292 + 262*z' + 104*z'^2 + 16*z'^3 }-> s10 :|: s10 >= 0, s10 <= 1 + 0 + (1 + (1 + (z' - 2))), z = 1 + 0, z' - 2 >= 0 gcd(z, z') -{ 16 + 181*z + 58*z*z' + 24*z*z'^2 + 80*z^2 + 24*z^2*z' + 16*z^3 + 180*z' + 80*z'^2 + 16*z'^3 }-> s11 :|: s11 >= 0, s11 <= 1 + (1 + (z - 2)) + (1 + (1 + (z' - 2))), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0 gcd(z, z') -{ 15 + 180*z + 58*z*z' + 24*z*z'^2 + 80*z^2 + 24*z^2*z' + 16*z^3 + 180*z' + 80*z'^2 + 16*z'^3 }-> s12 :|: s12 >= 0, s12 <= 1 + (z - 1) + (1 + (z' - 1)), z - 1 >= 0, z' - 1 >= 0 gcd(z, z') -{ 292 + 262*z + 104*z^2 + 16*z^3 }-> s9 :|: s9 >= 0, s9 <= 1 + (z - 1) + (1 + 0), z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0 if_gcd(z, z', z'') -{ 4 + 64*z'' + 26*z''^2 + 4*z''^3 }-> s13 :|: s13 >= 0, s13 <= 0 + (1 + (z'' - 1)), z = 2, z'' - 1 >= 0, z' = 1 + 0 if_gcd(z, z', z'') -{ 7 + 64*s7 + 52*s7*z'' + 12*s7*z''^2 + 26*s7^2 + 12*s7^2*z'' + 4*s7^3 + 3*z' + z'*z'' + 65*z'' + 26*z''^2 + 4*z''^3 }-> s14 :|: s14 >= 0, s14 <= s7 + (1 + (z'' - 1)), s7 >= 0, s7 <= 1 + (z' - 2), s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 3 + 64*z'' + 26*z''^2 + 4*z''^3 }-> s15 :|: s15 >= 0, s15 <= 0 + (1 + (z'' - 1)), z = 2, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 4 + 64*z' + 26*z'^2 + 4*z'^3 }-> s16 :|: s16 >= 0, s16 <= 0 + (1 + (z' - 1)), z = 1, z' - 1 >= 0, z'' = 1 + 0 if_gcd(z, z', z'') -{ 7 + 64*s8 + 52*s8*z' + 12*s8*z'^2 + 26*s8^2 + 12*s8^2*z' + 4*s8^3 + 65*z' + z'*z'' + 26*z'^2 + 4*z'^3 + 3*z'' }-> s17 :|: s17 >= 0, s17 <= s8 + (1 + (z' - 1)), s8 >= 0, s8 <= 1 + (z'' - 2), s2 >= 0, s2 <= 2, z = 1, z' - 1 >= 0, z'' - 2 >= 0 if_gcd(z, z', z'') -{ 3 + 64*z' + 26*z'^2 + 4*z'^3 }-> s18 :|: s18 >= 0, s18 <= 0 + (1 + (z' - 1)), z = 1, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0 if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s6 :|: s6 >= 0, s6 <= z' - 1, z = 1, z' - 1 >= 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') -{ 8 + 4*z }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), z - 1 >= 0, z' = 0 minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s5 :|: s5 >= 0, s5 <= 1 + (z - 1), z - 1 >= 0, z' >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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] minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z] if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z'] gcd: runtime: O(n^3) [2 + 64*z + 52*z*z' + 12*z*z'^2 + 26*z^2 + 12*z^2*z' + 4*z^3 + 64*z' + 26*z'^2 + 4*z'^3], size: O(n^1) [z + z'] if_gcd: runtime: O(n^3) [14 + 180*z' + 58*z'*z'' + 24*z'*z''^2 + 80*z'^2 + 24*z'^2*z'' + 16*z'^3 + 180*z'' + 80*z''^2 + 16*z''^3], size: O(n^1) [z' + z''] encArg: runtime: O(n^4) [3224 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4], size: O(n^1) [2 + 2*z] encode_if_gcd: runtime: O(n^4) [11918 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 18144*z' + 1000*z'*z'' + 192*z'*z''^2 + 23720*z'^2 + 192*z'^2*z'' + 12976*z'^3 + 2688*z'^4 + 18144*z'' + 23720*z''^2 + 12976*z''^3 + 2688*z''^4], size: O(n^1) [4 + 2*z' + 2*z''] encode_gcd: runtime: O(n^4) [7378 + 16880*z + 592*z*z' + 96*z*z'^2 + 23120*z^2 + 96*z^2*z' + 12880*z^3 + 2688*z^4 + 16880*z' + 23120*z'^2 + 12880*z'^3 + 2688*z'^4], size: O(n^1) [4 + 2*z + 2*z'] encode_if_minus: runtime: O(n^4) [9692 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15964*z' + 4*z'*z'' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 + 15958*z'' + 22824*z''^2 + 12848*z''^3 + 2688*z''^4], size: O(n^1) [2 + 2*z'] encode_minus: runtime: O(n^4) [6472 + 15964*z + 4*z*z' + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15960*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4], size: O(n^1) [2 + 2*z] encode_le: runtime: O(n^4) [6452 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15954*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4], size: O(1) [2] ---------------------------------------- (93) 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: 3 + 2*z ---------------------------------------- (94) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 6450 + s21 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s22 :|: s20 >= 0, s20 <= 2 * x_1 + 2, s21 >= 0, s21 <= 2 * x_2 + 2, s22 >= 0, s22 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 6456 + 4*s23 + s23*s24 + 2*s24 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s25 :|: s23 >= 0, s23 <= 2 * x_1 + 2, s24 >= 0, s24 <= 2 * x_2 + 2, s25 >= 0, s25 <= s23, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 9678 + 4*s27 + s27*s28 + s28 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 + 15952*x_3 + 22824*x_3^2 + 12848*x_3^3 + 2688*x_3^4 }-> s29 :|: s26 >= 0, s26 <= 2 * x_1 + 2, s27 >= 0, s27 <= 2 * x_2 + 2, s28 >= 0, s28 <= 2 * x_3 + 2, s29 >= 0, s29 <= s27, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 6450 + 64*s30 + 52*s30*s31 + 12*s30*s31^2 + 26*s30^2 + 12*s30^2*s31 + 4*s30^3 + 64*s31 + 26*s31^2 + 4*s31^3 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s32 :|: s30 >= 0, s30 <= 2 * x_1 + 2, s31 >= 0, s31 <= 2 * x_2 + 2, s32 >= 0, s32 <= s30 + s31, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 9686 + 180*s34 + 58*s34*s35 + 24*s34*s35^2 + 80*s34^2 + 24*s34^2*s35 + 16*s34^3 + 180*s35 + 80*s35^2 + 16*s35^3 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 + 15952*x_3 + 22824*x_3^2 + 12848*x_3^3 + 2688*x_3^4 }-> s36 :|: s33 >= 0, s33 <= 2 * x_1 + 2, s34 >= 0, s34 <= 2 * x_2 + 2, s35 >= 0, s35 <= 2 * x_3 + 2, s36 >= 0, s36 <= s34 + s35, 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) -{ -64 + -1904*z + 408*z^2 + 2096*z^3 + 2688*z^4 }-> 1 + s19 :|: s19 >= 0, s19 <= 2 * (z - 1) + 2, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gcd(z, z') -{ 6450 + 64*s48 + 52*s48*s49 + 12*s48*s49^2 + 26*s48^2 + 12*s48^2*s49 + 4*s48^3 + 64*s49 + 26*s49^2 + 4*s49^3 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s50 :|: s48 >= 0, s48 <= 2 * z + 2, s49 >= 0, s49 <= 2 * z' + 2, s50 >= 0, s50 <= s48 + s49, z >= 0, z' >= 0 encode_gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 9686 + 180*s52 + 58*s52*s53 + 24*s52*s53^2 + 80*s52^2 + 24*s52^2*s53 + 16*s52^3 + 180*s53 + 80*s53^2 + 16*s53^3 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 + 15952*z'' + 22824*z''^2 + 12848*z''^3 + 2688*z''^4 }-> s54 :|: s51 >= 0, s51 <= 2 * z + 2, s52 >= 0, s52 <= 2 * z' + 2, s53 >= 0, s53 <= 2 * z'' + 2, s54 >= 0, s54 <= s52 + s53, z >= 0, z'' >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_if_minus(z, z', z'') -{ 9678 + 4*s45 + s45*s46 + s46 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 + 15952*z'' + 22824*z''^2 + 12848*z''^3 + 2688*z''^4 }-> s47 :|: s44 >= 0, s44 <= 2 * z + 2, s45 >= 0, s45 <= 2 * z' + 2, s46 >= 0, s46 <= 2 * z'' + 2, s47 >= 0, s47 <= s45, z >= 0, z'' >= 0, z' >= 0 encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_le(z, z') -{ 6450 + s38 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s39 :|: s37 >= 0, s37 <= 2 * z + 2, s38 >= 0, s38 <= 2 * z' + 2, s39 >= 0, s39 <= 2, z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 6456 + 4*s41 + s41*s42 + 2*s42 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s43 :|: s41 >= 0, s41 <= 2 * z + 2, s42 >= 0, s42 <= 2 * z' + 2, s43 >= 0, s43 <= s41, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 3224 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 }-> 1 + s40 :|: s40 >= 0, s40 <= 2 * z + 2, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gcd(z, z') -{ 292 + 262*z' + 104*z'^2 + 16*z'^3 }-> s10 :|: s10 >= 0, s10 <= 1 + 0 + (1 + (1 + (z' - 2))), z = 1 + 0, z' - 2 >= 0 gcd(z, z') -{ 16 + 181*z + 58*z*z' + 24*z*z'^2 + 80*z^2 + 24*z^2*z' + 16*z^3 + 180*z' + 80*z'^2 + 16*z'^3 }-> s11 :|: s11 >= 0, s11 <= 1 + (1 + (z - 2)) + (1 + (1 + (z' - 2))), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0 gcd(z, z') -{ 15 + 180*z + 58*z*z' + 24*z*z'^2 + 80*z^2 + 24*z^2*z' + 16*z^3 + 180*z' + 80*z'^2 + 16*z'^3 }-> s12 :|: s12 >= 0, s12 <= 1 + (z - 1) + (1 + (z' - 1)), z - 1 >= 0, z' - 1 >= 0 gcd(z, z') -{ 292 + 262*z + 104*z^2 + 16*z^3 }-> s9 :|: s9 >= 0, s9 <= 1 + (z - 1) + (1 + 0), z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0 if_gcd(z, z', z'') -{ 4 + 64*z'' + 26*z''^2 + 4*z''^3 }-> s13 :|: s13 >= 0, s13 <= 0 + (1 + (z'' - 1)), z = 2, z'' - 1 >= 0, z' = 1 + 0 if_gcd(z, z', z'') -{ 7 + 64*s7 + 52*s7*z'' + 12*s7*z''^2 + 26*s7^2 + 12*s7^2*z'' + 4*s7^3 + 3*z' + z'*z'' + 65*z'' + 26*z''^2 + 4*z''^3 }-> s14 :|: s14 >= 0, s14 <= s7 + (1 + (z'' - 1)), s7 >= 0, s7 <= 1 + (z' - 2), s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 3 + 64*z'' + 26*z''^2 + 4*z''^3 }-> s15 :|: s15 >= 0, s15 <= 0 + (1 + (z'' - 1)), z = 2, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 4 + 64*z' + 26*z'^2 + 4*z'^3 }-> s16 :|: s16 >= 0, s16 <= 0 + (1 + (z' - 1)), z = 1, z' - 1 >= 0, z'' = 1 + 0 if_gcd(z, z', z'') -{ 7 + 64*s8 + 52*s8*z' + 12*s8*z'^2 + 26*s8^2 + 12*s8^2*z' + 4*s8^3 + 65*z' + z'*z'' + 26*z'^2 + 4*z'^3 + 3*z'' }-> s17 :|: s17 >= 0, s17 <= s8 + (1 + (z' - 1)), s8 >= 0, s8 <= 1 + (z'' - 2), s2 >= 0, s2 <= 2, z = 1, z' - 1 >= 0, z'' - 2 >= 0 if_gcd(z, z', z'') -{ 3 + 64*z' + 26*z'^2 + 4*z'^3 }-> s18 :|: s18 >= 0, s18 <= 0 + (1 + (z' - 1)), z = 1, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0 if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s6 :|: s6 >= 0, s6 <= z' - 1, z = 1, z' - 1 >= 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') -{ 8 + 4*z }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), z - 1 >= 0, z' = 0 minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s5 :|: s5 >= 0, s5 <= 1 + (z - 1), z - 1 >= 0, z' >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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] minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z] if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z'] gcd: runtime: O(n^3) [2 + 64*z + 52*z*z' + 12*z*z'^2 + 26*z^2 + 12*z^2*z' + 4*z^3 + 64*z' + 26*z'^2 + 4*z'^3], size: O(n^1) [z + z'] if_gcd: runtime: O(n^3) [14 + 180*z' + 58*z'*z'' + 24*z'*z''^2 + 80*z'^2 + 24*z'^2*z'' + 16*z'^3 + 180*z'' + 80*z''^2 + 16*z''^3], size: O(n^1) [z' + z''] encArg: runtime: O(n^4) [3224 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4], size: O(n^1) [2 + 2*z] encode_if_gcd: runtime: O(n^4) [11918 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 18144*z' + 1000*z'*z'' + 192*z'*z''^2 + 23720*z'^2 + 192*z'^2*z'' + 12976*z'^3 + 2688*z'^4 + 18144*z'' + 23720*z''^2 + 12976*z''^3 + 2688*z''^4], size: O(n^1) [4 + 2*z' + 2*z''] encode_gcd: runtime: O(n^4) [7378 + 16880*z + 592*z*z' + 96*z*z'^2 + 23120*z^2 + 96*z^2*z' + 12880*z^3 + 2688*z^4 + 16880*z' + 23120*z'^2 + 12880*z'^3 + 2688*z'^4], size: O(n^1) [4 + 2*z + 2*z'] encode_if_minus: runtime: O(n^4) [9692 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15964*z' + 4*z'*z'' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 + 15958*z'' + 22824*z''^2 + 12848*z''^3 + 2688*z''^4], size: O(n^1) [2 + 2*z'] encode_minus: runtime: O(n^4) [6472 + 15964*z + 4*z*z' + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15960*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4], size: O(n^1) [2 + 2*z] encode_le: runtime: O(n^4) [6452 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15954*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4], size: O(1) [2] encode_s: runtime: ?, size: O(n^1) [3 + 2*z] ---------------------------------------- (95) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: encode_s after applying outer abstraction to obtain an ITS, resulting in: O(n^4) with polynomial bound: 3224 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 ---------------------------------------- (96) Obligation: Complexity RNTS consisting of the following rules: encArg(z) -{ 6450 + s21 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s22 :|: s20 >= 0, s20 <= 2 * x_1 + 2, s21 >= 0, s21 <= 2 * x_2 + 2, s22 >= 0, s22 <= 2, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 6456 + 4*s23 + s23*s24 + 2*s24 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s25 :|: s23 >= 0, s23 <= 2 * x_1 + 2, s24 >= 0, s24 <= 2 * x_2 + 2, s25 >= 0, s25 <= s23, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 9678 + 4*s27 + s27*s28 + s28 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 + 15952*x_3 + 22824*x_3^2 + 12848*x_3^3 + 2688*x_3^4 }-> s29 :|: s26 >= 0, s26 <= 2 * x_1 + 2, s27 >= 0, s27 <= 2 * x_2 + 2, s28 >= 0, s28 <= 2 * x_3 + 2, s29 >= 0, s29 <= s27, x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z) -{ 6450 + 64*s30 + 52*s30*s31 + 12*s30*s31^2 + 26*s30^2 + 12*s30^2*s31 + 4*s30^3 + 64*s31 + 26*s31^2 + 4*s31^3 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 }-> s32 :|: s30 >= 0, s30 <= 2 * x_1 + 2, s31 >= 0, s31 <= 2 * x_2 + 2, s32 >= 0, s32 <= s30 + s31, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0 encArg(z) -{ 9686 + 180*s34 + 58*s34*s35 + 24*s34*s35^2 + 80*s34^2 + 24*s34^2*s35 + 16*s34^3 + 180*s35 + 80*s35^2 + 16*s35^3 + 15952*x_1 + 22824*x_1^2 + 12848*x_1^3 + 2688*x_1^4 + 15952*x_2 + 22824*x_2^2 + 12848*x_2^3 + 2688*x_2^4 + 15952*x_3 + 22824*x_3^2 + 12848*x_3^3 + 2688*x_3^4 }-> s36 :|: s33 >= 0, s33 <= 2 * x_1 + 2, s34 >= 0, s34 <= 2 * x_2 + 2, s35 >= 0, s35 <= 2 * x_3 + 2, s36 >= 0, s36 <= s34 + s35, 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) -{ -64 + -1904*z + 408*z^2 + 2096*z^3 + 2688*z^4 }-> 1 + s19 :|: s19 >= 0, s19 <= 2 * (z - 1) + 2, z - 1 >= 0 encode_0 -{ 0 }-> 0 :|: encode_false -{ 0 }-> 1 :|: encode_false -{ 0 }-> 0 :|: encode_gcd(z, z') -{ 6450 + 64*s48 + 52*s48*s49 + 12*s48*s49^2 + 26*s48^2 + 12*s48^2*s49 + 4*s48^3 + 64*s49 + 26*s49^2 + 4*s49^3 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s50 :|: s48 >= 0, s48 <= 2 * z + 2, s49 >= 0, s49 <= 2 * z' + 2, s50 >= 0, s50 <= s48 + s49, z >= 0, z' >= 0 encode_gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 9686 + 180*s52 + 58*s52*s53 + 24*s52*s53^2 + 80*s52^2 + 24*s52^2*s53 + 16*s52^3 + 180*s53 + 80*s53^2 + 16*s53^3 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 + 15952*z'' + 22824*z''^2 + 12848*z''^3 + 2688*z''^4 }-> s54 :|: s51 >= 0, s51 <= 2 * z + 2, s52 >= 0, s52 <= 2 * z' + 2, s53 >= 0, s53 <= 2 * z'' + 2, s54 >= 0, s54 <= s52 + s53, z >= 0, z'' >= 0, z' >= 0 encode_if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_if_minus(z, z', z'') -{ 9678 + 4*s45 + s45*s46 + s46 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 + 15952*z'' + 22824*z''^2 + 12848*z''^3 + 2688*z''^4 }-> s47 :|: s44 >= 0, s44 <= 2 * z + 2, s45 >= 0, s45 <= 2 * z' + 2, s46 >= 0, s46 <= 2 * z'' + 2, s47 >= 0, s47 <= s45, z >= 0, z'' >= 0, z' >= 0 encode_if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 encode_le(z, z') -{ 6450 + s38 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s39 :|: s37 >= 0, s37 <= 2 * z + 2, s38 >= 0, s38 <= 2 * z' + 2, s39 >= 0, s39 <= 2, z >= 0, z' >= 0 encode_le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_minus(z, z') -{ 6456 + 4*s41 + s41*s42 + 2*s42 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15952*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 }-> s43 :|: s41 >= 0, s41 <= 2 * z + 2, s42 >= 0, s42 <= 2 * z' + 2, s43 >= 0, s43 <= s41, z >= 0, z' >= 0 encode_minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 encode_s(z) -{ 0 }-> 0 :|: z >= 0 encode_s(z) -{ 3224 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 }-> 1 + s40 :|: s40 >= 0, s40 <= 2 * z + 2, z >= 0 encode_true -{ 0 }-> 2 :|: encode_true -{ 0 }-> 0 :|: gcd(z, z') -{ 292 + 262*z' + 104*z'^2 + 16*z'^3 }-> s10 :|: s10 >= 0, s10 <= 1 + 0 + (1 + (1 + (z' - 2))), z = 1 + 0, z' - 2 >= 0 gcd(z, z') -{ 16 + 181*z + 58*z*z' + 24*z*z'^2 + 80*z^2 + 24*z^2*z' + 16*z^3 + 180*z' + 80*z'^2 + 16*z'^3 }-> s11 :|: s11 >= 0, s11 <= 1 + (1 + (z - 2)) + (1 + (1 + (z' - 2))), s'' >= 0, s'' <= 2, z - 2 >= 0, z' - 2 >= 0 gcd(z, z') -{ 15 + 180*z + 58*z*z' + 24*z*z'^2 + 80*z^2 + 24*z^2*z' + 16*z^3 + 180*z' + 80*z'^2 + 16*z'^3 }-> s12 :|: s12 >= 0, s12 <= 1 + (z - 1) + (1 + (z' - 1)), z - 1 >= 0, z' - 1 >= 0 gcd(z, z') -{ 292 + 262*z + 104*z^2 + 16*z^3 }-> s9 :|: s9 >= 0, s9 <= 1 + (z - 1) + (1 + 0), z - 1 >= 0, z' = 1 + 0 gcd(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 gcd(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 gcd(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0 if_gcd(z, z', z'') -{ 4 + 64*z'' + 26*z''^2 + 4*z''^3 }-> s13 :|: s13 >= 0, s13 <= 0 + (1 + (z'' - 1)), z = 2, z'' - 1 >= 0, z' = 1 + 0 if_gcd(z, z', z'') -{ 7 + 64*s7 + 52*s7*z'' + 12*s7*z''^2 + 26*s7^2 + 12*s7^2*z'' + 4*s7^3 + 3*z' + z'*z'' + 65*z'' + 26*z''^2 + 4*z''^3 }-> s14 :|: s14 >= 0, s14 <= s7 + (1 + (z'' - 1)), s7 >= 0, s7 <= 1 + (z' - 2), s1 >= 0, s1 <= 2, z = 2, z' - 2 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 3 + 64*z'' + 26*z''^2 + 4*z''^3 }-> s15 :|: s15 >= 0, s15 <= 0 + (1 + (z'' - 1)), z = 2, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 4 + 64*z' + 26*z'^2 + 4*z'^3 }-> s16 :|: s16 >= 0, s16 <= 0 + (1 + (z' - 1)), z = 1, z' - 1 >= 0, z'' = 1 + 0 if_gcd(z, z', z'') -{ 7 + 64*s8 + 52*s8*z' + 12*s8*z'^2 + 26*s8^2 + 12*s8^2*z' + 4*s8^3 + 65*z' + z'*z'' + 26*z'^2 + 4*z'^3 + 3*z'' }-> s17 :|: s17 >= 0, s17 <= s8 + (1 + (z' - 1)), s8 >= 0, s8 <= 1 + (z'' - 2), s2 >= 0, s2 <= 2, z = 1, z' - 1 >= 0, z'' - 2 >= 0 if_gcd(z, z', z'') -{ 3 + 64*z' + 26*z'^2 + 4*z'^3 }-> s18 :|: s18 >= 0, s18 <= 0 + (1 + (z' - 1)), z = 1, z' - 1 >= 0, z'' - 1 >= 0 if_gcd(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' - 1 >= 0, z'' >= 0 if_minus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 if_minus(z, z', z'') -{ 5 + 4*z' + z'*z'' + z'' }-> 1 + s6 :|: s6 >= 0, s6 <= z' - 1, z = 1, z' - 1 >= 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') -{ 8 + 4*z }-> s3 :|: s3 >= 0, s3 <= 1 + (z - 1), z - 1 >= 0, z' = 0 minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s4 :|: s4 >= 0, s4 <= 1 + (z - 1), s' >= 0, s' <= 2, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 7 + 4*z + z*z' + z' }-> s5 :|: s5 >= 0, s5 <= 1 + (z - 1), z - 1 >= 0, z' >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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] minus: runtime: O(n^2) [8 + 4*z + z*z' + 2*z'], size: O(n^1) [z] if_minus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z'] gcd: runtime: O(n^3) [2 + 64*z + 52*z*z' + 12*z*z'^2 + 26*z^2 + 12*z^2*z' + 4*z^3 + 64*z' + 26*z'^2 + 4*z'^3], size: O(n^1) [z + z'] if_gcd: runtime: O(n^3) [14 + 180*z' + 58*z'*z'' + 24*z'*z''^2 + 80*z'^2 + 24*z'^2*z'' + 16*z'^3 + 180*z'' + 80*z''^2 + 16*z''^3], size: O(n^1) [z' + z''] encArg: runtime: O(n^4) [3224 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4], size: O(n^1) [2 + 2*z] encode_if_gcd: runtime: O(n^4) [11918 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 18144*z' + 1000*z'*z'' + 192*z'*z''^2 + 23720*z'^2 + 192*z'^2*z'' + 12976*z'^3 + 2688*z'^4 + 18144*z'' + 23720*z''^2 + 12976*z''^3 + 2688*z''^4], size: O(n^1) [4 + 2*z' + 2*z''] encode_gcd: runtime: O(n^4) [7378 + 16880*z + 592*z*z' + 96*z*z'^2 + 23120*z^2 + 96*z^2*z' + 12880*z^3 + 2688*z^4 + 16880*z' + 23120*z'^2 + 12880*z'^3 + 2688*z'^4], size: O(n^1) [4 + 2*z + 2*z'] encode_if_minus: runtime: O(n^4) [9692 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15964*z' + 4*z'*z'' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4 + 15958*z'' + 22824*z''^2 + 12848*z''^3 + 2688*z''^4], size: O(n^1) [2 + 2*z'] encode_minus: runtime: O(n^4) [6472 + 15964*z + 4*z*z' + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15960*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4], size: O(n^1) [2 + 2*z] encode_le: runtime: O(n^4) [6452 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4 + 15954*z' + 22824*z'^2 + 12848*z'^3 + 2688*z'^4], size: O(1) [2] encode_s: runtime: O(n^4) [3224 + 15952*z + 22824*z^2 + 12848*z^3 + 2688*z^4], size: O(n^1) [3 + 2*z] ---------------------------------------- (97) FinalProof (FINISHED) Computed overall runtime complexity ---------------------------------------- (98) BOUNDS(1, n^4) ---------------------------------------- (99) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (100) 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: le(0', y) -> true le(s(x), 0') -> false le(s(x), s(y)) -> le(x, y) minus(0', y) -> 0' minus(s(x), y) -> if_minus(le(s(x), y), s(x), y) if_minus(true, s(x), y) -> 0' if_minus(false, s(x), y) -> s(minus(x, y)) gcd(0', y) -> y gcd(s(x), 0') -> s(x) gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y)) if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x)) The (relative) TRS S consists of the following rules: encArg(0') -> 0' encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(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_minus(x_1, x_2, x_3)) -> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_gcd(x_1, x_2)) -> gcd(encArg(x_1), encArg(x_2)) encArg(cons_if_gcd(x_1, x_2, x_3)) -> if_gcd(encArg(x_1), encArg(x_2), encArg(x_3)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_if_minus(x_1, x_2, x_3) -> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) encode_gcd(x_1, x_2) -> gcd(encArg(x_1), encArg(x_2)) encode_if_gcd(x_1, x_2, x_3) -> if_gcd(encArg(x_1), encArg(x_2), encArg(x_3)) Rewrite Strategy: INNERMOST ---------------------------------------- (101) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (102) Obligation: Innermost TRS: Rules: le(0', y) -> true le(s(x), 0') -> false le(s(x), s(y)) -> le(x, y) minus(0', y) -> 0' minus(s(x), y) -> if_minus(le(s(x), y), s(x), y) if_minus(true, s(x), y) -> 0' if_minus(false, s(x), y) -> s(minus(x, y)) gcd(0', y) -> y gcd(s(x), 0') -> s(x) gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y)) if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x)) encArg(0') -> 0' encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false 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_minus(x_1, x_2, x_3)) -> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_gcd(x_1, x_2)) -> gcd(encArg(x_1), encArg(x_2)) encArg(cons_if_gcd(x_1, x_2, x_3)) -> if_gcd(encArg(x_1), encArg(x_2), encArg(x_3)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_if_minus(x_1, x_2, x_3) -> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) encode_gcd(x_1, x_2) -> gcd(encArg(x_1), encArg(x_2)) encode_if_gcd(x_1, x_2, x_3) -> if_gcd(encArg(x_1), encArg(x_2), encArg(x_3)) Types: le :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd 0' :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd true :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd s :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd false :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd minus :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd if_minus :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd gcd :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd if_gcd :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encArg :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd cons_le :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd cons_minus :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd cons_if_minus :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd cons_gcd :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd cons_if_gcd :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encode_le :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encode_0 :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encode_true :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encode_s :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encode_false :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encode_minus :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encode_if_minus :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encode_gcd :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encode_if_gcd :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd hole_0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd1_4 :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd gen_0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd2_4 :: Nat -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd ---------------------------------------- (103) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: le, minus, if_minus, gcd, if_gcd, encArg They will be analysed ascendingly in the following order: le < minus le < gcd le < encArg minus = if_minus minus < if_gcd minus < encArg if_minus < encArg gcd = if_gcd gcd < encArg if_gcd < encArg ---------------------------------------- (104) Obligation: Innermost TRS: Rules: le(0', y) -> true le(s(x), 0') -> false le(s(x), s(y)) -> le(x, y) minus(0', y) -> 0' minus(s(x), y) -> if_minus(le(s(x), y), s(x), y) if_minus(true, s(x), y) -> 0' if_minus(false, s(x), y) -> s(minus(x, y)) gcd(0', y) -> y gcd(s(x), 0') -> s(x) gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y)) if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x)) encArg(0') -> 0' encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false 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_minus(x_1, x_2, x_3)) -> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_gcd(x_1, x_2)) -> gcd(encArg(x_1), encArg(x_2)) encArg(cons_if_gcd(x_1, x_2, x_3)) -> if_gcd(encArg(x_1), encArg(x_2), encArg(x_3)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_if_minus(x_1, x_2, x_3) -> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) encode_gcd(x_1, x_2) -> gcd(encArg(x_1), encArg(x_2)) encode_if_gcd(x_1, x_2, x_3) -> if_gcd(encArg(x_1), encArg(x_2), encArg(x_3)) Types: le :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd 0' :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd true :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd s :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd false :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd minus :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd if_minus :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd gcd :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd if_gcd :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encArg :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd cons_le :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd cons_minus :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd cons_if_minus :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd cons_gcd :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd cons_if_gcd :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encode_le :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encode_0 :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encode_true :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encode_s :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encode_false :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encode_minus :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encode_if_minus :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encode_gcd :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encode_if_gcd :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd hole_0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd1_4 :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd gen_0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd2_4 :: Nat -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd Generator Equations: gen_0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd2_4(0) <=> 0' gen_0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd2_4(+(x, 1)) <=> s(gen_0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd2_4(x)) The following defined symbols remain to be analysed: le, minus, if_minus, gcd, if_gcd, encArg They will be analysed ascendingly in the following order: le < minus le < gcd le < encArg minus = if_minus minus < if_gcd minus < encArg if_minus < encArg gcd = if_gcd gcd < encArg if_gcd < encArg ---------------------------------------- (105) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: le(gen_0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd2_4(n4_4), gen_0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd2_4(n4_4)) -> true, rt in Omega(1 + n4_4) Induction Base: le(gen_0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd2_4(0), gen_0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd2_4(0)) ->_R^Omega(1) true Induction Step: le(gen_0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd2_4(+(n4_4, 1)), gen_0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd2_4(+(n4_4, 1))) ->_R^Omega(1) le(gen_0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd2_4(n4_4), gen_0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd2_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). ---------------------------------------- (106) Complex Obligation (BEST) ---------------------------------------- (107) Obligation: Proved the lower bound n^1 for the following obligation: Innermost TRS: Rules: le(0', y) -> true le(s(x), 0') -> false le(s(x), s(y)) -> le(x, y) minus(0', y) -> 0' minus(s(x), y) -> if_minus(le(s(x), y), s(x), y) if_minus(true, s(x), y) -> 0' if_minus(false, s(x), y) -> s(minus(x, y)) gcd(0', y) -> y gcd(s(x), 0') -> s(x) gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y)) if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x)) encArg(0') -> 0' encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false 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_minus(x_1, x_2, x_3)) -> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_gcd(x_1, x_2)) -> gcd(encArg(x_1), encArg(x_2)) encArg(cons_if_gcd(x_1, x_2, x_3)) -> if_gcd(encArg(x_1), encArg(x_2), encArg(x_3)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_if_minus(x_1, x_2, x_3) -> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) encode_gcd(x_1, x_2) -> gcd(encArg(x_1), encArg(x_2)) encode_if_gcd(x_1, x_2, x_3) -> if_gcd(encArg(x_1), encArg(x_2), encArg(x_3)) Types: le :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd 0' :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd true :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd s :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd false :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd minus :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd if_minus :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd gcd :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd if_gcd :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encArg :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd cons_le :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd cons_minus :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd cons_if_minus :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd cons_gcd :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd cons_if_gcd :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encode_le :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encode_0 :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encode_true :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encode_s :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encode_false :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encode_minus :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encode_if_minus :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encode_gcd :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encode_if_gcd :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd hole_0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd1_4 :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd gen_0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd2_4 :: Nat -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd Generator Equations: gen_0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd2_4(0) <=> 0' gen_0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd2_4(+(x, 1)) <=> s(gen_0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd2_4(x)) The following defined symbols remain to be analysed: le, minus, if_minus, gcd, if_gcd, encArg They will be analysed ascendingly in the following order: le < minus le < gcd le < encArg minus = if_minus minus < if_gcd minus < encArg if_minus < encArg gcd = if_gcd gcd < encArg if_gcd < encArg ---------------------------------------- (108) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (109) BOUNDS(n^1, INF) ---------------------------------------- (110) Obligation: Innermost TRS: Rules: le(0', y) -> true le(s(x), 0') -> false le(s(x), s(y)) -> le(x, y) minus(0', y) -> 0' minus(s(x), y) -> if_minus(le(s(x), y), s(x), y) if_minus(true, s(x), y) -> 0' if_minus(false, s(x), y) -> s(minus(x, y)) gcd(0', y) -> y gcd(s(x), 0') -> s(x) gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y)) if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x)) encArg(0') -> 0' encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false 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_minus(x_1, x_2, x_3)) -> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_gcd(x_1, x_2)) -> gcd(encArg(x_1), encArg(x_2)) encArg(cons_if_gcd(x_1, x_2, x_3)) -> if_gcd(encArg(x_1), encArg(x_2), encArg(x_3)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_if_minus(x_1, x_2, x_3) -> if_minus(encArg(x_1), encArg(x_2), encArg(x_3)) encode_gcd(x_1, x_2) -> gcd(encArg(x_1), encArg(x_2)) encode_if_gcd(x_1, x_2, x_3) -> if_gcd(encArg(x_1), encArg(x_2), encArg(x_3)) Types: le :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd 0' :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd true :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd s :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd false :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd minus :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd if_minus :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd gcd :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd if_gcd :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encArg :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd cons_le :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd cons_minus :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd cons_if_minus :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd cons_gcd :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd cons_if_gcd :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encode_le :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encode_0 :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encode_true :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encode_s :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encode_false :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encode_minus :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encode_if_minus :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encode_gcd :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd encode_if_gcd :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd hole_0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd1_4 :: 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd gen_0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd2_4 :: Nat -> 0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd Lemmas: le(gen_0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd2_4(n4_4), gen_0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd2_4(n4_4)) -> true, rt in Omega(1 + n4_4) Generator Equations: gen_0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd2_4(0) <=> 0' gen_0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd2_4(+(x, 1)) <=> s(gen_0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd2_4(x)) The following defined symbols remain to be analysed: if_minus, minus, gcd, if_gcd, encArg They will be analysed ascendingly in the following order: minus = if_minus minus < if_gcd minus < encArg if_minus < encArg gcd = if_gcd gcd < encArg if_gcd < encArg ---------------------------------------- (111) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd2_4(n830_4)) -> gen_0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd2_4(n830_4), rt in Omega(0) Induction Base: encArg(gen_0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd2_4(0)) ->_R^Omega(0) 0' Induction Step: encArg(gen_0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd2_4(+(n830_4, 1))) ->_R^Omega(0) s(encArg(gen_0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd2_4(n830_4))) ->_IH s(gen_0':true:s:false:cons_le:cons_minus:cons_if_minus:cons_gcd:cons_if_gcd2_4(c831_4)) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (112) BOUNDS(1, INF)