WORST_CASE(?, O(n^3)) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, n^3). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 284 ms] (4) CpxRelTRS (5) NonCtorToCtorProof [UPPER BOUND(ID), 0 ms] (6) CpxRelTRS (7) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxRelTRS (9) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CpxWeightedTrs (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxTypedWeightedTrs (13) CompletionProof [UPPER BOUND(ID), 0 ms] (14) CpxTypedWeightedCompleteTrs (15) NarrowingProof [BOTH BOUNDS(ID, ID), 4046 ms] (16) CpxTypedWeightedCompleteTrs (17) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (18) CpxRNTS (19) InliningProof [UPPER BOUND(ID), 403 ms] (20) CpxRNTS (21) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (22) CpxRNTS (23) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 353 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 117 ms] (30) CpxRNTS (31) ResultPropagationProof [UPPER BOUND(ID), 2 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 66 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 12 ms] (36) CpxRNTS (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 69 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] (42) CpxRNTS (43) ResultPropagationProof [UPPER BOUND(ID), 3 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 260 ms] (46) CpxRNTS (47) IntTrsBoundProof [UPPER BOUND(ID), 52 ms] (48) CpxRNTS (49) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (50) CpxRNTS (51) IntTrsBoundProof [UPPER BOUND(ID), 119 ms] (52) CpxRNTS (53) IntTrsBoundProof [UPPER BOUND(ID), 55 ms] (54) CpxRNTS (55) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (56) CpxRNTS (57) IntTrsBoundProof [UPPER BOUND(ID), 916 ms] (58) CpxRNTS (59) IntTrsBoundProof [UPPER BOUND(ID), 448 ms] (60) CpxRNTS (61) ResultPropagationProof [UPPER BOUND(ID), 1 ms] (62) CpxRNTS (63) IntTrsBoundProof [UPPER BOUND(ID), 2804 ms] (64) CpxRNTS (65) IntTrsBoundProof [UPPER BOUND(ID), 1226 ms] (66) CpxRNTS (67) ResultPropagationProof [UPPER BOUND(ID), 10 ms] (68) CpxRNTS (69) IntTrsBoundProof [UPPER BOUND(ID), 407 ms] (70) CpxRNTS (71) IntTrsBoundProof [UPPER BOUND(ID), 119 ms] (72) CpxRNTS (73) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (74) CpxRNTS (75) IntTrsBoundProof [UPPER BOUND(ID), 127 ms] (76) CpxRNTS (77) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] (78) CpxRNTS (79) ResultPropagationProof [UPPER BOUND(ID), 3 ms] (80) CpxRNTS (81) IntTrsBoundProof [UPPER BOUND(ID), 249 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), 200 ms] (88) CpxRNTS (89) IntTrsBoundProof [UPPER BOUND(ID), 3 ms] (90) CpxRNTS (91) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (92) CpxRNTS (93) IntTrsBoundProof [UPPER BOUND(ID), 318 ms] (94) CpxRNTS (95) IntTrsBoundProof [UPPER BOUND(ID), 289 ms] (96) CpxRNTS (97) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (98) CpxRNTS (99) IntTrsBoundProof [UPPER BOUND(ID), 128 ms] (100) CpxRNTS (101) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] (102) CpxRNTS (103) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (104) CpxRNTS (105) IntTrsBoundProof [UPPER BOUND(ID), 208 ms] (106) CpxRNTS (107) IntTrsBoundProof [UPPER BOUND(ID), 2 ms] (108) CpxRNTS (109) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (110) CpxRNTS (111) IntTrsBoundProof [UPPER BOUND(ID), 626 ms] (112) CpxRNTS (113) IntTrsBoundProof [UPPER BOUND(ID), 158 ms] (114) CpxRNTS (115) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (116) CpxRNTS (117) IntTrsBoundProof [UPPER BOUND(ID), 55 ms] (118) CpxRNTS (119) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] (120) CpxRNTS (121) FinalProof [FINISHED, 0 ms] (122) BOUNDS(1, n^3) ---------------------------------------- (0) Obligation: The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(1, n^3). The TRS R consists of the following rules: f(j(x, y), y) -> g(f(x, k(y))) f(x, h1(y, z)) -> h2(0, x, h1(y, z)) g(h2(x, y, h1(z, u))) -> h2(s(x), y, h1(z, u)) h2(x, j(y, h1(z, u)), h1(z, u)) -> h2(s(x), y, h1(s(z), u)) i(f(x, h(y))) -> y i(h2(s(x), y, h1(x, z))) -> z k(h(x)) -> h1(0, x) k(h1(x, y)) -> h1(s(x), y) S is empty. Rewrite Strategy: FULL ---------------------------------------- (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(j(x_1, x_2)) -> j(encArg(x_1), encArg(x_2)) encArg(h1(x_1, x_2)) -> h1(encArg(x_1), encArg(x_2)) encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(h(x_1)) -> h(encArg(x_1)) encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(cons_g(x_1)) -> g(encArg(x_1)) encArg(cons_h2(x_1, x_2, x_3)) -> h2(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_i(x_1)) -> i(encArg(x_1)) encArg(cons_k(x_1)) -> k(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_j(x_1, x_2) -> j(encArg(x_1), encArg(x_2)) encode_g(x_1) -> g(encArg(x_1)) encode_k(x_1) -> k(encArg(x_1)) encode_h1(x_1, x_2) -> h1(encArg(x_1), encArg(x_2)) encode_h2(x_1, x_2, x_3) -> h2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_i(x_1) -> i(encArg(x_1)) encode_h(x_1) -> h(encArg(x_1)) ---------------------------------------- (2) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, n^3). The TRS R consists of the following rules: f(j(x, y), y) -> g(f(x, k(y))) f(x, h1(y, z)) -> h2(0, x, h1(y, z)) g(h2(x, y, h1(z, u))) -> h2(s(x), y, h1(z, u)) h2(x, j(y, h1(z, u)), h1(z, u)) -> h2(s(x), y, h1(s(z), u)) i(f(x, h(y))) -> y i(h2(s(x), y, h1(x, z))) -> z k(h(x)) -> h1(0, x) k(h1(x, y)) -> h1(s(x), y) The (relative) TRS S consists of the following rules: encArg(j(x_1, x_2)) -> j(encArg(x_1), encArg(x_2)) encArg(h1(x_1, x_2)) -> h1(encArg(x_1), encArg(x_2)) encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(h(x_1)) -> h(encArg(x_1)) encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(cons_g(x_1)) -> g(encArg(x_1)) encArg(cons_h2(x_1, x_2, x_3)) -> h2(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_i(x_1)) -> i(encArg(x_1)) encArg(cons_k(x_1)) -> k(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_j(x_1, x_2) -> j(encArg(x_1), encArg(x_2)) encode_g(x_1) -> g(encArg(x_1)) encode_k(x_1) -> k(encArg(x_1)) encode_h1(x_1, x_2) -> h1(encArg(x_1), encArg(x_2)) encode_h2(x_1, x_2, x_3) -> h2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_i(x_1) -> i(encArg(x_1)) encode_h(x_1) -> h(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, n^3). The TRS R consists of the following rules: f(j(x, y), y) -> g(f(x, k(y))) f(x, h1(y, z)) -> h2(0, x, h1(y, z)) g(h2(x, y, h1(z, u))) -> h2(s(x), y, h1(z, u)) h2(x, j(y, h1(z, u)), h1(z, u)) -> h2(s(x), y, h1(s(z), u)) i(f(x, h(y))) -> y i(h2(s(x), y, h1(x, z))) -> z k(h(x)) -> h1(0, x) k(h1(x, y)) -> h1(s(x), y) The (relative) TRS S consists of the following rules: encArg(j(x_1, x_2)) -> j(encArg(x_1), encArg(x_2)) encArg(h1(x_1, x_2)) -> h1(encArg(x_1), encArg(x_2)) encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(h(x_1)) -> h(encArg(x_1)) encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(cons_g(x_1)) -> g(encArg(x_1)) encArg(cons_h2(x_1, x_2, x_3)) -> h2(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_i(x_1)) -> i(encArg(x_1)) encArg(cons_k(x_1)) -> k(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_j(x_1, x_2) -> j(encArg(x_1), encArg(x_2)) encode_g(x_1) -> g(encArg(x_1)) encode_k(x_1) -> k(encArg(x_1)) encode_h1(x_1, x_2) -> h1(encArg(x_1), encArg(x_2)) encode_h2(x_1, x_2, x_3) -> h2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_i(x_1) -> i(encArg(x_1)) encode_h(x_1) -> h(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (5) NonCtorToCtorProof (UPPER BOUND(ID)) transformed non-ctor to ctor-system ---------------------------------------- (6) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, n^3). The TRS R consists of the following rules: f(j(x, y), y) -> g(f(x, k(y))) f(x, h1(y, z)) -> h2(0, x, h1(y, z)) h2(x, j(y, h1(z, u)), h1(z, u)) -> h2(s(x), y, h1(s(z), u)) k(h(x)) -> h1(0, x) k(h1(x, y)) -> h1(s(x), y) i(c_h2(s(x), y, h1(x, z))) -> z g(c_h2(x, y, h1(z, u))) -> h2(s(x), y, h1(z, u)) i(c_f(x, h(y))) -> y The (relative) TRS S consists of the following rules: encArg(j(x_1, x_2)) -> j(encArg(x_1), encArg(x_2)) encArg(h1(x_1, x_2)) -> h1(encArg(x_1), encArg(x_2)) encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(h(x_1)) -> h(encArg(x_1)) encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(cons_g(x_1)) -> g(encArg(x_1)) encArg(cons_h2(x_1, x_2, x_3)) -> h2(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_i(x_1)) -> i(encArg(x_1)) encArg(cons_k(x_1)) -> k(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_j(x_1, x_2) -> j(encArg(x_1), encArg(x_2)) encode_g(x_1) -> g(encArg(x_1)) encode_k(x_1) -> k(encArg(x_1)) encode_h1(x_1, x_2) -> h1(encArg(x_1), encArg(x_2)) encode_h2(x_1, x_2, x_3) -> h2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_i(x_1) -> i(encArg(x_1)) encode_h(x_1) -> h(encArg(x_1)) f(x0, x1) -> c_f(x0, x1) h2(x0, x1, x2) -> c_h2(x0, x1, x2) Rewrite Strategy: FULL ---------------------------------------- (7) RcToIrcProof (BOTH BOUNDS(ID, ID)) Converted rc-obligation to irc-obligation. As the TRS is a non-duplicating overlay system, we have rc = irc. ---------------------------------------- (8) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^3). The TRS R consists of the following rules: f(j(x, y), y) -> g(f(x, k(y))) f(x, h1(y, z)) -> h2(0, x, h1(y, z)) h2(x, j(y, h1(z, u)), h1(z, u)) -> h2(s(x), y, h1(s(z), u)) k(h(x)) -> h1(0, x) k(h1(x, y)) -> h1(s(x), y) i(c_h2(s(x), y, h1(x, z))) -> z g(c_h2(x, y, h1(z, u))) -> h2(s(x), y, h1(z, u)) i(c_f(x, h(y))) -> y The (relative) TRS S consists of the following rules: encArg(j(x_1, x_2)) -> j(encArg(x_1), encArg(x_2)) encArg(h1(x_1, x_2)) -> h1(encArg(x_1), encArg(x_2)) encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(h(x_1)) -> h(encArg(x_1)) encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(cons_g(x_1)) -> g(encArg(x_1)) encArg(cons_h2(x_1, x_2, x_3)) -> h2(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_i(x_1)) -> i(encArg(x_1)) encArg(cons_k(x_1)) -> k(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_j(x_1, x_2) -> j(encArg(x_1), encArg(x_2)) encode_g(x_1) -> g(encArg(x_1)) encode_k(x_1) -> k(encArg(x_1)) encode_h1(x_1, x_2) -> h1(encArg(x_1), encArg(x_2)) encode_h2(x_1, x_2, x_3) -> h2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_i(x_1) -> i(encArg(x_1)) encode_h(x_1) -> h(encArg(x_1)) f(x0, x1) -> c_f(x0, x1) h2(x0, x1, x2) -> c_h2(x0, x1, x2) Rewrite Strategy: INNERMOST ---------------------------------------- (9) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (10) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^3). The TRS R consists of the following rules: f(j(x, y), y) -> g(f(x, k(y))) [1] f(x, h1(y, z)) -> h2(0, x, h1(y, z)) [1] h2(x, j(y, h1(z, u)), h1(z, u)) -> h2(s(x), y, h1(s(z), u)) [1] k(h(x)) -> h1(0, x) [1] k(h1(x, y)) -> h1(s(x), y) [1] i(c_h2(s(x), y, h1(x, z))) -> z [1] g(c_h2(x, y, h1(z, u))) -> h2(s(x), y, h1(z, u)) [1] i(c_f(x, h(y))) -> y [1] encArg(j(x_1, x_2)) -> j(encArg(x_1), encArg(x_2)) [0] encArg(h1(x_1, x_2)) -> h1(encArg(x_1), encArg(x_2)) [0] encArg(0) -> 0 [0] encArg(s(x_1)) -> s(encArg(x_1)) [0] encArg(h(x_1)) -> h(encArg(x_1)) [0] encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0] encArg(cons_g(x_1)) -> g(encArg(x_1)) [0] encArg(cons_h2(x_1, x_2, x_3)) -> h2(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encArg(cons_i(x_1)) -> i(encArg(x_1)) [0] encArg(cons_k(x_1)) -> k(encArg(x_1)) [0] encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0] encode_j(x_1, x_2) -> j(encArg(x_1), encArg(x_2)) [0] encode_g(x_1) -> g(encArg(x_1)) [0] encode_k(x_1) -> k(encArg(x_1)) [0] encode_h1(x_1, x_2) -> h1(encArg(x_1), encArg(x_2)) [0] encode_h2(x_1, x_2, x_3) -> h2(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encode_0 -> 0 [0] encode_s(x_1) -> s(encArg(x_1)) [0] encode_i(x_1) -> i(encArg(x_1)) [0] encode_h(x_1) -> h(encArg(x_1)) [0] f(x0, x1) -> c_f(x0, x1) [0] h2(x0, x1, x2) -> c_h2(x0, x1, x2) [0] Rewrite Strategy: INNERMOST ---------------------------------------- (11) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (12) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: f(j(x, y), y) -> g(f(x, k(y))) [1] f(x, h1(y, z)) -> h2(0, x, h1(y, z)) [1] h2(x, j(y, h1(z, u)), h1(z, u)) -> h2(s(x), y, h1(s(z), u)) [1] k(h(x)) -> h1(0, x) [1] k(h1(x, y)) -> h1(s(x), y) [1] i(c_h2(s(x), y, h1(x, z))) -> z [1] g(c_h2(x, y, h1(z, u))) -> h2(s(x), y, h1(z, u)) [1] i(c_f(x, h(y))) -> y [1] encArg(j(x_1, x_2)) -> j(encArg(x_1), encArg(x_2)) [0] encArg(h1(x_1, x_2)) -> h1(encArg(x_1), encArg(x_2)) [0] encArg(0) -> 0 [0] encArg(s(x_1)) -> s(encArg(x_1)) [0] encArg(h(x_1)) -> h(encArg(x_1)) [0] encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0] encArg(cons_g(x_1)) -> g(encArg(x_1)) [0] encArg(cons_h2(x_1, x_2, x_3)) -> h2(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encArg(cons_i(x_1)) -> i(encArg(x_1)) [0] encArg(cons_k(x_1)) -> k(encArg(x_1)) [0] encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0] encode_j(x_1, x_2) -> j(encArg(x_1), encArg(x_2)) [0] encode_g(x_1) -> g(encArg(x_1)) [0] encode_k(x_1) -> k(encArg(x_1)) [0] encode_h1(x_1, x_2) -> h1(encArg(x_1), encArg(x_2)) [0] encode_h2(x_1, x_2, x_3) -> h2(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encode_0 -> 0 [0] encode_s(x_1) -> s(encArg(x_1)) [0] encode_i(x_1) -> i(encArg(x_1)) [0] encode_h(x_1) -> h(encArg(x_1)) [0] f(x0, x1) -> c_f(x0, x1) [0] h2(x0, x1, x2) -> c_h2(x0, x1, x2) [0] The TRS has the following type information: f :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k j :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k g :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k k :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k h1 :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k h2 :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k 0 :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k s :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k h :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k i :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k c_h2 :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k c_f :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k encArg :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k cons_f :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k cons_g :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k cons_h2 :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k cons_i :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k cons_k :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k encode_f :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k encode_j :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k encode_g :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k encode_k :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k encode_h1 :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k encode_h2 :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k encode_0 :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k encode_s :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k encode_i :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k encode_h :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k Rewrite Strategy: INNERMOST ---------------------------------------- (13) 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: k_1 i_1 g_1 encArg_1 encode_f_2 encode_j_2 encode_g_1 encode_k_1 encode_h1_2 encode_h2_3 encode_0 encode_s_1 encode_i_1 encode_h_1 f_2 h2_3 Due to the following rules being added: encArg(v0) -> 0 [0] encode_f(v0, v1) -> 0 [0] encode_j(v0, v1) -> 0 [0] encode_g(v0) -> 0 [0] encode_k(v0) -> 0 [0] encode_h1(v0, v1) -> 0 [0] encode_h2(v0, v1, v2) -> 0 [0] encode_0 -> 0 [0] encode_s(v0) -> 0 [0] encode_i(v0) -> 0 [0] encode_h(v0) -> 0 [0] f(v0, v1) -> 0 [0] h2(v0, v1, v2) -> 0 [0] k(v0) -> 0 [0] i(v0) -> 0 [0] g(v0) -> 0 [0] And the following fresh constants: none ---------------------------------------- (14) 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: f(j(x, y), y) -> g(f(x, k(y))) [1] f(x, h1(y, z)) -> h2(0, x, h1(y, z)) [1] h2(x, j(y, h1(z, u)), h1(z, u)) -> h2(s(x), y, h1(s(z), u)) [1] k(h(x)) -> h1(0, x) [1] k(h1(x, y)) -> h1(s(x), y) [1] i(c_h2(s(x), y, h1(x, z))) -> z [1] g(c_h2(x, y, h1(z, u))) -> h2(s(x), y, h1(z, u)) [1] i(c_f(x, h(y))) -> y [1] encArg(j(x_1, x_2)) -> j(encArg(x_1), encArg(x_2)) [0] encArg(h1(x_1, x_2)) -> h1(encArg(x_1), encArg(x_2)) [0] encArg(0) -> 0 [0] encArg(s(x_1)) -> s(encArg(x_1)) [0] encArg(h(x_1)) -> h(encArg(x_1)) [0] encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0] encArg(cons_g(x_1)) -> g(encArg(x_1)) [0] encArg(cons_h2(x_1, x_2, x_3)) -> h2(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encArg(cons_i(x_1)) -> i(encArg(x_1)) [0] encArg(cons_k(x_1)) -> k(encArg(x_1)) [0] encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0] encode_j(x_1, x_2) -> j(encArg(x_1), encArg(x_2)) [0] encode_g(x_1) -> g(encArg(x_1)) [0] encode_k(x_1) -> k(encArg(x_1)) [0] encode_h1(x_1, x_2) -> h1(encArg(x_1), encArg(x_2)) [0] encode_h2(x_1, x_2, x_3) -> h2(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encode_0 -> 0 [0] encode_s(x_1) -> s(encArg(x_1)) [0] encode_i(x_1) -> i(encArg(x_1)) [0] encode_h(x_1) -> h(encArg(x_1)) [0] f(x0, x1) -> c_f(x0, x1) [0] h2(x0, x1, x2) -> c_h2(x0, x1, x2) [0] encArg(v0) -> 0 [0] encode_f(v0, v1) -> 0 [0] encode_j(v0, v1) -> 0 [0] encode_g(v0) -> 0 [0] encode_k(v0) -> 0 [0] encode_h1(v0, v1) -> 0 [0] encode_h2(v0, v1, v2) -> 0 [0] encode_0 -> 0 [0] encode_s(v0) -> 0 [0] encode_i(v0) -> 0 [0] encode_h(v0) -> 0 [0] f(v0, v1) -> 0 [0] h2(v0, v1, v2) -> 0 [0] k(v0) -> 0 [0] i(v0) -> 0 [0] g(v0) -> 0 [0] The TRS has the following type information: f :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k j :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k g :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k k :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k h1 :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k h2 :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k 0 :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k s :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k h :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k i :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k c_h2 :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k c_f :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k encArg :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k cons_f :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k cons_g :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k cons_h2 :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k cons_i :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k cons_k :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k encode_f :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k encode_j :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k encode_g :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k encode_k :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k encode_h1 :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k encode_h2 :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k encode_0 :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k encode_s :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k encode_i :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k encode_h :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k Rewrite Strategy: INNERMOST ---------------------------------------- (15) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (16) 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: f(j(x, h(x')), h(x')) -> g(f(x, h1(0, x'))) [2] f(j(x, h1(x'', y')), h1(x'', y')) -> g(f(x, h1(s(x''), y'))) [2] f(j(x, y), y) -> g(f(x, 0)) [1] f(x, h1(y, z)) -> h2(0, x, h1(y, z)) [1] h2(x, j(y, h1(z, u)), h1(z, u)) -> h2(s(x), y, h1(s(z), u)) [1] k(h(x)) -> h1(0, x) [1] k(h1(x, y)) -> h1(s(x), y) [1] i(c_h2(s(x), y, h1(x, z))) -> z [1] g(c_h2(x, y, h1(z, u))) -> h2(s(x), y, h1(z, u)) [1] i(c_f(x, h(y))) -> y [1] encArg(j(x_1, x_2)) -> j(encArg(x_1), encArg(x_2)) [0] encArg(h1(x_1, x_2)) -> h1(encArg(x_1), encArg(x_2)) [0] encArg(0) -> 0 [0] encArg(s(x_1)) -> s(encArg(x_1)) [0] encArg(h(x_1)) -> h(encArg(x_1)) [0] encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0] encArg(cons_g(j(x_1107, x_247))) -> g(j(encArg(x_1107), encArg(x_247))) [0] encArg(cons_g(h1(x_1108, x_248))) -> g(h1(encArg(x_1108), encArg(x_248))) [0] encArg(cons_g(0)) -> g(0) [0] encArg(cons_g(s(x_1109))) -> g(s(encArg(x_1109))) [0] encArg(cons_g(h(x_1110))) -> g(h(encArg(x_1110))) [0] encArg(cons_g(cons_f(x_1111, x_249))) -> g(f(encArg(x_1111), encArg(x_249))) [0] encArg(cons_g(cons_g(x_1112))) -> g(g(encArg(x_1112))) [0] encArg(cons_g(cons_h2(x_1113, x_250, x_311))) -> g(h2(encArg(x_1113), encArg(x_250), encArg(x_311))) [0] encArg(cons_g(cons_i(x_1114))) -> g(i(encArg(x_1114))) [0] encArg(cons_g(cons_k(x_1115))) -> g(k(encArg(x_1115))) [0] encArg(cons_g(x_1)) -> g(0) [0] encArg(cons_h2(x_1, x_2, x_3)) -> h2(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encArg(cons_i(j(x_11313, x_2583))) -> i(j(encArg(x_11313), encArg(x_2583))) [0] encArg(cons_i(h1(x_11314, x_2584))) -> i(h1(encArg(x_11314), encArg(x_2584))) [0] encArg(cons_i(0)) -> i(0) [0] encArg(cons_i(s(x_11315))) -> i(s(encArg(x_11315))) [0] encArg(cons_i(h(x_11316))) -> i(h(encArg(x_11316))) [0] encArg(cons_i(cons_f(x_11317, x_2585))) -> i(f(encArg(x_11317), encArg(x_2585))) [0] encArg(cons_i(cons_g(x_11318))) -> i(g(encArg(x_11318))) [0] encArg(cons_i(cons_h2(x_11319, x_2586, x_3145))) -> i(h2(encArg(x_11319), encArg(x_2586), encArg(x_3145))) [0] encArg(cons_i(cons_i(x_11320))) -> i(i(encArg(x_11320))) [0] encArg(cons_i(cons_k(x_11321))) -> i(k(encArg(x_11321))) [0] encArg(cons_i(x_1)) -> i(0) [0] encArg(cons_k(j(x_11322, x_2587))) -> k(j(encArg(x_11322), encArg(x_2587))) [0] encArg(cons_k(h1(x_11323, x_2588))) -> k(h1(encArg(x_11323), encArg(x_2588))) [0] encArg(cons_k(0)) -> k(0) [0] encArg(cons_k(s(x_11324))) -> k(s(encArg(x_11324))) [0] encArg(cons_k(h(x_11325))) -> k(h(encArg(x_11325))) [0] encArg(cons_k(cons_f(x_11326, x_2589))) -> k(f(encArg(x_11326), encArg(x_2589))) [0] encArg(cons_k(cons_g(x_11327))) -> k(g(encArg(x_11327))) [0] encArg(cons_k(cons_h2(x_11328, x_2590, x_3146))) -> k(h2(encArg(x_11328), encArg(x_2590), encArg(x_3146))) [0] encArg(cons_k(cons_i(x_11329))) -> k(i(encArg(x_11329))) [0] encArg(cons_k(cons_k(x_11330))) -> k(k(encArg(x_11330))) [0] encArg(cons_k(x_1)) -> k(0) [0] encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0] encode_j(x_1, x_2) -> j(encArg(x_1), encArg(x_2)) [0] encode_g(j(x_11439, x_2639)) -> g(j(encArg(x_11439), encArg(x_2639))) [0] encode_g(h1(x_11440, x_2640)) -> g(h1(encArg(x_11440), encArg(x_2640))) [0] encode_g(0) -> g(0) [0] encode_g(s(x_11441)) -> g(s(encArg(x_11441))) [0] encode_g(h(x_11442)) -> g(h(encArg(x_11442))) [0] encode_g(cons_f(x_11443, x_2641)) -> g(f(encArg(x_11443), encArg(x_2641))) [0] encode_g(cons_g(x_11444)) -> g(g(encArg(x_11444))) [0] encode_g(cons_h2(x_11445, x_2642, x_3159)) -> g(h2(encArg(x_11445), encArg(x_2642), encArg(x_3159))) [0] encode_g(cons_i(x_11446)) -> g(i(encArg(x_11446))) [0] encode_g(cons_k(x_11447)) -> g(k(encArg(x_11447))) [0] encode_g(x_1) -> g(0) [0] encode_k(j(x_11448, x_2643)) -> k(j(encArg(x_11448), encArg(x_2643))) [0] encode_k(h1(x_11449, x_2644)) -> k(h1(encArg(x_11449), encArg(x_2644))) [0] encode_k(0) -> k(0) [0] encode_k(s(x_11450)) -> k(s(encArg(x_11450))) [0] encode_k(h(x_11451)) -> k(h(encArg(x_11451))) [0] encode_k(cons_f(x_11452, x_2645)) -> k(f(encArg(x_11452), encArg(x_2645))) [0] encode_k(cons_g(x_11453)) -> k(g(encArg(x_11453))) [0] encode_k(cons_h2(x_11454, x_2646, x_3160)) -> k(h2(encArg(x_11454), encArg(x_2646), encArg(x_3160))) [0] encode_k(cons_i(x_11455)) -> k(i(encArg(x_11455))) [0] encode_k(cons_k(x_11456)) -> k(k(encArg(x_11456))) [0] encode_k(x_1) -> k(0) [0] encode_h1(x_1, x_2) -> h1(encArg(x_1), encArg(x_2)) [0] encode_h2(x_1, x_2, x_3) -> h2(encArg(x_1), encArg(x_2), encArg(x_3)) [0] encode_0 -> 0 [0] encode_s(x_1) -> s(encArg(x_1)) [0] encode_i(j(x_12654, x_21179)) -> i(j(encArg(x_12654), encArg(x_21179))) [0] encode_i(h1(x_12655, x_21180)) -> i(h1(encArg(x_12655), encArg(x_21180))) [0] encode_i(0) -> i(0) [0] encode_i(s(x_12656)) -> i(s(encArg(x_12656))) [0] encode_i(h(x_12657)) -> i(h(encArg(x_12657))) [0] encode_i(cons_f(x_12658, x_21181)) -> i(f(encArg(x_12658), encArg(x_21181))) [0] encode_i(cons_g(x_12659)) -> i(g(encArg(x_12659))) [0] encode_i(cons_h2(x_12660, x_21182, x_3294)) -> i(h2(encArg(x_12660), encArg(x_21182), encArg(x_3294))) [0] encode_i(cons_i(x_12661)) -> i(i(encArg(x_12661))) [0] encode_i(cons_k(x_12662)) -> i(k(encArg(x_12662))) [0] encode_i(x_1) -> i(0) [0] encode_h(x_1) -> h(encArg(x_1)) [0] f(x0, x1) -> c_f(x0, x1) [0] h2(x0, x1, x2) -> c_h2(x0, x1, x2) [0] encArg(v0) -> 0 [0] encode_f(v0, v1) -> 0 [0] encode_j(v0, v1) -> 0 [0] encode_g(v0) -> 0 [0] encode_k(v0) -> 0 [0] encode_h1(v0, v1) -> 0 [0] encode_h2(v0, v1, v2) -> 0 [0] encode_0 -> 0 [0] encode_s(v0) -> 0 [0] encode_i(v0) -> 0 [0] encode_h(v0) -> 0 [0] f(v0, v1) -> 0 [0] h2(v0, v1, v2) -> 0 [0] k(v0) -> 0 [0] i(v0) -> 0 [0] g(v0) -> 0 [0] The TRS has the following type information: f :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k j :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k g :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k k :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k h1 :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k h2 :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k 0 :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k s :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k h :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k i :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k c_h2 :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k c_f :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k encArg :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k cons_f :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k cons_g :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k cons_h2 :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k cons_i :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k cons_k :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k encode_f :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k encode_j :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k encode_g :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k encode_k :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k encode_h1 :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k encode_h2 :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k encode_0 :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k encode_s :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k encode_i :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k encode_h :: j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k -> j:h1:0:s:h:c_h2:c_f:cons_f:cons_g:cons_h2:cons_i:cons_k Rewrite Strategy: INNERMOST ---------------------------------------- (17) 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 ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: encArg(z') -{ 0 }-> k(k(encArg(x_11330))) :|: x_11330 >= 0, z' = 1 + (1 + x_11330) encArg(z') -{ 0 }-> k(i(encArg(x_11329))) :|: x_11329 >= 0, z' = 1 + (1 + x_11329) encArg(z') -{ 0 }-> k(h2(encArg(x_11328), encArg(x_2590), encArg(x_3146))) :|: z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ 0 }-> k(g(encArg(x_11327))) :|: z' = 1 + (1 + x_11327), x_11327 >= 0 encArg(z') -{ 0 }-> k(f(encArg(x_11326), encArg(x_2589))) :|: x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ 0 }-> k(0) :|: z' = 1 + 0 encArg(z') -{ 0 }-> k(0) :|: x_1 >= 0, z' = 1 + x_1 encArg(z') -{ 0 }-> k(1 + encArg(x_11324)) :|: z' = 1 + (1 + x_11324), x_11324 >= 0 encArg(z') -{ 0 }-> k(1 + encArg(x_11325)) :|: z' = 1 + (1 + x_11325), x_11325 >= 0 encArg(z') -{ 0 }-> k(1 + encArg(x_11322) + encArg(x_2587)) :|: x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 0 }-> k(1 + encArg(x_11323) + encArg(x_2588)) :|: z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ 0 }-> i(k(encArg(x_11321))) :|: z' = 1 + (1 + x_11321), x_11321 >= 0 encArg(z') -{ 0 }-> i(i(encArg(x_11320))) :|: x_11320 >= 0, z' = 1 + (1 + x_11320) encArg(z') -{ 0 }-> i(h2(encArg(x_11319), encArg(x_2586), encArg(x_3145))) :|: z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ 0 }-> i(g(encArg(x_11318))) :|: z' = 1 + (1 + x_11318), x_11318 >= 0 encArg(z') -{ 0 }-> i(f(encArg(x_11317), encArg(x_2585))) :|: x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ 0 }-> i(0) :|: z' = 1 + 0 encArg(z') -{ 0 }-> i(0) :|: x_1 >= 0, z' = 1 + x_1 encArg(z') -{ 0 }-> i(1 + encArg(x_11315)) :|: z' = 1 + (1 + x_11315), x_11315 >= 0 encArg(z') -{ 0 }-> i(1 + encArg(x_11316)) :|: z' = 1 + (1 + x_11316), x_11316 >= 0 encArg(z') -{ 0 }-> i(1 + encArg(x_11313) + encArg(x_2583)) :|: x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 0 }-> i(1 + encArg(x_11314) + encArg(x_2584)) :|: z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ 0 }-> h2(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 }-> g(k(encArg(x_1115))) :|: z' = 1 + (1 + x_1115), x_1115 >= 0 encArg(z') -{ 0 }-> g(i(encArg(x_1114))) :|: x_1114 >= 0, z' = 1 + (1 + x_1114) encArg(z') -{ 0 }-> g(h2(encArg(x_1113), encArg(x_250), encArg(x_311))) :|: x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 0 }-> g(g(encArg(x_1112))) :|: x_1112 >= 0, z' = 1 + (1 + x_1112) encArg(z') -{ 0 }-> g(f(encArg(x_1111), encArg(x_249))) :|: z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ 0 }-> g(0) :|: z' = 1 + 0 encArg(z') -{ 0 }-> g(0) :|: x_1 >= 0, z' = 1 + x_1 encArg(z') -{ 0 }-> g(1 + encArg(x_1109)) :|: x_1109 >= 0, z' = 1 + (1 + x_1109) encArg(z') -{ 0 }-> g(1 + encArg(x_1110)) :|: z' = 1 + (1 + x_1110), x_1110 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1107) + encArg(x_247)) :|: z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1108) + encArg(x_248)) :|: x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 encArg(z') -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z' = 1 + x_1 encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z' = x_1, x_2 >= 0, z'' = x_2 encode_f(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 encode_g(z') -{ 0 }-> g(k(encArg(x_11447))) :|: z' = 1 + x_11447, x_11447 >= 0 encode_g(z') -{ 0 }-> g(i(encArg(x_11446))) :|: z' = 1 + x_11446, x_11446 >= 0 encode_g(z') -{ 0 }-> g(h2(encArg(x_11445), encArg(x_2642), encArg(x_3159))) :|: x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> g(g(encArg(x_11444))) :|: x_11444 >= 0, z' = 1 + x_11444 encode_g(z') -{ 0 }-> g(f(encArg(x_11443), encArg(x_2641))) :|: x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ 0 }-> g(0) :|: z' = 0 encode_g(z') -{ 0 }-> g(0) :|: x_1 >= 0, z' = x_1 encode_g(z') -{ 0 }-> g(1 + encArg(x_11441)) :|: z' = 1 + x_11441, x_11441 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11442)) :|: z' = 1 + x_11442, x_11442 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11439) + encArg(x_2639)) :|: x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11440) + encArg(x_2640)) :|: x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 encode_h(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 encode_h(z') -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z' = x_1 encode_h1(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 encode_h1(z', z'') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, z' = x_1, x_2 >= 0, z'' = x_2 encode_h2(z', z'', z1) -{ 0 }-> h2(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z1 = x_3, z' = x_1, x_3 >= 0, x_2 >= 0, z'' = x_2 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0 encode_i(z') -{ 0 }-> i(k(encArg(x_12662))) :|: z' = 1 + x_12662, x_12662 >= 0 encode_i(z') -{ 0 }-> i(i(encArg(x_12661))) :|: z' = 1 + x_12661, x_12661 >= 0 encode_i(z') -{ 0 }-> i(h2(encArg(x_12660), encArg(x_21182), encArg(x_3294))) :|: z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ 0 }-> i(g(encArg(x_12659))) :|: z' = 1 + x_12659, x_12659 >= 0 encode_i(z') -{ 0 }-> i(f(encArg(x_12658), encArg(x_21181))) :|: x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ 0 }-> i(0) :|: z' = 0 encode_i(z') -{ 0 }-> i(0) :|: x_1 >= 0, z' = x_1 encode_i(z') -{ 0 }-> i(1 + encArg(x_12656)) :|: x_12656 >= 0, z' = 1 + x_12656 encode_i(z') -{ 0 }-> i(1 + encArg(x_12657)) :|: x_12657 >= 0, z' = 1 + x_12657 encode_i(z') -{ 0 }-> i(1 + encArg(x_12654) + encArg(x_21179)) :|: x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(x_12655) + encArg(x_21180)) :|: z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 encode_j(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 encode_j(z', z'') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, z' = x_1, x_2 >= 0, z'' = x_2 encode_k(z') -{ 0 }-> k(k(encArg(x_11456))) :|: x_11456 >= 0, z' = 1 + x_11456 encode_k(z') -{ 0 }-> k(i(encArg(x_11455))) :|: x_11455 >= 0, z' = 1 + x_11455 encode_k(z') -{ 0 }-> k(h2(encArg(x_11454), encArg(x_2646), encArg(x_3160))) :|: x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ 0 }-> k(g(encArg(x_11453))) :|: x_11453 >= 0, z' = 1 + x_11453 encode_k(z') -{ 0 }-> k(f(encArg(x_11452), encArg(x_2645))) :|: x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ 0 }-> k(0) :|: z' = 0 encode_k(z') -{ 0 }-> k(0) :|: x_1 >= 0, z' = x_1 encode_k(z') -{ 0 }-> k(1 + encArg(x_11450)) :|: x_11450 >= 0, z' = 1 + x_11450 encode_k(z') -{ 0 }-> k(1 + encArg(x_11451)) :|: z' = 1 + x_11451, x_11451 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11448) + encArg(x_2643)) :|: z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11449) + encArg(x_2644)) :|: x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 encode_s(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 encode_s(z') -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z' = x_1 f(z', z'') -{ 1 }-> h2(0, x, 1 + y + z) :|: z >= 0, z' = x, x >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 1 }-> g(f(x, 0)) :|: z'' = y, z' = 1 + x + y, x >= 0, y >= 0 f(z', z'') -{ 2 }-> g(f(x, 1 + 0 + x')) :|: z' = 1 + x + (1 + x'), z'' = 1 + x', x >= 0, x' >= 0 f(z', z'') -{ 2 }-> g(f(x, 1 + (1 + x'') + y')) :|: z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 f(z', z'') -{ 0 }-> 1 + x0 + x1 :|: z'' = x1, x0 >= 0, x1 >= 0, z' = x0 g(z') -{ 1 }-> h2(1 + x, y, 1 + z + u) :|: z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 h2(z', z'', z1) -{ 1 }-> h2(1 + x, y, 1 + (1 + z) + u) :|: z >= 0, z' = x, x >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0 h2(z', z'', z1) -{ 0 }-> 1 + x0 + x1 + x2 :|: z'' = x1, x0 >= 0, x1 >= 0, z1 = x2, x2 >= 0, z' = x0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 k(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 k(z') -{ 1 }-> 1 + 0 + x :|: z' = 1 + x, x >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 ---------------------------------------- (19) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: k(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 k(z') -{ 1 }-> 1 + 0 + x :|: z' = 1 + x, x >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: encArg(z') -{ 0 }-> k(k(encArg(x_11330))) :|: x_11330 >= 0, z' = 1 + (1 + x_11330) encArg(z') -{ 0 }-> k(i(encArg(x_11329))) :|: x_11329 >= 0, z' = 1 + (1 + x_11329) encArg(z') -{ 0 }-> k(h2(encArg(x_11328), encArg(x_2590), encArg(x_3146))) :|: z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ 0 }-> k(g(encArg(x_11327))) :|: z' = 1 + (1 + x_11327), x_11327 >= 0 encArg(z') -{ 0 }-> k(f(encArg(x_11326), encArg(x_2589))) :|: x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ 0 }-> k(1 + encArg(x_11324)) :|: z' = 1 + (1 + x_11324), x_11324 >= 0 encArg(z') -{ 0 }-> k(1 + encArg(x_11325)) :|: z' = 1 + (1 + x_11325), x_11325 >= 0 encArg(z') -{ 0 }-> k(1 + encArg(x_11322) + encArg(x_2587)) :|: x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 0 }-> k(1 + encArg(x_11323) + encArg(x_2588)) :|: z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ 0 }-> i(k(encArg(x_11321))) :|: z' = 1 + (1 + x_11321), x_11321 >= 0 encArg(z') -{ 0 }-> i(i(encArg(x_11320))) :|: x_11320 >= 0, z' = 1 + (1 + x_11320) encArg(z') -{ 0 }-> i(h2(encArg(x_11319), encArg(x_2586), encArg(x_3145))) :|: z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ 0 }-> i(g(encArg(x_11318))) :|: z' = 1 + (1 + x_11318), x_11318 >= 0 encArg(z') -{ 0 }-> i(f(encArg(x_11317), encArg(x_2585))) :|: x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ 0 }-> i(1 + encArg(x_11315)) :|: z' = 1 + (1 + x_11315), x_11315 >= 0 encArg(z') -{ 0 }-> i(1 + encArg(x_11316)) :|: z' = 1 + (1 + x_11316), x_11316 >= 0 encArg(z') -{ 0 }-> i(1 + encArg(x_11313) + encArg(x_2583)) :|: x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 0 }-> i(1 + encArg(x_11314) + encArg(x_2584)) :|: z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ 0 }-> h2(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 }-> g(k(encArg(x_1115))) :|: z' = 1 + (1 + x_1115), x_1115 >= 0 encArg(z') -{ 0 }-> g(i(encArg(x_1114))) :|: x_1114 >= 0, z' = 1 + (1 + x_1114) encArg(z') -{ 0 }-> g(h2(encArg(x_1113), encArg(x_250), encArg(x_311))) :|: x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 0 }-> g(g(encArg(x_1112))) :|: x_1112 >= 0, z' = 1 + (1 + x_1112) encArg(z') -{ 0 }-> g(f(encArg(x_1111), encArg(x_249))) :|: z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ 0 }-> g(0) :|: z' = 1 + 0 encArg(z') -{ 0 }-> g(0) :|: x_1 >= 0, z' = 1 + x_1 encArg(z') -{ 0 }-> g(1 + encArg(x_1109)) :|: x_1109 >= 0, z' = 1 + (1 + x_1109) encArg(z') -{ 0 }-> g(1 + encArg(x_1110)) :|: z' = 1 + (1 + x_1110), x_1110 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1107) + encArg(x_247)) :|: z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1108) + encArg(x_248)) :|: x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: x_1 >= 0, z' = 1 + x_1, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z' = 1 + x_1 encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z' = x_1, x_2 >= 0, z'' = x_2 encode_f(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 encode_g(z') -{ 0 }-> g(k(encArg(x_11447))) :|: z' = 1 + x_11447, x_11447 >= 0 encode_g(z') -{ 0 }-> g(i(encArg(x_11446))) :|: z' = 1 + x_11446, x_11446 >= 0 encode_g(z') -{ 0 }-> g(h2(encArg(x_11445), encArg(x_2642), encArg(x_3159))) :|: x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> g(g(encArg(x_11444))) :|: x_11444 >= 0, z' = 1 + x_11444 encode_g(z') -{ 0 }-> g(f(encArg(x_11443), encArg(x_2641))) :|: x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ 0 }-> g(0) :|: z' = 0 encode_g(z') -{ 0 }-> g(0) :|: x_1 >= 0, z' = x_1 encode_g(z') -{ 0 }-> g(1 + encArg(x_11441)) :|: z' = 1 + x_11441, x_11441 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11442)) :|: z' = 1 + x_11442, x_11442 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11439) + encArg(x_2639)) :|: x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11440) + encArg(x_2640)) :|: x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 encode_h(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 encode_h(z') -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z' = x_1 encode_h1(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 encode_h1(z', z'') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, z' = x_1, x_2 >= 0, z'' = x_2 encode_h2(z', z'', z1) -{ 0 }-> h2(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z1 = x_3, z' = x_1, x_3 >= 0, x_2 >= 0, z'' = x_2 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0 encode_i(z') -{ 0 }-> i(k(encArg(x_12662))) :|: z' = 1 + x_12662, x_12662 >= 0 encode_i(z') -{ 0 }-> i(i(encArg(x_12661))) :|: z' = 1 + x_12661, x_12661 >= 0 encode_i(z') -{ 0 }-> i(h2(encArg(x_12660), encArg(x_21182), encArg(x_3294))) :|: z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ 0 }-> i(g(encArg(x_12659))) :|: z' = 1 + x_12659, x_12659 >= 0 encode_i(z') -{ 0 }-> i(f(encArg(x_12658), encArg(x_21181))) :|: x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(x_12656)) :|: x_12656 >= 0, z' = 1 + x_12656 encode_i(z') -{ 0 }-> i(1 + encArg(x_12657)) :|: x_12657 >= 0, z' = 1 + x_12657 encode_i(z') -{ 0 }-> i(1 + encArg(x_12654) + encArg(x_21179)) :|: x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(x_12655) + encArg(x_21180)) :|: z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: x_1 >= 0, z' = x_1, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 encode_j(z', z'') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, z' = x_1, x_2 >= 0, z'' = x_2 encode_k(z') -{ 0 }-> k(k(encArg(x_11456))) :|: x_11456 >= 0, z' = 1 + x_11456 encode_k(z') -{ 0 }-> k(i(encArg(x_11455))) :|: x_11455 >= 0, z' = 1 + x_11455 encode_k(z') -{ 0 }-> k(h2(encArg(x_11454), encArg(x_2646), encArg(x_3160))) :|: x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ 0 }-> k(g(encArg(x_11453))) :|: x_11453 >= 0, z' = 1 + x_11453 encode_k(z') -{ 0 }-> k(f(encArg(x_11452), encArg(x_2645))) :|: x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11450)) :|: x_11450 >= 0, z' = 1 + x_11450 encode_k(z') -{ 0 }-> k(1 + encArg(x_11451)) :|: z' = 1 + x_11451, x_11451 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11448) + encArg(x_2643)) :|: z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11449) + encArg(x_2644)) :|: x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: x_1 >= 0, z' = x_1, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 encode_s(z') -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z' = x_1 f(z', z'') -{ 1 }-> h2(0, x, 1 + y + z) :|: z >= 0, z' = x, x >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 1 }-> g(f(x, 0)) :|: z'' = y, z' = 1 + x + y, x >= 0, y >= 0 f(z', z'') -{ 2 }-> g(f(x, 1 + 0 + x')) :|: z' = 1 + x + (1 + x'), z'' = 1 + x', x >= 0, x' >= 0 f(z', z'') -{ 2 }-> g(f(x, 1 + (1 + x'') + y')) :|: z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 f(z', z'') -{ 0 }-> 1 + x0 + x1 :|: z'' = x1, x0 >= 0, x1 >= 0, z' = x0 g(z') -{ 1 }-> h2(1 + x, y, 1 + z + u) :|: z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 h2(z', z'', z1) -{ 1 }-> h2(1 + x, y, 1 + (1 + z) + u) :|: z >= 0, z' = x, x >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0 h2(z', z'', z1) -{ 0 }-> 1 + x0 + x1 + x2 :|: z'' = x1, x0 >= 0, x1 >= 0, z1 = x2, x2 >= 0, z' = x0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 k(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0 k(z') -{ 1 }-> 1 + 0 + x :|: z' = 1 + x, x >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 ---------------------------------------- (21) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: encArg(z') -{ 0 }-> k(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(h2(encArg(x_11328), encArg(x_2590), encArg(x_3146))) :|: z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ 0 }-> k(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(f(encArg(x_11326), encArg(x_2589))) :|: x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ 0 }-> k(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(1 + encArg(x_11322) + encArg(x_2587)) :|: x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 0 }-> k(1 + encArg(x_11323) + encArg(x_2588)) :|: z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ 0 }-> i(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(h2(encArg(x_11319), encArg(x_2586), encArg(x_3145))) :|: z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ 0 }-> i(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(f(encArg(x_11317), encArg(x_2585))) :|: x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ 0 }-> i(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(1 + encArg(x_11313) + encArg(x_2583)) :|: x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 0 }-> i(1 + encArg(x_11314) + encArg(x_2584)) :|: z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ 0 }-> h2(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 }-> g(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(h2(encArg(x_1113), encArg(x_250), encArg(x_311))) :|: x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 0 }-> g(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(f(encArg(x_1111), encArg(x_249))) :|: z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ 0 }-> g(0) :|: z' = 1 + 0 encArg(z') -{ 0 }-> g(0) :|: z' - 1 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1107) + encArg(x_247)) :|: z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1108) + encArg(x_248)) :|: x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0 encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 0 }-> f(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ 0 }-> g(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(h2(encArg(x_11445), encArg(x_2642), encArg(x_3159))) :|: x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> g(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(f(encArg(x_11443), encArg(x_2641))) :|: x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ 0 }-> g(0) :|: z' = 0 encode_g(z') -{ 0 }-> g(0) :|: z' >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11439) + encArg(x_2639)) :|: x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11440) + encArg(x_2640)) :|: x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> h2(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 0 }-> i(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(h2(encArg(x_12660), encArg(x_21182), encArg(x_3294))) :|: z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ 0 }-> i(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(f(encArg(x_12658), encArg(x_21181))) :|: x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(x_12654) + encArg(x_21179)) :|: x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(x_12655) + encArg(x_21180)) :|: z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0 encode_k(z') -{ 0 }-> k(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(h2(encArg(x_11454), encArg(x_2646), encArg(x_3160))) :|: x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ 0 }-> k(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(f(encArg(x_11452), encArg(x_2645))) :|: x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11448) + encArg(x_2643)) :|: z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11449) + encArg(x_2644)) :|: x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0 f(z', z'') -{ 1 }-> h2(0, z', 1 + y + z) :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 1 }-> g(f(x, 0)) :|: z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 2 }-> g(f(x, 1 + 0 + (z'' - 1))) :|: z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 2 }-> g(f(x, 1 + (1 + x'') + y')) :|: z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 }-> h2(1 + x, y, 1 + z + u) :|: z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 }-> h2(1 + z', y, 1 + (1 + z) + u) :|: z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 ---------------------------------------- (23) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { h2 } { encode_0 } { i } { k } { g } { f } { encArg } { encode_k } { encode_h } { encode_h2 } { encode_h1 } { encode_g } { encode_f } { encode_j } { encode_i } { encode_s } ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: encArg(z') -{ 0 }-> k(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(h2(encArg(x_11328), encArg(x_2590), encArg(x_3146))) :|: z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ 0 }-> k(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(f(encArg(x_11326), encArg(x_2589))) :|: x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ 0 }-> k(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(1 + encArg(x_11322) + encArg(x_2587)) :|: x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 0 }-> k(1 + encArg(x_11323) + encArg(x_2588)) :|: z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ 0 }-> i(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(h2(encArg(x_11319), encArg(x_2586), encArg(x_3145))) :|: z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ 0 }-> i(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(f(encArg(x_11317), encArg(x_2585))) :|: x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ 0 }-> i(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(1 + encArg(x_11313) + encArg(x_2583)) :|: x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 0 }-> i(1 + encArg(x_11314) + encArg(x_2584)) :|: z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ 0 }-> h2(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 }-> g(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(h2(encArg(x_1113), encArg(x_250), encArg(x_311))) :|: x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 0 }-> g(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(f(encArg(x_1111), encArg(x_249))) :|: z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ 0 }-> g(0) :|: z' = 1 + 0 encArg(z') -{ 0 }-> g(0) :|: z' - 1 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1107) + encArg(x_247)) :|: z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1108) + encArg(x_248)) :|: x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0 encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 0 }-> f(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ 0 }-> g(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(h2(encArg(x_11445), encArg(x_2642), encArg(x_3159))) :|: x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> g(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(f(encArg(x_11443), encArg(x_2641))) :|: x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ 0 }-> g(0) :|: z' = 0 encode_g(z') -{ 0 }-> g(0) :|: z' >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11439) + encArg(x_2639)) :|: x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11440) + encArg(x_2640)) :|: x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> h2(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 0 }-> i(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(h2(encArg(x_12660), encArg(x_21182), encArg(x_3294))) :|: z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ 0 }-> i(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(f(encArg(x_12658), encArg(x_21181))) :|: x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(x_12654) + encArg(x_21179)) :|: x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(x_12655) + encArg(x_21180)) :|: z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0 encode_k(z') -{ 0 }-> k(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(h2(encArg(x_11454), encArg(x_2646), encArg(x_3160))) :|: x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ 0 }-> k(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(f(encArg(x_11452), encArg(x_2645))) :|: x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11448) + encArg(x_2643)) :|: z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11449) + encArg(x_2644)) :|: x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0 f(z', z'') -{ 1 }-> h2(0, z', 1 + y + z) :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 1 }-> g(f(x, 0)) :|: z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 2 }-> g(f(x, 1 + 0 + (z'' - 1))) :|: z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 2 }-> g(f(x, 1 + (1 + x'') + y')) :|: z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 }-> h2(1 + x, y, 1 + z + u) :|: z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 }-> h2(1 + z', y, 1 + (1 + z) + u) :|: z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {h2}, {encode_0}, {i}, {k}, {g}, {f}, {encArg}, {encode_k}, {encode_h}, {encode_h2}, {encode_h1}, {encode_g}, {encode_f}, {encode_j}, {encode_i}, {encode_s} ---------------------------------------- (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 }-> k(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(h2(encArg(x_11328), encArg(x_2590), encArg(x_3146))) :|: z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ 0 }-> k(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(f(encArg(x_11326), encArg(x_2589))) :|: x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ 0 }-> k(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(1 + encArg(x_11322) + encArg(x_2587)) :|: x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 0 }-> k(1 + encArg(x_11323) + encArg(x_2588)) :|: z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ 0 }-> i(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(h2(encArg(x_11319), encArg(x_2586), encArg(x_3145))) :|: z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ 0 }-> i(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(f(encArg(x_11317), encArg(x_2585))) :|: x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ 0 }-> i(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(1 + encArg(x_11313) + encArg(x_2583)) :|: x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 0 }-> i(1 + encArg(x_11314) + encArg(x_2584)) :|: z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ 0 }-> h2(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 }-> g(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(h2(encArg(x_1113), encArg(x_250), encArg(x_311))) :|: x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 0 }-> g(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(f(encArg(x_1111), encArg(x_249))) :|: z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ 0 }-> g(0) :|: z' = 1 + 0 encArg(z') -{ 0 }-> g(0) :|: z' - 1 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1107) + encArg(x_247)) :|: z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1108) + encArg(x_248)) :|: x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0 encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 0 }-> f(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ 0 }-> g(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(h2(encArg(x_11445), encArg(x_2642), encArg(x_3159))) :|: x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> g(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(f(encArg(x_11443), encArg(x_2641))) :|: x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ 0 }-> g(0) :|: z' = 0 encode_g(z') -{ 0 }-> g(0) :|: z' >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11439) + encArg(x_2639)) :|: x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11440) + encArg(x_2640)) :|: x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> h2(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 0 }-> i(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(h2(encArg(x_12660), encArg(x_21182), encArg(x_3294))) :|: z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ 0 }-> i(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(f(encArg(x_12658), encArg(x_21181))) :|: x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(x_12654) + encArg(x_21179)) :|: x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(x_12655) + encArg(x_21180)) :|: z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0 encode_k(z') -{ 0 }-> k(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(h2(encArg(x_11454), encArg(x_2646), encArg(x_3160))) :|: x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ 0 }-> k(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(f(encArg(x_11452), encArg(x_2645))) :|: x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11448) + encArg(x_2643)) :|: z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11449) + encArg(x_2644)) :|: x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0 f(z', z'') -{ 1 }-> h2(0, z', 1 + y + z) :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 1 }-> g(f(x, 0)) :|: z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 2 }-> g(f(x, 1 + 0 + (z'' - 1))) :|: z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 2 }-> g(f(x, 1 + (1 + x'') + y')) :|: z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 }-> h2(1 + x, y, 1 + z + u) :|: z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 }-> h2(1 + z', y, 1 + (1 + z) + u) :|: z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {h2}, {encode_0}, {i}, {k}, {g}, {f}, {encArg}, {encode_k}, {encode_h}, {encode_h2}, {encode_h1}, {encode_g}, {encode_f}, {encode_j}, {encode_i}, {encode_s} ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: h2 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z' + z'' + z1 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: encArg(z') -{ 0 }-> k(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(h2(encArg(x_11328), encArg(x_2590), encArg(x_3146))) :|: z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ 0 }-> k(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(f(encArg(x_11326), encArg(x_2589))) :|: x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ 0 }-> k(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(1 + encArg(x_11322) + encArg(x_2587)) :|: x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 0 }-> k(1 + encArg(x_11323) + encArg(x_2588)) :|: z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ 0 }-> i(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(h2(encArg(x_11319), encArg(x_2586), encArg(x_3145))) :|: z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ 0 }-> i(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(f(encArg(x_11317), encArg(x_2585))) :|: x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ 0 }-> i(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(1 + encArg(x_11313) + encArg(x_2583)) :|: x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 0 }-> i(1 + encArg(x_11314) + encArg(x_2584)) :|: z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ 0 }-> h2(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 }-> g(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(h2(encArg(x_1113), encArg(x_250), encArg(x_311))) :|: x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 0 }-> g(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(f(encArg(x_1111), encArg(x_249))) :|: z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ 0 }-> g(0) :|: z' = 1 + 0 encArg(z') -{ 0 }-> g(0) :|: z' - 1 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1107) + encArg(x_247)) :|: z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1108) + encArg(x_248)) :|: x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0 encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 0 }-> f(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ 0 }-> g(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(h2(encArg(x_11445), encArg(x_2642), encArg(x_3159))) :|: x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> g(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(f(encArg(x_11443), encArg(x_2641))) :|: x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ 0 }-> g(0) :|: z' = 0 encode_g(z') -{ 0 }-> g(0) :|: z' >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11439) + encArg(x_2639)) :|: x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11440) + encArg(x_2640)) :|: x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> h2(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 0 }-> i(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(h2(encArg(x_12660), encArg(x_21182), encArg(x_3294))) :|: z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ 0 }-> i(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(f(encArg(x_12658), encArg(x_21181))) :|: x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(x_12654) + encArg(x_21179)) :|: x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(x_12655) + encArg(x_21180)) :|: z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0 encode_k(z') -{ 0 }-> k(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(h2(encArg(x_11454), encArg(x_2646), encArg(x_3160))) :|: x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ 0 }-> k(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(f(encArg(x_11452), encArg(x_2645))) :|: x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11448) + encArg(x_2643)) :|: z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11449) + encArg(x_2644)) :|: x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0 f(z', z'') -{ 1 }-> h2(0, z', 1 + y + z) :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 1 }-> g(f(x, 0)) :|: z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 2 }-> g(f(x, 1 + 0 + (z'' - 1))) :|: z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 2 }-> g(f(x, 1 + (1 + x'') + y')) :|: z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 }-> h2(1 + x, y, 1 + z + u) :|: z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 }-> h2(1 + z', y, 1 + (1 + z) + u) :|: z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {h2}, {encode_0}, {i}, {k}, {g}, {f}, {encArg}, {encode_k}, {encode_h}, {encode_h2}, {encode_h1}, {encode_g}, {encode_f}, {encode_j}, {encode_i}, {encode_s} Previous analysis results are: h2: runtime: ?, size: O(n^1) [1 + z' + z'' + z1] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: h2 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z'' ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: encArg(z') -{ 0 }-> k(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(h2(encArg(x_11328), encArg(x_2590), encArg(x_3146))) :|: z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ 0 }-> k(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(f(encArg(x_11326), encArg(x_2589))) :|: x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ 0 }-> k(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(1 + encArg(x_11322) + encArg(x_2587)) :|: x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 0 }-> k(1 + encArg(x_11323) + encArg(x_2588)) :|: z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ 0 }-> i(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(h2(encArg(x_11319), encArg(x_2586), encArg(x_3145))) :|: z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ 0 }-> i(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(f(encArg(x_11317), encArg(x_2585))) :|: x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ 0 }-> i(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(1 + encArg(x_11313) + encArg(x_2583)) :|: x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 0 }-> i(1 + encArg(x_11314) + encArg(x_2584)) :|: z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ 0 }-> h2(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 }-> g(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(h2(encArg(x_1113), encArg(x_250), encArg(x_311))) :|: x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 0 }-> g(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(f(encArg(x_1111), encArg(x_249))) :|: z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ 0 }-> g(0) :|: z' = 1 + 0 encArg(z') -{ 0 }-> g(0) :|: z' - 1 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1107) + encArg(x_247)) :|: z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1108) + encArg(x_248)) :|: x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0 encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 0 }-> f(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ 0 }-> g(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(h2(encArg(x_11445), encArg(x_2642), encArg(x_3159))) :|: x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> g(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(f(encArg(x_11443), encArg(x_2641))) :|: x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ 0 }-> g(0) :|: z' = 0 encode_g(z') -{ 0 }-> g(0) :|: z' >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11439) + encArg(x_2639)) :|: x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11440) + encArg(x_2640)) :|: x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> h2(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 0 }-> i(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(h2(encArg(x_12660), encArg(x_21182), encArg(x_3294))) :|: z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ 0 }-> i(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(f(encArg(x_12658), encArg(x_21181))) :|: x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(x_12654) + encArg(x_21179)) :|: x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(x_12655) + encArg(x_21180)) :|: z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0 encode_k(z') -{ 0 }-> k(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(h2(encArg(x_11454), encArg(x_2646), encArg(x_3160))) :|: x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ 0 }-> k(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(f(encArg(x_11452), encArg(x_2645))) :|: x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11448) + encArg(x_2643)) :|: z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11449) + encArg(x_2644)) :|: x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0 f(z', z'') -{ 1 }-> h2(0, z', 1 + y + z) :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 1 }-> g(f(x, 0)) :|: z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 2 }-> g(f(x, 1 + 0 + (z'' - 1))) :|: z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 2 }-> g(f(x, 1 + (1 + x'') + y')) :|: z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 }-> h2(1 + x, y, 1 + z + u) :|: z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 }-> h2(1 + z', y, 1 + (1 + z) + u) :|: z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {encode_0}, {i}, {k}, {g}, {f}, {encArg}, {encode_k}, {encode_h}, {encode_h2}, {encode_h1}, {encode_g}, {encode_f}, {encode_j}, {encode_i}, {encode_s} Previous analysis results are: h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1] ---------------------------------------- (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 }-> k(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(h2(encArg(x_11328), encArg(x_2590), encArg(x_3146))) :|: z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ 0 }-> k(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(f(encArg(x_11326), encArg(x_2589))) :|: x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ 0 }-> k(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(1 + encArg(x_11322) + encArg(x_2587)) :|: x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 0 }-> k(1 + encArg(x_11323) + encArg(x_2588)) :|: z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ 0 }-> i(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(h2(encArg(x_11319), encArg(x_2586), encArg(x_3145))) :|: z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ 0 }-> i(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(f(encArg(x_11317), encArg(x_2585))) :|: x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ 0 }-> i(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(1 + encArg(x_11313) + encArg(x_2583)) :|: x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 0 }-> i(1 + encArg(x_11314) + encArg(x_2584)) :|: z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ 0 }-> h2(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 }-> g(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(h2(encArg(x_1113), encArg(x_250), encArg(x_311))) :|: x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 0 }-> g(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(f(encArg(x_1111), encArg(x_249))) :|: z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ 0 }-> g(0) :|: z' = 1 + 0 encArg(z') -{ 0 }-> g(0) :|: z' - 1 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1107) + encArg(x_247)) :|: z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1108) + encArg(x_248)) :|: x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0 encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 0 }-> f(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ 0 }-> g(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(h2(encArg(x_11445), encArg(x_2642), encArg(x_3159))) :|: x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> g(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(f(encArg(x_11443), encArg(x_2641))) :|: x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ 0 }-> g(0) :|: z' = 0 encode_g(z') -{ 0 }-> g(0) :|: z' >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11439) + encArg(x_2639)) :|: x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11440) + encArg(x_2640)) :|: x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> h2(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 0 }-> i(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(h2(encArg(x_12660), encArg(x_21182), encArg(x_3294))) :|: z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ 0 }-> i(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(f(encArg(x_12658), encArg(x_21181))) :|: x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(x_12654) + encArg(x_21179)) :|: x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(x_12655) + encArg(x_21180)) :|: z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0 encode_k(z') -{ 0 }-> k(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(h2(encArg(x_11454), encArg(x_2646), encArg(x_3160))) :|: x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ 0 }-> k(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(f(encArg(x_11452), encArg(x_2645))) :|: x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11448) + encArg(x_2643)) :|: z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11449) + encArg(x_2644)) :|: x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0 f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 1 }-> g(f(x, 0)) :|: z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 2 }-> g(f(x, 1 + 0 + (z'' - 1))) :|: z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 2 }-> g(f(x, 1 + (1 + x'') + y')) :|: z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {encode_0}, {i}, {k}, {g}, {f}, {encArg}, {encode_k}, {encode_h}, {encode_h2}, {encode_h1}, {encode_g}, {encode_f}, {encode_j}, {encode_i}, {encode_s} Previous analysis results are: h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1] ---------------------------------------- (33) 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 ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: encArg(z') -{ 0 }-> k(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(h2(encArg(x_11328), encArg(x_2590), encArg(x_3146))) :|: z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ 0 }-> k(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(f(encArg(x_11326), encArg(x_2589))) :|: x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ 0 }-> k(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(1 + encArg(x_11322) + encArg(x_2587)) :|: x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 0 }-> k(1 + encArg(x_11323) + encArg(x_2588)) :|: z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ 0 }-> i(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(h2(encArg(x_11319), encArg(x_2586), encArg(x_3145))) :|: z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ 0 }-> i(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(f(encArg(x_11317), encArg(x_2585))) :|: x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ 0 }-> i(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(1 + encArg(x_11313) + encArg(x_2583)) :|: x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 0 }-> i(1 + encArg(x_11314) + encArg(x_2584)) :|: z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ 0 }-> h2(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 }-> g(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(h2(encArg(x_1113), encArg(x_250), encArg(x_311))) :|: x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 0 }-> g(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(f(encArg(x_1111), encArg(x_249))) :|: z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ 0 }-> g(0) :|: z' = 1 + 0 encArg(z') -{ 0 }-> g(0) :|: z' - 1 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1107) + encArg(x_247)) :|: z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1108) + encArg(x_248)) :|: x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0 encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 0 }-> f(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ 0 }-> g(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(h2(encArg(x_11445), encArg(x_2642), encArg(x_3159))) :|: x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> g(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(f(encArg(x_11443), encArg(x_2641))) :|: x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ 0 }-> g(0) :|: z' = 0 encode_g(z') -{ 0 }-> g(0) :|: z' >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11439) + encArg(x_2639)) :|: x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11440) + encArg(x_2640)) :|: x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> h2(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 0 }-> i(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(h2(encArg(x_12660), encArg(x_21182), encArg(x_3294))) :|: z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ 0 }-> i(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(f(encArg(x_12658), encArg(x_21181))) :|: x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(x_12654) + encArg(x_21179)) :|: x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(x_12655) + encArg(x_21180)) :|: z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0 encode_k(z') -{ 0 }-> k(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(h2(encArg(x_11454), encArg(x_2646), encArg(x_3160))) :|: x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ 0 }-> k(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(f(encArg(x_11452), encArg(x_2645))) :|: x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11448) + encArg(x_2643)) :|: z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11449) + encArg(x_2644)) :|: x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0 f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 1 }-> g(f(x, 0)) :|: z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 2 }-> g(f(x, 1 + 0 + (z'' - 1))) :|: z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 2 }-> g(f(x, 1 + (1 + x'') + y')) :|: z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {encode_0}, {i}, {k}, {g}, {f}, {encArg}, {encode_k}, {encode_h}, {encode_h2}, {encode_h1}, {encode_g}, {encode_f}, {encode_j}, {encode_i}, {encode_s} Previous analysis results are: h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1] encode_0: runtime: ?, size: O(1) [0] ---------------------------------------- (35) 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 ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: encArg(z') -{ 0 }-> k(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(h2(encArg(x_11328), encArg(x_2590), encArg(x_3146))) :|: z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ 0 }-> k(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(f(encArg(x_11326), encArg(x_2589))) :|: x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ 0 }-> k(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(1 + encArg(x_11322) + encArg(x_2587)) :|: x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 0 }-> k(1 + encArg(x_11323) + encArg(x_2588)) :|: z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ 0 }-> i(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(h2(encArg(x_11319), encArg(x_2586), encArg(x_3145))) :|: z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ 0 }-> i(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(f(encArg(x_11317), encArg(x_2585))) :|: x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ 0 }-> i(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(1 + encArg(x_11313) + encArg(x_2583)) :|: x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 0 }-> i(1 + encArg(x_11314) + encArg(x_2584)) :|: z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ 0 }-> h2(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 }-> g(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(h2(encArg(x_1113), encArg(x_250), encArg(x_311))) :|: x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 0 }-> g(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(f(encArg(x_1111), encArg(x_249))) :|: z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ 0 }-> g(0) :|: z' = 1 + 0 encArg(z') -{ 0 }-> g(0) :|: z' - 1 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1107) + encArg(x_247)) :|: z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1108) + encArg(x_248)) :|: x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0 encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 0 }-> f(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ 0 }-> g(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(h2(encArg(x_11445), encArg(x_2642), encArg(x_3159))) :|: x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> g(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(f(encArg(x_11443), encArg(x_2641))) :|: x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ 0 }-> g(0) :|: z' = 0 encode_g(z') -{ 0 }-> g(0) :|: z' >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11439) + encArg(x_2639)) :|: x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11440) + encArg(x_2640)) :|: x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> h2(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 0 }-> i(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(h2(encArg(x_12660), encArg(x_21182), encArg(x_3294))) :|: z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ 0 }-> i(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(f(encArg(x_12658), encArg(x_21181))) :|: x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(x_12654) + encArg(x_21179)) :|: x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(x_12655) + encArg(x_21180)) :|: z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0 encode_k(z') -{ 0 }-> k(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(h2(encArg(x_11454), encArg(x_2646), encArg(x_3160))) :|: x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ 0 }-> k(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(f(encArg(x_11452), encArg(x_2645))) :|: x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11448) + encArg(x_2643)) :|: z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11449) + encArg(x_2644)) :|: x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0 f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 1 }-> g(f(x, 0)) :|: z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 2 }-> g(f(x, 1 + 0 + (z'' - 1))) :|: z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 2 }-> g(f(x, 1 + (1 + x'') + y')) :|: z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {i}, {k}, {g}, {f}, {encArg}, {encode_k}, {encode_h}, {encode_h2}, {encode_h1}, {encode_g}, {encode_f}, {encode_j}, {encode_i}, {encode_s} Previous analysis results are: h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1] encode_0: runtime: O(1) [0], size: O(1) [0] ---------------------------------------- (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 }-> k(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(h2(encArg(x_11328), encArg(x_2590), encArg(x_3146))) :|: z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ 0 }-> k(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(f(encArg(x_11326), encArg(x_2589))) :|: x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ 0 }-> k(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(1 + encArg(x_11322) + encArg(x_2587)) :|: x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 0 }-> k(1 + encArg(x_11323) + encArg(x_2588)) :|: z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ 0 }-> i(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(h2(encArg(x_11319), encArg(x_2586), encArg(x_3145))) :|: z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ 0 }-> i(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(f(encArg(x_11317), encArg(x_2585))) :|: x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ 0 }-> i(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(1 + encArg(x_11313) + encArg(x_2583)) :|: x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 0 }-> i(1 + encArg(x_11314) + encArg(x_2584)) :|: z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ 0 }-> h2(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 }-> g(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(h2(encArg(x_1113), encArg(x_250), encArg(x_311))) :|: x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 0 }-> g(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(f(encArg(x_1111), encArg(x_249))) :|: z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ 0 }-> g(0) :|: z' = 1 + 0 encArg(z') -{ 0 }-> g(0) :|: z' - 1 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1107) + encArg(x_247)) :|: z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1108) + encArg(x_248)) :|: x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0 encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 0 }-> f(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ 0 }-> g(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(h2(encArg(x_11445), encArg(x_2642), encArg(x_3159))) :|: x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> g(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(f(encArg(x_11443), encArg(x_2641))) :|: x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ 0 }-> g(0) :|: z' = 0 encode_g(z') -{ 0 }-> g(0) :|: z' >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11439) + encArg(x_2639)) :|: x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11440) + encArg(x_2640)) :|: x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> h2(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 0 }-> i(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(h2(encArg(x_12660), encArg(x_21182), encArg(x_3294))) :|: z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ 0 }-> i(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(f(encArg(x_12658), encArg(x_21181))) :|: x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(x_12654) + encArg(x_21179)) :|: x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(x_12655) + encArg(x_21180)) :|: z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0 encode_k(z') -{ 0 }-> k(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(h2(encArg(x_11454), encArg(x_2646), encArg(x_3160))) :|: x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ 0 }-> k(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(f(encArg(x_11452), encArg(x_2645))) :|: x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11448) + encArg(x_2643)) :|: z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11449) + encArg(x_2644)) :|: x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0 f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 1 }-> g(f(x, 0)) :|: z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 2 }-> g(f(x, 1 + 0 + (z'' - 1))) :|: z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 2 }-> g(f(x, 1 + (1 + x'') + y')) :|: z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {i}, {k}, {g}, {f}, {encArg}, {encode_k}, {encode_h}, {encode_h2}, {encode_h1}, {encode_g}, {encode_f}, {encode_j}, {encode_i}, {encode_s} Previous analysis results are: h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1] encode_0: runtime: O(1) [0], size: O(1) [0] ---------------------------------------- (39) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: i after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: encArg(z') -{ 0 }-> k(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(h2(encArg(x_11328), encArg(x_2590), encArg(x_3146))) :|: z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ 0 }-> k(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(f(encArg(x_11326), encArg(x_2589))) :|: x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ 0 }-> k(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(1 + encArg(x_11322) + encArg(x_2587)) :|: x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 0 }-> k(1 + encArg(x_11323) + encArg(x_2588)) :|: z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ 0 }-> i(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(h2(encArg(x_11319), encArg(x_2586), encArg(x_3145))) :|: z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ 0 }-> i(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(f(encArg(x_11317), encArg(x_2585))) :|: x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ 0 }-> i(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(1 + encArg(x_11313) + encArg(x_2583)) :|: x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 0 }-> i(1 + encArg(x_11314) + encArg(x_2584)) :|: z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ 0 }-> h2(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 }-> g(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(h2(encArg(x_1113), encArg(x_250), encArg(x_311))) :|: x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 0 }-> g(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(f(encArg(x_1111), encArg(x_249))) :|: z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ 0 }-> g(0) :|: z' = 1 + 0 encArg(z') -{ 0 }-> g(0) :|: z' - 1 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1107) + encArg(x_247)) :|: z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1108) + encArg(x_248)) :|: x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0 encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 0 }-> f(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ 0 }-> g(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(h2(encArg(x_11445), encArg(x_2642), encArg(x_3159))) :|: x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> g(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(f(encArg(x_11443), encArg(x_2641))) :|: x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ 0 }-> g(0) :|: z' = 0 encode_g(z') -{ 0 }-> g(0) :|: z' >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11439) + encArg(x_2639)) :|: x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11440) + encArg(x_2640)) :|: x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> h2(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 0 }-> i(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(h2(encArg(x_12660), encArg(x_21182), encArg(x_3294))) :|: z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ 0 }-> i(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(f(encArg(x_12658), encArg(x_21181))) :|: x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(x_12654) + encArg(x_21179)) :|: x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(x_12655) + encArg(x_21180)) :|: z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0 encode_k(z') -{ 0 }-> k(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(h2(encArg(x_11454), encArg(x_2646), encArg(x_3160))) :|: x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ 0 }-> k(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(f(encArg(x_11452), encArg(x_2645))) :|: x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11448) + encArg(x_2643)) :|: z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11449) + encArg(x_2644)) :|: x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0 f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 1 }-> g(f(x, 0)) :|: z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 2 }-> g(f(x, 1 + 0 + (z'' - 1))) :|: z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 2 }-> g(f(x, 1 + (1 + x'') + y')) :|: z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {i}, {k}, {g}, {f}, {encArg}, {encode_k}, {encode_h}, {encode_h2}, {encode_h1}, {encode_g}, {encode_f}, {encode_j}, {encode_i}, {encode_s} Previous analysis results are: h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1] encode_0: runtime: O(1) [0], size: O(1) [0] i: runtime: ?, size: O(n^1) [z'] ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: i after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: encArg(z') -{ 0 }-> k(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(h2(encArg(x_11328), encArg(x_2590), encArg(x_3146))) :|: z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ 0 }-> k(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(f(encArg(x_11326), encArg(x_2589))) :|: x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ 0 }-> k(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(1 + encArg(x_11322) + encArg(x_2587)) :|: x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 0 }-> k(1 + encArg(x_11323) + encArg(x_2588)) :|: z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ 0 }-> i(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(h2(encArg(x_11319), encArg(x_2586), encArg(x_3145))) :|: z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ 0 }-> i(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(f(encArg(x_11317), encArg(x_2585))) :|: x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ 0 }-> i(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(1 + encArg(x_11313) + encArg(x_2583)) :|: x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 0 }-> i(1 + encArg(x_11314) + encArg(x_2584)) :|: z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ 0 }-> h2(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 }-> g(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(h2(encArg(x_1113), encArg(x_250), encArg(x_311))) :|: x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 0 }-> g(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(f(encArg(x_1111), encArg(x_249))) :|: z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ 0 }-> g(0) :|: z' = 1 + 0 encArg(z') -{ 0 }-> g(0) :|: z' - 1 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1107) + encArg(x_247)) :|: z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1108) + encArg(x_248)) :|: x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0 encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 0 }-> f(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ 0 }-> g(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(h2(encArg(x_11445), encArg(x_2642), encArg(x_3159))) :|: x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> g(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(f(encArg(x_11443), encArg(x_2641))) :|: x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ 0 }-> g(0) :|: z' = 0 encode_g(z') -{ 0 }-> g(0) :|: z' >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11439) + encArg(x_2639)) :|: x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11440) + encArg(x_2640)) :|: x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> h2(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 0 }-> i(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(h2(encArg(x_12660), encArg(x_21182), encArg(x_3294))) :|: z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ 0 }-> i(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(f(encArg(x_12658), encArg(x_21181))) :|: x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(x_12654) + encArg(x_21179)) :|: x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(x_12655) + encArg(x_21180)) :|: z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0 encode_k(z') -{ 0 }-> k(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(h2(encArg(x_11454), encArg(x_2646), encArg(x_3160))) :|: x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ 0 }-> k(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(f(encArg(x_11452), encArg(x_2645))) :|: x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11448) + encArg(x_2643)) :|: z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11449) + encArg(x_2644)) :|: x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0 f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 1 }-> g(f(x, 0)) :|: z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 2 }-> g(f(x, 1 + 0 + (z'' - 1))) :|: z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 2 }-> g(f(x, 1 + (1 + x'') + y')) :|: z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {k}, {g}, {f}, {encArg}, {encode_k}, {encode_h}, {encode_h2}, {encode_h1}, {encode_g}, {encode_f}, {encode_j}, {encode_i}, {encode_s} Previous analysis results are: h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1] encode_0: runtime: O(1) [0], size: O(1) [0] i: runtime: O(1) [1], size: O(n^1) [z'] ---------------------------------------- (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 }-> k(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(h2(encArg(x_11328), encArg(x_2590), encArg(x_3146))) :|: z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ 0 }-> k(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(f(encArg(x_11326), encArg(x_2589))) :|: x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ 0 }-> k(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(1 + encArg(x_11322) + encArg(x_2587)) :|: x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 0 }-> k(1 + encArg(x_11323) + encArg(x_2588)) :|: z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ 0 }-> i(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(h2(encArg(x_11319), encArg(x_2586), encArg(x_3145))) :|: z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ 0 }-> i(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(f(encArg(x_11317), encArg(x_2585))) :|: x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ 0 }-> i(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(1 + encArg(x_11313) + encArg(x_2583)) :|: x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 0 }-> i(1 + encArg(x_11314) + encArg(x_2584)) :|: z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ 0 }-> h2(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 }-> g(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(h2(encArg(x_1113), encArg(x_250), encArg(x_311))) :|: x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 0 }-> g(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(f(encArg(x_1111), encArg(x_249))) :|: z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ 0 }-> g(0) :|: z' = 1 + 0 encArg(z') -{ 0 }-> g(0) :|: z' - 1 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1107) + encArg(x_247)) :|: z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1108) + encArg(x_248)) :|: x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0 encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 0 }-> f(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ 0 }-> g(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(h2(encArg(x_11445), encArg(x_2642), encArg(x_3159))) :|: x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> g(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(f(encArg(x_11443), encArg(x_2641))) :|: x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ 0 }-> g(0) :|: z' = 0 encode_g(z') -{ 0 }-> g(0) :|: z' >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11439) + encArg(x_2639)) :|: x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11440) + encArg(x_2640)) :|: x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> h2(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 0 }-> i(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(h2(encArg(x_12660), encArg(x_21182), encArg(x_3294))) :|: z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ 0 }-> i(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(f(encArg(x_12658), encArg(x_21181))) :|: x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(x_12654) + encArg(x_21179)) :|: x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(x_12655) + encArg(x_21180)) :|: z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0 encode_k(z') -{ 0 }-> k(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(h2(encArg(x_11454), encArg(x_2646), encArg(x_3160))) :|: x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ 0 }-> k(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(f(encArg(x_11452), encArg(x_2645))) :|: x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11448) + encArg(x_2643)) :|: z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11449) + encArg(x_2644)) :|: x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0 f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 1 }-> g(f(x, 0)) :|: z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 2 }-> g(f(x, 1 + 0 + (z'' - 1))) :|: z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 2 }-> g(f(x, 1 + (1 + x'') + y')) :|: z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {k}, {g}, {f}, {encArg}, {encode_k}, {encode_h}, {encode_h2}, {encode_h1}, {encode_g}, {encode_f}, {encode_j}, {encode_i}, {encode_s} Previous analysis results are: h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1] encode_0: runtime: O(1) [0], size: O(1) [0] i: runtime: O(1) [1], size: O(n^1) [z'] ---------------------------------------- (45) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: k after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z' ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: encArg(z') -{ 0 }-> k(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(h2(encArg(x_11328), encArg(x_2590), encArg(x_3146))) :|: z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ 0 }-> k(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(f(encArg(x_11326), encArg(x_2589))) :|: x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ 0 }-> k(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(1 + encArg(x_11322) + encArg(x_2587)) :|: x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 0 }-> k(1 + encArg(x_11323) + encArg(x_2588)) :|: z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ 0 }-> i(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(h2(encArg(x_11319), encArg(x_2586), encArg(x_3145))) :|: z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ 0 }-> i(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(f(encArg(x_11317), encArg(x_2585))) :|: x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ 0 }-> i(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(1 + encArg(x_11313) + encArg(x_2583)) :|: x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 0 }-> i(1 + encArg(x_11314) + encArg(x_2584)) :|: z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ 0 }-> h2(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 }-> g(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(h2(encArg(x_1113), encArg(x_250), encArg(x_311))) :|: x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 0 }-> g(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(f(encArg(x_1111), encArg(x_249))) :|: z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ 0 }-> g(0) :|: z' = 1 + 0 encArg(z') -{ 0 }-> g(0) :|: z' - 1 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1107) + encArg(x_247)) :|: z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1108) + encArg(x_248)) :|: x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0 encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 0 }-> f(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ 0 }-> g(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(h2(encArg(x_11445), encArg(x_2642), encArg(x_3159))) :|: x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> g(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(f(encArg(x_11443), encArg(x_2641))) :|: x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ 0 }-> g(0) :|: z' = 0 encode_g(z') -{ 0 }-> g(0) :|: z' >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11439) + encArg(x_2639)) :|: x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11440) + encArg(x_2640)) :|: x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> h2(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 0 }-> i(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(h2(encArg(x_12660), encArg(x_21182), encArg(x_3294))) :|: z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ 0 }-> i(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(f(encArg(x_12658), encArg(x_21181))) :|: x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(x_12654) + encArg(x_21179)) :|: x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(x_12655) + encArg(x_21180)) :|: z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0 encode_k(z') -{ 0 }-> k(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(h2(encArg(x_11454), encArg(x_2646), encArg(x_3160))) :|: x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ 0 }-> k(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(f(encArg(x_11452), encArg(x_2645))) :|: x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11448) + encArg(x_2643)) :|: z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11449) + encArg(x_2644)) :|: x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0 f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 1 }-> g(f(x, 0)) :|: z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 2 }-> g(f(x, 1 + 0 + (z'' - 1))) :|: z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 2 }-> g(f(x, 1 + (1 + x'') + y')) :|: z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {k}, {g}, {f}, {encArg}, {encode_k}, {encode_h}, {encode_h2}, {encode_h1}, {encode_g}, {encode_f}, {encode_j}, {encode_i}, {encode_s} Previous analysis results are: h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1] encode_0: runtime: O(1) [0], size: O(1) [0] i: runtime: O(1) [1], size: O(n^1) [z'] k: runtime: ?, size: O(n^1) [1 + z'] ---------------------------------------- (47) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: k after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: encArg(z') -{ 0 }-> k(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(h2(encArg(x_11328), encArg(x_2590), encArg(x_3146))) :|: z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ 0 }-> k(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(f(encArg(x_11326), encArg(x_2589))) :|: x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ 0 }-> k(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(1 + encArg(x_11322) + encArg(x_2587)) :|: x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 0 }-> k(1 + encArg(x_11323) + encArg(x_2588)) :|: z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ 0 }-> i(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(h2(encArg(x_11319), encArg(x_2586), encArg(x_3145))) :|: z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ 0 }-> i(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(f(encArg(x_11317), encArg(x_2585))) :|: x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ 0 }-> i(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(1 + encArg(x_11313) + encArg(x_2583)) :|: x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 0 }-> i(1 + encArg(x_11314) + encArg(x_2584)) :|: z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ 0 }-> h2(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 }-> g(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(h2(encArg(x_1113), encArg(x_250), encArg(x_311))) :|: x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 0 }-> g(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(f(encArg(x_1111), encArg(x_249))) :|: z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ 0 }-> g(0) :|: z' = 1 + 0 encArg(z') -{ 0 }-> g(0) :|: z' - 1 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1107) + encArg(x_247)) :|: z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1108) + encArg(x_248)) :|: x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0 encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 0 }-> f(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ 0 }-> g(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(h2(encArg(x_11445), encArg(x_2642), encArg(x_3159))) :|: x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> g(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(f(encArg(x_11443), encArg(x_2641))) :|: x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ 0 }-> g(0) :|: z' = 0 encode_g(z') -{ 0 }-> g(0) :|: z' >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11439) + encArg(x_2639)) :|: x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11440) + encArg(x_2640)) :|: x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> h2(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 0 }-> i(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(h2(encArg(x_12660), encArg(x_21182), encArg(x_3294))) :|: z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ 0 }-> i(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(f(encArg(x_12658), encArg(x_21181))) :|: x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(x_12654) + encArg(x_21179)) :|: x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(x_12655) + encArg(x_21180)) :|: z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0 encode_k(z') -{ 0 }-> k(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(h2(encArg(x_11454), encArg(x_2646), encArg(x_3160))) :|: x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ 0 }-> k(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(f(encArg(x_11452), encArg(x_2645))) :|: x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11448) + encArg(x_2643)) :|: z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11449) + encArg(x_2644)) :|: x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0 f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 1 }-> g(f(x, 0)) :|: z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 2 }-> g(f(x, 1 + 0 + (z'' - 1))) :|: z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 2 }-> g(f(x, 1 + (1 + x'') + y')) :|: z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {g}, {f}, {encArg}, {encode_k}, {encode_h}, {encode_h2}, {encode_h1}, {encode_g}, {encode_f}, {encode_j}, {encode_i}, {encode_s} Previous analysis results are: h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1] encode_0: runtime: O(1) [0], size: O(1) [0] i: runtime: O(1) [1], size: O(n^1) [z'] k: runtime: O(1) [1], size: O(n^1) [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 }-> k(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(h2(encArg(x_11328), encArg(x_2590), encArg(x_3146))) :|: z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ 0 }-> k(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(f(encArg(x_11326), encArg(x_2589))) :|: x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ 0 }-> k(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(1 + encArg(x_11322) + encArg(x_2587)) :|: x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 0 }-> k(1 + encArg(x_11323) + encArg(x_2588)) :|: z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ 0 }-> i(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(h2(encArg(x_11319), encArg(x_2586), encArg(x_3145))) :|: z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ 0 }-> i(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(f(encArg(x_11317), encArg(x_2585))) :|: x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ 0 }-> i(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(1 + encArg(x_11313) + encArg(x_2583)) :|: x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 0 }-> i(1 + encArg(x_11314) + encArg(x_2584)) :|: z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ 0 }-> h2(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 }-> g(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(h2(encArg(x_1113), encArg(x_250), encArg(x_311))) :|: x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 0 }-> g(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(f(encArg(x_1111), encArg(x_249))) :|: z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ 0 }-> g(0) :|: z' = 1 + 0 encArg(z') -{ 0 }-> g(0) :|: z' - 1 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1107) + encArg(x_247)) :|: z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1108) + encArg(x_248)) :|: x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0 encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 0 }-> f(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ 0 }-> g(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(h2(encArg(x_11445), encArg(x_2642), encArg(x_3159))) :|: x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> g(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(f(encArg(x_11443), encArg(x_2641))) :|: x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ 0 }-> g(0) :|: z' = 0 encode_g(z') -{ 0 }-> g(0) :|: z' >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11439) + encArg(x_2639)) :|: x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11440) + encArg(x_2640)) :|: x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> h2(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 0 }-> i(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(h2(encArg(x_12660), encArg(x_21182), encArg(x_3294))) :|: z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ 0 }-> i(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(f(encArg(x_12658), encArg(x_21181))) :|: x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(x_12654) + encArg(x_21179)) :|: x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(x_12655) + encArg(x_21180)) :|: z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0 encode_k(z') -{ 0 }-> k(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(h2(encArg(x_11454), encArg(x_2646), encArg(x_3160))) :|: x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ 0 }-> k(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(f(encArg(x_11452), encArg(x_2645))) :|: x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11448) + encArg(x_2643)) :|: z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11449) + encArg(x_2644)) :|: x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0 f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 1 }-> g(f(x, 0)) :|: z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 2 }-> g(f(x, 1 + 0 + (z'' - 1))) :|: z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 2 }-> g(f(x, 1 + (1 + x'') + y')) :|: z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {g}, {f}, {encArg}, {encode_k}, {encode_h}, {encode_h2}, {encode_h1}, {encode_g}, {encode_f}, {encode_j}, {encode_i}, {encode_s} Previous analysis results are: h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1] encode_0: runtime: O(1) [0], size: O(1) [0] i: runtime: O(1) [1], size: O(n^1) [z'] k: runtime: O(1) [1], size: O(n^1) [1 + z'] ---------------------------------------- (51) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: g after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z' ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: encArg(z') -{ 0 }-> k(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(h2(encArg(x_11328), encArg(x_2590), encArg(x_3146))) :|: z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ 0 }-> k(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(f(encArg(x_11326), encArg(x_2589))) :|: x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ 0 }-> k(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(1 + encArg(x_11322) + encArg(x_2587)) :|: x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 0 }-> k(1 + encArg(x_11323) + encArg(x_2588)) :|: z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ 0 }-> i(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(h2(encArg(x_11319), encArg(x_2586), encArg(x_3145))) :|: z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ 0 }-> i(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(f(encArg(x_11317), encArg(x_2585))) :|: x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ 0 }-> i(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(1 + encArg(x_11313) + encArg(x_2583)) :|: x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 0 }-> i(1 + encArg(x_11314) + encArg(x_2584)) :|: z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ 0 }-> h2(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 }-> g(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(h2(encArg(x_1113), encArg(x_250), encArg(x_311))) :|: x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 0 }-> g(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(f(encArg(x_1111), encArg(x_249))) :|: z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ 0 }-> g(0) :|: z' = 1 + 0 encArg(z') -{ 0 }-> g(0) :|: z' - 1 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1107) + encArg(x_247)) :|: z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1108) + encArg(x_248)) :|: x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0 encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 0 }-> f(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ 0 }-> g(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(h2(encArg(x_11445), encArg(x_2642), encArg(x_3159))) :|: x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> g(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(f(encArg(x_11443), encArg(x_2641))) :|: x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ 0 }-> g(0) :|: z' = 0 encode_g(z') -{ 0 }-> g(0) :|: z' >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11439) + encArg(x_2639)) :|: x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11440) + encArg(x_2640)) :|: x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> h2(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 0 }-> i(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(h2(encArg(x_12660), encArg(x_21182), encArg(x_3294))) :|: z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ 0 }-> i(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(f(encArg(x_12658), encArg(x_21181))) :|: x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(x_12654) + encArg(x_21179)) :|: x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(x_12655) + encArg(x_21180)) :|: z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0 encode_k(z') -{ 0 }-> k(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(h2(encArg(x_11454), encArg(x_2646), encArg(x_3160))) :|: x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ 0 }-> k(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(f(encArg(x_11452), encArg(x_2645))) :|: x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11448) + encArg(x_2643)) :|: z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11449) + encArg(x_2644)) :|: x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0 f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 1 }-> g(f(x, 0)) :|: z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 2 }-> g(f(x, 1 + 0 + (z'' - 1))) :|: z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 2 }-> g(f(x, 1 + (1 + x'') + y')) :|: z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {g}, {f}, {encArg}, {encode_k}, {encode_h}, {encode_h2}, {encode_h1}, {encode_g}, {encode_f}, {encode_j}, {encode_i}, {encode_s} Previous analysis results are: h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1] encode_0: runtime: O(1) [0], size: O(1) [0] i: runtime: O(1) [1], size: O(n^1) [z'] k: runtime: O(1) [1], size: O(n^1) [1 + z'] g: runtime: ?, size: O(n^1) [1 + z'] ---------------------------------------- (53) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: g after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z' ---------------------------------------- (54) Obligation: Complexity RNTS consisting of the following rules: encArg(z') -{ 0 }-> k(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(h2(encArg(x_11328), encArg(x_2590), encArg(x_3146))) :|: z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ 0 }-> k(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(f(encArg(x_11326), encArg(x_2589))) :|: x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ 0 }-> k(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(1 + encArg(x_11322) + encArg(x_2587)) :|: x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 0 }-> k(1 + encArg(x_11323) + encArg(x_2588)) :|: z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ 0 }-> i(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(h2(encArg(x_11319), encArg(x_2586), encArg(x_3145))) :|: z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ 0 }-> i(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(f(encArg(x_11317), encArg(x_2585))) :|: x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ 0 }-> i(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(1 + encArg(x_11313) + encArg(x_2583)) :|: x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 0 }-> i(1 + encArg(x_11314) + encArg(x_2584)) :|: z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ 0 }-> h2(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 }-> g(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(h2(encArg(x_1113), encArg(x_250), encArg(x_311))) :|: x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 0 }-> g(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(f(encArg(x_1111), encArg(x_249))) :|: z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ 0 }-> g(0) :|: z' = 1 + 0 encArg(z') -{ 0 }-> g(0) :|: z' - 1 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1107) + encArg(x_247)) :|: z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1108) + encArg(x_248)) :|: x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0 encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 0 }-> f(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ 0 }-> g(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(h2(encArg(x_11445), encArg(x_2642), encArg(x_3159))) :|: x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> g(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(f(encArg(x_11443), encArg(x_2641))) :|: x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ 0 }-> g(0) :|: z' = 0 encode_g(z') -{ 0 }-> g(0) :|: z' >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11439) + encArg(x_2639)) :|: x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11440) + encArg(x_2640)) :|: x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> h2(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 0 }-> i(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(h2(encArg(x_12660), encArg(x_21182), encArg(x_3294))) :|: z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ 0 }-> i(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(f(encArg(x_12658), encArg(x_21181))) :|: x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(x_12654) + encArg(x_21179)) :|: x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(x_12655) + encArg(x_21180)) :|: z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0 encode_k(z') -{ 0 }-> k(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(h2(encArg(x_11454), encArg(x_2646), encArg(x_3160))) :|: x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ 0 }-> k(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(f(encArg(x_11452), encArg(x_2645))) :|: x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11448) + encArg(x_2643)) :|: z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11449) + encArg(x_2644)) :|: x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0 f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 1 }-> g(f(x, 0)) :|: z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 2 }-> g(f(x, 1 + 0 + (z'' - 1))) :|: z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 2 }-> g(f(x, 1 + (1 + x'') + y')) :|: z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {f}, {encArg}, {encode_k}, {encode_h}, {encode_h2}, {encode_h1}, {encode_g}, {encode_f}, {encode_j}, {encode_i}, {encode_s} Previous analysis results are: h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1] encode_0: runtime: O(1) [0], size: O(1) [0] i: runtime: O(1) [1], size: O(n^1) [z'] k: runtime: O(1) [1], size: O(n^1) [1 + z'] g: runtime: O(n^1) [1 + z'], size: O(n^1) [1 + 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') -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z' = 1 + 0 encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0 + 1, z' - 1 >= 0 encArg(z') -{ 0 }-> k(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(h2(encArg(x_11328), encArg(x_2590), encArg(x_3146))) :|: z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ 0 }-> k(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(f(encArg(x_11326), encArg(x_2589))) :|: x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ 0 }-> k(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(1 + encArg(x_11322) + encArg(x_2587)) :|: x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 0 }-> k(1 + encArg(x_11323) + encArg(x_2588)) :|: z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ 0 }-> i(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(h2(encArg(x_11319), encArg(x_2586), encArg(x_3145))) :|: z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ 0 }-> i(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(f(encArg(x_11317), encArg(x_2585))) :|: x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ 0 }-> i(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(1 + encArg(x_11313) + encArg(x_2583)) :|: x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 0 }-> i(1 + encArg(x_11314) + encArg(x_2584)) :|: z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ 0 }-> h2(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 }-> g(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(h2(encArg(x_1113), encArg(x_250), encArg(x_311))) :|: x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 0 }-> g(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(f(encArg(x_1111), encArg(x_249))) :|: z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1107) + encArg(x_247)) :|: z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1108) + encArg(x_248)) :|: x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0 encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 0 }-> f(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0 + 1, z' = 0 encode_g(z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 1, z' >= 0 encode_g(z') -{ 0 }-> g(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(h2(encArg(x_11445), encArg(x_2642), encArg(x_3159))) :|: x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> g(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(f(encArg(x_11443), encArg(x_2641))) :|: x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11439) + encArg(x_2639)) :|: x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11440) + encArg(x_2640)) :|: x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> h2(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 0 }-> i(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(h2(encArg(x_12660), encArg(x_21182), encArg(x_3294))) :|: z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ 0 }-> i(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(f(encArg(x_12658), encArg(x_21181))) :|: x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(x_12654) + encArg(x_21179)) :|: x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(x_12655) + encArg(x_21180)) :|: z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0 encode_k(z') -{ 0 }-> k(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(h2(encArg(x_11454), encArg(x_2646), encArg(x_3160))) :|: x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ 0 }-> k(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(f(encArg(x_11452), encArg(x_2645))) :|: x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11448) + encArg(x_2643)) :|: z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11449) + encArg(x_2644)) :|: x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0 f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 1 }-> g(f(x, 0)) :|: z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 2 }-> g(f(x, 1 + 0 + (z'' - 1))) :|: z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 2 }-> g(f(x, 1 + (1 + x'') + y')) :|: z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {f}, {encArg}, {encode_k}, {encode_h}, {encode_h2}, {encode_h1}, {encode_g}, {encode_f}, {encode_j}, {encode_i}, {encode_s} Previous analysis results are: h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1] encode_0: runtime: O(1) [0], size: O(1) [0] i: runtime: O(1) [1], size: O(n^1) [z'] k: runtime: O(1) [1], size: O(n^1) [1 + z'] g: runtime: O(n^1) [1 + z'], size: O(n^1) [1 + z'] ---------------------------------------- (57) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: f after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z' + z'' ---------------------------------------- (58) Obligation: Complexity RNTS consisting of the following rules: encArg(z') -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z' = 1 + 0 encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0 + 1, z' - 1 >= 0 encArg(z') -{ 0 }-> k(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(h2(encArg(x_11328), encArg(x_2590), encArg(x_3146))) :|: z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ 0 }-> k(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(f(encArg(x_11326), encArg(x_2589))) :|: x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ 0 }-> k(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(1 + encArg(x_11322) + encArg(x_2587)) :|: x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 0 }-> k(1 + encArg(x_11323) + encArg(x_2588)) :|: z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ 0 }-> i(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(h2(encArg(x_11319), encArg(x_2586), encArg(x_3145))) :|: z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ 0 }-> i(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(f(encArg(x_11317), encArg(x_2585))) :|: x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ 0 }-> i(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(1 + encArg(x_11313) + encArg(x_2583)) :|: x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 0 }-> i(1 + encArg(x_11314) + encArg(x_2584)) :|: z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ 0 }-> h2(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 }-> g(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(h2(encArg(x_1113), encArg(x_250), encArg(x_311))) :|: x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 0 }-> g(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(f(encArg(x_1111), encArg(x_249))) :|: z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1107) + encArg(x_247)) :|: z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1108) + encArg(x_248)) :|: x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0 encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 0 }-> f(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0 + 1, z' = 0 encode_g(z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 1, z' >= 0 encode_g(z') -{ 0 }-> g(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(h2(encArg(x_11445), encArg(x_2642), encArg(x_3159))) :|: x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> g(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(f(encArg(x_11443), encArg(x_2641))) :|: x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11439) + encArg(x_2639)) :|: x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11440) + encArg(x_2640)) :|: x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> h2(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 0 }-> i(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(h2(encArg(x_12660), encArg(x_21182), encArg(x_3294))) :|: z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ 0 }-> i(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(f(encArg(x_12658), encArg(x_21181))) :|: x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(x_12654) + encArg(x_21179)) :|: x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(x_12655) + encArg(x_21180)) :|: z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0 encode_k(z') -{ 0 }-> k(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(h2(encArg(x_11454), encArg(x_2646), encArg(x_3160))) :|: x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ 0 }-> k(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(f(encArg(x_11452), encArg(x_2645))) :|: x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11448) + encArg(x_2643)) :|: z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11449) + encArg(x_2644)) :|: x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0 f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 1 }-> g(f(x, 0)) :|: z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 2 }-> g(f(x, 1 + 0 + (z'' - 1))) :|: z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 2 }-> g(f(x, 1 + (1 + x'') + y')) :|: z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {f}, {encArg}, {encode_k}, {encode_h}, {encode_h2}, {encode_h1}, {encode_g}, {encode_f}, {encode_j}, {encode_i}, {encode_s} Previous analysis results are: h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1] encode_0: runtime: O(1) [0], size: O(1) [0] i: runtime: O(1) [1], size: O(n^1) [z'] k: runtime: O(1) [1], size: O(n^1) [1 + z'] g: runtime: O(n^1) [1 + z'], size: O(n^1) [1 + z'] f: runtime: ?, size: O(n^1) [1 + z' + z''] ---------------------------------------- (59) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: f after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 8 + 6*z' + 2*z'*z'' + z'^2 + 4*z'' + z''^2 ---------------------------------------- (60) Obligation: Complexity RNTS consisting of the following rules: encArg(z') -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z' = 1 + 0 encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0 + 1, z' - 1 >= 0 encArg(z') -{ 0 }-> k(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(h2(encArg(x_11328), encArg(x_2590), encArg(x_3146))) :|: z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ 0 }-> k(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(f(encArg(x_11326), encArg(x_2589))) :|: x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ 0 }-> k(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(1 + encArg(x_11322) + encArg(x_2587)) :|: x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 0 }-> k(1 + encArg(x_11323) + encArg(x_2588)) :|: z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ 0 }-> i(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(h2(encArg(x_11319), encArg(x_2586), encArg(x_3145))) :|: z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ 0 }-> i(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(f(encArg(x_11317), encArg(x_2585))) :|: x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ 0 }-> i(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(1 + encArg(x_11313) + encArg(x_2583)) :|: x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 0 }-> i(1 + encArg(x_11314) + encArg(x_2584)) :|: z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ 0 }-> h2(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 }-> g(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(h2(encArg(x_1113), encArg(x_250), encArg(x_311))) :|: x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 0 }-> g(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(f(encArg(x_1111), encArg(x_249))) :|: z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1107) + encArg(x_247)) :|: z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1108) + encArg(x_248)) :|: x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0 encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 0 }-> f(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0 + 1, z' = 0 encode_g(z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 1, z' >= 0 encode_g(z') -{ 0 }-> g(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(h2(encArg(x_11445), encArg(x_2642), encArg(x_3159))) :|: x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> g(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(f(encArg(x_11443), encArg(x_2641))) :|: x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11439) + encArg(x_2639)) :|: x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11440) + encArg(x_2640)) :|: x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> h2(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 0 }-> i(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(h2(encArg(x_12660), encArg(x_21182), encArg(x_3294))) :|: z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ 0 }-> i(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(f(encArg(x_12658), encArg(x_21181))) :|: x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(x_12654) + encArg(x_21179)) :|: x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(x_12655) + encArg(x_21180)) :|: z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0 encode_k(z') -{ 0 }-> k(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(h2(encArg(x_11454), encArg(x_2646), encArg(x_3160))) :|: x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ 0 }-> k(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(f(encArg(x_11452), encArg(x_2645))) :|: x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11448) + encArg(x_2643)) :|: z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11449) + encArg(x_2644)) :|: x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0 f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 1 }-> g(f(x, 0)) :|: z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 2 }-> g(f(x, 1 + 0 + (z'' - 1))) :|: z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 2 }-> g(f(x, 1 + (1 + x'') + y')) :|: z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {encArg}, {encode_k}, {encode_h}, {encode_h2}, {encode_h1}, {encode_g}, {encode_f}, {encode_j}, {encode_i}, {encode_s} Previous analysis results are: h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1] encode_0: runtime: O(1) [0], size: O(1) [0] i: runtime: O(1) [1], size: O(n^1) [z'] k: runtime: O(1) [1], size: O(n^1) [1 + z'] g: runtime: O(n^1) [1 + z'], size: O(n^1) [1 + z'] f: runtime: O(n^2) [8 + 6*z' + 2*z'*z'' + z'^2 + 4*z'' + z''^2], size: O(n^1) [1 + z' + 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') -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z' = 1 + 0 encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0 + 1, z' - 1 >= 0 encArg(z') -{ 0 }-> k(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(h2(encArg(x_11328), encArg(x_2590), encArg(x_3146))) :|: z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ 0 }-> k(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(f(encArg(x_11326), encArg(x_2589))) :|: x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ 0 }-> k(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(1 + encArg(x_11322) + encArg(x_2587)) :|: x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 0 }-> k(1 + encArg(x_11323) + encArg(x_2588)) :|: z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ 0 }-> i(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(h2(encArg(x_11319), encArg(x_2586), encArg(x_3145))) :|: z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ 0 }-> i(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(f(encArg(x_11317), encArg(x_2585))) :|: x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ 0 }-> i(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(1 + encArg(x_11313) + encArg(x_2583)) :|: x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 0 }-> i(1 + encArg(x_11314) + encArg(x_2584)) :|: z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ 0 }-> h2(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 }-> g(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(h2(encArg(x_1113), encArg(x_250), encArg(x_311))) :|: x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 0 }-> g(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(f(encArg(x_1111), encArg(x_249))) :|: z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1107) + encArg(x_247)) :|: z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1108) + encArg(x_248)) :|: x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0 encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 0 }-> f(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0 + 1, z' = 0 encode_g(z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 1, z' >= 0 encode_g(z') -{ 0 }-> g(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(h2(encArg(x_11445), encArg(x_2642), encArg(x_3159))) :|: x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> g(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(f(encArg(x_11443), encArg(x_2641))) :|: x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11439) + encArg(x_2639)) :|: x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11440) + encArg(x_2640)) :|: x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> h2(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 0 }-> i(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(h2(encArg(x_12660), encArg(x_21182), encArg(x_3294))) :|: z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ 0 }-> i(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(f(encArg(x_12658), encArg(x_21181))) :|: x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(x_12654) + encArg(x_21179)) :|: x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(x_12655) + encArg(x_21180)) :|: z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0 encode_k(z') -{ 0 }-> k(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(h2(encArg(x_11454), encArg(x_2646), encArg(x_3160))) :|: x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ 0 }-> k(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(f(encArg(x_11452), encArg(x_2645))) :|: x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11448) + encArg(x_2643)) :|: z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11449) + encArg(x_2644)) :|: x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0 f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 10 + s9 + 6*x + x^2 }-> s10 :|: s9 >= 0, s9 <= x + 0 + 1, s10 >= 0, s10 <= s9 + 1, z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 11 + s5 + 6*x + 2*x*z'' + x^2 + 4*z'' + z''^2 }-> s6 :|: s5 >= 0, s5 <= x + (1 + 0 + (z'' - 1)) + 1, s6 >= 0, s6 <= s5 + 1, z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 23 + s7 + 10*x + 2*x*x'' + 2*x*y' + x^2 + 8*x'' + 2*x''*y' + x''^2 + 8*y' + y'^2 }-> s8 :|: s7 >= 0, s7 <= x + (1 + (1 + x'') + y') + 1, s8 >= 0, s8 <= s7 + 1, z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {encArg}, {encode_k}, {encode_h}, {encode_h2}, {encode_h1}, {encode_g}, {encode_f}, {encode_j}, {encode_i}, {encode_s} Previous analysis results are: h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1] encode_0: runtime: O(1) [0], size: O(1) [0] i: runtime: O(1) [1], size: O(n^1) [z'] k: runtime: O(1) [1], size: O(n^1) [1 + z'] g: runtime: O(n^1) [1 + z'], size: O(n^1) [1 + z'] f: runtime: O(n^2) [8 + 6*z' + 2*z'*z'' + z'^2 + 4*z'' + z''^2], size: O(n^1) [1 + z' + z''] ---------------------------------------- (63) 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: 1 + z' ---------------------------------------- (64) Obligation: Complexity RNTS consisting of the following rules: encArg(z') -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z' = 1 + 0 encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0 + 1, z' - 1 >= 0 encArg(z') -{ 0 }-> k(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(h2(encArg(x_11328), encArg(x_2590), encArg(x_3146))) :|: z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ 0 }-> k(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(f(encArg(x_11326), encArg(x_2589))) :|: x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ 0 }-> k(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(1 + encArg(x_11322) + encArg(x_2587)) :|: x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 0 }-> k(1 + encArg(x_11323) + encArg(x_2588)) :|: z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ 0 }-> i(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(h2(encArg(x_11319), encArg(x_2586), encArg(x_3145))) :|: z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ 0 }-> i(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(f(encArg(x_11317), encArg(x_2585))) :|: x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ 0 }-> i(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(1 + encArg(x_11313) + encArg(x_2583)) :|: x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 0 }-> i(1 + encArg(x_11314) + encArg(x_2584)) :|: z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ 0 }-> h2(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 }-> g(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(h2(encArg(x_1113), encArg(x_250), encArg(x_311))) :|: x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 0 }-> g(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(f(encArg(x_1111), encArg(x_249))) :|: z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1107) + encArg(x_247)) :|: z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1108) + encArg(x_248)) :|: x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0 encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 0 }-> f(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0 + 1, z' = 0 encode_g(z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 1, z' >= 0 encode_g(z') -{ 0 }-> g(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(h2(encArg(x_11445), encArg(x_2642), encArg(x_3159))) :|: x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> g(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(f(encArg(x_11443), encArg(x_2641))) :|: x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11439) + encArg(x_2639)) :|: x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11440) + encArg(x_2640)) :|: x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> h2(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 0 }-> i(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(h2(encArg(x_12660), encArg(x_21182), encArg(x_3294))) :|: z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ 0 }-> i(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(f(encArg(x_12658), encArg(x_21181))) :|: x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(x_12654) + encArg(x_21179)) :|: x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(x_12655) + encArg(x_21180)) :|: z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0 encode_k(z') -{ 0 }-> k(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(h2(encArg(x_11454), encArg(x_2646), encArg(x_3160))) :|: x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ 0 }-> k(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(f(encArg(x_11452), encArg(x_2645))) :|: x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11448) + encArg(x_2643)) :|: z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11449) + encArg(x_2644)) :|: x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0 f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 10 + s9 + 6*x + x^2 }-> s10 :|: s9 >= 0, s9 <= x + 0 + 1, s10 >= 0, s10 <= s9 + 1, z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 11 + s5 + 6*x + 2*x*z'' + x^2 + 4*z'' + z''^2 }-> s6 :|: s5 >= 0, s5 <= x + (1 + 0 + (z'' - 1)) + 1, s6 >= 0, s6 <= s5 + 1, z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 23 + s7 + 10*x + 2*x*x'' + 2*x*y' + x^2 + 8*x'' + 2*x''*y' + x''^2 + 8*y' + y'^2 }-> s8 :|: s7 >= 0, s7 <= x + (1 + (1 + x'') + y') + 1, s8 >= 0, s8 <= s7 + 1, z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {encArg}, {encode_k}, {encode_h}, {encode_h2}, {encode_h1}, {encode_g}, {encode_f}, {encode_j}, {encode_i}, {encode_s} Previous analysis results are: h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1] encode_0: runtime: O(1) [0], size: O(1) [0] i: runtime: O(1) [1], size: O(n^1) [z'] k: runtime: O(1) [1], size: O(n^1) [1 + z'] g: runtime: O(n^1) [1 + z'], size: O(n^1) [1 + z'] f: runtime: O(n^2) [8 + 6*z' + 2*z'*z'' + z'^2 + 4*z'' + z''^2], size: O(n^1) [1 + z' + z''] encArg: runtime: ?, size: O(n^1) [1 + z'] ---------------------------------------- (65) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: encArg after applying outer abstraction to obtain an ITS, resulting in: O(n^3) with polynomial bound: 126 + 467*z' + 283*z'^2 + 48*z'^3 ---------------------------------------- (66) Obligation: Complexity RNTS consisting of the following rules: encArg(z') -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z' = 1 + 0 encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0 + 1, z' - 1 >= 0 encArg(z') -{ 0 }-> k(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(h2(encArg(x_11328), encArg(x_2590), encArg(x_3146))) :|: z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ 0 }-> k(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(f(encArg(x_11326), encArg(x_2589))) :|: x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ 0 }-> k(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> k(1 + encArg(x_11322) + encArg(x_2587)) :|: x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 0 }-> k(1 + encArg(x_11323) + encArg(x_2588)) :|: z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ 0 }-> i(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(h2(encArg(x_11319), encArg(x_2586), encArg(x_3145))) :|: z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ 0 }-> i(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(f(encArg(x_11317), encArg(x_2585))) :|: x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ 0 }-> i(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> i(1 + encArg(x_11313) + encArg(x_2583)) :|: x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 0 }-> i(1 + encArg(x_11314) + encArg(x_2584)) :|: z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ 0 }-> h2(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 }-> g(k(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(i(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(h2(encArg(x_1113), encArg(x_250), encArg(x_311))) :|: x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 0 }-> g(g(encArg(z' - 2))) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(f(encArg(x_1111), encArg(x_249))) :|: z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(z' - 2)) :|: z' - 2 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1107) + encArg(x_247)) :|: z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 0 }-> g(1 + encArg(x_1108) + encArg(x_248)) :|: x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 1 + encArg(z' - 1) :|: z' - 1 >= 0 encArg(z') -{ 0 }-> 1 + encArg(x_1) + encArg(x_2) :|: x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 0 }-> f(encArg(z'), encArg(z'')) :|: z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0 + 1, z' = 0 encode_g(z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 1, z' >= 0 encode_g(z') -{ 0 }-> g(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(h2(encArg(x_11445), encArg(x_2642), encArg(x_3159))) :|: x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> g(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(f(encArg(x_11443), encArg(x_2641))) :|: x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11439) + encArg(x_2639)) :|: x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 0 }-> g(1 + encArg(x_11440) + encArg(x_2640)) :|: x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> h2(encArg(z'), encArg(z''), encArg(z1)) :|: z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 0 }-> i(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(h2(encArg(x_12660), encArg(x_21182), encArg(x_3294))) :|: z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ 0 }-> i(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(f(encArg(x_12658), encArg(x_21181))) :|: x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(x_12654) + encArg(x_21179)) :|: x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 0 }-> i(1 + encArg(x_12655) + encArg(x_21180)) :|: z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 0 }-> 1 + encArg(z') + encArg(z'') :|: z' >= 0, z'' >= 0 encode_k(z') -{ 0 }-> k(k(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(i(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(h2(encArg(x_11454), encArg(x_2646), encArg(x_3160))) :|: x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ 0 }-> k(g(encArg(z' - 1))) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(f(encArg(x_11452), encArg(x_2645))) :|: x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(z' - 1)) :|: z' - 1 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11448) + encArg(x_2643)) :|: z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 0 }-> k(1 + encArg(x_11449) + encArg(x_2644)) :|: x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 0 }-> 1 + encArg(z') :|: z' >= 0 f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 10 + s9 + 6*x + x^2 }-> s10 :|: s9 >= 0, s9 <= x + 0 + 1, s10 >= 0, s10 <= s9 + 1, z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 11 + s5 + 6*x + 2*x*z'' + x^2 + 4*z'' + z''^2 }-> s6 :|: s5 >= 0, s5 <= x + (1 + 0 + (z'' - 1)) + 1, s6 >= 0, s6 <= s5 + 1, z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 23 + s7 + 10*x + 2*x*x'' + 2*x*y' + x^2 + 8*x'' + 2*x''*y' + x''^2 + 8*y' + y'^2 }-> s8 :|: s7 >= 0, s7 <= x + (1 + (1 + x'') + y') + 1, s8 >= 0, s8 <= s7 + 1, z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {encode_k}, {encode_h}, {encode_h2}, {encode_h1}, {encode_g}, {encode_f}, {encode_j}, {encode_i}, {encode_s} Previous analysis results are: h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1] encode_0: runtime: O(1) [0], size: O(1) [0] i: runtime: O(1) [1], size: O(n^1) [z'] k: runtime: O(1) [1], size: O(n^1) [1 + z'] g: runtime: O(n^1) [1 + z'], size: O(n^1) [1 + z'] f: runtime: O(n^2) [8 + 6*z' + 2*z'*z'' + z'^2 + 4*z'' + z''^2], size: O(n^1) [1 + z' + z''] encArg: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [1 + 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') -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z' = 1 + 0 encArg(z') -{ -58 + s127 + -89*z' + -5*z'^2 + 48*z'^3 }-> s128 :|: s126 >= 0, s126 <= z' - 2 + 1, s127 >= 0, s127 <= s126, s128 >= 0, s128 <= s127 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11313 + 283*x_11313^2 + 48*x_11313^3 + 467*x_2583 + 283*x_2583^2 + 48*x_2583^3 }-> s131 :|: s129 >= 0, s129 <= x_11313 + 1, s130 >= 0, s130 <= x_2583 + 1, s131 >= 0, s131 <= 1 + s129 + s130, x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 253 + 467*x_11314 + 283*x_11314^2 + 48*x_11314^3 + 467*x_2584 + 283*x_2584^2 + 48*x_2584^3 }-> s134 :|: s132 >= 0, s132 <= x_11314 + 1, s133 >= 0, s133 <= x_2584 + 1, s134 >= 0, s134 <= 1 + s132 + s133, z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s136 :|: s135 >= 0, s135 <= z' - 2 + 1, s136 >= 0, s136 <= 1 + s135, z' - 2 >= 0 encArg(z') -{ 261 + 6*s137 + 2*s137*s138 + s137^2 + 4*s138 + s138^2 + 467*x_11317 + 283*x_11317^2 + 48*x_11317^3 + 467*x_2585 + 283*x_2585^2 + 48*x_2585^3 }-> s140 :|: s137 >= 0, s137 <= x_11317 + 1, s138 >= 0, s138 <= x_2585 + 1, s139 >= 0, s139 <= s137 + s138 + 1, s140 >= 0, s140 <= s139, x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ -58 + s141 + -89*z' + -5*z'^2 + 48*z'^3 }-> s143 :|: s141 >= 0, s141 <= z' - 2 + 1, s142 >= 0, s142 <= s141 + 1, s143 >= 0, s143 <= s142, z' - 2 >= 0 encArg(z') -{ 379 + s145 + 467*x_11319 + 283*x_11319^2 + 48*x_11319^3 + 467*x_2586 + 283*x_2586^2 + 48*x_2586^3 + 467*x_3145 + 283*x_3145^2 + 48*x_3145^3 }-> s148 :|: s144 >= 0, s144 <= x_11319 + 1, s145 >= 0, s145 <= x_2586 + 1, s146 >= 0, s146 <= x_3145 + 1, s147 >= 0, s147 <= s144 + s145 + s146 + 1, s148 >= 0, s148 <= s147, z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s151 :|: s149 >= 0, s149 <= z' - 2 + 1, s150 >= 0, s150 <= s149, s151 >= 0, s151 <= s150, z' - 2 >= 0 encArg(z') -{ 260 + 6*s14 + 2*s14*s15 + s14^2 + 4*s15 + s15^2 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> s16 :|: s14 >= 0, s14 <= x_1 + 1, s15 >= 0, s15 <= x_2 + 1, s16 >= 0, s16 <= s14 + s15 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s180 :|: s178 >= 0, s178 <= z' - 2 + 1, s179 >= 0, s179 <= s178 + 1, s180 >= 0, s180 <= s179, z' - 2 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s183 :|: s181 >= 0, s181 <= z' - 2 + 1, s182 >= 0, s182 <= s181, s183 >= 0, s183 <= s182 + 1, z' - 2 >= 0 encArg(z') -{ 254 + s17 + s18 + 467*x_1107 + 283*x_1107^2 + 48*x_1107^3 + 467*x_247 + 283*x_247^2 + 48*x_247^3 }-> s19 :|: s17 >= 0, s17 <= x_1107 + 1, s18 >= 0, s18 <= x_247 + 1, s19 >= 0, s19 <= 1 + s17 + s18 + 1, z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0 + 1, z' - 1 >= 0 encArg(z') -{ 254 + s20 + s21 + 467*x_1108 + 283*x_1108^2 + 48*x_1108^3 + 467*x_248 + 283*x_248^2 + 48*x_248^3 }-> s22 :|: s20 >= 0, s20 <= x_1108 + 1, s21 >= 0, s21 <= x_248 + 1, s22 >= 0, s22 <= 1 + s20 + s21 + 1, x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ -58 + s23 + -89*z' + -5*z'^2 + 48*z'^3 }-> s24 :|: s23 >= 0, s23 <= z' - 2 + 1, s24 >= 0, s24 <= 1 + s23 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s25 + 2*s25*s26 + s25^2 + 4*s26 + s26^2 + s27 + 467*x_1111 + 283*x_1111^2 + 48*x_1111^3 + 467*x_249 + 283*x_249^2 + 48*x_249^3 }-> s28 :|: s25 >= 0, s25 <= x_1111 + 1, s26 >= 0, s26 <= x_249 + 1, s27 >= 0, s27 <= s25 + s26 + 1, s28 >= 0, s28 <= s27 + 1, z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ -58 + s29 + s30 + -89*z' + -5*z'^2 + 48*z'^3 }-> s31 :|: s29 >= 0, s29 <= z' - 2 + 1, s30 >= 0, s30 <= s29 + 1, s31 >= 0, s31 <= s30 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s33 + s35 + 467*x_1113 + 283*x_1113^2 + 48*x_1113^3 + 467*x_250 + 283*x_250^2 + 48*x_250^3 + 467*x_311 + 283*x_311^2 + 48*x_311^3 }-> s36 :|: s32 >= 0, s32 <= x_1113 + 1, s33 >= 0, s33 <= x_250 + 1, s34 >= 0, s34 <= x_311 + 1, s35 >= 0, s35 <= s32 + s33 + s34 + 1, s36 >= 0, s36 <= s35 + 1, x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 378 + s38 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 + 467*x_3 + 283*x_3^2 + 48*x_3^3 }-> s40 :|: s37 >= 0, s37 <= x_1 + 1, s38 >= 0, s38 <= x_2 + 1, s39 >= 0, s39 <= x_3 + 1, s40 >= 0, s40 <= s37 + s38 + s39 + 1, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z') -{ -58 + s75 + -89*z' + -5*z'^2 + 48*z'^3 }-> s76 :|: s74 >= 0, s74 <= z' - 2 + 1, s75 >= 0, s75 <= s74 + 1, s76 >= 0, s76 <= s75 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11322 + 283*x_11322^2 + 48*x_11322^3 + 467*x_2587 + 283*x_2587^2 + 48*x_2587^3 }-> s79 :|: s77 >= 0, s77 <= x_11322 + 1, s78 >= 0, s78 <= x_2587 + 1, s79 >= 0, s79 <= 1 + s77 + s78 + 1, x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 253 + 467*x_11323 + 283*x_11323^2 + 48*x_11323^3 + 467*x_2588 + 283*x_2588^2 + 48*x_2588^3 }-> s82 :|: s80 >= 0, s80 <= x_11323 + 1, s81 >= 0, s81 <= x_2588 + 1, s82 >= 0, s82 <= 1 + s80 + s81 + 1, z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s84 :|: s83 >= 0, s83 <= z' - 2 + 1, s84 >= 0, s84 <= 1 + s83 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s85 + 2*s85*s86 + s85^2 + 4*s86 + s86^2 + 467*x_11326 + 283*x_11326^2 + 48*x_11326^3 + 467*x_2589 + 283*x_2589^2 + 48*x_2589^3 }-> s88 :|: s85 >= 0, s85 <= x_11326 + 1, s86 >= 0, s86 <= x_2589 + 1, s87 >= 0, s87 <= s85 + s86 + 1, s88 >= 0, s88 <= s87 + 1, x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ -58 + s89 + -89*z' + -5*z'^2 + 48*z'^3 }-> s91 :|: s89 >= 0, s89 <= z' - 2 + 1, s90 >= 0, s90 <= s89 + 1, s91 >= 0, s91 <= s90 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s93 + 467*x_11328 + 283*x_11328^2 + 48*x_11328^3 + 467*x_2590 + 283*x_2590^2 + 48*x_2590^3 + 467*x_3146 + 283*x_3146^2 + 48*x_3146^3 }-> s96 :|: s92 >= 0, s92 <= x_11328 + 1, s93 >= 0, s93 <= x_2590 + 1, s94 >= 0, s94 <= x_3146 + 1, s95 >= 0, s95 <= s92 + s93 + s94 + 1, s96 >= 0, s96 <= s95 + 1, z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s99 :|: s97 >= 0, s97 <= z' - 2 + 1, s98 >= 0, s98 <= s97 + 1, s99 >= 0, s99 <= s98 + 1, z' - 2 >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ -106 + 45*z' + 139*z'^2 + 48*z'^3 }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0 encArg(z') -{ 252 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> 1 + s11 + s12 :|: s11 >= 0, s11 <= x_1 + 1, s12 >= 0, s12 <= x_2 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 260 + 6*s41 + 2*s41*s42 + s41^2 + 4*s42 + s42^2 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> s43 :|: s41 >= 0, s41 <= z' + 1, s42 >= 0, s42 <= z'' + 1, s43 >= 0, s43 <= s41 + s42 + 1, z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ -104 + s101 + 45*z' + 139*z'^2 + 48*z'^3 }-> s102 :|: s100 >= 0, s100 <= z' - 1 + 1, s101 >= 0, s101 <= s100 + 1, s102 >= 0, s102 <= s101 + 1, z' - 1 >= 0 encode_g(z') -{ -104 + s153 + 45*z' + 139*z'^2 + 48*z'^3 }-> s154 :|: s152 >= 0, s152 <= z' - 1 + 1, s153 >= 0, s153 <= s152, s154 >= 0, s154 <= s153 + 1, z' - 1 >= 0 encode_g(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0 + 1, z' = 0 encode_g(z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 1, z' >= 0 encode_g(z') -{ 254 + s46 + s47 + 467*x_11439 + 283*x_11439^2 + 48*x_11439^3 + 467*x_2639 + 283*x_2639^2 + 48*x_2639^3 }-> s48 :|: s46 >= 0, s46 <= x_11439 + 1, s47 >= 0, s47 <= x_2639 + 1, s48 >= 0, s48 <= 1 + s46 + s47 + 1, x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 254 + s49 + s50 + 467*x_11440 + 283*x_11440^2 + 48*x_11440^3 + 467*x_2640 + 283*x_2640^2 + 48*x_2640^3 }-> s51 :|: s49 >= 0, s49 <= x_11440 + 1, s50 >= 0, s50 <= x_2640 + 1, s51 >= 0, s51 <= 1 + s49 + s50 + 1, x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ -104 + s52 + 45*z' + 139*z'^2 + 48*z'^3 }-> s53 :|: s52 >= 0, s52 <= z' - 1 + 1, s53 >= 0, s53 <= 1 + s52 + 1, z' - 1 >= 0 encode_g(z') -{ 261 + 6*s54 + 2*s54*s55 + s54^2 + 4*s55 + s55^2 + s56 + 467*x_11443 + 283*x_11443^2 + 48*x_11443^3 + 467*x_2641 + 283*x_2641^2 + 48*x_2641^3 }-> s57 :|: s54 >= 0, s54 <= x_11443 + 1, s55 >= 0, s55 <= x_2641 + 1, s56 >= 0, s56 <= s54 + s55 + 1, s57 >= 0, s57 <= s56 + 1, x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ -104 + s58 + s59 + 45*z' + 139*z'^2 + 48*z'^3 }-> s60 :|: s58 >= 0, s58 <= z' - 1 + 1, s59 >= 0, s59 <= s58 + 1, s60 >= 0, s60 <= s59 + 1, z' - 1 >= 0 encode_g(z') -{ 379 + s62 + s64 + 467*x_11445 + 283*x_11445^2 + 48*x_11445^3 + 467*x_2642 + 283*x_2642^2 + 48*x_2642^3 + 467*x_3159 + 283*x_3159^2 + 48*x_3159^3 }-> s65 :|: s61 >= 0, s61 <= x_11445 + 1, s62 >= 0, s62 <= x_2642 + 1, s63 >= 0, s63 <= x_3159 + 1, s64 >= 0, s64 <= s61 + s62 + s63 + 1, s65 >= 0, s65 <= s64 + 1, x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s73 :|: s73 >= 0, s73 <= z' + 1, z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s66 + s67 :|: s66 >= 0, s66 <= z' + 1, s67 >= 0, s67 <= z'' + 1, z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 378 + s69 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 + 467*z1 + 283*z1^2 + 48*z1^3 }-> s71 :|: s68 >= 0, s68 <= z' + 1, s69 >= 0, s69 <= z'' + 1, s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s68 + s69 + s70 + 1, z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 253 + 467*x_12654 + 283*x_12654^2 + 48*x_12654^3 + 467*x_21179 + 283*x_21179^2 + 48*x_21179^3 }-> s157 :|: s155 >= 0, s155 <= x_12654 + 1, s156 >= 0, s156 <= x_21179 + 1, s157 >= 0, s157 <= 1 + s155 + s156, x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 253 + 467*x_12655 + 283*x_12655^2 + 48*x_12655^3 + 467*x_21180 + 283*x_21180^2 + 48*x_21180^3 }-> s160 :|: s158 >= 0, s158 <= x_12655 + 1, s159 >= 0, s159 <= x_21180 + 1, s160 >= 0, s160 <= 1 + s158 + s159, z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s162 :|: s161 >= 0, s161 <= z' - 1 + 1, s162 >= 0, s162 <= 1 + s161, z' - 1 >= 0 encode_i(z') -{ 261 + 6*s163 + 2*s163*s164 + s163^2 + 4*s164 + s164^2 + 467*x_12658 + 283*x_12658^2 + 48*x_12658^3 + 467*x_21181 + 283*x_21181^2 + 48*x_21181^3 }-> s166 :|: s163 >= 0, s163 <= x_12658 + 1, s164 >= 0, s164 <= x_21181 + 1, s165 >= 0, s165 <= s163 + s164 + 1, s166 >= 0, s166 <= s165, x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ -104 + s167 + 45*z' + 139*z'^2 + 48*z'^3 }-> s169 :|: s167 >= 0, s167 <= z' - 1 + 1, s168 >= 0, s168 <= s167 + 1, s169 >= 0, s169 <= s168, z' - 1 >= 0 encode_i(z') -{ 379 + s171 + 467*x_12660 + 283*x_12660^2 + 48*x_12660^3 + 467*x_21182 + 283*x_21182^2 + 48*x_21182^3 + 467*x_3294 + 283*x_3294^2 + 48*x_3294^3 }-> s174 :|: s170 >= 0, s170 <= x_12660 + 1, s171 >= 0, s171 <= x_21182 + 1, s172 >= 0, s172 <= x_3294 + 1, s173 >= 0, s173 <= s170 + s171 + s172 + 1, s174 >= 0, s174 <= s173, z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s177 :|: s175 >= 0, s175 <= z' - 1 + 1, s176 >= 0, s176 <= s175, s177 >= 0, s177 <= s176, z' - 1 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s189 :|: s187 >= 0, s187 <= z' - 1 + 1, s188 >= 0, s188 <= s187 + 1, s189 >= 0, s189 <= s188, z' - 1 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s44 + s45 :|: s44 >= 0, s44 <= z' + 1, s45 >= 0, s45 <= z'' + 1, z' >= 0, z'' >= 0 encode_k(z') -{ 253 + 467*x_11448 + 283*x_11448^2 + 48*x_11448^3 + 467*x_2643 + 283*x_2643^2 + 48*x_2643^3 }-> s105 :|: s103 >= 0, s103 <= x_11448 + 1, s104 >= 0, s104 <= x_2643 + 1, s105 >= 0, s105 <= 1 + s103 + s104 + 1, z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 253 + 467*x_11449 + 283*x_11449^2 + 48*x_11449^3 + 467*x_2644 + 283*x_2644^2 + 48*x_2644^3 }-> s108 :|: s106 >= 0, s106 <= x_11449 + 1, s107 >= 0, s107 <= x_2644 + 1, s108 >= 0, s108 <= 1 + s106 + s107 + 1, x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s110 :|: s109 >= 0, s109 <= z' - 1 + 1, s110 >= 0, s110 <= 1 + s109 + 1, z' - 1 >= 0 encode_k(z') -{ 261 + 6*s111 + 2*s111*s112 + s111^2 + 4*s112 + s112^2 + 467*x_11452 + 283*x_11452^2 + 48*x_11452^3 + 467*x_2645 + 283*x_2645^2 + 48*x_2645^3 }-> s114 :|: s111 >= 0, s111 <= x_11452 + 1, s112 >= 0, s112 <= x_2645 + 1, s113 >= 0, s113 <= s111 + s112 + 1, s114 >= 0, s114 <= s113 + 1, x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ -104 + s115 + 45*z' + 139*z'^2 + 48*z'^3 }-> s117 :|: s115 >= 0, s115 <= z' - 1 + 1, s116 >= 0, s116 <= s115 + 1, s117 >= 0, s117 <= s116 + 1, z' - 1 >= 0 encode_k(z') -{ 379 + s119 + 467*x_11454 + 283*x_11454^2 + 48*x_11454^3 + 467*x_2646 + 283*x_2646^2 + 48*x_2646^3 + 467*x_3160 + 283*x_3160^2 + 48*x_3160^3 }-> s122 :|: s118 >= 0, s118 <= x_11454 + 1, s119 >= 0, s119 <= x_2646 + 1, s120 >= 0, s120 <= x_3160 + 1, s121 >= 0, s121 <= s118 + s119 + s120 + 1, s122 >= 0, s122 <= s121 + 1, x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s125 :|: s123 >= 0, s123 <= z' - 1 + 1, s124 >= 0, s124 <= s123 + 1, s125 >= 0, s125 <= s124 + 1, z' - 1 >= 0 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s186 :|: s184 >= 0, s184 <= z' - 1 + 1, s185 >= 0, s185 <= s184, s186 >= 0, s186 <= s185 + 1, z' - 1 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s72 :|: s72 >= 0, s72 <= z' + 1, z' >= 0 f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 10 + s9 + 6*x + x^2 }-> s10 :|: s9 >= 0, s9 <= x + 0 + 1, s10 >= 0, s10 <= s9 + 1, z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 11 + s5 + 6*x + 2*x*z'' + x^2 + 4*z'' + z''^2 }-> s6 :|: s5 >= 0, s5 <= x + (1 + 0 + (z'' - 1)) + 1, s6 >= 0, s6 <= s5 + 1, z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 23 + s7 + 10*x + 2*x*x'' + 2*x*y' + x^2 + 8*x'' + 2*x''*y' + x''^2 + 8*y' + y'^2 }-> s8 :|: s7 >= 0, s7 <= x + (1 + (1 + x'') + y') + 1, s8 >= 0, s8 <= s7 + 1, z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {encode_k}, {encode_h}, {encode_h2}, {encode_h1}, {encode_g}, {encode_f}, {encode_j}, {encode_i}, {encode_s} Previous analysis results are: h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1] encode_0: runtime: O(1) [0], size: O(1) [0] i: runtime: O(1) [1], size: O(n^1) [z'] k: runtime: O(1) [1], size: O(n^1) [1 + z'] g: runtime: O(n^1) [1 + z'], size: O(n^1) [1 + z'] f: runtime: O(n^2) [8 + 6*z' + 2*z'*z'' + z'^2 + 4*z'' + z''^2], size: O(n^1) [1 + z' + z''] encArg: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [1 + z'] ---------------------------------------- (69) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: encode_k after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 4 + z' ---------------------------------------- (70) Obligation: Complexity RNTS consisting of the following rules: encArg(z') -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z' = 1 + 0 encArg(z') -{ -58 + s127 + -89*z' + -5*z'^2 + 48*z'^3 }-> s128 :|: s126 >= 0, s126 <= z' - 2 + 1, s127 >= 0, s127 <= s126, s128 >= 0, s128 <= s127 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11313 + 283*x_11313^2 + 48*x_11313^3 + 467*x_2583 + 283*x_2583^2 + 48*x_2583^3 }-> s131 :|: s129 >= 0, s129 <= x_11313 + 1, s130 >= 0, s130 <= x_2583 + 1, s131 >= 0, s131 <= 1 + s129 + s130, x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 253 + 467*x_11314 + 283*x_11314^2 + 48*x_11314^3 + 467*x_2584 + 283*x_2584^2 + 48*x_2584^3 }-> s134 :|: s132 >= 0, s132 <= x_11314 + 1, s133 >= 0, s133 <= x_2584 + 1, s134 >= 0, s134 <= 1 + s132 + s133, z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s136 :|: s135 >= 0, s135 <= z' - 2 + 1, s136 >= 0, s136 <= 1 + s135, z' - 2 >= 0 encArg(z') -{ 261 + 6*s137 + 2*s137*s138 + s137^2 + 4*s138 + s138^2 + 467*x_11317 + 283*x_11317^2 + 48*x_11317^3 + 467*x_2585 + 283*x_2585^2 + 48*x_2585^3 }-> s140 :|: s137 >= 0, s137 <= x_11317 + 1, s138 >= 0, s138 <= x_2585 + 1, s139 >= 0, s139 <= s137 + s138 + 1, s140 >= 0, s140 <= s139, x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ -58 + s141 + -89*z' + -5*z'^2 + 48*z'^3 }-> s143 :|: s141 >= 0, s141 <= z' - 2 + 1, s142 >= 0, s142 <= s141 + 1, s143 >= 0, s143 <= s142, z' - 2 >= 0 encArg(z') -{ 379 + s145 + 467*x_11319 + 283*x_11319^2 + 48*x_11319^3 + 467*x_2586 + 283*x_2586^2 + 48*x_2586^3 + 467*x_3145 + 283*x_3145^2 + 48*x_3145^3 }-> s148 :|: s144 >= 0, s144 <= x_11319 + 1, s145 >= 0, s145 <= x_2586 + 1, s146 >= 0, s146 <= x_3145 + 1, s147 >= 0, s147 <= s144 + s145 + s146 + 1, s148 >= 0, s148 <= s147, z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s151 :|: s149 >= 0, s149 <= z' - 2 + 1, s150 >= 0, s150 <= s149, s151 >= 0, s151 <= s150, z' - 2 >= 0 encArg(z') -{ 260 + 6*s14 + 2*s14*s15 + s14^2 + 4*s15 + s15^2 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> s16 :|: s14 >= 0, s14 <= x_1 + 1, s15 >= 0, s15 <= x_2 + 1, s16 >= 0, s16 <= s14 + s15 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s180 :|: s178 >= 0, s178 <= z' - 2 + 1, s179 >= 0, s179 <= s178 + 1, s180 >= 0, s180 <= s179, z' - 2 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s183 :|: s181 >= 0, s181 <= z' - 2 + 1, s182 >= 0, s182 <= s181, s183 >= 0, s183 <= s182 + 1, z' - 2 >= 0 encArg(z') -{ 254 + s17 + s18 + 467*x_1107 + 283*x_1107^2 + 48*x_1107^3 + 467*x_247 + 283*x_247^2 + 48*x_247^3 }-> s19 :|: s17 >= 0, s17 <= x_1107 + 1, s18 >= 0, s18 <= x_247 + 1, s19 >= 0, s19 <= 1 + s17 + s18 + 1, z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0 + 1, z' - 1 >= 0 encArg(z') -{ 254 + s20 + s21 + 467*x_1108 + 283*x_1108^2 + 48*x_1108^3 + 467*x_248 + 283*x_248^2 + 48*x_248^3 }-> s22 :|: s20 >= 0, s20 <= x_1108 + 1, s21 >= 0, s21 <= x_248 + 1, s22 >= 0, s22 <= 1 + s20 + s21 + 1, x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ -58 + s23 + -89*z' + -5*z'^2 + 48*z'^3 }-> s24 :|: s23 >= 0, s23 <= z' - 2 + 1, s24 >= 0, s24 <= 1 + s23 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s25 + 2*s25*s26 + s25^2 + 4*s26 + s26^2 + s27 + 467*x_1111 + 283*x_1111^2 + 48*x_1111^3 + 467*x_249 + 283*x_249^2 + 48*x_249^3 }-> s28 :|: s25 >= 0, s25 <= x_1111 + 1, s26 >= 0, s26 <= x_249 + 1, s27 >= 0, s27 <= s25 + s26 + 1, s28 >= 0, s28 <= s27 + 1, z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ -58 + s29 + s30 + -89*z' + -5*z'^2 + 48*z'^3 }-> s31 :|: s29 >= 0, s29 <= z' - 2 + 1, s30 >= 0, s30 <= s29 + 1, s31 >= 0, s31 <= s30 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s33 + s35 + 467*x_1113 + 283*x_1113^2 + 48*x_1113^3 + 467*x_250 + 283*x_250^2 + 48*x_250^3 + 467*x_311 + 283*x_311^2 + 48*x_311^3 }-> s36 :|: s32 >= 0, s32 <= x_1113 + 1, s33 >= 0, s33 <= x_250 + 1, s34 >= 0, s34 <= x_311 + 1, s35 >= 0, s35 <= s32 + s33 + s34 + 1, s36 >= 0, s36 <= s35 + 1, x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 378 + s38 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 + 467*x_3 + 283*x_3^2 + 48*x_3^3 }-> s40 :|: s37 >= 0, s37 <= x_1 + 1, s38 >= 0, s38 <= x_2 + 1, s39 >= 0, s39 <= x_3 + 1, s40 >= 0, s40 <= s37 + s38 + s39 + 1, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z') -{ -58 + s75 + -89*z' + -5*z'^2 + 48*z'^3 }-> s76 :|: s74 >= 0, s74 <= z' - 2 + 1, s75 >= 0, s75 <= s74 + 1, s76 >= 0, s76 <= s75 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11322 + 283*x_11322^2 + 48*x_11322^3 + 467*x_2587 + 283*x_2587^2 + 48*x_2587^3 }-> s79 :|: s77 >= 0, s77 <= x_11322 + 1, s78 >= 0, s78 <= x_2587 + 1, s79 >= 0, s79 <= 1 + s77 + s78 + 1, x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 253 + 467*x_11323 + 283*x_11323^2 + 48*x_11323^3 + 467*x_2588 + 283*x_2588^2 + 48*x_2588^3 }-> s82 :|: s80 >= 0, s80 <= x_11323 + 1, s81 >= 0, s81 <= x_2588 + 1, s82 >= 0, s82 <= 1 + s80 + s81 + 1, z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s84 :|: s83 >= 0, s83 <= z' - 2 + 1, s84 >= 0, s84 <= 1 + s83 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s85 + 2*s85*s86 + s85^2 + 4*s86 + s86^2 + 467*x_11326 + 283*x_11326^2 + 48*x_11326^3 + 467*x_2589 + 283*x_2589^2 + 48*x_2589^3 }-> s88 :|: s85 >= 0, s85 <= x_11326 + 1, s86 >= 0, s86 <= x_2589 + 1, s87 >= 0, s87 <= s85 + s86 + 1, s88 >= 0, s88 <= s87 + 1, x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ -58 + s89 + -89*z' + -5*z'^2 + 48*z'^3 }-> s91 :|: s89 >= 0, s89 <= z' - 2 + 1, s90 >= 0, s90 <= s89 + 1, s91 >= 0, s91 <= s90 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s93 + 467*x_11328 + 283*x_11328^2 + 48*x_11328^3 + 467*x_2590 + 283*x_2590^2 + 48*x_2590^3 + 467*x_3146 + 283*x_3146^2 + 48*x_3146^3 }-> s96 :|: s92 >= 0, s92 <= x_11328 + 1, s93 >= 0, s93 <= x_2590 + 1, s94 >= 0, s94 <= x_3146 + 1, s95 >= 0, s95 <= s92 + s93 + s94 + 1, s96 >= 0, s96 <= s95 + 1, z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s99 :|: s97 >= 0, s97 <= z' - 2 + 1, s98 >= 0, s98 <= s97 + 1, s99 >= 0, s99 <= s98 + 1, z' - 2 >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ -106 + 45*z' + 139*z'^2 + 48*z'^3 }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0 encArg(z') -{ 252 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> 1 + s11 + s12 :|: s11 >= 0, s11 <= x_1 + 1, s12 >= 0, s12 <= x_2 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 260 + 6*s41 + 2*s41*s42 + s41^2 + 4*s42 + s42^2 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> s43 :|: s41 >= 0, s41 <= z' + 1, s42 >= 0, s42 <= z'' + 1, s43 >= 0, s43 <= s41 + s42 + 1, z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ -104 + s101 + 45*z' + 139*z'^2 + 48*z'^3 }-> s102 :|: s100 >= 0, s100 <= z' - 1 + 1, s101 >= 0, s101 <= s100 + 1, s102 >= 0, s102 <= s101 + 1, z' - 1 >= 0 encode_g(z') -{ -104 + s153 + 45*z' + 139*z'^2 + 48*z'^3 }-> s154 :|: s152 >= 0, s152 <= z' - 1 + 1, s153 >= 0, s153 <= s152, s154 >= 0, s154 <= s153 + 1, z' - 1 >= 0 encode_g(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0 + 1, z' = 0 encode_g(z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 1, z' >= 0 encode_g(z') -{ 254 + s46 + s47 + 467*x_11439 + 283*x_11439^2 + 48*x_11439^3 + 467*x_2639 + 283*x_2639^2 + 48*x_2639^3 }-> s48 :|: s46 >= 0, s46 <= x_11439 + 1, s47 >= 0, s47 <= x_2639 + 1, s48 >= 0, s48 <= 1 + s46 + s47 + 1, x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 254 + s49 + s50 + 467*x_11440 + 283*x_11440^2 + 48*x_11440^3 + 467*x_2640 + 283*x_2640^2 + 48*x_2640^3 }-> s51 :|: s49 >= 0, s49 <= x_11440 + 1, s50 >= 0, s50 <= x_2640 + 1, s51 >= 0, s51 <= 1 + s49 + s50 + 1, x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ -104 + s52 + 45*z' + 139*z'^2 + 48*z'^3 }-> s53 :|: s52 >= 0, s52 <= z' - 1 + 1, s53 >= 0, s53 <= 1 + s52 + 1, z' - 1 >= 0 encode_g(z') -{ 261 + 6*s54 + 2*s54*s55 + s54^2 + 4*s55 + s55^2 + s56 + 467*x_11443 + 283*x_11443^2 + 48*x_11443^3 + 467*x_2641 + 283*x_2641^2 + 48*x_2641^3 }-> s57 :|: s54 >= 0, s54 <= x_11443 + 1, s55 >= 0, s55 <= x_2641 + 1, s56 >= 0, s56 <= s54 + s55 + 1, s57 >= 0, s57 <= s56 + 1, x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ -104 + s58 + s59 + 45*z' + 139*z'^2 + 48*z'^3 }-> s60 :|: s58 >= 0, s58 <= z' - 1 + 1, s59 >= 0, s59 <= s58 + 1, s60 >= 0, s60 <= s59 + 1, z' - 1 >= 0 encode_g(z') -{ 379 + s62 + s64 + 467*x_11445 + 283*x_11445^2 + 48*x_11445^3 + 467*x_2642 + 283*x_2642^2 + 48*x_2642^3 + 467*x_3159 + 283*x_3159^2 + 48*x_3159^3 }-> s65 :|: s61 >= 0, s61 <= x_11445 + 1, s62 >= 0, s62 <= x_2642 + 1, s63 >= 0, s63 <= x_3159 + 1, s64 >= 0, s64 <= s61 + s62 + s63 + 1, s65 >= 0, s65 <= s64 + 1, x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s73 :|: s73 >= 0, s73 <= z' + 1, z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s66 + s67 :|: s66 >= 0, s66 <= z' + 1, s67 >= 0, s67 <= z'' + 1, z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 378 + s69 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 + 467*z1 + 283*z1^2 + 48*z1^3 }-> s71 :|: s68 >= 0, s68 <= z' + 1, s69 >= 0, s69 <= z'' + 1, s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s68 + s69 + s70 + 1, z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 253 + 467*x_12654 + 283*x_12654^2 + 48*x_12654^3 + 467*x_21179 + 283*x_21179^2 + 48*x_21179^3 }-> s157 :|: s155 >= 0, s155 <= x_12654 + 1, s156 >= 0, s156 <= x_21179 + 1, s157 >= 0, s157 <= 1 + s155 + s156, x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 253 + 467*x_12655 + 283*x_12655^2 + 48*x_12655^3 + 467*x_21180 + 283*x_21180^2 + 48*x_21180^3 }-> s160 :|: s158 >= 0, s158 <= x_12655 + 1, s159 >= 0, s159 <= x_21180 + 1, s160 >= 0, s160 <= 1 + s158 + s159, z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s162 :|: s161 >= 0, s161 <= z' - 1 + 1, s162 >= 0, s162 <= 1 + s161, z' - 1 >= 0 encode_i(z') -{ 261 + 6*s163 + 2*s163*s164 + s163^2 + 4*s164 + s164^2 + 467*x_12658 + 283*x_12658^2 + 48*x_12658^3 + 467*x_21181 + 283*x_21181^2 + 48*x_21181^3 }-> s166 :|: s163 >= 0, s163 <= x_12658 + 1, s164 >= 0, s164 <= x_21181 + 1, s165 >= 0, s165 <= s163 + s164 + 1, s166 >= 0, s166 <= s165, x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ -104 + s167 + 45*z' + 139*z'^2 + 48*z'^3 }-> s169 :|: s167 >= 0, s167 <= z' - 1 + 1, s168 >= 0, s168 <= s167 + 1, s169 >= 0, s169 <= s168, z' - 1 >= 0 encode_i(z') -{ 379 + s171 + 467*x_12660 + 283*x_12660^2 + 48*x_12660^3 + 467*x_21182 + 283*x_21182^2 + 48*x_21182^3 + 467*x_3294 + 283*x_3294^2 + 48*x_3294^3 }-> s174 :|: s170 >= 0, s170 <= x_12660 + 1, s171 >= 0, s171 <= x_21182 + 1, s172 >= 0, s172 <= x_3294 + 1, s173 >= 0, s173 <= s170 + s171 + s172 + 1, s174 >= 0, s174 <= s173, z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s177 :|: s175 >= 0, s175 <= z' - 1 + 1, s176 >= 0, s176 <= s175, s177 >= 0, s177 <= s176, z' - 1 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s189 :|: s187 >= 0, s187 <= z' - 1 + 1, s188 >= 0, s188 <= s187 + 1, s189 >= 0, s189 <= s188, z' - 1 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s44 + s45 :|: s44 >= 0, s44 <= z' + 1, s45 >= 0, s45 <= z'' + 1, z' >= 0, z'' >= 0 encode_k(z') -{ 253 + 467*x_11448 + 283*x_11448^2 + 48*x_11448^3 + 467*x_2643 + 283*x_2643^2 + 48*x_2643^3 }-> s105 :|: s103 >= 0, s103 <= x_11448 + 1, s104 >= 0, s104 <= x_2643 + 1, s105 >= 0, s105 <= 1 + s103 + s104 + 1, z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 253 + 467*x_11449 + 283*x_11449^2 + 48*x_11449^3 + 467*x_2644 + 283*x_2644^2 + 48*x_2644^3 }-> s108 :|: s106 >= 0, s106 <= x_11449 + 1, s107 >= 0, s107 <= x_2644 + 1, s108 >= 0, s108 <= 1 + s106 + s107 + 1, x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s110 :|: s109 >= 0, s109 <= z' - 1 + 1, s110 >= 0, s110 <= 1 + s109 + 1, z' - 1 >= 0 encode_k(z') -{ 261 + 6*s111 + 2*s111*s112 + s111^2 + 4*s112 + s112^2 + 467*x_11452 + 283*x_11452^2 + 48*x_11452^3 + 467*x_2645 + 283*x_2645^2 + 48*x_2645^3 }-> s114 :|: s111 >= 0, s111 <= x_11452 + 1, s112 >= 0, s112 <= x_2645 + 1, s113 >= 0, s113 <= s111 + s112 + 1, s114 >= 0, s114 <= s113 + 1, x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ -104 + s115 + 45*z' + 139*z'^2 + 48*z'^3 }-> s117 :|: s115 >= 0, s115 <= z' - 1 + 1, s116 >= 0, s116 <= s115 + 1, s117 >= 0, s117 <= s116 + 1, z' - 1 >= 0 encode_k(z') -{ 379 + s119 + 467*x_11454 + 283*x_11454^2 + 48*x_11454^3 + 467*x_2646 + 283*x_2646^2 + 48*x_2646^3 + 467*x_3160 + 283*x_3160^2 + 48*x_3160^3 }-> s122 :|: s118 >= 0, s118 <= x_11454 + 1, s119 >= 0, s119 <= x_2646 + 1, s120 >= 0, s120 <= x_3160 + 1, s121 >= 0, s121 <= s118 + s119 + s120 + 1, s122 >= 0, s122 <= s121 + 1, x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s125 :|: s123 >= 0, s123 <= z' - 1 + 1, s124 >= 0, s124 <= s123 + 1, s125 >= 0, s125 <= s124 + 1, z' - 1 >= 0 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s186 :|: s184 >= 0, s184 <= z' - 1 + 1, s185 >= 0, s185 <= s184, s186 >= 0, s186 <= s185 + 1, z' - 1 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s72 :|: s72 >= 0, s72 <= z' + 1, z' >= 0 f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 10 + s9 + 6*x + x^2 }-> s10 :|: s9 >= 0, s9 <= x + 0 + 1, s10 >= 0, s10 <= s9 + 1, z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 11 + s5 + 6*x + 2*x*z'' + x^2 + 4*z'' + z''^2 }-> s6 :|: s5 >= 0, s5 <= x + (1 + 0 + (z'' - 1)) + 1, s6 >= 0, s6 <= s5 + 1, z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 23 + s7 + 10*x + 2*x*x'' + 2*x*y' + x^2 + 8*x'' + 2*x''*y' + x''^2 + 8*y' + y'^2 }-> s8 :|: s7 >= 0, s7 <= x + (1 + (1 + x'') + y') + 1, s8 >= 0, s8 <= s7 + 1, z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {encode_k}, {encode_h}, {encode_h2}, {encode_h1}, {encode_g}, {encode_f}, {encode_j}, {encode_i}, {encode_s} Previous analysis results are: h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1] encode_0: runtime: O(1) [0], size: O(1) [0] i: runtime: O(1) [1], size: O(n^1) [z'] k: runtime: O(1) [1], size: O(n^1) [1 + z'] g: runtime: O(n^1) [1 + z'], size: O(n^1) [1 + z'] f: runtime: O(n^2) [8 + 6*z' + 2*z'*z'' + z'^2 + 4*z'' + z''^2], size: O(n^1) [1 + z' + z''] encArg: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [1 + z'] encode_k: runtime: ?, size: O(n^1) [4 + z'] ---------------------------------------- (71) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: encode_k after applying outer abstraction to obtain an ITS, resulting in: O(n^3) with polynomial bound: 1161 + 4403*z' + 3107*z'^2 + 624*z'^3 ---------------------------------------- (72) Obligation: Complexity RNTS consisting of the following rules: encArg(z') -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z' = 1 + 0 encArg(z') -{ -58 + s127 + -89*z' + -5*z'^2 + 48*z'^3 }-> s128 :|: s126 >= 0, s126 <= z' - 2 + 1, s127 >= 0, s127 <= s126, s128 >= 0, s128 <= s127 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11313 + 283*x_11313^2 + 48*x_11313^3 + 467*x_2583 + 283*x_2583^2 + 48*x_2583^3 }-> s131 :|: s129 >= 0, s129 <= x_11313 + 1, s130 >= 0, s130 <= x_2583 + 1, s131 >= 0, s131 <= 1 + s129 + s130, x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 253 + 467*x_11314 + 283*x_11314^2 + 48*x_11314^3 + 467*x_2584 + 283*x_2584^2 + 48*x_2584^3 }-> s134 :|: s132 >= 0, s132 <= x_11314 + 1, s133 >= 0, s133 <= x_2584 + 1, s134 >= 0, s134 <= 1 + s132 + s133, z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s136 :|: s135 >= 0, s135 <= z' - 2 + 1, s136 >= 0, s136 <= 1 + s135, z' - 2 >= 0 encArg(z') -{ 261 + 6*s137 + 2*s137*s138 + s137^2 + 4*s138 + s138^2 + 467*x_11317 + 283*x_11317^2 + 48*x_11317^3 + 467*x_2585 + 283*x_2585^2 + 48*x_2585^3 }-> s140 :|: s137 >= 0, s137 <= x_11317 + 1, s138 >= 0, s138 <= x_2585 + 1, s139 >= 0, s139 <= s137 + s138 + 1, s140 >= 0, s140 <= s139, x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ -58 + s141 + -89*z' + -5*z'^2 + 48*z'^3 }-> s143 :|: s141 >= 0, s141 <= z' - 2 + 1, s142 >= 0, s142 <= s141 + 1, s143 >= 0, s143 <= s142, z' - 2 >= 0 encArg(z') -{ 379 + s145 + 467*x_11319 + 283*x_11319^2 + 48*x_11319^3 + 467*x_2586 + 283*x_2586^2 + 48*x_2586^3 + 467*x_3145 + 283*x_3145^2 + 48*x_3145^3 }-> s148 :|: s144 >= 0, s144 <= x_11319 + 1, s145 >= 0, s145 <= x_2586 + 1, s146 >= 0, s146 <= x_3145 + 1, s147 >= 0, s147 <= s144 + s145 + s146 + 1, s148 >= 0, s148 <= s147, z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s151 :|: s149 >= 0, s149 <= z' - 2 + 1, s150 >= 0, s150 <= s149, s151 >= 0, s151 <= s150, z' - 2 >= 0 encArg(z') -{ 260 + 6*s14 + 2*s14*s15 + s14^2 + 4*s15 + s15^2 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> s16 :|: s14 >= 0, s14 <= x_1 + 1, s15 >= 0, s15 <= x_2 + 1, s16 >= 0, s16 <= s14 + s15 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s180 :|: s178 >= 0, s178 <= z' - 2 + 1, s179 >= 0, s179 <= s178 + 1, s180 >= 0, s180 <= s179, z' - 2 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s183 :|: s181 >= 0, s181 <= z' - 2 + 1, s182 >= 0, s182 <= s181, s183 >= 0, s183 <= s182 + 1, z' - 2 >= 0 encArg(z') -{ 254 + s17 + s18 + 467*x_1107 + 283*x_1107^2 + 48*x_1107^3 + 467*x_247 + 283*x_247^2 + 48*x_247^3 }-> s19 :|: s17 >= 0, s17 <= x_1107 + 1, s18 >= 0, s18 <= x_247 + 1, s19 >= 0, s19 <= 1 + s17 + s18 + 1, z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0 + 1, z' - 1 >= 0 encArg(z') -{ 254 + s20 + s21 + 467*x_1108 + 283*x_1108^2 + 48*x_1108^3 + 467*x_248 + 283*x_248^2 + 48*x_248^3 }-> s22 :|: s20 >= 0, s20 <= x_1108 + 1, s21 >= 0, s21 <= x_248 + 1, s22 >= 0, s22 <= 1 + s20 + s21 + 1, x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ -58 + s23 + -89*z' + -5*z'^2 + 48*z'^3 }-> s24 :|: s23 >= 0, s23 <= z' - 2 + 1, s24 >= 0, s24 <= 1 + s23 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s25 + 2*s25*s26 + s25^2 + 4*s26 + s26^2 + s27 + 467*x_1111 + 283*x_1111^2 + 48*x_1111^3 + 467*x_249 + 283*x_249^2 + 48*x_249^3 }-> s28 :|: s25 >= 0, s25 <= x_1111 + 1, s26 >= 0, s26 <= x_249 + 1, s27 >= 0, s27 <= s25 + s26 + 1, s28 >= 0, s28 <= s27 + 1, z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ -58 + s29 + s30 + -89*z' + -5*z'^2 + 48*z'^3 }-> s31 :|: s29 >= 0, s29 <= z' - 2 + 1, s30 >= 0, s30 <= s29 + 1, s31 >= 0, s31 <= s30 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s33 + s35 + 467*x_1113 + 283*x_1113^2 + 48*x_1113^3 + 467*x_250 + 283*x_250^2 + 48*x_250^3 + 467*x_311 + 283*x_311^2 + 48*x_311^3 }-> s36 :|: s32 >= 0, s32 <= x_1113 + 1, s33 >= 0, s33 <= x_250 + 1, s34 >= 0, s34 <= x_311 + 1, s35 >= 0, s35 <= s32 + s33 + s34 + 1, s36 >= 0, s36 <= s35 + 1, x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 378 + s38 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 + 467*x_3 + 283*x_3^2 + 48*x_3^3 }-> s40 :|: s37 >= 0, s37 <= x_1 + 1, s38 >= 0, s38 <= x_2 + 1, s39 >= 0, s39 <= x_3 + 1, s40 >= 0, s40 <= s37 + s38 + s39 + 1, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z') -{ -58 + s75 + -89*z' + -5*z'^2 + 48*z'^3 }-> s76 :|: s74 >= 0, s74 <= z' - 2 + 1, s75 >= 0, s75 <= s74 + 1, s76 >= 0, s76 <= s75 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11322 + 283*x_11322^2 + 48*x_11322^3 + 467*x_2587 + 283*x_2587^2 + 48*x_2587^3 }-> s79 :|: s77 >= 0, s77 <= x_11322 + 1, s78 >= 0, s78 <= x_2587 + 1, s79 >= 0, s79 <= 1 + s77 + s78 + 1, x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 253 + 467*x_11323 + 283*x_11323^2 + 48*x_11323^3 + 467*x_2588 + 283*x_2588^2 + 48*x_2588^3 }-> s82 :|: s80 >= 0, s80 <= x_11323 + 1, s81 >= 0, s81 <= x_2588 + 1, s82 >= 0, s82 <= 1 + s80 + s81 + 1, z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s84 :|: s83 >= 0, s83 <= z' - 2 + 1, s84 >= 0, s84 <= 1 + s83 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s85 + 2*s85*s86 + s85^2 + 4*s86 + s86^2 + 467*x_11326 + 283*x_11326^2 + 48*x_11326^3 + 467*x_2589 + 283*x_2589^2 + 48*x_2589^3 }-> s88 :|: s85 >= 0, s85 <= x_11326 + 1, s86 >= 0, s86 <= x_2589 + 1, s87 >= 0, s87 <= s85 + s86 + 1, s88 >= 0, s88 <= s87 + 1, x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ -58 + s89 + -89*z' + -5*z'^2 + 48*z'^3 }-> s91 :|: s89 >= 0, s89 <= z' - 2 + 1, s90 >= 0, s90 <= s89 + 1, s91 >= 0, s91 <= s90 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s93 + 467*x_11328 + 283*x_11328^2 + 48*x_11328^3 + 467*x_2590 + 283*x_2590^2 + 48*x_2590^3 + 467*x_3146 + 283*x_3146^2 + 48*x_3146^3 }-> s96 :|: s92 >= 0, s92 <= x_11328 + 1, s93 >= 0, s93 <= x_2590 + 1, s94 >= 0, s94 <= x_3146 + 1, s95 >= 0, s95 <= s92 + s93 + s94 + 1, s96 >= 0, s96 <= s95 + 1, z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s99 :|: s97 >= 0, s97 <= z' - 2 + 1, s98 >= 0, s98 <= s97 + 1, s99 >= 0, s99 <= s98 + 1, z' - 2 >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ -106 + 45*z' + 139*z'^2 + 48*z'^3 }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0 encArg(z') -{ 252 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> 1 + s11 + s12 :|: s11 >= 0, s11 <= x_1 + 1, s12 >= 0, s12 <= x_2 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 260 + 6*s41 + 2*s41*s42 + s41^2 + 4*s42 + s42^2 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> s43 :|: s41 >= 0, s41 <= z' + 1, s42 >= 0, s42 <= z'' + 1, s43 >= 0, s43 <= s41 + s42 + 1, z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ -104 + s101 + 45*z' + 139*z'^2 + 48*z'^3 }-> s102 :|: s100 >= 0, s100 <= z' - 1 + 1, s101 >= 0, s101 <= s100 + 1, s102 >= 0, s102 <= s101 + 1, z' - 1 >= 0 encode_g(z') -{ -104 + s153 + 45*z' + 139*z'^2 + 48*z'^3 }-> s154 :|: s152 >= 0, s152 <= z' - 1 + 1, s153 >= 0, s153 <= s152, s154 >= 0, s154 <= s153 + 1, z' - 1 >= 0 encode_g(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0 + 1, z' = 0 encode_g(z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 1, z' >= 0 encode_g(z') -{ 254 + s46 + s47 + 467*x_11439 + 283*x_11439^2 + 48*x_11439^3 + 467*x_2639 + 283*x_2639^2 + 48*x_2639^3 }-> s48 :|: s46 >= 0, s46 <= x_11439 + 1, s47 >= 0, s47 <= x_2639 + 1, s48 >= 0, s48 <= 1 + s46 + s47 + 1, x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 254 + s49 + s50 + 467*x_11440 + 283*x_11440^2 + 48*x_11440^3 + 467*x_2640 + 283*x_2640^2 + 48*x_2640^3 }-> s51 :|: s49 >= 0, s49 <= x_11440 + 1, s50 >= 0, s50 <= x_2640 + 1, s51 >= 0, s51 <= 1 + s49 + s50 + 1, x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ -104 + s52 + 45*z' + 139*z'^2 + 48*z'^3 }-> s53 :|: s52 >= 0, s52 <= z' - 1 + 1, s53 >= 0, s53 <= 1 + s52 + 1, z' - 1 >= 0 encode_g(z') -{ 261 + 6*s54 + 2*s54*s55 + s54^2 + 4*s55 + s55^2 + s56 + 467*x_11443 + 283*x_11443^2 + 48*x_11443^3 + 467*x_2641 + 283*x_2641^2 + 48*x_2641^3 }-> s57 :|: s54 >= 0, s54 <= x_11443 + 1, s55 >= 0, s55 <= x_2641 + 1, s56 >= 0, s56 <= s54 + s55 + 1, s57 >= 0, s57 <= s56 + 1, x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ -104 + s58 + s59 + 45*z' + 139*z'^2 + 48*z'^3 }-> s60 :|: s58 >= 0, s58 <= z' - 1 + 1, s59 >= 0, s59 <= s58 + 1, s60 >= 0, s60 <= s59 + 1, z' - 1 >= 0 encode_g(z') -{ 379 + s62 + s64 + 467*x_11445 + 283*x_11445^2 + 48*x_11445^3 + 467*x_2642 + 283*x_2642^2 + 48*x_2642^3 + 467*x_3159 + 283*x_3159^2 + 48*x_3159^3 }-> s65 :|: s61 >= 0, s61 <= x_11445 + 1, s62 >= 0, s62 <= x_2642 + 1, s63 >= 0, s63 <= x_3159 + 1, s64 >= 0, s64 <= s61 + s62 + s63 + 1, s65 >= 0, s65 <= s64 + 1, x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s73 :|: s73 >= 0, s73 <= z' + 1, z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s66 + s67 :|: s66 >= 0, s66 <= z' + 1, s67 >= 0, s67 <= z'' + 1, z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 378 + s69 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 + 467*z1 + 283*z1^2 + 48*z1^3 }-> s71 :|: s68 >= 0, s68 <= z' + 1, s69 >= 0, s69 <= z'' + 1, s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s68 + s69 + s70 + 1, z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 253 + 467*x_12654 + 283*x_12654^2 + 48*x_12654^3 + 467*x_21179 + 283*x_21179^2 + 48*x_21179^3 }-> s157 :|: s155 >= 0, s155 <= x_12654 + 1, s156 >= 0, s156 <= x_21179 + 1, s157 >= 0, s157 <= 1 + s155 + s156, x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 253 + 467*x_12655 + 283*x_12655^2 + 48*x_12655^3 + 467*x_21180 + 283*x_21180^2 + 48*x_21180^3 }-> s160 :|: s158 >= 0, s158 <= x_12655 + 1, s159 >= 0, s159 <= x_21180 + 1, s160 >= 0, s160 <= 1 + s158 + s159, z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s162 :|: s161 >= 0, s161 <= z' - 1 + 1, s162 >= 0, s162 <= 1 + s161, z' - 1 >= 0 encode_i(z') -{ 261 + 6*s163 + 2*s163*s164 + s163^2 + 4*s164 + s164^2 + 467*x_12658 + 283*x_12658^2 + 48*x_12658^3 + 467*x_21181 + 283*x_21181^2 + 48*x_21181^3 }-> s166 :|: s163 >= 0, s163 <= x_12658 + 1, s164 >= 0, s164 <= x_21181 + 1, s165 >= 0, s165 <= s163 + s164 + 1, s166 >= 0, s166 <= s165, x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ -104 + s167 + 45*z' + 139*z'^2 + 48*z'^3 }-> s169 :|: s167 >= 0, s167 <= z' - 1 + 1, s168 >= 0, s168 <= s167 + 1, s169 >= 0, s169 <= s168, z' - 1 >= 0 encode_i(z') -{ 379 + s171 + 467*x_12660 + 283*x_12660^2 + 48*x_12660^3 + 467*x_21182 + 283*x_21182^2 + 48*x_21182^3 + 467*x_3294 + 283*x_3294^2 + 48*x_3294^3 }-> s174 :|: s170 >= 0, s170 <= x_12660 + 1, s171 >= 0, s171 <= x_21182 + 1, s172 >= 0, s172 <= x_3294 + 1, s173 >= 0, s173 <= s170 + s171 + s172 + 1, s174 >= 0, s174 <= s173, z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s177 :|: s175 >= 0, s175 <= z' - 1 + 1, s176 >= 0, s176 <= s175, s177 >= 0, s177 <= s176, z' - 1 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s189 :|: s187 >= 0, s187 <= z' - 1 + 1, s188 >= 0, s188 <= s187 + 1, s189 >= 0, s189 <= s188, z' - 1 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s44 + s45 :|: s44 >= 0, s44 <= z' + 1, s45 >= 0, s45 <= z'' + 1, z' >= 0, z'' >= 0 encode_k(z') -{ 253 + 467*x_11448 + 283*x_11448^2 + 48*x_11448^3 + 467*x_2643 + 283*x_2643^2 + 48*x_2643^3 }-> s105 :|: s103 >= 0, s103 <= x_11448 + 1, s104 >= 0, s104 <= x_2643 + 1, s105 >= 0, s105 <= 1 + s103 + s104 + 1, z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 253 + 467*x_11449 + 283*x_11449^2 + 48*x_11449^3 + 467*x_2644 + 283*x_2644^2 + 48*x_2644^3 }-> s108 :|: s106 >= 0, s106 <= x_11449 + 1, s107 >= 0, s107 <= x_2644 + 1, s108 >= 0, s108 <= 1 + s106 + s107 + 1, x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s110 :|: s109 >= 0, s109 <= z' - 1 + 1, s110 >= 0, s110 <= 1 + s109 + 1, z' - 1 >= 0 encode_k(z') -{ 261 + 6*s111 + 2*s111*s112 + s111^2 + 4*s112 + s112^2 + 467*x_11452 + 283*x_11452^2 + 48*x_11452^3 + 467*x_2645 + 283*x_2645^2 + 48*x_2645^3 }-> s114 :|: s111 >= 0, s111 <= x_11452 + 1, s112 >= 0, s112 <= x_2645 + 1, s113 >= 0, s113 <= s111 + s112 + 1, s114 >= 0, s114 <= s113 + 1, x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ -104 + s115 + 45*z' + 139*z'^2 + 48*z'^3 }-> s117 :|: s115 >= 0, s115 <= z' - 1 + 1, s116 >= 0, s116 <= s115 + 1, s117 >= 0, s117 <= s116 + 1, z' - 1 >= 0 encode_k(z') -{ 379 + s119 + 467*x_11454 + 283*x_11454^2 + 48*x_11454^3 + 467*x_2646 + 283*x_2646^2 + 48*x_2646^3 + 467*x_3160 + 283*x_3160^2 + 48*x_3160^3 }-> s122 :|: s118 >= 0, s118 <= x_11454 + 1, s119 >= 0, s119 <= x_2646 + 1, s120 >= 0, s120 <= x_3160 + 1, s121 >= 0, s121 <= s118 + s119 + s120 + 1, s122 >= 0, s122 <= s121 + 1, x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s125 :|: s123 >= 0, s123 <= z' - 1 + 1, s124 >= 0, s124 <= s123 + 1, s125 >= 0, s125 <= s124 + 1, z' - 1 >= 0 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s186 :|: s184 >= 0, s184 <= z' - 1 + 1, s185 >= 0, s185 <= s184, s186 >= 0, s186 <= s185 + 1, z' - 1 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s72 :|: s72 >= 0, s72 <= z' + 1, z' >= 0 f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 10 + s9 + 6*x + x^2 }-> s10 :|: s9 >= 0, s9 <= x + 0 + 1, s10 >= 0, s10 <= s9 + 1, z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 11 + s5 + 6*x + 2*x*z'' + x^2 + 4*z'' + z''^2 }-> s6 :|: s5 >= 0, s5 <= x + (1 + 0 + (z'' - 1)) + 1, s6 >= 0, s6 <= s5 + 1, z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 23 + s7 + 10*x + 2*x*x'' + 2*x*y' + x^2 + 8*x'' + 2*x''*y' + x''^2 + 8*y' + y'^2 }-> s8 :|: s7 >= 0, s7 <= x + (1 + (1 + x'') + y') + 1, s8 >= 0, s8 <= s7 + 1, z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {encode_h}, {encode_h2}, {encode_h1}, {encode_g}, {encode_f}, {encode_j}, {encode_i}, {encode_s} Previous analysis results are: h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1] encode_0: runtime: O(1) [0], size: O(1) [0] i: runtime: O(1) [1], size: O(n^1) [z'] k: runtime: O(1) [1], size: O(n^1) [1 + z'] g: runtime: O(n^1) [1 + z'], size: O(n^1) [1 + z'] f: runtime: O(n^2) [8 + 6*z' + 2*z'*z'' + z'^2 + 4*z'' + z''^2], size: O(n^1) [1 + z' + z''] encArg: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [1 + z'] encode_k: runtime: O(n^3) [1161 + 4403*z' + 3107*z'^2 + 624*z'^3], size: O(n^1) [4 + 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') -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z' = 1 + 0 encArg(z') -{ -58 + s127 + -89*z' + -5*z'^2 + 48*z'^3 }-> s128 :|: s126 >= 0, s126 <= z' - 2 + 1, s127 >= 0, s127 <= s126, s128 >= 0, s128 <= s127 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11313 + 283*x_11313^2 + 48*x_11313^3 + 467*x_2583 + 283*x_2583^2 + 48*x_2583^3 }-> s131 :|: s129 >= 0, s129 <= x_11313 + 1, s130 >= 0, s130 <= x_2583 + 1, s131 >= 0, s131 <= 1 + s129 + s130, x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 253 + 467*x_11314 + 283*x_11314^2 + 48*x_11314^3 + 467*x_2584 + 283*x_2584^2 + 48*x_2584^3 }-> s134 :|: s132 >= 0, s132 <= x_11314 + 1, s133 >= 0, s133 <= x_2584 + 1, s134 >= 0, s134 <= 1 + s132 + s133, z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s136 :|: s135 >= 0, s135 <= z' - 2 + 1, s136 >= 0, s136 <= 1 + s135, z' - 2 >= 0 encArg(z') -{ 261 + 6*s137 + 2*s137*s138 + s137^2 + 4*s138 + s138^2 + 467*x_11317 + 283*x_11317^2 + 48*x_11317^3 + 467*x_2585 + 283*x_2585^2 + 48*x_2585^3 }-> s140 :|: s137 >= 0, s137 <= x_11317 + 1, s138 >= 0, s138 <= x_2585 + 1, s139 >= 0, s139 <= s137 + s138 + 1, s140 >= 0, s140 <= s139, x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ -58 + s141 + -89*z' + -5*z'^2 + 48*z'^3 }-> s143 :|: s141 >= 0, s141 <= z' - 2 + 1, s142 >= 0, s142 <= s141 + 1, s143 >= 0, s143 <= s142, z' - 2 >= 0 encArg(z') -{ 379 + s145 + 467*x_11319 + 283*x_11319^2 + 48*x_11319^3 + 467*x_2586 + 283*x_2586^2 + 48*x_2586^3 + 467*x_3145 + 283*x_3145^2 + 48*x_3145^3 }-> s148 :|: s144 >= 0, s144 <= x_11319 + 1, s145 >= 0, s145 <= x_2586 + 1, s146 >= 0, s146 <= x_3145 + 1, s147 >= 0, s147 <= s144 + s145 + s146 + 1, s148 >= 0, s148 <= s147, z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s151 :|: s149 >= 0, s149 <= z' - 2 + 1, s150 >= 0, s150 <= s149, s151 >= 0, s151 <= s150, z' - 2 >= 0 encArg(z') -{ 260 + 6*s14 + 2*s14*s15 + s14^2 + 4*s15 + s15^2 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> s16 :|: s14 >= 0, s14 <= x_1 + 1, s15 >= 0, s15 <= x_2 + 1, s16 >= 0, s16 <= s14 + s15 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s180 :|: s178 >= 0, s178 <= z' - 2 + 1, s179 >= 0, s179 <= s178 + 1, s180 >= 0, s180 <= s179, z' - 2 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s183 :|: s181 >= 0, s181 <= z' - 2 + 1, s182 >= 0, s182 <= s181, s183 >= 0, s183 <= s182 + 1, z' - 2 >= 0 encArg(z') -{ 254 + s17 + s18 + 467*x_1107 + 283*x_1107^2 + 48*x_1107^3 + 467*x_247 + 283*x_247^2 + 48*x_247^3 }-> s19 :|: s17 >= 0, s17 <= x_1107 + 1, s18 >= 0, s18 <= x_247 + 1, s19 >= 0, s19 <= 1 + s17 + s18 + 1, z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0 + 1, z' - 1 >= 0 encArg(z') -{ 254 + s20 + s21 + 467*x_1108 + 283*x_1108^2 + 48*x_1108^3 + 467*x_248 + 283*x_248^2 + 48*x_248^3 }-> s22 :|: s20 >= 0, s20 <= x_1108 + 1, s21 >= 0, s21 <= x_248 + 1, s22 >= 0, s22 <= 1 + s20 + s21 + 1, x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ -58 + s23 + -89*z' + -5*z'^2 + 48*z'^3 }-> s24 :|: s23 >= 0, s23 <= z' - 2 + 1, s24 >= 0, s24 <= 1 + s23 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s25 + 2*s25*s26 + s25^2 + 4*s26 + s26^2 + s27 + 467*x_1111 + 283*x_1111^2 + 48*x_1111^3 + 467*x_249 + 283*x_249^2 + 48*x_249^3 }-> s28 :|: s25 >= 0, s25 <= x_1111 + 1, s26 >= 0, s26 <= x_249 + 1, s27 >= 0, s27 <= s25 + s26 + 1, s28 >= 0, s28 <= s27 + 1, z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ -58 + s29 + s30 + -89*z' + -5*z'^2 + 48*z'^3 }-> s31 :|: s29 >= 0, s29 <= z' - 2 + 1, s30 >= 0, s30 <= s29 + 1, s31 >= 0, s31 <= s30 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s33 + s35 + 467*x_1113 + 283*x_1113^2 + 48*x_1113^3 + 467*x_250 + 283*x_250^2 + 48*x_250^3 + 467*x_311 + 283*x_311^2 + 48*x_311^3 }-> s36 :|: s32 >= 0, s32 <= x_1113 + 1, s33 >= 0, s33 <= x_250 + 1, s34 >= 0, s34 <= x_311 + 1, s35 >= 0, s35 <= s32 + s33 + s34 + 1, s36 >= 0, s36 <= s35 + 1, x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 378 + s38 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 + 467*x_3 + 283*x_3^2 + 48*x_3^3 }-> s40 :|: s37 >= 0, s37 <= x_1 + 1, s38 >= 0, s38 <= x_2 + 1, s39 >= 0, s39 <= x_3 + 1, s40 >= 0, s40 <= s37 + s38 + s39 + 1, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z') -{ -58 + s75 + -89*z' + -5*z'^2 + 48*z'^3 }-> s76 :|: s74 >= 0, s74 <= z' - 2 + 1, s75 >= 0, s75 <= s74 + 1, s76 >= 0, s76 <= s75 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11322 + 283*x_11322^2 + 48*x_11322^3 + 467*x_2587 + 283*x_2587^2 + 48*x_2587^3 }-> s79 :|: s77 >= 0, s77 <= x_11322 + 1, s78 >= 0, s78 <= x_2587 + 1, s79 >= 0, s79 <= 1 + s77 + s78 + 1, x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 253 + 467*x_11323 + 283*x_11323^2 + 48*x_11323^3 + 467*x_2588 + 283*x_2588^2 + 48*x_2588^3 }-> s82 :|: s80 >= 0, s80 <= x_11323 + 1, s81 >= 0, s81 <= x_2588 + 1, s82 >= 0, s82 <= 1 + s80 + s81 + 1, z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s84 :|: s83 >= 0, s83 <= z' - 2 + 1, s84 >= 0, s84 <= 1 + s83 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s85 + 2*s85*s86 + s85^2 + 4*s86 + s86^2 + 467*x_11326 + 283*x_11326^2 + 48*x_11326^3 + 467*x_2589 + 283*x_2589^2 + 48*x_2589^3 }-> s88 :|: s85 >= 0, s85 <= x_11326 + 1, s86 >= 0, s86 <= x_2589 + 1, s87 >= 0, s87 <= s85 + s86 + 1, s88 >= 0, s88 <= s87 + 1, x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ -58 + s89 + -89*z' + -5*z'^2 + 48*z'^3 }-> s91 :|: s89 >= 0, s89 <= z' - 2 + 1, s90 >= 0, s90 <= s89 + 1, s91 >= 0, s91 <= s90 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s93 + 467*x_11328 + 283*x_11328^2 + 48*x_11328^3 + 467*x_2590 + 283*x_2590^2 + 48*x_2590^3 + 467*x_3146 + 283*x_3146^2 + 48*x_3146^3 }-> s96 :|: s92 >= 0, s92 <= x_11328 + 1, s93 >= 0, s93 <= x_2590 + 1, s94 >= 0, s94 <= x_3146 + 1, s95 >= 0, s95 <= s92 + s93 + s94 + 1, s96 >= 0, s96 <= s95 + 1, z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s99 :|: s97 >= 0, s97 <= z' - 2 + 1, s98 >= 0, s98 <= s97 + 1, s99 >= 0, s99 <= s98 + 1, z' - 2 >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ -106 + 45*z' + 139*z'^2 + 48*z'^3 }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0 encArg(z') -{ 252 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> 1 + s11 + s12 :|: s11 >= 0, s11 <= x_1 + 1, s12 >= 0, s12 <= x_2 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 260 + 6*s41 + 2*s41*s42 + s41^2 + 4*s42 + s42^2 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> s43 :|: s41 >= 0, s41 <= z' + 1, s42 >= 0, s42 <= z'' + 1, s43 >= 0, s43 <= s41 + s42 + 1, z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ -104 + s101 + 45*z' + 139*z'^2 + 48*z'^3 }-> s102 :|: s100 >= 0, s100 <= z' - 1 + 1, s101 >= 0, s101 <= s100 + 1, s102 >= 0, s102 <= s101 + 1, z' - 1 >= 0 encode_g(z') -{ -104 + s153 + 45*z' + 139*z'^2 + 48*z'^3 }-> s154 :|: s152 >= 0, s152 <= z' - 1 + 1, s153 >= 0, s153 <= s152, s154 >= 0, s154 <= s153 + 1, z' - 1 >= 0 encode_g(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0 + 1, z' = 0 encode_g(z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 1, z' >= 0 encode_g(z') -{ 254 + s46 + s47 + 467*x_11439 + 283*x_11439^2 + 48*x_11439^3 + 467*x_2639 + 283*x_2639^2 + 48*x_2639^3 }-> s48 :|: s46 >= 0, s46 <= x_11439 + 1, s47 >= 0, s47 <= x_2639 + 1, s48 >= 0, s48 <= 1 + s46 + s47 + 1, x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 254 + s49 + s50 + 467*x_11440 + 283*x_11440^2 + 48*x_11440^3 + 467*x_2640 + 283*x_2640^2 + 48*x_2640^3 }-> s51 :|: s49 >= 0, s49 <= x_11440 + 1, s50 >= 0, s50 <= x_2640 + 1, s51 >= 0, s51 <= 1 + s49 + s50 + 1, x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ -104 + s52 + 45*z' + 139*z'^2 + 48*z'^3 }-> s53 :|: s52 >= 0, s52 <= z' - 1 + 1, s53 >= 0, s53 <= 1 + s52 + 1, z' - 1 >= 0 encode_g(z') -{ 261 + 6*s54 + 2*s54*s55 + s54^2 + 4*s55 + s55^2 + s56 + 467*x_11443 + 283*x_11443^2 + 48*x_11443^3 + 467*x_2641 + 283*x_2641^2 + 48*x_2641^3 }-> s57 :|: s54 >= 0, s54 <= x_11443 + 1, s55 >= 0, s55 <= x_2641 + 1, s56 >= 0, s56 <= s54 + s55 + 1, s57 >= 0, s57 <= s56 + 1, x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ -104 + s58 + s59 + 45*z' + 139*z'^2 + 48*z'^3 }-> s60 :|: s58 >= 0, s58 <= z' - 1 + 1, s59 >= 0, s59 <= s58 + 1, s60 >= 0, s60 <= s59 + 1, z' - 1 >= 0 encode_g(z') -{ 379 + s62 + s64 + 467*x_11445 + 283*x_11445^2 + 48*x_11445^3 + 467*x_2642 + 283*x_2642^2 + 48*x_2642^3 + 467*x_3159 + 283*x_3159^2 + 48*x_3159^3 }-> s65 :|: s61 >= 0, s61 <= x_11445 + 1, s62 >= 0, s62 <= x_2642 + 1, s63 >= 0, s63 <= x_3159 + 1, s64 >= 0, s64 <= s61 + s62 + s63 + 1, s65 >= 0, s65 <= s64 + 1, x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s73 :|: s73 >= 0, s73 <= z' + 1, z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s66 + s67 :|: s66 >= 0, s66 <= z' + 1, s67 >= 0, s67 <= z'' + 1, z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 378 + s69 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 + 467*z1 + 283*z1^2 + 48*z1^3 }-> s71 :|: s68 >= 0, s68 <= z' + 1, s69 >= 0, s69 <= z'' + 1, s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s68 + s69 + s70 + 1, z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 253 + 467*x_12654 + 283*x_12654^2 + 48*x_12654^3 + 467*x_21179 + 283*x_21179^2 + 48*x_21179^3 }-> s157 :|: s155 >= 0, s155 <= x_12654 + 1, s156 >= 0, s156 <= x_21179 + 1, s157 >= 0, s157 <= 1 + s155 + s156, x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 253 + 467*x_12655 + 283*x_12655^2 + 48*x_12655^3 + 467*x_21180 + 283*x_21180^2 + 48*x_21180^3 }-> s160 :|: s158 >= 0, s158 <= x_12655 + 1, s159 >= 0, s159 <= x_21180 + 1, s160 >= 0, s160 <= 1 + s158 + s159, z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s162 :|: s161 >= 0, s161 <= z' - 1 + 1, s162 >= 0, s162 <= 1 + s161, z' - 1 >= 0 encode_i(z') -{ 261 + 6*s163 + 2*s163*s164 + s163^2 + 4*s164 + s164^2 + 467*x_12658 + 283*x_12658^2 + 48*x_12658^3 + 467*x_21181 + 283*x_21181^2 + 48*x_21181^3 }-> s166 :|: s163 >= 0, s163 <= x_12658 + 1, s164 >= 0, s164 <= x_21181 + 1, s165 >= 0, s165 <= s163 + s164 + 1, s166 >= 0, s166 <= s165, x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ -104 + s167 + 45*z' + 139*z'^2 + 48*z'^3 }-> s169 :|: s167 >= 0, s167 <= z' - 1 + 1, s168 >= 0, s168 <= s167 + 1, s169 >= 0, s169 <= s168, z' - 1 >= 0 encode_i(z') -{ 379 + s171 + 467*x_12660 + 283*x_12660^2 + 48*x_12660^3 + 467*x_21182 + 283*x_21182^2 + 48*x_21182^3 + 467*x_3294 + 283*x_3294^2 + 48*x_3294^3 }-> s174 :|: s170 >= 0, s170 <= x_12660 + 1, s171 >= 0, s171 <= x_21182 + 1, s172 >= 0, s172 <= x_3294 + 1, s173 >= 0, s173 <= s170 + s171 + s172 + 1, s174 >= 0, s174 <= s173, z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s177 :|: s175 >= 0, s175 <= z' - 1 + 1, s176 >= 0, s176 <= s175, s177 >= 0, s177 <= s176, z' - 1 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s189 :|: s187 >= 0, s187 <= z' - 1 + 1, s188 >= 0, s188 <= s187 + 1, s189 >= 0, s189 <= s188, z' - 1 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s44 + s45 :|: s44 >= 0, s44 <= z' + 1, s45 >= 0, s45 <= z'' + 1, z' >= 0, z'' >= 0 encode_k(z') -{ 253 + 467*x_11448 + 283*x_11448^2 + 48*x_11448^3 + 467*x_2643 + 283*x_2643^2 + 48*x_2643^3 }-> s105 :|: s103 >= 0, s103 <= x_11448 + 1, s104 >= 0, s104 <= x_2643 + 1, s105 >= 0, s105 <= 1 + s103 + s104 + 1, z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 253 + 467*x_11449 + 283*x_11449^2 + 48*x_11449^3 + 467*x_2644 + 283*x_2644^2 + 48*x_2644^3 }-> s108 :|: s106 >= 0, s106 <= x_11449 + 1, s107 >= 0, s107 <= x_2644 + 1, s108 >= 0, s108 <= 1 + s106 + s107 + 1, x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s110 :|: s109 >= 0, s109 <= z' - 1 + 1, s110 >= 0, s110 <= 1 + s109 + 1, z' - 1 >= 0 encode_k(z') -{ 261 + 6*s111 + 2*s111*s112 + s111^2 + 4*s112 + s112^2 + 467*x_11452 + 283*x_11452^2 + 48*x_11452^3 + 467*x_2645 + 283*x_2645^2 + 48*x_2645^3 }-> s114 :|: s111 >= 0, s111 <= x_11452 + 1, s112 >= 0, s112 <= x_2645 + 1, s113 >= 0, s113 <= s111 + s112 + 1, s114 >= 0, s114 <= s113 + 1, x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ -104 + s115 + 45*z' + 139*z'^2 + 48*z'^3 }-> s117 :|: s115 >= 0, s115 <= z' - 1 + 1, s116 >= 0, s116 <= s115 + 1, s117 >= 0, s117 <= s116 + 1, z' - 1 >= 0 encode_k(z') -{ 379 + s119 + 467*x_11454 + 283*x_11454^2 + 48*x_11454^3 + 467*x_2646 + 283*x_2646^2 + 48*x_2646^3 + 467*x_3160 + 283*x_3160^2 + 48*x_3160^3 }-> s122 :|: s118 >= 0, s118 <= x_11454 + 1, s119 >= 0, s119 <= x_2646 + 1, s120 >= 0, s120 <= x_3160 + 1, s121 >= 0, s121 <= s118 + s119 + s120 + 1, s122 >= 0, s122 <= s121 + 1, x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s125 :|: s123 >= 0, s123 <= z' - 1 + 1, s124 >= 0, s124 <= s123 + 1, s125 >= 0, s125 <= s124 + 1, z' - 1 >= 0 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s186 :|: s184 >= 0, s184 <= z' - 1 + 1, s185 >= 0, s185 <= s184, s186 >= 0, s186 <= s185 + 1, z' - 1 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s72 :|: s72 >= 0, s72 <= z' + 1, z' >= 0 f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 10 + s9 + 6*x + x^2 }-> s10 :|: s9 >= 0, s9 <= x + 0 + 1, s10 >= 0, s10 <= s9 + 1, z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 11 + s5 + 6*x + 2*x*z'' + x^2 + 4*z'' + z''^2 }-> s6 :|: s5 >= 0, s5 <= x + (1 + 0 + (z'' - 1)) + 1, s6 >= 0, s6 <= s5 + 1, z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 23 + s7 + 10*x + 2*x*x'' + 2*x*y' + x^2 + 8*x'' + 2*x''*y' + x''^2 + 8*y' + y'^2 }-> s8 :|: s7 >= 0, s7 <= x + (1 + (1 + x'') + y') + 1, s8 >= 0, s8 <= s7 + 1, z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {encode_h}, {encode_h2}, {encode_h1}, {encode_g}, {encode_f}, {encode_j}, {encode_i}, {encode_s} Previous analysis results are: h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1] encode_0: runtime: O(1) [0], size: O(1) [0] i: runtime: O(1) [1], size: O(n^1) [z'] k: runtime: O(1) [1], size: O(n^1) [1 + z'] g: runtime: O(n^1) [1 + z'], size: O(n^1) [1 + z'] f: runtime: O(n^2) [8 + 6*z' + 2*z'*z'' + z'^2 + 4*z'' + z''^2], size: O(n^1) [1 + z' + z''] encArg: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [1 + z'] encode_k: runtime: O(n^3) [1161 + 4403*z' + 3107*z'^2 + 624*z'^3], size: O(n^1) [4 + z'] ---------------------------------------- (75) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: encode_h after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z' ---------------------------------------- (76) Obligation: Complexity RNTS consisting of the following rules: encArg(z') -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z' = 1 + 0 encArg(z') -{ -58 + s127 + -89*z' + -5*z'^2 + 48*z'^3 }-> s128 :|: s126 >= 0, s126 <= z' - 2 + 1, s127 >= 0, s127 <= s126, s128 >= 0, s128 <= s127 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11313 + 283*x_11313^2 + 48*x_11313^3 + 467*x_2583 + 283*x_2583^2 + 48*x_2583^3 }-> s131 :|: s129 >= 0, s129 <= x_11313 + 1, s130 >= 0, s130 <= x_2583 + 1, s131 >= 0, s131 <= 1 + s129 + s130, x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 253 + 467*x_11314 + 283*x_11314^2 + 48*x_11314^3 + 467*x_2584 + 283*x_2584^2 + 48*x_2584^3 }-> s134 :|: s132 >= 0, s132 <= x_11314 + 1, s133 >= 0, s133 <= x_2584 + 1, s134 >= 0, s134 <= 1 + s132 + s133, z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s136 :|: s135 >= 0, s135 <= z' - 2 + 1, s136 >= 0, s136 <= 1 + s135, z' - 2 >= 0 encArg(z') -{ 261 + 6*s137 + 2*s137*s138 + s137^2 + 4*s138 + s138^2 + 467*x_11317 + 283*x_11317^2 + 48*x_11317^3 + 467*x_2585 + 283*x_2585^2 + 48*x_2585^3 }-> s140 :|: s137 >= 0, s137 <= x_11317 + 1, s138 >= 0, s138 <= x_2585 + 1, s139 >= 0, s139 <= s137 + s138 + 1, s140 >= 0, s140 <= s139, x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ -58 + s141 + -89*z' + -5*z'^2 + 48*z'^3 }-> s143 :|: s141 >= 0, s141 <= z' - 2 + 1, s142 >= 0, s142 <= s141 + 1, s143 >= 0, s143 <= s142, z' - 2 >= 0 encArg(z') -{ 379 + s145 + 467*x_11319 + 283*x_11319^2 + 48*x_11319^3 + 467*x_2586 + 283*x_2586^2 + 48*x_2586^3 + 467*x_3145 + 283*x_3145^2 + 48*x_3145^3 }-> s148 :|: s144 >= 0, s144 <= x_11319 + 1, s145 >= 0, s145 <= x_2586 + 1, s146 >= 0, s146 <= x_3145 + 1, s147 >= 0, s147 <= s144 + s145 + s146 + 1, s148 >= 0, s148 <= s147, z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s151 :|: s149 >= 0, s149 <= z' - 2 + 1, s150 >= 0, s150 <= s149, s151 >= 0, s151 <= s150, z' - 2 >= 0 encArg(z') -{ 260 + 6*s14 + 2*s14*s15 + s14^2 + 4*s15 + s15^2 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> s16 :|: s14 >= 0, s14 <= x_1 + 1, s15 >= 0, s15 <= x_2 + 1, s16 >= 0, s16 <= s14 + s15 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s180 :|: s178 >= 0, s178 <= z' - 2 + 1, s179 >= 0, s179 <= s178 + 1, s180 >= 0, s180 <= s179, z' - 2 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s183 :|: s181 >= 0, s181 <= z' - 2 + 1, s182 >= 0, s182 <= s181, s183 >= 0, s183 <= s182 + 1, z' - 2 >= 0 encArg(z') -{ 254 + s17 + s18 + 467*x_1107 + 283*x_1107^2 + 48*x_1107^3 + 467*x_247 + 283*x_247^2 + 48*x_247^3 }-> s19 :|: s17 >= 0, s17 <= x_1107 + 1, s18 >= 0, s18 <= x_247 + 1, s19 >= 0, s19 <= 1 + s17 + s18 + 1, z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0 + 1, z' - 1 >= 0 encArg(z') -{ 254 + s20 + s21 + 467*x_1108 + 283*x_1108^2 + 48*x_1108^3 + 467*x_248 + 283*x_248^2 + 48*x_248^3 }-> s22 :|: s20 >= 0, s20 <= x_1108 + 1, s21 >= 0, s21 <= x_248 + 1, s22 >= 0, s22 <= 1 + s20 + s21 + 1, x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ -58 + s23 + -89*z' + -5*z'^2 + 48*z'^3 }-> s24 :|: s23 >= 0, s23 <= z' - 2 + 1, s24 >= 0, s24 <= 1 + s23 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s25 + 2*s25*s26 + s25^2 + 4*s26 + s26^2 + s27 + 467*x_1111 + 283*x_1111^2 + 48*x_1111^3 + 467*x_249 + 283*x_249^2 + 48*x_249^3 }-> s28 :|: s25 >= 0, s25 <= x_1111 + 1, s26 >= 0, s26 <= x_249 + 1, s27 >= 0, s27 <= s25 + s26 + 1, s28 >= 0, s28 <= s27 + 1, z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ -58 + s29 + s30 + -89*z' + -5*z'^2 + 48*z'^3 }-> s31 :|: s29 >= 0, s29 <= z' - 2 + 1, s30 >= 0, s30 <= s29 + 1, s31 >= 0, s31 <= s30 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s33 + s35 + 467*x_1113 + 283*x_1113^2 + 48*x_1113^3 + 467*x_250 + 283*x_250^2 + 48*x_250^3 + 467*x_311 + 283*x_311^2 + 48*x_311^3 }-> s36 :|: s32 >= 0, s32 <= x_1113 + 1, s33 >= 0, s33 <= x_250 + 1, s34 >= 0, s34 <= x_311 + 1, s35 >= 0, s35 <= s32 + s33 + s34 + 1, s36 >= 0, s36 <= s35 + 1, x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 378 + s38 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 + 467*x_3 + 283*x_3^2 + 48*x_3^3 }-> s40 :|: s37 >= 0, s37 <= x_1 + 1, s38 >= 0, s38 <= x_2 + 1, s39 >= 0, s39 <= x_3 + 1, s40 >= 0, s40 <= s37 + s38 + s39 + 1, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z') -{ -58 + s75 + -89*z' + -5*z'^2 + 48*z'^3 }-> s76 :|: s74 >= 0, s74 <= z' - 2 + 1, s75 >= 0, s75 <= s74 + 1, s76 >= 0, s76 <= s75 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11322 + 283*x_11322^2 + 48*x_11322^3 + 467*x_2587 + 283*x_2587^2 + 48*x_2587^3 }-> s79 :|: s77 >= 0, s77 <= x_11322 + 1, s78 >= 0, s78 <= x_2587 + 1, s79 >= 0, s79 <= 1 + s77 + s78 + 1, x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 253 + 467*x_11323 + 283*x_11323^2 + 48*x_11323^3 + 467*x_2588 + 283*x_2588^2 + 48*x_2588^3 }-> s82 :|: s80 >= 0, s80 <= x_11323 + 1, s81 >= 0, s81 <= x_2588 + 1, s82 >= 0, s82 <= 1 + s80 + s81 + 1, z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s84 :|: s83 >= 0, s83 <= z' - 2 + 1, s84 >= 0, s84 <= 1 + s83 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s85 + 2*s85*s86 + s85^2 + 4*s86 + s86^2 + 467*x_11326 + 283*x_11326^2 + 48*x_11326^3 + 467*x_2589 + 283*x_2589^2 + 48*x_2589^3 }-> s88 :|: s85 >= 0, s85 <= x_11326 + 1, s86 >= 0, s86 <= x_2589 + 1, s87 >= 0, s87 <= s85 + s86 + 1, s88 >= 0, s88 <= s87 + 1, x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ -58 + s89 + -89*z' + -5*z'^2 + 48*z'^3 }-> s91 :|: s89 >= 0, s89 <= z' - 2 + 1, s90 >= 0, s90 <= s89 + 1, s91 >= 0, s91 <= s90 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s93 + 467*x_11328 + 283*x_11328^2 + 48*x_11328^3 + 467*x_2590 + 283*x_2590^2 + 48*x_2590^3 + 467*x_3146 + 283*x_3146^2 + 48*x_3146^3 }-> s96 :|: s92 >= 0, s92 <= x_11328 + 1, s93 >= 0, s93 <= x_2590 + 1, s94 >= 0, s94 <= x_3146 + 1, s95 >= 0, s95 <= s92 + s93 + s94 + 1, s96 >= 0, s96 <= s95 + 1, z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s99 :|: s97 >= 0, s97 <= z' - 2 + 1, s98 >= 0, s98 <= s97 + 1, s99 >= 0, s99 <= s98 + 1, z' - 2 >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ -106 + 45*z' + 139*z'^2 + 48*z'^3 }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0 encArg(z') -{ 252 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> 1 + s11 + s12 :|: s11 >= 0, s11 <= x_1 + 1, s12 >= 0, s12 <= x_2 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 260 + 6*s41 + 2*s41*s42 + s41^2 + 4*s42 + s42^2 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> s43 :|: s41 >= 0, s41 <= z' + 1, s42 >= 0, s42 <= z'' + 1, s43 >= 0, s43 <= s41 + s42 + 1, z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ -104 + s101 + 45*z' + 139*z'^2 + 48*z'^3 }-> s102 :|: s100 >= 0, s100 <= z' - 1 + 1, s101 >= 0, s101 <= s100 + 1, s102 >= 0, s102 <= s101 + 1, z' - 1 >= 0 encode_g(z') -{ -104 + s153 + 45*z' + 139*z'^2 + 48*z'^3 }-> s154 :|: s152 >= 0, s152 <= z' - 1 + 1, s153 >= 0, s153 <= s152, s154 >= 0, s154 <= s153 + 1, z' - 1 >= 0 encode_g(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0 + 1, z' = 0 encode_g(z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 1, z' >= 0 encode_g(z') -{ 254 + s46 + s47 + 467*x_11439 + 283*x_11439^2 + 48*x_11439^3 + 467*x_2639 + 283*x_2639^2 + 48*x_2639^3 }-> s48 :|: s46 >= 0, s46 <= x_11439 + 1, s47 >= 0, s47 <= x_2639 + 1, s48 >= 0, s48 <= 1 + s46 + s47 + 1, x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 254 + s49 + s50 + 467*x_11440 + 283*x_11440^2 + 48*x_11440^3 + 467*x_2640 + 283*x_2640^2 + 48*x_2640^3 }-> s51 :|: s49 >= 0, s49 <= x_11440 + 1, s50 >= 0, s50 <= x_2640 + 1, s51 >= 0, s51 <= 1 + s49 + s50 + 1, x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ -104 + s52 + 45*z' + 139*z'^2 + 48*z'^3 }-> s53 :|: s52 >= 0, s52 <= z' - 1 + 1, s53 >= 0, s53 <= 1 + s52 + 1, z' - 1 >= 0 encode_g(z') -{ 261 + 6*s54 + 2*s54*s55 + s54^2 + 4*s55 + s55^2 + s56 + 467*x_11443 + 283*x_11443^2 + 48*x_11443^3 + 467*x_2641 + 283*x_2641^2 + 48*x_2641^3 }-> s57 :|: s54 >= 0, s54 <= x_11443 + 1, s55 >= 0, s55 <= x_2641 + 1, s56 >= 0, s56 <= s54 + s55 + 1, s57 >= 0, s57 <= s56 + 1, x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ -104 + s58 + s59 + 45*z' + 139*z'^2 + 48*z'^3 }-> s60 :|: s58 >= 0, s58 <= z' - 1 + 1, s59 >= 0, s59 <= s58 + 1, s60 >= 0, s60 <= s59 + 1, z' - 1 >= 0 encode_g(z') -{ 379 + s62 + s64 + 467*x_11445 + 283*x_11445^2 + 48*x_11445^3 + 467*x_2642 + 283*x_2642^2 + 48*x_2642^3 + 467*x_3159 + 283*x_3159^2 + 48*x_3159^3 }-> s65 :|: s61 >= 0, s61 <= x_11445 + 1, s62 >= 0, s62 <= x_2642 + 1, s63 >= 0, s63 <= x_3159 + 1, s64 >= 0, s64 <= s61 + s62 + s63 + 1, s65 >= 0, s65 <= s64 + 1, x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s73 :|: s73 >= 0, s73 <= z' + 1, z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s66 + s67 :|: s66 >= 0, s66 <= z' + 1, s67 >= 0, s67 <= z'' + 1, z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 378 + s69 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 + 467*z1 + 283*z1^2 + 48*z1^3 }-> s71 :|: s68 >= 0, s68 <= z' + 1, s69 >= 0, s69 <= z'' + 1, s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s68 + s69 + s70 + 1, z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 253 + 467*x_12654 + 283*x_12654^2 + 48*x_12654^3 + 467*x_21179 + 283*x_21179^2 + 48*x_21179^3 }-> s157 :|: s155 >= 0, s155 <= x_12654 + 1, s156 >= 0, s156 <= x_21179 + 1, s157 >= 0, s157 <= 1 + s155 + s156, x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 253 + 467*x_12655 + 283*x_12655^2 + 48*x_12655^3 + 467*x_21180 + 283*x_21180^2 + 48*x_21180^3 }-> s160 :|: s158 >= 0, s158 <= x_12655 + 1, s159 >= 0, s159 <= x_21180 + 1, s160 >= 0, s160 <= 1 + s158 + s159, z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s162 :|: s161 >= 0, s161 <= z' - 1 + 1, s162 >= 0, s162 <= 1 + s161, z' - 1 >= 0 encode_i(z') -{ 261 + 6*s163 + 2*s163*s164 + s163^2 + 4*s164 + s164^2 + 467*x_12658 + 283*x_12658^2 + 48*x_12658^3 + 467*x_21181 + 283*x_21181^2 + 48*x_21181^3 }-> s166 :|: s163 >= 0, s163 <= x_12658 + 1, s164 >= 0, s164 <= x_21181 + 1, s165 >= 0, s165 <= s163 + s164 + 1, s166 >= 0, s166 <= s165, x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ -104 + s167 + 45*z' + 139*z'^2 + 48*z'^3 }-> s169 :|: s167 >= 0, s167 <= z' - 1 + 1, s168 >= 0, s168 <= s167 + 1, s169 >= 0, s169 <= s168, z' - 1 >= 0 encode_i(z') -{ 379 + s171 + 467*x_12660 + 283*x_12660^2 + 48*x_12660^3 + 467*x_21182 + 283*x_21182^2 + 48*x_21182^3 + 467*x_3294 + 283*x_3294^2 + 48*x_3294^3 }-> s174 :|: s170 >= 0, s170 <= x_12660 + 1, s171 >= 0, s171 <= x_21182 + 1, s172 >= 0, s172 <= x_3294 + 1, s173 >= 0, s173 <= s170 + s171 + s172 + 1, s174 >= 0, s174 <= s173, z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s177 :|: s175 >= 0, s175 <= z' - 1 + 1, s176 >= 0, s176 <= s175, s177 >= 0, s177 <= s176, z' - 1 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s189 :|: s187 >= 0, s187 <= z' - 1 + 1, s188 >= 0, s188 <= s187 + 1, s189 >= 0, s189 <= s188, z' - 1 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s44 + s45 :|: s44 >= 0, s44 <= z' + 1, s45 >= 0, s45 <= z'' + 1, z' >= 0, z'' >= 0 encode_k(z') -{ 253 + 467*x_11448 + 283*x_11448^2 + 48*x_11448^3 + 467*x_2643 + 283*x_2643^2 + 48*x_2643^3 }-> s105 :|: s103 >= 0, s103 <= x_11448 + 1, s104 >= 0, s104 <= x_2643 + 1, s105 >= 0, s105 <= 1 + s103 + s104 + 1, z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 253 + 467*x_11449 + 283*x_11449^2 + 48*x_11449^3 + 467*x_2644 + 283*x_2644^2 + 48*x_2644^3 }-> s108 :|: s106 >= 0, s106 <= x_11449 + 1, s107 >= 0, s107 <= x_2644 + 1, s108 >= 0, s108 <= 1 + s106 + s107 + 1, x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s110 :|: s109 >= 0, s109 <= z' - 1 + 1, s110 >= 0, s110 <= 1 + s109 + 1, z' - 1 >= 0 encode_k(z') -{ 261 + 6*s111 + 2*s111*s112 + s111^2 + 4*s112 + s112^2 + 467*x_11452 + 283*x_11452^2 + 48*x_11452^3 + 467*x_2645 + 283*x_2645^2 + 48*x_2645^3 }-> s114 :|: s111 >= 0, s111 <= x_11452 + 1, s112 >= 0, s112 <= x_2645 + 1, s113 >= 0, s113 <= s111 + s112 + 1, s114 >= 0, s114 <= s113 + 1, x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ -104 + s115 + 45*z' + 139*z'^2 + 48*z'^3 }-> s117 :|: s115 >= 0, s115 <= z' - 1 + 1, s116 >= 0, s116 <= s115 + 1, s117 >= 0, s117 <= s116 + 1, z' - 1 >= 0 encode_k(z') -{ 379 + s119 + 467*x_11454 + 283*x_11454^2 + 48*x_11454^3 + 467*x_2646 + 283*x_2646^2 + 48*x_2646^3 + 467*x_3160 + 283*x_3160^2 + 48*x_3160^3 }-> s122 :|: s118 >= 0, s118 <= x_11454 + 1, s119 >= 0, s119 <= x_2646 + 1, s120 >= 0, s120 <= x_3160 + 1, s121 >= 0, s121 <= s118 + s119 + s120 + 1, s122 >= 0, s122 <= s121 + 1, x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s125 :|: s123 >= 0, s123 <= z' - 1 + 1, s124 >= 0, s124 <= s123 + 1, s125 >= 0, s125 <= s124 + 1, z' - 1 >= 0 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s186 :|: s184 >= 0, s184 <= z' - 1 + 1, s185 >= 0, s185 <= s184, s186 >= 0, s186 <= s185 + 1, z' - 1 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s72 :|: s72 >= 0, s72 <= z' + 1, z' >= 0 f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 10 + s9 + 6*x + x^2 }-> s10 :|: s9 >= 0, s9 <= x + 0 + 1, s10 >= 0, s10 <= s9 + 1, z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 11 + s5 + 6*x + 2*x*z'' + x^2 + 4*z'' + z''^2 }-> s6 :|: s5 >= 0, s5 <= x + (1 + 0 + (z'' - 1)) + 1, s6 >= 0, s6 <= s5 + 1, z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 23 + s7 + 10*x + 2*x*x'' + 2*x*y' + x^2 + 8*x'' + 2*x''*y' + x''^2 + 8*y' + y'^2 }-> s8 :|: s7 >= 0, s7 <= x + (1 + (1 + x'') + y') + 1, s8 >= 0, s8 <= s7 + 1, z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {encode_h}, {encode_h2}, {encode_h1}, {encode_g}, {encode_f}, {encode_j}, {encode_i}, {encode_s} Previous analysis results are: h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1] encode_0: runtime: O(1) [0], size: O(1) [0] i: runtime: O(1) [1], size: O(n^1) [z'] k: runtime: O(1) [1], size: O(n^1) [1 + z'] g: runtime: O(n^1) [1 + z'], size: O(n^1) [1 + z'] f: runtime: O(n^2) [8 + 6*z' + 2*z'*z'' + z'^2 + 4*z'' + z''^2], size: O(n^1) [1 + z' + z''] encArg: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [1 + z'] encode_k: runtime: O(n^3) [1161 + 4403*z' + 3107*z'^2 + 624*z'^3], size: O(n^1) [4 + z'] encode_h: runtime: ?, size: O(n^1) [2 + z'] ---------------------------------------- (77) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: encode_h after applying outer abstraction to obtain an ITS, resulting in: O(n^3) with polynomial bound: 126 + 467*z' + 283*z'^2 + 48*z'^3 ---------------------------------------- (78) Obligation: Complexity RNTS consisting of the following rules: encArg(z') -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z' = 1 + 0 encArg(z') -{ -58 + s127 + -89*z' + -5*z'^2 + 48*z'^3 }-> s128 :|: s126 >= 0, s126 <= z' - 2 + 1, s127 >= 0, s127 <= s126, s128 >= 0, s128 <= s127 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11313 + 283*x_11313^2 + 48*x_11313^3 + 467*x_2583 + 283*x_2583^2 + 48*x_2583^3 }-> s131 :|: s129 >= 0, s129 <= x_11313 + 1, s130 >= 0, s130 <= x_2583 + 1, s131 >= 0, s131 <= 1 + s129 + s130, x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 253 + 467*x_11314 + 283*x_11314^2 + 48*x_11314^3 + 467*x_2584 + 283*x_2584^2 + 48*x_2584^3 }-> s134 :|: s132 >= 0, s132 <= x_11314 + 1, s133 >= 0, s133 <= x_2584 + 1, s134 >= 0, s134 <= 1 + s132 + s133, z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s136 :|: s135 >= 0, s135 <= z' - 2 + 1, s136 >= 0, s136 <= 1 + s135, z' - 2 >= 0 encArg(z') -{ 261 + 6*s137 + 2*s137*s138 + s137^2 + 4*s138 + s138^2 + 467*x_11317 + 283*x_11317^2 + 48*x_11317^3 + 467*x_2585 + 283*x_2585^2 + 48*x_2585^3 }-> s140 :|: s137 >= 0, s137 <= x_11317 + 1, s138 >= 0, s138 <= x_2585 + 1, s139 >= 0, s139 <= s137 + s138 + 1, s140 >= 0, s140 <= s139, x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ -58 + s141 + -89*z' + -5*z'^2 + 48*z'^3 }-> s143 :|: s141 >= 0, s141 <= z' - 2 + 1, s142 >= 0, s142 <= s141 + 1, s143 >= 0, s143 <= s142, z' - 2 >= 0 encArg(z') -{ 379 + s145 + 467*x_11319 + 283*x_11319^2 + 48*x_11319^3 + 467*x_2586 + 283*x_2586^2 + 48*x_2586^3 + 467*x_3145 + 283*x_3145^2 + 48*x_3145^3 }-> s148 :|: s144 >= 0, s144 <= x_11319 + 1, s145 >= 0, s145 <= x_2586 + 1, s146 >= 0, s146 <= x_3145 + 1, s147 >= 0, s147 <= s144 + s145 + s146 + 1, s148 >= 0, s148 <= s147, z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s151 :|: s149 >= 0, s149 <= z' - 2 + 1, s150 >= 0, s150 <= s149, s151 >= 0, s151 <= s150, z' - 2 >= 0 encArg(z') -{ 260 + 6*s14 + 2*s14*s15 + s14^2 + 4*s15 + s15^2 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> s16 :|: s14 >= 0, s14 <= x_1 + 1, s15 >= 0, s15 <= x_2 + 1, s16 >= 0, s16 <= s14 + s15 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s180 :|: s178 >= 0, s178 <= z' - 2 + 1, s179 >= 0, s179 <= s178 + 1, s180 >= 0, s180 <= s179, z' - 2 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s183 :|: s181 >= 0, s181 <= z' - 2 + 1, s182 >= 0, s182 <= s181, s183 >= 0, s183 <= s182 + 1, z' - 2 >= 0 encArg(z') -{ 254 + s17 + s18 + 467*x_1107 + 283*x_1107^2 + 48*x_1107^3 + 467*x_247 + 283*x_247^2 + 48*x_247^3 }-> s19 :|: s17 >= 0, s17 <= x_1107 + 1, s18 >= 0, s18 <= x_247 + 1, s19 >= 0, s19 <= 1 + s17 + s18 + 1, z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0 + 1, z' - 1 >= 0 encArg(z') -{ 254 + s20 + s21 + 467*x_1108 + 283*x_1108^2 + 48*x_1108^3 + 467*x_248 + 283*x_248^2 + 48*x_248^3 }-> s22 :|: s20 >= 0, s20 <= x_1108 + 1, s21 >= 0, s21 <= x_248 + 1, s22 >= 0, s22 <= 1 + s20 + s21 + 1, x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ -58 + s23 + -89*z' + -5*z'^2 + 48*z'^3 }-> s24 :|: s23 >= 0, s23 <= z' - 2 + 1, s24 >= 0, s24 <= 1 + s23 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s25 + 2*s25*s26 + s25^2 + 4*s26 + s26^2 + s27 + 467*x_1111 + 283*x_1111^2 + 48*x_1111^3 + 467*x_249 + 283*x_249^2 + 48*x_249^3 }-> s28 :|: s25 >= 0, s25 <= x_1111 + 1, s26 >= 0, s26 <= x_249 + 1, s27 >= 0, s27 <= s25 + s26 + 1, s28 >= 0, s28 <= s27 + 1, z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ -58 + s29 + s30 + -89*z' + -5*z'^2 + 48*z'^3 }-> s31 :|: s29 >= 0, s29 <= z' - 2 + 1, s30 >= 0, s30 <= s29 + 1, s31 >= 0, s31 <= s30 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s33 + s35 + 467*x_1113 + 283*x_1113^2 + 48*x_1113^3 + 467*x_250 + 283*x_250^2 + 48*x_250^3 + 467*x_311 + 283*x_311^2 + 48*x_311^3 }-> s36 :|: s32 >= 0, s32 <= x_1113 + 1, s33 >= 0, s33 <= x_250 + 1, s34 >= 0, s34 <= x_311 + 1, s35 >= 0, s35 <= s32 + s33 + s34 + 1, s36 >= 0, s36 <= s35 + 1, x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 378 + s38 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 + 467*x_3 + 283*x_3^2 + 48*x_3^3 }-> s40 :|: s37 >= 0, s37 <= x_1 + 1, s38 >= 0, s38 <= x_2 + 1, s39 >= 0, s39 <= x_3 + 1, s40 >= 0, s40 <= s37 + s38 + s39 + 1, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z') -{ -58 + s75 + -89*z' + -5*z'^2 + 48*z'^3 }-> s76 :|: s74 >= 0, s74 <= z' - 2 + 1, s75 >= 0, s75 <= s74 + 1, s76 >= 0, s76 <= s75 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11322 + 283*x_11322^2 + 48*x_11322^3 + 467*x_2587 + 283*x_2587^2 + 48*x_2587^3 }-> s79 :|: s77 >= 0, s77 <= x_11322 + 1, s78 >= 0, s78 <= x_2587 + 1, s79 >= 0, s79 <= 1 + s77 + s78 + 1, x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 253 + 467*x_11323 + 283*x_11323^2 + 48*x_11323^3 + 467*x_2588 + 283*x_2588^2 + 48*x_2588^3 }-> s82 :|: s80 >= 0, s80 <= x_11323 + 1, s81 >= 0, s81 <= x_2588 + 1, s82 >= 0, s82 <= 1 + s80 + s81 + 1, z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s84 :|: s83 >= 0, s83 <= z' - 2 + 1, s84 >= 0, s84 <= 1 + s83 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s85 + 2*s85*s86 + s85^2 + 4*s86 + s86^2 + 467*x_11326 + 283*x_11326^2 + 48*x_11326^3 + 467*x_2589 + 283*x_2589^2 + 48*x_2589^3 }-> s88 :|: s85 >= 0, s85 <= x_11326 + 1, s86 >= 0, s86 <= x_2589 + 1, s87 >= 0, s87 <= s85 + s86 + 1, s88 >= 0, s88 <= s87 + 1, x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ -58 + s89 + -89*z' + -5*z'^2 + 48*z'^3 }-> s91 :|: s89 >= 0, s89 <= z' - 2 + 1, s90 >= 0, s90 <= s89 + 1, s91 >= 0, s91 <= s90 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s93 + 467*x_11328 + 283*x_11328^2 + 48*x_11328^3 + 467*x_2590 + 283*x_2590^2 + 48*x_2590^3 + 467*x_3146 + 283*x_3146^2 + 48*x_3146^3 }-> s96 :|: s92 >= 0, s92 <= x_11328 + 1, s93 >= 0, s93 <= x_2590 + 1, s94 >= 0, s94 <= x_3146 + 1, s95 >= 0, s95 <= s92 + s93 + s94 + 1, s96 >= 0, s96 <= s95 + 1, z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s99 :|: s97 >= 0, s97 <= z' - 2 + 1, s98 >= 0, s98 <= s97 + 1, s99 >= 0, s99 <= s98 + 1, z' - 2 >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ -106 + 45*z' + 139*z'^2 + 48*z'^3 }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0 encArg(z') -{ 252 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> 1 + s11 + s12 :|: s11 >= 0, s11 <= x_1 + 1, s12 >= 0, s12 <= x_2 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 260 + 6*s41 + 2*s41*s42 + s41^2 + 4*s42 + s42^2 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> s43 :|: s41 >= 0, s41 <= z' + 1, s42 >= 0, s42 <= z'' + 1, s43 >= 0, s43 <= s41 + s42 + 1, z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ -104 + s101 + 45*z' + 139*z'^2 + 48*z'^3 }-> s102 :|: s100 >= 0, s100 <= z' - 1 + 1, s101 >= 0, s101 <= s100 + 1, s102 >= 0, s102 <= s101 + 1, z' - 1 >= 0 encode_g(z') -{ -104 + s153 + 45*z' + 139*z'^2 + 48*z'^3 }-> s154 :|: s152 >= 0, s152 <= z' - 1 + 1, s153 >= 0, s153 <= s152, s154 >= 0, s154 <= s153 + 1, z' - 1 >= 0 encode_g(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0 + 1, z' = 0 encode_g(z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 1, z' >= 0 encode_g(z') -{ 254 + s46 + s47 + 467*x_11439 + 283*x_11439^2 + 48*x_11439^3 + 467*x_2639 + 283*x_2639^2 + 48*x_2639^3 }-> s48 :|: s46 >= 0, s46 <= x_11439 + 1, s47 >= 0, s47 <= x_2639 + 1, s48 >= 0, s48 <= 1 + s46 + s47 + 1, x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 254 + s49 + s50 + 467*x_11440 + 283*x_11440^2 + 48*x_11440^3 + 467*x_2640 + 283*x_2640^2 + 48*x_2640^3 }-> s51 :|: s49 >= 0, s49 <= x_11440 + 1, s50 >= 0, s50 <= x_2640 + 1, s51 >= 0, s51 <= 1 + s49 + s50 + 1, x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ -104 + s52 + 45*z' + 139*z'^2 + 48*z'^3 }-> s53 :|: s52 >= 0, s52 <= z' - 1 + 1, s53 >= 0, s53 <= 1 + s52 + 1, z' - 1 >= 0 encode_g(z') -{ 261 + 6*s54 + 2*s54*s55 + s54^2 + 4*s55 + s55^2 + s56 + 467*x_11443 + 283*x_11443^2 + 48*x_11443^3 + 467*x_2641 + 283*x_2641^2 + 48*x_2641^3 }-> s57 :|: s54 >= 0, s54 <= x_11443 + 1, s55 >= 0, s55 <= x_2641 + 1, s56 >= 0, s56 <= s54 + s55 + 1, s57 >= 0, s57 <= s56 + 1, x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ -104 + s58 + s59 + 45*z' + 139*z'^2 + 48*z'^3 }-> s60 :|: s58 >= 0, s58 <= z' - 1 + 1, s59 >= 0, s59 <= s58 + 1, s60 >= 0, s60 <= s59 + 1, z' - 1 >= 0 encode_g(z') -{ 379 + s62 + s64 + 467*x_11445 + 283*x_11445^2 + 48*x_11445^3 + 467*x_2642 + 283*x_2642^2 + 48*x_2642^3 + 467*x_3159 + 283*x_3159^2 + 48*x_3159^3 }-> s65 :|: s61 >= 0, s61 <= x_11445 + 1, s62 >= 0, s62 <= x_2642 + 1, s63 >= 0, s63 <= x_3159 + 1, s64 >= 0, s64 <= s61 + s62 + s63 + 1, s65 >= 0, s65 <= s64 + 1, x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s73 :|: s73 >= 0, s73 <= z' + 1, z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s66 + s67 :|: s66 >= 0, s66 <= z' + 1, s67 >= 0, s67 <= z'' + 1, z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 378 + s69 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 + 467*z1 + 283*z1^2 + 48*z1^3 }-> s71 :|: s68 >= 0, s68 <= z' + 1, s69 >= 0, s69 <= z'' + 1, s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s68 + s69 + s70 + 1, z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 253 + 467*x_12654 + 283*x_12654^2 + 48*x_12654^3 + 467*x_21179 + 283*x_21179^2 + 48*x_21179^3 }-> s157 :|: s155 >= 0, s155 <= x_12654 + 1, s156 >= 0, s156 <= x_21179 + 1, s157 >= 0, s157 <= 1 + s155 + s156, x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 253 + 467*x_12655 + 283*x_12655^2 + 48*x_12655^3 + 467*x_21180 + 283*x_21180^2 + 48*x_21180^3 }-> s160 :|: s158 >= 0, s158 <= x_12655 + 1, s159 >= 0, s159 <= x_21180 + 1, s160 >= 0, s160 <= 1 + s158 + s159, z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s162 :|: s161 >= 0, s161 <= z' - 1 + 1, s162 >= 0, s162 <= 1 + s161, z' - 1 >= 0 encode_i(z') -{ 261 + 6*s163 + 2*s163*s164 + s163^2 + 4*s164 + s164^2 + 467*x_12658 + 283*x_12658^2 + 48*x_12658^3 + 467*x_21181 + 283*x_21181^2 + 48*x_21181^3 }-> s166 :|: s163 >= 0, s163 <= x_12658 + 1, s164 >= 0, s164 <= x_21181 + 1, s165 >= 0, s165 <= s163 + s164 + 1, s166 >= 0, s166 <= s165, x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ -104 + s167 + 45*z' + 139*z'^2 + 48*z'^3 }-> s169 :|: s167 >= 0, s167 <= z' - 1 + 1, s168 >= 0, s168 <= s167 + 1, s169 >= 0, s169 <= s168, z' - 1 >= 0 encode_i(z') -{ 379 + s171 + 467*x_12660 + 283*x_12660^2 + 48*x_12660^3 + 467*x_21182 + 283*x_21182^2 + 48*x_21182^3 + 467*x_3294 + 283*x_3294^2 + 48*x_3294^3 }-> s174 :|: s170 >= 0, s170 <= x_12660 + 1, s171 >= 0, s171 <= x_21182 + 1, s172 >= 0, s172 <= x_3294 + 1, s173 >= 0, s173 <= s170 + s171 + s172 + 1, s174 >= 0, s174 <= s173, z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s177 :|: s175 >= 0, s175 <= z' - 1 + 1, s176 >= 0, s176 <= s175, s177 >= 0, s177 <= s176, z' - 1 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s189 :|: s187 >= 0, s187 <= z' - 1 + 1, s188 >= 0, s188 <= s187 + 1, s189 >= 0, s189 <= s188, z' - 1 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s44 + s45 :|: s44 >= 0, s44 <= z' + 1, s45 >= 0, s45 <= z'' + 1, z' >= 0, z'' >= 0 encode_k(z') -{ 253 + 467*x_11448 + 283*x_11448^2 + 48*x_11448^3 + 467*x_2643 + 283*x_2643^2 + 48*x_2643^3 }-> s105 :|: s103 >= 0, s103 <= x_11448 + 1, s104 >= 0, s104 <= x_2643 + 1, s105 >= 0, s105 <= 1 + s103 + s104 + 1, z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 253 + 467*x_11449 + 283*x_11449^2 + 48*x_11449^3 + 467*x_2644 + 283*x_2644^2 + 48*x_2644^3 }-> s108 :|: s106 >= 0, s106 <= x_11449 + 1, s107 >= 0, s107 <= x_2644 + 1, s108 >= 0, s108 <= 1 + s106 + s107 + 1, x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s110 :|: s109 >= 0, s109 <= z' - 1 + 1, s110 >= 0, s110 <= 1 + s109 + 1, z' - 1 >= 0 encode_k(z') -{ 261 + 6*s111 + 2*s111*s112 + s111^2 + 4*s112 + s112^2 + 467*x_11452 + 283*x_11452^2 + 48*x_11452^3 + 467*x_2645 + 283*x_2645^2 + 48*x_2645^3 }-> s114 :|: s111 >= 0, s111 <= x_11452 + 1, s112 >= 0, s112 <= x_2645 + 1, s113 >= 0, s113 <= s111 + s112 + 1, s114 >= 0, s114 <= s113 + 1, x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ -104 + s115 + 45*z' + 139*z'^2 + 48*z'^3 }-> s117 :|: s115 >= 0, s115 <= z' - 1 + 1, s116 >= 0, s116 <= s115 + 1, s117 >= 0, s117 <= s116 + 1, z' - 1 >= 0 encode_k(z') -{ 379 + s119 + 467*x_11454 + 283*x_11454^2 + 48*x_11454^3 + 467*x_2646 + 283*x_2646^2 + 48*x_2646^3 + 467*x_3160 + 283*x_3160^2 + 48*x_3160^3 }-> s122 :|: s118 >= 0, s118 <= x_11454 + 1, s119 >= 0, s119 <= x_2646 + 1, s120 >= 0, s120 <= x_3160 + 1, s121 >= 0, s121 <= s118 + s119 + s120 + 1, s122 >= 0, s122 <= s121 + 1, x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s125 :|: s123 >= 0, s123 <= z' - 1 + 1, s124 >= 0, s124 <= s123 + 1, s125 >= 0, s125 <= s124 + 1, z' - 1 >= 0 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s186 :|: s184 >= 0, s184 <= z' - 1 + 1, s185 >= 0, s185 <= s184, s186 >= 0, s186 <= s185 + 1, z' - 1 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s72 :|: s72 >= 0, s72 <= z' + 1, z' >= 0 f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 10 + s9 + 6*x + x^2 }-> s10 :|: s9 >= 0, s9 <= x + 0 + 1, s10 >= 0, s10 <= s9 + 1, z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 11 + s5 + 6*x + 2*x*z'' + x^2 + 4*z'' + z''^2 }-> s6 :|: s5 >= 0, s5 <= x + (1 + 0 + (z'' - 1)) + 1, s6 >= 0, s6 <= s5 + 1, z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 23 + s7 + 10*x + 2*x*x'' + 2*x*y' + x^2 + 8*x'' + 2*x''*y' + x''^2 + 8*y' + y'^2 }-> s8 :|: s7 >= 0, s7 <= x + (1 + (1 + x'') + y') + 1, s8 >= 0, s8 <= s7 + 1, z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {encode_h2}, {encode_h1}, {encode_g}, {encode_f}, {encode_j}, {encode_i}, {encode_s} Previous analysis results are: h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1] encode_0: runtime: O(1) [0], size: O(1) [0] i: runtime: O(1) [1], size: O(n^1) [z'] k: runtime: O(1) [1], size: O(n^1) [1 + z'] g: runtime: O(n^1) [1 + z'], size: O(n^1) [1 + z'] f: runtime: O(n^2) [8 + 6*z' + 2*z'*z'' + z'^2 + 4*z'' + z''^2], size: O(n^1) [1 + z' + z''] encArg: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [1 + z'] encode_k: runtime: O(n^3) [1161 + 4403*z' + 3107*z'^2 + 624*z'^3], size: O(n^1) [4 + z'] encode_h: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [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') -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z' = 1 + 0 encArg(z') -{ -58 + s127 + -89*z' + -5*z'^2 + 48*z'^3 }-> s128 :|: s126 >= 0, s126 <= z' - 2 + 1, s127 >= 0, s127 <= s126, s128 >= 0, s128 <= s127 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11313 + 283*x_11313^2 + 48*x_11313^3 + 467*x_2583 + 283*x_2583^2 + 48*x_2583^3 }-> s131 :|: s129 >= 0, s129 <= x_11313 + 1, s130 >= 0, s130 <= x_2583 + 1, s131 >= 0, s131 <= 1 + s129 + s130, x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 253 + 467*x_11314 + 283*x_11314^2 + 48*x_11314^3 + 467*x_2584 + 283*x_2584^2 + 48*x_2584^3 }-> s134 :|: s132 >= 0, s132 <= x_11314 + 1, s133 >= 0, s133 <= x_2584 + 1, s134 >= 0, s134 <= 1 + s132 + s133, z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s136 :|: s135 >= 0, s135 <= z' - 2 + 1, s136 >= 0, s136 <= 1 + s135, z' - 2 >= 0 encArg(z') -{ 261 + 6*s137 + 2*s137*s138 + s137^2 + 4*s138 + s138^2 + 467*x_11317 + 283*x_11317^2 + 48*x_11317^3 + 467*x_2585 + 283*x_2585^2 + 48*x_2585^3 }-> s140 :|: s137 >= 0, s137 <= x_11317 + 1, s138 >= 0, s138 <= x_2585 + 1, s139 >= 0, s139 <= s137 + s138 + 1, s140 >= 0, s140 <= s139, x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ -58 + s141 + -89*z' + -5*z'^2 + 48*z'^3 }-> s143 :|: s141 >= 0, s141 <= z' - 2 + 1, s142 >= 0, s142 <= s141 + 1, s143 >= 0, s143 <= s142, z' - 2 >= 0 encArg(z') -{ 379 + s145 + 467*x_11319 + 283*x_11319^2 + 48*x_11319^3 + 467*x_2586 + 283*x_2586^2 + 48*x_2586^3 + 467*x_3145 + 283*x_3145^2 + 48*x_3145^3 }-> s148 :|: s144 >= 0, s144 <= x_11319 + 1, s145 >= 0, s145 <= x_2586 + 1, s146 >= 0, s146 <= x_3145 + 1, s147 >= 0, s147 <= s144 + s145 + s146 + 1, s148 >= 0, s148 <= s147, z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s151 :|: s149 >= 0, s149 <= z' - 2 + 1, s150 >= 0, s150 <= s149, s151 >= 0, s151 <= s150, z' - 2 >= 0 encArg(z') -{ 260 + 6*s14 + 2*s14*s15 + s14^2 + 4*s15 + s15^2 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> s16 :|: s14 >= 0, s14 <= x_1 + 1, s15 >= 0, s15 <= x_2 + 1, s16 >= 0, s16 <= s14 + s15 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s180 :|: s178 >= 0, s178 <= z' - 2 + 1, s179 >= 0, s179 <= s178 + 1, s180 >= 0, s180 <= s179, z' - 2 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s183 :|: s181 >= 0, s181 <= z' - 2 + 1, s182 >= 0, s182 <= s181, s183 >= 0, s183 <= s182 + 1, z' - 2 >= 0 encArg(z') -{ 254 + s17 + s18 + 467*x_1107 + 283*x_1107^2 + 48*x_1107^3 + 467*x_247 + 283*x_247^2 + 48*x_247^3 }-> s19 :|: s17 >= 0, s17 <= x_1107 + 1, s18 >= 0, s18 <= x_247 + 1, s19 >= 0, s19 <= 1 + s17 + s18 + 1, z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0 + 1, z' - 1 >= 0 encArg(z') -{ 254 + s20 + s21 + 467*x_1108 + 283*x_1108^2 + 48*x_1108^3 + 467*x_248 + 283*x_248^2 + 48*x_248^3 }-> s22 :|: s20 >= 0, s20 <= x_1108 + 1, s21 >= 0, s21 <= x_248 + 1, s22 >= 0, s22 <= 1 + s20 + s21 + 1, x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ -58 + s23 + -89*z' + -5*z'^2 + 48*z'^3 }-> s24 :|: s23 >= 0, s23 <= z' - 2 + 1, s24 >= 0, s24 <= 1 + s23 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s25 + 2*s25*s26 + s25^2 + 4*s26 + s26^2 + s27 + 467*x_1111 + 283*x_1111^2 + 48*x_1111^3 + 467*x_249 + 283*x_249^2 + 48*x_249^3 }-> s28 :|: s25 >= 0, s25 <= x_1111 + 1, s26 >= 0, s26 <= x_249 + 1, s27 >= 0, s27 <= s25 + s26 + 1, s28 >= 0, s28 <= s27 + 1, z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ -58 + s29 + s30 + -89*z' + -5*z'^2 + 48*z'^3 }-> s31 :|: s29 >= 0, s29 <= z' - 2 + 1, s30 >= 0, s30 <= s29 + 1, s31 >= 0, s31 <= s30 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s33 + s35 + 467*x_1113 + 283*x_1113^2 + 48*x_1113^3 + 467*x_250 + 283*x_250^2 + 48*x_250^3 + 467*x_311 + 283*x_311^2 + 48*x_311^3 }-> s36 :|: s32 >= 0, s32 <= x_1113 + 1, s33 >= 0, s33 <= x_250 + 1, s34 >= 0, s34 <= x_311 + 1, s35 >= 0, s35 <= s32 + s33 + s34 + 1, s36 >= 0, s36 <= s35 + 1, x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 378 + s38 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 + 467*x_3 + 283*x_3^2 + 48*x_3^3 }-> s40 :|: s37 >= 0, s37 <= x_1 + 1, s38 >= 0, s38 <= x_2 + 1, s39 >= 0, s39 <= x_3 + 1, s40 >= 0, s40 <= s37 + s38 + s39 + 1, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z') -{ -58 + s75 + -89*z' + -5*z'^2 + 48*z'^3 }-> s76 :|: s74 >= 0, s74 <= z' - 2 + 1, s75 >= 0, s75 <= s74 + 1, s76 >= 0, s76 <= s75 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11322 + 283*x_11322^2 + 48*x_11322^3 + 467*x_2587 + 283*x_2587^2 + 48*x_2587^3 }-> s79 :|: s77 >= 0, s77 <= x_11322 + 1, s78 >= 0, s78 <= x_2587 + 1, s79 >= 0, s79 <= 1 + s77 + s78 + 1, x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 253 + 467*x_11323 + 283*x_11323^2 + 48*x_11323^3 + 467*x_2588 + 283*x_2588^2 + 48*x_2588^3 }-> s82 :|: s80 >= 0, s80 <= x_11323 + 1, s81 >= 0, s81 <= x_2588 + 1, s82 >= 0, s82 <= 1 + s80 + s81 + 1, z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s84 :|: s83 >= 0, s83 <= z' - 2 + 1, s84 >= 0, s84 <= 1 + s83 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s85 + 2*s85*s86 + s85^2 + 4*s86 + s86^2 + 467*x_11326 + 283*x_11326^2 + 48*x_11326^3 + 467*x_2589 + 283*x_2589^2 + 48*x_2589^3 }-> s88 :|: s85 >= 0, s85 <= x_11326 + 1, s86 >= 0, s86 <= x_2589 + 1, s87 >= 0, s87 <= s85 + s86 + 1, s88 >= 0, s88 <= s87 + 1, x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ -58 + s89 + -89*z' + -5*z'^2 + 48*z'^3 }-> s91 :|: s89 >= 0, s89 <= z' - 2 + 1, s90 >= 0, s90 <= s89 + 1, s91 >= 0, s91 <= s90 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s93 + 467*x_11328 + 283*x_11328^2 + 48*x_11328^3 + 467*x_2590 + 283*x_2590^2 + 48*x_2590^3 + 467*x_3146 + 283*x_3146^2 + 48*x_3146^3 }-> s96 :|: s92 >= 0, s92 <= x_11328 + 1, s93 >= 0, s93 <= x_2590 + 1, s94 >= 0, s94 <= x_3146 + 1, s95 >= 0, s95 <= s92 + s93 + s94 + 1, s96 >= 0, s96 <= s95 + 1, z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s99 :|: s97 >= 0, s97 <= z' - 2 + 1, s98 >= 0, s98 <= s97 + 1, s99 >= 0, s99 <= s98 + 1, z' - 2 >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ -106 + 45*z' + 139*z'^2 + 48*z'^3 }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0 encArg(z') -{ 252 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> 1 + s11 + s12 :|: s11 >= 0, s11 <= x_1 + 1, s12 >= 0, s12 <= x_2 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 260 + 6*s41 + 2*s41*s42 + s41^2 + 4*s42 + s42^2 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> s43 :|: s41 >= 0, s41 <= z' + 1, s42 >= 0, s42 <= z'' + 1, s43 >= 0, s43 <= s41 + s42 + 1, z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ -104 + s101 + 45*z' + 139*z'^2 + 48*z'^3 }-> s102 :|: s100 >= 0, s100 <= z' - 1 + 1, s101 >= 0, s101 <= s100 + 1, s102 >= 0, s102 <= s101 + 1, z' - 1 >= 0 encode_g(z') -{ -104 + s153 + 45*z' + 139*z'^2 + 48*z'^3 }-> s154 :|: s152 >= 0, s152 <= z' - 1 + 1, s153 >= 0, s153 <= s152, s154 >= 0, s154 <= s153 + 1, z' - 1 >= 0 encode_g(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0 + 1, z' = 0 encode_g(z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 1, z' >= 0 encode_g(z') -{ 254 + s46 + s47 + 467*x_11439 + 283*x_11439^2 + 48*x_11439^3 + 467*x_2639 + 283*x_2639^2 + 48*x_2639^3 }-> s48 :|: s46 >= 0, s46 <= x_11439 + 1, s47 >= 0, s47 <= x_2639 + 1, s48 >= 0, s48 <= 1 + s46 + s47 + 1, x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 254 + s49 + s50 + 467*x_11440 + 283*x_11440^2 + 48*x_11440^3 + 467*x_2640 + 283*x_2640^2 + 48*x_2640^3 }-> s51 :|: s49 >= 0, s49 <= x_11440 + 1, s50 >= 0, s50 <= x_2640 + 1, s51 >= 0, s51 <= 1 + s49 + s50 + 1, x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ -104 + s52 + 45*z' + 139*z'^2 + 48*z'^3 }-> s53 :|: s52 >= 0, s52 <= z' - 1 + 1, s53 >= 0, s53 <= 1 + s52 + 1, z' - 1 >= 0 encode_g(z') -{ 261 + 6*s54 + 2*s54*s55 + s54^2 + 4*s55 + s55^2 + s56 + 467*x_11443 + 283*x_11443^2 + 48*x_11443^3 + 467*x_2641 + 283*x_2641^2 + 48*x_2641^3 }-> s57 :|: s54 >= 0, s54 <= x_11443 + 1, s55 >= 0, s55 <= x_2641 + 1, s56 >= 0, s56 <= s54 + s55 + 1, s57 >= 0, s57 <= s56 + 1, x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ -104 + s58 + s59 + 45*z' + 139*z'^2 + 48*z'^3 }-> s60 :|: s58 >= 0, s58 <= z' - 1 + 1, s59 >= 0, s59 <= s58 + 1, s60 >= 0, s60 <= s59 + 1, z' - 1 >= 0 encode_g(z') -{ 379 + s62 + s64 + 467*x_11445 + 283*x_11445^2 + 48*x_11445^3 + 467*x_2642 + 283*x_2642^2 + 48*x_2642^3 + 467*x_3159 + 283*x_3159^2 + 48*x_3159^3 }-> s65 :|: s61 >= 0, s61 <= x_11445 + 1, s62 >= 0, s62 <= x_2642 + 1, s63 >= 0, s63 <= x_3159 + 1, s64 >= 0, s64 <= s61 + s62 + s63 + 1, s65 >= 0, s65 <= s64 + 1, x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s73 :|: s73 >= 0, s73 <= z' + 1, z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s66 + s67 :|: s66 >= 0, s66 <= z' + 1, s67 >= 0, s67 <= z'' + 1, z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 378 + s69 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 + 467*z1 + 283*z1^2 + 48*z1^3 }-> s71 :|: s68 >= 0, s68 <= z' + 1, s69 >= 0, s69 <= z'' + 1, s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s68 + s69 + s70 + 1, z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 253 + 467*x_12654 + 283*x_12654^2 + 48*x_12654^3 + 467*x_21179 + 283*x_21179^2 + 48*x_21179^3 }-> s157 :|: s155 >= 0, s155 <= x_12654 + 1, s156 >= 0, s156 <= x_21179 + 1, s157 >= 0, s157 <= 1 + s155 + s156, x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 253 + 467*x_12655 + 283*x_12655^2 + 48*x_12655^3 + 467*x_21180 + 283*x_21180^2 + 48*x_21180^3 }-> s160 :|: s158 >= 0, s158 <= x_12655 + 1, s159 >= 0, s159 <= x_21180 + 1, s160 >= 0, s160 <= 1 + s158 + s159, z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s162 :|: s161 >= 0, s161 <= z' - 1 + 1, s162 >= 0, s162 <= 1 + s161, z' - 1 >= 0 encode_i(z') -{ 261 + 6*s163 + 2*s163*s164 + s163^2 + 4*s164 + s164^2 + 467*x_12658 + 283*x_12658^2 + 48*x_12658^3 + 467*x_21181 + 283*x_21181^2 + 48*x_21181^3 }-> s166 :|: s163 >= 0, s163 <= x_12658 + 1, s164 >= 0, s164 <= x_21181 + 1, s165 >= 0, s165 <= s163 + s164 + 1, s166 >= 0, s166 <= s165, x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ -104 + s167 + 45*z' + 139*z'^2 + 48*z'^3 }-> s169 :|: s167 >= 0, s167 <= z' - 1 + 1, s168 >= 0, s168 <= s167 + 1, s169 >= 0, s169 <= s168, z' - 1 >= 0 encode_i(z') -{ 379 + s171 + 467*x_12660 + 283*x_12660^2 + 48*x_12660^3 + 467*x_21182 + 283*x_21182^2 + 48*x_21182^3 + 467*x_3294 + 283*x_3294^2 + 48*x_3294^3 }-> s174 :|: s170 >= 0, s170 <= x_12660 + 1, s171 >= 0, s171 <= x_21182 + 1, s172 >= 0, s172 <= x_3294 + 1, s173 >= 0, s173 <= s170 + s171 + s172 + 1, s174 >= 0, s174 <= s173, z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s177 :|: s175 >= 0, s175 <= z' - 1 + 1, s176 >= 0, s176 <= s175, s177 >= 0, s177 <= s176, z' - 1 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s189 :|: s187 >= 0, s187 <= z' - 1 + 1, s188 >= 0, s188 <= s187 + 1, s189 >= 0, s189 <= s188, z' - 1 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s44 + s45 :|: s44 >= 0, s44 <= z' + 1, s45 >= 0, s45 <= z'' + 1, z' >= 0, z'' >= 0 encode_k(z') -{ 253 + 467*x_11448 + 283*x_11448^2 + 48*x_11448^3 + 467*x_2643 + 283*x_2643^2 + 48*x_2643^3 }-> s105 :|: s103 >= 0, s103 <= x_11448 + 1, s104 >= 0, s104 <= x_2643 + 1, s105 >= 0, s105 <= 1 + s103 + s104 + 1, z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 253 + 467*x_11449 + 283*x_11449^2 + 48*x_11449^3 + 467*x_2644 + 283*x_2644^2 + 48*x_2644^3 }-> s108 :|: s106 >= 0, s106 <= x_11449 + 1, s107 >= 0, s107 <= x_2644 + 1, s108 >= 0, s108 <= 1 + s106 + s107 + 1, x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s110 :|: s109 >= 0, s109 <= z' - 1 + 1, s110 >= 0, s110 <= 1 + s109 + 1, z' - 1 >= 0 encode_k(z') -{ 261 + 6*s111 + 2*s111*s112 + s111^2 + 4*s112 + s112^2 + 467*x_11452 + 283*x_11452^2 + 48*x_11452^3 + 467*x_2645 + 283*x_2645^2 + 48*x_2645^3 }-> s114 :|: s111 >= 0, s111 <= x_11452 + 1, s112 >= 0, s112 <= x_2645 + 1, s113 >= 0, s113 <= s111 + s112 + 1, s114 >= 0, s114 <= s113 + 1, x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ -104 + s115 + 45*z' + 139*z'^2 + 48*z'^3 }-> s117 :|: s115 >= 0, s115 <= z' - 1 + 1, s116 >= 0, s116 <= s115 + 1, s117 >= 0, s117 <= s116 + 1, z' - 1 >= 0 encode_k(z') -{ 379 + s119 + 467*x_11454 + 283*x_11454^2 + 48*x_11454^3 + 467*x_2646 + 283*x_2646^2 + 48*x_2646^3 + 467*x_3160 + 283*x_3160^2 + 48*x_3160^3 }-> s122 :|: s118 >= 0, s118 <= x_11454 + 1, s119 >= 0, s119 <= x_2646 + 1, s120 >= 0, s120 <= x_3160 + 1, s121 >= 0, s121 <= s118 + s119 + s120 + 1, s122 >= 0, s122 <= s121 + 1, x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s125 :|: s123 >= 0, s123 <= z' - 1 + 1, s124 >= 0, s124 <= s123 + 1, s125 >= 0, s125 <= s124 + 1, z' - 1 >= 0 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s186 :|: s184 >= 0, s184 <= z' - 1 + 1, s185 >= 0, s185 <= s184, s186 >= 0, s186 <= s185 + 1, z' - 1 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s72 :|: s72 >= 0, s72 <= z' + 1, z' >= 0 f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 10 + s9 + 6*x + x^2 }-> s10 :|: s9 >= 0, s9 <= x + 0 + 1, s10 >= 0, s10 <= s9 + 1, z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 11 + s5 + 6*x + 2*x*z'' + x^2 + 4*z'' + z''^2 }-> s6 :|: s5 >= 0, s5 <= x + (1 + 0 + (z'' - 1)) + 1, s6 >= 0, s6 <= s5 + 1, z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 23 + s7 + 10*x + 2*x*x'' + 2*x*y' + x^2 + 8*x'' + 2*x''*y' + x''^2 + 8*y' + y'^2 }-> s8 :|: s7 >= 0, s7 <= x + (1 + (1 + x'') + y') + 1, s8 >= 0, s8 <= s7 + 1, z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {encode_h2}, {encode_h1}, {encode_g}, {encode_f}, {encode_j}, {encode_i}, {encode_s} Previous analysis results are: h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1] encode_0: runtime: O(1) [0], size: O(1) [0] i: runtime: O(1) [1], size: O(n^1) [z'] k: runtime: O(1) [1], size: O(n^1) [1 + z'] g: runtime: O(n^1) [1 + z'], size: O(n^1) [1 + z'] f: runtime: O(n^2) [8 + 6*z' + 2*z'*z'' + z'^2 + 4*z'' + z''^2], size: O(n^1) [1 + z' + z''] encArg: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [1 + z'] encode_k: runtime: O(n^3) [1161 + 4403*z' + 3107*z'^2 + 624*z'^3], size: O(n^1) [4 + z'] encode_h: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [2 + z'] ---------------------------------------- (81) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: encode_h2 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 4 + z' + z'' + z1 ---------------------------------------- (82) Obligation: Complexity RNTS consisting of the following rules: encArg(z') -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z' = 1 + 0 encArg(z') -{ -58 + s127 + -89*z' + -5*z'^2 + 48*z'^3 }-> s128 :|: s126 >= 0, s126 <= z' - 2 + 1, s127 >= 0, s127 <= s126, s128 >= 0, s128 <= s127 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11313 + 283*x_11313^2 + 48*x_11313^3 + 467*x_2583 + 283*x_2583^2 + 48*x_2583^3 }-> s131 :|: s129 >= 0, s129 <= x_11313 + 1, s130 >= 0, s130 <= x_2583 + 1, s131 >= 0, s131 <= 1 + s129 + s130, x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 253 + 467*x_11314 + 283*x_11314^2 + 48*x_11314^3 + 467*x_2584 + 283*x_2584^2 + 48*x_2584^3 }-> s134 :|: s132 >= 0, s132 <= x_11314 + 1, s133 >= 0, s133 <= x_2584 + 1, s134 >= 0, s134 <= 1 + s132 + s133, z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s136 :|: s135 >= 0, s135 <= z' - 2 + 1, s136 >= 0, s136 <= 1 + s135, z' - 2 >= 0 encArg(z') -{ 261 + 6*s137 + 2*s137*s138 + s137^2 + 4*s138 + s138^2 + 467*x_11317 + 283*x_11317^2 + 48*x_11317^3 + 467*x_2585 + 283*x_2585^2 + 48*x_2585^3 }-> s140 :|: s137 >= 0, s137 <= x_11317 + 1, s138 >= 0, s138 <= x_2585 + 1, s139 >= 0, s139 <= s137 + s138 + 1, s140 >= 0, s140 <= s139, x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ -58 + s141 + -89*z' + -5*z'^2 + 48*z'^3 }-> s143 :|: s141 >= 0, s141 <= z' - 2 + 1, s142 >= 0, s142 <= s141 + 1, s143 >= 0, s143 <= s142, z' - 2 >= 0 encArg(z') -{ 379 + s145 + 467*x_11319 + 283*x_11319^2 + 48*x_11319^3 + 467*x_2586 + 283*x_2586^2 + 48*x_2586^3 + 467*x_3145 + 283*x_3145^2 + 48*x_3145^3 }-> s148 :|: s144 >= 0, s144 <= x_11319 + 1, s145 >= 0, s145 <= x_2586 + 1, s146 >= 0, s146 <= x_3145 + 1, s147 >= 0, s147 <= s144 + s145 + s146 + 1, s148 >= 0, s148 <= s147, z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s151 :|: s149 >= 0, s149 <= z' - 2 + 1, s150 >= 0, s150 <= s149, s151 >= 0, s151 <= s150, z' - 2 >= 0 encArg(z') -{ 260 + 6*s14 + 2*s14*s15 + s14^2 + 4*s15 + s15^2 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> s16 :|: s14 >= 0, s14 <= x_1 + 1, s15 >= 0, s15 <= x_2 + 1, s16 >= 0, s16 <= s14 + s15 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s180 :|: s178 >= 0, s178 <= z' - 2 + 1, s179 >= 0, s179 <= s178 + 1, s180 >= 0, s180 <= s179, z' - 2 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s183 :|: s181 >= 0, s181 <= z' - 2 + 1, s182 >= 0, s182 <= s181, s183 >= 0, s183 <= s182 + 1, z' - 2 >= 0 encArg(z') -{ 254 + s17 + s18 + 467*x_1107 + 283*x_1107^2 + 48*x_1107^3 + 467*x_247 + 283*x_247^2 + 48*x_247^3 }-> s19 :|: s17 >= 0, s17 <= x_1107 + 1, s18 >= 0, s18 <= x_247 + 1, s19 >= 0, s19 <= 1 + s17 + s18 + 1, z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0 + 1, z' - 1 >= 0 encArg(z') -{ 254 + s20 + s21 + 467*x_1108 + 283*x_1108^2 + 48*x_1108^3 + 467*x_248 + 283*x_248^2 + 48*x_248^3 }-> s22 :|: s20 >= 0, s20 <= x_1108 + 1, s21 >= 0, s21 <= x_248 + 1, s22 >= 0, s22 <= 1 + s20 + s21 + 1, x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ -58 + s23 + -89*z' + -5*z'^2 + 48*z'^3 }-> s24 :|: s23 >= 0, s23 <= z' - 2 + 1, s24 >= 0, s24 <= 1 + s23 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s25 + 2*s25*s26 + s25^2 + 4*s26 + s26^2 + s27 + 467*x_1111 + 283*x_1111^2 + 48*x_1111^3 + 467*x_249 + 283*x_249^2 + 48*x_249^3 }-> s28 :|: s25 >= 0, s25 <= x_1111 + 1, s26 >= 0, s26 <= x_249 + 1, s27 >= 0, s27 <= s25 + s26 + 1, s28 >= 0, s28 <= s27 + 1, z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ -58 + s29 + s30 + -89*z' + -5*z'^2 + 48*z'^3 }-> s31 :|: s29 >= 0, s29 <= z' - 2 + 1, s30 >= 0, s30 <= s29 + 1, s31 >= 0, s31 <= s30 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s33 + s35 + 467*x_1113 + 283*x_1113^2 + 48*x_1113^3 + 467*x_250 + 283*x_250^2 + 48*x_250^3 + 467*x_311 + 283*x_311^2 + 48*x_311^3 }-> s36 :|: s32 >= 0, s32 <= x_1113 + 1, s33 >= 0, s33 <= x_250 + 1, s34 >= 0, s34 <= x_311 + 1, s35 >= 0, s35 <= s32 + s33 + s34 + 1, s36 >= 0, s36 <= s35 + 1, x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 378 + s38 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 + 467*x_3 + 283*x_3^2 + 48*x_3^3 }-> s40 :|: s37 >= 0, s37 <= x_1 + 1, s38 >= 0, s38 <= x_2 + 1, s39 >= 0, s39 <= x_3 + 1, s40 >= 0, s40 <= s37 + s38 + s39 + 1, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z') -{ -58 + s75 + -89*z' + -5*z'^2 + 48*z'^3 }-> s76 :|: s74 >= 0, s74 <= z' - 2 + 1, s75 >= 0, s75 <= s74 + 1, s76 >= 0, s76 <= s75 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11322 + 283*x_11322^2 + 48*x_11322^3 + 467*x_2587 + 283*x_2587^2 + 48*x_2587^3 }-> s79 :|: s77 >= 0, s77 <= x_11322 + 1, s78 >= 0, s78 <= x_2587 + 1, s79 >= 0, s79 <= 1 + s77 + s78 + 1, x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 253 + 467*x_11323 + 283*x_11323^2 + 48*x_11323^3 + 467*x_2588 + 283*x_2588^2 + 48*x_2588^3 }-> s82 :|: s80 >= 0, s80 <= x_11323 + 1, s81 >= 0, s81 <= x_2588 + 1, s82 >= 0, s82 <= 1 + s80 + s81 + 1, z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s84 :|: s83 >= 0, s83 <= z' - 2 + 1, s84 >= 0, s84 <= 1 + s83 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s85 + 2*s85*s86 + s85^2 + 4*s86 + s86^2 + 467*x_11326 + 283*x_11326^2 + 48*x_11326^3 + 467*x_2589 + 283*x_2589^2 + 48*x_2589^3 }-> s88 :|: s85 >= 0, s85 <= x_11326 + 1, s86 >= 0, s86 <= x_2589 + 1, s87 >= 0, s87 <= s85 + s86 + 1, s88 >= 0, s88 <= s87 + 1, x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ -58 + s89 + -89*z' + -5*z'^2 + 48*z'^3 }-> s91 :|: s89 >= 0, s89 <= z' - 2 + 1, s90 >= 0, s90 <= s89 + 1, s91 >= 0, s91 <= s90 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s93 + 467*x_11328 + 283*x_11328^2 + 48*x_11328^3 + 467*x_2590 + 283*x_2590^2 + 48*x_2590^3 + 467*x_3146 + 283*x_3146^2 + 48*x_3146^3 }-> s96 :|: s92 >= 0, s92 <= x_11328 + 1, s93 >= 0, s93 <= x_2590 + 1, s94 >= 0, s94 <= x_3146 + 1, s95 >= 0, s95 <= s92 + s93 + s94 + 1, s96 >= 0, s96 <= s95 + 1, z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s99 :|: s97 >= 0, s97 <= z' - 2 + 1, s98 >= 0, s98 <= s97 + 1, s99 >= 0, s99 <= s98 + 1, z' - 2 >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ -106 + 45*z' + 139*z'^2 + 48*z'^3 }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0 encArg(z') -{ 252 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> 1 + s11 + s12 :|: s11 >= 0, s11 <= x_1 + 1, s12 >= 0, s12 <= x_2 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 260 + 6*s41 + 2*s41*s42 + s41^2 + 4*s42 + s42^2 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> s43 :|: s41 >= 0, s41 <= z' + 1, s42 >= 0, s42 <= z'' + 1, s43 >= 0, s43 <= s41 + s42 + 1, z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ -104 + s101 + 45*z' + 139*z'^2 + 48*z'^3 }-> s102 :|: s100 >= 0, s100 <= z' - 1 + 1, s101 >= 0, s101 <= s100 + 1, s102 >= 0, s102 <= s101 + 1, z' - 1 >= 0 encode_g(z') -{ -104 + s153 + 45*z' + 139*z'^2 + 48*z'^3 }-> s154 :|: s152 >= 0, s152 <= z' - 1 + 1, s153 >= 0, s153 <= s152, s154 >= 0, s154 <= s153 + 1, z' - 1 >= 0 encode_g(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0 + 1, z' = 0 encode_g(z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 1, z' >= 0 encode_g(z') -{ 254 + s46 + s47 + 467*x_11439 + 283*x_11439^2 + 48*x_11439^3 + 467*x_2639 + 283*x_2639^2 + 48*x_2639^3 }-> s48 :|: s46 >= 0, s46 <= x_11439 + 1, s47 >= 0, s47 <= x_2639 + 1, s48 >= 0, s48 <= 1 + s46 + s47 + 1, x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 254 + s49 + s50 + 467*x_11440 + 283*x_11440^2 + 48*x_11440^3 + 467*x_2640 + 283*x_2640^2 + 48*x_2640^3 }-> s51 :|: s49 >= 0, s49 <= x_11440 + 1, s50 >= 0, s50 <= x_2640 + 1, s51 >= 0, s51 <= 1 + s49 + s50 + 1, x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ -104 + s52 + 45*z' + 139*z'^2 + 48*z'^3 }-> s53 :|: s52 >= 0, s52 <= z' - 1 + 1, s53 >= 0, s53 <= 1 + s52 + 1, z' - 1 >= 0 encode_g(z') -{ 261 + 6*s54 + 2*s54*s55 + s54^2 + 4*s55 + s55^2 + s56 + 467*x_11443 + 283*x_11443^2 + 48*x_11443^3 + 467*x_2641 + 283*x_2641^2 + 48*x_2641^3 }-> s57 :|: s54 >= 0, s54 <= x_11443 + 1, s55 >= 0, s55 <= x_2641 + 1, s56 >= 0, s56 <= s54 + s55 + 1, s57 >= 0, s57 <= s56 + 1, x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ -104 + s58 + s59 + 45*z' + 139*z'^2 + 48*z'^3 }-> s60 :|: s58 >= 0, s58 <= z' - 1 + 1, s59 >= 0, s59 <= s58 + 1, s60 >= 0, s60 <= s59 + 1, z' - 1 >= 0 encode_g(z') -{ 379 + s62 + s64 + 467*x_11445 + 283*x_11445^2 + 48*x_11445^3 + 467*x_2642 + 283*x_2642^2 + 48*x_2642^3 + 467*x_3159 + 283*x_3159^2 + 48*x_3159^3 }-> s65 :|: s61 >= 0, s61 <= x_11445 + 1, s62 >= 0, s62 <= x_2642 + 1, s63 >= 0, s63 <= x_3159 + 1, s64 >= 0, s64 <= s61 + s62 + s63 + 1, s65 >= 0, s65 <= s64 + 1, x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s73 :|: s73 >= 0, s73 <= z' + 1, z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s66 + s67 :|: s66 >= 0, s66 <= z' + 1, s67 >= 0, s67 <= z'' + 1, z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 378 + s69 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 + 467*z1 + 283*z1^2 + 48*z1^3 }-> s71 :|: s68 >= 0, s68 <= z' + 1, s69 >= 0, s69 <= z'' + 1, s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s68 + s69 + s70 + 1, z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 253 + 467*x_12654 + 283*x_12654^2 + 48*x_12654^3 + 467*x_21179 + 283*x_21179^2 + 48*x_21179^3 }-> s157 :|: s155 >= 0, s155 <= x_12654 + 1, s156 >= 0, s156 <= x_21179 + 1, s157 >= 0, s157 <= 1 + s155 + s156, x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 253 + 467*x_12655 + 283*x_12655^2 + 48*x_12655^3 + 467*x_21180 + 283*x_21180^2 + 48*x_21180^3 }-> s160 :|: s158 >= 0, s158 <= x_12655 + 1, s159 >= 0, s159 <= x_21180 + 1, s160 >= 0, s160 <= 1 + s158 + s159, z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s162 :|: s161 >= 0, s161 <= z' - 1 + 1, s162 >= 0, s162 <= 1 + s161, z' - 1 >= 0 encode_i(z') -{ 261 + 6*s163 + 2*s163*s164 + s163^2 + 4*s164 + s164^2 + 467*x_12658 + 283*x_12658^2 + 48*x_12658^3 + 467*x_21181 + 283*x_21181^2 + 48*x_21181^3 }-> s166 :|: s163 >= 0, s163 <= x_12658 + 1, s164 >= 0, s164 <= x_21181 + 1, s165 >= 0, s165 <= s163 + s164 + 1, s166 >= 0, s166 <= s165, x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ -104 + s167 + 45*z' + 139*z'^2 + 48*z'^3 }-> s169 :|: s167 >= 0, s167 <= z' - 1 + 1, s168 >= 0, s168 <= s167 + 1, s169 >= 0, s169 <= s168, z' - 1 >= 0 encode_i(z') -{ 379 + s171 + 467*x_12660 + 283*x_12660^2 + 48*x_12660^3 + 467*x_21182 + 283*x_21182^2 + 48*x_21182^3 + 467*x_3294 + 283*x_3294^2 + 48*x_3294^3 }-> s174 :|: s170 >= 0, s170 <= x_12660 + 1, s171 >= 0, s171 <= x_21182 + 1, s172 >= 0, s172 <= x_3294 + 1, s173 >= 0, s173 <= s170 + s171 + s172 + 1, s174 >= 0, s174 <= s173, z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s177 :|: s175 >= 0, s175 <= z' - 1 + 1, s176 >= 0, s176 <= s175, s177 >= 0, s177 <= s176, z' - 1 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s189 :|: s187 >= 0, s187 <= z' - 1 + 1, s188 >= 0, s188 <= s187 + 1, s189 >= 0, s189 <= s188, z' - 1 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s44 + s45 :|: s44 >= 0, s44 <= z' + 1, s45 >= 0, s45 <= z'' + 1, z' >= 0, z'' >= 0 encode_k(z') -{ 253 + 467*x_11448 + 283*x_11448^2 + 48*x_11448^3 + 467*x_2643 + 283*x_2643^2 + 48*x_2643^3 }-> s105 :|: s103 >= 0, s103 <= x_11448 + 1, s104 >= 0, s104 <= x_2643 + 1, s105 >= 0, s105 <= 1 + s103 + s104 + 1, z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 253 + 467*x_11449 + 283*x_11449^2 + 48*x_11449^3 + 467*x_2644 + 283*x_2644^2 + 48*x_2644^3 }-> s108 :|: s106 >= 0, s106 <= x_11449 + 1, s107 >= 0, s107 <= x_2644 + 1, s108 >= 0, s108 <= 1 + s106 + s107 + 1, x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s110 :|: s109 >= 0, s109 <= z' - 1 + 1, s110 >= 0, s110 <= 1 + s109 + 1, z' - 1 >= 0 encode_k(z') -{ 261 + 6*s111 + 2*s111*s112 + s111^2 + 4*s112 + s112^2 + 467*x_11452 + 283*x_11452^2 + 48*x_11452^3 + 467*x_2645 + 283*x_2645^2 + 48*x_2645^3 }-> s114 :|: s111 >= 0, s111 <= x_11452 + 1, s112 >= 0, s112 <= x_2645 + 1, s113 >= 0, s113 <= s111 + s112 + 1, s114 >= 0, s114 <= s113 + 1, x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ -104 + s115 + 45*z' + 139*z'^2 + 48*z'^3 }-> s117 :|: s115 >= 0, s115 <= z' - 1 + 1, s116 >= 0, s116 <= s115 + 1, s117 >= 0, s117 <= s116 + 1, z' - 1 >= 0 encode_k(z') -{ 379 + s119 + 467*x_11454 + 283*x_11454^2 + 48*x_11454^3 + 467*x_2646 + 283*x_2646^2 + 48*x_2646^3 + 467*x_3160 + 283*x_3160^2 + 48*x_3160^3 }-> s122 :|: s118 >= 0, s118 <= x_11454 + 1, s119 >= 0, s119 <= x_2646 + 1, s120 >= 0, s120 <= x_3160 + 1, s121 >= 0, s121 <= s118 + s119 + s120 + 1, s122 >= 0, s122 <= s121 + 1, x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s125 :|: s123 >= 0, s123 <= z' - 1 + 1, s124 >= 0, s124 <= s123 + 1, s125 >= 0, s125 <= s124 + 1, z' - 1 >= 0 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s186 :|: s184 >= 0, s184 <= z' - 1 + 1, s185 >= 0, s185 <= s184, s186 >= 0, s186 <= s185 + 1, z' - 1 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s72 :|: s72 >= 0, s72 <= z' + 1, z' >= 0 f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 10 + s9 + 6*x + x^2 }-> s10 :|: s9 >= 0, s9 <= x + 0 + 1, s10 >= 0, s10 <= s9 + 1, z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 11 + s5 + 6*x + 2*x*z'' + x^2 + 4*z'' + z''^2 }-> s6 :|: s5 >= 0, s5 <= x + (1 + 0 + (z'' - 1)) + 1, s6 >= 0, s6 <= s5 + 1, z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 23 + s7 + 10*x + 2*x*x'' + 2*x*y' + x^2 + 8*x'' + 2*x''*y' + x''^2 + 8*y' + y'^2 }-> s8 :|: s7 >= 0, s7 <= x + (1 + (1 + x'') + y') + 1, s8 >= 0, s8 <= s7 + 1, z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {encode_h2}, {encode_h1}, {encode_g}, {encode_f}, {encode_j}, {encode_i}, {encode_s} Previous analysis results are: h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1] encode_0: runtime: O(1) [0], size: O(1) [0] i: runtime: O(1) [1], size: O(n^1) [z'] k: runtime: O(1) [1], size: O(n^1) [1 + z'] g: runtime: O(n^1) [1 + z'], size: O(n^1) [1 + z'] f: runtime: O(n^2) [8 + 6*z' + 2*z'*z'' + z'^2 + 4*z'' + z''^2], size: O(n^1) [1 + z' + z''] encArg: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [1 + z'] encode_k: runtime: O(n^3) [1161 + 4403*z' + 3107*z'^2 + 624*z'^3], size: O(n^1) [4 + z'] encode_h: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [2 + z'] encode_h2: runtime: ?, size: O(n^1) [4 + z' + z'' + z1] ---------------------------------------- (83) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: encode_h2 after applying outer abstraction to obtain an ITS, resulting in: O(n^3) with polynomial bound: 379 + 467*z' + 283*z'^2 + 48*z'^3 + 468*z'' + 283*z''^2 + 48*z''^3 + 467*z1 + 283*z1^2 + 48*z1^3 ---------------------------------------- (84) Obligation: Complexity RNTS consisting of the following rules: encArg(z') -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z' = 1 + 0 encArg(z') -{ -58 + s127 + -89*z' + -5*z'^2 + 48*z'^3 }-> s128 :|: s126 >= 0, s126 <= z' - 2 + 1, s127 >= 0, s127 <= s126, s128 >= 0, s128 <= s127 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11313 + 283*x_11313^2 + 48*x_11313^3 + 467*x_2583 + 283*x_2583^2 + 48*x_2583^3 }-> s131 :|: s129 >= 0, s129 <= x_11313 + 1, s130 >= 0, s130 <= x_2583 + 1, s131 >= 0, s131 <= 1 + s129 + s130, x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 253 + 467*x_11314 + 283*x_11314^2 + 48*x_11314^3 + 467*x_2584 + 283*x_2584^2 + 48*x_2584^3 }-> s134 :|: s132 >= 0, s132 <= x_11314 + 1, s133 >= 0, s133 <= x_2584 + 1, s134 >= 0, s134 <= 1 + s132 + s133, z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s136 :|: s135 >= 0, s135 <= z' - 2 + 1, s136 >= 0, s136 <= 1 + s135, z' - 2 >= 0 encArg(z') -{ 261 + 6*s137 + 2*s137*s138 + s137^2 + 4*s138 + s138^2 + 467*x_11317 + 283*x_11317^2 + 48*x_11317^3 + 467*x_2585 + 283*x_2585^2 + 48*x_2585^3 }-> s140 :|: s137 >= 0, s137 <= x_11317 + 1, s138 >= 0, s138 <= x_2585 + 1, s139 >= 0, s139 <= s137 + s138 + 1, s140 >= 0, s140 <= s139, x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ -58 + s141 + -89*z' + -5*z'^2 + 48*z'^3 }-> s143 :|: s141 >= 0, s141 <= z' - 2 + 1, s142 >= 0, s142 <= s141 + 1, s143 >= 0, s143 <= s142, z' - 2 >= 0 encArg(z') -{ 379 + s145 + 467*x_11319 + 283*x_11319^2 + 48*x_11319^3 + 467*x_2586 + 283*x_2586^2 + 48*x_2586^3 + 467*x_3145 + 283*x_3145^2 + 48*x_3145^3 }-> s148 :|: s144 >= 0, s144 <= x_11319 + 1, s145 >= 0, s145 <= x_2586 + 1, s146 >= 0, s146 <= x_3145 + 1, s147 >= 0, s147 <= s144 + s145 + s146 + 1, s148 >= 0, s148 <= s147, z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s151 :|: s149 >= 0, s149 <= z' - 2 + 1, s150 >= 0, s150 <= s149, s151 >= 0, s151 <= s150, z' - 2 >= 0 encArg(z') -{ 260 + 6*s14 + 2*s14*s15 + s14^2 + 4*s15 + s15^2 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> s16 :|: s14 >= 0, s14 <= x_1 + 1, s15 >= 0, s15 <= x_2 + 1, s16 >= 0, s16 <= s14 + s15 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s180 :|: s178 >= 0, s178 <= z' - 2 + 1, s179 >= 0, s179 <= s178 + 1, s180 >= 0, s180 <= s179, z' - 2 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s183 :|: s181 >= 0, s181 <= z' - 2 + 1, s182 >= 0, s182 <= s181, s183 >= 0, s183 <= s182 + 1, z' - 2 >= 0 encArg(z') -{ 254 + s17 + s18 + 467*x_1107 + 283*x_1107^2 + 48*x_1107^3 + 467*x_247 + 283*x_247^2 + 48*x_247^3 }-> s19 :|: s17 >= 0, s17 <= x_1107 + 1, s18 >= 0, s18 <= x_247 + 1, s19 >= 0, s19 <= 1 + s17 + s18 + 1, z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0 + 1, z' - 1 >= 0 encArg(z') -{ 254 + s20 + s21 + 467*x_1108 + 283*x_1108^2 + 48*x_1108^3 + 467*x_248 + 283*x_248^2 + 48*x_248^3 }-> s22 :|: s20 >= 0, s20 <= x_1108 + 1, s21 >= 0, s21 <= x_248 + 1, s22 >= 0, s22 <= 1 + s20 + s21 + 1, x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ -58 + s23 + -89*z' + -5*z'^2 + 48*z'^3 }-> s24 :|: s23 >= 0, s23 <= z' - 2 + 1, s24 >= 0, s24 <= 1 + s23 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s25 + 2*s25*s26 + s25^2 + 4*s26 + s26^2 + s27 + 467*x_1111 + 283*x_1111^2 + 48*x_1111^3 + 467*x_249 + 283*x_249^2 + 48*x_249^3 }-> s28 :|: s25 >= 0, s25 <= x_1111 + 1, s26 >= 0, s26 <= x_249 + 1, s27 >= 0, s27 <= s25 + s26 + 1, s28 >= 0, s28 <= s27 + 1, z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ -58 + s29 + s30 + -89*z' + -5*z'^2 + 48*z'^3 }-> s31 :|: s29 >= 0, s29 <= z' - 2 + 1, s30 >= 0, s30 <= s29 + 1, s31 >= 0, s31 <= s30 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s33 + s35 + 467*x_1113 + 283*x_1113^2 + 48*x_1113^3 + 467*x_250 + 283*x_250^2 + 48*x_250^3 + 467*x_311 + 283*x_311^2 + 48*x_311^3 }-> s36 :|: s32 >= 0, s32 <= x_1113 + 1, s33 >= 0, s33 <= x_250 + 1, s34 >= 0, s34 <= x_311 + 1, s35 >= 0, s35 <= s32 + s33 + s34 + 1, s36 >= 0, s36 <= s35 + 1, x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 378 + s38 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 + 467*x_3 + 283*x_3^2 + 48*x_3^3 }-> s40 :|: s37 >= 0, s37 <= x_1 + 1, s38 >= 0, s38 <= x_2 + 1, s39 >= 0, s39 <= x_3 + 1, s40 >= 0, s40 <= s37 + s38 + s39 + 1, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z') -{ -58 + s75 + -89*z' + -5*z'^2 + 48*z'^3 }-> s76 :|: s74 >= 0, s74 <= z' - 2 + 1, s75 >= 0, s75 <= s74 + 1, s76 >= 0, s76 <= s75 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11322 + 283*x_11322^2 + 48*x_11322^3 + 467*x_2587 + 283*x_2587^2 + 48*x_2587^3 }-> s79 :|: s77 >= 0, s77 <= x_11322 + 1, s78 >= 0, s78 <= x_2587 + 1, s79 >= 0, s79 <= 1 + s77 + s78 + 1, x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 253 + 467*x_11323 + 283*x_11323^2 + 48*x_11323^3 + 467*x_2588 + 283*x_2588^2 + 48*x_2588^3 }-> s82 :|: s80 >= 0, s80 <= x_11323 + 1, s81 >= 0, s81 <= x_2588 + 1, s82 >= 0, s82 <= 1 + s80 + s81 + 1, z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s84 :|: s83 >= 0, s83 <= z' - 2 + 1, s84 >= 0, s84 <= 1 + s83 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s85 + 2*s85*s86 + s85^2 + 4*s86 + s86^2 + 467*x_11326 + 283*x_11326^2 + 48*x_11326^3 + 467*x_2589 + 283*x_2589^2 + 48*x_2589^3 }-> s88 :|: s85 >= 0, s85 <= x_11326 + 1, s86 >= 0, s86 <= x_2589 + 1, s87 >= 0, s87 <= s85 + s86 + 1, s88 >= 0, s88 <= s87 + 1, x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ -58 + s89 + -89*z' + -5*z'^2 + 48*z'^3 }-> s91 :|: s89 >= 0, s89 <= z' - 2 + 1, s90 >= 0, s90 <= s89 + 1, s91 >= 0, s91 <= s90 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s93 + 467*x_11328 + 283*x_11328^2 + 48*x_11328^3 + 467*x_2590 + 283*x_2590^2 + 48*x_2590^3 + 467*x_3146 + 283*x_3146^2 + 48*x_3146^3 }-> s96 :|: s92 >= 0, s92 <= x_11328 + 1, s93 >= 0, s93 <= x_2590 + 1, s94 >= 0, s94 <= x_3146 + 1, s95 >= 0, s95 <= s92 + s93 + s94 + 1, s96 >= 0, s96 <= s95 + 1, z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s99 :|: s97 >= 0, s97 <= z' - 2 + 1, s98 >= 0, s98 <= s97 + 1, s99 >= 0, s99 <= s98 + 1, z' - 2 >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ -106 + 45*z' + 139*z'^2 + 48*z'^3 }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0 encArg(z') -{ 252 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> 1 + s11 + s12 :|: s11 >= 0, s11 <= x_1 + 1, s12 >= 0, s12 <= x_2 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 260 + 6*s41 + 2*s41*s42 + s41^2 + 4*s42 + s42^2 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> s43 :|: s41 >= 0, s41 <= z' + 1, s42 >= 0, s42 <= z'' + 1, s43 >= 0, s43 <= s41 + s42 + 1, z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ -104 + s101 + 45*z' + 139*z'^2 + 48*z'^3 }-> s102 :|: s100 >= 0, s100 <= z' - 1 + 1, s101 >= 0, s101 <= s100 + 1, s102 >= 0, s102 <= s101 + 1, z' - 1 >= 0 encode_g(z') -{ -104 + s153 + 45*z' + 139*z'^2 + 48*z'^3 }-> s154 :|: s152 >= 0, s152 <= z' - 1 + 1, s153 >= 0, s153 <= s152, s154 >= 0, s154 <= s153 + 1, z' - 1 >= 0 encode_g(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0 + 1, z' = 0 encode_g(z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 1, z' >= 0 encode_g(z') -{ 254 + s46 + s47 + 467*x_11439 + 283*x_11439^2 + 48*x_11439^3 + 467*x_2639 + 283*x_2639^2 + 48*x_2639^3 }-> s48 :|: s46 >= 0, s46 <= x_11439 + 1, s47 >= 0, s47 <= x_2639 + 1, s48 >= 0, s48 <= 1 + s46 + s47 + 1, x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 254 + s49 + s50 + 467*x_11440 + 283*x_11440^2 + 48*x_11440^3 + 467*x_2640 + 283*x_2640^2 + 48*x_2640^3 }-> s51 :|: s49 >= 0, s49 <= x_11440 + 1, s50 >= 0, s50 <= x_2640 + 1, s51 >= 0, s51 <= 1 + s49 + s50 + 1, x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ -104 + s52 + 45*z' + 139*z'^2 + 48*z'^3 }-> s53 :|: s52 >= 0, s52 <= z' - 1 + 1, s53 >= 0, s53 <= 1 + s52 + 1, z' - 1 >= 0 encode_g(z') -{ 261 + 6*s54 + 2*s54*s55 + s54^2 + 4*s55 + s55^2 + s56 + 467*x_11443 + 283*x_11443^2 + 48*x_11443^3 + 467*x_2641 + 283*x_2641^2 + 48*x_2641^3 }-> s57 :|: s54 >= 0, s54 <= x_11443 + 1, s55 >= 0, s55 <= x_2641 + 1, s56 >= 0, s56 <= s54 + s55 + 1, s57 >= 0, s57 <= s56 + 1, x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ -104 + s58 + s59 + 45*z' + 139*z'^2 + 48*z'^3 }-> s60 :|: s58 >= 0, s58 <= z' - 1 + 1, s59 >= 0, s59 <= s58 + 1, s60 >= 0, s60 <= s59 + 1, z' - 1 >= 0 encode_g(z') -{ 379 + s62 + s64 + 467*x_11445 + 283*x_11445^2 + 48*x_11445^3 + 467*x_2642 + 283*x_2642^2 + 48*x_2642^3 + 467*x_3159 + 283*x_3159^2 + 48*x_3159^3 }-> s65 :|: s61 >= 0, s61 <= x_11445 + 1, s62 >= 0, s62 <= x_2642 + 1, s63 >= 0, s63 <= x_3159 + 1, s64 >= 0, s64 <= s61 + s62 + s63 + 1, s65 >= 0, s65 <= s64 + 1, x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s73 :|: s73 >= 0, s73 <= z' + 1, z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s66 + s67 :|: s66 >= 0, s66 <= z' + 1, s67 >= 0, s67 <= z'' + 1, z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 378 + s69 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 + 467*z1 + 283*z1^2 + 48*z1^3 }-> s71 :|: s68 >= 0, s68 <= z' + 1, s69 >= 0, s69 <= z'' + 1, s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s68 + s69 + s70 + 1, z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 253 + 467*x_12654 + 283*x_12654^2 + 48*x_12654^3 + 467*x_21179 + 283*x_21179^2 + 48*x_21179^3 }-> s157 :|: s155 >= 0, s155 <= x_12654 + 1, s156 >= 0, s156 <= x_21179 + 1, s157 >= 0, s157 <= 1 + s155 + s156, x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 253 + 467*x_12655 + 283*x_12655^2 + 48*x_12655^3 + 467*x_21180 + 283*x_21180^2 + 48*x_21180^3 }-> s160 :|: s158 >= 0, s158 <= x_12655 + 1, s159 >= 0, s159 <= x_21180 + 1, s160 >= 0, s160 <= 1 + s158 + s159, z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s162 :|: s161 >= 0, s161 <= z' - 1 + 1, s162 >= 0, s162 <= 1 + s161, z' - 1 >= 0 encode_i(z') -{ 261 + 6*s163 + 2*s163*s164 + s163^2 + 4*s164 + s164^2 + 467*x_12658 + 283*x_12658^2 + 48*x_12658^3 + 467*x_21181 + 283*x_21181^2 + 48*x_21181^3 }-> s166 :|: s163 >= 0, s163 <= x_12658 + 1, s164 >= 0, s164 <= x_21181 + 1, s165 >= 0, s165 <= s163 + s164 + 1, s166 >= 0, s166 <= s165, x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ -104 + s167 + 45*z' + 139*z'^2 + 48*z'^3 }-> s169 :|: s167 >= 0, s167 <= z' - 1 + 1, s168 >= 0, s168 <= s167 + 1, s169 >= 0, s169 <= s168, z' - 1 >= 0 encode_i(z') -{ 379 + s171 + 467*x_12660 + 283*x_12660^2 + 48*x_12660^3 + 467*x_21182 + 283*x_21182^2 + 48*x_21182^3 + 467*x_3294 + 283*x_3294^2 + 48*x_3294^3 }-> s174 :|: s170 >= 0, s170 <= x_12660 + 1, s171 >= 0, s171 <= x_21182 + 1, s172 >= 0, s172 <= x_3294 + 1, s173 >= 0, s173 <= s170 + s171 + s172 + 1, s174 >= 0, s174 <= s173, z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s177 :|: s175 >= 0, s175 <= z' - 1 + 1, s176 >= 0, s176 <= s175, s177 >= 0, s177 <= s176, z' - 1 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s189 :|: s187 >= 0, s187 <= z' - 1 + 1, s188 >= 0, s188 <= s187 + 1, s189 >= 0, s189 <= s188, z' - 1 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s44 + s45 :|: s44 >= 0, s44 <= z' + 1, s45 >= 0, s45 <= z'' + 1, z' >= 0, z'' >= 0 encode_k(z') -{ 253 + 467*x_11448 + 283*x_11448^2 + 48*x_11448^3 + 467*x_2643 + 283*x_2643^2 + 48*x_2643^3 }-> s105 :|: s103 >= 0, s103 <= x_11448 + 1, s104 >= 0, s104 <= x_2643 + 1, s105 >= 0, s105 <= 1 + s103 + s104 + 1, z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 253 + 467*x_11449 + 283*x_11449^2 + 48*x_11449^3 + 467*x_2644 + 283*x_2644^2 + 48*x_2644^3 }-> s108 :|: s106 >= 0, s106 <= x_11449 + 1, s107 >= 0, s107 <= x_2644 + 1, s108 >= 0, s108 <= 1 + s106 + s107 + 1, x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s110 :|: s109 >= 0, s109 <= z' - 1 + 1, s110 >= 0, s110 <= 1 + s109 + 1, z' - 1 >= 0 encode_k(z') -{ 261 + 6*s111 + 2*s111*s112 + s111^2 + 4*s112 + s112^2 + 467*x_11452 + 283*x_11452^2 + 48*x_11452^3 + 467*x_2645 + 283*x_2645^2 + 48*x_2645^3 }-> s114 :|: s111 >= 0, s111 <= x_11452 + 1, s112 >= 0, s112 <= x_2645 + 1, s113 >= 0, s113 <= s111 + s112 + 1, s114 >= 0, s114 <= s113 + 1, x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ -104 + s115 + 45*z' + 139*z'^2 + 48*z'^3 }-> s117 :|: s115 >= 0, s115 <= z' - 1 + 1, s116 >= 0, s116 <= s115 + 1, s117 >= 0, s117 <= s116 + 1, z' - 1 >= 0 encode_k(z') -{ 379 + s119 + 467*x_11454 + 283*x_11454^2 + 48*x_11454^3 + 467*x_2646 + 283*x_2646^2 + 48*x_2646^3 + 467*x_3160 + 283*x_3160^2 + 48*x_3160^3 }-> s122 :|: s118 >= 0, s118 <= x_11454 + 1, s119 >= 0, s119 <= x_2646 + 1, s120 >= 0, s120 <= x_3160 + 1, s121 >= 0, s121 <= s118 + s119 + s120 + 1, s122 >= 0, s122 <= s121 + 1, x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s125 :|: s123 >= 0, s123 <= z' - 1 + 1, s124 >= 0, s124 <= s123 + 1, s125 >= 0, s125 <= s124 + 1, z' - 1 >= 0 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s186 :|: s184 >= 0, s184 <= z' - 1 + 1, s185 >= 0, s185 <= s184, s186 >= 0, s186 <= s185 + 1, z' - 1 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s72 :|: s72 >= 0, s72 <= z' + 1, z' >= 0 f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 10 + s9 + 6*x + x^2 }-> s10 :|: s9 >= 0, s9 <= x + 0 + 1, s10 >= 0, s10 <= s9 + 1, z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 11 + s5 + 6*x + 2*x*z'' + x^2 + 4*z'' + z''^2 }-> s6 :|: s5 >= 0, s5 <= x + (1 + 0 + (z'' - 1)) + 1, s6 >= 0, s6 <= s5 + 1, z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 23 + s7 + 10*x + 2*x*x'' + 2*x*y' + x^2 + 8*x'' + 2*x''*y' + x''^2 + 8*y' + y'^2 }-> s8 :|: s7 >= 0, s7 <= x + (1 + (1 + x'') + y') + 1, s8 >= 0, s8 <= s7 + 1, z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {encode_h1}, {encode_g}, {encode_f}, {encode_j}, {encode_i}, {encode_s} Previous analysis results are: h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1] encode_0: runtime: O(1) [0], size: O(1) [0] i: runtime: O(1) [1], size: O(n^1) [z'] k: runtime: O(1) [1], size: O(n^1) [1 + z'] g: runtime: O(n^1) [1 + z'], size: O(n^1) [1 + z'] f: runtime: O(n^2) [8 + 6*z' + 2*z'*z'' + z'^2 + 4*z'' + z''^2], size: O(n^1) [1 + z' + z''] encArg: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [1 + z'] encode_k: runtime: O(n^3) [1161 + 4403*z' + 3107*z'^2 + 624*z'^3], size: O(n^1) [4 + z'] encode_h: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [2 + z'] encode_h2: runtime: O(n^3) [379 + 467*z' + 283*z'^2 + 48*z'^3 + 468*z'' + 283*z''^2 + 48*z''^3 + 467*z1 + 283*z1^2 + 48*z1^3], size: O(n^1) [4 + z' + z'' + z1] ---------------------------------------- (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') -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z' = 1 + 0 encArg(z') -{ -58 + s127 + -89*z' + -5*z'^2 + 48*z'^3 }-> s128 :|: s126 >= 0, s126 <= z' - 2 + 1, s127 >= 0, s127 <= s126, s128 >= 0, s128 <= s127 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11313 + 283*x_11313^2 + 48*x_11313^3 + 467*x_2583 + 283*x_2583^2 + 48*x_2583^3 }-> s131 :|: s129 >= 0, s129 <= x_11313 + 1, s130 >= 0, s130 <= x_2583 + 1, s131 >= 0, s131 <= 1 + s129 + s130, x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 253 + 467*x_11314 + 283*x_11314^2 + 48*x_11314^3 + 467*x_2584 + 283*x_2584^2 + 48*x_2584^3 }-> s134 :|: s132 >= 0, s132 <= x_11314 + 1, s133 >= 0, s133 <= x_2584 + 1, s134 >= 0, s134 <= 1 + s132 + s133, z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s136 :|: s135 >= 0, s135 <= z' - 2 + 1, s136 >= 0, s136 <= 1 + s135, z' - 2 >= 0 encArg(z') -{ 261 + 6*s137 + 2*s137*s138 + s137^2 + 4*s138 + s138^2 + 467*x_11317 + 283*x_11317^2 + 48*x_11317^3 + 467*x_2585 + 283*x_2585^2 + 48*x_2585^3 }-> s140 :|: s137 >= 0, s137 <= x_11317 + 1, s138 >= 0, s138 <= x_2585 + 1, s139 >= 0, s139 <= s137 + s138 + 1, s140 >= 0, s140 <= s139, x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ -58 + s141 + -89*z' + -5*z'^2 + 48*z'^3 }-> s143 :|: s141 >= 0, s141 <= z' - 2 + 1, s142 >= 0, s142 <= s141 + 1, s143 >= 0, s143 <= s142, z' - 2 >= 0 encArg(z') -{ 379 + s145 + 467*x_11319 + 283*x_11319^2 + 48*x_11319^3 + 467*x_2586 + 283*x_2586^2 + 48*x_2586^3 + 467*x_3145 + 283*x_3145^2 + 48*x_3145^3 }-> s148 :|: s144 >= 0, s144 <= x_11319 + 1, s145 >= 0, s145 <= x_2586 + 1, s146 >= 0, s146 <= x_3145 + 1, s147 >= 0, s147 <= s144 + s145 + s146 + 1, s148 >= 0, s148 <= s147, z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s151 :|: s149 >= 0, s149 <= z' - 2 + 1, s150 >= 0, s150 <= s149, s151 >= 0, s151 <= s150, z' - 2 >= 0 encArg(z') -{ 260 + 6*s14 + 2*s14*s15 + s14^2 + 4*s15 + s15^2 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> s16 :|: s14 >= 0, s14 <= x_1 + 1, s15 >= 0, s15 <= x_2 + 1, s16 >= 0, s16 <= s14 + s15 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s180 :|: s178 >= 0, s178 <= z' - 2 + 1, s179 >= 0, s179 <= s178 + 1, s180 >= 0, s180 <= s179, z' - 2 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s183 :|: s181 >= 0, s181 <= z' - 2 + 1, s182 >= 0, s182 <= s181, s183 >= 0, s183 <= s182 + 1, z' - 2 >= 0 encArg(z') -{ 254 + s17 + s18 + 467*x_1107 + 283*x_1107^2 + 48*x_1107^3 + 467*x_247 + 283*x_247^2 + 48*x_247^3 }-> s19 :|: s17 >= 0, s17 <= x_1107 + 1, s18 >= 0, s18 <= x_247 + 1, s19 >= 0, s19 <= 1 + s17 + s18 + 1, z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0 + 1, z' - 1 >= 0 encArg(z') -{ 254 + s20 + s21 + 467*x_1108 + 283*x_1108^2 + 48*x_1108^3 + 467*x_248 + 283*x_248^2 + 48*x_248^3 }-> s22 :|: s20 >= 0, s20 <= x_1108 + 1, s21 >= 0, s21 <= x_248 + 1, s22 >= 0, s22 <= 1 + s20 + s21 + 1, x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ -58 + s23 + -89*z' + -5*z'^2 + 48*z'^3 }-> s24 :|: s23 >= 0, s23 <= z' - 2 + 1, s24 >= 0, s24 <= 1 + s23 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s25 + 2*s25*s26 + s25^2 + 4*s26 + s26^2 + s27 + 467*x_1111 + 283*x_1111^2 + 48*x_1111^3 + 467*x_249 + 283*x_249^2 + 48*x_249^3 }-> s28 :|: s25 >= 0, s25 <= x_1111 + 1, s26 >= 0, s26 <= x_249 + 1, s27 >= 0, s27 <= s25 + s26 + 1, s28 >= 0, s28 <= s27 + 1, z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ -58 + s29 + s30 + -89*z' + -5*z'^2 + 48*z'^3 }-> s31 :|: s29 >= 0, s29 <= z' - 2 + 1, s30 >= 0, s30 <= s29 + 1, s31 >= 0, s31 <= s30 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s33 + s35 + 467*x_1113 + 283*x_1113^2 + 48*x_1113^3 + 467*x_250 + 283*x_250^2 + 48*x_250^3 + 467*x_311 + 283*x_311^2 + 48*x_311^3 }-> s36 :|: s32 >= 0, s32 <= x_1113 + 1, s33 >= 0, s33 <= x_250 + 1, s34 >= 0, s34 <= x_311 + 1, s35 >= 0, s35 <= s32 + s33 + s34 + 1, s36 >= 0, s36 <= s35 + 1, x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 378 + s38 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 + 467*x_3 + 283*x_3^2 + 48*x_3^3 }-> s40 :|: s37 >= 0, s37 <= x_1 + 1, s38 >= 0, s38 <= x_2 + 1, s39 >= 0, s39 <= x_3 + 1, s40 >= 0, s40 <= s37 + s38 + s39 + 1, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z') -{ -58 + s75 + -89*z' + -5*z'^2 + 48*z'^3 }-> s76 :|: s74 >= 0, s74 <= z' - 2 + 1, s75 >= 0, s75 <= s74 + 1, s76 >= 0, s76 <= s75 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11322 + 283*x_11322^2 + 48*x_11322^3 + 467*x_2587 + 283*x_2587^2 + 48*x_2587^3 }-> s79 :|: s77 >= 0, s77 <= x_11322 + 1, s78 >= 0, s78 <= x_2587 + 1, s79 >= 0, s79 <= 1 + s77 + s78 + 1, x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 253 + 467*x_11323 + 283*x_11323^2 + 48*x_11323^3 + 467*x_2588 + 283*x_2588^2 + 48*x_2588^3 }-> s82 :|: s80 >= 0, s80 <= x_11323 + 1, s81 >= 0, s81 <= x_2588 + 1, s82 >= 0, s82 <= 1 + s80 + s81 + 1, z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s84 :|: s83 >= 0, s83 <= z' - 2 + 1, s84 >= 0, s84 <= 1 + s83 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s85 + 2*s85*s86 + s85^2 + 4*s86 + s86^2 + 467*x_11326 + 283*x_11326^2 + 48*x_11326^3 + 467*x_2589 + 283*x_2589^2 + 48*x_2589^3 }-> s88 :|: s85 >= 0, s85 <= x_11326 + 1, s86 >= 0, s86 <= x_2589 + 1, s87 >= 0, s87 <= s85 + s86 + 1, s88 >= 0, s88 <= s87 + 1, x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ -58 + s89 + -89*z' + -5*z'^2 + 48*z'^3 }-> s91 :|: s89 >= 0, s89 <= z' - 2 + 1, s90 >= 0, s90 <= s89 + 1, s91 >= 0, s91 <= s90 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s93 + 467*x_11328 + 283*x_11328^2 + 48*x_11328^3 + 467*x_2590 + 283*x_2590^2 + 48*x_2590^3 + 467*x_3146 + 283*x_3146^2 + 48*x_3146^3 }-> s96 :|: s92 >= 0, s92 <= x_11328 + 1, s93 >= 0, s93 <= x_2590 + 1, s94 >= 0, s94 <= x_3146 + 1, s95 >= 0, s95 <= s92 + s93 + s94 + 1, s96 >= 0, s96 <= s95 + 1, z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s99 :|: s97 >= 0, s97 <= z' - 2 + 1, s98 >= 0, s98 <= s97 + 1, s99 >= 0, s99 <= s98 + 1, z' - 2 >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ -106 + 45*z' + 139*z'^2 + 48*z'^3 }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0 encArg(z') -{ 252 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> 1 + s11 + s12 :|: s11 >= 0, s11 <= x_1 + 1, s12 >= 0, s12 <= x_2 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 260 + 6*s41 + 2*s41*s42 + s41^2 + 4*s42 + s42^2 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> s43 :|: s41 >= 0, s41 <= z' + 1, s42 >= 0, s42 <= z'' + 1, s43 >= 0, s43 <= s41 + s42 + 1, z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ -104 + s101 + 45*z' + 139*z'^2 + 48*z'^3 }-> s102 :|: s100 >= 0, s100 <= z' - 1 + 1, s101 >= 0, s101 <= s100 + 1, s102 >= 0, s102 <= s101 + 1, z' - 1 >= 0 encode_g(z') -{ -104 + s153 + 45*z' + 139*z'^2 + 48*z'^3 }-> s154 :|: s152 >= 0, s152 <= z' - 1 + 1, s153 >= 0, s153 <= s152, s154 >= 0, s154 <= s153 + 1, z' - 1 >= 0 encode_g(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0 + 1, z' = 0 encode_g(z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 1, z' >= 0 encode_g(z') -{ 254 + s46 + s47 + 467*x_11439 + 283*x_11439^2 + 48*x_11439^3 + 467*x_2639 + 283*x_2639^2 + 48*x_2639^3 }-> s48 :|: s46 >= 0, s46 <= x_11439 + 1, s47 >= 0, s47 <= x_2639 + 1, s48 >= 0, s48 <= 1 + s46 + s47 + 1, x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 254 + s49 + s50 + 467*x_11440 + 283*x_11440^2 + 48*x_11440^3 + 467*x_2640 + 283*x_2640^2 + 48*x_2640^3 }-> s51 :|: s49 >= 0, s49 <= x_11440 + 1, s50 >= 0, s50 <= x_2640 + 1, s51 >= 0, s51 <= 1 + s49 + s50 + 1, x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ -104 + s52 + 45*z' + 139*z'^2 + 48*z'^3 }-> s53 :|: s52 >= 0, s52 <= z' - 1 + 1, s53 >= 0, s53 <= 1 + s52 + 1, z' - 1 >= 0 encode_g(z') -{ 261 + 6*s54 + 2*s54*s55 + s54^2 + 4*s55 + s55^2 + s56 + 467*x_11443 + 283*x_11443^2 + 48*x_11443^3 + 467*x_2641 + 283*x_2641^2 + 48*x_2641^3 }-> s57 :|: s54 >= 0, s54 <= x_11443 + 1, s55 >= 0, s55 <= x_2641 + 1, s56 >= 0, s56 <= s54 + s55 + 1, s57 >= 0, s57 <= s56 + 1, x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ -104 + s58 + s59 + 45*z' + 139*z'^2 + 48*z'^3 }-> s60 :|: s58 >= 0, s58 <= z' - 1 + 1, s59 >= 0, s59 <= s58 + 1, s60 >= 0, s60 <= s59 + 1, z' - 1 >= 0 encode_g(z') -{ 379 + s62 + s64 + 467*x_11445 + 283*x_11445^2 + 48*x_11445^3 + 467*x_2642 + 283*x_2642^2 + 48*x_2642^3 + 467*x_3159 + 283*x_3159^2 + 48*x_3159^3 }-> s65 :|: s61 >= 0, s61 <= x_11445 + 1, s62 >= 0, s62 <= x_2642 + 1, s63 >= 0, s63 <= x_3159 + 1, s64 >= 0, s64 <= s61 + s62 + s63 + 1, s65 >= 0, s65 <= s64 + 1, x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s73 :|: s73 >= 0, s73 <= z' + 1, z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s66 + s67 :|: s66 >= 0, s66 <= z' + 1, s67 >= 0, s67 <= z'' + 1, z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 378 + s69 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 + 467*z1 + 283*z1^2 + 48*z1^3 }-> s71 :|: s68 >= 0, s68 <= z' + 1, s69 >= 0, s69 <= z'' + 1, s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s68 + s69 + s70 + 1, z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 253 + 467*x_12654 + 283*x_12654^2 + 48*x_12654^3 + 467*x_21179 + 283*x_21179^2 + 48*x_21179^3 }-> s157 :|: s155 >= 0, s155 <= x_12654 + 1, s156 >= 0, s156 <= x_21179 + 1, s157 >= 0, s157 <= 1 + s155 + s156, x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 253 + 467*x_12655 + 283*x_12655^2 + 48*x_12655^3 + 467*x_21180 + 283*x_21180^2 + 48*x_21180^3 }-> s160 :|: s158 >= 0, s158 <= x_12655 + 1, s159 >= 0, s159 <= x_21180 + 1, s160 >= 0, s160 <= 1 + s158 + s159, z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s162 :|: s161 >= 0, s161 <= z' - 1 + 1, s162 >= 0, s162 <= 1 + s161, z' - 1 >= 0 encode_i(z') -{ 261 + 6*s163 + 2*s163*s164 + s163^2 + 4*s164 + s164^2 + 467*x_12658 + 283*x_12658^2 + 48*x_12658^3 + 467*x_21181 + 283*x_21181^2 + 48*x_21181^3 }-> s166 :|: s163 >= 0, s163 <= x_12658 + 1, s164 >= 0, s164 <= x_21181 + 1, s165 >= 0, s165 <= s163 + s164 + 1, s166 >= 0, s166 <= s165, x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ -104 + s167 + 45*z' + 139*z'^2 + 48*z'^3 }-> s169 :|: s167 >= 0, s167 <= z' - 1 + 1, s168 >= 0, s168 <= s167 + 1, s169 >= 0, s169 <= s168, z' - 1 >= 0 encode_i(z') -{ 379 + s171 + 467*x_12660 + 283*x_12660^2 + 48*x_12660^3 + 467*x_21182 + 283*x_21182^2 + 48*x_21182^3 + 467*x_3294 + 283*x_3294^2 + 48*x_3294^3 }-> s174 :|: s170 >= 0, s170 <= x_12660 + 1, s171 >= 0, s171 <= x_21182 + 1, s172 >= 0, s172 <= x_3294 + 1, s173 >= 0, s173 <= s170 + s171 + s172 + 1, s174 >= 0, s174 <= s173, z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s177 :|: s175 >= 0, s175 <= z' - 1 + 1, s176 >= 0, s176 <= s175, s177 >= 0, s177 <= s176, z' - 1 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s189 :|: s187 >= 0, s187 <= z' - 1 + 1, s188 >= 0, s188 <= s187 + 1, s189 >= 0, s189 <= s188, z' - 1 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s44 + s45 :|: s44 >= 0, s44 <= z' + 1, s45 >= 0, s45 <= z'' + 1, z' >= 0, z'' >= 0 encode_k(z') -{ 253 + 467*x_11448 + 283*x_11448^2 + 48*x_11448^3 + 467*x_2643 + 283*x_2643^2 + 48*x_2643^3 }-> s105 :|: s103 >= 0, s103 <= x_11448 + 1, s104 >= 0, s104 <= x_2643 + 1, s105 >= 0, s105 <= 1 + s103 + s104 + 1, z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 253 + 467*x_11449 + 283*x_11449^2 + 48*x_11449^3 + 467*x_2644 + 283*x_2644^2 + 48*x_2644^3 }-> s108 :|: s106 >= 0, s106 <= x_11449 + 1, s107 >= 0, s107 <= x_2644 + 1, s108 >= 0, s108 <= 1 + s106 + s107 + 1, x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s110 :|: s109 >= 0, s109 <= z' - 1 + 1, s110 >= 0, s110 <= 1 + s109 + 1, z' - 1 >= 0 encode_k(z') -{ 261 + 6*s111 + 2*s111*s112 + s111^2 + 4*s112 + s112^2 + 467*x_11452 + 283*x_11452^2 + 48*x_11452^3 + 467*x_2645 + 283*x_2645^2 + 48*x_2645^3 }-> s114 :|: s111 >= 0, s111 <= x_11452 + 1, s112 >= 0, s112 <= x_2645 + 1, s113 >= 0, s113 <= s111 + s112 + 1, s114 >= 0, s114 <= s113 + 1, x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ -104 + s115 + 45*z' + 139*z'^2 + 48*z'^3 }-> s117 :|: s115 >= 0, s115 <= z' - 1 + 1, s116 >= 0, s116 <= s115 + 1, s117 >= 0, s117 <= s116 + 1, z' - 1 >= 0 encode_k(z') -{ 379 + s119 + 467*x_11454 + 283*x_11454^2 + 48*x_11454^3 + 467*x_2646 + 283*x_2646^2 + 48*x_2646^3 + 467*x_3160 + 283*x_3160^2 + 48*x_3160^3 }-> s122 :|: s118 >= 0, s118 <= x_11454 + 1, s119 >= 0, s119 <= x_2646 + 1, s120 >= 0, s120 <= x_3160 + 1, s121 >= 0, s121 <= s118 + s119 + s120 + 1, s122 >= 0, s122 <= s121 + 1, x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s125 :|: s123 >= 0, s123 <= z' - 1 + 1, s124 >= 0, s124 <= s123 + 1, s125 >= 0, s125 <= s124 + 1, z' - 1 >= 0 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s186 :|: s184 >= 0, s184 <= z' - 1 + 1, s185 >= 0, s185 <= s184, s186 >= 0, s186 <= s185 + 1, z' - 1 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s72 :|: s72 >= 0, s72 <= z' + 1, z' >= 0 f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 10 + s9 + 6*x + x^2 }-> s10 :|: s9 >= 0, s9 <= x + 0 + 1, s10 >= 0, s10 <= s9 + 1, z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 11 + s5 + 6*x + 2*x*z'' + x^2 + 4*z'' + z''^2 }-> s6 :|: s5 >= 0, s5 <= x + (1 + 0 + (z'' - 1)) + 1, s6 >= 0, s6 <= s5 + 1, z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 23 + s7 + 10*x + 2*x*x'' + 2*x*y' + x^2 + 8*x'' + 2*x''*y' + x''^2 + 8*y' + y'^2 }-> s8 :|: s7 >= 0, s7 <= x + (1 + (1 + x'') + y') + 1, s8 >= 0, s8 <= s7 + 1, z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {encode_h1}, {encode_g}, {encode_f}, {encode_j}, {encode_i}, {encode_s} Previous analysis results are: h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1] encode_0: runtime: O(1) [0], size: O(1) [0] i: runtime: O(1) [1], size: O(n^1) [z'] k: runtime: O(1) [1], size: O(n^1) [1 + z'] g: runtime: O(n^1) [1 + z'], size: O(n^1) [1 + z'] f: runtime: O(n^2) [8 + 6*z' + 2*z'*z'' + z'^2 + 4*z'' + z''^2], size: O(n^1) [1 + z' + z''] encArg: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [1 + z'] encode_k: runtime: O(n^3) [1161 + 4403*z' + 3107*z'^2 + 624*z'^3], size: O(n^1) [4 + z'] encode_h: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [2 + z'] encode_h2: runtime: O(n^3) [379 + 467*z' + 283*z'^2 + 48*z'^3 + 468*z'' + 283*z''^2 + 48*z''^3 + 467*z1 + 283*z1^2 + 48*z1^3], size: O(n^1) [4 + z' + z'' + z1] ---------------------------------------- (87) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: encode_h1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 3 + z' + z'' ---------------------------------------- (88) Obligation: Complexity RNTS consisting of the following rules: encArg(z') -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z' = 1 + 0 encArg(z') -{ -58 + s127 + -89*z' + -5*z'^2 + 48*z'^3 }-> s128 :|: s126 >= 0, s126 <= z' - 2 + 1, s127 >= 0, s127 <= s126, s128 >= 0, s128 <= s127 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11313 + 283*x_11313^2 + 48*x_11313^3 + 467*x_2583 + 283*x_2583^2 + 48*x_2583^3 }-> s131 :|: s129 >= 0, s129 <= x_11313 + 1, s130 >= 0, s130 <= x_2583 + 1, s131 >= 0, s131 <= 1 + s129 + s130, x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 253 + 467*x_11314 + 283*x_11314^2 + 48*x_11314^3 + 467*x_2584 + 283*x_2584^2 + 48*x_2584^3 }-> s134 :|: s132 >= 0, s132 <= x_11314 + 1, s133 >= 0, s133 <= x_2584 + 1, s134 >= 0, s134 <= 1 + s132 + s133, z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s136 :|: s135 >= 0, s135 <= z' - 2 + 1, s136 >= 0, s136 <= 1 + s135, z' - 2 >= 0 encArg(z') -{ 261 + 6*s137 + 2*s137*s138 + s137^2 + 4*s138 + s138^2 + 467*x_11317 + 283*x_11317^2 + 48*x_11317^3 + 467*x_2585 + 283*x_2585^2 + 48*x_2585^3 }-> s140 :|: s137 >= 0, s137 <= x_11317 + 1, s138 >= 0, s138 <= x_2585 + 1, s139 >= 0, s139 <= s137 + s138 + 1, s140 >= 0, s140 <= s139, x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ -58 + s141 + -89*z' + -5*z'^2 + 48*z'^3 }-> s143 :|: s141 >= 0, s141 <= z' - 2 + 1, s142 >= 0, s142 <= s141 + 1, s143 >= 0, s143 <= s142, z' - 2 >= 0 encArg(z') -{ 379 + s145 + 467*x_11319 + 283*x_11319^2 + 48*x_11319^3 + 467*x_2586 + 283*x_2586^2 + 48*x_2586^3 + 467*x_3145 + 283*x_3145^2 + 48*x_3145^3 }-> s148 :|: s144 >= 0, s144 <= x_11319 + 1, s145 >= 0, s145 <= x_2586 + 1, s146 >= 0, s146 <= x_3145 + 1, s147 >= 0, s147 <= s144 + s145 + s146 + 1, s148 >= 0, s148 <= s147, z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s151 :|: s149 >= 0, s149 <= z' - 2 + 1, s150 >= 0, s150 <= s149, s151 >= 0, s151 <= s150, z' - 2 >= 0 encArg(z') -{ 260 + 6*s14 + 2*s14*s15 + s14^2 + 4*s15 + s15^2 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> s16 :|: s14 >= 0, s14 <= x_1 + 1, s15 >= 0, s15 <= x_2 + 1, s16 >= 0, s16 <= s14 + s15 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s180 :|: s178 >= 0, s178 <= z' - 2 + 1, s179 >= 0, s179 <= s178 + 1, s180 >= 0, s180 <= s179, z' - 2 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s183 :|: s181 >= 0, s181 <= z' - 2 + 1, s182 >= 0, s182 <= s181, s183 >= 0, s183 <= s182 + 1, z' - 2 >= 0 encArg(z') -{ 254 + s17 + s18 + 467*x_1107 + 283*x_1107^2 + 48*x_1107^3 + 467*x_247 + 283*x_247^2 + 48*x_247^3 }-> s19 :|: s17 >= 0, s17 <= x_1107 + 1, s18 >= 0, s18 <= x_247 + 1, s19 >= 0, s19 <= 1 + s17 + s18 + 1, z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0 + 1, z' - 1 >= 0 encArg(z') -{ 254 + s20 + s21 + 467*x_1108 + 283*x_1108^2 + 48*x_1108^3 + 467*x_248 + 283*x_248^2 + 48*x_248^3 }-> s22 :|: s20 >= 0, s20 <= x_1108 + 1, s21 >= 0, s21 <= x_248 + 1, s22 >= 0, s22 <= 1 + s20 + s21 + 1, x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ -58 + s23 + -89*z' + -5*z'^2 + 48*z'^3 }-> s24 :|: s23 >= 0, s23 <= z' - 2 + 1, s24 >= 0, s24 <= 1 + s23 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s25 + 2*s25*s26 + s25^2 + 4*s26 + s26^2 + s27 + 467*x_1111 + 283*x_1111^2 + 48*x_1111^3 + 467*x_249 + 283*x_249^2 + 48*x_249^3 }-> s28 :|: s25 >= 0, s25 <= x_1111 + 1, s26 >= 0, s26 <= x_249 + 1, s27 >= 0, s27 <= s25 + s26 + 1, s28 >= 0, s28 <= s27 + 1, z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ -58 + s29 + s30 + -89*z' + -5*z'^2 + 48*z'^3 }-> s31 :|: s29 >= 0, s29 <= z' - 2 + 1, s30 >= 0, s30 <= s29 + 1, s31 >= 0, s31 <= s30 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s33 + s35 + 467*x_1113 + 283*x_1113^2 + 48*x_1113^3 + 467*x_250 + 283*x_250^2 + 48*x_250^3 + 467*x_311 + 283*x_311^2 + 48*x_311^3 }-> s36 :|: s32 >= 0, s32 <= x_1113 + 1, s33 >= 0, s33 <= x_250 + 1, s34 >= 0, s34 <= x_311 + 1, s35 >= 0, s35 <= s32 + s33 + s34 + 1, s36 >= 0, s36 <= s35 + 1, x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 378 + s38 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 + 467*x_3 + 283*x_3^2 + 48*x_3^3 }-> s40 :|: s37 >= 0, s37 <= x_1 + 1, s38 >= 0, s38 <= x_2 + 1, s39 >= 0, s39 <= x_3 + 1, s40 >= 0, s40 <= s37 + s38 + s39 + 1, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z') -{ -58 + s75 + -89*z' + -5*z'^2 + 48*z'^3 }-> s76 :|: s74 >= 0, s74 <= z' - 2 + 1, s75 >= 0, s75 <= s74 + 1, s76 >= 0, s76 <= s75 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11322 + 283*x_11322^2 + 48*x_11322^3 + 467*x_2587 + 283*x_2587^2 + 48*x_2587^3 }-> s79 :|: s77 >= 0, s77 <= x_11322 + 1, s78 >= 0, s78 <= x_2587 + 1, s79 >= 0, s79 <= 1 + s77 + s78 + 1, x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 253 + 467*x_11323 + 283*x_11323^2 + 48*x_11323^3 + 467*x_2588 + 283*x_2588^2 + 48*x_2588^3 }-> s82 :|: s80 >= 0, s80 <= x_11323 + 1, s81 >= 0, s81 <= x_2588 + 1, s82 >= 0, s82 <= 1 + s80 + s81 + 1, z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s84 :|: s83 >= 0, s83 <= z' - 2 + 1, s84 >= 0, s84 <= 1 + s83 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s85 + 2*s85*s86 + s85^2 + 4*s86 + s86^2 + 467*x_11326 + 283*x_11326^2 + 48*x_11326^3 + 467*x_2589 + 283*x_2589^2 + 48*x_2589^3 }-> s88 :|: s85 >= 0, s85 <= x_11326 + 1, s86 >= 0, s86 <= x_2589 + 1, s87 >= 0, s87 <= s85 + s86 + 1, s88 >= 0, s88 <= s87 + 1, x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ -58 + s89 + -89*z' + -5*z'^2 + 48*z'^3 }-> s91 :|: s89 >= 0, s89 <= z' - 2 + 1, s90 >= 0, s90 <= s89 + 1, s91 >= 0, s91 <= s90 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s93 + 467*x_11328 + 283*x_11328^2 + 48*x_11328^3 + 467*x_2590 + 283*x_2590^2 + 48*x_2590^3 + 467*x_3146 + 283*x_3146^2 + 48*x_3146^3 }-> s96 :|: s92 >= 0, s92 <= x_11328 + 1, s93 >= 0, s93 <= x_2590 + 1, s94 >= 0, s94 <= x_3146 + 1, s95 >= 0, s95 <= s92 + s93 + s94 + 1, s96 >= 0, s96 <= s95 + 1, z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s99 :|: s97 >= 0, s97 <= z' - 2 + 1, s98 >= 0, s98 <= s97 + 1, s99 >= 0, s99 <= s98 + 1, z' - 2 >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ -106 + 45*z' + 139*z'^2 + 48*z'^3 }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0 encArg(z') -{ 252 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> 1 + s11 + s12 :|: s11 >= 0, s11 <= x_1 + 1, s12 >= 0, s12 <= x_2 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 260 + 6*s41 + 2*s41*s42 + s41^2 + 4*s42 + s42^2 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> s43 :|: s41 >= 0, s41 <= z' + 1, s42 >= 0, s42 <= z'' + 1, s43 >= 0, s43 <= s41 + s42 + 1, z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ -104 + s101 + 45*z' + 139*z'^2 + 48*z'^3 }-> s102 :|: s100 >= 0, s100 <= z' - 1 + 1, s101 >= 0, s101 <= s100 + 1, s102 >= 0, s102 <= s101 + 1, z' - 1 >= 0 encode_g(z') -{ -104 + s153 + 45*z' + 139*z'^2 + 48*z'^3 }-> s154 :|: s152 >= 0, s152 <= z' - 1 + 1, s153 >= 0, s153 <= s152, s154 >= 0, s154 <= s153 + 1, z' - 1 >= 0 encode_g(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0 + 1, z' = 0 encode_g(z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 1, z' >= 0 encode_g(z') -{ 254 + s46 + s47 + 467*x_11439 + 283*x_11439^2 + 48*x_11439^3 + 467*x_2639 + 283*x_2639^2 + 48*x_2639^3 }-> s48 :|: s46 >= 0, s46 <= x_11439 + 1, s47 >= 0, s47 <= x_2639 + 1, s48 >= 0, s48 <= 1 + s46 + s47 + 1, x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 254 + s49 + s50 + 467*x_11440 + 283*x_11440^2 + 48*x_11440^3 + 467*x_2640 + 283*x_2640^2 + 48*x_2640^3 }-> s51 :|: s49 >= 0, s49 <= x_11440 + 1, s50 >= 0, s50 <= x_2640 + 1, s51 >= 0, s51 <= 1 + s49 + s50 + 1, x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ -104 + s52 + 45*z' + 139*z'^2 + 48*z'^3 }-> s53 :|: s52 >= 0, s52 <= z' - 1 + 1, s53 >= 0, s53 <= 1 + s52 + 1, z' - 1 >= 0 encode_g(z') -{ 261 + 6*s54 + 2*s54*s55 + s54^2 + 4*s55 + s55^2 + s56 + 467*x_11443 + 283*x_11443^2 + 48*x_11443^3 + 467*x_2641 + 283*x_2641^2 + 48*x_2641^3 }-> s57 :|: s54 >= 0, s54 <= x_11443 + 1, s55 >= 0, s55 <= x_2641 + 1, s56 >= 0, s56 <= s54 + s55 + 1, s57 >= 0, s57 <= s56 + 1, x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ -104 + s58 + s59 + 45*z' + 139*z'^2 + 48*z'^3 }-> s60 :|: s58 >= 0, s58 <= z' - 1 + 1, s59 >= 0, s59 <= s58 + 1, s60 >= 0, s60 <= s59 + 1, z' - 1 >= 0 encode_g(z') -{ 379 + s62 + s64 + 467*x_11445 + 283*x_11445^2 + 48*x_11445^3 + 467*x_2642 + 283*x_2642^2 + 48*x_2642^3 + 467*x_3159 + 283*x_3159^2 + 48*x_3159^3 }-> s65 :|: s61 >= 0, s61 <= x_11445 + 1, s62 >= 0, s62 <= x_2642 + 1, s63 >= 0, s63 <= x_3159 + 1, s64 >= 0, s64 <= s61 + s62 + s63 + 1, s65 >= 0, s65 <= s64 + 1, x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s73 :|: s73 >= 0, s73 <= z' + 1, z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s66 + s67 :|: s66 >= 0, s66 <= z' + 1, s67 >= 0, s67 <= z'' + 1, z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 378 + s69 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 + 467*z1 + 283*z1^2 + 48*z1^3 }-> s71 :|: s68 >= 0, s68 <= z' + 1, s69 >= 0, s69 <= z'' + 1, s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s68 + s69 + s70 + 1, z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 253 + 467*x_12654 + 283*x_12654^2 + 48*x_12654^3 + 467*x_21179 + 283*x_21179^2 + 48*x_21179^3 }-> s157 :|: s155 >= 0, s155 <= x_12654 + 1, s156 >= 0, s156 <= x_21179 + 1, s157 >= 0, s157 <= 1 + s155 + s156, x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 253 + 467*x_12655 + 283*x_12655^2 + 48*x_12655^3 + 467*x_21180 + 283*x_21180^2 + 48*x_21180^3 }-> s160 :|: s158 >= 0, s158 <= x_12655 + 1, s159 >= 0, s159 <= x_21180 + 1, s160 >= 0, s160 <= 1 + s158 + s159, z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s162 :|: s161 >= 0, s161 <= z' - 1 + 1, s162 >= 0, s162 <= 1 + s161, z' - 1 >= 0 encode_i(z') -{ 261 + 6*s163 + 2*s163*s164 + s163^2 + 4*s164 + s164^2 + 467*x_12658 + 283*x_12658^2 + 48*x_12658^3 + 467*x_21181 + 283*x_21181^2 + 48*x_21181^3 }-> s166 :|: s163 >= 0, s163 <= x_12658 + 1, s164 >= 0, s164 <= x_21181 + 1, s165 >= 0, s165 <= s163 + s164 + 1, s166 >= 0, s166 <= s165, x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ -104 + s167 + 45*z' + 139*z'^2 + 48*z'^3 }-> s169 :|: s167 >= 0, s167 <= z' - 1 + 1, s168 >= 0, s168 <= s167 + 1, s169 >= 0, s169 <= s168, z' - 1 >= 0 encode_i(z') -{ 379 + s171 + 467*x_12660 + 283*x_12660^2 + 48*x_12660^3 + 467*x_21182 + 283*x_21182^2 + 48*x_21182^3 + 467*x_3294 + 283*x_3294^2 + 48*x_3294^3 }-> s174 :|: s170 >= 0, s170 <= x_12660 + 1, s171 >= 0, s171 <= x_21182 + 1, s172 >= 0, s172 <= x_3294 + 1, s173 >= 0, s173 <= s170 + s171 + s172 + 1, s174 >= 0, s174 <= s173, z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s177 :|: s175 >= 0, s175 <= z' - 1 + 1, s176 >= 0, s176 <= s175, s177 >= 0, s177 <= s176, z' - 1 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s189 :|: s187 >= 0, s187 <= z' - 1 + 1, s188 >= 0, s188 <= s187 + 1, s189 >= 0, s189 <= s188, z' - 1 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s44 + s45 :|: s44 >= 0, s44 <= z' + 1, s45 >= 0, s45 <= z'' + 1, z' >= 0, z'' >= 0 encode_k(z') -{ 253 + 467*x_11448 + 283*x_11448^2 + 48*x_11448^3 + 467*x_2643 + 283*x_2643^2 + 48*x_2643^3 }-> s105 :|: s103 >= 0, s103 <= x_11448 + 1, s104 >= 0, s104 <= x_2643 + 1, s105 >= 0, s105 <= 1 + s103 + s104 + 1, z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 253 + 467*x_11449 + 283*x_11449^2 + 48*x_11449^3 + 467*x_2644 + 283*x_2644^2 + 48*x_2644^3 }-> s108 :|: s106 >= 0, s106 <= x_11449 + 1, s107 >= 0, s107 <= x_2644 + 1, s108 >= 0, s108 <= 1 + s106 + s107 + 1, x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s110 :|: s109 >= 0, s109 <= z' - 1 + 1, s110 >= 0, s110 <= 1 + s109 + 1, z' - 1 >= 0 encode_k(z') -{ 261 + 6*s111 + 2*s111*s112 + s111^2 + 4*s112 + s112^2 + 467*x_11452 + 283*x_11452^2 + 48*x_11452^3 + 467*x_2645 + 283*x_2645^2 + 48*x_2645^3 }-> s114 :|: s111 >= 0, s111 <= x_11452 + 1, s112 >= 0, s112 <= x_2645 + 1, s113 >= 0, s113 <= s111 + s112 + 1, s114 >= 0, s114 <= s113 + 1, x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ -104 + s115 + 45*z' + 139*z'^2 + 48*z'^3 }-> s117 :|: s115 >= 0, s115 <= z' - 1 + 1, s116 >= 0, s116 <= s115 + 1, s117 >= 0, s117 <= s116 + 1, z' - 1 >= 0 encode_k(z') -{ 379 + s119 + 467*x_11454 + 283*x_11454^2 + 48*x_11454^3 + 467*x_2646 + 283*x_2646^2 + 48*x_2646^3 + 467*x_3160 + 283*x_3160^2 + 48*x_3160^3 }-> s122 :|: s118 >= 0, s118 <= x_11454 + 1, s119 >= 0, s119 <= x_2646 + 1, s120 >= 0, s120 <= x_3160 + 1, s121 >= 0, s121 <= s118 + s119 + s120 + 1, s122 >= 0, s122 <= s121 + 1, x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s125 :|: s123 >= 0, s123 <= z' - 1 + 1, s124 >= 0, s124 <= s123 + 1, s125 >= 0, s125 <= s124 + 1, z' - 1 >= 0 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s186 :|: s184 >= 0, s184 <= z' - 1 + 1, s185 >= 0, s185 <= s184, s186 >= 0, s186 <= s185 + 1, z' - 1 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s72 :|: s72 >= 0, s72 <= z' + 1, z' >= 0 f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 10 + s9 + 6*x + x^2 }-> s10 :|: s9 >= 0, s9 <= x + 0 + 1, s10 >= 0, s10 <= s9 + 1, z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 11 + s5 + 6*x + 2*x*z'' + x^2 + 4*z'' + z''^2 }-> s6 :|: s5 >= 0, s5 <= x + (1 + 0 + (z'' - 1)) + 1, s6 >= 0, s6 <= s5 + 1, z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 23 + s7 + 10*x + 2*x*x'' + 2*x*y' + x^2 + 8*x'' + 2*x''*y' + x''^2 + 8*y' + y'^2 }-> s8 :|: s7 >= 0, s7 <= x + (1 + (1 + x'') + y') + 1, s8 >= 0, s8 <= s7 + 1, z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {encode_h1}, {encode_g}, {encode_f}, {encode_j}, {encode_i}, {encode_s} Previous analysis results are: h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1] encode_0: runtime: O(1) [0], size: O(1) [0] i: runtime: O(1) [1], size: O(n^1) [z'] k: runtime: O(1) [1], size: O(n^1) [1 + z'] g: runtime: O(n^1) [1 + z'], size: O(n^1) [1 + z'] f: runtime: O(n^2) [8 + 6*z' + 2*z'*z'' + z'^2 + 4*z'' + z''^2], size: O(n^1) [1 + z' + z''] encArg: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [1 + z'] encode_k: runtime: O(n^3) [1161 + 4403*z' + 3107*z'^2 + 624*z'^3], size: O(n^1) [4 + z'] encode_h: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [2 + z'] encode_h2: runtime: O(n^3) [379 + 467*z' + 283*z'^2 + 48*z'^3 + 468*z'' + 283*z''^2 + 48*z''^3 + 467*z1 + 283*z1^2 + 48*z1^3], size: O(n^1) [4 + z' + z'' + z1] encode_h1: runtime: ?, size: O(n^1) [3 + z' + z''] ---------------------------------------- (89) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: encode_h1 after applying outer abstraction to obtain an ITS, resulting in: O(n^3) with polynomial bound: 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 ---------------------------------------- (90) Obligation: Complexity RNTS consisting of the following rules: encArg(z') -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z' = 1 + 0 encArg(z') -{ -58 + s127 + -89*z' + -5*z'^2 + 48*z'^3 }-> s128 :|: s126 >= 0, s126 <= z' - 2 + 1, s127 >= 0, s127 <= s126, s128 >= 0, s128 <= s127 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11313 + 283*x_11313^2 + 48*x_11313^3 + 467*x_2583 + 283*x_2583^2 + 48*x_2583^3 }-> s131 :|: s129 >= 0, s129 <= x_11313 + 1, s130 >= 0, s130 <= x_2583 + 1, s131 >= 0, s131 <= 1 + s129 + s130, x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 253 + 467*x_11314 + 283*x_11314^2 + 48*x_11314^3 + 467*x_2584 + 283*x_2584^2 + 48*x_2584^3 }-> s134 :|: s132 >= 0, s132 <= x_11314 + 1, s133 >= 0, s133 <= x_2584 + 1, s134 >= 0, s134 <= 1 + s132 + s133, z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s136 :|: s135 >= 0, s135 <= z' - 2 + 1, s136 >= 0, s136 <= 1 + s135, z' - 2 >= 0 encArg(z') -{ 261 + 6*s137 + 2*s137*s138 + s137^2 + 4*s138 + s138^2 + 467*x_11317 + 283*x_11317^2 + 48*x_11317^3 + 467*x_2585 + 283*x_2585^2 + 48*x_2585^3 }-> s140 :|: s137 >= 0, s137 <= x_11317 + 1, s138 >= 0, s138 <= x_2585 + 1, s139 >= 0, s139 <= s137 + s138 + 1, s140 >= 0, s140 <= s139, x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ -58 + s141 + -89*z' + -5*z'^2 + 48*z'^3 }-> s143 :|: s141 >= 0, s141 <= z' - 2 + 1, s142 >= 0, s142 <= s141 + 1, s143 >= 0, s143 <= s142, z' - 2 >= 0 encArg(z') -{ 379 + s145 + 467*x_11319 + 283*x_11319^2 + 48*x_11319^3 + 467*x_2586 + 283*x_2586^2 + 48*x_2586^3 + 467*x_3145 + 283*x_3145^2 + 48*x_3145^3 }-> s148 :|: s144 >= 0, s144 <= x_11319 + 1, s145 >= 0, s145 <= x_2586 + 1, s146 >= 0, s146 <= x_3145 + 1, s147 >= 0, s147 <= s144 + s145 + s146 + 1, s148 >= 0, s148 <= s147, z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s151 :|: s149 >= 0, s149 <= z' - 2 + 1, s150 >= 0, s150 <= s149, s151 >= 0, s151 <= s150, z' - 2 >= 0 encArg(z') -{ 260 + 6*s14 + 2*s14*s15 + s14^2 + 4*s15 + s15^2 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> s16 :|: s14 >= 0, s14 <= x_1 + 1, s15 >= 0, s15 <= x_2 + 1, s16 >= 0, s16 <= s14 + s15 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s180 :|: s178 >= 0, s178 <= z' - 2 + 1, s179 >= 0, s179 <= s178 + 1, s180 >= 0, s180 <= s179, z' - 2 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s183 :|: s181 >= 0, s181 <= z' - 2 + 1, s182 >= 0, s182 <= s181, s183 >= 0, s183 <= s182 + 1, z' - 2 >= 0 encArg(z') -{ 254 + s17 + s18 + 467*x_1107 + 283*x_1107^2 + 48*x_1107^3 + 467*x_247 + 283*x_247^2 + 48*x_247^3 }-> s19 :|: s17 >= 0, s17 <= x_1107 + 1, s18 >= 0, s18 <= x_247 + 1, s19 >= 0, s19 <= 1 + s17 + s18 + 1, z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0 + 1, z' - 1 >= 0 encArg(z') -{ 254 + s20 + s21 + 467*x_1108 + 283*x_1108^2 + 48*x_1108^3 + 467*x_248 + 283*x_248^2 + 48*x_248^3 }-> s22 :|: s20 >= 0, s20 <= x_1108 + 1, s21 >= 0, s21 <= x_248 + 1, s22 >= 0, s22 <= 1 + s20 + s21 + 1, x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ -58 + s23 + -89*z' + -5*z'^2 + 48*z'^3 }-> s24 :|: s23 >= 0, s23 <= z' - 2 + 1, s24 >= 0, s24 <= 1 + s23 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s25 + 2*s25*s26 + s25^2 + 4*s26 + s26^2 + s27 + 467*x_1111 + 283*x_1111^2 + 48*x_1111^3 + 467*x_249 + 283*x_249^2 + 48*x_249^3 }-> s28 :|: s25 >= 0, s25 <= x_1111 + 1, s26 >= 0, s26 <= x_249 + 1, s27 >= 0, s27 <= s25 + s26 + 1, s28 >= 0, s28 <= s27 + 1, z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ -58 + s29 + s30 + -89*z' + -5*z'^2 + 48*z'^3 }-> s31 :|: s29 >= 0, s29 <= z' - 2 + 1, s30 >= 0, s30 <= s29 + 1, s31 >= 0, s31 <= s30 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s33 + s35 + 467*x_1113 + 283*x_1113^2 + 48*x_1113^3 + 467*x_250 + 283*x_250^2 + 48*x_250^3 + 467*x_311 + 283*x_311^2 + 48*x_311^3 }-> s36 :|: s32 >= 0, s32 <= x_1113 + 1, s33 >= 0, s33 <= x_250 + 1, s34 >= 0, s34 <= x_311 + 1, s35 >= 0, s35 <= s32 + s33 + s34 + 1, s36 >= 0, s36 <= s35 + 1, x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 378 + s38 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 + 467*x_3 + 283*x_3^2 + 48*x_3^3 }-> s40 :|: s37 >= 0, s37 <= x_1 + 1, s38 >= 0, s38 <= x_2 + 1, s39 >= 0, s39 <= x_3 + 1, s40 >= 0, s40 <= s37 + s38 + s39 + 1, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z') -{ -58 + s75 + -89*z' + -5*z'^2 + 48*z'^3 }-> s76 :|: s74 >= 0, s74 <= z' - 2 + 1, s75 >= 0, s75 <= s74 + 1, s76 >= 0, s76 <= s75 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11322 + 283*x_11322^2 + 48*x_11322^3 + 467*x_2587 + 283*x_2587^2 + 48*x_2587^3 }-> s79 :|: s77 >= 0, s77 <= x_11322 + 1, s78 >= 0, s78 <= x_2587 + 1, s79 >= 0, s79 <= 1 + s77 + s78 + 1, x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 253 + 467*x_11323 + 283*x_11323^2 + 48*x_11323^3 + 467*x_2588 + 283*x_2588^2 + 48*x_2588^3 }-> s82 :|: s80 >= 0, s80 <= x_11323 + 1, s81 >= 0, s81 <= x_2588 + 1, s82 >= 0, s82 <= 1 + s80 + s81 + 1, z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s84 :|: s83 >= 0, s83 <= z' - 2 + 1, s84 >= 0, s84 <= 1 + s83 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s85 + 2*s85*s86 + s85^2 + 4*s86 + s86^2 + 467*x_11326 + 283*x_11326^2 + 48*x_11326^3 + 467*x_2589 + 283*x_2589^2 + 48*x_2589^3 }-> s88 :|: s85 >= 0, s85 <= x_11326 + 1, s86 >= 0, s86 <= x_2589 + 1, s87 >= 0, s87 <= s85 + s86 + 1, s88 >= 0, s88 <= s87 + 1, x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ -58 + s89 + -89*z' + -5*z'^2 + 48*z'^3 }-> s91 :|: s89 >= 0, s89 <= z' - 2 + 1, s90 >= 0, s90 <= s89 + 1, s91 >= 0, s91 <= s90 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s93 + 467*x_11328 + 283*x_11328^2 + 48*x_11328^3 + 467*x_2590 + 283*x_2590^2 + 48*x_2590^3 + 467*x_3146 + 283*x_3146^2 + 48*x_3146^3 }-> s96 :|: s92 >= 0, s92 <= x_11328 + 1, s93 >= 0, s93 <= x_2590 + 1, s94 >= 0, s94 <= x_3146 + 1, s95 >= 0, s95 <= s92 + s93 + s94 + 1, s96 >= 0, s96 <= s95 + 1, z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s99 :|: s97 >= 0, s97 <= z' - 2 + 1, s98 >= 0, s98 <= s97 + 1, s99 >= 0, s99 <= s98 + 1, z' - 2 >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ -106 + 45*z' + 139*z'^2 + 48*z'^3 }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0 encArg(z') -{ 252 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> 1 + s11 + s12 :|: s11 >= 0, s11 <= x_1 + 1, s12 >= 0, s12 <= x_2 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 260 + 6*s41 + 2*s41*s42 + s41^2 + 4*s42 + s42^2 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> s43 :|: s41 >= 0, s41 <= z' + 1, s42 >= 0, s42 <= z'' + 1, s43 >= 0, s43 <= s41 + s42 + 1, z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ -104 + s101 + 45*z' + 139*z'^2 + 48*z'^3 }-> s102 :|: s100 >= 0, s100 <= z' - 1 + 1, s101 >= 0, s101 <= s100 + 1, s102 >= 0, s102 <= s101 + 1, z' - 1 >= 0 encode_g(z') -{ -104 + s153 + 45*z' + 139*z'^2 + 48*z'^3 }-> s154 :|: s152 >= 0, s152 <= z' - 1 + 1, s153 >= 0, s153 <= s152, s154 >= 0, s154 <= s153 + 1, z' - 1 >= 0 encode_g(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0 + 1, z' = 0 encode_g(z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 1, z' >= 0 encode_g(z') -{ 254 + s46 + s47 + 467*x_11439 + 283*x_11439^2 + 48*x_11439^3 + 467*x_2639 + 283*x_2639^2 + 48*x_2639^3 }-> s48 :|: s46 >= 0, s46 <= x_11439 + 1, s47 >= 0, s47 <= x_2639 + 1, s48 >= 0, s48 <= 1 + s46 + s47 + 1, x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 254 + s49 + s50 + 467*x_11440 + 283*x_11440^2 + 48*x_11440^3 + 467*x_2640 + 283*x_2640^2 + 48*x_2640^3 }-> s51 :|: s49 >= 0, s49 <= x_11440 + 1, s50 >= 0, s50 <= x_2640 + 1, s51 >= 0, s51 <= 1 + s49 + s50 + 1, x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ -104 + s52 + 45*z' + 139*z'^2 + 48*z'^3 }-> s53 :|: s52 >= 0, s52 <= z' - 1 + 1, s53 >= 0, s53 <= 1 + s52 + 1, z' - 1 >= 0 encode_g(z') -{ 261 + 6*s54 + 2*s54*s55 + s54^2 + 4*s55 + s55^2 + s56 + 467*x_11443 + 283*x_11443^2 + 48*x_11443^3 + 467*x_2641 + 283*x_2641^2 + 48*x_2641^3 }-> s57 :|: s54 >= 0, s54 <= x_11443 + 1, s55 >= 0, s55 <= x_2641 + 1, s56 >= 0, s56 <= s54 + s55 + 1, s57 >= 0, s57 <= s56 + 1, x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ -104 + s58 + s59 + 45*z' + 139*z'^2 + 48*z'^3 }-> s60 :|: s58 >= 0, s58 <= z' - 1 + 1, s59 >= 0, s59 <= s58 + 1, s60 >= 0, s60 <= s59 + 1, z' - 1 >= 0 encode_g(z') -{ 379 + s62 + s64 + 467*x_11445 + 283*x_11445^2 + 48*x_11445^3 + 467*x_2642 + 283*x_2642^2 + 48*x_2642^3 + 467*x_3159 + 283*x_3159^2 + 48*x_3159^3 }-> s65 :|: s61 >= 0, s61 <= x_11445 + 1, s62 >= 0, s62 <= x_2642 + 1, s63 >= 0, s63 <= x_3159 + 1, s64 >= 0, s64 <= s61 + s62 + s63 + 1, s65 >= 0, s65 <= s64 + 1, x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s73 :|: s73 >= 0, s73 <= z' + 1, z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s66 + s67 :|: s66 >= 0, s66 <= z' + 1, s67 >= 0, s67 <= z'' + 1, z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 378 + s69 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 + 467*z1 + 283*z1^2 + 48*z1^3 }-> s71 :|: s68 >= 0, s68 <= z' + 1, s69 >= 0, s69 <= z'' + 1, s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s68 + s69 + s70 + 1, z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 253 + 467*x_12654 + 283*x_12654^2 + 48*x_12654^3 + 467*x_21179 + 283*x_21179^2 + 48*x_21179^3 }-> s157 :|: s155 >= 0, s155 <= x_12654 + 1, s156 >= 0, s156 <= x_21179 + 1, s157 >= 0, s157 <= 1 + s155 + s156, x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 253 + 467*x_12655 + 283*x_12655^2 + 48*x_12655^3 + 467*x_21180 + 283*x_21180^2 + 48*x_21180^3 }-> s160 :|: s158 >= 0, s158 <= x_12655 + 1, s159 >= 0, s159 <= x_21180 + 1, s160 >= 0, s160 <= 1 + s158 + s159, z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s162 :|: s161 >= 0, s161 <= z' - 1 + 1, s162 >= 0, s162 <= 1 + s161, z' - 1 >= 0 encode_i(z') -{ 261 + 6*s163 + 2*s163*s164 + s163^2 + 4*s164 + s164^2 + 467*x_12658 + 283*x_12658^2 + 48*x_12658^3 + 467*x_21181 + 283*x_21181^2 + 48*x_21181^3 }-> s166 :|: s163 >= 0, s163 <= x_12658 + 1, s164 >= 0, s164 <= x_21181 + 1, s165 >= 0, s165 <= s163 + s164 + 1, s166 >= 0, s166 <= s165, x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ -104 + s167 + 45*z' + 139*z'^2 + 48*z'^3 }-> s169 :|: s167 >= 0, s167 <= z' - 1 + 1, s168 >= 0, s168 <= s167 + 1, s169 >= 0, s169 <= s168, z' - 1 >= 0 encode_i(z') -{ 379 + s171 + 467*x_12660 + 283*x_12660^2 + 48*x_12660^3 + 467*x_21182 + 283*x_21182^2 + 48*x_21182^3 + 467*x_3294 + 283*x_3294^2 + 48*x_3294^3 }-> s174 :|: s170 >= 0, s170 <= x_12660 + 1, s171 >= 0, s171 <= x_21182 + 1, s172 >= 0, s172 <= x_3294 + 1, s173 >= 0, s173 <= s170 + s171 + s172 + 1, s174 >= 0, s174 <= s173, z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s177 :|: s175 >= 0, s175 <= z' - 1 + 1, s176 >= 0, s176 <= s175, s177 >= 0, s177 <= s176, z' - 1 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s189 :|: s187 >= 0, s187 <= z' - 1 + 1, s188 >= 0, s188 <= s187 + 1, s189 >= 0, s189 <= s188, z' - 1 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s44 + s45 :|: s44 >= 0, s44 <= z' + 1, s45 >= 0, s45 <= z'' + 1, z' >= 0, z'' >= 0 encode_k(z') -{ 253 + 467*x_11448 + 283*x_11448^2 + 48*x_11448^3 + 467*x_2643 + 283*x_2643^2 + 48*x_2643^3 }-> s105 :|: s103 >= 0, s103 <= x_11448 + 1, s104 >= 0, s104 <= x_2643 + 1, s105 >= 0, s105 <= 1 + s103 + s104 + 1, z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 253 + 467*x_11449 + 283*x_11449^2 + 48*x_11449^3 + 467*x_2644 + 283*x_2644^2 + 48*x_2644^3 }-> s108 :|: s106 >= 0, s106 <= x_11449 + 1, s107 >= 0, s107 <= x_2644 + 1, s108 >= 0, s108 <= 1 + s106 + s107 + 1, x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s110 :|: s109 >= 0, s109 <= z' - 1 + 1, s110 >= 0, s110 <= 1 + s109 + 1, z' - 1 >= 0 encode_k(z') -{ 261 + 6*s111 + 2*s111*s112 + s111^2 + 4*s112 + s112^2 + 467*x_11452 + 283*x_11452^2 + 48*x_11452^3 + 467*x_2645 + 283*x_2645^2 + 48*x_2645^3 }-> s114 :|: s111 >= 0, s111 <= x_11452 + 1, s112 >= 0, s112 <= x_2645 + 1, s113 >= 0, s113 <= s111 + s112 + 1, s114 >= 0, s114 <= s113 + 1, x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ -104 + s115 + 45*z' + 139*z'^2 + 48*z'^3 }-> s117 :|: s115 >= 0, s115 <= z' - 1 + 1, s116 >= 0, s116 <= s115 + 1, s117 >= 0, s117 <= s116 + 1, z' - 1 >= 0 encode_k(z') -{ 379 + s119 + 467*x_11454 + 283*x_11454^2 + 48*x_11454^3 + 467*x_2646 + 283*x_2646^2 + 48*x_2646^3 + 467*x_3160 + 283*x_3160^2 + 48*x_3160^3 }-> s122 :|: s118 >= 0, s118 <= x_11454 + 1, s119 >= 0, s119 <= x_2646 + 1, s120 >= 0, s120 <= x_3160 + 1, s121 >= 0, s121 <= s118 + s119 + s120 + 1, s122 >= 0, s122 <= s121 + 1, x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s125 :|: s123 >= 0, s123 <= z' - 1 + 1, s124 >= 0, s124 <= s123 + 1, s125 >= 0, s125 <= s124 + 1, z' - 1 >= 0 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s186 :|: s184 >= 0, s184 <= z' - 1 + 1, s185 >= 0, s185 <= s184, s186 >= 0, s186 <= s185 + 1, z' - 1 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s72 :|: s72 >= 0, s72 <= z' + 1, z' >= 0 f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 10 + s9 + 6*x + x^2 }-> s10 :|: s9 >= 0, s9 <= x + 0 + 1, s10 >= 0, s10 <= s9 + 1, z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 11 + s5 + 6*x + 2*x*z'' + x^2 + 4*z'' + z''^2 }-> s6 :|: s5 >= 0, s5 <= x + (1 + 0 + (z'' - 1)) + 1, s6 >= 0, s6 <= s5 + 1, z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 23 + s7 + 10*x + 2*x*x'' + 2*x*y' + x^2 + 8*x'' + 2*x''*y' + x''^2 + 8*y' + y'^2 }-> s8 :|: s7 >= 0, s7 <= x + (1 + (1 + x'') + y') + 1, s8 >= 0, s8 <= s7 + 1, z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {encode_g}, {encode_f}, {encode_j}, {encode_i}, {encode_s} Previous analysis results are: h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1] encode_0: runtime: O(1) [0], size: O(1) [0] i: runtime: O(1) [1], size: O(n^1) [z'] k: runtime: O(1) [1], size: O(n^1) [1 + z'] g: runtime: O(n^1) [1 + z'], size: O(n^1) [1 + z'] f: runtime: O(n^2) [8 + 6*z' + 2*z'*z'' + z'^2 + 4*z'' + z''^2], size: O(n^1) [1 + z' + z''] encArg: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [1 + z'] encode_k: runtime: O(n^3) [1161 + 4403*z' + 3107*z'^2 + 624*z'^3], size: O(n^1) [4 + z'] encode_h: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [2 + z'] encode_h2: runtime: O(n^3) [379 + 467*z' + 283*z'^2 + 48*z'^3 + 468*z'' + 283*z''^2 + 48*z''^3 + 467*z1 + 283*z1^2 + 48*z1^3], size: O(n^1) [4 + z' + z'' + z1] encode_h1: runtime: O(n^3) [252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3], size: O(n^1) [3 + z' + z''] ---------------------------------------- (91) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (92) Obligation: Complexity RNTS consisting of the following rules: encArg(z') -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z' = 1 + 0 encArg(z') -{ -58 + s127 + -89*z' + -5*z'^2 + 48*z'^3 }-> s128 :|: s126 >= 0, s126 <= z' - 2 + 1, s127 >= 0, s127 <= s126, s128 >= 0, s128 <= s127 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11313 + 283*x_11313^2 + 48*x_11313^3 + 467*x_2583 + 283*x_2583^2 + 48*x_2583^3 }-> s131 :|: s129 >= 0, s129 <= x_11313 + 1, s130 >= 0, s130 <= x_2583 + 1, s131 >= 0, s131 <= 1 + s129 + s130, x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 253 + 467*x_11314 + 283*x_11314^2 + 48*x_11314^3 + 467*x_2584 + 283*x_2584^2 + 48*x_2584^3 }-> s134 :|: s132 >= 0, s132 <= x_11314 + 1, s133 >= 0, s133 <= x_2584 + 1, s134 >= 0, s134 <= 1 + s132 + s133, z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s136 :|: s135 >= 0, s135 <= z' - 2 + 1, s136 >= 0, s136 <= 1 + s135, z' - 2 >= 0 encArg(z') -{ 261 + 6*s137 + 2*s137*s138 + s137^2 + 4*s138 + s138^2 + 467*x_11317 + 283*x_11317^2 + 48*x_11317^3 + 467*x_2585 + 283*x_2585^2 + 48*x_2585^3 }-> s140 :|: s137 >= 0, s137 <= x_11317 + 1, s138 >= 0, s138 <= x_2585 + 1, s139 >= 0, s139 <= s137 + s138 + 1, s140 >= 0, s140 <= s139, x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ -58 + s141 + -89*z' + -5*z'^2 + 48*z'^3 }-> s143 :|: s141 >= 0, s141 <= z' - 2 + 1, s142 >= 0, s142 <= s141 + 1, s143 >= 0, s143 <= s142, z' - 2 >= 0 encArg(z') -{ 379 + s145 + 467*x_11319 + 283*x_11319^2 + 48*x_11319^3 + 467*x_2586 + 283*x_2586^2 + 48*x_2586^3 + 467*x_3145 + 283*x_3145^2 + 48*x_3145^3 }-> s148 :|: s144 >= 0, s144 <= x_11319 + 1, s145 >= 0, s145 <= x_2586 + 1, s146 >= 0, s146 <= x_3145 + 1, s147 >= 0, s147 <= s144 + s145 + s146 + 1, s148 >= 0, s148 <= s147, z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s151 :|: s149 >= 0, s149 <= z' - 2 + 1, s150 >= 0, s150 <= s149, s151 >= 0, s151 <= s150, z' - 2 >= 0 encArg(z') -{ 260 + 6*s14 + 2*s14*s15 + s14^2 + 4*s15 + s15^2 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> s16 :|: s14 >= 0, s14 <= x_1 + 1, s15 >= 0, s15 <= x_2 + 1, s16 >= 0, s16 <= s14 + s15 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s180 :|: s178 >= 0, s178 <= z' - 2 + 1, s179 >= 0, s179 <= s178 + 1, s180 >= 0, s180 <= s179, z' - 2 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s183 :|: s181 >= 0, s181 <= z' - 2 + 1, s182 >= 0, s182 <= s181, s183 >= 0, s183 <= s182 + 1, z' - 2 >= 0 encArg(z') -{ 254 + s17 + s18 + 467*x_1107 + 283*x_1107^2 + 48*x_1107^3 + 467*x_247 + 283*x_247^2 + 48*x_247^3 }-> s19 :|: s17 >= 0, s17 <= x_1107 + 1, s18 >= 0, s18 <= x_247 + 1, s19 >= 0, s19 <= 1 + s17 + s18 + 1, z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0 + 1, z' - 1 >= 0 encArg(z') -{ 254 + s20 + s21 + 467*x_1108 + 283*x_1108^2 + 48*x_1108^3 + 467*x_248 + 283*x_248^2 + 48*x_248^3 }-> s22 :|: s20 >= 0, s20 <= x_1108 + 1, s21 >= 0, s21 <= x_248 + 1, s22 >= 0, s22 <= 1 + s20 + s21 + 1, x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ -58 + s23 + -89*z' + -5*z'^2 + 48*z'^3 }-> s24 :|: s23 >= 0, s23 <= z' - 2 + 1, s24 >= 0, s24 <= 1 + s23 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s25 + 2*s25*s26 + s25^2 + 4*s26 + s26^2 + s27 + 467*x_1111 + 283*x_1111^2 + 48*x_1111^3 + 467*x_249 + 283*x_249^2 + 48*x_249^3 }-> s28 :|: s25 >= 0, s25 <= x_1111 + 1, s26 >= 0, s26 <= x_249 + 1, s27 >= 0, s27 <= s25 + s26 + 1, s28 >= 0, s28 <= s27 + 1, z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ -58 + s29 + s30 + -89*z' + -5*z'^2 + 48*z'^3 }-> s31 :|: s29 >= 0, s29 <= z' - 2 + 1, s30 >= 0, s30 <= s29 + 1, s31 >= 0, s31 <= s30 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s33 + s35 + 467*x_1113 + 283*x_1113^2 + 48*x_1113^3 + 467*x_250 + 283*x_250^2 + 48*x_250^3 + 467*x_311 + 283*x_311^2 + 48*x_311^3 }-> s36 :|: s32 >= 0, s32 <= x_1113 + 1, s33 >= 0, s33 <= x_250 + 1, s34 >= 0, s34 <= x_311 + 1, s35 >= 0, s35 <= s32 + s33 + s34 + 1, s36 >= 0, s36 <= s35 + 1, x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 378 + s38 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 + 467*x_3 + 283*x_3^2 + 48*x_3^3 }-> s40 :|: s37 >= 0, s37 <= x_1 + 1, s38 >= 0, s38 <= x_2 + 1, s39 >= 0, s39 <= x_3 + 1, s40 >= 0, s40 <= s37 + s38 + s39 + 1, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z') -{ -58 + s75 + -89*z' + -5*z'^2 + 48*z'^3 }-> s76 :|: s74 >= 0, s74 <= z' - 2 + 1, s75 >= 0, s75 <= s74 + 1, s76 >= 0, s76 <= s75 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11322 + 283*x_11322^2 + 48*x_11322^3 + 467*x_2587 + 283*x_2587^2 + 48*x_2587^3 }-> s79 :|: s77 >= 0, s77 <= x_11322 + 1, s78 >= 0, s78 <= x_2587 + 1, s79 >= 0, s79 <= 1 + s77 + s78 + 1, x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 253 + 467*x_11323 + 283*x_11323^2 + 48*x_11323^3 + 467*x_2588 + 283*x_2588^2 + 48*x_2588^3 }-> s82 :|: s80 >= 0, s80 <= x_11323 + 1, s81 >= 0, s81 <= x_2588 + 1, s82 >= 0, s82 <= 1 + s80 + s81 + 1, z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s84 :|: s83 >= 0, s83 <= z' - 2 + 1, s84 >= 0, s84 <= 1 + s83 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s85 + 2*s85*s86 + s85^2 + 4*s86 + s86^2 + 467*x_11326 + 283*x_11326^2 + 48*x_11326^3 + 467*x_2589 + 283*x_2589^2 + 48*x_2589^3 }-> s88 :|: s85 >= 0, s85 <= x_11326 + 1, s86 >= 0, s86 <= x_2589 + 1, s87 >= 0, s87 <= s85 + s86 + 1, s88 >= 0, s88 <= s87 + 1, x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ -58 + s89 + -89*z' + -5*z'^2 + 48*z'^3 }-> s91 :|: s89 >= 0, s89 <= z' - 2 + 1, s90 >= 0, s90 <= s89 + 1, s91 >= 0, s91 <= s90 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s93 + 467*x_11328 + 283*x_11328^2 + 48*x_11328^3 + 467*x_2590 + 283*x_2590^2 + 48*x_2590^3 + 467*x_3146 + 283*x_3146^2 + 48*x_3146^3 }-> s96 :|: s92 >= 0, s92 <= x_11328 + 1, s93 >= 0, s93 <= x_2590 + 1, s94 >= 0, s94 <= x_3146 + 1, s95 >= 0, s95 <= s92 + s93 + s94 + 1, s96 >= 0, s96 <= s95 + 1, z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s99 :|: s97 >= 0, s97 <= z' - 2 + 1, s98 >= 0, s98 <= s97 + 1, s99 >= 0, s99 <= s98 + 1, z' - 2 >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ -106 + 45*z' + 139*z'^2 + 48*z'^3 }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0 encArg(z') -{ 252 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> 1 + s11 + s12 :|: s11 >= 0, s11 <= x_1 + 1, s12 >= 0, s12 <= x_2 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 260 + 6*s41 + 2*s41*s42 + s41^2 + 4*s42 + s42^2 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> s43 :|: s41 >= 0, s41 <= z' + 1, s42 >= 0, s42 <= z'' + 1, s43 >= 0, s43 <= s41 + s42 + 1, z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ -104 + s101 + 45*z' + 139*z'^2 + 48*z'^3 }-> s102 :|: s100 >= 0, s100 <= z' - 1 + 1, s101 >= 0, s101 <= s100 + 1, s102 >= 0, s102 <= s101 + 1, z' - 1 >= 0 encode_g(z') -{ -104 + s153 + 45*z' + 139*z'^2 + 48*z'^3 }-> s154 :|: s152 >= 0, s152 <= z' - 1 + 1, s153 >= 0, s153 <= s152, s154 >= 0, s154 <= s153 + 1, z' - 1 >= 0 encode_g(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0 + 1, z' = 0 encode_g(z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 1, z' >= 0 encode_g(z') -{ 254 + s46 + s47 + 467*x_11439 + 283*x_11439^2 + 48*x_11439^3 + 467*x_2639 + 283*x_2639^2 + 48*x_2639^3 }-> s48 :|: s46 >= 0, s46 <= x_11439 + 1, s47 >= 0, s47 <= x_2639 + 1, s48 >= 0, s48 <= 1 + s46 + s47 + 1, x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 254 + s49 + s50 + 467*x_11440 + 283*x_11440^2 + 48*x_11440^3 + 467*x_2640 + 283*x_2640^2 + 48*x_2640^3 }-> s51 :|: s49 >= 0, s49 <= x_11440 + 1, s50 >= 0, s50 <= x_2640 + 1, s51 >= 0, s51 <= 1 + s49 + s50 + 1, x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ -104 + s52 + 45*z' + 139*z'^2 + 48*z'^3 }-> s53 :|: s52 >= 0, s52 <= z' - 1 + 1, s53 >= 0, s53 <= 1 + s52 + 1, z' - 1 >= 0 encode_g(z') -{ 261 + 6*s54 + 2*s54*s55 + s54^2 + 4*s55 + s55^2 + s56 + 467*x_11443 + 283*x_11443^2 + 48*x_11443^3 + 467*x_2641 + 283*x_2641^2 + 48*x_2641^3 }-> s57 :|: s54 >= 0, s54 <= x_11443 + 1, s55 >= 0, s55 <= x_2641 + 1, s56 >= 0, s56 <= s54 + s55 + 1, s57 >= 0, s57 <= s56 + 1, x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ -104 + s58 + s59 + 45*z' + 139*z'^2 + 48*z'^3 }-> s60 :|: s58 >= 0, s58 <= z' - 1 + 1, s59 >= 0, s59 <= s58 + 1, s60 >= 0, s60 <= s59 + 1, z' - 1 >= 0 encode_g(z') -{ 379 + s62 + s64 + 467*x_11445 + 283*x_11445^2 + 48*x_11445^3 + 467*x_2642 + 283*x_2642^2 + 48*x_2642^3 + 467*x_3159 + 283*x_3159^2 + 48*x_3159^3 }-> s65 :|: s61 >= 0, s61 <= x_11445 + 1, s62 >= 0, s62 <= x_2642 + 1, s63 >= 0, s63 <= x_3159 + 1, s64 >= 0, s64 <= s61 + s62 + s63 + 1, s65 >= 0, s65 <= s64 + 1, x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s73 :|: s73 >= 0, s73 <= z' + 1, z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s66 + s67 :|: s66 >= 0, s66 <= z' + 1, s67 >= 0, s67 <= z'' + 1, z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 378 + s69 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 + 467*z1 + 283*z1^2 + 48*z1^3 }-> s71 :|: s68 >= 0, s68 <= z' + 1, s69 >= 0, s69 <= z'' + 1, s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s68 + s69 + s70 + 1, z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 253 + 467*x_12654 + 283*x_12654^2 + 48*x_12654^3 + 467*x_21179 + 283*x_21179^2 + 48*x_21179^3 }-> s157 :|: s155 >= 0, s155 <= x_12654 + 1, s156 >= 0, s156 <= x_21179 + 1, s157 >= 0, s157 <= 1 + s155 + s156, x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 253 + 467*x_12655 + 283*x_12655^2 + 48*x_12655^3 + 467*x_21180 + 283*x_21180^2 + 48*x_21180^3 }-> s160 :|: s158 >= 0, s158 <= x_12655 + 1, s159 >= 0, s159 <= x_21180 + 1, s160 >= 0, s160 <= 1 + s158 + s159, z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s162 :|: s161 >= 0, s161 <= z' - 1 + 1, s162 >= 0, s162 <= 1 + s161, z' - 1 >= 0 encode_i(z') -{ 261 + 6*s163 + 2*s163*s164 + s163^2 + 4*s164 + s164^2 + 467*x_12658 + 283*x_12658^2 + 48*x_12658^3 + 467*x_21181 + 283*x_21181^2 + 48*x_21181^3 }-> s166 :|: s163 >= 0, s163 <= x_12658 + 1, s164 >= 0, s164 <= x_21181 + 1, s165 >= 0, s165 <= s163 + s164 + 1, s166 >= 0, s166 <= s165, x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ -104 + s167 + 45*z' + 139*z'^2 + 48*z'^3 }-> s169 :|: s167 >= 0, s167 <= z' - 1 + 1, s168 >= 0, s168 <= s167 + 1, s169 >= 0, s169 <= s168, z' - 1 >= 0 encode_i(z') -{ 379 + s171 + 467*x_12660 + 283*x_12660^2 + 48*x_12660^3 + 467*x_21182 + 283*x_21182^2 + 48*x_21182^3 + 467*x_3294 + 283*x_3294^2 + 48*x_3294^3 }-> s174 :|: s170 >= 0, s170 <= x_12660 + 1, s171 >= 0, s171 <= x_21182 + 1, s172 >= 0, s172 <= x_3294 + 1, s173 >= 0, s173 <= s170 + s171 + s172 + 1, s174 >= 0, s174 <= s173, z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s177 :|: s175 >= 0, s175 <= z' - 1 + 1, s176 >= 0, s176 <= s175, s177 >= 0, s177 <= s176, z' - 1 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s189 :|: s187 >= 0, s187 <= z' - 1 + 1, s188 >= 0, s188 <= s187 + 1, s189 >= 0, s189 <= s188, z' - 1 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s44 + s45 :|: s44 >= 0, s44 <= z' + 1, s45 >= 0, s45 <= z'' + 1, z' >= 0, z'' >= 0 encode_k(z') -{ 253 + 467*x_11448 + 283*x_11448^2 + 48*x_11448^3 + 467*x_2643 + 283*x_2643^2 + 48*x_2643^3 }-> s105 :|: s103 >= 0, s103 <= x_11448 + 1, s104 >= 0, s104 <= x_2643 + 1, s105 >= 0, s105 <= 1 + s103 + s104 + 1, z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 253 + 467*x_11449 + 283*x_11449^2 + 48*x_11449^3 + 467*x_2644 + 283*x_2644^2 + 48*x_2644^3 }-> s108 :|: s106 >= 0, s106 <= x_11449 + 1, s107 >= 0, s107 <= x_2644 + 1, s108 >= 0, s108 <= 1 + s106 + s107 + 1, x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s110 :|: s109 >= 0, s109 <= z' - 1 + 1, s110 >= 0, s110 <= 1 + s109 + 1, z' - 1 >= 0 encode_k(z') -{ 261 + 6*s111 + 2*s111*s112 + s111^2 + 4*s112 + s112^2 + 467*x_11452 + 283*x_11452^2 + 48*x_11452^3 + 467*x_2645 + 283*x_2645^2 + 48*x_2645^3 }-> s114 :|: s111 >= 0, s111 <= x_11452 + 1, s112 >= 0, s112 <= x_2645 + 1, s113 >= 0, s113 <= s111 + s112 + 1, s114 >= 0, s114 <= s113 + 1, x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ -104 + s115 + 45*z' + 139*z'^2 + 48*z'^3 }-> s117 :|: s115 >= 0, s115 <= z' - 1 + 1, s116 >= 0, s116 <= s115 + 1, s117 >= 0, s117 <= s116 + 1, z' - 1 >= 0 encode_k(z') -{ 379 + s119 + 467*x_11454 + 283*x_11454^2 + 48*x_11454^3 + 467*x_2646 + 283*x_2646^2 + 48*x_2646^3 + 467*x_3160 + 283*x_3160^2 + 48*x_3160^3 }-> s122 :|: s118 >= 0, s118 <= x_11454 + 1, s119 >= 0, s119 <= x_2646 + 1, s120 >= 0, s120 <= x_3160 + 1, s121 >= 0, s121 <= s118 + s119 + s120 + 1, s122 >= 0, s122 <= s121 + 1, x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s125 :|: s123 >= 0, s123 <= z' - 1 + 1, s124 >= 0, s124 <= s123 + 1, s125 >= 0, s125 <= s124 + 1, z' - 1 >= 0 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s186 :|: s184 >= 0, s184 <= z' - 1 + 1, s185 >= 0, s185 <= s184, s186 >= 0, s186 <= s185 + 1, z' - 1 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s72 :|: s72 >= 0, s72 <= z' + 1, z' >= 0 f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 10 + s9 + 6*x + x^2 }-> s10 :|: s9 >= 0, s9 <= x + 0 + 1, s10 >= 0, s10 <= s9 + 1, z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 11 + s5 + 6*x + 2*x*z'' + x^2 + 4*z'' + z''^2 }-> s6 :|: s5 >= 0, s5 <= x + (1 + 0 + (z'' - 1)) + 1, s6 >= 0, s6 <= s5 + 1, z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 23 + s7 + 10*x + 2*x*x'' + 2*x*y' + x^2 + 8*x'' + 2*x''*y' + x''^2 + 8*y' + y'^2 }-> s8 :|: s7 >= 0, s7 <= x + (1 + (1 + x'') + y') + 1, s8 >= 0, s8 <= s7 + 1, z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {encode_g}, {encode_f}, {encode_j}, {encode_i}, {encode_s} Previous analysis results are: h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1] encode_0: runtime: O(1) [0], size: O(1) [0] i: runtime: O(1) [1], size: O(n^1) [z'] k: runtime: O(1) [1], size: O(n^1) [1 + z'] g: runtime: O(n^1) [1 + z'], size: O(n^1) [1 + z'] f: runtime: O(n^2) [8 + 6*z' + 2*z'*z'' + z'^2 + 4*z'' + z''^2], size: O(n^1) [1 + z' + z''] encArg: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [1 + z'] encode_k: runtime: O(n^3) [1161 + 4403*z' + 3107*z'^2 + 624*z'^3], size: O(n^1) [4 + z'] encode_h: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [2 + z'] encode_h2: runtime: O(n^3) [379 + 467*z' + 283*z'^2 + 48*z'^3 + 468*z'' + 283*z''^2 + 48*z''^3 + 467*z1 + 283*z1^2 + 48*z1^3], size: O(n^1) [4 + z' + z'' + z1] encode_h1: runtime: O(n^3) [252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3], size: O(n^1) [3 + z' + z''] ---------------------------------------- (93) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: encode_g after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 4 + z' ---------------------------------------- (94) Obligation: Complexity RNTS consisting of the following rules: encArg(z') -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z' = 1 + 0 encArg(z') -{ -58 + s127 + -89*z' + -5*z'^2 + 48*z'^3 }-> s128 :|: s126 >= 0, s126 <= z' - 2 + 1, s127 >= 0, s127 <= s126, s128 >= 0, s128 <= s127 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11313 + 283*x_11313^2 + 48*x_11313^3 + 467*x_2583 + 283*x_2583^2 + 48*x_2583^3 }-> s131 :|: s129 >= 0, s129 <= x_11313 + 1, s130 >= 0, s130 <= x_2583 + 1, s131 >= 0, s131 <= 1 + s129 + s130, x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 253 + 467*x_11314 + 283*x_11314^2 + 48*x_11314^3 + 467*x_2584 + 283*x_2584^2 + 48*x_2584^3 }-> s134 :|: s132 >= 0, s132 <= x_11314 + 1, s133 >= 0, s133 <= x_2584 + 1, s134 >= 0, s134 <= 1 + s132 + s133, z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s136 :|: s135 >= 0, s135 <= z' - 2 + 1, s136 >= 0, s136 <= 1 + s135, z' - 2 >= 0 encArg(z') -{ 261 + 6*s137 + 2*s137*s138 + s137^2 + 4*s138 + s138^2 + 467*x_11317 + 283*x_11317^2 + 48*x_11317^3 + 467*x_2585 + 283*x_2585^2 + 48*x_2585^3 }-> s140 :|: s137 >= 0, s137 <= x_11317 + 1, s138 >= 0, s138 <= x_2585 + 1, s139 >= 0, s139 <= s137 + s138 + 1, s140 >= 0, s140 <= s139, x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ -58 + s141 + -89*z' + -5*z'^2 + 48*z'^3 }-> s143 :|: s141 >= 0, s141 <= z' - 2 + 1, s142 >= 0, s142 <= s141 + 1, s143 >= 0, s143 <= s142, z' - 2 >= 0 encArg(z') -{ 379 + s145 + 467*x_11319 + 283*x_11319^2 + 48*x_11319^3 + 467*x_2586 + 283*x_2586^2 + 48*x_2586^3 + 467*x_3145 + 283*x_3145^2 + 48*x_3145^3 }-> s148 :|: s144 >= 0, s144 <= x_11319 + 1, s145 >= 0, s145 <= x_2586 + 1, s146 >= 0, s146 <= x_3145 + 1, s147 >= 0, s147 <= s144 + s145 + s146 + 1, s148 >= 0, s148 <= s147, z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s151 :|: s149 >= 0, s149 <= z' - 2 + 1, s150 >= 0, s150 <= s149, s151 >= 0, s151 <= s150, z' - 2 >= 0 encArg(z') -{ 260 + 6*s14 + 2*s14*s15 + s14^2 + 4*s15 + s15^2 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> s16 :|: s14 >= 0, s14 <= x_1 + 1, s15 >= 0, s15 <= x_2 + 1, s16 >= 0, s16 <= s14 + s15 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s180 :|: s178 >= 0, s178 <= z' - 2 + 1, s179 >= 0, s179 <= s178 + 1, s180 >= 0, s180 <= s179, z' - 2 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s183 :|: s181 >= 0, s181 <= z' - 2 + 1, s182 >= 0, s182 <= s181, s183 >= 0, s183 <= s182 + 1, z' - 2 >= 0 encArg(z') -{ 254 + s17 + s18 + 467*x_1107 + 283*x_1107^2 + 48*x_1107^3 + 467*x_247 + 283*x_247^2 + 48*x_247^3 }-> s19 :|: s17 >= 0, s17 <= x_1107 + 1, s18 >= 0, s18 <= x_247 + 1, s19 >= 0, s19 <= 1 + s17 + s18 + 1, z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0 + 1, z' - 1 >= 0 encArg(z') -{ 254 + s20 + s21 + 467*x_1108 + 283*x_1108^2 + 48*x_1108^3 + 467*x_248 + 283*x_248^2 + 48*x_248^3 }-> s22 :|: s20 >= 0, s20 <= x_1108 + 1, s21 >= 0, s21 <= x_248 + 1, s22 >= 0, s22 <= 1 + s20 + s21 + 1, x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ -58 + s23 + -89*z' + -5*z'^2 + 48*z'^3 }-> s24 :|: s23 >= 0, s23 <= z' - 2 + 1, s24 >= 0, s24 <= 1 + s23 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s25 + 2*s25*s26 + s25^2 + 4*s26 + s26^2 + s27 + 467*x_1111 + 283*x_1111^2 + 48*x_1111^3 + 467*x_249 + 283*x_249^2 + 48*x_249^3 }-> s28 :|: s25 >= 0, s25 <= x_1111 + 1, s26 >= 0, s26 <= x_249 + 1, s27 >= 0, s27 <= s25 + s26 + 1, s28 >= 0, s28 <= s27 + 1, z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ -58 + s29 + s30 + -89*z' + -5*z'^2 + 48*z'^3 }-> s31 :|: s29 >= 0, s29 <= z' - 2 + 1, s30 >= 0, s30 <= s29 + 1, s31 >= 0, s31 <= s30 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s33 + s35 + 467*x_1113 + 283*x_1113^2 + 48*x_1113^3 + 467*x_250 + 283*x_250^2 + 48*x_250^3 + 467*x_311 + 283*x_311^2 + 48*x_311^3 }-> s36 :|: s32 >= 0, s32 <= x_1113 + 1, s33 >= 0, s33 <= x_250 + 1, s34 >= 0, s34 <= x_311 + 1, s35 >= 0, s35 <= s32 + s33 + s34 + 1, s36 >= 0, s36 <= s35 + 1, x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 378 + s38 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 + 467*x_3 + 283*x_3^2 + 48*x_3^3 }-> s40 :|: s37 >= 0, s37 <= x_1 + 1, s38 >= 0, s38 <= x_2 + 1, s39 >= 0, s39 <= x_3 + 1, s40 >= 0, s40 <= s37 + s38 + s39 + 1, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z') -{ -58 + s75 + -89*z' + -5*z'^2 + 48*z'^3 }-> s76 :|: s74 >= 0, s74 <= z' - 2 + 1, s75 >= 0, s75 <= s74 + 1, s76 >= 0, s76 <= s75 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11322 + 283*x_11322^2 + 48*x_11322^3 + 467*x_2587 + 283*x_2587^2 + 48*x_2587^3 }-> s79 :|: s77 >= 0, s77 <= x_11322 + 1, s78 >= 0, s78 <= x_2587 + 1, s79 >= 0, s79 <= 1 + s77 + s78 + 1, x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 253 + 467*x_11323 + 283*x_11323^2 + 48*x_11323^3 + 467*x_2588 + 283*x_2588^2 + 48*x_2588^3 }-> s82 :|: s80 >= 0, s80 <= x_11323 + 1, s81 >= 0, s81 <= x_2588 + 1, s82 >= 0, s82 <= 1 + s80 + s81 + 1, z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s84 :|: s83 >= 0, s83 <= z' - 2 + 1, s84 >= 0, s84 <= 1 + s83 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s85 + 2*s85*s86 + s85^2 + 4*s86 + s86^2 + 467*x_11326 + 283*x_11326^2 + 48*x_11326^3 + 467*x_2589 + 283*x_2589^2 + 48*x_2589^3 }-> s88 :|: s85 >= 0, s85 <= x_11326 + 1, s86 >= 0, s86 <= x_2589 + 1, s87 >= 0, s87 <= s85 + s86 + 1, s88 >= 0, s88 <= s87 + 1, x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ -58 + s89 + -89*z' + -5*z'^2 + 48*z'^3 }-> s91 :|: s89 >= 0, s89 <= z' - 2 + 1, s90 >= 0, s90 <= s89 + 1, s91 >= 0, s91 <= s90 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s93 + 467*x_11328 + 283*x_11328^2 + 48*x_11328^3 + 467*x_2590 + 283*x_2590^2 + 48*x_2590^3 + 467*x_3146 + 283*x_3146^2 + 48*x_3146^3 }-> s96 :|: s92 >= 0, s92 <= x_11328 + 1, s93 >= 0, s93 <= x_2590 + 1, s94 >= 0, s94 <= x_3146 + 1, s95 >= 0, s95 <= s92 + s93 + s94 + 1, s96 >= 0, s96 <= s95 + 1, z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s99 :|: s97 >= 0, s97 <= z' - 2 + 1, s98 >= 0, s98 <= s97 + 1, s99 >= 0, s99 <= s98 + 1, z' - 2 >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ -106 + 45*z' + 139*z'^2 + 48*z'^3 }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0 encArg(z') -{ 252 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> 1 + s11 + s12 :|: s11 >= 0, s11 <= x_1 + 1, s12 >= 0, s12 <= x_2 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 260 + 6*s41 + 2*s41*s42 + s41^2 + 4*s42 + s42^2 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> s43 :|: s41 >= 0, s41 <= z' + 1, s42 >= 0, s42 <= z'' + 1, s43 >= 0, s43 <= s41 + s42 + 1, z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ -104 + s101 + 45*z' + 139*z'^2 + 48*z'^3 }-> s102 :|: s100 >= 0, s100 <= z' - 1 + 1, s101 >= 0, s101 <= s100 + 1, s102 >= 0, s102 <= s101 + 1, z' - 1 >= 0 encode_g(z') -{ -104 + s153 + 45*z' + 139*z'^2 + 48*z'^3 }-> s154 :|: s152 >= 0, s152 <= z' - 1 + 1, s153 >= 0, s153 <= s152, s154 >= 0, s154 <= s153 + 1, z' - 1 >= 0 encode_g(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0 + 1, z' = 0 encode_g(z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 1, z' >= 0 encode_g(z') -{ 254 + s46 + s47 + 467*x_11439 + 283*x_11439^2 + 48*x_11439^3 + 467*x_2639 + 283*x_2639^2 + 48*x_2639^3 }-> s48 :|: s46 >= 0, s46 <= x_11439 + 1, s47 >= 0, s47 <= x_2639 + 1, s48 >= 0, s48 <= 1 + s46 + s47 + 1, x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 254 + s49 + s50 + 467*x_11440 + 283*x_11440^2 + 48*x_11440^3 + 467*x_2640 + 283*x_2640^2 + 48*x_2640^3 }-> s51 :|: s49 >= 0, s49 <= x_11440 + 1, s50 >= 0, s50 <= x_2640 + 1, s51 >= 0, s51 <= 1 + s49 + s50 + 1, x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ -104 + s52 + 45*z' + 139*z'^2 + 48*z'^3 }-> s53 :|: s52 >= 0, s52 <= z' - 1 + 1, s53 >= 0, s53 <= 1 + s52 + 1, z' - 1 >= 0 encode_g(z') -{ 261 + 6*s54 + 2*s54*s55 + s54^2 + 4*s55 + s55^2 + s56 + 467*x_11443 + 283*x_11443^2 + 48*x_11443^3 + 467*x_2641 + 283*x_2641^2 + 48*x_2641^3 }-> s57 :|: s54 >= 0, s54 <= x_11443 + 1, s55 >= 0, s55 <= x_2641 + 1, s56 >= 0, s56 <= s54 + s55 + 1, s57 >= 0, s57 <= s56 + 1, x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ -104 + s58 + s59 + 45*z' + 139*z'^2 + 48*z'^3 }-> s60 :|: s58 >= 0, s58 <= z' - 1 + 1, s59 >= 0, s59 <= s58 + 1, s60 >= 0, s60 <= s59 + 1, z' - 1 >= 0 encode_g(z') -{ 379 + s62 + s64 + 467*x_11445 + 283*x_11445^2 + 48*x_11445^3 + 467*x_2642 + 283*x_2642^2 + 48*x_2642^3 + 467*x_3159 + 283*x_3159^2 + 48*x_3159^3 }-> s65 :|: s61 >= 0, s61 <= x_11445 + 1, s62 >= 0, s62 <= x_2642 + 1, s63 >= 0, s63 <= x_3159 + 1, s64 >= 0, s64 <= s61 + s62 + s63 + 1, s65 >= 0, s65 <= s64 + 1, x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s73 :|: s73 >= 0, s73 <= z' + 1, z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s66 + s67 :|: s66 >= 0, s66 <= z' + 1, s67 >= 0, s67 <= z'' + 1, z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 378 + s69 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 + 467*z1 + 283*z1^2 + 48*z1^3 }-> s71 :|: s68 >= 0, s68 <= z' + 1, s69 >= 0, s69 <= z'' + 1, s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s68 + s69 + s70 + 1, z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 253 + 467*x_12654 + 283*x_12654^2 + 48*x_12654^3 + 467*x_21179 + 283*x_21179^2 + 48*x_21179^3 }-> s157 :|: s155 >= 0, s155 <= x_12654 + 1, s156 >= 0, s156 <= x_21179 + 1, s157 >= 0, s157 <= 1 + s155 + s156, x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 253 + 467*x_12655 + 283*x_12655^2 + 48*x_12655^3 + 467*x_21180 + 283*x_21180^2 + 48*x_21180^3 }-> s160 :|: s158 >= 0, s158 <= x_12655 + 1, s159 >= 0, s159 <= x_21180 + 1, s160 >= 0, s160 <= 1 + s158 + s159, z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s162 :|: s161 >= 0, s161 <= z' - 1 + 1, s162 >= 0, s162 <= 1 + s161, z' - 1 >= 0 encode_i(z') -{ 261 + 6*s163 + 2*s163*s164 + s163^2 + 4*s164 + s164^2 + 467*x_12658 + 283*x_12658^2 + 48*x_12658^3 + 467*x_21181 + 283*x_21181^2 + 48*x_21181^3 }-> s166 :|: s163 >= 0, s163 <= x_12658 + 1, s164 >= 0, s164 <= x_21181 + 1, s165 >= 0, s165 <= s163 + s164 + 1, s166 >= 0, s166 <= s165, x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ -104 + s167 + 45*z' + 139*z'^2 + 48*z'^3 }-> s169 :|: s167 >= 0, s167 <= z' - 1 + 1, s168 >= 0, s168 <= s167 + 1, s169 >= 0, s169 <= s168, z' - 1 >= 0 encode_i(z') -{ 379 + s171 + 467*x_12660 + 283*x_12660^2 + 48*x_12660^3 + 467*x_21182 + 283*x_21182^2 + 48*x_21182^3 + 467*x_3294 + 283*x_3294^2 + 48*x_3294^3 }-> s174 :|: s170 >= 0, s170 <= x_12660 + 1, s171 >= 0, s171 <= x_21182 + 1, s172 >= 0, s172 <= x_3294 + 1, s173 >= 0, s173 <= s170 + s171 + s172 + 1, s174 >= 0, s174 <= s173, z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s177 :|: s175 >= 0, s175 <= z' - 1 + 1, s176 >= 0, s176 <= s175, s177 >= 0, s177 <= s176, z' - 1 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s189 :|: s187 >= 0, s187 <= z' - 1 + 1, s188 >= 0, s188 <= s187 + 1, s189 >= 0, s189 <= s188, z' - 1 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s44 + s45 :|: s44 >= 0, s44 <= z' + 1, s45 >= 0, s45 <= z'' + 1, z' >= 0, z'' >= 0 encode_k(z') -{ 253 + 467*x_11448 + 283*x_11448^2 + 48*x_11448^3 + 467*x_2643 + 283*x_2643^2 + 48*x_2643^3 }-> s105 :|: s103 >= 0, s103 <= x_11448 + 1, s104 >= 0, s104 <= x_2643 + 1, s105 >= 0, s105 <= 1 + s103 + s104 + 1, z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 253 + 467*x_11449 + 283*x_11449^2 + 48*x_11449^3 + 467*x_2644 + 283*x_2644^2 + 48*x_2644^3 }-> s108 :|: s106 >= 0, s106 <= x_11449 + 1, s107 >= 0, s107 <= x_2644 + 1, s108 >= 0, s108 <= 1 + s106 + s107 + 1, x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s110 :|: s109 >= 0, s109 <= z' - 1 + 1, s110 >= 0, s110 <= 1 + s109 + 1, z' - 1 >= 0 encode_k(z') -{ 261 + 6*s111 + 2*s111*s112 + s111^2 + 4*s112 + s112^2 + 467*x_11452 + 283*x_11452^2 + 48*x_11452^3 + 467*x_2645 + 283*x_2645^2 + 48*x_2645^3 }-> s114 :|: s111 >= 0, s111 <= x_11452 + 1, s112 >= 0, s112 <= x_2645 + 1, s113 >= 0, s113 <= s111 + s112 + 1, s114 >= 0, s114 <= s113 + 1, x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ -104 + s115 + 45*z' + 139*z'^2 + 48*z'^3 }-> s117 :|: s115 >= 0, s115 <= z' - 1 + 1, s116 >= 0, s116 <= s115 + 1, s117 >= 0, s117 <= s116 + 1, z' - 1 >= 0 encode_k(z') -{ 379 + s119 + 467*x_11454 + 283*x_11454^2 + 48*x_11454^3 + 467*x_2646 + 283*x_2646^2 + 48*x_2646^3 + 467*x_3160 + 283*x_3160^2 + 48*x_3160^3 }-> s122 :|: s118 >= 0, s118 <= x_11454 + 1, s119 >= 0, s119 <= x_2646 + 1, s120 >= 0, s120 <= x_3160 + 1, s121 >= 0, s121 <= s118 + s119 + s120 + 1, s122 >= 0, s122 <= s121 + 1, x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s125 :|: s123 >= 0, s123 <= z' - 1 + 1, s124 >= 0, s124 <= s123 + 1, s125 >= 0, s125 <= s124 + 1, z' - 1 >= 0 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s186 :|: s184 >= 0, s184 <= z' - 1 + 1, s185 >= 0, s185 <= s184, s186 >= 0, s186 <= s185 + 1, z' - 1 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s72 :|: s72 >= 0, s72 <= z' + 1, z' >= 0 f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 10 + s9 + 6*x + x^2 }-> s10 :|: s9 >= 0, s9 <= x + 0 + 1, s10 >= 0, s10 <= s9 + 1, z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 11 + s5 + 6*x + 2*x*z'' + x^2 + 4*z'' + z''^2 }-> s6 :|: s5 >= 0, s5 <= x + (1 + 0 + (z'' - 1)) + 1, s6 >= 0, s6 <= s5 + 1, z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 23 + s7 + 10*x + 2*x*x'' + 2*x*y' + x^2 + 8*x'' + 2*x''*y' + x''^2 + 8*y' + y'^2 }-> s8 :|: s7 >= 0, s7 <= x + (1 + (1 + x'') + y') + 1, s8 >= 0, s8 <= s7 + 1, z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {encode_g}, {encode_f}, {encode_j}, {encode_i}, {encode_s} Previous analysis results are: h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1] encode_0: runtime: O(1) [0], size: O(1) [0] i: runtime: O(1) [1], size: O(n^1) [z'] k: runtime: O(1) [1], size: O(n^1) [1 + z'] g: runtime: O(n^1) [1 + z'], size: O(n^1) [1 + z'] f: runtime: O(n^2) [8 + 6*z' + 2*z'*z'' + z'^2 + 4*z'' + z''^2], size: O(n^1) [1 + z' + z''] encArg: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [1 + z'] encode_k: runtime: O(n^3) [1161 + 4403*z' + 3107*z'^2 + 624*z'^3], size: O(n^1) [4 + z'] encode_h: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [2 + z'] encode_h2: runtime: O(n^3) [379 + 467*z' + 283*z'^2 + 48*z'^3 + 468*z'' + 283*z''^2 + 48*z''^3 + 467*z1 + 283*z1^2 + 48*z1^3], size: O(n^1) [4 + z' + z'' + z1] encode_h1: runtime: O(n^3) [252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3], size: O(n^1) [3 + z' + z''] encode_g: runtime: ?, size: O(n^1) [4 + z'] ---------------------------------------- (95) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: encode_g after applying outer abstraction to obtain an ITS, resulting in: O(n^3) with polynomial bound: 1171 + 4416*z' + 3107*z'^2 + 624*z'^3 ---------------------------------------- (96) Obligation: Complexity RNTS consisting of the following rules: encArg(z') -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z' = 1 + 0 encArg(z') -{ -58 + s127 + -89*z' + -5*z'^2 + 48*z'^3 }-> s128 :|: s126 >= 0, s126 <= z' - 2 + 1, s127 >= 0, s127 <= s126, s128 >= 0, s128 <= s127 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11313 + 283*x_11313^2 + 48*x_11313^3 + 467*x_2583 + 283*x_2583^2 + 48*x_2583^3 }-> s131 :|: s129 >= 0, s129 <= x_11313 + 1, s130 >= 0, s130 <= x_2583 + 1, s131 >= 0, s131 <= 1 + s129 + s130, x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 253 + 467*x_11314 + 283*x_11314^2 + 48*x_11314^3 + 467*x_2584 + 283*x_2584^2 + 48*x_2584^3 }-> s134 :|: s132 >= 0, s132 <= x_11314 + 1, s133 >= 0, s133 <= x_2584 + 1, s134 >= 0, s134 <= 1 + s132 + s133, z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s136 :|: s135 >= 0, s135 <= z' - 2 + 1, s136 >= 0, s136 <= 1 + s135, z' - 2 >= 0 encArg(z') -{ 261 + 6*s137 + 2*s137*s138 + s137^2 + 4*s138 + s138^2 + 467*x_11317 + 283*x_11317^2 + 48*x_11317^3 + 467*x_2585 + 283*x_2585^2 + 48*x_2585^3 }-> s140 :|: s137 >= 0, s137 <= x_11317 + 1, s138 >= 0, s138 <= x_2585 + 1, s139 >= 0, s139 <= s137 + s138 + 1, s140 >= 0, s140 <= s139, x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ -58 + s141 + -89*z' + -5*z'^2 + 48*z'^3 }-> s143 :|: s141 >= 0, s141 <= z' - 2 + 1, s142 >= 0, s142 <= s141 + 1, s143 >= 0, s143 <= s142, z' - 2 >= 0 encArg(z') -{ 379 + s145 + 467*x_11319 + 283*x_11319^2 + 48*x_11319^3 + 467*x_2586 + 283*x_2586^2 + 48*x_2586^3 + 467*x_3145 + 283*x_3145^2 + 48*x_3145^3 }-> s148 :|: s144 >= 0, s144 <= x_11319 + 1, s145 >= 0, s145 <= x_2586 + 1, s146 >= 0, s146 <= x_3145 + 1, s147 >= 0, s147 <= s144 + s145 + s146 + 1, s148 >= 0, s148 <= s147, z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s151 :|: s149 >= 0, s149 <= z' - 2 + 1, s150 >= 0, s150 <= s149, s151 >= 0, s151 <= s150, z' - 2 >= 0 encArg(z') -{ 260 + 6*s14 + 2*s14*s15 + s14^2 + 4*s15 + s15^2 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> s16 :|: s14 >= 0, s14 <= x_1 + 1, s15 >= 0, s15 <= x_2 + 1, s16 >= 0, s16 <= s14 + s15 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s180 :|: s178 >= 0, s178 <= z' - 2 + 1, s179 >= 0, s179 <= s178 + 1, s180 >= 0, s180 <= s179, z' - 2 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s183 :|: s181 >= 0, s181 <= z' - 2 + 1, s182 >= 0, s182 <= s181, s183 >= 0, s183 <= s182 + 1, z' - 2 >= 0 encArg(z') -{ 254 + s17 + s18 + 467*x_1107 + 283*x_1107^2 + 48*x_1107^3 + 467*x_247 + 283*x_247^2 + 48*x_247^3 }-> s19 :|: s17 >= 0, s17 <= x_1107 + 1, s18 >= 0, s18 <= x_247 + 1, s19 >= 0, s19 <= 1 + s17 + s18 + 1, z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0 + 1, z' - 1 >= 0 encArg(z') -{ 254 + s20 + s21 + 467*x_1108 + 283*x_1108^2 + 48*x_1108^3 + 467*x_248 + 283*x_248^2 + 48*x_248^3 }-> s22 :|: s20 >= 0, s20 <= x_1108 + 1, s21 >= 0, s21 <= x_248 + 1, s22 >= 0, s22 <= 1 + s20 + s21 + 1, x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ -58 + s23 + -89*z' + -5*z'^2 + 48*z'^3 }-> s24 :|: s23 >= 0, s23 <= z' - 2 + 1, s24 >= 0, s24 <= 1 + s23 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s25 + 2*s25*s26 + s25^2 + 4*s26 + s26^2 + s27 + 467*x_1111 + 283*x_1111^2 + 48*x_1111^3 + 467*x_249 + 283*x_249^2 + 48*x_249^3 }-> s28 :|: s25 >= 0, s25 <= x_1111 + 1, s26 >= 0, s26 <= x_249 + 1, s27 >= 0, s27 <= s25 + s26 + 1, s28 >= 0, s28 <= s27 + 1, z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ -58 + s29 + s30 + -89*z' + -5*z'^2 + 48*z'^3 }-> s31 :|: s29 >= 0, s29 <= z' - 2 + 1, s30 >= 0, s30 <= s29 + 1, s31 >= 0, s31 <= s30 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s33 + s35 + 467*x_1113 + 283*x_1113^2 + 48*x_1113^3 + 467*x_250 + 283*x_250^2 + 48*x_250^3 + 467*x_311 + 283*x_311^2 + 48*x_311^3 }-> s36 :|: s32 >= 0, s32 <= x_1113 + 1, s33 >= 0, s33 <= x_250 + 1, s34 >= 0, s34 <= x_311 + 1, s35 >= 0, s35 <= s32 + s33 + s34 + 1, s36 >= 0, s36 <= s35 + 1, x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 378 + s38 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 + 467*x_3 + 283*x_3^2 + 48*x_3^3 }-> s40 :|: s37 >= 0, s37 <= x_1 + 1, s38 >= 0, s38 <= x_2 + 1, s39 >= 0, s39 <= x_3 + 1, s40 >= 0, s40 <= s37 + s38 + s39 + 1, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z') -{ -58 + s75 + -89*z' + -5*z'^2 + 48*z'^3 }-> s76 :|: s74 >= 0, s74 <= z' - 2 + 1, s75 >= 0, s75 <= s74 + 1, s76 >= 0, s76 <= s75 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11322 + 283*x_11322^2 + 48*x_11322^3 + 467*x_2587 + 283*x_2587^2 + 48*x_2587^3 }-> s79 :|: s77 >= 0, s77 <= x_11322 + 1, s78 >= 0, s78 <= x_2587 + 1, s79 >= 0, s79 <= 1 + s77 + s78 + 1, x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 253 + 467*x_11323 + 283*x_11323^2 + 48*x_11323^3 + 467*x_2588 + 283*x_2588^2 + 48*x_2588^3 }-> s82 :|: s80 >= 0, s80 <= x_11323 + 1, s81 >= 0, s81 <= x_2588 + 1, s82 >= 0, s82 <= 1 + s80 + s81 + 1, z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s84 :|: s83 >= 0, s83 <= z' - 2 + 1, s84 >= 0, s84 <= 1 + s83 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s85 + 2*s85*s86 + s85^2 + 4*s86 + s86^2 + 467*x_11326 + 283*x_11326^2 + 48*x_11326^3 + 467*x_2589 + 283*x_2589^2 + 48*x_2589^3 }-> s88 :|: s85 >= 0, s85 <= x_11326 + 1, s86 >= 0, s86 <= x_2589 + 1, s87 >= 0, s87 <= s85 + s86 + 1, s88 >= 0, s88 <= s87 + 1, x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ -58 + s89 + -89*z' + -5*z'^2 + 48*z'^3 }-> s91 :|: s89 >= 0, s89 <= z' - 2 + 1, s90 >= 0, s90 <= s89 + 1, s91 >= 0, s91 <= s90 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s93 + 467*x_11328 + 283*x_11328^2 + 48*x_11328^3 + 467*x_2590 + 283*x_2590^2 + 48*x_2590^3 + 467*x_3146 + 283*x_3146^2 + 48*x_3146^3 }-> s96 :|: s92 >= 0, s92 <= x_11328 + 1, s93 >= 0, s93 <= x_2590 + 1, s94 >= 0, s94 <= x_3146 + 1, s95 >= 0, s95 <= s92 + s93 + s94 + 1, s96 >= 0, s96 <= s95 + 1, z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s99 :|: s97 >= 0, s97 <= z' - 2 + 1, s98 >= 0, s98 <= s97 + 1, s99 >= 0, s99 <= s98 + 1, z' - 2 >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ -106 + 45*z' + 139*z'^2 + 48*z'^3 }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0 encArg(z') -{ 252 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> 1 + s11 + s12 :|: s11 >= 0, s11 <= x_1 + 1, s12 >= 0, s12 <= x_2 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 260 + 6*s41 + 2*s41*s42 + s41^2 + 4*s42 + s42^2 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> s43 :|: s41 >= 0, s41 <= z' + 1, s42 >= 0, s42 <= z'' + 1, s43 >= 0, s43 <= s41 + s42 + 1, z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ -104 + s101 + 45*z' + 139*z'^2 + 48*z'^3 }-> s102 :|: s100 >= 0, s100 <= z' - 1 + 1, s101 >= 0, s101 <= s100 + 1, s102 >= 0, s102 <= s101 + 1, z' - 1 >= 0 encode_g(z') -{ -104 + s153 + 45*z' + 139*z'^2 + 48*z'^3 }-> s154 :|: s152 >= 0, s152 <= z' - 1 + 1, s153 >= 0, s153 <= s152, s154 >= 0, s154 <= s153 + 1, z' - 1 >= 0 encode_g(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0 + 1, z' = 0 encode_g(z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 1, z' >= 0 encode_g(z') -{ 254 + s46 + s47 + 467*x_11439 + 283*x_11439^2 + 48*x_11439^3 + 467*x_2639 + 283*x_2639^2 + 48*x_2639^3 }-> s48 :|: s46 >= 0, s46 <= x_11439 + 1, s47 >= 0, s47 <= x_2639 + 1, s48 >= 0, s48 <= 1 + s46 + s47 + 1, x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 254 + s49 + s50 + 467*x_11440 + 283*x_11440^2 + 48*x_11440^3 + 467*x_2640 + 283*x_2640^2 + 48*x_2640^3 }-> s51 :|: s49 >= 0, s49 <= x_11440 + 1, s50 >= 0, s50 <= x_2640 + 1, s51 >= 0, s51 <= 1 + s49 + s50 + 1, x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ -104 + s52 + 45*z' + 139*z'^2 + 48*z'^3 }-> s53 :|: s52 >= 0, s52 <= z' - 1 + 1, s53 >= 0, s53 <= 1 + s52 + 1, z' - 1 >= 0 encode_g(z') -{ 261 + 6*s54 + 2*s54*s55 + s54^2 + 4*s55 + s55^2 + s56 + 467*x_11443 + 283*x_11443^2 + 48*x_11443^3 + 467*x_2641 + 283*x_2641^2 + 48*x_2641^3 }-> s57 :|: s54 >= 0, s54 <= x_11443 + 1, s55 >= 0, s55 <= x_2641 + 1, s56 >= 0, s56 <= s54 + s55 + 1, s57 >= 0, s57 <= s56 + 1, x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ -104 + s58 + s59 + 45*z' + 139*z'^2 + 48*z'^3 }-> s60 :|: s58 >= 0, s58 <= z' - 1 + 1, s59 >= 0, s59 <= s58 + 1, s60 >= 0, s60 <= s59 + 1, z' - 1 >= 0 encode_g(z') -{ 379 + s62 + s64 + 467*x_11445 + 283*x_11445^2 + 48*x_11445^3 + 467*x_2642 + 283*x_2642^2 + 48*x_2642^3 + 467*x_3159 + 283*x_3159^2 + 48*x_3159^3 }-> s65 :|: s61 >= 0, s61 <= x_11445 + 1, s62 >= 0, s62 <= x_2642 + 1, s63 >= 0, s63 <= x_3159 + 1, s64 >= 0, s64 <= s61 + s62 + s63 + 1, s65 >= 0, s65 <= s64 + 1, x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s73 :|: s73 >= 0, s73 <= z' + 1, z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s66 + s67 :|: s66 >= 0, s66 <= z' + 1, s67 >= 0, s67 <= z'' + 1, z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 378 + s69 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 + 467*z1 + 283*z1^2 + 48*z1^3 }-> s71 :|: s68 >= 0, s68 <= z' + 1, s69 >= 0, s69 <= z'' + 1, s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s68 + s69 + s70 + 1, z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 253 + 467*x_12654 + 283*x_12654^2 + 48*x_12654^3 + 467*x_21179 + 283*x_21179^2 + 48*x_21179^3 }-> s157 :|: s155 >= 0, s155 <= x_12654 + 1, s156 >= 0, s156 <= x_21179 + 1, s157 >= 0, s157 <= 1 + s155 + s156, x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 253 + 467*x_12655 + 283*x_12655^2 + 48*x_12655^3 + 467*x_21180 + 283*x_21180^2 + 48*x_21180^3 }-> s160 :|: s158 >= 0, s158 <= x_12655 + 1, s159 >= 0, s159 <= x_21180 + 1, s160 >= 0, s160 <= 1 + s158 + s159, z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s162 :|: s161 >= 0, s161 <= z' - 1 + 1, s162 >= 0, s162 <= 1 + s161, z' - 1 >= 0 encode_i(z') -{ 261 + 6*s163 + 2*s163*s164 + s163^2 + 4*s164 + s164^2 + 467*x_12658 + 283*x_12658^2 + 48*x_12658^3 + 467*x_21181 + 283*x_21181^2 + 48*x_21181^3 }-> s166 :|: s163 >= 0, s163 <= x_12658 + 1, s164 >= 0, s164 <= x_21181 + 1, s165 >= 0, s165 <= s163 + s164 + 1, s166 >= 0, s166 <= s165, x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ -104 + s167 + 45*z' + 139*z'^2 + 48*z'^3 }-> s169 :|: s167 >= 0, s167 <= z' - 1 + 1, s168 >= 0, s168 <= s167 + 1, s169 >= 0, s169 <= s168, z' - 1 >= 0 encode_i(z') -{ 379 + s171 + 467*x_12660 + 283*x_12660^2 + 48*x_12660^3 + 467*x_21182 + 283*x_21182^2 + 48*x_21182^3 + 467*x_3294 + 283*x_3294^2 + 48*x_3294^3 }-> s174 :|: s170 >= 0, s170 <= x_12660 + 1, s171 >= 0, s171 <= x_21182 + 1, s172 >= 0, s172 <= x_3294 + 1, s173 >= 0, s173 <= s170 + s171 + s172 + 1, s174 >= 0, s174 <= s173, z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s177 :|: s175 >= 0, s175 <= z' - 1 + 1, s176 >= 0, s176 <= s175, s177 >= 0, s177 <= s176, z' - 1 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s189 :|: s187 >= 0, s187 <= z' - 1 + 1, s188 >= 0, s188 <= s187 + 1, s189 >= 0, s189 <= s188, z' - 1 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s44 + s45 :|: s44 >= 0, s44 <= z' + 1, s45 >= 0, s45 <= z'' + 1, z' >= 0, z'' >= 0 encode_k(z') -{ 253 + 467*x_11448 + 283*x_11448^2 + 48*x_11448^3 + 467*x_2643 + 283*x_2643^2 + 48*x_2643^3 }-> s105 :|: s103 >= 0, s103 <= x_11448 + 1, s104 >= 0, s104 <= x_2643 + 1, s105 >= 0, s105 <= 1 + s103 + s104 + 1, z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 253 + 467*x_11449 + 283*x_11449^2 + 48*x_11449^3 + 467*x_2644 + 283*x_2644^2 + 48*x_2644^3 }-> s108 :|: s106 >= 0, s106 <= x_11449 + 1, s107 >= 0, s107 <= x_2644 + 1, s108 >= 0, s108 <= 1 + s106 + s107 + 1, x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s110 :|: s109 >= 0, s109 <= z' - 1 + 1, s110 >= 0, s110 <= 1 + s109 + 1, z' - 1 >= 0 encode_k(z') -{ 261 + 6*s111 + 2*s111*s112 + s111^2 + 4*s112 + s112^2 + 467*x_11452 + 283*x_11452^2 + 48*x_11452^3 + 467*x_2645 + 283*x_2645^2 + 48*x_2645^3 }-> s114 :|: s111 >= 0, s111 <= x_11452 + 1, s112 >= 0, s112 <= x_2645 + 1, s113 >= 0, s113 <= s111 + s112 + 1, s114 >= 0, s114 <= s113 + 1, x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ -104 + s115 + 45*z' + 139*z'^2 + 48*z'^3 }-> s117 :|: s115 >= 0, s115 <= z' - 1 + 1, s116 >= 0, s116 <= s115 + 1, s117 >= 0, s117 <= s116 + 1, z' - 1 >= 0 encode_k(z') -{ 379 + s119 + 467*x_11454 + 283*x_11454^2 + 48*x_11454^3 + 467*x_2646 + 283*x_2646^2 + 48*x_2646^3 + 467*x_3160 + 283*x_3160^2 + 48*x_3160^3 }-> s122 :|: s118 >= 0, s118 <= x_11454 + 1, s119 >= 0, s119 <= x_2646 + 1, s120 >= 0, s120 <= x_3160 + 1, s121 >= 0, s121 <= s118 + s119 + s120 + 1, s122 >= 0, s122 <= s121 + 1, x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s125 :|: s123 >= 0, s123 <= z' - 1 + 1, s124 >= 0, s124 <= s123 + 1, s125 >= 0, s125 <= s124 + 1, z' - 1 >= 0 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s186 :|: s184 >= 0, s184 <= z' - 1 + 1, s185 >= 0, s185 <= s184, s186 >= 0, s186 <= s185 + 1, z' - 1 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s72 :|: s72 >= 0, s72 <= z' + 1, z' >= 0 f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 10 + s9 + 6*x + x^2 }-> s10 :|: s9 >= 0, s9 <= x + 0 + 1, s10 >= 0, s10 <= s9 + 1, z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 11 + s5 + 6*x + 2*x*z'' + x^2 + 4*z'' + z''^2 }-> s6 :|: s5 >= 0, s5 <= x + (1 + 0 + (z'' - 1)) + 1, s6 >= 0, s6 <= s5 + 1, z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 23 + s7 + 10*x + 2*x*x'' + 2*x*y' + x^2 + 8*x'' + 2*x''*y' + x''^2 + 8*y' + y'^2 }-> s8 :|: s7 >= 0, s7 <= x + (1 + (1 + x'') + y') + 1, s8 >= 0, s8 <= s7 + 1, z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {encode_f}, {encode_j}, {encode_i}, {encode_s} Previous analysis results are: h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1] encode_0: runtime: O(1) [0], size: O(1) [0] i: runtime: O(1) [1], size: O(n^1) [z'] k: runtime: O(1) [1], size: O(n^1) [1 + z'] g: runtime: O(n^1) [1 + z'], size: O(n^1) [1 + z'] f: runtime: O(n^2) [8 + 6*z' + 2*z'*z'' + z'^2 + 4*z'' + z''^2], size: O(n^1) [1 + z' + z''] encArg: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [1 + z'] encode_k: runtime: O(n^3) [1161 + 4403*z' + 3107*z'^2 + 624*z'^3], size: O(n^1) [4 + z'] encode_h: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [2 + z'] encode_h2: runtime: O(n^3) [379 + 467*z' + 283*z'^2 + 48*z'^3 + 468*z'' + 283*z''^2 + 48*z''^3 + 467*z1 + 283*z1^2 + 48*z1^3], size: O(n^1) [4 + z' + z'' + z1] encode_h1: runtime: O(n^3) [252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3], size: O(n^1) [3 + z' + z''] encode_g: runtime: O(n^3) [1171 + 4416*z' + 3107*z'^2 + 624*z'^3], size: O(n^1) [4 + z'] ---------------------------------------- (97) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (98) Obligation: Complexity RNTS consisting of the following rules: encArg(z') -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z' = 1 + 0 encArg(z') -{ -58 + s127 + -89*z' + -5*z'^2 + 48*z'^3 }-> s128 :|: s126 >= 0, s126 <= z' - 2 + 1, s127 >= 0, s127 <= s126, s128 >= 0, s128 <= s127 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11313 + 283*x_11313^2 + 48*x_11313^3 + 467*x_2583 + 283*x_2583^2 + 48*x_2583^3 }-> s131 :|: s129 >= 0, s129 <= x_11313 + 1, s130 >= 0, s130 <= x_2583 + 1, s131 >= 0, s131 <= 1 + s129 + s130, x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 253 + 467*x_11314 + 283*x_11314^2 + 48*x_11314^3 + 467*x_2584 + 283*x_2584^2 + 48*x_2584^3 }-> s134 :|: s132 >= 0, s132 <= x_11314 + 1, s133 >= 0, s133 <= x_2584 + 1, s134 >= 0, s134 <= 1 + s132 + s133, z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s136 :|: s135 >= 0, s135 <= z' - 2 + 1, s136 >= 0, s136 <= 1 + s135, z' - 2 >= 0 encArg(z') -{ 261 + 6*s137 + 2*s137*s138 + s137^2 + 4*s138 + s138^2 + 467*x_11317 + 283*x_11317^2 + 48*x_11317^3 + 467*x_2585 + 283*x_2585^2 + 48*x_2585^3 }-> s140 :|: s137 >= 0, s137 <= x_11317 + 1, s138 >= 0, s138 <= x_2585 + 1, s139 >= 0, s139 <= s137 + s138 + 1, s140 >= 0, s140 <= s139, x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ -58 + s141 + -89*z' + -5*z'^2 + 48*z'^3 }-> s143 :|: s141 >= 0, s141 <= z' - 2 + 1, s142 >= 0, s142 <= s141 + 1, s143 >= 0, s143 <= s142, z' - 2 >= 0 encArg(z') -{ 379 + s145 + 467*x_11319 + 283*x_11319^2 + 48*x_11319^3 + 467*x_2586 + 283*x_2586^2 + 48*x_2586^3 + 467*x_3145 + 283*x_3145^2 + 48*x_3145^3 }-> s148 :|: s144 >= 0, s144 <= x_11319 + 1, s145 >= 0, s145 <= x_2586 + 1, s146 >= 0, s146 <= x_3145 + 1, s147 >= 0, s147 <= s144 + s145 + s146 + 1, s148 >= 0, s148 <= s147, z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s151 :|: s149 >= 0, s149 <= z' - 2 + 1, s150 >= 0, s150 <= s149, s151 >= 0, s151 <= s150, z' - 2 >= 0 encArg(z') -{ 260 + 6*s14 + 2*s14*s15 + s14^2 + 4*s15 + s15^2 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> s16 :|: s14 >= 0, s14 <= x_1 + 1, s15 >= 0, s15 <= x_2 + 1, s16 >= 0, s16 <= s14 + s15 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s180 :|: s178 >= 0, s178 <= z' - 2 + 1, s179 >= 0, s179 <= s178 + 1, s180 >= 0, s180 <= s179, z' - 2 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s183 :|: s181 >= 0, s181 <= z' - 2 + 1, s182 >= 0, s182 <= s181, s183 >= 0, s183 <= s182 + 1, z' - 2 >= 0 encArg(z') -{ 254 + s17 + s18 + 467*x_1107 + 283*x_1107^2 + 48*x_1107^3 + 467*x_247 + 283*x_247^2 + 48*x_247^3 }-> s19 :|: s17 >= 0, s17 <= x_1107 + 1, s18 >= 0, s18 <= x_247 + 1, s19 >= 0, s19 <= 1 + s17 + s18 + 1, z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0 + 1, z' - 1 >= 0 encArg(z') -{ 254 + s20 + s21 + 467*x_1108 + 283*x_1108^2 + 48*x_1108^3 + 467*x_248 + 283*x_248^2 + 48*x_248^3 }-> s22 :|: s20 >= 0, s20 <= x_1108 + 1, s21 >= 0, s21 <= x_248 + 1, s22 >= 0, s22 <= 1 + s20 + s21 + 1, x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ -58 + s23 + -89*z' + -5*z'^2 + 48*z'^3 }-> s24 :|: s23 >= 0, s23 <= z' - 2 + 1, s24 >= 0, s24 <= 1 + s23 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s25 + 2*s25*s26 + s25^2 + 4*s26 + s26^2 + s27 + 467*x_1111 + 283*x_1111^2 + 48*x_1111^3 + 467*x_249 + 283*x_249^2 + 48*x_249^3 }-> s28 :|: s25 >= 0, s25 <= x_1111 + 1, s26 >= 0, s26 <= x_249 + 1, s27 >= 0, s27 <= s25 + s26 + 1, s28 >= 0, s28 <= s27 + 1, z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ -58 + s29 + s30 + -89*z' + -5*z'^2 + 48*z'^3 }-> s31 :|: s29 >= 0, s29 <= z' - 2 + 1, s30 >= 0, s30 <= s29 + 1, s31 >= 0, s31 <= s30 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s33 + s35 + 467*x_1113 + 283*x_1113^2 + 48*x_1113^3 + 467*x_250 + 283*x_250^2 + 48*x_250^3 + 467*x_311 + 283*x_311^2 + 48*x_311^3 }-> s36 :|: s32 >= 0, s32 <= x_1113 + 1, s33 >= 0, s33 <= x_250 + 1, s34 >= 0, s34 <= x_311 + 1, s35 >= 0, s35 <= s32 + s33 + s34 + 1, s36 >= 0, s36 <= s35 + 1, x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 378 + s38 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 + 467*x_3 + 283*x_3^2 + 48*x_3^3 }-> s40 :|: s37 >= 0, s37 <= x_1 + 1, s38 >= 0, s38 <= x_2 + 1, s39 >= 0, s39 <= x_3 + 1, s40 >= 0, s40 <= s37 + s38 + s39 + 1, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z') -{ -58 + s75 + -89*z' + -5*z'^2 + 48*z'^3 }-> s76 :|: s74 >= 0, s74 <= z' - 2 + 1, s75 >= 0, s75 <= s74 + 1, s76 >= 0, s76 <= s75 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11322 + 283*x_11322^2 + 48*x_11322^3 + 467*x_2587 + 283*x_2587^2 + 48*x_2587^3 }-> s79 :|: s77 >= 0, s77 <= x_11322 + 1, s78 >= 0, s78 <= x_2587 + 1, s79 >= 0, s79 <= 1 + s77 + s78 + 1, x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 253 + 467*x_11323 + 283*x_11323^2 + 48*x_11323^3 + 467*x_2588 + 283*x_2588^2 + 48*x_2588^3 }-> s82 :|: s80 >= 0, s80 <= x_11323 + 1, s81 >= 0, s81 <= x_2588 + 1, s82 >= 0, s82 <= 1 + s80 + s81 + 1, z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s84 :|: s83 >= 0, s83 <= z' - 2 + 1, s84 >= 0, s84 <= 1 + s83 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s85 + 2*s85*s86 + s85^2 + 4*s86 + s86^2 + 467*x_11326 + 283*x_11326^2 + 48*x_11326^3 + 467*x_2589 + 283*x_2589^2 + 48*x_2589^3 }-> s88 :|: s85 >= 0, s85 <= x_11326 + 1, s86 >= 0, s86 <= x_2589 + 1, s87 >= 0, s87 <= s85 + s86 + 1, s88 >= 0, s88 <= s87 + 1, x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ -58 + s89 + -89*z' + -5*z'^2 + 48*z'^3 }-> s91 :|: s89 >= 0, s89 <= z' - 2 + 1, s90 >= 0, s90 <= s89 + 1, s91 >= 0, s91 <= s90 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s93 + 467*x_11328 + 283*x_11328^2 + 48*x_11328^3 + 467*x_2590 + 283*x_2590^2 + 48*x_2590^3 + 467*x_3146 + 283*x_3146^2 + 48*x_3146^3 }-> s96 :|: s92 >= 0, s92 <= x_11328 + 1, s93 >= 0, s93 <= x_2590 + 1, s94 >= 0, s94 <= x_3146 + 1, s95 >= 0, s95 <= s92 + s93 + s94 + 1, s96 >= 0, s96 <= s95 + 1, z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s99 :|: s97 >= 0, s97 <= z' - 2 + 1, s98 >= 0, s98 <= s97 + 1, s99 >= 0, s99 <= s98 + 1, z' - 2 >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ -106 + 45*z' + 139*z'^2 + 48*z'^3 }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0 encArg(z') -{ 252 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> 1 + s11 + s12 :|: s11 >= 0, s11 <= x_1 + 1, s12 >= 0, s12 <= x_2 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 260 + 6*s41 + 2*s41*s42 + s41^2 + 4*s42 + s42^2 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> s43 :|: s41 >= 0, s41 <= z' + 1, s42 >= 0, s42 <= z'' + 1, s43 >= 0, s43 <= s41 + s42 + 1, z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ -104 + s101 + 45*z' + 139*z'^2 + 48*z'^3 }-> s102 :|: s100 >= 0, s100 <= z' - 1 + 1, s101 >= 0, s101 <= s100 + 1, s102 >= 0, s102 <= s101 + 1, z' - 1 >= 0 encode_g(z') -{ -104 + s153 + 45*z' + 139*z'^2 + 48*z'^3 }-> s154 :|: s152 >= 0, s152 <= z' - 1 + 1, s153 >= 0, s153 <= s152, s154 >= 0, s154 <= s153 + 1, z' - 1 >= 0 encode_g(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0 + 1, z' = 0 encode_g(z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 1, z' >= 0 encode_g(z') -{ 254 + s46 + s47 + 467*x_11439 + 283*x_11439^2 + 48*x_11439^3 + 467*x_2639 + 283*x_2639^2 + 48*x_2639^3 }-> s48 :|: s46 >= 0, s46 <= x_11439 + 1, s47 >= 0, s47 <= x_2639 + 1, s48 >= 0, s48 <= 1 + s46 + s47 + 1, x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 254 + s49 + s50 + 467*x_11440 + 283*x_11440^2 + 48*x_11440^3 + 467*x_2640 + 283*x_2640^2 + 48*x_2640^3 }-> s51 :|: s49 >= 0, s49 <= x_11440 + 1, s50 >= 0, s50 <= x_2640 + 1, s51 >= 0, s51 <= 1 + s49 + s50 + 1, x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ -104 + s52 + 45*z' + 139*z'^2 + 48*z'^3 }-> s53 :|: s52 >= 0, s52 <= z' - 1 + 1, s53 >= 0, s53 <= 1 + s52 + 1, z' - 1 >= 0 encode_g(z') -{ 261 + 6*s54 + 2*s54*s55 + s54^2 + 4*s55 + s55^2 + s56 + 467*x_11443 + 283*x_11443^2 + 48*x_11443^3 + 467*x_2641 + 283*x_2641^2 + 48*x_2641^3 }-> s57 :|: s54 >= 0, s54 <= x_11443 + 1, s55 >= 0, s55 <= x_2641 + 1, s56 >= 0, s56 <= s54 + s55 + 1, s57 >= 0, s57 <= s56 + 1, x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ -104 + s58 + s59 + 45*z' + 139*z'^2 + 48*z'^3 }-> s60 :|: s58 >= 0, s58 <= z' - 1 + 1, s59 >= 0, s59 <= s58 + 1, s60 >= 0, s60 <= s59 + 1, z' - 1 >= 0 encode_g(z') -{ 379 + s62 + s64 + 467*x_11445 + 283*x_11445^2 + 48*x_11445^3 + 467*x_2642 + 283*x_2642^2 + 48*x_2642^3 + 467*x_3159 + 283*x_3159^2 + 48*x_3159^3 }-> s65 :|: s61 >= 0, s61 <= x_11445 + 1, s62 >= 0, s62 <= x_2642 + 1, s63 >= 0, s63 <= x_3159 + 1, s64 >= 0, s64 <= s61 + s62 + s63 + 1, s65 >= 0, s65 <= s64 + 1, x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s73 :|: s73 >= 0, s73 <= z' + 1, z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s66 + s67 :|: s66 >= 0, s66 <= z' + 1, s67 >= 0, s67 <= z'' + 1, z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 378 + s69 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 + 467*z1 + 283*z1^2 + 48*z1^3 }-> s71 :|: s68 >= 0, s68 <= z' + 1, s69 >= 0, s69 <= z'' + 1, s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s68 + s69 + s70 + 1, z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 253 + 467*x_12654 + 283*x_12654^2 + 48*x_12654^3 + 467*x_21179 + 283*x_21179^2 + 48*x_21179^3 }-> s157 :|: s155 >= 0, s155 <= x_12654 + 1, s156 >= 0, s156 <= x_21179 + 1, s157 >= 0, s157 <= 1 + s155 + s156, x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 253 + 467*x_12655 + 283*x_12655^2 + 48*x_12655^3 + 467*x_21180 + 283*x_21180^2 + 48*x_21180^3 }-> s160 :|: s158 >= 0, s158 <= x_12655 + 1, s159 >= 0, s159 <= x_21180 + 1, s160 >= 0, s160 <= 1 + s158 + s159, z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s162 :|: s161 >= 0, s161 <= z' - 1 + 1, s162 >= 0, s162 <= 1 + s161, z' - 1 >= 0 encode_i(z') -{ 261 + 6*s163 + 2*s163*s164 + s163^2 + 4*s164 + s164^2 + 467*x_12658 + 283*x_12658^2 + 48*x_12658^3 + 467*x_21181 + 283*x_21181^2 + 48*x_21181^3 }-> s166 :|: s163 >= 0, s163 <= x_12658 + 1, s164 >= 0, s164 <= x_21181 + 1, s165 >= 0, s165 <= s163 + s164 + 1, s166 >= 0, s166 <= s165, x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ -104 + s167 + 45*z' + 139*z'^2 + 48*z'^3 }-> s169 :|: s167 >= 0, s167 <= z' - 1 + 1, s168 >= 0, s168 <= s167 + 1, s169 >= 0, s169 <= s168, z' - 1 >= 0 encode_i(z') -{ 379 + s171 + 467*x_12660 + 283*x_12660^2 + 48*x_12660^3 + 467*x_21182 + 283*x_21182^2 + 48*x_21182^3 + 467*x_3294 + 283*x_3294^2 + 48*x_3294^3 }-> s174 :|: s170 >= 0, s170 <= x_12660 + 1, s171 >= 0, s171 <= x_21182 + 1, s172 >= 0, s172 <= x_3294 + 1, s173 >= 0, s173 <= s170 + s171 + s172 + 1, s174 >= 0, s174 <= s173, z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s177 :|: s175 >= 0, s175 <= z' - 1 + 1, s176 >= 0, s176 <= s175, s177 >= 0, s177 <= s176, z' - 1 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s189 :|: s187 >= 0, s187 <= z' - 1 + 1, s188 >= 0, s188 <= s187 + 1, s189 >= 0, s189 <= s188, z' - 1 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s44 + s45 :|: s44 >= 0, s44 <= z' + 1, s45 >= 0, s45 <= z'' + 1, z' >= 0, z'' >= 0 encode_k(z') -{ 253 + 467*x_11448 + 283*x_11448^2 + 48*x_11448^3 + 467*x_2643 + 283*x_2643^2 + 48*x_2643^3 }-> s105 :|: s103 >= 0, s103 <= x_11448 + 1, s104 >= 0, s104 <= x_2643 + 1, s105 >= 0, s105 <= 1 + s103 + s104 + 1, z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 253 + 467*x_11449 + 283*x_11449^2 + 48*x_11449^3 + 467*x_2644 + 283*x_2644^2 + 48*x_2644^3 }-> s108 :|: s106 >= 0, s106 <= x_11449 + 1, s107 >= 0, s107 <= x_2644 + 1, s108 >= 0, s108 <= 1 + s106 + s107 + 1, x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s110 :|: s109 >= 0, s109 <= z' - 1 + 1, s110 >= 0, s110 <= 1 + s109 + 1, z' - 1 >= 0 encode_k(z') -{ 261 + 6*s111 + 2*s111*s112 + s111^2 + 4*s112 + s112^2 + 467*x_11452 + 283*x_11452^2 + 48*x_11452^3 + 467*x_2645 + 283*x_2645^2 + 48*x_2645^3 }-> s114 :|: s111 >= 0, s111 <= x_11452 + 1, s112 >= 0, s112 <= x_2645 + 1, s113 >= 0, s113 <= s111 + s112 + 1, s114 >= 0, s114 <= s113 + 1, x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ -104 + s115 + 45*z' + 139*z'^2 + 48*z'^3 }-> s117 :|: s115 >= 0, s115 <= z' - 1 + 1, s116 >= 0, s116 <= s115 + 1, s117 >= 0, s117 <= s116 + 1, z' - 1 >= 0 encode_k(z') -{ 379 + s119 + 467*x_11454 + 283*x_11454^2 + 48*x_11454^3 + 467*x_2646 + 283*x_2646^2 + 48*x_2646^3 + 467*x_3160 + 283*x_3160^2 + 48*x_3160^3 }-> s122 :|: s118 >= 0, s118 <= x_11454 + 1, s119 >= 0, s119 <= x_2646 + 1, s120 >= 0, s120 <= x_3160 + 1, s121 >= 0, s121 <= s118 + s119 + s120 + 1, s122 >= 0, s122 <= s121 + 1, x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s125 :|: s123 >= 0, s123 <= z' - 1 + 1, s124 >= 0, s124 <= s123 + 1, s125 >= 0, s125 <= s124 + 1, z' - 1 >= 0 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s186 :|: s184 >= 0, s184 <= z' - 1 + 1, s185 >= 0, s185 <= s184, s186 >= 0, s186 <= s185 + 1, z' - 1 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s72 :|: s72 >= 0, s72 <= z' + 1, z' >= 0 f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 10 + s9 + 6*x + x^2 }-> s10 :|: s9 >= 0, s9 <= x + 0 + 1, s10 >= 0, s10 <= s9 + 1, z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 11 + s5 + 6*x + 2*x*z'' + x^2 + 4*z'' + z''^2 }-> s6 :|: s5 >= 0, s5 <= x + (1 + 0 + (z'' - 1)) + 1, s6 >= 0, s6 <= s5 + 1, z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 23 + s7 + 10*x + 2*x*x'' + 2*x*y' + x^2 + 8*x'' + 2*x''*y' + x''^2 + 8*y' + y'^2 }-> s8 :|: s7 >= 0, s7 <= x + (1 + (1 + x'') + y') + 1, s8 >= 0, s8 <= s7 + 1, z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {encode_f}, {encode_j}, {encode_i}, {encode_s} Previous analysis results are: h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1] encode_0: runtime: O(1) [0], size: O(1) [0] i: runtime: O(1) [1], size: O(n^1) [z'] k: runtime: O(1) [1], size: O(n^1) [1 + z'] g: runtime: O(n^1) [1 + z'], size: O(n^1) [1 + z'] f: runtime: O(n^2) [8 + 6*z' + 2*z'*z'' + z'^2 + 4*z'' + z''^2], size: O(n^1) [1 + z' + z''] encArg: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [1 + z'] encode_k: runtime: O(n^3) [1161 + 4403*z' + 3107*z'^2 + 624*z'^3], size: O(n^1) [4 + z'] encode_h: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [2 + z'] encode_h2: runtime: O(n^3) [379 + 467*z' + 283*z'^2 + 48*z'^3 + 468*z'' + 283*z''^2 + 48*z''^3 + 467*z1 + 283*z1^2 + 48*z1^3], size: O(n^1) [4 + z' + z'' + z1] encode_h1: runtime: O(n^3) [252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3], size: O(n^1) [3 + z' + z''] encode_g: runtime: O(n^3) [1171 + 4416*z' + 3107*z'^2 + 624*z'^3], size: O(n^1) [4 + z'] ---------------------------------------- (99) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: encode_f after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 3 + z' + z'' ---------------------------------------- (100) Obligation: Complexity RNTS consisting of the following rules: encArg(z') -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z' = 1 + 0 encArg(z') -{ -58 + s127 + -89*z' + -5*z'^2 + 48*z'^3 }-> s128 :|: s126 >= 0, s126 <= z' - 2 + 1, s127 >= 0, s127 <= s126, s128 >= 0, s128 <= s127 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11313 + 283*x_11313^2 + 48*x_11313^3 + 467*x_2583 + 283*x_2583^2 + 48*x_2583^3 }-> s131 :|: s129 >= 0, s129 <= x_11313 + 1, s130 >= 0, s130 <= x_2583 + 1, s131 >= 0, s131 <= 1 + s129 + s130, x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 253 + 467*x_11314 + 283*x_11314^2 + 48*x_11314^3 + 467*x_2584 + 283*x_2584^2 + 48*x_2584^3 }-> s134 :|: s132 >= 0, s132 <= x_11314 + 1, s133 >= 0, s133 <= x_2584 + 1, s134 >= 0, s134 <= 1 + s132 + s133, z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s136 :|: s135 >= 0, s135 <= z' - 2 + 1, s136 >= 0, s136 <= 1 + s135, z' - 2 >= 0 encArg(z') -{ 261 + 6*s137 + 2*s137*s138 + s137^2 + 4*s138 + s138^2 + 467*x_11317 + 283*x_11317^2 + 48*x_11317^3 + 467*x_2585 + 283*x_2585^2 + 48*x_2585^3 }-> s140 :|: s137 >= 0, s137 <= x_11317 + 1, s138 >= 0, s138 <= x_2585 + 1, s139 >= 0, s139 <= s137 + s138 + 1, s140 >= 0, s140 <= s139, x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ -58 + s141 + -89*z' + -5*z'^2 + 48*z'^3 }-> s143 :|: s141 >= 0, s141 <= z' - 2 + 1, s142 >= 0, s142 <= s141 + 1, s143 >= 0, s143 <= s142, z' - 2 >= 0 encArg(z') -{ 379 + s145 + 467*x_11319 + 283*x_11319^2 + 48*x_11319^3 + 467*x_2586 + 283*x_2586^2 + 48*x_2586^3 + 467*x_3145 + 283*x_3145^2 + 48*x_3145^3 }-> s148 :|: s144 >= 0, s144 <= x_11319 + 1, s145 >= 0, s145 <= x_2586 + 1, s146 >= 0, s146 <= x_3145 + 1, s147 >= 0, s147 <= s144 + s145 + s146 + 1, s148 >= 0, s148 <= s147, z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s151 :|: s149 >= 0, s149 <= z' - 2 + 1, s150 >= 0, s150 <= s149, s151 >= 0, s151 <= s150, z' - 2 >= 0 encArg(z') -{ 260 + 6*s14 + 2*s14*s15 + s14^2 + 4*s15 + s15^2 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> s16 :|: s14 >= 0, s14 <= x_1 + 1, s15 >= 0, s15 <= x_2 + 1, s16 >= 0, s16 <= s14 + s15 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s180 :|: s178 >= 0, s178 <= z' - 2 + 1, s179 >= 0, s179 <= s178 + 1, s180 >= 0, s180 <= s179, z' - 2 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s183 :|: s181 >= 0, s181 <= z' - 2 + 1, s182 >= 0, s182 <= s181, s183 >= 0, s183 <= s182 + 1, z' - 2 >= 0 encArg(z') -{ 254 + s17 + s18 + 467*x_1107 + 283*x_1107^2 + 48*x_1107^3 + 467*x_247 + 283*x_247^2 + 48*x_247^3 }-> s19 :|: s17 >= 0, s17 <= x_1107 + 1, s18 >= 0, s18 <= x_247 + 1, s19 >= 0, s19 <= 1 + s17 + s18 + 1, z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0 + 1, z' - 1 >= 0 encArg(z') -{ 254 + s20 + s21 + 467*x_1108 + 283*x_1108^2 + 48*x_1108^3 + 467*x_248 + 283*x_248^2 + 48*x_248^3 }-> s22 :|: s20 >= 0, s20 <= x_1108 + 1, s21 >= 0, s21 <= x_248 + 1, s22 >= 0, s22 <= 1 + s20 + s21 + 1, x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ -58 + s23 + -89*z' + -5*z'^2 + 48*z'^3 }-> s24 :|: s23 >= 0, s23 <= z' - 2 + 1, s24 >= 0, s24 <= 1 + s23 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s25 + 2*s25*s26 + s25^2 + 4*s26 + s26^2 + s27 + 467*x_1111 + 283*x_1111^2 + 48*x_1111^3 + 467*x_249 + 283*x_249^2 + 48*x_249^3 }-> s28 :|: s25 >= 0, s25 <= x_1111 + 1, s26 >= 0, s26 <= x_249 + 1, s27 >= 0, s27 <= s25 + s26 + 1, s28 >= 0, s28 <= s27 + 1, z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ -58 + s29 + s30 + -89*z' + -5*z'^2 + 48*z'^3 }-> s31 :|: s29 >= 0, s29 <= z' - 2 + 1, s30 >= 0, s30 <= s29 + 1, s31 >= 0, s31 <= s30 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s33 + s35 + 467*x_1113 + 283*x_1113^2 + 48*x_1113^3 + 467*x_250 + 283*x_250^2 + 48*x_250^3 + 467*x_311 + 283*x_311^2 + 48*x_311^3 }-> s36 :|: s32 >= 0, s32 <= x_1113 + 1, s33 >= 0, s33 <= x_250 + 1, s34 >= 0, s34 <= x_311 + 1, s35 >= 0, s35 <= s32 + s33 + s34 + 1, s36 >= 0, s36 <= s35 + 1, x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 378 + s38 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 + 467*x_3 + 283*x_3^2 + 48*x_3^3 }-> s40 :|: s37 >= 0, s37 <= x_1 + 1, s38 >= 0, s38 <= x_2 + 1, s39 >= 0, s39 <= x_3 + 1, s40 >= 0, s40 <= s37 + s38 + s39 + 1, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z') -{ -58 + s75 + -89*z' + -5*z'^2 + 48*z'^3 }-> s76 :|: s74 >= 0, s74 <= z' - 2 + 1, s75 >= 0, s75 <= s74 + 1, s76 >= 0, s76 <= s75 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11322 + 283*x_11322^2 + 48*x_11322^3 + 467*x_2587 + 283*x_2587^2 + 48*x_2587^3 }-> s79 :|: s77 >= 0, s77 <= x_11322 + 1, s78 >= 0, s78 <= x_2587 + 1, s79 >= 0, s79 <= 1 + s77 + s78 + 1, x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 253 + 467*x_11323 + 283*x_11323^2 + 48*x_11323^3 + 467*x_2588 + 283*x_2588^2 + 48*x_2588^3 }-> s82 :|: s80 >= 0, s80 <= x_11323 + 1, s81 >= 0, s81 <= x_2588 + 1, s82 >= 0, s82 <= 1 + s80 + s81 + 1, z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s84 :|: s83 >= 0, s83 <= z' - 2 + 1, s84 >= 0, s84 <= 1 + s83 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s85 + 2*s85*s86 + s85^2 + 4*s86 + s86^2 + 467*x_11326 + 283*x_11326^2 + 48*x_11326^3 + 467*x_2589 + 283*x_2589^2 + 48*x_2589^3 }-> s88 :|: s85 >= 0, s85 <= x_11326 + 1, s86 >= 0, s86 <= x_2589 + 1, s87 >= 0, s87 <= s85 + s86 + 1, s88 >= 0, s88 <= s87 + 1, x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ -58 + s89 + -89*z' + -5*z'^2 + 48*z'^3 }-> s91 :|: s89 >= 0, s89 <= z' - 2 + 1, s90 >= 0, s90 <= s89 + 1, s91 >= 0, s91 <= s90 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s93 + 467*x_11328 + 283*x_11328^2 + 48*x_11328^3 + 467*x_2590 + 283*x_2590^2 + 48*x_2590^3 + 467*x_3146 + 283*x_3146^2 + 48*x_3146^3 }-> s96 :|: s92 >= 0, s92 <= x_11328 + 1, s93 >= 0, s93 <= x_2590 + 1, s94 >= 0, s94 <= x_3146 + 1, s95 >= 0, s95 <= s92 + s93 + s94 + 1, s96 >= 0, s96 <= s95 + 1, z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s99 :|: s97 >= 0, s97 <= z' - 2 + 1, s98 >= 0, s98 <= s97 + 1, s99 >= 0, s99 <= s98 + 1, z' - 2 >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ -106 + 45*z' + 139*z'^2 + 48*z'^3 }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0 encArg(z') -{ 252 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> 1 + s11 + s12 :|: s11 >= 0, s11 <= x_1 + 1, s12 >= 0, s12 <= x_2 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 260 + 6*s41 + 2*s41*s42 + s41^2 + 4*s42 + s42^2 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> s43 :|: s41 >= 0, s41 <= z' + 1, s42 >= 0, s42 <= z'' + 1, s43 >= 0, s43 <= s41 + s42 + 1, z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ -104 + s101 + 45*z' + 139*z'^2 + 48*z'^3 }-> s102 :|: s100 >= 0, s100 <= z' - 1 + 1, s101 >= 0, s101 <= s100 + 1, s102 >= 0, s102 <= s101 + 1, z' - 1 >= 0 encode_g(z') -{ -104 + s153 + 45*z' + 139*z'^2 + 48*z'^3 }-> s154 :|: s152 >= 0, s152 <= z' - 1 + 1, s153 >= 0, s153 <= s152, s154 >= 0, s154 <= s153 + 1, z' - 1 >= 0 encode_g(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0 + 1, z' = 0 encode_g(z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 1, z' >= 0 encode_g(z') -{ 254 + s46 + s47 + 467*x_11439 + 283*x_11439^2 + 48*x_11439^3 + 467*x_2639 + 283*x_2639^2 + 48*x_2639^3 }-> s48 :|: s46 >= 0, s46 <= x_11439 + 1, s47 >= 0, s47 <= x_2639 + 1, s48 >= 0, s48 <= 1 + s46 + s47 + 1, x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 254 + s49 + s50 + 467*x_11440 + 283*x_11440^2 + 48*x_11440^3 + 467*x_2640 + 283*x_2640^2 + 48*x_2640^3 }-> s51 :|: s49 >= 0, s49 <= x_11440 + 1, s50 >= 0, s50 <= x_2640 + 1, s51 >= 0, s51 <= 1 + s49 + s50 + 1, x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ -104 + s52 + 45*z' + 139*z'^2 + 48*z'^3 }-> s53 :|: s52 >= 0, s52 <= z' - 1 + 1, s53 >= 0, s53 <= 1 + s52 + 1, z' - 1 >= 0 encode_g(z') -{ 261 + 6*s54 + 2*s54*s55 + s54^2 + 4*s55 + s55^2 + s56 + 467*x_11443 + 283*x_11443^2 + 48*x_11443^3 + 467*x_2641 + 283*x_2641^2 + 48*x_2641^3 }-> s57 :|: s54 >= 0, s54 <= x_11443 + 1, s55 >= 0, s55 <= x_2641 + 1, s56 >= 0, s56 <= s54 + s55 + 1, s57 >= 0, s57 <= s56 + 1, x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ -104 + s58 + s59 + 45*z' + 139*z'^2 + 48*z'^3 }-> s60 :|: s58 >= 0, s58 <= z' - 1 + 1, s59 >= 0, s59 <= s58 + 1, s60 >= 0, s60 <= s59 + 1, z' - 1 >= 0 encode_g(z') -{ 379 + s62 + s64 + 467*x_11445 + 283*x_11445^2 + 48*x_11445^3 + 467*x_2642 + 283*x_2642^2 + 48*x_2642^3 + 467*x_3159 + 283*x_3159^2 + 48*x_3159^3 }-> s65 :|: s61 >= 0, s61 <= x_11445 + 1, s62 >= 0, s62 <= x_2642 + 1, s63 >= 0, s63 <= x_3159 + 1, s64 >= 0, s64 <= s61 + s62 + s63 + 1, s65 >= 0, s65 <= s64 + 1, x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s73 :|: s73 >= 0, s73 <= z' + 1, z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s66 + s67 :|: s66 >= 0, s66 <= z' + 1, s67 >= 0, s67 <= z'' + 1, z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 378 + s69 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 + 467*z1 + 283*z1^2 + 48*z1^3 }-> s71 :|: s68 >= 0, s68 <= z' + 1, s69 >= 0, s69 <= z'' + 1, s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s68 + s69 + s70 + 1, z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 253 + 467*x_12654 + 283*x_12654^2 + 48*x_12654^3 + 467*x_21179 + 283*x_21179^2 + 48*x_21179^3 }-> s157 :|: s155 >= 0, s155 <= x_12654 + 1, s156 >= 0, s156 <= x_21179 + 1, s157 >= 0, s157 <= 1 + s155 + s156, x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 253 + 467*x_12655 + 283*x_12655^2 + 48*x_12655^3 + 467*x_21180 + 283*x_21180^2 + 48*x_21180^3 }-> s160 :|: s158 >= 0, s158 <= x_12655 + 1, s159 >= 0, s159 <= x_21180 + 1, s160 >= 0, s160 <= 1 + s158 + s159, z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s162 :|: s161 >= 0, s161 <= z' - 1 + 1, s162 >= 0, s162 <= 1 + s161, z' - 1 >= 0 encode_i(z') -{ 261 + 6*s163 + 2*s163*s164 + s163^2 + 4*s164 + s164^2 + 467*x_12658 + 283*x_12658^2 + 48*x_12658^3 + 467*x_21181 + 283*x_21181^2 + 48*x_21181^3 }-> s166 :|: s163 >= 0, s163 <= x_12658 + 1, s164 >= 0, s164 <= x_21181 + 1, s165 >= 0, s165 <= s163 + s164 + 1, s166 >= 0, s166 <= s165, x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ -104 + s167 + 45*z' + 139*z'^2 + 48*z'^3 }-> s169 :|: s167 >= 0, s167 <= z' - 1 + 1, s168 >= 0, s168 <= s167 + 1, s169 >= 0, s169 <= s168, z' - 1 >= 0 encode_i(z') -{ 379 + s171 + 467*x_12660 + 283*x_12660^2 + 48*x_12660^3 + 467*x_21182 + 283*x_21182^2 + 48*x_21182^3 + 467*x_3294 + 283*x_3294^2 + 48*x_3294^3 }-> s174 :|: s170 >= 0, s170 <= x_12660 + 1, s171 >= 0, s171 <= x_21182 + 1, s172 >= 0, s172 <= x_3294 + 1, s173 >= 0, s173 <= s170 + s171 + s172 + 1, s174 >= 0, s174 <= s173, z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s177 :|: s175 >= 0, s175 <= z' - 1 + 1, s176 >= 0, s176 <= s175, s177 >= 0, s177 <= s176, z' - 1 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s189 :|: s187 >= 0, s187 <= z' - 1 + 1, s188 >= 0, s188 <= s187 + 1, s189 >= 0, s189 <= s188, z' - 1 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s44 + s45 :|: s44 >= 0, s44 <= z' + 1, s45 >= 0, s45 <= z'' + 1, z' >= 0, z'' >= 0 encode_k(z') -{ 253 + 467*x_11448 + 283*x_11448^2 + 48*x_11448^3 + 467*x_2643 + 283*x_2643^2 + 48*x_2643^3 }-> s105 :|: s103 >= 0, s103 <= x_11448 + 1, s104 >= 0, s104 <= x_2643 + 1, s105 >= 0, s105 <= 1 + s103 + s104 + 1, z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 253 + 467*x_11449 + 283*x_11449^2 + 48*x_11449^3 + 467*x_2644 + 283*x_2644^2 + 48*x_2644^3 }-> s108 :|: s106 >= 0, s106 <= x_11449 + 1, s107 >= 0, s107 <= x_2644 + 1, s108 >= 0, s108 <= 1 + s106 + s107 + 1, x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s110 :|: s109 >= 0, s109 <= z' - 1 + 1, s110 >= 0, s110 <= 1 + s109 + 1, z' - 1 >= 0 encode_k(z') -{ 261 + 6*s111 + 2*s111*s112 + s111^2 + 4*s112 + s112^2 + 467*x_11452 + 283*x_11452^2 + 48*x_11452^3 + 467*x_2645 + 283*x_2645^2 + 48*x_2645^3 }-> s114 :|: s111 >= 0, s111 <= x_11452 + 1, s112 >= 0, s112 <= x_2645 + 1, s113 >= 0, s113 <= s111 + s112 + 1, s114 >= 0, s114 <= s113 + 1, x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ -104 + s115 + 45*z' + 139*z'^2 + 48*z'^3 }-> s117 :|: s115 >= 0, s115 <= z' - 1 + 1, s116 >= 0, s116 <= s115 + 1, s117 >= 0, s117 <= s116 + 1, z' - 1 >= 0 encode_k(z') -{ 379 + s119 + 467*x_11454 + 283*x_11454^2 + 48*x_11454^3 + 467*x_2646 + 283*x_2646^2 + 48*x_2646^3 + 467*x_3160 + 283*x_3160^2 + 48*x_3160^3 }-> s122 :|: s118 >= 0, s118 <= x_11454 + 1, s119 >= 0, s119 <= x_2646 + 1, s120 >= 0, s120 <= x_3160 + 1, s121 >= 0, s121 <= s118 + s119 + s120 + 1, s122 >= 0, s122 <= s121 + 1, x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s125 :|: s123 >= 0, s123 <= z' - 1 + 1, s124 >= 0, s124 <= s123 + 1, s125 >= 0, s125 <= s124 + 1, z' - 1 >= 0 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s186 :|: s184 >= 0, s184 <= z' - 1 + 1, s185 >= 0, s185 <= s184, s186 >= 0, s186 <= s185 + 1, z' - 1 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s72 :|: s72 >= 0, s72 <= z' + 1, z' >= 0 f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 10 + s9 + 6*x + x^2 }-> s10 :|: s9 >= 0, s9 <= x + 0 + 1, s10 >= 0, s10 <= s9 + 1, z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 11 + s5 + 6*x + 2*x*z'' + x^2 + 4*z'' + z''^2 }-> s6 :|: s5 >= 0, s5 <= x + (1 + 0 + (z'' - 1)) + 1, s6 >= 0, s6 <= s5 + 1, z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 23 + s7 + 10*x + 2*x*x'' + 2*x*y' + x^2 + 8*x'' + 2*x''*y' + x''^2 + 8*y' + y'^2 }-> s8 :|: s7 >= 0, s7 <= x + (1 + (1 + x'') + y') + 1, s8 >= 0, s8 <= s7 + 1, z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {encode_f}, {encode_j}, {encode_i}, {encode_s} Previous analysis results are: h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1] encode_0: runtime: O(1) [0], size: O(1) [0] i: runtime: O(1) [1], size: O(n^1) [z'] k: runtime: O(1) [1], size: O(n^1) [1 + z'] g: runtime: O(n^1) [1 + z'], size: O(n^1) [1 + z'] f: runtime: O(n^2) [8 + 6*z' + 2*z'*z'' + z'^2 + 4*z'' + z''^2], size: O(n^1) [1 + z' + z''] encArg: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [1 + z'] encode_k: runtime: O(n^3) [1161 + 4403*z' + 3107*z'^2 + 624*z'^3], size: O(n^1) [4 + z'] encode_h: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [2 + z'] encode_h2: runtime: O(n^3) [379 + 467*z' + 283*z'^2 + 48*z'^3 + 468*z'' + 283*z''^2 + 48*z''^3 + 467*z1 + 283*z1^2 + 48*z1^3], size: O(n^1) [4 + z' + z'' + z1] encode_h1: runtime: O(n^3) [252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3], size: O(n^1) [3 + z' + z''] encode_g: runtime: O(n^3) [1171 + 4416*z' + 3107*z'^2 + 624*z'^3], size: O(n^1) [4 + z'] encode_f: runtime: ?, size: O(n^1) [3 + z' + z''] ---------------------------------------- (101) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: encode_f after applying outer abstraction to obtain an ITS, resulting in: O(n^3) with polynomial bound: 274 + 477*z' + 2*z'*z'' + 284*z'^2 + 48*z'^3 + 475*z'' + 284*z''^2 + 48*z''^3 ---------------------------------------- (102) Obligation: Complexity RNTS consisting of the following rules: encArg(z') -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z' = 1 + 0 encArg(z') -{ -58 + s127 + -89*z' + -5*z'^2 + 48*z'^3 }-> s128 :|: s126 >= 0, s126 <= z' - 2 + 1, s127 >= 0, s127 <= s126, s128 >= 0, s128 <= s127 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11313 + 283*x_11313^2 + 48*x_11313^3 + 467*x_2583 + 283*x_2583^2 + 48*x_2583^3 }-> s131 :|: s129 >= 0, s129 <= x_11313 + 1, s130 >= 0, s130 <= x_2583 + 1, s131 >= 0, s131 <= 1 + s129 + s130, x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 253 + 467*x_11314 + 283*x_11314^2 + 48*x_11314^3 + 467*x_2584 + 283*x_2584^2 + 48*x_2584^3 }-> s134 :|: s132 >= 0, s132 <= x_11314 + 1, s133 >= 0, s133 <= x_2584 + 1, s134 >= 0, s134 <= 1 + s132 + s133, z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s136 :|: s135 >= 0, s135 <= z' - 2 + 1, s136 >= 0, s136 <= 1 + s135, z' - 2 >= 0 encArg(z') -{ 261 + 6*s137 + 2*s137*s138 + s137^2 + 4*s138 + s138^2 + 467*x_11317 + 283*x_11317^2 + 48*x_11317^3 + 467*x_2585 + 283*x_2585^2 + 48*x_2585^3 }-> s140 :|: s137 >= 0, s137 <= x_11317 + 1, s138 >= 0, s138 <= x_2585 + 1, s139 >= 0, s139 <= s137 + s138 + 1, s140 >= 0, s140 <= s139, x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ -58 + s141 + -89*z' + -5*z'^2 + 48*z'^3 }-> s143 :|: s141 >= 0, s141 <= z' - 2 + 1, s142 >= 0, s142 <= s141 + 1, s143 >= 0, s143 <= s142, z' - 2 >= 0 encArg(z') -{ 379 + s145 + 467*x_11319 + 283*x_11319^2 + 48*x_11319^3 + 467*x_2586 + 283*x_2586^2 + 48*x_2586^3 + 467*x_3145 + 283*x_3145^2 + 48*x_3145^3 }-> s148 :|: s144 >= 0, s144 <= x_11319 + 1, s145 >= 0, s145 <= x_2586 + 1, s146 >= 0, s146 <= x_3145 + 1, s147 >= 0, s147 <= s144 + s145 + s146 + 1, s148 >= 0, s148 <= s147, z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s151 :|: s149 >= 0, s149 <= z' - 2 + 1, s150 >= 0, s150 <= s149, s151 >= 0, s151 <= s150, z' - 2 >= 0 encArg(z') -{ 260 + 6*s14 + 2*s14*s15 + s14^2 + 4*s15 + s15^2 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> s16 :|: s14 >= 0, s14 <= x_1 + 1, s15 >= 0, s15 <= x_2 + 1, s16 >= 0, s16 <= s14 + s15 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s180 :|: s178 >= 0, s178 <= z' - 2 + 1, s179 >= 0, s179 <= s178 + 1, s180 >= 0, s180 <= s179, z' - 2 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s183 :|: s181 >= 0, s181 <= z' - 2 + 1, s182 >= 0, s182 <= s181, s183 >= 0, s183 <= s182 + 1, z' - 2 >= 0 encArg(z') -{ 254 + s17 + s18 + 467*x_1107 + 283*x_1107^2 + 48*x_1107^3 + 467*x_247 + 283*x_247^2 + 48*x_247^3 }-> s19 :|: s17 >= 0, s17 <= x_1107 + 1, s18 >= 0, s18 <= x_247 + 1, s19 >= 0, s19 <= 1 + s17 + s18 + 1, z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0 + 1, z' - 1 >= 0 encArg(z') -{ 254 + s20 + s21 + 467*x_1108 + 283*x_1108^2 + 48*x_1108^3 + 467*x_248 + 283*x_248^2 + 48*x_248^3 }-> s22 :|: s20 >= 0, s20 <= x_1108 + 1, s21 >= 0, s21 <= x_248 + 1, s22 >= 0, s22 <= 1 + s20 + s21 + 1, x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ -58 + s23 + -89*z' + -5*z'^2 + 48*z'^3 }-> s24 :|: s23 >= 0, s23 <= z' - 2 + 1, s24 >= 0, s24 <= 1 + s23 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s25 + 2*s25*s26 + s25^2 + 4*s26 + s26^2 + s27 + 467*x_1111 + 283*x_1111^2 + 48*x_1111^3 + 467*x_249 + 283*x_249^2 + 48*x_249^3 }-> s28 :|: s25 >= 0, s25 <= x_1111 + 1, s26 >= 0, s26 <= x_249 + 1, s27 >= 0, s27 <= s25 + s26 + 1, s28 >= 0, s28 <= s27 + 1, z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ -58 + s29 + s30 + -89*z' + -5*z'^2 + 48*z'^3 }-> s31 :|: s29 >= 0, s29 <= z' - 2 + 1, s30 >= 0, s30 <= s29 + 1, s31 >= 0, s31 <= s30 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s33 + s35 + 467*x_1113 + 283*x_1113^2 + 48*x_1113^3 + 467*x_250 + 283*x_250^2 + 48*x_250^3 + 467*x_311 + 283*x_311^2 + 48*x_311^3 }-> s36 :|: s32 >= 0, s32 <= x_1113 + 1, s33 >= 0, s33 <= x_250 + 1, s34 >= 0, s34 <= x_311 + 1, s35 >= 0, s35 <= s32 + s33 + s34 + 1, s36 >= 0, s36 <= s35 + 1, x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 378 + s38 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 + 467*x_3 + 283*x_3^2 + 48*x_3^3 }-> s40 :|: s37 >= 0, s37 <= x_1 + 1, s38 >= 0, s38 <= x_2 + 1, s39 >= 0, s39 <= x_3 + 1, s40 >= 0, s40 <= s37 + s38 + s39 + 1, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z') -{ -58 + s75 + -89*z' + -5*z'^2 + 48*z'^3 }-> s76 :|: s74 >= 0, s74 <= z' - 2 + 1, s75 >= 0, s75 <= s74 + 1, s76 >= 0, s76 <= s75 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11322 + 283*x_11322^2 + 48*x_11322^3 + 467*x_2587 + 283*x_2587^2 + 48*x_2587^3 }-> s79 :|: s77 >= 0, s77 <= x_11322 + 1, s78 >= 0, s78 <= x_2587 + 1, s79 >= 0, s79 <= 1 + s77 + s78 + 1, x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 253 + 467*x_11323 + 283*x_11323^2 + 48*x_11323^3 + 467*x_2588 + 283*x_2588^2 + 48*x_2588^3 }-> s82 :|: s80 >= 0, s80 <= x_11323 + 1, s81 >= 0, s81 <= x_2588 + 1, s82 >= 0, s82 <= 1 + s80 + s81 + 1, z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s84 :|: s83 >= 0, s83 <= z' - 2 + 1, s84 >= 0, s84 <= 1 + s83 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s85 + 2*s85*s86 + s85^2 + 4*s86 + s86^2 + 467*x_11326 + 283*x_11326^2 + 48*x_11326^3 + 467*x_2589 + 283*x_2589^2 + 48*x_2589^3 }-> s88 :|: s85 >= 0, s85 <= x_11326 + 1, s86 >= 0, s86 <= x_2589 + 1, s87 >= 0, s87 <= s85 + s86 + 1, s88 >= 0, s88 <= s87 + 1, x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ -58 + s89 + -89*z' + -5*z'^2 + 48*z'^3 }-> s91 :|: s89 >= 0, s89 <= z' - 2 + 1, s90 >= 0, s90 <= s89 + 1, s91 >= 0, s91 <= s90 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s93 + 467*x_11328 + 283*x_11328^2 + 48*x_11328^3 + 467*x_2590 + 283*x_2590^2 + 48*x_2590^3 + 467*x_3146 + 283*x_3146^2 + 48*x_3146^3 }-> s96 :|: s92 >= 0, s92 <= x_11328 + 1, s93 >= 0, s93 <= x_2590 + 1, s94 >= 0, s94 <= x_3146 + 1, s95 >= 0, s95 <= s92 + s93 + s94 + 1, s96 >= 0, s96 <= s95 + 1, z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s99 :|: s97 >= 0, s97 <= z' - 2 + 1, s98 >= 0, s98 <= s97 + 1, s99 >= 0, s99 <= s98 + 1, z' - 2 >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ -106 + 45*z' + 139*z'^2 + 48*z'^3 }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0 encArg(z') -{ 252 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> 1 + s11 + s12 :|: s11 >= 0, s11 <= x_1 + 1, s12 >= 0, s12 <= x_2 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 260 + 6*s41 + 2*s41*s42 + s41^2 + 4*s42 + s42^2 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> s43 :|: s41 >= 0, s41 <= z' + 1, s42 >= 0, s42 <= z'' + 1, s43 >= 0, s43 <= s41 + s42 + 1, z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ -104 + s101 + 45*z' + 139*z'^2 + 48*z'^3 }-> s102 :|: s100 >= 0, s100 <= z' - 1 + 1, s101 >= 0, s101 <= s100 + 1, s102 >= 0, s102 <= s101 + 1, z' - 1 >= 0 encode_g(z') -{ -104 + s153 + 45*z' + 139*z'^2 + 48*z'^3 }-> s154 :|: s152 >= 0, s152 <= z' - 1 + 1, s153 >= 0, s153 <= s152, s154 >= 0, s154 <= s153 + 1, z' - 1 >= 0 encode_g(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0 + 1, z' = 0 encode_g(z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 1, z' >= 0 encode_g(z') -{ 254 + s46 + s47 + 467*x_11439 + 283*x_11439^2 + 48*x_11439^3 + 467*x_2639 + 283*x_2639^2 + 48*x_2639^3 }-> s48 :|: s46 >= 0, s46 <= x_11439 + 1, s47 >= 0, s47 <= x_2639 + 1, s48 >= 0, s48 <= 1 + s46 + s47 + 1, x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 254 + s49 + s50 + 467*x_11440 + 283*x_11440^2 + 48*x_11440^3 + 467*x_2640 + 283*x_2640^2 + 48*x_2640^3 }-> s51 :|: s49 >= 0, s49 <= x_11440 + 1, s50 >= 0, s50 <= x_2640 + 1, s51 >= 0, s51 <= 1 + s49 + s50 + 1, x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ -104 + s52 + 45*z' + 139*z'^2 + 48*z'^3 }-> s53 :|: s52 >= 0, s52 <= z' - 1 + 1, s53 >= 0, s53 <= 1 + s52 + 1, z' - 1 >= 0 encode_g(z') -{ 261 + 6*s54 + 2*s54*s55 + s54^2 + 4*s55 + s55^2 + s56 + 467*x_11443 + 283*x_11443^2 + 48*x_11443^3 + 467*x_2641 + 283*x_2641^2 + 48*x_2641^3 }-> s57 :|: s54 >= 0, s54 <= x_11443 + 1, s55 >= 0, s55 <= x_2641 + 1, s56 >= 0, s56 <= s54 + s55 + 1, s57 >= 0, s57 <= s56 + 1, x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ -104 + s58 + s59 + 45*z' + 139*z'^2 + 48*z'^3 }-> s60 :|: s58 >= 0, s58 <= z' - 1 + 1, s59 >= 0, s59 <= s58 + 1, s60 >= 0, s60 <= s59 + 1, z' - 1 >= 0 encode_g(z') -{ 379 + s62 + s64 + 467*x_11445 + 283*x_11445^2 + 48*x_11445^3 + 467*x_2642 + 283*x_2642^2 + 48*x_2642^3 + 467*x_3159 + 283*x_3159^2 + 48*x_3159^3 }-> s65 :|: s61 >= 0, s61 <= x_11445 + 1, s62 >= 0, s62 <= x_2642 + 1, s63 >= 0, s63 <= x_3159 + 1, s64 >= 0, s64 <= s61 + s62 + s63 + 1, s65 >= 0, s65 <= s64 + 1, x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s73 :|: s73 >= 0, s73 <= z' + 1, z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s66 + s67 :|: s66 >= 0, s66 <= z' + 1, s67 >= 0, s67 <= z'' + 1, z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 378 + s69 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 + 467*z1 + 283*z1^2 + 48*z1^3 }-> s71 :|: s68 >= 0, s68 <= z' + 1, s69 >= 0, s69 <= z'' + 1, s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s68 + s69 + s70 + 1, z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 253 + 467*x_12654 + 283*x_12654^2 + 48*x_12654^3 + 467*x_21179 + 283*x_21179^2 + 48*x_21179^3 }-> s157 :|: s155 >= 0, s155 <= x_12654 + 1, s156 >= 0, s156 <= x_21179 + 1, s157 >= 0, s157 <= 1 + s155 + s156, x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 253 + 467*x_12655 + 283*x_12655^2 + 48*x_12655^3 + 467*x_21180 + 283*x_21180^2 + 48*x_21180^3 }-> s160 :|: s158 >= 0, s158 <= x_12655 + 1, s159 >= 0, s159 <= x_21180 + 1, s160 >= 0, s160 <= 1 + s158 + s159, z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s162 :|: s161 >= 0, s161 <= z' - 1 + 1, s162 >= 0, s162 <= 1 + s161, z' - 1 >= 0 encode_i(z') -{ 261 + 6*s163 + 2*s163*s164 + s163^2 + 4*s164 + s164^2 + 467*x_12658 + 283*x_12658^2 + 48*x_12658^3 + 467*x_21181 + 283*x_21181^2 + 48*x_21181^3 }-> s166 :|: s163 >= 0, s163 <= x_12658 + 1, s164 >= 0, s164 <= x_21181 + 1, s165 >= 0, s165 <= s163 + s164 + 1, s166 >= 0, s166 <= s165, x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ -104 + s167 + 45*z' + 139*z'^2 + 48*z'^3 }-> s169 :|: s167 >= 0, s167 <= z' - 1 + 1, s168 >= 0, s168 <= s167 + 1, s169 >= 0, s169 <= s168, z' - 1 >= 0 encode_i(z') -{ 379 + s171 + 467*x_12660 + 283*x_12660^2 + 48*x_12660^3 + 467*x_21182 + 283*x_21182^2 + 48*x_21182^3 + 467*x_3294 + 283*x_3294^2 + 48*x_3294^3 }-> s174 :|: s170 >= 0, s170 <= x_12660 + 1, s171 >= 0, s171 <= x_21182 + 1, s172 >= 0, s172 <= x_3294 + 1, s173 >= 0, s173 <= s170 + s171 + s172 + 1, s174 >= 0, s174 <= s173, z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s177 :|: s175 >= 0, s175 <= z' - 1 + 1, s176 >= 0, s176 <= s175, s177 >= 0, s177 <= s176, z' - 1 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s189 :|: s187 >= 0, s187 <= z' - 1 + 1, s188 >= 0, s188 <= s187 + 1, s189 >= 0, s189 <= s188, z' - 1 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s44 + s45 :|: s44 >= 0, s44 <= z' + 1, s45 >= 0, s45 <= z'' + 1, z' >= 0, z'' >= 0 encode_k(z') -{ 253 + 467*x_11448 + 283*x_11448^2 + 48*x_11448^3 + 467*x_2643 + 283*x_2643^2 + 48*x_2643^3 }-> s105 :|: s103 >= 0, s103 <= x_11448 + 1, s104 >= 0, s104 <= x_2643 + 1, s105 >= 0, s105 <= 1 + s103 + s104 + 1, z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 253 + 467*x_11449 + 283*x_11449^2 + 48*x_11449^3 + 467*x_2644 + 283*x_2644^2 + 48*x_2644^3 }-> s108 :|: s106 >= 0, s106 <= x_11449 + 1, s107 >= 0, s107 <= x_2644 + 1, s108 >= 0, s108 <= 1 + s106 + s107 + 1, x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s110 :|: s109 >= 0, s109 <= z' - 1 + 1, s110 >= 0, s110 <= 1 + s109 + 1, z' - 1 >= 0 encode_k(z') -{ 261 + 6*s111 + 2*s111*s112 + s111^2 + 4*s112 + s112^2 + 467*x_11452 + 283*x_11452^2 + 48*x_11452^3 + 467*x_2645 + 283*x_2645^2 + 48*x_2645^3 }-> s114 :|: s111 >= 0, s111 <= x_11452 + 1, s112 >= 0, s112 <= x_2645 + 1, s113 >= 0, s113 <= s111 + s112 + 1, s114 >= 0, s114 <= s113 + 1, x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ -104 + s115 + 45*z' + 139*z'^2 + 48*z'^3 }-> s117 :|: s115 >= 0, s115 <= z' - 1 + 1, s116 >= 0, s116 <= s115 + 1, s117 >= 0, s117 <= s116 + 1, z' - 1 >= 0 encode_k(z') -{ 379 + s119 + 467*x_11454 + 283*x_11454^2 + 48*x_11454^3 + 467*x_2646 + 283*x_2646^2 + 48*x_2646^3 + 467*x_3160 + 283*x_3160^2 + 48*x_3160^3 }-> s122 :|: s118 >= 0, s118 <= x_11454 + 1, s119 >= 0, s119 <= x_2646 + 1, s120 >= 0, s120 <= x_3160 + 1, s121 >= 0, s121 <= s118 + s119 + s120 + 1, s122 >= 0, s122 <= s121 + 1, x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s125 :|: s123 >= 0, s123 <= z' - 1 + 1, s124 >= 0, s124 <= s123 + 1, s125 >= 0, s125 <= s124 + 1, z' - 1 >= 0 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s186 :|: s184 >= 0, s184 <= z' - 1 + 1, s185 >= 0, s185 <= s184, s186 >= 0, s186 <= s185 + 1, z' - 1 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s72 :|: s72 >= 0, s72 <= z' + 1, z' >= 0 f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 10 + s9 + 6*x + x^2 }-> s10 :|: s9 >= 0, s9 <= x + 0 + 1, s10 >= 0, s10 <= s9 + 1, z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 11 + s5 + 6*x + 2*x*z'' + x^2 + 4*z'' + z''^2 }-> s6 :|: s5 >= 0, s5 <= x + (1 + 0 + (z'' - 1)) + 1, s6 >= 0, s6 <= s5 + 1, z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 23 + s7 + 10*x + 2*x*x'' + 2*x*y' + x^2 + 8*x'' + 2*x''*y' + x''^2 + 8*y' + y'^2 }-> s8 :|: s7 >= 0, s7 <= x + (1 + (1 + x'') + y') + 1, s8 >= 0, s8 <= s7 + 1, z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {encode_j}, {encode_i}, {encode_s} Previous analysis results are: h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1] encode_0: runtime: O(1) [0], size: O(1) [0] i: runtime: O(1) [1], size: O(n^1) [z'] k: runtime: O(1) [1], size: O(n^1) [1 + z'] g: runtime: O(n^1) [1 + z'], size: O(n^1) [1 + z'] f: runtime: O(n^2) [8 + 6*z' + 2*z'*z'' + z'^2 + 4*z'' + z''^2], size: O(n^1) [1 + z' + z''] encArg: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [1 + z'] encode_k: runtime: O(n^3) [1161 + 4403*z' + 3107*z'^2 + 624*z'^3], size: O(n^1) [4 + z'] encode_h: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [2 + z'] encode_h2: runtime: O(n^3) [379 + 467*z' + 283*z'^2 + 48*z'^3 + 468*z'' + 283*z''^2 + 48*z''^3 + 467*z1 + 283*z1^2 + 48*z1^3], size: O(n^1) [4 + z' + z'' + z1] encode_h1: runtime: O(n^3) [252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3], size: O(n^1) [3 + z' + z''] encode_g: runtime: O(n^3) [1171 + 4416*z' + 3107*z'^2 + 624*z'^3], size: O(n^1) [4 + z'] encode_f: runtime: O(n^3) [274 + 477*z' + 2*z'*z'' + 284*z'^2 + 48*z'^3 + 475*z'' + 284*z''^2 + 48*z''^3], size: O(n^1) [3 + z' + z''] ---------------------------------------- (103) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (104) Obligation: Complexity RNTS consisting of the following rules: encArg(z') -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z' = 1 + 0 encArg(z') -{ -58 + s127 + -89*z' + -5*z'^2 + 48*z'^3 }-> s128 :|: s126 >= 0, s126 <= z' - 2 + 1, s127 >= 0, s127 <= s126, s128 >= 0, s128 <= s127 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11313 + 283*x_11313^2 + 48*x_11313^3 + 467*x_2583 + 283*x_2583^2 + 48*x_2583^3 }-> s131 :|: s129 >= 0, s129 <= x_11313 + 1, s130 >= 0, s130 <= x_2583 + 1, s131 >= 0, s131 <= 1 + s129 + s130, x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 253 + 467*x_11314 + 283*x_11314^2 + 48*x_11314^3 + 467*x_2584 + 283*x_2584^2 + 48*x_2584^3 }-> s134 :|: s132 >= 0, s132 <= x_11314 + 1, s133 >= 0, s133 <= x_2584 + 1, s134 >= 0, s134 <= 1 + s132 + s133, z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s136 :|: s135 >= 0, s135 <= z' - 2 + 1, s136 >= 0, s136 <= 1 + s135, z' - 2 >= 0 encArg(z') -{ 261 + 6*s137 + 2*s137*s138 + s137^2 + 4*s138 + s138^2 + 467*x_11317 + 283*x_11317^2 + 48*x_11317^3 + 467*x_2585 + 283*x_2585^2 + 48*x_2585^3 }-> s140 :|: s137 >= 0, s137 <= x_11317 + 1, s138 >= 0, s138 <= x_2585 + 1, s139 >= 0, s139 <= s137 + s138 + 1, s140 >= 0, s140 <= s139, x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ -58 + s141 + -89*z' + -5*z'^2 + 48*z'^3 }-> s143 :|: s141 >= 0, s141 <= z' - 2 + 1, s142 >= 0, s142 <= s141 + 1, s143 >= 0, s143 <= s142, z' - 2 >= 0 encArg(z') -{ 379 + s145 + 467*x_11319 + 283*x_11319^2 + 48*x_11319^3 + 467*x_2586 + 283*x_2586^2 + 48*x_2586^3 + 467*x_3145 + 283*x_3145^2 + 48*x_3145^3 }-> s148 :|: s144 >= 0, s144 <= x_11319 + 1, s145 >= 0, s145 <= x_2586 + 1, s146 >= 0, s146 <= x_3145 + 1, s147 >= 0, s147 <= s144 + s145 + s146 + 1, s148 >= 0, s148 <= s147, z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s151 :|: s149 >= 0, s149 <= z' - 2 + 1, s150 >= 0, s150 <= s149, s151 >= 0, s151 <= s150, z' - 2 >= 0 encArg(z') -{ 260 + 6*s14 + 2*s14*s15 + s14^2 + 4*s15 + s15^2 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> s16 :|: s14 >= 0, s14 <= x_1 + 1, s15 >= 0, s15 <= x_2 + 1, s16 >= 0, s16 <= s14 + s15 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s180 :|: s178 >= 0, s178 <= z' - 2 + 1, s179 >= 0, s179 <= s178 + 1, s180 >= 0, s180 <= s179, z' - 2 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s183 :|: s181 >= 0, s181 <= z' - 2 + 1, s182 >= 0, s182 <= s181, s183 >= 0, s183 <= s182 + 1, z' - 2 >= 0 encArg(z') -{ 254 + s17 + s18 + 467*x_1107 + 283*x_1107^2 + 48*x_1107^3 + 467*x_247 + 283*x_247^2 + 48*x_247^3 }-> s19 :|: s17 >= 0, s17 <= x_1107 + 1, s18 >= 0, s18 <= x_247 + 1, s19 >= 0, s19 <= 1 + s17 + s18 + 1, z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0 + 1, z' - 1 >= 0 encArg(z') -{ 254 + s20 + s21 + 467*x_1108 + 283*x_1108^2 + 48*x_1108^3 + 467*x_248 + 283*x_248^2 + 48*x_248^3 }-> s22 :|: s20 >= 0, s20 <= x_1108 + 1, s21 >= 0, s21 <= x_248 + 1, s22 >= 0, s22 <= 1 + s20 + s21 + 1, x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ -58 + s23 + -89*z' + -5*z'^2 + 48*z'^3 }-> s24 :|: s23 >= 0, s23 <= z' - 2 + 1, s24 >= 0, s24 <= 1 + s23 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s25 + 2*s25*s26 + s25^2 + 4*s26 + s26^2 + s27 + 467*x_1111 + 283*x_1111^2 + 48*x_1111^3 + 467*x_249 + 283*x_249^2 + 48*x_249^3 }-> s28 :|: s25 >= 0, s25 <= x_1111 + 1, s26 >= 0, s26 <= x_249 + 1, s27 >= 0, s27 <= s25 + s26 + 1, s28 >= 0, s28 <= s27 + 1, z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ -58 + s29 + s30 + -89*z' + -5*z'^2 + 48*z'^3 }-> s31 :|: s29 >= 0, s29 <= z' - 2 + 1, s30 >= 0, s30 <= s29 + 1, s31 >= 0, s31 <= s30 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s33 + s35 + 467*x_1113 + 283*x_1113^2 + 48*x_1113^3 + 467*x_250 + 283*x_250^2 + 48*x_250^3 + 467*x_311 + 283*x_311^2 + 48*x_311^3 }-> s36 :|: s32 >= 0, s32 <= x_1113 + 1, s33 >= 0, s33 <= x_250 + 1, s34 >= 0, s34 <= x_311 + 1, s35 >= 0, s35 <= s32 + s33 + s34 + 1, s36 >= 0, s36 <= s35 + 1, x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 378 + s38 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 + 467*x_3 + 283*x_3^2 + 48*x_3^3 }-> s40 :|: s37 >= 0, s37 <= x_1 + 1, s38 >= 0, s38 <= x_2 + 1, s39 >= 0, s39 <= x_3 + 1, s40 >= 0, s40 <= s37 + s38 + s39 + 1, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z') -{ -58 + s75 + -89*z' + -5*z'^2 + 48*z'^3 }-> s76 :|: s74 >= 0, s74 <= z' - 2 + 1, s75 >= 0, s75 <= s74 + 1, s76 >= 0, s76 <= s75 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11322 + 283*x_11322^2 + 48*x_11322^3 + 467*x_2587 + 283*x_2587^2 + 48*x_2587^3 }-> s79 :|: s77 >= 0, s77 <= x_11322 + 1, s78 >= 0, s78 <= x_2587 + 1, s79 >= 0, s79 <= 1 + s77 + s78 + 1, x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 253 + 467*x_11323 + 283*x_11323^2 + 48*x_11323^3 + 467*x_2588 + 283*x_2588^2 + 48*x_2588^3 }-> s82 :|: s80 >= 0, s80 <= x_11323 + 1, s81 >= 0, s81 <= x_2588 + 1, s82 >= 0, s82 <= 1 + s80 + s81 + 1, z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s84 :|: s83 >= 0, s83 <= z' - 2 + 1, s84 >= 0, s84 <= 1 + s83 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s85 + 2*s85*s86 + s85^2 + 4*s86 + s86^2 + 467*x_11326 + 283*x_11326^2 + 48*x_11326^3 + 467*x_2589 + 283*x_2589^2 + 48*x_2589^3 }-> s88 :|: s85 >= 0, s85 <= x_11326 + 1, s86 >= 0, s86 <= x_2589 + 1, s87 >= 0, s87 <= s85 + s86 + 1, s88 >= 0, s88 <= s87 + 1, x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ -58 + s89 + -89*z' + -5*z'^2 + 48*z'^3 }-> s91 :|: s89 >= 0, s89 <= z' - 2 + 1, s90 >= 0, s90 <= s89 + 1, s91 >= 0, s91 <= s90 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s93 + 467*x_11328 + 283*x_11328^2 + 48*x_11328^3 + 467*x_2590 + 283*x_2590^2 + 48*x_2590^3 + 467*x_3146 + 283*x_3146^2 + 48*x_3146^3 }-> s96 :|: s92 >= 0, s92 <= x_11328 + 1, s93 >= 0, s93 <= x_2590 + 1, s94 >= 0, s94 <= x_3146 + 1, s95 >= 0, s95 <= s92 + s93 + s94 + 1, s96 >= 0, s96 <= s95 + 1, z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s99 :|: s97 >= 0, s97 <= z' - 2 + 1, s98 >= 0, s98 <= s97 + 1, s99 >= 0, s99 <= s98 + 1, z' - 2 >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ -106 + 45*z' + 139*z'^2 + 48*z'^3 }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0 encArg(z') -{ 252 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> 1 + s11 + s12 :|: s11 >= 0, s11 <= x_1 + 1, s12 >= 0, s12 <= x_2 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 260 + 6*s41 + 2*s41*s42 + s41^2 + 4*s42 + s42^2 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> s43 :|: s41 >= 0, s41 <= z' + 1, s42 >= 0, s42 <= z'' + 1, s43 >= 0, s43 <= s41 + s42 + 1, z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ -104 + s101 + 45*z' + 139*z'^2 + 48*z'^3 }-> s102 :|: s100 >= 0, s100 <= z' - 1 + 1, s101 >= 0, s101 <= s100 + 1, s102 >= 0, s102 <= s101 + 1, z' - 1 >= 0 encode_g(z') -{ -104 + s153 + 45*z' + 139*z'^2 + 48*z'^3 }-> s154 :|: s152 >= 0, s152 <= z' - 1 + 1, s153 >= 0, s153 <= s152, s154 >= 0, s154 <= s153 + 1, z' - 1 >= 0 encode_g(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0 + 1, z' = 0 encode_g(z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 1, z' >= 0 encode_g(z') -{ 254 + s46 + s47 + 467*x_11439 + 283*x_11439^2 + 48*x_11439^3 + 467*x_2639 + 283*x_2639^2 + 48*x_2639^3 }-> s48 :|: s46 >= 0, s46 <= x_11439 + 1, s47 >= 0, s47 <= x_2639 + 1, s48 >= 0, s48 <= 1 + s46 + s47 + 1, x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 254 + s49 + s50 + 467*x_11440 + 283*x_11440^2 + 48*x_11440^3 + 467*x_2640 + 283*x_2640^2 + 48*x_2640^3 }-> s51 :|: s49 >= 0, s49 <= x_11440 + 1, s50 >= 0, s50 <= x_2640 + 1, s51 >= 0, s51 <= 1 + s49 + s50 + 1, x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ -104 + s52 + 45*z' + 139*z'^2 + 48*z'^3 }-> s53 :|: s52 >= 0, s52 <= z' - 1 + 1, s53 >= 0, s53 <= 1 + s52 + 1, z' - 1 >= 0 encode_g(z') -{ 261 + 6*s54 + 2*s54*s55 + s54^2 + 4*s55 + s55^2 + s56 + 467*x_11443 + 283*x_11443^2 + 48*x_11443^3 + 467*x_2641 + 283*x_2641^2 + 48*x_2641^3 }-> s57 :|: s54 >= 0, s54 <= x_11443 + 1, s55 >= 0, s55 <= x_2641 + 1, s56 >= 0, s56 <= s54 + s55 + 1, s57 >= 0, s57 <= s56 + 1, x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ -104 + s58 + s59 + 45*z' + 139*z'^2 + 48*z'^3 }-> s60 :|: s58 >= 0, s58 <= z' - 1 + 1, s59 >= 0, s59 <= s58 + 1, s60 >= 0, s60 <= s59 + 1, z' - 1 >= 0 encode_g(z') -{ 379 + s62 + s64 + 467*x_11445 + 283*x_11445^2 + 48*x_11445^3 + 467*x_2642 + 283*x_2642^2 + 48*x_2642^3 + 467*x_3159 + 283*x_3159^2 + 48*x_3159^3 }-> s65 :|: s61 >= 0, s61 <= x_11445 + 1, s62 >= 0, s62 <= x_2642 + 1, s63 >= 0, s63 <= x_3159 + 1, s64 >= 0, s64 <= s61 + s62 + s63 + 1, s65 >= 0, s65 <= s64 + 1, x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s73 :|: s73 >= 0, s73 <= z' + 1, z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s66 + s67 :|: s66 >= 0, s66 <= z' + 1, s67 >= 0, s67 <= z'' + 1, z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 378 + s69 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 + 467*z1 + 283*z1^2 + 48*z1^3 }-> s71 :|: s68 >= 0, s68 <= z' + 1, s69 >= 0, s69 <= z'' + 1, s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s68 + s69 + s70 + 1, z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 253 + 467*x_12654 + 283*x_12654^2 + 48*x_12654^3 + 467*x_21179 + 283*x_21179^2 + 48*x_21179^3 }-> s157 :|: s155 >= 0, s155 <= x_12654 + 1, s156 >= 0, s156 <= x_21179 + 1, s157 >= 0, s157 <= 1 + s155 + s156, x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 253 + 467*x_12655 + 283*x_12655^2 + 48*x_12655^3 + 467*x_21180 + 283*x_21180^2 + 48*x_21180^3 }-> s160 :|: s158 >= 0, s158 <= x_12655 + 1, s159 >= 0, s159 <= x_21180 + 1, s160 >= 0, s160 <= 1 + s158 + s159, z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s162 :|: s161 >= 0, s161 <= z' - 1 + 1, s162 >= 0, s162 <= 1 + s161, z' - 1 >= 0 encode_i(z') -{ 261 + 6*s163 + 2*s163*s164 + s163^2 + 4*s164 + s164^2 + 467*x_12658 + 283*x_12658^2 + 48*x_12658^3 + 467*x_21181 + 283*x_21181^2 + 48*x_21181^3 }-> s166 :|: s163 >= 0, s163 <= x_12658 + 1, s164 >= 0, s164 <= x_21181 + 1, s165 >= 0, s165 <= s163 + s164 + 1, s166 >= 0, s166 <= s165, x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ -104 + s167 + 45*z' + 139*z'^2 + 48*z'^3 }-> s169 :|: s167 >= 0, s167 <= z' - 1 + 1, s168 >= 0, s168 <= s167 + 1, s169 >= 0, s169 <= s168, z' - 1 >= 0 encode_i(z') -{ 379 + s171 + 467*x_12660 + 283*x_12660^2 + 48*x_12660^3 + 467*x_21182 + 283*x_21182^2 + 48*x_21182^3 + 467*x_3294 + 283*x_3294^2 + 48*x_3294^3 }-> s174 :|: s170 >= 0, s170 <= x_12660 + 1, s171 >= 0, s171 <= x_21182 + 1, s172 >= 0, s172 <= x_3294 + 1, s173 >= 0, s173 <= s170 + s171 + s172 + 1, s174 >= 0, s174 <= s173, z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s177 :|: s175 >= 0, s175 <= z' - 1 + 1, s176 >= 0, s176 <= s175, s177 >= 0, s177 <= s176, z' - 1 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s189 :|: s187 >= 0, s187 <= z' - 1 + 1, s188 >= 0, s188 <= s187 + 1, s189 >= 0, s189 <= s188, z' - 1 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s44 + s45 :|: s44 >= 0, s44 <= z' + 1, s45 >= 0, s45 <= z'' + 1, z' >= 0, z'' >= 0 encode_k(z') -{ 253 + 467*x_11448 + 283*x_11448^2 + 48*x_11448^3 + 467*x_2643 + 283*x_2643^2 + 48*x_2643^3 }-> s105 :|: s103 >= 0, s103 <= x_11448 + 1, s104 >= 0, s104 <= x_2643 + 1, s105 >= 0, s105 <= 1 + s103 + s104 + 1, z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 253 + 467*x_11449 + 283*x_11449^2 + 48*x_11449^3 + 467*x_2644 + 283*x_2644^2 + 48*x_2644^3 }-> s108 :|: s106 >= 0, s106 <= x_11449 + 1, s107 >= 0, s107 <= x_2644 + 1, s108 >= 0, s108 <= 1 + s106 + s107 + 1, x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s110 :|: s109 >= 0, s109 <= z' - 1 + 1, s110 >= 0, s110 <= 1 + s109 + 1, z' - 1 >= 0 encode_k(z') -{ 261 + 6*s111 + 2*s111*s112 + s111^2 + 4*s112 + s112^2 + 467*x_11452 + 283*x_11452^2 + 48*x_11452^3 + 467*x_2645 + 283*x_2645^2 + 48*x_2645^3 }-> s114 :|: s111 >= 0, s111 <= x_11452 + 1, s112 >= 0, s112 <= x_2645 + 1, s113 >= 0, s113 <= s111 + s112 + 1, s114 >= 0, s114 <= s113 + 1, x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ -104 + s115 + 45*z' + 139*z'^2 + 48*z'^3 }-> s117 :|: s115 >= 0, s115 <= z' - 1 + 1, s116 >= 0, s116 <= s115 + 1, s117 >= 0, s117 <= s116 + 1, z' - 1 >= 0 encode_k(z') -{ 379 + s119 + 467*x_11454 + 283*x_11454^2 + 48*x_11454^3 + 467*x_2646 + 283*x_2646^2 + 48*x_2646^3 + 467*x_3160 + 283*x_3160^2 + 48*x_3160^3 }-> s122 :|: s118 >= 0, s118 <= x_11454 + 1, s119 >= 0, s119 <= x_2646 + 1, s120 >= 0, s120 <= x_3160 + 1, s121 >= 0, s121 <= s118 + s119 + s120 + 1, s122 >= 0, s122 <= s121 + 1, x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s125 :|: s123 >= 0, s123 <= z' - 1 + 1, s124 >= 0, s124 <= s123 + 1, s125 >= 0, s125 <= s124 + 1, z' - 1 >= 0 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s186 :|: s184 >= 0, s184 <= z' - 1 + 1, s185 >= 0, s185 <= s184, s186 >= 0, s186 <= s185 + 1, z' - 1 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s72 :|: s72 >= 0, s72 <= z' + 1, z' >= 0 f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 10 + s9 + 6*x + x^2 }-> s10 :|: s9 >= 0, s9 <= x + 0 + 1, s10 >= 0, s10 <= s9 + 1, z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 11 + s5 + 6*x + 2*x*z'' + x^2 + 4*z'' + z''^2 }-> s6 :|: s5 >= 0, s5 <= x + (1 + 0 + (z'' - 1)) + 1, s6 >= 0, s6 <= s5 + 1, z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 23 + s7 + 10*x + 2*x*x'' + 2*x*y' + x^2 + 8*x'' + 2*x''*y' + x''^2 + 8*y' + y'^2 }-> s8 :|: s7 >= 0, s7 <= x + (1 + (1 + x'') + y') + 1, s8 >= 0, s8 <= s7 + 1, z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {encode_j}, {encode_i}, {encode_s} Previous analysis results are: h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1] encode_0: runtime: O(1) [0], size: O(1) [0] i: runtime: O(1) [1], size: O(n^1) [z'] k: runtime: O(1) [1], size: O(n^1) [1 + z'] g: runtime: O(n^1) [1 + z'], size: O(n^1) [1 + z'] f: runtime: O(n^2) [8 + 6*z' + 2*z'*z'' + z'^2 + 4*z'' + z''^2], size: O(n^1) [1 + z' + z''] encArg: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [1 + z'] encode_k: runtime: O(n^3) [1161 + 4403*z' + 3107*z'^2 + 624*z'^3], size: O(n^1) [4 + z'] encode_h: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [2 + z'] encode_h2: runtime: O(n^3) [379 + 467*z' + 283*z'^2 + 48*z'^3 + 468*z'' + 283*z''^2 + 48*z''^3 + 467*z1 + 283*z1^2 + 48*z1^3], size: O(n^1) [4 + z' + z'' + z1] encode_h1: runtime: O(n^3) [252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3], size: O(n^1) [3 + z' + z''] encode_g: runtime: O(n^3) [1171 + 4416*z' + 3107*z'^2 + 624*z'^3], size: O(n^1) [4 + z'] encode_f: runtime: O(n^3) [274 + 477*z' + 2*z'*z'' + 284*z'^2 + 48*z'^3 + 475*z'' + 284*z''^2 + 48*z''^3], size: O(n^1) [3 + z' + z''] ---------------------------------------- (105) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: encode_j after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 3 + z' + z'' ---------------------------------------- (106) Obligation: Complexity RNTS consisting of the following rules: encArg(z') -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z' = 1 + 0 encArg(z') -{ -58 + s127 + -89*z' + -5*z'^2 + 48*z'^3 }-> s128 :|: s126 >= 0, s126 <= z' - 2 + 1, s127 >= 0, s127 <= s126, s128 >= 0, s128 <= s127 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11313 + 283*x_11313^2 + 48*x_11313^3 + 467*x_2583 + 283*x_2583^2 + 48*x_2583^3 }-> s131 :|: s129 >= 0, s129 <= x_11313 + 1, s130 >= 0, s130 <= x_2583 + 1, s131 >= 0, s131 <= 1 + s129 + s130, x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 253 + 467*x_11314 + 283*x_11314^2 + 48*x_11314^3 + 467*x_2584 + 283*x_2584^2 + 48*x_2584^3 }-> s134 :|: s132 >= 0, s132 <= x_11314 + 1, s133 >= 0, s133 <= x_2584 + 1, s134 >= 0, s134 <= 1 + s132 + s133, z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s136 :|: s135 >= 0, s135 <= z' - 2 + 1, s136 >= 0, s136 <= 1 + s135, z' - 2 >= 0 encArg(z') -{ 261 + 6*s137 + 2*s137*s138 + s137^2 + 4*s138 + s138^2 + 467*x_11317 + 283*x_11317^2 + 48*x_11317^3 + 467*x_2585 + 283*x_2585^2 + 48*x_2585^3 }-> s140 :|: s137 >= 0, s137 <= x_11317 + 1, s138 >= 0, s138 <= x_2585 + 1, s139 >= 0, s139 <= s137 + s138 + 1, s140 >= 0, s140 <= s139, x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ -58 + s141 + -89*z' + -5*z'^2 + 48*z'^3 }-> s143 :|: s141 >= 0, s141 <= z' - 2 + 1, s142 >= 0, s142 <= s141 + 1, s143 >= 0, s143 <= s142, z' - 2 >= 0 encArg(z') -{ 379 + s145 + 467*x_11319 + 283*x_11319^2 + 48*x_11319^3 + 467*x_2586 + 283*x_2586^2 + 48*x_2586^3 + 467*x_3145 + 283*x_3145^2 + 48*x_3145^3 }-> s148 :|: s144 >= 0, s144 <= x_11319 + 1, s145 >= 0, s145 <= x_2586 + 1, s146 >= 0, s146 <= x_3145 + 1, s147 >= 0, s147 <= s144 + s145 + s146 + 1, s148 >= 0, s148 <= s147, z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s151 :|: s149 >= 0, s149 <= z' - 2 + 1, s150 >= 0, s150 <= s149, s151 >= 0, s151 <= s150, z' - 2 >= 0 encArg(z') -{ 260 + 6*s14 + 2*s14*s15 + s14^2 + 4*s15 + s15^2 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> s16 :|: s14 >= 0, s14 <= x_1 + 1, s15 >= 0, s15 <= x_2 + 1, s16 >= 0, s16 <= s14 + s15 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s180 :|: s178 >= 0, s178 <= z' - 2 + 1, s179 >= 0, s179 <= s178 + 1, s180 >= 0, s180 <= s179, z' - 2 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s183 :|: s181 >= 0, s181 <= z' - 2 + 1, s182 >= 0, s182 <= s181, s183 >= 0, s183 <= s182 + 1, z' - 2 >= 0 encArg(z') -{ 254 + s17 + s18 + 467*x_1107 + 283*x_1107^2 + 48*x_1107^3 + 467*x_247 + 283*x_247^2 + 48*x_247^3 }-> s19 :|: s17 >= 0, s17 <= x_1107 + 1, s18 >= 0, s18 <= x_247 + 1, s19 >= 0, s19 <= 1 + s17 + s18 + 1, z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0 + 1, z' - 1 >= 0 encArg(z') -{ 254 + s20 + s21 + 467*x_1108 + 283*x_1108^2 + 48*x_1108^3 + 467*x_248 + 283*x_248^2 + 48*x_248^3 }-> s22 :|: s20 >= 0, s20 <= x_1108 + 1, s21 >= 0, s21 <= x_248 + 1, s22 >= 0, s22 <= 1 + s20 + s21 + 1, x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ -58 + s23 + -89*z' + -5*z'^2 + 48*z'^3 }-> s24 :|: s23 >= 0, s23 <= z' - 2 + 1, s24 >= 0, s24 <= 1 + s23 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s25 + 2*s25*s26 + s25^2 + 4*s26 + s26^2 + s27 + 467*x_1111 + 283*x_1111^2 + 48*x_1111^3 + 467*x_249 + 283*x_249^2 + 48*x_249^3 }-> s28 :|: s25 >= 0, s25 <= x_1111 + 1, s26 >= 0, s26 <= x_249 + 1, s27 >= 0, s27 <= s25 + s26 + 1, s28 >= 0, s28 <= s27 + 1, z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ -58 + s29 + s30 + -89*z' + -5*z'^2 + 48*z'^3 }-> s31 :|: s29 >= 0, s29 <= z' - 2 + 1, s30 >= 0, s30 <= s29 + 1, s31 >= 0, s31 <= s30 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s33 + s35 + 467*x_1113 + 283*x_1113^2 + 48*x_1113^3 + 467*x_250 + 283*x_250^2 + 48*x_250^3 + 467*x_311 + 283*x_311^2 + 48*x_311^3 }-> s36 :|: s32 >= 0, s32 <= x_1113 + 1, s33 >= 0, s33 <= x_250 + 1, s34 >= 0, s34 <= x_311 + 1, s35 >= 0, s35 <= s32 + s33 + s34 + 1, s36 >= 0, s36 <= s35 + 1, x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 378 + s38 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 + 467*x_3 + 283*x_3^2 + 48*x_3^3 }-> s40 :|: s37 >= 0, s37 <= x_1 + 1, s38 >= 0, s38 <= x_2 + 1, s39 >= 0, s39 <= x_3 + 1, s40 >= 0, s40 <= s37 + s38 + s39 + 1, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z') -{ -58 + s75 + -89*z' + -5*z'^2 + 48*z'^3 }-> s76 :|: s74 >= 0, s74 <= z' - 2 + 1, s75 >= 0, s75 <= s74 + 1, s76 >= 0, s76 <= s75 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11322 + 283*x_11322^2 + 48*x_11322^3 + 467*x_2587 + 283*x_2587^2 + 48*x_2587^3 }-> s79 :|: s77 >= 0, s77 <= x_11322 + 1, s78 >= 0, s78 <= x_2587 + 1, s79 >= 0, s79 <= 1 + s77 + s78 + 1, x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 253 + 467*x_11323 + 283*x_11323^2 + 48*x_11323^3 + 467*x_2588 + 283*x_2588^2 + 48*x_2588^3 }-> s82 :|: s80 >= 0, s80 <= x_11323 + 1, s81 >= 0, s81 <= x_2588 + 1, s82 >= 0, s82 <= 1 + s80 + s81 + 1, z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s84 :|: s83 >= 0, s83 <= z' - 2 + 1, s84 >= 0, s84 <= 1 + s83 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s85 + 2*s85*s86 + s85^2 + 4*s86 + s86^2 + 467*x_11326 + 283*x_11326^2 + 48*x_11326^3 + 467*x_2589 + 283*x_2589^2 + 48*x_2589^3 }-> s88 :|: s85 >= 0, s85 <= x_11326 + 1, s86 >= 0, s86 <= x_2589 + 1, s87 >= 0, s87 <= s85 + s86 + 1, s88 >= 0, s88 <= s87 + 1, x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ -58 + s89 + -89*z' + -5*z'^2 + 48*z'^3 }-> s91 :|: s89 >= 0, s89 <= z' - 2 + 1, s90 >= 0, s90 <= s89 + 1, s91 >= 0, s91 <= s90 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s93 + 467*x_11328 + 283*x_11328^2 + 48*x_11328^3 + 467*x_2590 + 283*x_2590^2 + 48*x_2590^3 + 467*x_3146 + 283*x_3146^2 + 48*x_3146^3 }-> s96 :|: s92 >= 0, s92 <= x_11328 + 1, s93 >= 0, s93 <= x_2590 + 1, s94 >= 0, s94 <= x_3146 + 1, s95 >= 0, s95 <= s92 + s93 + s94 + 1, s96 >= 0, s96 <= s95 + 1, z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s99 :|: s97 >= 0, s97 <= z' - 2 + 1, s98 >= 0, s98 <= s97 + 1, s99 >= 0, s99 <= s98 + 1, z' - 2 >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ -106 + 45*z' + 139*z'^2 + 48*z'^3 }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0 encArg(z') -{ 252 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> 1 + s11 + s12 :|: s11 >= 0, s11 <= x_1 + 1, s12 >= 0, s12 <= x_2 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 260 + 6*s41 + 2*s41*s42 + s41^2 + 4*s42 + s42^2 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> s43 :|: s41 >= 0, s41 <= z' + 1, s42 >= 0, s42 <= z'' + 1, s43 >= 0, s43 <= s41 + s42 + 1, z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ -104 + s101 + 45*z' + 139*z'^2 + 48*z'^3 }-> s102 :|: s100 >= 0, s100 <= z' - 1 + 1, s101 >= 0, s101 <= s100 + 1, s102 >= 0, s102 <= s101 + 1, z' - 1 >= 0 encode_g(z') -{ -104 + s153 + 45*z' + 139*z'^2 + 48*z'^3 }-> s154 :|: s152 >= 0, s152 <= z' - 1 + 1, s153 >= 0, s153 <= s152, s154 >= 0, s154 <= s153 + 1, z' - 1 >= 0 encode_g(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0 + 1, z' = 0 encode_g(z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 1, z' >= 0 encode_g(z') -{ 254 + s46 + s47 + 467*x_11439 + 283*x_11439^2 + 48*x_11439^3 + 467*x_2639 + 283*x_2639^2 + 48*x_2639^3 }-> s48 :|: s46 >= 0, s46 <= x_11439 + 1, s47 >= 0, s47 <= x_2639 + 1, s48 >= 0, s48 <= 1 + s46 + s47 + 1, x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 254 + s49 + s50 + 467*x_11440 + 283*x_11440^2 + 48*x_11440^3 + 467*x_2640 + 283*x_2640^2 + 48*x_2640^3 }-> s51 :|: s49 >= 0, s49 <= x_11440 + 1, s50 >= 0, s50 <= x_2640 + 1, s51 >= 0, s51 <= 1 + s49 + s50 + 1, x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ -104 + s52 + 45*z' + 139*z'^2 + 48*z'^3 }-> s53 :|: s52 >= 0, s52 <= z' - 1 + 1, s53 >= 0, s53 <= 1 + s52 + 1, z' - 1 >= 0 encode_g(z') -{ 261 + 6*s54 + 2*s54*s55 + s54^2 + 4*s55 + s55^2 + s56 + 467*x_11443 + 283*x_11443^2 + 48*x_11443^3 + 467*x_2641 + 283*x_2641^2 + 48*x_2641^3 }-> s57 :|: s54 >= 0, s54 <= x_11443 + 1, s55 >= 0, s55 <= x_2641 + 1, s56 >= 0, s56 <= s54 + s55 + 1, s57 >= 0, s57 <= s56 + 1, x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ -104 + s58 + s59 + 45*z' + 139*z'^2 + 48*z'^3 }-> s60 :|: s58 >= 0, s58 <= z' - 1 + 1, s59 >= 0, s59 <= s58 + 1, s60 >= 0, s60 <= s59 + 1, z' - 1 >= 0 encode_g(z') -{ 379 + s62 + s64 + 467*x_11445 + 283*x_11445^2 + 48*x_11445^3 + 467*x_2642 + 283*x_2642^2 + 48*x_2642^3 + 467*x_3159 + 283*x_3159^2 + 48*x_3159^3 }-> s65 :|: s61 >= 0, s61 <= x_11445 + 1, s62 >= 0, s62 <= x_2642 + 1, s63 >= 0, s63 <= x_3159 + 1, s64 >= 0, s64 <= s61 + s62 + s63 + 1, s65 >= 0, s65 <= s64 + 1, x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s73 :|: s73 >= 0, s73 <= z' + 1, z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s66 + s67 :|: s66 >= 0, s66 <= z' + 1, s67 >= 0, s67 <= z'' + 1, z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 378 + s69 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 + 467*z1 + 283*z1^2 + 48*z1^3 }-> s71 :|: s68 >= 0, s68 <= z' + 1, s69 >= 0, s69 <= z'' + 1, s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s68 + s69 + s70 + 1, z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 253 + 467*x_12654 + 283*x_12654^2 + 48*x_12654^3 + 467*x_21179 + 283*x_21179^2 + 48*x_21179^3 }-> s157 :|: s155 >= 0, s155 <= x_12654 + 1, s156 >= 0, s156 <= x_21179 + 1, s157 >= 0, s157 <= 1 + s155 + s156, x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 253 + 467*x_12655 + 283*x_12655^2 + 48*x_12655^3 + 467*x_21180 + 283*x_21180^2 + 48*x_21180^3 }-> s160 :|: s158 >= 0, s158 <= x_12655 + 1, s159 >= 0, s159 <= x_21180 + 1, s160 >= 0, s160 <= 1 + s158 + s159, z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s162 :|: s161 >= 0, s161 <= z' - 1 + 1, s162 >= 0, s162 <= 1 + s161, z' - 1 >= 0 encode_i(z') -{ 261 + 6*s163 + 2*s163*s164 + s163^2 + 4*s164 + s164^2 + 467*x_12658 + 283*x_12658^2 + 48*x_12658^3 + 467*x_21181 + 283*x_21181^2 + 48*x_21181^3 }-> s166 :|: s163 >= 0, s163 <= x_12658 + 1, s164 >= 0, s164 <= x_21181 + 1, s165 >= 0, s165 <= s163 + s164 + 1, s166 >= 0, s166 <= s165, x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ -104 + s167 + 45*z' + 139*z'^2 + 48*z'^3 }-> s169 :|: s167 >= 0, s167 <= z' - 1 + 1, s168 >= 0, s168 <= s167 + 1, s169 >= 0, s169 <= s168, z' - 1 >= 0 encode_i(z') -{ 379 + s171 + 467*x_12660 + 283*x_12660^2 + 48*x_12660^3 + 467*x_21182 + 283*x_21182^2 + 48*x_21182^3 + 467*x_3294 + 283*x_3294^2 + 48*x_3294^3 }-> s174 :|: s170 >= 0, s170 <= x_12660 + 1, s171 >= 0, s171 <= x_21182 + 1, s172 >= 0, s172 <= x_3294 + 1, s173 >= 0, s173 <= s170 + s171 + s172 + 1, s174 >= 0, s174 <= s173, z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s177 :|: s175 >= 0, s175 <= z' - 1 + 1, s176 >= 0, s176 <= s175, s177 >= 0, s177 <= s176, z' - 1 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s189 :|: s187 >= 0, s187 <= z' - 1 + 1, s188 >= 0, s188 <= s187 + 1, s189 >= 0, s189 <= s188, z' - 1 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s44 + s45 :|: s44 >= 0, s44 <= z' + 1, s45 >= 0, s45 <= z'' + 1, z' >= 0, z'' >= 0 encode_k(z') -{ 253 + 467*x_11448 + 283*x_11448^2 + 48*x_11448^3 + 467*x_2643 + 283*x_2643^2 + 48*x_2643^3 }-> s105 :|: s103 >= 0, s103 <= x_11448 + 1, s104 >= 0, s104 <= x_2643 + 1, s105 >= 0, s105 <= 1 + s103 + s104 + 1, z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 253 + 467*x_11449 + 283*x_11449^2 + 48*x_11449^3 + 467*x_2644 + 283*x_2644^2 + 48*x_2644^3 }-> s108 :|: s106 >= 0, s106 <= x_11449 + 1, s107 >= 0, s107 <= x_2644 + 1, s108 >= 0, s108 <= 1 + s106 + s107 + 1, x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s110 :|: s109 >= 0, s109 <= z' - 1 + 1, s110 >= 0, s110 <= 1 + s109 + 1, z' - 1 >= 0 encode_k(z') -{ 261 + 6*s111 + 2*s111*s112 + s111^2 + 4*s112 + s112^2 + 467*x_11452 + 283*x_11452^2 + 48*x_11452^3 + 467*x_2645 + 283*x_2645^2 + 48*x_2645^3 }-> s114 :|: s111 >= 0, s111 <= x_11452 + 1, s112 >= 0, s112 <= x_2645 + 1, s113 >= 0, s113 <= s111 + s112 + 1, s114 >= 0, s114 <= s113 + 1, x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ -104 + s115 + 45*z' + 139*z'^2 + 48*z'^3 }-> s117 :|: s115 >= 0, s115 <= z' - 1 + 1, s116 >= 0, s116 <= s115 + 1, s117 >= 0, s117 <= s116 + 1, z' - 1 >= 0 encode_k(z') -{ 379 + s119 + 467*x_11454 + 283*x_11454^2 + 48*x_11454^3 + 467*x_2646 + 283*x_2646^2 + 48*x_2646^3 + 467*x_3160 + 283*x_3160^2 + 48*x_3160^3 }-> s122 :|: s118 >= 0, s118 <= x_11454 + 1, s119 >= 0, s119 <= x_2646 + 1, s120 >= 0, s120 <= x_3160 + 1, s121 >= 0, s121 <= s118 + s119 + s120 + 1, s122 >= 0, s122 <= s121 + 1, x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s125 :|: s123 >= 0, s123 <= z' - 1 + 1, s124 >= 0, s124 <= s123 + 1, s125 >= 0, s125 <= s124 + 1, z' - 1 >= 0 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s186 :|: s184 >= 0, s184 <= z' - 1 + 1, s185 >= 0, s185 <= s184, s186 >= 0, s186 <= s185 + 1, z' - 1 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s72 :|: s72 >= 0, s72 <= z' + 1, z' >= 0 f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 10 + s9 + 6*x + x^2 }-> s10 :|: s9 >= 0, s9 <= x + 0 + 1, s10 >= 0, s10 <= s9 + 1, z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 11 + s5 + 6*x + 2*x*z'' + x^2 + 4*z'' + z''^2 }-> s6 :|: s5 >= 0, s5 <= x + (1 + 0 + (z'' - 1)) + 1, s6 >= 0, s6 <= s5 + 1, z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 23 + s7 + 10*x + 2*x*x'' + 2*x*y' + x^2 + 8*x'' + 2*x''*y' + x''^2 + 8*y' + y'^2 }-> s8 :|: s7 >= 0, s7 <= x + (1 + (1 + x'') + y') + 1, s8 >= 0, s8 <= s7 + 1, z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {encode_j}, {encode_i}, {encode_s} Previous analysis results are: h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1] encode_0: runtime: O(1) [0], size: O(1) [0] i: runtime: O(1) [1], size: O(n^1) [z'] k: runtime: O(1) [1], size: O(n^1) [1 + z'] g: runtime: O(n^1) [1 + z'], size: O(n^1) [1 + z'] f: runtime: O(n^2) [8 + 6*z' + 2*z'*z'' + z'^2 + 4*z'' + z''^2], size: O(n^1) [1 + z' + z''] encArg: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [1 + z'] encode_k: runtime: O(n^3) [1161 + 4403*z' + 3107*z'^2 + 624*z'^3], size: O(n^1) [4 + z'] encode_h: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [2 + z'] encode_h2: runtime: O(n^3) [379 + 467*z' + 283*z'^2 + 48*z'^3 + 468*z'' + 283*z''^2 + 48*z''^3 + 467*z1 + 283*z1^2 + 48*z1^3], size: O(n^1) [4 + z' + z'' + z1] encode_h1: runtime: O(n^3) [252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3], size: O(n^1) [3 + z' + z''] encode_g: runtime: O(n^3) [1171 + 4416*z' + 3107*z'^2 + 624*z'^3], size: O(n^1) [4 + z'] encode_f: runtime: O(n^3) [274 + 477*z' + 2*z'*z'' + 284*z'^2 + 48*z'^3 + 475*z'' + 284*z''^2 + 48*z''^3], size: O(n^1) [3 + z' + z''] encode_j: runtime: ?, size: O(n^1) [3 + z' + z''] ---------------------------------------- (107) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: encode_j after applying outer abstraction to obtain an ITS, resulting in: O(n^3) with polynomial bound: 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 ---------------------------------------- (108) Obligation: Complexity RNTS consisting of the following rules: encArg(z') -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z' = 1 + 0 encArg(z') -{ -58 + s127 + -89*z' + -5*z'^2 + 48*z'^3 }-> s128 :|: s126 >= 0, s126 <= z' - 2 + 1, s127 >= 0, s127 <= s126, s128 >= 0, s128 <= s127 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11313 + 283*x_11313^2 + 48*x_11313^3 + 467*x_2583 + 283*x_2583^2 + 48*x_2583^3 }-> s131 :|: s129 >= 0, s129 <= x_11313 + 1, s130 >= 0, s130 <= x_2583 + 1, s131 >= 0, s131 <= 1 + s129 + s130, x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 253 + 467*x_11314 + 283*x_11314^2 + 48*x_11314^3 + 467*x_2584 + 283*x_2584^2 + 48*x_2584^3 }-> s134 :|: s132 >= 0, s132 <= x_11314 + 1, s133 >= 0, s133 <= x_2584 + 1, s134 >= 0, s134 <= 1 + s132 + s133, z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s136 :|: s135 >= 0, s135 <= z' - 2 + 1, s136 >= 0, s136 <= 1 + s135, z' - 2 >= 0 encArg(z') -{ 261 + 6*s137 + 2*s137*s138 + s137^2 + 4*s138 + s138^2 + 467*x_11317 + 283*x_11317^2 + 48*x_11317^3 + 467*x_2585 + 283*x_2585^2 + 48*x_2585^3 }-> s140 :|: s137 >= 0, s137 <= x_11317 + 1, s138 >= 0, s138 <= x_2585 + 1, s139 >= 0, s139 <= s137 + s138 + 1, s140 >= 0, s140 <= s139, x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ -58 + s141 + -89*z' + -5*z'^2 + 48*z'^3 }-> s143 :|: s141 >= 0, s141 <= z' - 2 + 1, s142 >= 0, s142 <= s141 + 1, s143 >= 0, s143 <= s142, z' - 2 >= 0 encArg(z') -{ 379 + s145 + 467*x_11319 + 283*x_11319^2 + 48*x_11319^3 + 467*x_2586 + 283*x_2586^2 + 48*x_2586^3 + 467*x_3145 + 283*x_3145^2 + 48*x_3145^3 }-> s148 :|: s144 >= 0, s144 <= x_11319 + 1, s145 >= 0, s145 <= x_2586 + 1, s146 >= 0, s146 <= x_3145 + 1, s147 >= 0, s147 <= s144 + s145 + s146 + 1, s148 >= 0, s148 <= s147, z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s151 :|: s149 >= 0, s149 <= z' - 2 + 1, s150 >= 0, s150 <= s149, s151 >= 0, s151 <= s150, z' - 2 >= 0 encArg(z') -{ 260 + 6*s14 + 2*s14*s15 + s14^2 + 4*s15 + s15^2 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> s16 :|: s14 >= 0, s14 <= x_1 + 1, s15 >= 0, s15 <= x_2 + 1, s16 >= 0, s16 <= s14 + s15 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s180 :|: s178 >= 0, s178 <= z' - 2 + 1, s179 >= 0, s179 <= s178 + 1, s180 >= 0, s180 <= s179, z' - 2 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s183 :|: s181 >= 0, s181 <= z' - 2 + 1, s182 >= 0, s182 <= s181, s183 >= 0, s183 <= s182 + 1, z' - 2 >= 0 encArg(z') -{ 254 + s17 + s18 + 467*x_1107 + 283*x_1107^2 + 48*x_1107^3 + 467*x_247 + 283*x_247^2 + 48*x_247^3 }-> s19 :|: s17 >= 0, s17 <= x_1107 + 1, s18 >= 0, s18 <= x_247 + 1, s19 >= 0, s19 <= 1 + s17 + s18 + 1, z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0 + 1, z' - 1 >= 0 encArg(z') -{ 254 + s20 + s21 + 467*x_1108 + 283*x_1108^2 + 48*x_1108^3 + 467*x_248 + 283*x_248^2 + 48*x_248^3 }-> s22 :|: s20 >= 0, s20 <= x_1108 + 1, s21 >= 0, s21 <= x_248 + 1, s22 >= 0, s22 <= 1 + s20 + s21 + 1, x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ -58 + s23 + -89*z' + -5*z'^2 + 48*z'^3 }-> s24 :|: s23 >= 0, s23 <= z' - 2 + 1, s24 >= 0, s24 <= 1 + s23 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s25 + 2*s25*s26 + s25^2 + 4*s26 + s26^2 + s27 + 467*x_1111 + 283*x_1111^2 + 48*x_1111^3 + 467*x_249 + 283*x_249^2 + 48*x_249^3 }-> s28 :|: s25 >= 0, s25 <= x_1111 + 1, s26 >= 0, s26 <= x_249 + 1, s27 >= 0, s27 <= s25 + s26 + 1, s28 >= 0, s28 <= s27 + 1, z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ -58 + s29 + s30 + -89*z' + -5*z'^2 + 48*z'^3 }-> s31 :|: s29 >= 0, s29 <= z' - 2 + 1, s30 >= 0, s30 <= s29 + 1, s31 >= 0, s31 <= s30 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s33 + s35 + 467*x_1113 + 283*x_1113^2 + 48*x_1113^3 + 467*x_250 + 283*x_250^2 + 48*x_250^3 + 467*x_311 + 283*x_311^2 + 48*x_311^3 }-> s36 :|: s32 >= 0, s32 <= x_1113 + 1, s33 >= 0, s33 <= x_250 + 1, s34 >= 0, s34 <= x_311 + 1, s35 >= 0, s35 <= s32 + s33 + s34 + 1, s36 >= 0, s36 <= s35 + 1, x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 378 + s38 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 + 467*x_3 + 283*x_3^2 + 48*x_3^3 }-> s40 :|: s37 >= 0, s37 <= x_1 + 1, s38 >= 0, s38 <= x_2 + 1, s39 >= 0, s39 <= x_3 + 1, s40 >= 0, s40 <= s37 + s38 + s39 + 1, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z') -{ -58 + s75 + -89*z' + -5*z'^2 + 48*z'^3 }-> s76 :|: s74 >= 0, s74 <= z' - 2 + 1, s75 >= 0, s75 <= s74 + 1, s76 >= 0, s76 <= s75 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11322 + 283*x_11322^2 + 48*x_11322^3 + 467*x_2587 + 283*x_2587^2 + 48*x_2587^3 }-> s79 :|: s77 >= 0, s77 <= x_11322 + 1, s78 >= 0, s78 <= x_2587 + 1, s79 >= 0, s79 <= 1 + s77 + s78 + 1, x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 253 + 467*x_11323 + 283*x_11323^2 + 48*x_11323^3 + 467*x_2588 + 283*x_2588^2 + 48*x_2588^3 }-> s82 :|: s80 >= 0, s80 <= x_11323 + 1, s81 >= 0, s81 <= x_2588 + 1, s82 >= 0, s82 <= 1 + s80 + s81 + 1, z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s84 :|: s83 >= 0, s83 <= z' - 2 + 1, s84 >= 0, s84 <= 1 + s83 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s85 + 2*s85*s86 + s85^2 + 4*s86 + s86^2 + 467*x_11326 + 283*x_11326^2 + 48*x_11326^3 + 467*x_2589 + 283*x_2589^2 + 48*x_2589^3 }-> s88 :|: s85 >= 0, s85 <= x_11326 + 1, s86 >= 0, s86 <= x_2589 + 1, s87 >= 0, s87 <= s85 + s86 + 1, s88 >= 0, s88 <= s87 + 1, x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ -58 + s89 + -89*z' + -5*z'^2 + 48*z'^3 }-> s91 :|: s89 >= 0, s89 <= z' - 2 + 1, s90 >= 0, s90 <= s89 + 1, s91 >= 0, s91 <= s90 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s93 + 467*x_11328 + 283*x_11328^2 + 48*x_11328^3 + 467*x_2590 + 283*x_2590^2 + 48*x_2590^3 + 467*x_3146 + 283*x_3146^2 + 48*x_3146^3 }-> s96 :|: s92 >= 0, s92 <= x_11328 + 1, s93 >= 0, s93 <= x_2590 + 1, s94 >= 0, s94 <= x_3146 + 1, s95 >= 0, s95 <= s92 + s93 + s94 + 1, s96 >= 0, s96 <= s95 + 1, z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s99 :|: s97 >= 0, s97 <= z' - 2 + 1, s98 >= 0, s98 <= s97 + 1, s99 >= 0, s99 <= s98 + 1, z' - 2 >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ -106 + 45*z' + 139*z'^2 + 48*z'^3 }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0 encArg(z') -{ 252 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> 1 + s11 + s12 :|: s11 >= 0, s11 <= x_1 + 1, s12 >= 0, s12 <= x_2 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 260 + 6*s41 + 2*s41*s42 + s41^2 + 4*s42 + s42^2 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> s43 :|: s41 >= 0, s41 <= z' + 1, s42 >= 0, s42 <= z'' + 1, s43 >= 0, s43 <= s41 + s42 + 1, z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ -104 + s101 + 45*z' + 139*z'^2 + 48*z'^3 }-> s102 :|: s100 >= 0, s100 <= z' - 1 + 1, s101 >= 0, s101 <= s100 + 1, s102 >= 0, s102 <= s101 + 1, z' - 1 >= 0 encode_g(z') -{ -104 + s153 + 45*z' + 139*z'^2 + 48*z'^3 }-> s154 :|: s152 >= 0, s152 <= z' - 1 + 1, s153 >= 0, s153 <= s152, s154 >= 0, s154 <= s153 + 1, z' - 1 >= 0 encode_g(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0 + 1, z' = 0 encode_g(z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 1, z' >= 0 encode_g(z') -{ 254 + s46 + s47 + 467*x_11439 + 283*x_11439^2 + 48*x_11439^3 + 467*x_2639 + 283*x_2639^2 + 48*x_2639^3 }-> s48 :|: s46 >= 0, s46 <= x_11439 + 1, s47 >= 0, s47 <= x_2639 + 1, s48 >= 0, s48 <= 1 + s46 + s47 + 1, x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 254 + s49 + s50 + 467*x_11440 + 283*x_11440^2 + 48*x_11440^3 + 467*x_2640 + 283*x_2640^2 + 48*x_2640^3 }-> s51 :|: s49 >= 0, s49 <= x_11440 + 1, s50 >= 0, s50 <= x_2640 + 1, s51 >= 0, s51 <= 1 + s49 + s50 + 1, x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ -104 + s52 + 45*z' + 139*z'^2 + 48*z'^3 }-> s53 :|: s52 >= 0, s52 <= z' - 1 + 1, s53 >= 0, s53 <= 1 + s52 + 1, z' - 1 >= 0 encode_g(z') -{ 261 + 6*s54 + 2*s54*s55 + s54^2 + 4*s55 + s55^2 + s56 + 467*x_11443 + 283*x_11443^2 + 48*x_11443^3 + 467*x_2641 + 283*x_2641^2 + 48*x_2641^3 }-> s57 :|: s54 >= 0, s54 <= x_11443 + 1, s55 >= 0, s55 <= x_2641 + 1, s56 >= 0, s56 <= s54 + s55 + 1, s57 >= 0, s57 <= s56 + 1, x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ -104 + s58 + s59 + 45*z' + 139*z'^2 + 48*z'^3 }-> s60 :|: s58 >= 0, s58 <= z' - 1 + 1, s59 >= 0, s59 <= s58 + 1, s60 >= 0, s60 <= s59 + 1, z' - 1 >= 0 encode_g(z') -{ 379 + s62 + s64 + 467*x_11445 + 283*x_11445^2 + 48*x_11445^3 + 467*x_2642 + 283*x_2642^2 + 48*x_2642^3 + 467*x_3159 + 283*x_3159^2 + 48*x_3159^3 }-> s65 :|: s61 >= 0, s61 <= x_11445 + 1, s62 >= 0, s62 <= x_2642 + 1, s63 >= 0, s63 <= x_3159 + 1, s64 >= 0, s64 <= s61 + s62 + s63 + 1, s65 >= 0, s65 <= s64 + 1, x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s73 :|: s73 >= 0, s73 <= z' + 1, z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s66 + s67 :|: s66 >= 0, s66 <= z' + 1, s67 >= 0, s67 <= z'' + 1, z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 378 + s69 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 + 467*z1 + 283*z1^2 + 48*z1^3 }-> s71 :|: s68 >= 0, s68 <= z' + 1, s69 >= 0, s69 <= z'' + 1, s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s68 + s69 + s70 + 1, z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 253 + 467*x_12654 + 283*x_12654^2 + 48*x_12654^3 + 467*x_21179 + 283*x_21179^2 + 48*x_21179^3 }-> s157 :|: s155 >= 0, s155 <= x_12654 + 1, s156 >= 0, s156 <= x_21179 + 1, s157 >= 0, s157 <= 1 + s155 + s156, x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 253 + 467*x_12655 + 283*x_12655^2 + 48*x_12655^3 + 467*x_21180 + 283*x_21180^2 + 48*x_21180^3 }-> s160 :|: s158 >= 0, s158 <= x_12655 + 1, s159 >= 0, s159 <= x_21180 + 1, s160 >= 0, s160 <= 1 + s158 + s159, z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s162 :|: s161 >= 0, s161 <= z' - 1 + 1, s162 >= 0, s162 <= 1 + s161, z' - 1 >= 0 encode_i(z') -{ 261 + 6*s163 + 2*s163*s164 + s163^2 + 4*s164 + s164^2 + 467*x_12658 + 283*x_12658^2 + 48*x_12658^3 + 467*x_21181 + 283*x_21181^2 + 48*x_21181^3 }-> s166 :|: s163 >= 0, s163 <= x_12658 + 1, s164 >= 0, s164 <= x_21181 + 1, s165 >= 0, s165 <= s163 + s164 + 1, s166 >= 0, s166 <= s165, x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ -104 + s167 + 45*z' + 139*z'^2 + 48*z'^3 }-> s169 :|: s167 >= 0, s167 <= z' - 1 + 1, s168 >= 0, s168 <= s167 + 1, s169 >= 0, s169 <= s168, z' - 1 >= 0 encode_i(z') -{ 379 + s171 + 467*x_12660 + 283*x_12660^2 + 48*x_12660^3 + 467*x_21182 + 283*x_21182^2 + 48*x_21182^3 + 467*x_3294 + 283*x_3294^2 + 48*x_3294^3 }-> s174 :|: s170 >= 0, s170 <= x_12660 + 1, s171 >= 0, s171 <= x_21182 + 1, s172 >= 0, s172 <= x_3294 + 1, s173 >= 0, s173 <= s170 + s171 + s172 + 1, s174 >= 0, s174 <= s173, z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s177 :|: s175 >= 0, s175 <= z' - 1 + 1, s176 >= 0, s176 <= s175, s177 >= 0, s177 <= s176, z' - 1 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s189 :|: s187 >= 0, s187 <= z' - 1 + 1, s188 >= 0, s188 <= s187 + 1, s189 >= 0, s189 <= s188, z' - 1 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s44 + s45 :|: s44 >= 0, s44 <= z' + 1, s45 >= 0, s45 <= z'' + 1, z' >= 0, z'' >= 0 encode_k(z') -{ 253 + 467*x_11448 + 283*x_11448^2 + 48*x_11448^3 + 467*x_2643 + 283*x_2643^2 + 48*x_2643^3 }-> s105 :|: s103 >= 0, s103 <= x_11448 + 1, s104 >= 0, s104 <= x_2643 + 1, s105 >= 0, s105 <= 1 + s103 + s104 + 1, z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 253 + 467*x_11449 + 283*x_11449^2 + 48*x_11449^3 + 467*x_2644 + 283*x_2644^2 + 48*x_2644^3 }-> s108 :|: s106 >= 0, s106 <= x_11449 + 1, s107 >= 0, s107 <= x_2644 + 1, s108 >= 0, s108 <= 1 + s106 + s107 + 1, x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s110 :|: s109 >= 0, s109 <= z' - 1 + 1, s110 >= 0, s110 <= 1 + s109 + 1, z' - 1 >= 0 encode_k(z') -{ 261 + 6*s111 + 2*s111*s112 + s111^2 + 4*s112 + s112^2 + 467*x_11452 + 283*x_11452^2 + 48*x_11452^3 + 467*x_2645 + 283*x_2645^2 + 48*x_2645^3 }-> s114 :|: s111 >= 0, s111 <= x_11452 + 1, s112 >= 0, s112 <= x_2645 + 1, s113 >= 0, s113 <= s111 + s112 + 1, s114 >= 0, s114 <= s113 + 1, x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ -104 + s115 + 45*z' + 139*z'^2 + 48*z'^3 }-> s117 :|: s115 >= 0, s115 <= z' - 1 + 1, s116 >= 0, s116 <= s115 + 1, s117 >= 0, s117 <= s116 + 1, z' - 1 >= 0 encode_k(z') -{ 379 + s119 + 467*x_11454 + 283*x_11454^2 + 48*x_11454^3 + 467*x_2646 + 283*x_2646^2 + 48*x_2646^3 + 467*x_3160 + 283*x_3160^2 + 48*x_3160^3 }-> s122 :|: s118 >= 0, s118 <= x_11454 + 1, s119 >= 0, s119 <= x_2646 + 1, s120 >= 0, s120 <= x_3160 + 1, s121 >= 0, s121 <= s118 + s119 + s120 + 1, s122 >= 0, s122 <= s121 + 1, x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s125 :|: s123 >= 0, s123 <= z' - 1 + 1, s124 >= 0, s124 <= s123 + 1, s125 >= 0, s125 <= s124 + 1, z' - 1 >= 0 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s186 :|: s184 >= 0, s184 <= z' - 1 + 1, s185 >= 0, s185 <= s184, s186 >= 0, s186 <= s185 + 1, z' - 1 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s72 :|: s72 >= 0, s72 <= z' + 1, z' >= 0 f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 10 + s9 + 6*x + x^2 }-> s10 :|: s9 >= 0, s9 <= x + 0 + 1, s10 >= 0, s10 <= s9 + 1, z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 11 + s5 + 6*x + 2*x*z'' + x^2 + 4*z'' + z''^2 }-> s6 :|: s5 >= 0, s5 <= x + (1 + 0 + (z'' - 1)) + 1, s6 >= 0, s6 <= s5 + 1, z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 23 + s7 + 10*x + 2*x*x'' + 2*x*y' + x^2 + 8*x'' + 2*x''*y' + x''^2 + 8*y' + y'^2 }-> s8 :|: s7 >= 0, s7 <= x + (1 + (1 + x'') + y') + 1, s8 >= 0, s8 <= s7 + 1, z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {encode_i}, {encode_s} Previous analysis results are: h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1] encode_0: runtime: O(1) [0], size: O(1) [0] i: runtime: O(1) [1], size: O(n^1) [z'] k: runtime: O(1) [1], size: O(n^1) [1 + z'] g: runtime: O(n^1) [1 + z'], size: O(n^1) [1 + z'] f: runtime: O(n^2) [8 + 6*z' + 2*z'*z'' + z'^2 + 4*z'' + z''^2], size: O(n^1) [1 + z' + z''] encArg: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [1 + z'] encode_k: runtime: O(n^3) [1161 + 4403*z' + 3107*z'^2 + 624*z'^3], size: O(n^1) [4 + z'] encode_h: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [2 + z'] encode_h2: runtime: O(n^3) [379 + 467*z' + 283*z'^2 + 48*z'^3 + 468*z'' + 283*z''^2 + 48*z''^3 + 467*z1 + 283*z1^2 + 48*z1^3], size: O(n^1) [4 + z' + z'' + z1] encode_h1: runtime: O(n^3) [252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3], size: O(n^1) [3 + z' + z''] encode_g: runtime: O(n^3) [1171 + 4416*z' + 3107*z'^2 + 624*z'^3], size: O(n^1) [4 + z'] encode_f: runtime: O(n^3) [274 + 477*z' + 2*z'*z'' + 284*z'^2 + 48*z'^3 + 475*z'' + 284*z''^2 + 48*z''^3], size: O(n^1) [3 + z' + z''] encode_j: runtime: O(n^3) [252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3], size: O(n^1) [3 + z' + z''] ---------------------------------------- (109) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (110) Obligation: Complexity RNTS consisting of the following rules: encArg(z') -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z' = 1 + 0 encArg(z') -{ -58 + s127 + -89*z' + -5*z'^2 + 48*z'^3 }-> s128 :|: s126 >= 0, s126 <= z' - 2 + 1, s127 >= 0, s127 <= s126, s128 >= 0, s128 <= s127 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11313 + 283*x_11313^2 + 48*x_11313^3 + 467*x_2583 + 283*x_2583^2 + 48*x_2583^3 }-> s131 :|: s129 >= 0, s129 <= x_11313 + 1, s130 >= 0, s130 <= x_2583 + 1, s131 >= 0, s131 <= 1 + s129 + s130, x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 253 + 467*x_11314 + 283*x_11314^2 + 48*x_11314^3 + 467*x_2584 + 283*x_2584^2 + 48*x_2584^3 }-> s134 :|: s132 >= 0, s132 <= x_11314 + 1, s133 >= 0, s133 <= x_2584 + 1, s134 >= 0, s134 <= 1 + s132 + s133, z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s136 :|: s135 >= 0, s135 <= z' - 2 + 1, s136 >= 0, s136 <= 1 + s135, z' - 2 >= 0 encArg(z') -{ 261 + 6*s137 + 2*s137*s138 + s137^2 + 4*s138 + s138^2 + 467*x_11317 + 283*x_11317^2 + 48*x_11317^3 + 467*x_2585 + 283*x_2585^2 + 48*x_2585^3 }-> s140 :|: s137 >= 0, s137 <= x_11317 + 1, s138 >= 0, s138 <= x_2585 + 1, s139 >= 0, s139 <= s137 + s138 + 1, s140 >= 0, s140 <= s139, x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ -58 + s141 + -89*z' + -5*z'^2 + 48*z'^3 }-> s143 :|: s141 >= 0, s141 <= z' - 2 + 1, s142 >= 0, s142 <= s141 + 1, s143 >= 0, s143 <= s142, z' - 2 >= 0 encArg(z') -{ 379 + s145 + 467*x_11319 + 283*x_11319^2 + 48*x_11319^3 + 467*x_2586 + 283*x_2586^2 + 48*x_2586^3 + 467*x_3145 + 283*x_3145^2 + 48*x_3145^3 }-> s148 :|: s144 >= 0, s144 <= x_11319 + 1, s145 >= 0, s145 <= x_2586 + 1, s146 >= 0, s146 <= x_3145 + 1, s147 >= 0, s147 <= s144 + s145 + s146 + 1, s148 >= 0, s148 <= s147, z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s151 :|: s149 >= 0, s149 <= z' - 2 + 1, s150 >= 0, s150 <= s149, s151 >= 0, s151 <= s150, z' - 2 >= 0 encArg(z') -{ 260 + 6*s14 + 2*s14*s15 + s14^2 + 4*s15 + s15^2 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> s16 :|: s14 >= 0, s14 <= x_1 + 1, s15 >= 0, s15 <= x_2 + 1, s16 >= 0, s16 <= s14 + s15 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s180 :|: s178 >= 0, s178 <= z' - 2 + 1, s179 >= 0, s179 <= s178 + 1, s180 >= 0, s180 <= s179, z' - 2 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s183 :|: s181 >= 0, s181 <= z' - 2 + 1, s182 >= 0, s182 <= s181, s183 >= 0, s183 <= s182 + 1, z' - 2 >= 0 encArg(z') -{ 254 + s17 + s18 + 467*x_1107 + 283*x_1107^2 + 48*x_1107^3 + 467*x_247 + 283*x_247^2 + 48*x_247^3 }-> s19 :|: s17 >= 0, s17 <= x_1107 + 1, s18 >= 0, s18 <= x_247 + 1, s19 >= 0, s19 <= 1 + s17 + s18 + 1, z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0 + 1, z' - 1 >= 0 encArg(z') -{ 254 + s20 + s21 + 467*x_1108 + 283*x_1108^2 + 48*x_1108^3 + 467*x_248 + 283*x_248^2 + 48*x_248^3 }-> s22 :|: s20 >= 0, s20 <= x_1108 + 1, s21 >= 0, s21 <= x_248 + 1, s22 >= 0, s22 <= 1 + s20 + s21 + 1, x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ -58 + s23 + -89*z' + -5*z'^2 + 48*z'^3 }-> s24 :|: s23 >= 0, s23 <= z' - 2 + 1, s24 >= 0, s24 <= 1 + s23 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s25 + 2*s25*s26 + s25^2 + 4*s26 + s26^2 + s27 + 467*x_1111 + 283*x_1111^2 + 48*x_1111^3 + 467*x_249 + 283*x_249^2 + 48*x_249^3 }-> s28 :|: s25 >= 0, s25 <= x_1111 + 1, s26 >= 0, s26 <= x_249 + 1, s27 >= 0, s27 <= s25 + s26 + 1, s28 >= 0, s28 <= s27 + 1, z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ -58 + s29 + s30 + -89*z' + -5*z'^2 + 48*z'^3 }-> s31 :|: s29 >= 0, s29 <= z' - 2 + 1, s30 >= 0, s30 <= s29 + 1, s31 >= 0, s31 <= s30 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s33 + s35 + 467*x_1113 + 283*x_1113^2 + 48*x_1113^3 + 467*x_250 + 283*x_250^2 + 48*x_250^3 + 467*x_311 + 283*x_311^2 + 48*x_311^3 }-> s36 :|: s32 >= 0, s32 <= x_1113 + 1, s33 >= 0, s33 <= x_250 + 1, s34 >= 0, s34 <= x_311 + 1, s35 >= 0, s35 <= s32 + s33 + s34 + 1, s36 >= 0, s36 <= s35 + 1, x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 378 + s38 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 + 467*x_3 + 283*x_3^2 + 48*x_3^3 }-> s40 :|: s37 >= 0, s37 <= x_1 + 1, s38 >= 0, s38 <= x_2 + 1, s39 >= 0, s39 <= x_3 + 1, s40 >= 0, s40 <= s37 + s38 + s39 + 1, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z') -{ -58 + s75 + -89*z' + -5*z'^2 + 48*z'^3 }-> s76 :|: s74 >= 0, s74 <= z' - 2 + 1, s75 >= 0, s75 <= s74 + 1, s76 >= 0, s76 <= s75 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11322 + 283*x_11322^2 + 48*x_11322^3 + 467*x_2587 + 283*x_2587^2 + 48*x_2587^3 }-> s79 :|: s77 >= 0, s77 <= x_11322 + 1, s78 >= 0, s78 <= x_2587 + 1, s79 >= 0, s79 <= 1 + s77 + s78 + 1, x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 253 + 467*x_11323 + 283*x_11323^2 + 48*x_11323^3 + 467*x_2588 + 283*x_2588^2 + 48*x_2588^3 }-> s82 :|: s80 >= 0, s80 <= x_11323 + 1, s81 >= 0, s81 <= x_2588 + 1, s82 >= 0, s82 <= 1 + s80 + s81 + 1, z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s84 :|: s83 >= 0, s83 <= z' - 2 + 1, s84 >= 0, s84 <= 1 + s83 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s85 + 2*s85*s86 + s85^2 + 4*s86 + s86^2 + 467*x_11326 + 283*x_11326^2 + 48*x_11326^3 + 467*x_2589 + 283*x_2589^2 + 48*x_2589^3 }-> s88 :|: s85 >= 0, s85 <= x_11326 + 1, s86 >= 0, s86 <= x_2589 + 1, s87 >= 0, s87 <= s85 + s86 + 1, s88 >= 0, s88 <= s87 + 1, x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ -58 + s89 + -89*z' + -5*z'^2 + 48*z'^3 }-> s91 :|: s89 >= 0, s89 <= z' - 2 + 1, s90 >= 0, s90 <= s89 + 1, s91 >= 0, s91 <= s90 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s93 + 467*x_11328 + 283*x_11328^2 + 48*x_11328^3 + 467*x_2590 + 283*x_2590^2 + 48*x_2590^3 + 467*x_3146 + 283*x_3146^2 + 48*x_3146^3 }-> s96 :|: s92 >= 0, s92 <= x_11328 + 1, s93 >= 0, s93 <= x_2590 + 1, s94 >= 0, s94 <= x_3146 + 1, s95 >= 0, s95 <= s92 + s93 + s94 + 1, s96 >= 0, s96 <= s95 + 1, z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s99 :|: s97 >= 0, s97 <= z' - 2 + 1, s98 >= 0, s98 <= s97 + 1, s99 >= 0, s99 <= s98 + 1, z' - 2 >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ -106 + 45*z' + 139*z'^2 + 48*z'^3 }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0 encArg(z') -{ 252 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> 1 + s11 + s12 :|: s11 >= 0, s11 <= x_1 + 1, s12 >= 0, s12 <= x_2 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 260 + 6*s41 + 2*s41*s42 + s41^2 + 4*s42 + s42^2 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> s43 :|: s41 >= 0, s41 <= z' + 1, s42 >= 0, s42 <= z'' + 1, s43 >= 0, s43 <= s41 + s42 + 1, z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ -104 + s101 + 45*z' + 139*z'^2 + 48*z'^3 }-> s102 :|: s100 >= 0, s100 <= z' - 1 + 1, s101 >= 0, s101 <= s100 + 1, s102 >= 0, s102 <= s101 + 1, z' - 1 >= 0 encode_g(z') -{ -104 + s153 + 45*z' + 139*z'^2 + 48*z'^3 }-> s154 :|: s152 >= 0, s152 <= z' - 1 + 1, s153 >= 0, s153 <= s152, s154 >= 0, s154 <= s153 + 1, z' - 1 >= 0 encode_g(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0 + 1, z' = 0 encode_g(z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 1, z' >= 0 encode_g(z') -{ 254 + s46 + s47 + 467*x_11439 + 283*x_11439^2 + 48*x_11439^3 + 467*x_2639 + 283*x_2639^2 + 48*x_2639^3 }-> s48 :|: s46 >= 0, s46 <= x_11439 + 1, s47 >= 0, s47 <= x_2639 + 1, s48 >= 0, s48 <= 1 + s46 + s47 + 1, x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 254 + s49 + s50 + 467*x_11440 + 283*x_11440^2 + 48*x_11440^3 + 467*x_2640 + 283*x_2640^2 + 48*x_2640^3 }-> s51 :|: s49 >= 0, s49 <= x_11440 + 1, s50 >= 0, s50 <= x_2640 + 1, s51 >= 0, s51 <= 1 + s49 + s50 + 1, x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ -104 + s52 + 45*z' + 139*z'^2 + 48*z'^3 }-> s53 :|: s52 >= 0, s52 <= z' - 1 + 1, s53 >= 0, s53 <= 1 + s52 + 1, z' - 1 >= 0 encode_g(z') -{ 261 + 6*s54 + 2*s54*s55 + s54^2 + 4*s55 + s55^2 + s56 + 467*x_11443 + 283*x_11443^2 + 48*x_11443^3 + 467*x_2641 + 283*x_2641^2 + 48*x_2641^3 }-> s57 :|: s54 >= 0, s54 <= x_11443 + 1, s55 >= 0, s55 <= x_2641 + 1, s56 >= 0, s56 <= s54 + s55 + 1, s57 >= 0, s57 <= s56 + 1, x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ -104 + s58 + s59 + 45*z' + 139*z'^2 + 48*z'^3 }-> s60 :|: s58 >= 0, s58 <= z' - 1 + 1, s59 >= 0, s59 <= s58 + 1, s60 >= 0, s60 <= s59 + 1, z' - 1 >= 0 encode_g(z') -{ 379 + s62 + s64 + 467*x_11445 + 283*x_11445^2 + 48*x_11445^3 + 467*x_2642 + 283*x_2642^2 + 48*x_2642^3 + 467*x_3159 + 283*x_3159^2 + 48*x_3159^3 }-> s65 :|: s61 >= 0, s61 <= x_11445 + 1, s62 >= 0, s62 <= x_2642 + 1, s63 >= 0, s63 <= x_3159 + 1, s64 >= 0, s64 <= s61 + s62 + s63 + 1, s65 >= 0, s65 <= s64 + 1, x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s73 :|: s73 >= 0, s73 <= z' + 1, z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s66 + s67 :|: s66 >= 0, s66 <= z' + 1, s67 >= 0, s67 <= z'' + 1, z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 378 + s69 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 + 467*z1 + 283*z1^2 + 48*z1^3 }-> s71 :|: s68 >= 0, s68 <= z' + 1, s69 >= 0, s69 <= z'' + 1, s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s68 + s69 + s70 + 1, z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 253 + 467*x_12654 + 283*x_12654^2 + 48*x_12654^3 + 467*x_21179 + 283*x_21179^2 + 48*x_21179^3 }-> s157 :|: s155 >= 0, s155 <= x_12654 + 1, s156 >= 0, s156 <= x_21179 + 1, s157 >= 0, s157 <= 1 + s155 + s156, x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 253 + 467*x_12655 + 283*x_12655^2 + 48*x_12655^3 + 467*x_21180 + 283*x_21180^2 + 48*x_21180^3 }-> s160 :|: s158 >= 0, s158 <= x_12655 + 1, s159 >= 0, s159 <= x_21180 + 1, s160 >= 0, s160 <= 1 + s158 + s159, z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s162 :|: s161 >= 0, s161 <= z' - 1 + 1, s162 >= 0, s162 <= 1 + s161, z' - 1 >= 0 encode_i(z') -{ 261 + 6*s163 + 2*s163*s164 + s163^2 + 4*s164 + s164^2 + 467*x_12658 + 283*x_12658^2 + 48*x_12658^3 + 467*x_21181 + 283*x_21181^2 + 48*x_21181^3 }-> s166 :|: s163 >= 0, s163 <= x_12658 + 1, s164 >= 0, s164 <= x_21181 + 1, s165 >= 0, s165 <= s163 + s164 + 1, s166 >= 0, s166 <= s165, x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ -104 + s167 + 45*z' + 139*z'^2 + 48*z'^3 }-> s169 :|: s167 >= 0, s167 <= z' - 1 + 1, s168 >= 0, s168 <= s167 + 1, s169 >= 0, s169 <= s168, z' - 1 >= 0 encode_i(z') -{ 379 + s171 + 467*x_12660 + 283*x_12660^2 + 48*x_12660^3 + 467*x_21182 + 283*x_21182^2 + 48*x_21182^3 + 467*x_3294 + 283*x_3294^2 + 48*x_3294^3 }-> s174 :|: s170 >= 0, s170 <= x_12660 + 1, s171 >= 0, s171 <= x_21182 + 1, s172 >= 0, s172 <= x_3294 + 1, s173 >= 0, s173 <= s170 + s171 + s172 + 1, s174 >= 0, s174 <= s173, z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s177 :|: s175 >= 0, s175 <= z' - 1 + 1, s176 >= 0, s176 <= s175, s177 >= 0, s177 <= s176, z' - 1 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s189 :|: s187 >= 0, s187 <= z' - 1 + 1, s188 >= 0, s188 <= s187 + 1, s189 >= 0, s189 <= s188, z' - 1 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s44 + s45 :|: s44 >= 0, s44 <= z' + 1, s45 >= 0, s45 <= z'' + 1, z' >= 0, z'' >= 0 encode_k(z') -{ 253 + 467*x_11448 + 283*x_11448^2 + 48*x_11448^3 + 467*x_2643 + 283*x_2643^2 + 48*x_2643^3 }-> s105 :|: s103 >= 0, s103 <= x_11448 + 1, s104 >= 0, s104 <= x_2643 + 1, s105 >= 0, s105 <= 1 + s103 + s104 + 1, z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 253 + 467*x_11449 + 283*x_11449^2 + 48*x_11449^3 + 467*x_2644 + 283*x_2644^2 + 48*x_2644^3 }-> s108 :|: s106 >= 0, s106 <= x_11449 + 1, s107 >= 0, s107 <= x_2644 + 1, s108 >= 0, s108 <= 1 + s106 + s107 + 1, x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s110 :|: s109 >= 0, s109 <= z' - 1 + 1, s110 >= 0, s110 <= 1 + s109 + 1, z' - 1 >= 0 encode_k(z') -{ 261 + 6*s111 + 2*s111*s112 + s111^2 + 4*s112 + s112^2 + 467*x_11452 + 283*x_11452^2 + 48*x_11452^3 + 467*x_2645 + 283*x_2645^2 + 48*x_2645^3 }-> s114 :|: s111 >= 0, s111 <= x_11452 + 1, s112 >= 0, s112 <= x_2645 + 1, s113 >= 0, s113 <= s111 + s112 + 1, s114 >= 0, s114 <= s113 + 1, x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ -104 + s115 + 45*z' + 139*z'^2 + 48*z'^3 }-> s117 :|: s115 >= 0, s115 <= z' - 1 + 1, s116 >= 0, s116 <= s115 + 1, s117 >= 0, s117 <= s116 + 1, z' - 1 >= 0 encode_k(z') -{ 379 + s119 + 467*x_11454 + 283*x_11454^2 + 48*x_11454^3 + 467*x_2646 + 283*x_2646^2 + 48*x_2646^3 + 467*x_3160 + 283*x_3160^2 + 48*x_3160^3 }-> s122 :|: s118 >= 0, s118 <= x_11454 + 1, s119 >= 0, s119 <= x_2646 + 1, s120 >= 0, s120 <= x_3160 + 1, s121 >= 0, s121 <= s118 + s119 + s120 + 1, s122 >= 0, s122 <= s121 + 1, x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s125 :|: s123 >= 0, s123 <= z' - 1 + 1, s124 >= 0, s124 <= s123 + 1, s125 >= 0, s125 <= s124 + 1, z' - 1 >= 0 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s186 :|: s184 >= 0, s184 <= z' - 1 + 1, s185 >= 0, s185 <= s184, s186 >= 0, s186 <= s185 + 1, z' - 1 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s72 :|: s72 >= 0, s72 <= z' + 1, z' >= 0 f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 10 + s9 + 6*x + x^2 }-> s10 :|: s9 >= 0, s9 <= x + 0 + 1, s10 >= 0, s10 <= s9 + 1, z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 11 + s5 + 6*x + 2*x*z'' + x^2 + 4*z'' + z''^2 }-> s6 :|: s5 >= 0, s5 <= x + (1 + 0 + (z'' - 1)) + 1, s6 >= 0, s6 <= s5 + 1, z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 23 + s7 + 10*x + 2*x*x'' + 2*x*y' + x^2 + 8*x'' + 2*x''*y' + x''^2 + 8*y' + y'^2 }-> s8 :|: s7 >= 0, s7 <= x + (1 + (1 + x'') + y') + 1, s8 >= 0, s8 <= s7 + 1, z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {encode_i}, {encode_s} Previous analysis results are: h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1] encode_0: runtime: O(1) [0], size: O(1) [0] i: runtime: O(1) [1], size: O(n^1) [z'] k: runtime: O(1) [1], size: O(n^1) [1 + z'] g: runtime: O(n^1) [1 + z'], size: O(n^1) [1 + z'] f: runtime: O(n^2) [8 + 6*z' + 2*z'*z'' + z'^2 + 4*z'' + z''^2], size: O(n^1) [1 + z' + z''] encArg: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [1 + z'] encode_k: runtime: O(n^3) [1161 + 4403*z' + 3107*z'^2 + 624*z'^3], size: O(n^1) [4 + z'] encode_h: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [2 + z'] encode_h2: runtime: O(n^3) [379 + 467*z' + 283*z'^2 + 48*z'^3 + 468*z'' + 283*z''^2 + 48*z''^3 + 467*z1 + 283*z1^2 + 48*z1^3], size: O(n^1) [4 + z' + z'' + z1] encode_h1: runtime: O(n^3) [252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3], size: O(n^1) [3 + z' + z''] encode_g: runtime: O(n^3) [1171 + 4416*z' + 3107*z'^2 + 624*z'^3], size: O(n^1) [4 + z'] encode_f: runtime: O(n^3) [274 + 477*z' + 2*z'*z'' + 284*z'^2 + 48*z'^3 + 475*z'' + 284*z''^2 + 48*z''^3], size: O(n^1) [3 + z' + z''] encode_j: runtime: O(n^3) [252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3], size: O(n^1) [3 + z' + z''] ---------------------------------------- (111) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: encode_i after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 3 + z' ---------------------------------------- (112) Obligation: Complexity RNTS consisting of the following rules: encArg(z') -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z' = 1 + 0 encArg(z') -{ -58 + s127 + -89*z' + -5*z'^2 + 48*z'^3 }-> s128 :|: s126 >= 0, s126 <= z' - 2 + 1, s127 >= 0, s127 <= s126, s128 >= 0, s128 <= s127 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11313 + 283*x_11313^2 + 48*x_11313^3 + 467*x_2583 + 283*x_2583^2 + 48*x_2583^3 }-> s131 :|: s129 >= 0, s129 <= x_11313 + 1, s130 >= 0, s130 <= x_2583 + 1, s131 >= 0, s131 <= 1 + s129 + s130, x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 253 + 467*x_11314 + 283*x_11314^2 + 48*x_11314^3 + 467*x_2584 + 283*x_2584^2 + 48*x_2584^3 }-> s134 :|: s132 >= 0, s132 <= x_11314 + 1, s133 >= 0, s133 <= x_2584 + 1, s134 >= 0, s134 <= 1 + s132 + s133, z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s136 :|: s135 >= 0, s135 <= z' - 2 + 1, s136 >= 0, s136 <= 1 + s135, z' - 2 >= 0 encArg(z') -{ 261 + 6*s137 + 2*s137*s138 + s137^2 + 4*s138 + s138^2 + 467*x_11317 + 283*x_11317^2 + 48*x_11317^3 + 467*x_2585 + 283*x_2585^2 + 48*x_2585^3 }-> s140 :|: s137 >= 0, s137 <= x_11317 + 1, s138 >= 0, s138 <= x_2585 + 1, s139 >= 0, s139 <= s137 + s138 + 1, s140 >= 0, s140 <= s139, x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ -58 + s141 + -89*z' + -5*z'^2 + 48*z'^3 }-> s143 :|: s141 >= 0, s141 <= z' - 2 + 1, s142 >= 0, s142 <= s141 + 1, s143 >= 0, s143 <= s142, z' - 2 >= 0 encArg(z') -{ 379 + s145 + 467*x_11319 + 283*x_11319^2 + 48*x_11319^3 + 467*x_2586 + 283*x_2586^2 + 48*x_2586^3 + 467*x_3145 + 283*x_3145^2 + 48*x_3145^3 }-> s148 :|: s144 >= 0, s144 <= x_11319 + 1, s145 >= 0, s145 <= x_2586 + 1, s146 >= 0, s146 <= x_3145 + 1, s147 >= 0, s147 <= s144 + s145 + s146 + 1, s148 >= 0, s148 <= s147, z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s151 :|: s149 >= 0, s149 <= z' - 2 + 1, s150 >= 0, s150 <= s149, s151 >= 0, s151 <= s150, z' - 2 >= 0 encArg(z') -{ 260 + 6*s14 + 2*s14*s15 + s14^2 + 4*s15 + s15^2 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> s16 :|: s14 >= 0, s14 <= x_1 + 1, s15 >= 0, s15 <= x_2 + 1, s16 >= 0, s16 <= s14 + s15 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s180 :|: s178 >= 0, s178 <= z' - 2 + 1, s179 >= 0, s179 <= s178 + 1, s180 >= 0, s180 <= s179, z' - 2 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s183 :|: s181 >= 0, s181 <= z' - 2 + 1, s182 >= 0, s182 <= s181, s183 >= 0, s183 <= s182 + 1, z' - 2 >= 0 encArg(z') -{ 254 + s17 + s18 + 467*x_1107 + 283*x_1107^2 + 48*x_1107^3 + 467*x_247 + 283*x_247^2 + 48*x_247^3 }-> s19 :|: s17 >= 0, s17 <= x_1107 + 1, s18 >= 0, s18 <= x_247 + 1, s19 >= 0, s19 <= 1 + s17 + s18 + 1, z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0 + 1, z' - 1 >= 0 encArg(z') -{ 254 + s20 + s21 + 467*x_1108 + 283*x_1108^2 + 48*x_1108^3 + 467*x_248 + 283*x_248^2 + 48*x_248^3 }-> s22 :|: s20 >= 0, s20 <= x_1108 + 1, s21 >= 0, s21 <= x_248 + 1, s22 >= 0, s22 <= 1 + s20 + s21 + 1, x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ -58 + s23 + -89*z' + -5*z'^2 + 48*z'^3 }-> s24 :|: s23 >= 0, s23 <= z' - 2 + 1, s24 >= 0, s24 <= 1 + s23 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s25 + 2*s25*s26 + s25^2 + 4*s26 + s26^2 + s27 + 467*x_1111 + 283*x_1111^2 + 48*x_1111^3 + 467*x_249 + 283*x_249^2 + 48*x_249^3 }-> s28 :|: s25 >= 0, s25 <= x_1111 + 1, s26 >= 0, s26 <= x_249 + 1, s27 >= 0, s27 <= s25 + s26 + 1, s28 >= 0, s28 <= s27 + 1, z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ -58 + s29 + s30 + -89*z' + -5*z'^2 + 48*z'^3 }-> s31 :|: s29 >= 0, s29 <= z' - 2 + 1, s30 >= 0, s30 <= s29 + 1, s31 >= 0, s31 <= s30 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s33 + s35 + 467*x_1113 + 283*x_1113^2 + 48*x_1113^3 + 467*x_250 + 283*x_250^2 + 48*x_250^3 + 467*x_311 + 283*x_311^2 + 48*x_311^3 }-> s36 :|: s32 >= 0, s32 <= x_1113 + 1, s33 >= 0, s33 <= x_250 + 1, s34 >= 0, s34 <= x_311 + 1, s35 >= 0, s35 <= s32 + s33 + s34 + 1, s36 >= 0, s36 <= s35 + 1, x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 378 + s38 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 + 467*x_3 + 283*x_3^2 + 48*x_3^3 }-> s40 :|: s37 >= 0, s37 <= x_1 + 1, s38 >= 0, s38 <= x_2 + 1, s39 >= 0, s39 <= x_3 + 1, s40 >= 0, s40 <= s37 + s38 + s39 + 1, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z') -{ -58 + s75 + -89*z' + -5*z'^2 + 48*z'^3 }-> s76 :|: s74 >= 0, s74 <= z' - 2 + 1, s75 >= 0, s75 <= s74 + 1, s76 >= 0, s76 <= s75 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11322 + 283*x_11322^2 + 48*x_11322^3 + 467*x_2587 + 283*x_2587^2 + 48*x_2587^3 }-> s79 :|: s77 >= 0, s77 <= x_11322 + 1, s78 >= 0, s78 <= x_2587 + 1, s79 >= 0, s79 <= 1 + s77 + s78 + 1, x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 253 + 467*x_11323 + 283*x_11323^2 + 48*x_11323^3 + 467*x_2588 + 283*x_2588^2 + 48*x_2588^3 }-> s82 :|: s80 >= 0, s80 <= x_11323 + 1, s81 >= 0, s81 <= x_2588 + 1, s82 >= 0, s82 <= 1 + s80 + s81 + 1, z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s84 :|: s83 >= 0, s83 <= z' - 2 + 1, s84 >= 0, s84 <= 1 + s83 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s85 + 2*s85*s86 + s85^2 + 4*s86 + s86^2 + 467*x_11326 + 283*x_11326^2 + 48*x_11326^3 + 467*x_2589 + 283*x_2589^2 + 48*x_2589^3 }-> s88 :|: s85 >= 0, s85 <= x_11326 + 1, s86 >= 0, s86 <= x_2589 + 1, s87 >= 0, s87 <= s85 + s86 + 1, s88 >= 0, s88 <= s87 + 1, x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ -58 + s89 + -89*z' + -5*z'^2 + 48*z'^3 }-> s91 :|: s89 >= 0, s89 <= z' - 2 + 1, s90 >= 0, s90 <= s89 + 1, s91 >= 0, s91 <= s90 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s93 + 467*x_11328 + 283*x_11328^2 + 48*x_11328^3 + 467*x_2590 + 283*x_2590^2 + 48*x_2590^3 + 467*x_3146 + 283*x_3146^2 + 48*x_3146^3 }-> s96 :|: s92 >= 0, s92 <= x_11328 + 1, s93 >= 0, s93 <= x_2590 + 1, s94 >= 0, s94 <= x_3146 + 1, s95 >= 0, s95 <= s92 + s93 + s94 + 1, s96 >= 0, s96 <= s95 + 1, z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s99 :|: s97 >= 0, s97 <= z' - 2 + 1, s98 >= 0, s98 <= s97 + 1, s99 >= 0, s99 <= s98 + 1, z' - 2 >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ -106 + 45*z' + 139*z'^2 + 48*z'^3 }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0 encArg(z') -{ 252 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> 1 + s11 + s12 :|: s11 >= 0, s11 <= x_1 + 1, s12 >= 0, s12 <= x_2 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 260 + 6*s41 + 2*s41*s42 + s41^2 + 4*s42 + s42^2 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> s43 :|: s41 >= 0, s41 <= z' + 1, s42 >= 0, s42 <= z'' + 1, s43 >= 0, s43 <= s41 + s42 + 1, z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ -104 + s101 + 45*z' + 139*z'^2 + 48*z'^3 }-> s102 :|: s100 >= 0, s100 <= z' - 1 + 1, s101 >= 0, s101 <= s100 + 1, s102 >= 0, s102 <= s101 + 1, z' - 1 >= 0 encode_g(z') -{ -104 + s153 + 45*z' + 139*z'^2 + 48*z'^3 }-> s154 :|: s152 >= 0, s152 <= z' - 1 + 1, s153 >= 0, s153 <= s152, s154 >= 0, s154 <= s153 + 1, z' - 1 >= 0 encode_g(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0 + 1, z' = 0 encode_g(z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 1, z' >= 0 encode_g(z') -{ 254 + s46 + s47 + 467*x_11439 + 283*x_11439^2 + 48*x_11439^3 + 467*x_2639 + 283*x_2639^2 + 48*x_2639^3 }-> s48 :|: s46 >= 0, s46 <= x_11439 + 1, s47 >= 0, s47 <= x_2639 + 1, s48 >= 0, s48 <= 1 + s46 + s47 + 1, x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 254 + s49 + s50 + 467*x_11440 + 283*x_11440^2 + 48*x_11440^3 + 467*x_2640 + 283*x_2640^2 + 48*x_2640^3 }-> s51 :|: s49 >= 0, s49 <= x_11440 + 1, s50 >= 0, s50 <= x_2640 + 1, s51 >= 0, s51 <= 1 + s49 + s50 + 1, x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ -104 + s52 + 45*z' + 139*z'^2 + 48*z'^3 }-> s53 :|: s52 >= 0, s52 <= z' - 1 + 1, s53 >= 0, s53 <= 1 + s52 + 1, z' - 1 >= 0 encode_g(z') -{ 261 + 6*s54 + 2*s54*s55 + s54^2 + 4*s55 + s55^2 + s56 + 467*x_11443 + 283*x_11443^2 + 48*x_11443^3 + 467*x_2641 + 283*x_2641^2 + 48*x_2641^3 }-> s57 :|: s54 >= 0, s54 <= x_11443 + 1, s55 >= 0, s55 <= x_2641 + 1, s56 >= 0, s56 <= s54 + s55 + 1, s57 >= 0, s57 <= s56 + 1, x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ -104 + s58 + s59 + 45*z' + 139*z'^2 + 48*z'^3 }-> s60 :|: s58 >= 0, s58 <= z' - 1 + 1, s59 >= 0, s59 <= s58 + 1, s60 >= 0, s60 <= s59 + 1, z' - 1 >= 0 encode_g(z') -{ 379 + s62 + s64 + 467*x_11445 + 283*x_11445^2 + 48*x_11445^3 + 467*x_2642 + 283*x_2642^2 + 48*x_2642^3 + 467*x_3159 + 283*x_3159^2 + 48*x_3159^3 }-> s65 :|: s61 >= 0, s61 <= x_11445 + 1, s62 >= 0, s62 <= x_2642 + 1, s63 >= 0, s63 <= x_3159 + 1, s64 >= 0, s64 <= s61 + s62 + s63 + 1, s65 >= 0, s65 <= s64 + 1, x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s73 :|: s73 >= 0, s73 <= z' + 1, z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s66 + s67 :|: s66 >= 0, s66 <= z' + 1, s67 >= 0, s67 <= z'' + 1, z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 378 + s69 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 + 467*z1 + 283*z1^2 + 48*z1^3 }-> s71 :|: s68 >= 0, s68 <= z' + 1, s69 >= 0, s69 <= z'' + 1, s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s68 + s69 + s70 + 1, z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 253 + 467*x_12654 + 283*x_12654^2 + 48*x_12654^3 + 467*x_21179 + 283*x_21179^2 + 48*x_21179^3 }-> s157 :|: s155 >= 0, s155 <= x_12654 + 1, s156 >= 0, s156 <= x_21179 + 1, s157 >= 0, s157 <= 1 + s155 + s156, x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 253 + 467*x_12655 + 283*x_12655^2 + 48*x_12655^3 + 467*x_21180 + 283*x_21180^2 + 48*x_21180^3 }-> s160 :|: s158 >= 0, s158 <= x_12655 + 1, s159 >= 0, s159 <= x_21180 + 1, s160 >= 0, s160 <= 1 + s158 + s159, z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s162 :|: s161 >= 0, s161 <= z' - 1 + 1, s162 >= 0, s162 <= 1 + s161, z' - 1 >= 0 encode_i(z') -{ 261 + 6*s163 + 2*s163*s164 + s163^2 + 4*s164 + s164^2 + 467*x_12658 + 283*x_12658^2 + 48*x_12658^3 + 467*x_21181 + 283*x_21181^2 + 48*x_21181^3 }-> s166 :|: s163 >= 0, s163 <= x_12658 + 1, s164 >= 0, s164 <= x_21181 + 1, s165 >= 0, s165 <= s163 + s164 + 1, s166 >= 0, s166 <= s165, x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ -104 + s167 + 45*z' + 139*z'^2 + 48*z'^3 }-> s169 :|: s167 >= 0, s167 <= z' - 1 + 1, s168 >= 0, s168 <= s167 + 1, s169 >= 0, s169 <= s168, z' - 1 >= 0 encode_i(z') -{ 379 + s171 + 467*x_12660 + 283*x_12660^2 + 48*x_12660^3 + 467*x_21182 + 283*x_21182^2 + 48*x_21182^3 + 467*x_3294 + 283*x_3294^2 + 48*x_3294^3 }-> s174 :|: s170 >= 0, s170 <= x_12660 + 1, s171 >= 0, s171 <= x_21182 + 1, s172 >= 0, s172 <= x_3294 + 1, s173 >= 0, s173 <= s170 + s171 + s172 + 1, s174 >= 0, s174 <= s173, z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s177 :|: s175 >= 0, s175 <= z' - 1 + 1, s176 >= 0, s176 <= s175, s177 >= 0, s177 <= s176, z' - 1 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s189 :|: s187 >= 0, s187 <= z' - 1 + 1, s188 >= 0, s188 <= s187 + 1, s189 >= 0, s189 <= s188, z' - 1 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s44 + s45 :|: s44 >= 0, s44 <= z' + 1, s45 >= 0, s45 <= z'' + 1, z' >= 0, z'' >= 0 encode_k(z') -{ 253 + 467*x_11448 + 283*x_11448^2 + 48*x_11448^3 + 467*x_2643 + 283*x_2643^2 + 48*x_2643^3 }-> s105 :|: s103 >= 0, s103 <= x_11448 + 1, s104 >= 0, s104 <= x_2643 + 1, s105 >= 0, s105 <= 1 + s103 + s104 + 1, z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 253 + 467*x_11449 + 283*x_11449^2 + 48*x_11449^3 + 467*x_2644 + 283*x_2644^2 + 48*x_2644^3 }-> s108 :|: s106 >= 0, s106 <= x_11449 + 1, s107 >= 0, s107 <= x_2644 + 1, s108 >= 0, s108 <= 1 + s106 + s107 + 1, x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s110 :|: s109 >= 0, s109 <= z' - 1 + 1, s110 >= 0, s110 <= 1 + s109 + 1, z' - 1 >= 0 encode_k(z') -{ 261 + 6*s111 + 2*s111*s112 + s111^2 + 4*s112 + s112^2 + 467*x_11452 + 283*x_11452^2 + 48*x_11452^3 + 467*x_2645 + 283*x_2645^2 + 48*x_2645^3 }-> s114 :|: s111 >= 0, s111 <= x_11452 + 1, s112 >= 0, s112 <= x_2645 + 1, s113 >= 0, s113 <= s111 + s112 + 1, s114 >= 0, s114 <= s113 + 1, x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ -104 + s115 + 45*z' + 139*z'^2 + 48*z'^3 }-> s117 :|: s115 >= 0, s115 <= z' - 1 + 1, s116 >= 0, s116 <= s115 + 1, s117 >= 0, s117 <= s116 + 1, z' - 1 >= 0 encode_k(z') -{ 379 + s119 + 467*x_11454 + 283*x_11454^2 + 48*x_11454^3 + 467*x_2646 + 283*x_2646^2 + 48*x_2646^3 + 467*x_3160 + 283*x_3160^2 + 48*x_3160^3 }-> s122 :|: s118 >= 0, s118 <= x_11454 + 1, s119 >= 0, s119 <= x_2646 + 1, s120 >= 0, s120 <= x_3160 + 1, s121 >= 0, s121 <= s118 + s119 + s120 + 1, s122 >= 0, s122 <= s121 + 1, x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s125 :|: s123 >= 0, s123 <= z' - 1 + 1, s124 >= 0, s124 <= s123 + 1, s125 >= 0, s125 <= s124 + 1, z' - 1 >= 0 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s186 :|: s184 >= 0, s184 <= z' - 1 + 1, s185 >= 0, s185 <= s184, s186 >= 0, s186 <= s185 + 1, z' - 1 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s72 :|: s72 >= 0, s72 <= z' + 1, z' >= 0 f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 10 + s9 + 6*x + x^2 }-> s10 :|: s9 >= 0, s9 <= x + 0 + 1, s10 >= 0, s10 <= s9 + 1, z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 11 + s5 + 6*x + 2*x*z'' + x^2 + 4*z'' + z''^2 }-> s6 :|: s5 >= 0, s5 <= x + (1 + 0 + (z'' - 1)) + 1, s6 >= 0, s6 <= s5 + 1, z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 23 + s7 + 10*x + 2*x*x'' + 2*x*y' + x^2 + 8*x'' + 2*x''*y' + x''^2 + 8*y' + y'^2 }-> s8 :|: s7 >= 0, s7 <= x + (1 + (1 + x'') + y') + 1, s8 >= 0, s8 <= s7 + 1, z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {encode_i}, {encode_s} Previous analysis results are: h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1] encode_0: runtime: O(1) [0], size: O(1) [0] i: runtime: O(1) [1], size: O(n^1) [z'] k: runtime: O(1) [1], size: O(n^1) [1 + z'] g: runtime: O(n^1) [1 + z'], size: O(n^1) [1 + z'] f: runtime: O(n^2) [8 + 6*z' + 2*z'*z'' + z'^2 + 4*z'' + z''^2], size: O(n^1) [1 + z' + z''] encArg: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [1 + z'] encode_k: runtime: O(n^3) [1161 + 4403*z' + 3107*z'^2 + 624*z'^3], size: O(n^1) [4 + z'] encode_h: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [2 + z'] encode_h2: runtime: O(n^3) [379 + 467*z' + 283*z'^2 + 48*z'^3 + 468*z'' + 283*z''^2 + 48*z''^3 + 467*z1 + 283*z1^2 + 48*z1^3], size: O(n^1) [4 + z' + z'' + z1] encode_h1: runtime: O(n^3) [252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3], size: O(n^1) [3 + z' + z''] encode_g: runtime: O(n^3) [1171 + 4416*z' + 3107*z'^2 + 624*z'^3], size: O(n^1) [4 + z'] encode_f: runtime: O(n^3) [274 + 477*z' + 2*z'*z'' + 284*z'^2 + 48*z'^3 + 475*z'' + 284*z''^2 + 48*z''^3], size: O(n^1) [3 + z' + z''] encode_j: runtime: O(n^3) [252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3], size: O(n^1) [3 + z' + z''] encode_i: runtime: ?, size: O(n^1) [3 + z'] ---------------------------------------- (113) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: encode_i after applying outer abstraction to obtain an ITS, resulting in: O(n^3) with polynomial bound: 1161 + 4403*z' + 3107*z'^2 + 624*z'^3 ---------------------------------------- (114) Obligation: Complexity RNTS consisting of the following rules: encArg(z') -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z' = 1 + 0 encArg(z') -{ -58 + s127 + -89*z' + -5*z'^2 + 48*z'^3 }-> s128 :|: s126 >= 0, s126 <= z' - 2 + 1, s127 >= 0, s127 <= s126, s128 >= 0, s128 <= s127 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11313 + 283*x_11313^2 + 48*x_11313^3 + 467*x_2583 + 283*x_2583^2 + 48*x_2583^3 }-> s131 :|: s129 >= 0, s129 <= x_11313 + 1, s130 >= 0, s130 <= x_2583 + 1, s131 >= 0, s131 <= 1 + s129 + s130, x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 253 + 467*x_11314 + 283*x_11314^2 + 48*x_11314^3 + 467*x_2584 + 283*x_2584^2 + 48*x_2584^3 }-> s134 :|: s132 >= 0, s132 <= x_11314 + 1, s133 >= 0, s133 <= x_2584 + 1, s134 >= 0, s134 <= 1 + s132 + s133, z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s136 :|: s135 >= 0, s135 <= z' - 2 + 1, s136 >= 0, s136 <= 1 + s135, z' - 2 >= 0 encArg(z') -{ 261 + 6*s137 + 2*s137*s138 + s137^2 + 4*s138 + s138^2 + 467*x_11317 + 283*x_11317^2 + 48*x_11317^3 + 467*x_2585 + 283*x_2585^2 + 48*x_2585^3 }-> s140 :|: s137 >= 0, s137 <= x_11317 + 1, s138 >= 0, s138 <= x_2585 + 1, s139 >= 0, s139 <= s137 + s138 + 1, s140 >= 0, s140 <= s139, x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ -58 + s141 + -89*z' + -5*z'^2 + 48*z'^3 }-> s143 :|: s141 >= 0, s141 <= z' - 2 + 1, s142 >= 0, s142 <= s141 + 1, s143 >= 0, s143 <= s142, z' - 2 >= 0 encArg(z') -{ 379 + s145 + 467*x_11319 + 283*x_11319^2 + 48*x_11319^3 + 467*x_2586 + 283*x_2586^2 + 48*x_2586^3 + 467*x_3145 + 283*x_3145^2 + 48*x_3145^3 }-> s148 :|: s144 >= 0, s144 <= x_11319 + 1, s145 >= 0, s145 <= x_2586 + 1, s146 >= 0, s146 <= x_3145 + 1, s147 >= 0, s147 <= s144 + s145 + s146 + 1, s148 >= 0, s148 <= s147, z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s151 :|: s149 >= 0, s149 <= z' - 2 + 1, s150 >= 0, s150 <= s149, s151 >= 0, s151 <= s150, z' - 2 >= 0 encArg(z') -{ 260 + 6*s14 + 2*s14*s15 + s14^2 + 4*s15 + s15^2 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> s16 :|: s14 >= 0, s14 <= x_1 + 1, s15 >= 0, s15 <= x_2 + 1, s16 >= 0, s16 <= s14 + s15 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s180 :|: s178 >= 0, s178 <= z' - 2 + 1, s179 >= 0, s179 <= s178 + 1, s180 >= 0, s180 <= s179, z' - 2 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s183 :|: s181 >= 0, s181 <= z' - 2 + 1, s182 >= 0, s182 <= s181, s183 >= 0, s183 <= s182 + 1, z' - 2 >= 0 encArg(z') -{ 254 + s17 + s18 + 467*x_1107 + 283*x_1107^2 + 48*x_1107^3 + 467*x_247 + 283*x_247^2 + 48*x_247^3 }-> s19 :|: s17 >= 0, s17 <= x_1107 + 1, s18 >= 0, s18 <= x_247 + 1, s19 >= 0, s19 <= 1 + s17 + s18 + 1, z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0 + 1, z' - 1 >= 0 encArg(z') -{ 254 + s20 + s21 + 467*x_1108 + 283*x_1108^2 + 48*x_1108^3 + 467*x_248 + 283*x_248^2 + 48*x_248^3 }-> s22 :|: s20 >= 0, s20 <= x_1108 + 1, s21 >= 0, s21 <= x_248 + 1, s22 >= 0, s22 <= 1 + s20 + s21 + 1, x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ -58 + s23 + -89*z' + -5*z'^2 + 48*z'^3 }-> s24 :|: s23 >= 0, s23 <= z' - 2 + 1, s24 >= 0, s24 <= 1 + s23 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s25 + 2*s25*s26 + s25^2 + 4*s26 + s26^2 + s27 + 467*x_1111 + 283*x_1111^2 + 48*x_1111^3 + 467*x_249 + 283*x_249^2 + 48*x_249^3 }-> s28 :|: s25 >= 0, s25 <= x_1111 + 1, s26 >= 0, s26 <= x_249 + 1, s27 >= 0, s27 <= s25 + s26 + 1, s28 >= 0, s28 <= s27 + 1, z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ -58 + s29 + s30 + -89*z' + -5*z'^2 + 48*z'^3 }-> s31 :|: s29 >= 0, s29 <= z' - 2 + 1, s30 >= 0, s30 <= s29 + 1, s31 >= 0, s31 <= s30 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s33 + s35 + 467*x_1113 + 283*x_1113^2 + 48*x_1113^3 + 467*x_250 + 283*x_250^2 + 48*x_250^3 + 467*x_311 + 283*x_311^2 + 48*x_311^3 }-> s36 :|: s32 >= 0, s32 <= x_1113 + 1, s33 >= 0, s33 <= x_250 + 1, s34 >= 0, s34 <= x_311 + 1, s35 >= 0, s35 <= s32 + s33 + s34 + 1, s36 >= 0, s36 <= s35 + 1, x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 378 + s38 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 + 467*x_3 + 283*x_3^2 + 48*x_3^3 }-> s40 :|: s37 >= 0, s37 <= x_1 + 1, s38 >= 0, s38 <= x_2 + 1, s39 >= 0, s39 <= x_3 + 1, s40 >= 0, s40 <= s37 + s38 + s39 + 1, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z') -{ -58 + s75 + -89*z' + -5*z'^2 + 48*z'^3 }-> s76 :|: s74 >= 0, s74 <= z' - 2 + 1, s75 >= 0, s75 <= s74 + 1, s76 >= 0, s76 <= s75 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11322 + 283*x_11322^2 + 48*x_11322^3 + 467*x_2587 + 283*x_2587^2 + 48*x_2587^3 }-> s79 :|: s77 >= 0, s77 <= x_11322 + 1, s78 >= 0, s78 <= x_2587 + 1, s79 >= 0, s79 <= 1 + s77 + s78 + 1, x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 253 + 467*x_11323 + 283*x_11323^2 + 48*x_11323^3 + 467*x_2588 + 283*x_2588^2 + 48*x_2588^3 }-> s82 :|: s80 >= 0, s80 <= x_11323 + 1, s81 >= 0, s81 <= x_2588 + 1, s82 >= 0, s82 <= 1 + s80 + s81 + 1, z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s84 :|: s83 >= 0, s83 <= z' - 2 + 1, s84 >= 0, s84 <= 1 + s83 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s85 + 2*s85*s86 + s85^2 + 4*s86 + s86^2 + 467*x_11326 + 283*x_11326^2 + 48*x_11326^3 + 467*x_2589 + 283*x_2589^2 + 48*x_2589^3 }-> s88 :|: s85 >= 0, s85 <= x_11326 + 1, s86 >= 0, s86 <= x_2589 + 1, s87 >= 0, s87 <= s85 + s86 + 1, s88 >= 0, s88 <= s87 + 1, x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ -58 + s89 + -89*z' + -5*z'^2 + 48*z'^3 }-> s91 :|: s89 >= 0, s89 <= z' - 2 + 1, s90 >= 0, s90 <= s89 + 1, s91 >= 0, s91 <= s90 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s93 + 467*x_11328 + 283*x_11328^2 + 48*x_11328^3 + 467*x_2590 + 283*x_2590^2 + 48*x_2590^3 + 467*x_3146 + 283*x_3146^2 + 48*x_3146^3 }-> s96 :|: s92 >= 0, s92 <= x_11328 + 1, s93 >= 0, s93 <= x_2590 + 1, s94 >= 0, s94 <= x_3146 + 1, s95 >= 0, s95 <= s92 + s93 + s94 + 1, s96 >= 0, s96 <= s95 + 1, z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s99 :|: s97 >= 0, s97 <= z' - 2 + 1, s98 >= 0, s98 <= s97 + 1, s99 >= 0, s99 <= s98 + 1, z' - 2 >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ -106 + 45*z' + 139*z'^2 + 48*z'^3 }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0 encArg(z') -{ 252 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> 1 + s11 + s12 :|: s11 >= 0, s11 <= x_1 + 1, s12 >= 0, s12 <= x_2 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 260 + 6*s41 + 2*s41*s42 + s41^2 + 4*s42 + s42^2 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> s43 :|: s41 >= 0, s41 <= z' + 1, s42 >= 0, s42 <= z'' + 1, s43 >= 0, s43 <= s41 + s42 + 1, z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ -104 + s101 + 45*z' + 139*z'^2 + 48*z'^3 }-> s102 :|: s100 >= 0, s100 <= z' - 1 + 1, s101 >= 0, s101 <= s100 + 1, s102 >= 0, s102 <= s101 + 1, z' - 1 >= 0 encode_g(z') -{ -104 + s153 + 45*z' + 139*z'^2 + 48*z'^3 }-> s154 :|: s152 >= 0, s152 <= z' - 1 + 1, s153 >= 0, s153 <= s152, s154 >= 0, s154 <= s153 + 1, z' - 1 >= 0 encode_g(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0 + 1, z' = 0 encode_g(z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 1, z' >= 0 encode_g(z') -{ 254 + s46 + s47 + 467*x_11439 + 283*x_11439^2 + 48*x_11439^3 + 467*x_2639 + 283*x_2639^2 + 48*x_2639^3 }-> s48 :|: s46 >= 0, s46 <= x_11439 + 1, s47 >= 0, s47 <= x_2639 + 1, s48 >= 0, s48 <= 1 + s46 + s47 + 1, x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 254 + s49 + s50 + 467*x_11440 + 283*x_11440^2 + 48*x_11440^3 + 467*x_2640 + 283*x_2640^2 + 48*x_2640^3 }-> s51 :|: s49 >= 0, s49 <= x_11440 + 1, s50 >= 0, s50 <= x_2640 + 1, s51 >= 0, s51 <= 1 + s49 + s50 + 1, x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ -104 + s52 + 45*z' + 139*z'^2 + 48*z'^3 }-> s53 :|: s52 >= 0, s52 <= z' - 1 + 1, s53 >= 0, s53 <= 1 + s52 + 1, z' - 1 >= 0 encode_g(z') -{ 261 + 6*s54 + 2*s54*s55 + s54^2 + 4*s55 + s55^2 + s56 + 467*x_11443 + 283*x_11443^2 + 48*x_11443^3 + 467*x_2641 + 283*x_2641^2 + 48*x_2641^3 }-> s57 :|: s54 >= 0, s54 <= x_11443 + 1, s55 >= 0, s55 <= x_2641 + 1, s56 >= 0, s56 <= s54 + s55 + 1, s57 >= 0, s57 <= s56 + 1, x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ -104 + s58 + s59 + 45*z' + 139*z'^2 + 48*z'^3 }-> s60 :|: s58 >= 0, s58 <= z' - 1 + 1, s59 >= 0, s59 <= s58 + 1, s60 >= 0, s60 <= s59 + 1, z' - 1 >= 0 encode_g(z') -{ 379 + s62 + s64 + 467*x_11445 + 283*x_11445^2 + 48*x_11445^3 + 467*x_2642 + 283*x_2642^2 + 48*x_2642^3 + 467*x_3159 + 283*x_3159^2 + 48*x_3159^3 }-> s65 :|: s61 >= 0, s61 <= x_11445 + 1, s62 >= 0, s62 <= x_2642 + 1, s63 >= 0, s63 <= x_3159 + 1, s64 >= 0, s64 <= s61 + s62 + s63 + 1, s65 >= 0, s65 <= s64 + 1, x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s73 :|: s73 >= 0, s73 <= z' + 1, z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s66 + s67 :|: s66 >= 0, s66 <= z' + 1, s67 >= 0, s67 <= z'' + 1, z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 378 + s69 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 + 467*z1 + 283*z1^2 + 48*z1^3 }-> s71 :|: s68 >= 0, s68 <= z' + 1, s69 >= 0, s69 <= z'' + 1, s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s68 + s69 + s70 + 1, z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 253 + 467*x_12654 + 283*x_12654^2 + 48*x_12654^3 + 467*x_21179 + 283*x_21179^2 + 48*x_21179^3 }-> s157 :|: s155 >= 0, s155 <= x_12654 + 1, s156 >= 0, s156 <= x_21179 + 1, s157 >= 0, s157 <= 1 + s155 + s156, x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 253 + 467*x_12655 + 283*x_12655^2 + 48*x_12655^3 + 467*x_21180 + 283*x_21180^2 + 48*x_21180^3 }-> s160 :|: s158 >= 0, s158 <= x_12655 + 1, s159 >= 0, s159 <= x_21180 + 1, s160 >= 0, s160 <= 1 + s158 + s159, z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s162 :|: s161 >= 0, s161 <= z' - 1 + 1, s162 >= 0, s162 <= 1 + s161, z' - 1 >= 0 encode_i(z') -{ 261 + 6*s163 + 2*s163*s164 + s163^2 + 4*s164 + s164^2 + 467*x_12658 + 283*x_12658^2 + 48*x_12658^3 + 467*x_21181 + 283*x_21181^2 + 48*x_21181^3 }-> s166 :|: s163 >= 0, s163 <= x_12658 + 1, s164 >= 0, s164 <= x_21181 + 1, s165 >= 0, s165 <= s163 + s164 + 1, s166 >= 0, s166 <= s165, x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ -104 + s167 + 45*z' + 139*z'^2 + 48*z'^3 }-> s169 :|: s167 >= 0, s167 <= z' - 1 + 1, s168 >= 0, s168 <= s167 + 1, s169 >= 0, s169 <= s168, z' - 1 >= 0 encode_i(z') -{ 379 + s171 + 467*x_12660 + 283*x_12660^2 + 48*x_12660^3 + 467*x_21182 + 283*x_21182^2 + 48*x_21182^3 + 467*x_3294 + 283*x_3294^2 + 48*x_3294^3 }-> s174 :|: s170 >= 0, s170 <= x_12660 + 1, s171 >= 0, s171 <= x_21182 + 1, s172 >= 0, s172 <= x_3294 + 1, s173 >= 0, s173 <= s170 + s171 + s172 + 1, s174 >= 0, s174 <= s173, z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s177 :|: s175 >= 0, s175 <= z' - 1 + 1, s176 >= 0, s176 <= s175, s177 >= 0, s177 <= s176, z' - 1 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s189 :|: s187 >= 0, s187 <= z' - 1 + 1, s188 >= 0, s188 <= s187 + 1, s189 >= 0, s189 <= s188, z' - 1 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s44 + s45 :|: s44 >= 0, s44 <= z' + 1, s45 >= 0, s45 <= z'' + 1, z' >= 0, z'' >= 0 encode_k(z') -{ 253 + 467*x_11448 + 283*x_11448^2 + 48*x_11448^3 + 467*x_2643 + 283*x_2643^2 + 48*x_2643^3 }-> s105 :|: s103 >= 0, s103 <= x_11448 + 1, s104 >= 0, s104 <= x_2643 + 1, s105 >= 0, s105 <= 1 + s103 + s104 + 1, z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 253 + 467*x_11449 + 283*x_11449^2 + 48*x_11449^3 + 467*x_2644 + 283*x_2644^2 + 48*x_2644^3 }-> s108 :|: s106 >= 0, s106 <= x_11449 + 1, s107 >= 0, s107 <= x_2644 + 1, s108 >= 0, s108 <= 1 + s106 + s107 + 1, x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s110 :|: s109 >= 0, s109 <= z' - 1 + 1, s110 >= 0, s110 <= 1 + s109 + 1, z' - 1 >= 0 encode_k(z') -{ 261 + 6*s111 + 2*s111*s112 + s111^2 + 4*s112 + s112^2 + 467*x_11452 + 283*x_11452^2 + 48*x_11452^3 + 467*x_2645 + 283*x_2645^2 + 48*x_2645^3 }-> s114 :|: s111 >= 0, s111 <= x_11452 + 1, s112 >= 0, s112 <= x_2645 + 1, s113 >= 0, s113 <= s111 + s112 + 1, s114 >= 0, s114 <= s113 + 1, x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ -104 + s115 + 45*z' + 139*z'^2 + 48*z'^3 }-> s117 :|: s115 >= 0, s115 <= z' - 1 + 1, s116 >= 0, s116 <= s115 + 1, s117 >= 0, s117 <= s116 + 1, z' - 1 >= 0 encode_k(z') -{ 379 + s119 + 467*x_11454 + 283*x_11454^2 + 48*x_11454^3 + 467*x_2646 + 283*x_2646^2 + 48*x_2646^3 + 467*x_3160 + 283*x_3160^2 + 48*x_3160^3 }-> s122 :|: s118 >= 0, s118 <= x_11454 + 1, s119 >= 0, s119 <= x_2646 + 1, s120 >= 0, s120 <= x_3160 + 1, s121 >= 0, s121 <= s118 + s119 + s120 + 1, s122 >= 0, s122 <= s121 + 1, x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s125 :|: s123 >= 0, s123 <= z' - 1 + 1, s124 >= 0, s124 <= s123 + 1, s125 >= 0, s125 <= s124 + 1, z' - 1 >= 0 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s186 :|: s184 >= 0, s184 <= z' - 1 + 1, s185 >= 0, s185 <= s184, s186 >= 0, s186 <= s185 + 1, z' - 1 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s72 :|: s72 >= 0, s72 <= z' + 1, z' >= 0 f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 10 + s9 + 6*x + x^2 }-> s10 :|: s9 >= 0, s9 <= x + 0 + 1, s10 >= 0, s10 <= s9 + 1, z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 11 + s5 + 6*x + 2*x*z'' + x^2 + 4*z'' + z''^2 }-> s6 :|: s5 >= 0, s5 <= x + (1 + 0 + (z'' - 1)) + 1, s6 >= 0, s6 <= s5 + 1, z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 23 + s7 + 10*x + 2*x*x'' + 2*x*y' + x^2 + 8*x'' + 2*x''*y' + x''^2 + 8*y' + y'^2 }-> s8 :|: s7 >= 0, s7 <= x + (1 + (1 + x'') + y') + 1, s8 >= 0, s8 <= s7 + 1, z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {encode_s} Previous analysis results are: h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1] encode_0: runtime: O(1) [0], size: O(1) [0] i: runtime: O(1) [1], size: O(n^1) [z'] k: runtime: O(1) [1], size: O(n^1) [1 + z'] g: runtime: O(n^1) [1 + z'], size: O(n^1) [1 + z'] f: runtime: O(n^2) [8 + 6*z' + 2*z'*z'' + z'^2 + 4*z'' + z''^2], size: O(n^1) [1 + z' + z''] encArg: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [1 + z'] encode_k: runtime: O(n^3) [1161 + 4403*z' + 3107*z'^2 + 624*z'^3], size: O(n^1) [4 + z'] encode_h: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [2 + z'] encode_h2: runtime: O(n^3) [379 + 467*z' + 283*z'^2 + 48*z'^3 + 468*z'' + 283*z''^2 + 48*z''^3 + 467*z1 + 283*z1^2 + 48*z1^3], size: O(n^1) [4 + z' + z'' + z1] encode_h1: runtime: O(n^3) [252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3], size: O(n^1) [3 + z' + z''] encode_g: runtime: O(n^3) [1171 + 4416*z' + 3107*z'^2 + 624*z'^3], size: O(n^1) [4 + z'] encode_f: runtime: O(n^3) [274 + 477*z' + 2*z'*z'' + 284*z'^2 + 48*z'^3 + 475*z'' + 284*z''^2 + 48*z''^3], size: O(n^1) [3 + z' + z''] encode_j: runtime: O(n^3) [252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3], size: O(n^1) [3 + z' + z''] encode_i: runtime: O(n^3) [1161 + 4403*z' + 3107*z'^2 + 624*z'^3], size: O(n^1) [3 + z'] ---------------------------------------- (115) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (116) Obligation: Complexity RNTS consisting of the following rules: encArg(z') -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z' = 1 + 0 encArg(z') -{ -58 + s127 + -89*z' + -5*z'^2 + 48*z'^3 }-> s128 :|: s126 >= 0, s126 <= z' - 2 + 1, s127 >= 0, s127 <= s126, s128 >= 0, s128 <= s127 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11313 + 283*x_11313^2 + 48*x_11313^3 + 467*x_2583 + 283*x_2583^2 + 48*x_2583^3 }-> s131 :|: s129 >= 0, s129 <= x_11313 + 1, s130 >= 0, s130 <= x_2583 + 1, s131 >= 0, s131 <= 1 + s129 + s130, x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 253 + 467*x_11314 + 283*x_11314^2 + 48*x_11314^3 + 467*x_2584 + 283*x_2584^2 + 48*x_2584^3 }-> s134 :|: s132 >= 0, s132 <= x_11314 + 1, s133 >= 0, s133 <= x_2584 + 1, s134 >= 0, s134 <= 1 + s132 + s133, z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s136 :|: s135 >= 0, s135 <= z' - 2 + 1, s136 >= 0, s136 <= 1 + s135, z' - 2 >= 0 encArg(z') -{ 261 + 6*s137 + 2*s137*s138 + s137^2 + 4*s138 + s138^2 + 467*x_11317 + 283*x_11317^2 + 48*x_11317^3 + 467*x_2585 + 283*x_2585^2 + 48*x_2585^3 }-> s140 :|: s137 >= 0, s137 <= x_11317 + 1, s138 >= 0, s138 <= x_2585 + 1, s139 >= 0, s139 <= s137 + s138 + 1, s140 >= 0, s140 <= s139, x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ -58 + s141 + -89*z' + -5*z'^2 + 48*z'^3 }-> s143 :|: s141 >= 0, s141 <= z' - 2 + 1, s142 >= 0, s142 <= s141 + 1, s143 >= 0, s143 <= s142, z' - 2 >= 0 encArg(z') -{ 379 + s145 + 467*x_11319 + 283*x_11319^2 + 48*x_11319^3 + 467*x_2586 + 283*x_2586^2 + 48*x_2586^3 + 467*x_3145 + 283*x_3145^2 + 48*x_3145^3 }-> s148 :|: s144 >= 0, s144 <= x_11319 + 1, s145 >= 0, s145 <= x_2586 + 1, s146 >= 0, s146 <= x_3145 + 1, s147 >= 0, s147 <= s144 + s145 + s146 + 1, s148 >= 0, s148 <= s147, z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s151 :|: s149 >= 0, s149 <= z' - 2 + 1, s150 >= 0, s150 <= s149, s151 >= 0, s151 <= s150, z' - 2 >= 0 encArg(z') -{ 260 + 6*s14 + 2*s14*s15 + s14^2 + 4*s15 + s15^2 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> s16 :|: s14 >= 0, s14 <= x_1 + 1, s15 >= 0, s15 <= x_2 + 1, s16 >= 0, s16 <= s14 + s15 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s180 :|: s178 >= 0, s178 <= z' - 2 + 1, s179 >= 0, s179 <= s178 + 1, s180 >= 0, s180 <= s179, z' - 2 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s183 :|: s181 >= 0, s181 <= z' - 2 + 1, s182 >= 0, s182 <= s181, s183 >= 0, s183 <= s182 + 1, z' - 2 >= 0 encArg(z') -{ 254 + s17 + s18 + 467*x_1107 + 283*x_1107^2 + 48*x_1107^3 + 467*x_247 + 283*x_247^2 + 48*x_247^3 }-> s19 :|: s17 >= 0, s17 <= x_1107 + 1, s18 >= 0, s18 <= x_247 + 1, s19 >= 0, s19 <= 1 + s17 + s18 + 1, z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0 + 1, z' - 1 >= 0 encArg(z') -{ 254 + s20 + s21 + 467*x_1108 + 283*x_1108^2 + 48*x_1108^3 + 467*x_248 + 283*x_248^2 + 48*x_248^3 }-> s22 :|: s20 >= 0, s20 <= x_1108 + 1, s21 >= 0, s21 <= x_248 + 1, s22 >= 0, s22 <= 1 + s20 + s21 + 1, x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ -58 + s23 + -89*z' + -5*z'^2 + 48*z'^3 }-> s24 :|: s23 >= 0, s23 <= z' - 2 + 1, s24 >= 0, s24 <= 1 + s23 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s25 + 2*s25*s26 + s25^2 + 4*s26 + s26^2 + s27 + 467*x_1111 + 283*x_1111^2 + 48*x_1111^3 + 467*x_249 + 283*x_249^2 + 48*x_249^3 }-> s28 :|: s25 >= 0, s25 <= x_1111 + 1, s26 >= 0, s26 <= x_249 + 1, s27 >= 0, s27 <= s25 + s26 + 1, s28 >= 0, s28 <= s27 + 1, z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ -58 + s29 + s30 + -89*z' + -5*z'^2 + 48*z'^3 }-> s31 :|: s29 >= 0, s29 <= z' - 2 + 1, s30 >= 0, s30 <= s29 + 1, s31 >= 0, s31 <= s30 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s33 + s35 + 467*x_1113 + 283*x_1113^2 + 48*x_1113^3 + 467*x_250 + 283*x_250^2 + 48*x_250^3 + 467*x_311 + 283*x_311^2 + 48*x_311^3 }-> s36 :|: s32 >= 0, s32 <= x_1113 + 1, s33 >= 0, s33 <= x_250 + 1, s34 >= 0, s34 <= x_311 + 1, s35 >= 0, s35 <= s32 + s33 + s34 + 1, s36 >= 0, s36 <= s35 + 1, x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 378 + s38 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 + 467*x_3 + 283*x_3^2 + 48*x_3^3 }-> s40 :|: s37 >= 0, s37 <= x_1 + 1, s38 >= 0, s38 <= x_2 + 1, s39 >= 0, s39 <= x_3 + 1, s40 >= 0, s40 <= s37 + s38 + s39 + 1, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z') -{ -58 + s75 + -89*z' + -5*z'^2 + 48*z'^3 }-> s76 :|: s74 >= 0, s74 <= z' - 2 + 1, s75 >= 0, s75 <= s74 + 1, s76 >= 0, s76 <= s75 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11322 + 283*x_11322^2 + 48*x_11322^3 + 467*x_2587 + 283*x_2587^2 + 48*x_2587^3 }-> s79 :|: s77 >= 0, s77 <= x_11322 + 1, s78 >= 0, s78 <= x_2587 + 1, s79 >= 0, s79 <= 1 + s77 + s78 + 1, x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 253 + 467*x_11323 + 283*x_11323^2 + 48*x_11323^3 + 467*x_2588 + 283*x_2588^2 + 48*x_2588^3 }-> s82 :|: s80 >= 0, s80 <= x_11323 + 1, s81 >= 0, s81 <= x_2588 + 1, s82 >= 0, s82 <= 1 + s80 + s81 + 1, z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s84 :|: s83 >= 0, s83 <= z' - 2 + 1, s84 >= 0, s84 <= 1 + s83 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s85 + 2*s85*s86 + s85^2 + 4*s86 + s86^2 + 467*x_11326 + 283*x_11326^2 + 48*x_11326^3 + 467*x_2589 + 283*x_2589^2 + 48*x_2589^3 }-> s88 :|: s85 >= 0, s85 <= x_11326 + 1, s86 >= 0, s86 <= x_2589 + 1, s87 >= 0, s87 <= s85 + s86 + 1, s88 >= 0, s88 <= s87 + 1, x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ -58 + s89 + -89*z' + -5*z'^2 + 48*z'^3 }-> s91 :|: s89 >= 0, s89 <= z' - 2 + 1, s90 >= 0, s90 <= s89 + 1, s91 >= 0, s91 <= s90 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s93 + 467*x_11328 + 283*x_11328^2 + 48*x_11328^3 + 467*x_2590 + 283*x_2590^2 + 48*x_2590^3 + 467*x_3146 + 283*x_3146^2 + 48*x_3146^3 }-> s96 :|: s92 >= 0, s92 <= x_11328 + 1, s93 >= 0, s93 <= x_2590 + 1, s94 >= 0, s94 <= x_3146 + 1, s95 >= 0, s95 <= s92 + s93 + s94 + 1, s96 >= 0, s96 <= s95 + 1, z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s99 :|: s97 >= 0, s97 <= z' - 2 + 1, s98 >= 0, s98 <= s97 + 1, s99 >= 0, s99 <= s98 + 1, z' - 2 >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ -106 + 45*z' + 139*z'^2 + 48*z'^3 }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0 encArg(z') -{ 252 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> 1 + s11 + s12 :|: s11 >= 0, s11 <= x_1 + 1, s12 >= 0, s12 <= x_2 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 260 + 6*s41 + 2*s41*s42 + s41^2 + 4*s42 + s42^2 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> s43 :|: s41 >= 0, s41 <= z' + 1, s42 >= 0, s42 <= z'' + 1, s43 >= 0, s43 <= s41 + s42 + 1, z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ -104 + s101 + 45*z' + 139*z'^2 + 48*z'^3 }-> s102 :|: s100 >= 0, s100 <= z' - 1 + 1, s101 >= 0, s101 <= s100 + 1, s102 >= 0, s102 <= s101 + 1, z' - 1 >= 0 encode_g(z') -{ -104 + s153 + 45*z' + 139*z'^2 + 48*z'^3 }-> s154 :|: s152 >= 0, s152 <= z' - 1 + 1, s153 >= 0, s153 <= s152, s154 >= 0, s154 <= s153 + 1, z' - 1 >= 0 encode_g(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0 + 1, z' = 0 encode_g(z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 1, z' >= 0 encode_g(z') -{ 254 + s46 + s47 + 467*x_11439 + 283*x_11439^2 + 48*x_11439^3 + 467*x_2639 + 283*x_2639^2 + 48*x_2639^3 }-> s48 :|: s46 >= 0, s46 <= x_11439 + 1, s47 >= 0, s47 <= x_2639 + 1, s48 >= 0, s48 <= 1 + s46 + s47 + 1, x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 254 + s49 + s50 + 467*x_11440 + 283*x_11440^2 + 48*x_11440^3 + 467*x_2640 + 283*x_2640^2 + 48*x_2640^3 }-> s51 :|: s49 >= 0, s49 <= x_11440 + 1, s50 >= 0, s50 <= x_2640 + 1, s51 >= 0, s51 <= 1 + s49 + s50 + 1, x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ -104 + s52 + 45*z' + 139*z'^2 + 48*z'^3 }-> s53 :|: s52 >= 0, s52 <= z' - 1 + 1, s53 >= 0, s53 <= 1 + s52 + 1, z' - 1 >= 0 encode_g(z') -{ 261 + 6*s54 + 2*s54*s55 + s54^2 + 4*s55 + s55^2 + s56 + 467*x_11443 + 283*x_11443^2 + 48*x_11443^3 + 467*x_2641 + 283*x_2641^2 + 48*x_2641^3 }-> s57 :|: s54 >= 0, s54 <= x_11443 + 1, s55 >= 0, s55 <= x_2641 + 1, s56 >= 0, s56 <= s54 + s55 + 1, s57 >= 0, s57 <= s56 + 1, x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ -104 + s58 + s59 + 45*z' + 139*z'^2 + 48*z'^3 }-> s60 :|: s58 >= 0, s58 <= z' - 1 + 1, s59 >= 0, s59 <= s58 + 1, s60 >= 0, s60 <= s59 + 1, z' - 1 >= 0 encode_g(z') -{ 379 + s62 + s64 + 467*x_11445 + 283*x_11445^2 + 48*x_11445^3 + 467*x_2642 + 283*x_2642^2 + 48*x_2642^3 + 467*x_3159 + 283*x_3159^2 + 48*x_3159^3 }-> s65 :|: s61 >= 0, s61 <= x_11445 + 1, s62 >= 0, s62 <= x_2642 + 1, s63 >= 0, s63 <= x_3159 + 1, s64 >= 0, s64 <= s61 + s62 + s63 + 1, s65 >= 0, s65 <= s64 + 1, x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s73 :|: s73 >= 0, s73 <= z' + 1, z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s66 + s67 :|: s66 >= 0, s66 <= z' + 1, s67 >= 0, s67 <= z'' + 1, z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 378 + s69 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 + 467*z1 + 283*z1^2 + 48*z1^3 }-> s71 :|: s68 >= 0, s68 <= z' + 1, s69 >= 0, s69 <= z'' + 1, s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s68 + s69 + s70 + 1, z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 253 + 467*x_12654 + 283*x_12654^2 + 48*x_12654^3 + 467*x_21179 + 283*x_21179^2 + 48*x_21179^3 }-> s157 :|: s155 >= 0, s155 <= x_12654 + 1, s156 >= 0, s156 <= x_21179 + 1, s157 >= 0, s157 <= 1 + s155 + s156, x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 253 + 467*x_12655 + 283*x_12655^2 + 48*x_12655^3 + 467*x_21180 + 283*x_21180^2 + 48*x_21180^3 }-> s160 :|: s158 >= 0, s158 <= x_12655 + 1, s159 >= 0, s159 <= x_21180 + 1, s160 >= 0, s160 <= 1 + s158 + s159, z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s162 :|: s161 >= 0, s161 <= z' - 1 + 1, s162 >= 0, s162 <= 1 + s161, z' - 1 >= 0 encode_i(z') -{ 261 + 6*s163 + 2*s163*s164 + s163^2 + 4*s164 + s164^2 + 467*x_12658 + 283*x_12658^2 + 48*x_12658^3 + 467*x_21181 + 283*x_21181^2 + 48*x_21181^3 }-> s166 :|: s163 >= 0, s163 <= x_12658 + 1, s164 >= 0, s164 <= x_21181 + 1, s165 >= 0, s165 <= s163 + s164 + 1, s166 >= 0, s166 <= s165, x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ -104 + s167 + 45*z' + 139*z'^2 + 48*z'^3 }-> s169 :|: s167 >= 0, s167 <= z' - 1 + 1, s168 >= 0, s168 <= s167 + 1, s169 >= 0, s169 <= s168, z' - 1 >= 0 encode_i(z') -{ 379 + s171 + 467*x_12660 + 283*x_12660^2 + 48*x_12660^3 + 467*x_21182 + 283*x_21182^2 + 48*x_21182^3 + 467*x_3294 + 283*x_3294^2 + 48*x_3294^3 }-> s174 :|: s170 >= 0, s170 <= x_12660 + 1, s171 >= 0, s171 <= x_21182 + 1, s172 >= 0, s172 <= x_3294 + 1, s173 >= 0, s173 <= s170 + s171 + s172 + 1, s174 >= 0, s174 <= s173, z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s177 :|: s175 >= 0, s175 <= z' - 1 + 1, s176 >= 0, s176 <= s175, s177 >= 0, s177 <= s176, z' - 1 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s189 :|: s187 >= 0, s187 <= z' - 1 + 1, s188 >= 0, s188 <= s187 + 1, s189 >= 0, s189 <= s188, z' - 1 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s44 + s45 :|: s44 >= 0, s44 <= z' + 1, s45 >= 0, s45 <= z'' + 1, z' >= 0, z'' >= 0 encode_k(z') -{ 253 + 467*x_11448 + 283*x_11448^2 + 48*x_11448^3 + 467*x_2643 + 283*x_2643^2 + 48*x_2643^3 }-> s105 :|: s103 >= 0, s103 <= x_11448 + 1, s104 >= 0, s104 <= x_2643 + 1, s105 >= 0, s105 <= 1 + s103 + s104 + 1, z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 253 + 467*x_11449 + 283*x_11449^2 + 48*x_11449^3 + 467*x_2644 + 283*x_2644^2 + 48*x_2644^3 }-> s108 :|: s106 >= 0, s106 <= x_11449 + 1, s107 >= 0, s107 <= x_2644 + 1, s108 >= 0, s108 <= 1 + s106 + s107 + 1, x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s110 :|: s109 >= 0, s109 <= z' - 1 + 1, s110 >= 0, s110 <= 1 + s109 + 1, z' - 1 >= 0 encode_k(z') -{ 261 + 6*s111 + 2*s111*s112 + s111^2 + 4*s112 + s112^2 + 467*x_11452 + 283*x_11452^2 + 48*x_11452^3 + 467*x_2645 + 283*x_2645^2 + 48*x_2645^3 }-> s114 :|: s111 >= 0, s111 <= x_11452 + 1, s112 >= 0, s112 <= x_2645 + 1, s113 >= 0, s113 <= s111 + s112 + 1, s114 >= 0, s114 <= s113 + 1, x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ -104 + s115 + 45*z' + 139*z'^2 + 48*z'^3 }-> s117 :|: s115 >= 0, s115 <= z' - 1 + 1, s116 >= 0, s116 <= s115 + 1, s117 >= 0, s117 <= s116 + 1, z' - 1 >= 0 encode_k(z') -{ 379 + s119 + 467*x_11454 + 283*x_11454^2 + 48*x_11454^3 + 467*x_2646 + 283*x_2646^2 + 48*x_2646^3 + 467*x_3160 + 283*x_3160^2 + 48*x_3160^3 }-> s122 :|: s118 >= 0, s118 <= x_11454 + 1, s119 >= 0, s119 <= x_2646 + 1, s120 >= 0, s120 <= x_3160 + 1, s121 >= 0, s121 <= s118 + s119 + s120 + 1, s122 >= 0, s122 <= s121 + 1, x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s125 :|: s123 >= 0, s123 <= z' - 1 + 1, s124 >= 0, s124 <= s123 + 1, s125 >= 0, s125 <= s124 + 1, z' - 1 >= 0 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s186 :|: s184 >= 0, s184 <= z' - 1 + 1, s185 >= 0, s185 <= s184, s186 >= 0, s186 <= s185 + 1, z' - 1 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s72 :|: s72 >= 0, s72 <= z' + 1, z' >= 0 f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 10 + s9 + 6*x + x^2 }-> s10 :|: s9 >= 0, s9 <= x + 0 + 1, s10 >= 0, s10 <= s9 + 1, z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 11 + s5 + 6*x + 2*x*z'' + x^2 + 4*z'' + z''^2 }-> s6 :|: s5 >= 0, s5 <= x + (1 + 0 + (z'' - 1)) + 1, s6 >= 0, s6 <= s5 + 1, z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 23 + s7 + 10*x + 2*x*x'' + 2*x*y' + x^2 + 8*x'' + 2*x''*y' + x''^2 + 8*y' + y'^2 }-> s8 :|: s7 >= 0, s7 <= x + (1 + (1 + x'') + y') + 1, s8 >= 0, s8 <= s7 + 1, z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {encode_s} Previous analysis results are: h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1] encode_0: runtime: O(1) [0], size: O(1) [0] i: runtime: O(1) [1], size: O(n^1) [z'] k: runtime: O(1) [1], size: O(n^1) [1 + z'] g: runtime: O(n^1) [1 + z'], size: O(n^1) [1 + z'] f: runtime: O(n^2) [8 + 6*z' + 2*z'*z'' + z'^2 + 4*z'' + z''^2], size: O(n^1) [1 + z' + z''] encArg: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [1 + z'] encode_k: runtime: O(n^3) [1161 + 4403*z' + 3107*z'^2 + 624*z'^3], size: O(n^1) [4 + z'] encode_h: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [2 + z'] encode_h2: runtime: O(n^3) [379 + 467*z' + 283*z'^2 + 48*z'^3 + 468*z'' + 283*z''^2 + 48*z''^3 + 467*z1 + 283*z1^2 + 48*z1^3], size: O(n^1) [4 + z' + z'' + z1] encode_h1: runtime: O(n^3) [252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3], size: O(n^1) [3 + z' + z''] encode_g: runtime: O(n^3) [1171 + 4416*z' + 3107*z'^2 + 624*z'^3], size: O(n^1) [4 + z'] encode_f: runtime: O(n^3) [274 + 477*z' + 2*z'*z'' + 284*z'^2 + 48*z'^3 + 475*z'' + 284*z''^2 + 48*z''^3], size: O(n^1) [3 + z' + z''] encode_j: runtime: O(n^3) [252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3], size: O(n^1) [3 + z' + z''] encode_i: runtime: O(n^3) [1161 + 4403*z' + 3107*z'^2 + 624*z'^3], size: O(n^1) [3 + z'] ---------------------------------------- (117) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: encode_s after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z' ---------------------------------------- (118) Obligation: Complexity RNTS consisting of the following rules: encArg(z') -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z' = 1 + 0 encArg(z') -{ -58 + s127 + -89*z' + -5*z'^2 + 48*z'^3 }-> s128 :|: s126 >= 0, s126 <= z' - 2 + 1, s127 >= 0, s127 <= s126, s128 >= 0, s128 <= s127 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11313 + 283*x_11313^2 + 48*x_11313^3 + 467*x_2583 + 283*x_2583^2 + 48*x_2583^3 }-> s131 :|: s129 >= 0, s129 <= x_11313 + 1, s130 >= 0, s130 <= x_2583 + 1, s131 >= 0, s131 <= 1 + s129 + s130, x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 253 + 467*x_11314 + 283*x_11314^2 + 48*x_11314^3 + 467*x_2584 + 283*x_2584^2 + 48*x_2584^3 }-> s134 :|: s132 >= 0, s132 <= x_11314 + 1, s133 >= 0, s133 <= x_2584 + 1, s134 >= 0, s134 <= 1 + s132 + s133, z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s136 :|: s135 >= 0, s135 <= z' - 2 + 1, s136 >= 0, s136 <= 1 + s135, z' - 2 >= 0 encArg(z') -{ 261 + 6*s137 + 2*s137*s138 + s137^2 + 4*s138 + s138^2 + 467*x_11317 + 283*x_11317^2 + 48*x_11317^3 + 467*x_2585 + 283*x_2585^2 + 48*x_2585^3 }-> s140 :|: s137 >= 0, s137 <= x_11317 + 1, s138 >= 0, s138 <= x_2585 + 1, s139 >= 0, s139 <= s137 + s138 + 1, s140 >= 0, s140 <= s139, x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ -58 + s141 + -89*z' + -5*z'^2 + 48*z'^3 }-> s143 :|: s141 >= 0, s141 <= z' - 2 + 1, s142 >= 0, s142 <= s141 + 1, s143 >= 0, s143 <= s142, z' - 2 >= 0 encArg(z') -{ 379 + s145 + 467*x_11319 + 283*x_11319^2 + 48*x_11319^3 + 467*x_2586 + 283*x_2586^2 + 48*x_2586^3 + 467*x_3145 + 283*x_3145^2 + 48*x_3145^3 }-> s148 :|: s144 >= 0, s144 <= x_11319 + 1, s145 >= 0, s145 <= x_2586 + 1, s146 >= 0, s146 <= x_3145 + 1, s147 >= 0, s147 <= s144 + s145 + s146 + 1, s148 >= 0, s148 <= s147, z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s151 :|: s149 >= 0, s149 <= z' - 2 + 1, s150 >= 0, s150 <= s149, s151 >= 0, s151 <= s150, z' - 2 >= 0 encArg(z') -{ 260 + 6*s14 + 2*s14*s15 + s14^2 + 4*s15 + s15^2 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> s16 :|: s14 >= 0, s14 <= x_1 + 1, s15 >= 0, s15 <= x_2 + 1, s16 >= 0, s16 <= s14 + s15 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s180 :|: s178 >= 0, s178 <= z' - 2 + 1, s179 >= 0, s179 <= s178 + 1, s180 >= 0, s180 <= s179, z' - 2 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s183 :|: s181 >= 0, s181 <= z' - 2 + 1, s182 >= 0, s182 <= s181, s183 >= 0, s183 <= s182 + 1, z' - 2 >= 0 encArg(z') -{ 254 + s17 + s18 + 467*x_1107 + 283*x_1107^2 + 48*x_1107^3 + 467*x_247 + 283*x_247^2 + 48*x_247^3 }-> s19 :|: s17 >= 0, s17 <= x_1107 + 1, s18 >= 0, s18 <= x_247 + 1, s19 >= 0, s19 <= 1 + s17 + s18 + 1, z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0 + 1, z' - 1 >= 0 encArg(z') -{ 254 + s20 + s21 + 467*x_1108 + 283*x_1108^2 + 48*x_1108^3 + 467*x_248 + 283*x_248^2 + 48*x_248^3 }-> s22 :|: s20 >= 0, s20 <= x_1108 + 1, s21 >= 0, s21 <= x_248 + 1, s22 >= 0, s22 <= 1 + s20 + s21 + 1, x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ -58 + s23 + -89*z' + -5*z'^2 + 48*z'^3 }-> s24 :|: s23 >= 0, s23 <= z' - 2 + 1, s24 >= 0, s24 <= 1 + s23 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s25 + 2*s25*s26 + s25^2 + 4*s26 + s26^2 + s27 + 467*x_1111 + 283*x_1111^2 + 48*x_1111^3 + 467*x_249 + 283*x_249^2 + 48*x_249^3 }-> s28 :|: s25 >= 0, s25 <= x_1111 + 1, s26 >= 0, s26 <= x_249 + 1, s27 >= 0, s27 <= s25 + s26 + 1, s28 >= 0, s28 <= s27 + 1, z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ -58 + s29 + s30 + -89*z' + -5*z'^2 + 48*z'^3 }-> s31 :|: s29 >= 0, s29 <= z' - 2 + 1, s30 >= 0, s30 <= s29 + 1, s31 >= 0, s31 <= s30 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s33 + s35 + 467*x_1113 + 283*x_1113^2 + 48*x_1113^3 + 467*x_250 + 283*x_250^2 + 48*x_250^3 + 467*x_311 + 283*x_311^2 + 48*x_311^3 }-> s36 :|: s32 >= 0, s32 <= x_1113 + 1, s33 >= 0, s33 <= x_250 + 1, s34 >= 0, s34 <= x_311 + 1, s35 >= 0, s35 <= s32 + s33 + s34 + 1, s36 >= 0, s36 <= s35 + 1, x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 378 + s38 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 + 467*x_3 + 283*x_3^2 + 48*x_3^3 }-> s40 :|: s37 >= 0, s37 <= x_1 + 1, s38 >= 0, s38 <= x_2 + 1, s39 >= 0, s39 <= x_3 + 1, s40 >= 0, s40 <= s37 + s38 + s39 + 1, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z') -{ -58 + s75 + -89*z' + -5*z'^2 + 48*z'^3 }-> s76 :|: s74 >= 0, s74 <= z' - 2 + 1, s75 >= 0, s75 <= s74 + 1, s76 >= 0, s76 <= s75 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11322 + 283*x_11322^2 + 48*x_11322^3 + 467*x_2587 + 283*x_2587^2 + 48*x_2587^3 }-> s79 :|: s77 >= 0, s77 <= x_11322 + 1, s78 >= 0, s78 <= x_2587 + 1, s79 >= 0, s79 <= 1 + s77 + s78 + 1, x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 253 + 467*x_11323 + 283*x_11323^2 + 48*x_11323^3 + 467*x_2588 + 283*x_2588^2 + 48*x_2588^3 }-> s82 :|: s80 >= 0, s80 <= x_11323 + 1, s81 >= 0, s81 <= x_2588 + 1, s82 >= 0, s82 <= 1 + s80 + s81 + 1, z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s84 :|: s83 >= 0, s83 <= z' - 2 + 1, s84 >= 0, s84 <= 1 + s83 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s85 + 2*s85*s86 + s85^2 + 4*s86 + s86^2 + 467*x_11326 + 283*x_11326^2 + 48*x_11326^3 + 467*x_2589 + 283*x_2589^2 + 48*x_2589^3 }-> s88 :|: s85 >= 0, s85 <= x_11326 + 1, s86 >= 0, s86 <= x_2589 + 1, s87 >= 0, s87 <= s85 + s86 + 1, s88 >= 0, s88 <= s87 + 1, x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ -58 + s89 + -89*z' + -5*z'^2 + 48*z'^3 }-> s91 :|: s89 >= 0, s89 <= z' - 2 + 1, s90 >= 0, s90 <= s89 + 1, s91 >= 0, s91 <= s90 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s93 + 467*x_11328 + 283*x_11328^2 + 48*x_11328^3 + 467*x_2590 + 283*x_2590^2 + 48*x_2590^3 + 467*x_3146 + 283*x_3146^2 + 48*x_3146^3 }-> s96 :|: s92 >= 0, s92 <= x_11328 + 1, s93 >= 0, s93 <= x_2590 + 1, s94 >= 0, s94 <= x_3146 + 1, s95 >= 0, s95 <= s92 + s93 + s94 + 1, s96 >= 0, s96 <= s95 + 1, z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s99 :|: s97 >= 0, s97 <= z' - 2 + 1, s98 >= 0, s98 <= s97 + 1, s99 >= 0, s99 <= s98 + 1, z' - 2 >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ -106 + 45*z' + 139*z'^2 + 48*z'^3 }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0 encArg(z') -{ 252 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> 1 + s11 + s12 :|: s11 >= 0, s11 <= x_1 + 1, s12 >= 0, s12 <= x_2 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 260 + 6*s41 + 2*s41*s42 + s41^2 + 4*s42 + s42^2 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> s43 :|: s41 >= 0, s41 <= z' + 1, s42 >= 0, s42 <= z'' + 1, s43 >= 0, s43 <= s41 + s42 + 1, z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ -104 + s101 + 45*z' + 139*z'^2 + 48*z'^3 }-> s102 :|: s100 >= 0, s100 <= z' - 1 + 1, s101 >= 0, s101 <= s100 + 1, s102 >= 0, s102 <= s101 + 1, z' - 1 >= 0 encode_g(z') -{ -104 + s153 + 45*z' + 139*z'^2 + 48*z'^3 }-> s154 :|: s152 >= 0, s152 <= z' - 1 + 1, s153 >= 0, s153 <= s152, s154 >= 0, s154 <= s153 + 1, z' - 1 >= 0 encode_g(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0 + 1, z' = 0 encode_g(z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 1, z' >= 0 encode_g(z') -{ 254 + s46 + s47 + 467*x_11439 + 283*x_11439^2 + 48*x_11439^3 + 467*x_2639 + 283*x_2639^2 + 48*x_2639^3 }-> s48 :|: s46 >= 0, s46 <= x_11439 + 1, s47 >= 0, s47 <= x_2639 + 1, s48 >= 0, s48 <= 1 + s46 + s47 + 1, x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 254 + s49 + s50 + 467*x_11440 + 283*x_11440^2 + 48*x_11440^3 + 467*x_2640 + 283*x_2640^2 + 48*x_2640^3 }-> s51 :|: s49 >= 0, s49 <= x_11440 + 1, s50 >= 0, s50 <= x_2640 + 1, s51 >= 0, s51 <= 1 + s49 + s50 + 1, x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ -104 + s52 + 45*z' + 139*z'^2 + 48*z'^3 }-> s53 :|: s52 >= 0, s52 <= z' - 1 + 1, s53 >= 0, s53 <= 1 + s52 + 1, z' - 1 >= 0 encode_g(z') -{ 261 + 6*s54 + 2*s54*s55 + s54^2 + 4*s55 + s55^2 + s56 + 467*x_11443 + 283*x_11443^2 + 48*x_11443^3 + 467*x_2641 + 283*x_2641^2 + 48*x_2641^3 }-> s57 :|: s54 >= 0, s54 <= x_11443 + 1, s55 >= 0, s55 <= x_2641 + 1, s56 >= 0, s56 <= s54 + s55 + 1, s57 >= 0, s57 <= s56 + 1, x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ -104 + s58 + s59 + 45*z' + 139*z'^2 + 48*z'^3 }-> s60 :|: s58 >= 0, s58 <= z' - 1 + 1, s59 >= 0, s59 <= s58 + 1, s60 >= 0, s60 <= s59 + 1, z' - 1 >= 0 encode_g(z') -{ 379 + s62 + s64 + 467*x_11445 + 283*x_11445^2 + 48*x_11445^3 + 467*x_2642 + 283*x_2642^2 + 48*x_2642^3 + 467*x_3159 + 283*x_3159^2 + 48*x_3159^3 }-> s65 :|: s61 >= 0, s61 <= x_11445 + 1, s62 >= 0, s62 <= x_2642 + 1, s63 >= 0, s63 <= x_3159 + 1, s64 >= 0, s64 <= s61 + s62 + s63 + 1, s65 >= 0, s65 <= s64 + 1, x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s73 :|: s73 >= 0, s73 <= z' + 1, z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s66 + s67 :|: s66 >= 0, s66 <= z' + 1, s67 >= 0, s67 <= z'' + 1, z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 378 + s69 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 + 467*z1 + 283*z1^2 + 48*z1^3 }-> s71 :|: s68 >= 0, s68 <= z' + 1, s69 >= 0, s69 <= z'' + 1, s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s68 + s69 + s70 + 1, z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 253 + 467*x_12654 + 283*x_12654^2 + 48*x_12654^3 + 467*x_21179 + 283*x_21179^2 + 48*x_21179^3 }-> s157 :|: s155 >= 0, s155 <= x_12654 + 1, s156 >= 0, s156 <= x_21179 + 1, s157 >= 0, s157 <= 1 + s155 + s156, x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 253 + 467*x_12655 + 283*x_12655^2 + 48*x_12655^3 + 467*x_21180 + 283*x_21180^2 + 48*x_21180^3 }-> s160 :|: s158 >= 0, s158 <= x_12655 + 1, s159 >= 0, s159 <= x_21180 + 1, s160 >= 0, s160 <= 1 + s158 + s159, z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s162 :|: s161 >= 0, s161 <= z' - 1 + 1, s162 >= 0, s162 <= 1 + s161, z' - 1 >= 0 encode_i(z') -{ 261 + 6*s163 + 2*s163*s164 + s163^2 + 4*s164 + s164^2 + 467*x_12658 + 283*x_12658^2 + 48*x_12658^3 + 467*x_21181 + 283*x_21181^2 + 48*x_21181^3 }-> s166 :|: s163 >= 0, s163 <= x_12658 + 1, s164 >= 0, s164 <= x_21181 + 1, s165 >= 0, s165 <= s163 + s164 + 1, s166 >= 0, s166 <= s165, x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ -104 + s167 + 45*z' + 139*z'^2 + 48*z'^3 }-> s169 :|: s167 >= 0, s167 <= z' - 1 + 1, s168 >= 0, s168 <= s167 + 1, s169 >= 0, s169 <= s168, z' - 1 >= 0 encode_i(z') -{ 379 + s171 + 467*x_12660 + 283*x_12660^2 + 48*x_12660^3 + 467*x_21182 + 283*x_21182^2 + 48*x_21182^3 + 467*x_3294 + 283*x_3294^2 + 48*x_3294^3 }-> s174 :|: s170 >= 0, s170 <= x_12660 + 1, s171 >= 0, s171 <= x_21182 + 1, s172 >= 0, s172 <= x_3294 + 1, s173 >= 0, s173 <= s170 + s171 + s172 + 1, s174 >= 0, s174 <= s173, z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s177 :|: s175 >= 0, s175 <= z' - 1 + 1, s176 >= 0, s176 <= s175, s177 >= 0, s177 <= s176, z' - 1 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s189 :|: s187 >= 0, s187 <= z' - 1 + 1, s188 >= 0, s188 <= s187 + 1, s189 >= 0, s189 <= s188, z' - 1 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s44 + s45 :|: s44 >= 0, s44 <= z' + 1, s45 >= 0, s45 <= z'' + 1, z' >= 0, z'' >= 0 encode_k(z') -{ 253 + 467*x_11448 + 283*x_11448^2 + 48*x_11448^3 + 467*x_2643 + 283*x_2643^2 + 48*x_2643^3 }-> s105 :|: s103 >= 0, s103 <= x_11448 + 1, s104 >= 0, s104 <= x_2643 + 1, s105 >= 0, s105 <= 1 + s103 + s104 + 1, z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 253 + 467*x_11449 + 283*x_11449^2 + 48*x_11449^3 + 467*x_2644 + 283*x_2644^2 + 48*x_2644^3 }-> s108 :|: s106 >= 0, s106 <= x_11449 + 1, s107 >= 0, s107 <= x_2644 + 1, s108 >= 0, s108 <= 1 + s106 + s107 + 1, x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s110 :|: s109 >= 0, s109 <= z' - 1 + 1, s110 >= 0, s110 <= 1 + s109 + 1, z' - 1 >= 0 encode_k(z') -{ 261 + 6*s111 + 2*s111*s112 + s111^2 + 4*s112 + s112^2 + 467*x_11452 + 283*x_11452^2 + 48*x_11452^3 + 467*x_2645 + 283*x_2645^2 + 48*x_2645^3 }-> s114 :|: s111 >= 0, s111 <= x_11452 + 1, s112 >= 0, s112 <= x_2645 + 1, s113 >= 0, s113 <= s111 + s112 + 1, s114 >= 0, s114 <= s113 + 1, x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ -104 + s115 + 45*z' + 139*z'^2 + 48*z'^3 }-> s117 :|: s115 >= 0, s115 <= z' - 1 + 1, s116 >= 0, s116 <= s115 + 1, s117 >= 0, s117 <= s116 + 1, z' - 1 >= 0 encode_k(z') -{ 379 + s119 + 467*x_11454 + 283*x_11454^2 + 48*x_11454^3 + 467*x_2646 + 283*x_2646^2 + 48*x_2646^3 + 467*x_3160 + 283*x_3160^2 + 48*x_3160^3 }-> s122 :|: s118 >= 0, s118 <= x_11454 + 1, s119 >= 0, s119 <= x_2646 + 1, s120 >= 0, s120 <= x_3160 + 1, s121 >= 0, s121 <= s118 + s119 + s120 + 1, s122 >= 0, s122 <= s121 + 1, x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s125 :|: s123 >= 0, s123 <= z' - 1 + 1, s124 >= 0, s124 <= s123 + 1, s125 >= 0, s125 <= s124 + 1, z' - 1 >= 0 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s186 :|: s184 >= 0, s184 <= z' - 1 + 1, s185 >= 0, s185 <= s184, s186 >= 0, s186 <= s185 + 1, z' - 1 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s72 :|: s72 >= 0, s72 <= z' + 1, z' >= 0 f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 10 + s9 + 6*x + x^2 }-> s10 :|: s9 >= 0, s9 <= x + 0 + 1, s10 >= 0, s10 <= s9 + 1, z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 11 + s5 + 6*x + 2*x*z'' + x^2 + 4*z'' + z''^2 }-> s6 :|: s5 >= 0, s5 <= x + (1 + 0 + (z'' - 1)) + 1, s6 >= 0, s6 <= s5 + 1, z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 23 + s7 + 10*x + 2*x*x'' + 2*x*y' + x^2 + 8*x'' + 2*x''*y' + x''^2 + 8*y' + y'^2 }-> s8 :|: s7 >= 0, s7 <= x + (1 + (1 + x'') + y') + 1, s8 >= 0, s8 <= s7 + 1, z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {encode_s} Previous analysis results are: h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1] encode_0: runtime: O(1) [0], size: O(1) [0] i: runtime: O(1) [1], size: O(n^1) [z'] k: runtime: O(1) [1], size: O(n^1) [1 + z'] g: runtime: O(n^1) [1 + z'], size: O(n^1) [1 + z'] f: runtime: O(n^2) [8 + 6*z' + 2*z'*z'' + z'^2 + 4*z'' + z''^2], size: O(n^1) [1 + z' + z''] encArg: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [1 + z'] encode_k: runtime: O(n^3) [1161 + 4403*z' + 3107*z'^2 + 624*z'^3], size: O(n^1) [4 + z'] encode_h: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [2 + z'] encode_h2: runtime: O(n^3) [379 + 467*z' + 283*z'^2 + 48*z'^3 + 468*z'' + 283*z''^2 + 48*z''^3 + 467*z1 + 283*z1^2 + 48*z1^3], size: O(n^1) [4 + z' + z'' + z1] encode_h1: runtime: O(n^3) [252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3], size: O(n^1) [3 + z' + z''] encode_g: runtime: O(n^3) [1171 + 4416*z' + 3107*z'^2 + 624*z'^3], size: O(n^1) [4 + z'] encode_f: runtime: O(n^3) [274 + 477*z' + 2*z'*z'' + 284*z'^2 + 48*z'^3 + 475*z'' + 284*z''^2 + 48*z''^3], size: O(n^1) [3 + z' + z''] encode_j: runtime: O(n^3) [252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3], size: O(n^1) [3 + z' + z''] encode_i: runtime: O(n^3) [1161 + 4403*z' + 3107*z'^2 + 624*z'^3], size: O(n^1) [3 + z'] encode_s: runtime: ?, size: O(n^1) [2 + z'] ---------------------------------------- (119) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: encode_s after applying outer abstraction to obtain an ITS, resulting in: O(n^3) with polynomial bound: 126 + 467*z' + 283*z'^2 + 48*z'^3 ---------------------------------------- (120) Obligation: Complexity RNTS consisting of the following rules: encArg(z') -{ 1 }-> s1 :|: s1 >= 0, s1 <= 0 + 1, z' = 1 + 0 encArg(z') -{ -58 + s127 + -89*z' + -5*z'^2 + 48*z'^3 }-> s128 :|: s126 >= 0, s126 <= z' - 2 + 1, s127 >= 0, s127 <= s126, s128 >= 0, s128 <= s127 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11313 + 283*x_11313^2 + 48*x_11313^3 + 467*x_2583 + 283*x_2583^2 + 48*x_2583^3 }-> s131 :|: s129 >= 0, s129 <= x_11313 + 1, s130 >= 0, s130 <= x_2583 + 1, s131 >= 0, s131 <= 1 + s129 + s130, x_11313 >= 0, x_2583 >= 0, z' = 1 + (1 + x_11313 + x_2583) encArg(z') -{ 253 + 467*x_11314 + 283*x_11314^2 + 48*x_11314^3 + 467*x_2584 + 283*x_2584^2 + 48*x_2584^3 }-> s134 :|: s132 >= 0, s132 <= x_11314 + 1, s133 >= 0, s133 <= x_2584 + 1, s134 >= 0, s134 <= 1 + s132 + s133, z' = 1 + (1 + x_11314 + x_2584), x_2584 >= 0, x_11314 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s136 :|: s135 >= 0, s135 <= z' - 2 + 1, s136 >= 0, s136 <= 1 + s135, z' - 2 >= 0 encArg(z') -{ 261 + 6*s137 + 2*s137*s138 + s137^2 + 4*s138 + s138^2 + 467*x_11317 + 283*x_11317^2 + 48*x_11317^3 + 467*x_2585 + 283*x_2585^2 + 48*x_2585^3 }-> s140 :|: s137 >= 0, s137 <= x_11317 + 1, s138 >= 0, s138 <= x_2585 + 1, s139 >= 0, s139 <= s137 + s138 + 1, s140 >= 0, s140 <= s139, x_11317 >= 0, x_2585 >= 0, z' = 1 + (1 + x_11317 + x_2585) encArg(z') -{ -58 + s141 + -89*z' + -5*z'^2 + 48*z'^3 }-> s143 :|: s141 >= 0, s141 <= z' - 2 + 1, s142 >= 0, s142 <= s141 + 1, s143 >= 0, s143 <= s142, z' - 2 >= 0 encArg(z') -{ 379 + s145 + 467*x_11319 + 283*x_11319^2 + 48*x_11319^3 + 467*x_2586 + 283*x_2586^2 + 48*x_2586^3 + 467*x_3145 + 283*x_3145^2 + 48*x_3145^3 }-> s148 :|: s144 >= 0, s144 <= x_11319 + 1, s145 >= 0, s145 <= x_2586 + 1, s146 >= 0, s146 <= x_3145 + 1, s147 >= 0, s147 <= s144 + s145 + s146 + 1, s148 >= 0, s148 <= s147, z' = 1 + (1 + x_11319 + x_2586 + x_3145), x_2586 >= 0, x_3145 >= 0, x_11319 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s151 :|: s149 >= 0, s149 <= z' - 2 + 1, s150 >= 0, s150 <= s149, s151 >= 0, s151 <= s150, z' - 2 >= 0 encArg(z') -{ 260 + 6*s14 + 2*s14*s15 + s14^2 + 4*s15 + s15^2 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> s16 :|: s14 >= 0, s14 <= x_1 + 1, s15 >= 0, s15 <= x_2 + 1, s16 >= 0, s16 <= s14 + s15 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s180 :|: s178 >= 0, s178 <= z' - 2 + 1, s179 >= 0, s179 <= s178 + 1, s180 >= 0, s180 <= s179, z' - 2 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s183 :|: s181 >= 0, s181 <= z' - 2 + 1, s182 >= 0, s182 <= s181, s183 >= 0, s183 <= s182 + 1, z' - 2 >= 0 encArg(z') -{ 254 + s17 + s18 + 467*x_1107 + 283*x_1107^2 + 48*x_1107^3 + 467*x_247 + 283*x_247^2 + 48*x_247^3 }-> s19 :|: s17 >= 0, s17 <= x_1107 + 1, s18 >= 0, s18 <= x_247 + 1, s19 >= 0, s19 <= 1 + s17 + s18 + 1, z' = 1 + (1 + x_1107 + x_247), x_1107 >= 0, x_247 >= 0 encArg(z') -{ 1 }-> s2 :|: s2 >= 0, s2 <= 0 + 1, z' - 1 >= 0 encArg(z') -{ 254 + s20 + s21 + 467*x_1108 + 283*x_1108^2 + 48*x_1108^3 + 467*x_248 + 283*x_248^2 + 48*x_248^3 }-> s22 :|: s20 >= 0, s20 <= x_1108 + 1, s21 >= 0, s21 <= x_248 + 1, s22 >= 0, s22 <= 1 + s20 + s21 + 1, x_1108 >= 0, z' = 1 + (1 + x_1108 + x_248), x_248 >= 0 encArg(z') -{ -58 + s23 + -89*z' + -5*z'^2 + 48*z'^3 }-> s24 :|: s23 >= 0, s23 <= z' - 2 + 1, s24 >= 0, s24 <= 1 + s23 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s25 + 2*s25*s26 + s25^2 + 4*s26 + s26^2 + s27 + 467*x_1111 + 283*x_1111^2 + 48*x_1111^3 + 467*x_249 + 283*x_249^2 + 48*x_249^3 }-> s28 :|: s25 >= 0, s25 <= x_1111 + 1, s26 >= 0, s26 <= x_249 + 1, s27 >= 0, s27 <= s25 + s26 + 1, s28 >= 0, s28 <= s27 + 1, z' = 1 + (1 + x_1111 + x_249), x_1111 >= 0, x_249 >= 0 encArg(z') -{ -58 + s29 + s30 + -89*z' + -5*z'^2 + 48*z'^3 }-> s31 :|: s29 >= 0, s29 <= z' - 2 + 1, s30 >= 0, s30 <= s29 + 1, s31 >= 0, s31 <= s30 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s33 + s35 + 467*x_1113 + 283*x_1113^2 + 48*x_1113^3 + 467*x_250 + 283*x_250^2 + 48*x_250^3 + 467*x_311 + 283*x_311^2 + 48*x_311^3 }-> s36 :|: s32 >= 0, s32 <= x_1113 + 1, s33 >= 0, s33 <= x_250 + 1, s34 >= 0, s34 <= x_311 + 1, s35 >= 0, s35 <= s32 + s33 + s34 + 1, s36 >= 0, s36 <= s35 + 1, x_250 >= 0, x_311 >= 0, x_1113 >= 0, z' = 1 + (1 + x_1113 + x_250 + x_311) encArg(z') -{ 378 + s38 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 + 467*x_3 + 283*x_3^2 + 48*x_3^3 }-> s40 :|: s37 >= 0, s37 <= x_1 + 1, s38 >= 0, s38 <= x_2 + 1, s39 >= 0, s39 <= x_3 + 1, s40 >= 0, s40 <= s37 + s38 + s39 + 1, x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0 encArg(z') -{ -58 + s75 + -89*z' + -5*z'^2 + 48*z'^3 }-> s76 :|: s74 >= 0, s74 <= z' - 2 + 1, s75 >= 0, s75 <= s74 + 1, s76 >= 0, s76 <= s75 + 1, z' - 2 >= 0 encArg(z') -{ 253 + 467*x_11322 + 283*x_11322^2 + 48*x_11322^3 + 467*x_2587 + 283*x_2587^2 + 48*x_2587^3 }-> s79 :|: s77 >= 0, s77 <= x_11322 + 1, s78 >= 0, s78 <= x_2587 + 1, s79 >= 0, s79 <= 1 + s77 + s78 + 1, x_2587 >= 0, x_11322 >= 0, z' = 1 + (1 + x_11322 + x_2587) encArg(z') -{ 253 + 467*x_11323 + 283*x_11323^2 + 48*x_11323^3 + 467*x_2588 + 283*x_2588^2 + 48*x_2588^3 }-> s82 :|: s80 >= 0, s80 <= x_11323 + 1, s81 >= 0, s81 <= x_2588 + 1, s82 >= 0, s82 <= 1 + s80 + s81 + 1, z' = 1 + (1 + x_11323 + x_2588), x_2588 >= 0, x_11323 >= 0 encArg(z') -{ -59 + -89*z' + -5*z'^2 + 48*z'^3 }-> s84 :|: s83 >= 0, s83 <= z' - 2 + 1, s84 >= 0, s84 <= 1 + s83 + 1, z' - 2 >= 0 encArg(z') -{ 261 + 6*s85 + 2*s85*s86 + s85^2 + 4*s86 + s86^2 + 467*x_11326 + 283*x_11326^2 + 48*x_11326^3 + 467*x_2589 + 283*x_2589^2 + 48*x_2589^3 }-> s88 :|: s85 >= 0, s85 <= x_11326 + 1, s86 >= 0, s86 <= x_2589 + 1, s87 >= 0, s87 <= s85 + s86 + 1, s88 >= 0, s88 <= s87 + 1, x_11326 >= 0, x_2589 >= 0, z' = 1 + (1 + x_11326 + x_2589) encArg(z') -{ -58 + s89 + -89*z' + -5*z'^2 + 48*z'^3 }-> s91 :|: s89 >= 0, s89 <= z' - 2 + 1, s90 >= 0, s90 <= s89 + 1, s91 >= 0, s91 <= s90 + 1, z' - 2 >= 0 encArg(z') -{ 379 + s93 + 467*x_11328 + 283*x_11328^2 + 48*x_11328^3 + 467*x_2590 + 283*x_2590^2 + 48*x_2590^3 + 467*x_3146 + 283*x_3146^2 + 48*x_3146^3 }-> s96 :|: s92 >= 0, s92 <= x_11328 + 1, s93 >= 0, s93 <= x_2590 + 1, s94 >= 0, s94 <= x_3146 + 1, s95 >= 0, s95 <= s92 + s93 + s94 + 1, s96 >= 0, s96 <= s95 + 1, z' = 1 + (1 + x_11328 + x_2590 + x_3146), x_11328 >= 0, x_2590 >= 0, x_3146 >= 0 encArg(z') -{ -58 + -89*z' + -5*z'^2 + 48*z'^3 }-> s99 :|: s97 >= 0, s97 <= z' - 2 + 1, s98 >= 0, s98 <= s97 + 1, s99 >= 0, s99 <= s98 + 1, z' - 2 >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 0 encArg(z') -{ 0 }-> 0 :|: z' >= 0 encArg(z') -{ 0 }-> 0 :|: z' = 1 + 0, v0 >= 0, 0 = v0 encArg(z') -{ 0 }-> 0 :|: z' - 1 >= 0, v0 >= 0, 0 = v0 encArg(z') -{ -106 + 45*z' + 139*z'^2 + 48*z'^3 }-> 1 + s13 :|: s13 >= 0, s13 <= z' - 1 + 1, z' - 1 >= 0 encArg(z') -{ 252 + 467*x_1 + 283*x_1^2 + 48*x_1^3 + 467*x_2 + 283*x_2^2 + 48*x_2^3 }-> 1 + s11 + s12 :|: s11 >= 0, s11 <= x_1 + 1, s12 >= 0, s12 <= x_2 + 1, x_1 >= 0, x_2 >= 0, z' = 1 + x_1 + x_2 encode_0 -{ 0 }-> 0 :|: encode_f(z', z'') -{ 260 + 6*s41 + 2*s41*s42 + s41^2 + 4*s42 + s42^2 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> s43 :|: s41 >= 0, s41 <= z' + 1, s42 >= 0, s42 <= z'' + 1, s43 >= 0, s43 <= s41 + s42 + 1, z' >= 0, z'' >= 0 encode_f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_g(z') -{ -104 + s101 + 45*z' + 139*z'^2 + 48*z'^3 }-> s102 :|: s100 >= 0, s100 <= z' - 1 + 1, s101 >= 0, s101 <= s100 + 1, s102 >= 0, s102 <= s101 + 1, z' - 1 >= 0 encode_g(z') -{ -104 + s153 + 45*z' + 139*z'^2 + 48*z'^3 }-> s154 :|: s152 >= 0, s152 <= z' - 1 + 1, s153 >= 0, s153 <= s152, s154 >= 0, s154 <= s153 + 1, z' - 1 >= 0 encode_g(z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 0 + 1, z' = 0 encode_g(z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 0 + 1, z' >= 0 encode_g(z') -{ 254 + s46 + s47 + 467*x_11439 + 283*x_11439^2 + 48*x_11439^3 + 467*x_2639 + 283*x_2639^2 + 48*x_2639^3 }-> s48 :|: s46 >= 0, s46 <= x_11439 + 1, s47 >= 0, s47 <= x_2639 + 1, s48 >= 0, s48 <= 1 + s46 + s47 + 1, x_2639 >= 0, z' = 1 + x_11439 + x_2639, x_11439 >= 0 encode_g(z') -{ 254 + s49 + s50 + 467*x_11440 + 283*x_11440^2 + 48*x_11440^3 + 467*x_2640 + 283*x_2640^2 + 48*x_2640^3 }-> s51 :|: s49 >= 0, s49 <= x_11440 + 1, s50 >= 0, s50 <= x_2640 + 1, s51 >= 0, s51 <= 1 + s49 + s50 + 1, x_2640 >= 0, x_11440 >= 0, z' = 1 + x_11440 + x_2640 encode_g(z') -{ -104 + s52 + 45*z' + 139*z'^2 + 48*z'^3 }-> s53 :|: s52 >= 0, s52 <= z' - 1 + 1, s53 >= 0, s53 <= 1 + s52 + 1, z' - 1 >= 0 encode_g(z') -{ 261 + 6*s54 + 2*s54*s55 + s54^2 + 4*s55 + s55^2 + s56 + 467*x_11443 + 283*x_11443^2 + 48*x_11443^3 + 467*x_2641 + 283*x_2641^2 + 48*x_2641^3 }-> s57 :|: s54 >= 0, s54 <= x_11443 + 1, s55 >= 0, s55 <= x_2641 + 1, s56 >= 0, s56 <= s54 + s55 + 1, s57 >= 0, s57 <= s56 + 1, x_11443 >= 0, z' = 1 + x_11443 + x_2641, x_2641 >= 0 encode_g(z') -{ -104 + s58 + s59 + 45*z' + 139*z'^2 + 48*z'^3 }-> s60 :|: s58 >= 0, s58 <= z' - 1 + 1, s59 >= 0, s59 <= s58 + 1, s60 >= 0, s60 <= s59 + 1, z' - 1 >= 0 encode_g(z') -{ 379 + s62 + s64 + 467*x_11445 + 283*x_11445^2 + 48*x_11445^3 + 467*x_2642 + 283*x_2642^2 + 48*x_2642^3 + 467*x_3159 + 283*x_3159^2 + 48*x_3159^3 }-> s65 :|: s61 >= 0, s61 <= x_11445 + 1, s62 >= 0, s62 <= x_2642 + 1, s63 >= 0, s63 <= x_3159 + 1, s64 >= 0, s64 <= s61 + s62 + s63 + 1, s65 >= 0, s65 <= s64 + 1, x_11445 >= 0, x_2642 >= 0, z' = 1 + x_11445 + x_2642 + x_3159, x_3159 >= 0 encode_g(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 0 }-> 0 :|: z' >= 0 encode_h(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s73 :|: s73 >= 0, s73 <= z' + 1, z' >= 0 encode_h1(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_h1(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s66 + s67 :|: s66 >= 0, s66 <= z' + 1, s67 >= 0, s67 <= z'' + 1, z' >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 378 + s69 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 + 467*z1 + 283*z1^2 + 48*z1^3 }-> s71 :|: s68 >= 0, s68 <= z' + 1, s69 >= 0, s69 <= z'' + 1, s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s68 + s69 + s70 + 1, z' >= 0, z1 >= 0, z'' >= 0 encode_h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 encode_i(z') -{ 253 + 467*x_12654 + 283*x_12654^2 + 48*x_12654^3 + 467*x_21179 + 283*x_21179^2 + 48*x_21179^3 }-> s157 :|: s155 >= 0, s155 <= x_12654 + 1, s156 >= 0, s156 <= x_21179 + 1, s157 >= 0, s157 <= 1 + s155 + s156, x_12654 >= 0, z' = 1 + x_12654 + x_21179, x_21179 >= 0 encode_i(z') -{ 253 + 467*x_12655 + 283*x_12655^2 + 48*x_12655^3 + 467*x_21180 + 283*x_21180^2 + 48*x_21180^3 }-> s160 :|: s158 >= 0, s158 <= x_12655 + 1, s159 >= 0, s159 <= x_21180 + 1, s160 >= 0, s160 <= 1 + s158 + s159, z' = 1 + x_12655 + x_21180, x_12655 >= 0, x_21180 >= 0 encode_i(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s162 :|: s161 >= 0, s161 <= z' - 1 + 1, s162 >= 0, s162 <= 1 + s161, z' - 1 >= 0 encode_i(z') -{ 261 + 6*s163 + 2*s163*s164 + s163^2 + 4*s164 + s164^2 + 467*x_12658 + 283*x_12658^2 + 48*x_12658^3 + 467*x_21181 + 283*x_21181^2 + 48*x_21181^3 }-> s166 :|: s163 >= 0, s163 <= x_12658 + 1, s164 >= 0, s164 <= x_21181 + 1, s165 >= 0, s165 <= s163 + s164 + 1, s166 >= 0, s166 <= s165, x_12658 >= 0, z' = 1 + x_12658 + x_21181, x_21181 >= 0 encode_i(z') -{ -104 + s167 + 45*z' + 139*z'^2 + 48*z'^3 }-> s169 :|: s167 >= 0, s167 <= z' - 1 + 1, s168 >= 0, s168 <= s167 + 1, s169 >= 0, s169 <= s168, z' - 1 >= 0 encode_i(z') -{ 379 + s171 + 467*x_12660 + 283*x_12660^2 + 48*x_12660^3 + 467*x_21182 + 283*x_21182^2 + 48*x_21182^3 + 467*x_3294 + 283*x_3294^2 + 48*x_3294^3 }-> s174 :|: s170 >= 0, s170 <= x_12660 + 1, s171 >= 0, s171 <= x_21182 + 1, s172 >= 0, s172 <= x_3294 + 1, s173 >= 0, s173 <= s170 + s171 + s172 + 1, s174 >= 0, s174 <= s173, z' = 1 + x_12660 + x_21182 + x_3294, x_21182 >= 0, x_12660 >= 0, x_3294 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s177 :|: s175 >= 0, s175 <= z' - 1 + 1, s176 >= 0, s176 <= s175, s177 >= 0, s177 <= s176, z' - 1 >= 0 encode_i(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s189 :|: s187 >= 0, s187 <= z' - 1 + 1, s188 >= 0, s188 <= s187 + 1, s189 >= 0, s189 <= s188, z' - 1 >= 0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0 encode_i(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_i(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_j(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 encode_j(z', z'') -{ 252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3 }-> 1 + s44 + s45 :|: s44 >= 0, s44 <= z' + 1, s45 >= 0, s45 <= z'' + 1, z' >= 0, z'' >= 0 encode_k(z') -{ 253 + 467*x_11448 + 283*x_11448^2 + 48*x_11448^3 + 467*x_2643 + 283*x_2643^2 + 48*x_2643^3 }-> s105 :|: s103 >= 0, s103 <= x_11448 + 1, s104 >= 0, s104 <= x_2643 + 1, s105 >= 0, s105 <= 1 + s103 + s104 + 1, z' = 1 + x_11448 + x_2643, x_2643 >= 0, x_11448 >= 0 encode_k(z') -{ 253 + 467*x_11449 + 283*x_11449^2 + 48*x_11449^3 + 467*x_2644 + 283*x_2644^2 + 48*x_2644^3 }-> s108 :|: s106 >= 0, s106 <= x_11449 + 1, s107 >= 0, s107 <= x_2644 + 1, s108 >= 0, s108 <= 1 + s106 + s107 + 1, x_2644 >= 0, z' = 1 + x_11449 + x_2644, x_11449 >= 0 encode_k(z') -{ -105 + 45*z' + 139*z'^2 + 48*z'^3 }-> s110 :|: s109 >= 0, s109 <= z' - 1 + 1, s110 >= 0, s110 <= 1 + s109 + 1, z' - 1 >= 0 encode_k(z') -{ 261 + 6*s111 + 2*s111*s112 + s111^2 + 4*s112 + s112^2 + 467*x_11452 + 283*x_11452^2 + 48*x_11452^3 + 467*x_2645 + 283*x_2645^2 + 48*x_2645^3 }-> s114 :|: s111 >= 0, s111 <= x_11452 + 1, s112 >= 0, s112 <= x_2645 + 1, s113 >= 0, s113 <= s111 + s112 + 1, s114 >= 0, s114 <= s113 + 1, x_2645 >= 0, z' = 1 + x_11452 + x_2645, x_11452 >= 0 encode_k(z') -{ -104 + s115 + 45*z' + 139*z'^2 + 48*z'^3 }-> s117 :|: s115 >= 0, s115 <= z' - 1 + 1, s116 >= 0, s116 <= s115 + 1, s117 >= 0, s117 <= s116 + 1, z' - 1 >= 0 encode_k(z') -{ 379 + s119 + 467*x_11454 + 283*x_11454^2 + 48*x_11454^3 + 467*x_2646 + 283*x_2646^2 + 48*x_2646^3 + 467*x_3160 + 283*x_3160^2 + 48*x_3160^3 }-> s122 :|: s118 >= 0, s118 <= x_11454 + 1, s119 >= 0, s119 <= x_2646 + 1, s120 >= 0, s120 <= x_3160 + 1, s121 >= 0, s121 <= s118 + s119 + s120 + 1, s122 >= 0, s122 <= s121 + 1, x_11454 >= 0, x_3160 >= 0, x_2646 >= 0, z' = 1 + x_11454 + x_2646 + x_3160 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s125 :|: s123 >= 0, s123 <= z' - 1 + 1, s124 >= 0, s124 <= s123 + 1, s125 >= 0, s125 <= s124 + 1, z' - 1 >= 0 encode_k(z') -{ -104 + 45*z' + 139*z'^2 + 48*z'^3 }-> s186 :|: s184 >= 0, s184 <= z' - 1 + 1, s185 >= 0, s185 <= s184, s186 >= 0, s186 <= s185 + 1, z' - 1 >= 0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0 encode_k(z') -{ 0 }-> 0 :|: z' = 0, v0 >= 0, 0 = v0 encode_k(z') -{ 0 }-> 0 :|: z' >= 0, v0 >= 0, 0 = v0 encode_s(z') -{ 0 }-> 0 :|: z' >= 0 encode_s(z') -{ 126 + 467*z' + 283*z'^2 + 48*z'^3 }-> 1 + s72 :|: s72 >= 0, s72 <= z' + 1, z' >= 0 f(z', z'') -{ 1 + z' }-> s :|: s >= 0, s <= 0 + z' + (1 + y + z) + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z f(z', z'') -{ 10 + s9 + 6*x + x^2 }-> s10 :|: s9 >= 0, s9 <= x + 0 + 1, s10 >= 0, s10 <= s9 + 1, z' = 1 + x + z'', x >= 0, z'' >= 0 f(z', z'') -{ 11 + s5 + 6*x + 2*x*z'' + x^2 + 4*z'' + z''^2 }-> s6 :|: s5 >= 0, s5 <= x + (1 + 0 + (z'' - 1)) + 1, s6 >= 0, s6 <= s5 + 1, z' = 1 + x + (1 + (z'' - 1)), x >= 0, z'' - 1 >= 0 f(z', z'') -{ 23 + s7 + 10*x + 2*x*x'' + 2*x*y' + x^2 + 8*x'' + 2*x''*y' + x''^2 + 8*y' + y'^2 }-> s8 :|: s7 >= 0, s7 <= x + (1 + (1 + x'') + y') + 1, s8 >= 0, s8 <= s7 + 1, z' = 1 + x + (1 + x'' + y'), x >= 0, z'' = 1 + x'' + y', y' >= 0, x'' >= 0 f(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 f(z', z'') -{ 0 }-> 1 + z' + z'' :|: z' >= 0, z'' >= 0 g(z') -{ 1 + y }-> s'' :|: s'' >= 0, s'' <= 1 + x + y + (1 + z + u) + 1, z >= 0, x >= 0, y >= 0, z' = 1 + x + y + (1 + z + u), u >= 0 g(z') -{ 0 }-> 0 :|: z' >= 0 h2(z', z'', z1) -{ 1 + y }-> s' :|: s' >= 0, s' <= 1 + z' + y + (1 + (1 + z) + u) + 1, z >= 0, z' >= 0, y >= 0, z1 = 1 + z + u, z'' = 1 + y + (1 + z + u), u >= 0 h2(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0 h2(z', z'', z1) -{ 0 }-> 1 + z' + z'' + z1 :|: z' >= 0, z'' >= 0, z1 >= 0 i(z') -{ 1 }-> y :|: x >= 0, y >= 0, z' = 1 + x + (1 + y) i(z') -{ 1 }-> z :|: z >= 0, x >= 0, y >= 0, z' = 1 + (1 + x) + y + (1 + x + z) i(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 0 }-> 0 :|: z' >= 0 k(z') -{ 1 }-> 1 + 0 + (z' - 1) :|: z' - 1 >= 0 k(z') -{ 1 }-> 1 + (1 + x) + y :|: z' = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: Previous analysis results are: h2: runtime: O(n^1) [z''], size: O(n^1) [1 + z' + z'' + z1] encode_0: runtime: O(1) [0], size: O(1) [0] i: runtime: O(1) [1], size: O(n^1) [z'] k: runtime: O(1) [1], size: O(n^1) [1 + z'] g: runtime: O(n^1) [1 + z'], size: O(n^1) [1 + z'] f: runtime: O(n^2) [8 + 6*z' + 2*z'*z'' + z'^2 + 4*z'' + z''^2], size: O(n^1) [1 + z' + z''] encArg: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [1 + z'] encode_k: runtime: O(n^3) [1161 + 4403*z' + 3107*z'^2 + 624*z'^3], size: O(n^1) [4 + z'] encode_h: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [2 + z'] encode_h2: runtime: O(n^3) [379 + 467*z' + 283*z'^2 + 48*z'^3 + 468*z'' + 283*z''^2 + 48*z''^3 + 467*z1 + 283*z1^2 + 48*z1^3], size: O(n^1) [4 + z' + z'' + z1] encode_h1: runtime: O(n^3) [252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3], size: O(n^1) [3 + z' + z''] encode_g: runtime: O(n^3) [1171 + 4416*z' + 3107*z'^2 + 624*z'^3], size: O(n^1) [4 + z'] encode_f: runtime: O(n^3) [274 + 477*z' + 2*z'*z'' + 284*z'^2 + 48*z'^3 + 475*z'' + 284*z''^2 + 48*z''^3], size: O(n^1) [3 + z' + z''] encode_j: runtime: O(n^3) [252 + 467*z' + 283*z'^2 + 48*z'^3 + 467*z'' + 283*z''^2 + 48*z''^3], size: O(n^1) [3 + z' + z''] encode_i: runtime: O(n^3) [1161 + 4403*z' + 3107*z'^2 + 624*z'^3], size: O(n^1) [3 + z'] encode_s: runtime: O(n^3) [126 + 467*z' + 283*z'^2 + 48*z'^3], size: O(n^1) [2 + z'] ---------------------------------------- (121) FinalProof (FINISHED) Computed overall runtime complexity ---------------------------------------- (122) BOUNDS(1, n^3)