676.15/291.59 WORST_CASE(Omega(n^1), O(n^10)) 676.38/291.62 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 676.38/291.62 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 676.38/291.62 676.38/291.62 676.38/291.62 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^10). 676.38/291.62 676.38/291.62 (0) CpxTRS 676.38/291.62 (1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] 676.38/291.62 (2) CpxWeightedTrs 676.38/291.62 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 676.38/291.62 (4) CpxTypedWeightedTrs 676.38/291.62 (5) CompletionProof [UPPER BOUND(ID), 0 ms] 676.38/291.62 (6) CpxTypedWeightedCompleteTrs 676.38/291.62 (7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] 676.38/291.62 (8) CpxTypedWeightedCompleteTrs 676.38/291.62 (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] 676.38/291.62 (10) CpxRNTS 676.38/291.62 (11) InliningProof [UPPER BOUND(ID), 0 ms] 676.38/291.62 (12) CpxRNTS 676.38/291.62 (13) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] 676.38/291.62 (14) CpxRNTS 676.38/291.62 (15) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] 676.38/291.62 (16) CpxRNTS 676.38/291.62 (17) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 676.38/291.62 (18) CpxRNTS 676.38/291.62 (19) IntTrsBoundProof [UPPER BOUND(ID), 124 ms] 676.38/291.62 (20) CpxRNTS 676.38/291.62 (21) IntTrsBoundProof [UPPER BOUND(ID), 35 ms] 676.38/291.62 (22) CpxRNTS 676.38/291.62 (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 676.38/291.62 (24) CpxRNTS 676.38/291.62 (25) IntTrsBoundProof [UPPER BOUND(ID), 57 ms] 676.38/291.62 (26) CpxRNTS 676.38/291.62 (27) IntTrsBoundProof [UPPER BOUND(ID), 4 ms] 676.38/291.62 (28) CpxRNTS 676.38/291.62 (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 676.38/291.62 (30) CpxRNTS 676.38/291.62 (31) IntTrsBoundProof [UPPER BOUND(ID), 87 ms] 676.38/291.62 (32) CpxRNTS 676.38/291.62 (33) IntTrsBoundProof [UPPER BOUND(ID), 52 ms] 676.38/291.62 (34) CpxRNTS 676.38/291.62 (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 676.38/291.62 (36) CpxRNTS 676.38/291.62 (37) IntTrsBoundProof [UPPER BOUND(ID), 87 ms] 676.38/291.62 (38) CpxRNTS 676.38/291.62 (39) IntTrsBoundProof [UPPER BOUND(ID), 92 ms] 676.38/291.62 (40) CpxRNTS 676.38/291.62 (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 676.38/291.62 (42) CpxRNTS 676.38/291.62 (43) IntTrsBoundProof [UPPER BOUND(ID), 86 ms] 676.38/291.62 (44) CpxRNTS 676.38/291.62 (45) IntTrsBoundProof [UPPER BOUND(ID), 3 ms] 676.38/291.62 (46) CpxRNTS 676.38/291.62 (47) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 676.38/291.62 (48) CpxRNTS 676.38/291.62 (49) IntTrsBoundProof [UPPER BOUND(ID), 159 ms] 676.38/291.62 (50) CpxRNTS 676.38/291.62 (51) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] 676.38/291.62 (52) CpxRNTS 676.38/291.62 (53) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 676.38/291.62 (54) CpxRNTS 676.38/291.62 (55) IntTrsBoundProof [UPPER BOUND(ID), 117 ms] 676.38/291.62 (56) CpxRNTS 676.38/291.62 (57) IntTrsBoundProof [UPPER BOUND(ID), 3 ms] 676.38/291.62 (58) CpxRNTS 676.38/291.62 (59) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 676.38/291.62 (60) CpxRNTS 676.38/291.62 (61) IntTrsBoundProof [UPPER BOUND(ID), 218 ms] 676.38/291.62 (62) CpxRNTS 676.38/291.62 (63) IntTrsBoundProof [UPPER BOUND(ID), 3 ms] 676.38/291.62 (64) CpxRNTS 676.38/291.62 (65) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 676.38/291.62 (66) CpxRNTS 676.38/291.62 (67) IntTrsBoundProof [UPPER BOUND(ID), 157 ms] 676.38/291.62 (68) CpxRNTS 676.38/291.62 (69) IntTrsBoundProof [UPPER BOUND(ID), 1 ms] 676.38/291.62 (70) CpxRNTS 676.38/291.62 (71) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 676.38/291.62 (72) CpxRNTS 676.38/291.62 (73) IntTrsBoundProof [UPPER BOUND(ID), 158 ms] 676.38/291.62 (74) CpxRNTS 676.38/291.62 (75) IntTrsBoundProof [UPPER BOUND(ID), 2 ms] 676.38/291.62 (76) CpxRNTS 676.38/291.62 (77) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 676.38/291.62 (78) CpxRNTS 676.38/291.62 (79) IntTrsBoundProof [UPPER BOUND(ID), 87 ms] 676.38/291.62 (80) CpxRNTS 676.38/291.62 (81) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] 676.38/291.62 (82) CpxRNTS 676.38/291.62 (83) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 676.38/291.62 (84) CpxRNTS 676.38/291.62 (85) IntTrsBoundProof [UPPER BOUND(ID), 238 ms] 676.38/291.62 (86) CpxRNTS 676.38/291.62 (87) IntTrsBoundProof [UPPER BOUND(ID), 33 ms] 676.38/291.62 (88) CpxRNTS 676.38/291.62 (89) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 676.38/291.62 (90) CpxRNTS 676.38/291.62 (91) IntTrsBoundProof [UPPER BOUND(ID), 7 ms] 676.38/291.62 (92) CpxRNTS 676.38/291.62 (93) IntTrsBoundProof [UPPER BOUND(ID), 3 ms] 676.38/291.62 (94) CpxRNTS 676.38/291.62 (95) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 676.38/291.62 (96) CpxRNTS 676.38/291.62 (97) IntTrsBoundProof [UPPER BOUND(ID), 147 ms] 676.38/291.62 (98) CpxRNTS 676.38/291.62 (99) IntTrsBoundProof [UPPER BOUND(ID), 32 ms] 676.38/291.62 (100) CpxRNTS 676.38/291.62 (101) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 676.38/291.62 (102) CpxRNTS 676.38/291.62 (103) IntTrsBoundProof [UPPER BOUND(ID), 86 ms] 676.38/291.62 (104) CpxRNTS 676.38/291.62 (105) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] 676.38/291.62 (106) CpxRNTS 676.38/291.62 (107) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 676.38/291.62 (108) CpxRNTS 676.38/291.62 (109) IntTrsBoundProof [UPPER BOUND(ID), 78 ms] 676.38/291.62 (110) CpxRNTS 676.38/291.62 (111) IntTrsBoundProof [UPPER BOUND(ID), 33 ms] 676.38/291.62 (112) CpxRNTS 676.38/291.62 (113) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 676.38/291.62 (114) CpxRNTS 676.38/291.62 (115) IntTrsBoundProof [UPPER BOUND(ID), 98 ms] 676.38/291.62 (116) CpxRNTS 676.38/291.62 (117) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] 676.38/291.62 (118) CpxRNTS 676.38/291.62 (119) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 676.38/291.62 (120) CpxRNTS 676.38/291.62 (121) IntTrsBoundProof [UPPER BOUND(ID), 178 ms] 676.38/291.62 (122) CpxRNTS 676.38/291.62 (123) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] 676.38/291.62 (124) CpxRNTS 676.38/291.62 (125) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 676.38/291.62 (126) CpxRNTS 676.38/291.62 (127) IntTrsBoundProof [UPPER BOUND(ID), 97 ms] 676.38/291.62 (128) CpxRNTS 676.38/291.62 (129) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] 676.38/291.62 (130) CpxRNTS 676.38/291.62 (131) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 676.38/291.62 (132) CpxRNTS 676.38/291.62 (133) IntTrsBoundProof [UPPER BOUND(ID), 86 ms] 676.38/291.62 (134) CpxRNTS 676.38/291.62 (135) IntTrsBoundProof [UPPER BOUND(ID), 3 ms] 676.38/291.62 (136) CpxRNTS 676.38/291.62 (137) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 676.38/291.62 (138) CpxRNTS 676.38/291.62 (139) IntTrsBoundProof [UPPER BOUND(ID), 147 ms] 676.38/291.62 (140) CpxRNTS 676.38/291.62 (141) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] 676.38/291.62 (142) CpxRNTS 676.38/291.62 (143) FinalProof [FINISHED, 0 ms] 676.38/291.62 (144) BOUNDS(1, n^10) 676.38/291.62 (145) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 676.38/291.62 (146) TRS for Loop Detection 676.38/291.62 (147) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 676.38/291.62 (148) BEST 676.38/291.62 (149) proven lower bound 676.38/291.62 (150) LowerBoundPropagationProof [FINISHED, 0 ms] 676.38/291.62 (151) BOUNDS(n^1, INF) 676.38/291.62 (152) TRS for Loop Detection 676.38/291.62 676.38/291.62 676.38/291.62 ---------------------------------------- 676.38/291.62 676.38/291.62 (0) 676.38/291.62 Obligation: 676.38/291.62 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^10). 676.38/291.62 676.38/291.62 676.38/291.62 The TRS R consists of the following rules: 676.38/291.62 676.38/291.62 f_0(x) -> a 676.38/291.62 f_1(x) -> g_1(x, x) 676.38/291.62 g_1(s(x), y) -> b(f_0(y), g_1(x, y)) 676.38/291.62 f_2(x) -> g_2(x, x) 676.38/291.62 g_2(s(x), y) -> b(f_1(y), g_2(x, y)) 676.38/291.62 f_3(x) -> g_3(x, x) 676.38/291.62 g_3(s(x), y) -> b(f_2(y), g_3(x, y)) 676.38/291.62 f_4(x) -> g_4(x, x) 676.38/291.62 g_4(s(x), y) -> b(f_3(y), g_4(x, y)) 676.38/291.62 f_5(x) -> g_5(x, x) 676.38/291.62 g_5(s(x), y) -> b(f_4(y), g_5(x, y)) 676.38/291.62 f_6(x) -> g_6(x, x) 676.38/291.62 g_6(s(x), y) -> b(f_5(y), g_6(x, y)) 676.38/291.62 f_7(x) -> g_7(x, x) 676.38/291.62 g_7(s(x), y) -> b(f_6(y), g_7(x, y)) 676.38/291.62 f_8(x) -> g_8(x, x) 676.38/291.62 g_8(s(x), y) -> b(f_7(y), g_8(x, y)) 676.38/291.62 f_9(x) -> g_9(x, x) 676.38/291.62 g_9(s(x), y) -> b(f_8(y), g_9(x, y)) 676.38/291.62 f_10(x) -> g_10(x, x) 676.38/291.62 g_10(s(x), y) -> b(f_9(y), g_10(x, y)) 676.38/291.62 676.38/291.62 S is empty. 676.38/291.62 Rewrite Strategy: INNERMOST 676.38/291.62 ---------------------------------------- 676.38/291.62 676.38/291.62 (1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) 676.38/291.62 Transformed relative TRS to weighted TRS 676.38/291.62 ---------------------------------------- 676.38/291.62 676.38/291.62 (2) 676.38/291.62 Obligation: 676.38/291.62 The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^10). 676.38/291.62 676.38/291.62 676.38/291.62 The TRS R consists of the following rules: 676.38/291.62 676.38/291.62 f_0(x) -> a [1] 676.38/291.62 f_1(x) -> g_1(x, x) [1] 676.38/291.62 g_1(s(x), y) -> b(f_0(y), g_1(x, y)) [1] 676.38/291.62 f_2(x) -> g_2(x, x) [1] 676.38/291.62 g_2(s(x), y) -> b(f_1(y), g_2(x, y)) [1] 676.38/291.62 f_3(x) -> g_3(x, x) [1] 676.38/291.62 g_3(s(x), y) -> b(f_2(y), g_3(x, y)) [1] 676.38/291.62 f_4(x) -> g_4(x, x) [1] 676.38/291.62 g_4(s(x), y) -> b(f_3(y), g_4(x, y)) [1] 676.38/291.62 f_5(x) -> g_5(x, x) [1] 676.38/291.62 g_5(s(x), y) -> b(f_4(y), g_5(x, y)) [1] 676.38/291.62 f_6(x) -> g_6(x, x) [1] 676.38/291.62 g_6(s(x), y) -> b(f_5(y), g_6(x, y)) [1] 676.38/291.62 f_7(x) -> g_7(x, x) [1] 676.38/291.62 g_7(s(x), y) -> b(f_6(y), g_7(x, y)) [1] 676.38/291.62 f_8(x) -> g_8(x, x) [1] 676.38/291.62 g_8(s(x), y) -> b(f_7(y), g_8(x, y)) [1] 676.38/291.62 f_9(x) -> g_9(x, x) [1] 676.38/291.62 g_9(s(x), y) -> b(f_8(y), g_9(x, y)) [1] 676.38/291.62 f_10(x) -> g_10(x, x) [1] 676.38/291.62 g_10(s(x), y) -> b(f_9(y), g_10(x, y)) [1] 676.38/291.62 676.38/291.62 Rewrite Strategy: INNERMOST 676.38/291.62 ---------------------------------------- 676.38/291.62 676.38/291.62 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 676.38/291.62 Infered types. 676.38/291.62 ---------------------------------------- 676.38/291.62 676.38/291.62 (4) 676.38/291.62 Obligation: 676.38/291.62 Runtime Complexity Weighted TRS with Types. 676.38/291.62 The TRS R consists of the following rules: 676.38/291.62 676.38/291.62 f_0(x) -> a [1] 676.38/291.62 f_1(x) -> g_1(x, x) [1] 676.38/291.62 g_1(s(x), y) -> b(f_0(y), g_1(x, y)) [1] 676.38/291.62 f_2(x) -> g_2(x, x) [1] 676.38/291.62 g_2(s(x), y) -> b(f_1(y), g_2(x, y)) [1] 676.38/291.62 f_3(x) -> g_3(x, x) [1] 676.38/291.62 g_3(s(x), y) -> b(f_2(y), g_3(x, y)) [1] 676.38/291.62 f_4(x) -> g_4(x, x) [1] 676.38/291.62 g_4(s(x), y) -> b(f_3(y), g_4(x, y)) [1] 676.38/291.62 f_5(x) -> g_5(x, x) [1] 676.38/291.62 g_5(s(x), y) -> b(f_4(y), g_5(x, y)) [1] 676.38/291.62 f_6(x) -> g_6(x, x) [1] 676.38/291.62 g_6(s(x), y) -> b(f_5(y), g_6(x, y)) [1] 676.38/291.62 f_7(x) -> g_7(x, x) [1] 676.38/291.62 g_7(s(x), y) -> b(f_6(y), g_7(x, y)) [1] 676.38/291.62 f_8(x) -> g_8(x, x) [1] 676.38/291.62 g_8(s(x), y) -> b(f_7(y), g_8(x, y)) [1] 676.38/291.62 f_9(x) -> g_9(x, x) [1] 676.38/291.62 g_9(s(x), y) -> b(f_8(y), g_9(x, y)) [1] 676.38/291.62 f_10(x) -> g_10(x, x) [1] 676.38/291.62 g_10(s(x), y) -> b(f_9(y), g_10(x, y)) [1] 676.38/291.62 676.38/291.62 The TRS has the following type information: 676.38/291.62 f_0 :: s -> a:b 676.38/291.62 a :: a:b 676.38/291.62 f_1 :: s -> a:b 676.38/291.62 g_1 :: s -> s -> a:b 676.38/291.62 s :: s -> s 676.38/291.62 b :: a:b -> a:b -> a:b 676.38/291.62 f_2 :: s -> a:b 676.38/291.62 g_2 :: s -> s -> a:b 676.38/291.62 f_3 :: s -> a:b 676.38/291.62 g_3 :: s -> s -> a:b 676.38/291.62 f_4 :: s -> a:b 676.38/291.62 g_4 :: s -> s -> a:b 676.38/291.62 f_5 :: s -> a:b 676.38/291.62 g_5 :: s -> s -> a:b 676.38/291.62 f_6 :: s -> a:b 676.38/291.62 g_6 :: s -> s -> a:b 676.38/291.62 f_7 :: s -> a:b 676.38/291.62 g_7 :: s -> s -> a:b 676.38/291.62 f_8 :: s -> a:b 676.38/291.62 g_8 :: s -> s -> a:b 676.38/291.62 f_9 :: s -> a:b 676.38/291.62 g_9 :: s -> s -> a:b 676.38/291.62 f_10 :: s -> a:b 676.38/291.62 g_10 :: s -> s -> a:b 676.38/291.62 676.38/291.62 Rewrite Strategy: INNERMOST 676.38/291.62 ---------------------------------------- 676.38/291.62 676.38/291.62 (5) CompletionProof (UPPER BOUND(ID)) 676.38/291.62 The transformation into a RNTS is sound, since: 676.38/291.62 676.38/291.62 (a) The obligation is a constructor system where every type has a constant constructor, 676.38/291.62 676.38/291.62 (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: 676.38/291.62 676.38/291.62 f_0_1 676.38/291.62 f_1_1 676.38/291.62 g_1_2 676.38/291.62 f_2_1 676.38/291.62 g_2_2 676.38/291.62 f_3_1 676.38/291.62 g_3_2 676.38/291.62 f_4_1 676.38/291.62 g_4_2 676.38/291.62 f_5_1 676.38/291.62 g_5_2 676.38/291.62 f_6_1 676.38/291.62 g_6_2 676.38/291.62 f_7_1 676.38/291.62 g_7_2 676.38/291.62 f_8_1 676.38/291.62 g_8_2 676.38/291.62 f_9_1 676.38/291.62 g_9_2 676.38/291.62 f_10_1 676.38/291.62 g_10_2 676.38/291.62 676.38/291.62 (c) The following functions are completely defined: 676.38/291.62 none 676.38/291.62 676.38/291.62 Due to the following rules being added: 676.38/291.62 none 676.38/291.62 676.38/291.62 And the following fresh constants: const 676.38/291.62 676.38/291.62 ---------------------------------------- 676.38/291.62 676.38/291.62 (6) 676.38/291.62 Obligation: 676.38/291.62 Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: 676.38/291.62 676.38/291.62 Runtime Complexity Weighted TRS with Types. 676.38/291.62 The TRS R consists of the following rules: 676.38/291.62 676.38/291.62 f_0(x) -> a [1] 676.38/291.62 f_1(x) -> g_1(x, x) [1] 676.38/291.62 g_1(s(x), y) -> b(f_0(y), g_1(x, y)) [1] 676.38/291.62 f_2(x) -> g_2(x, x) [1] 676.38/291.62 g_2(s(x), y) -> b(f_1(y), g_2(x, y)) [1] 676.38/291.62 f_3(x) -> g_3(x, x) [1] 676.38/291.62 g_3(s(x), y) -> b(f_2(y), g_3(x, y)) [1] 676.38/291.62 f_4(x) -> g_4(x, x) [1] 676.38/291.62 g_4(s(x), y) -> b(f_3(y), g_4(x, y)) [1] 676.38/291.62 f_5(x) -> g_5(x, x) [1] 676.38/291.62 g_5(s(x), y) -> b(f_4(y), g_5(x, y)) [1] 676.38/291.62 f_6(x) -> g_6(x, x) [1] 676.38/291.62 g_6(s(x), y) -> b(f_5(y), g_6(x, y)) [1] 676.38/291.62 f_7(x) -> g_7(x, x) [1] 676.38/291.62 g_7(s(x), y) -> b(f_6(y), g_7(x, y)) [1] 676.38/291.62 f_8(x) -> g_8(x, x) [1] 676.38/291.62 g_8(s(x), y) -> b(f_7(y), g_8(x, y)) [1] 676.38/291.62 f_9(x) -> g_9(x, x) [1] 676.38/291.62 g_9(s(x), y) -> b(f_8(y), g_9(x, y)) [1] 676.38/291.62 f_10(x) -> g_10(x, x) [1] 676.38/291.62 g_10(s(x), y) -> b(f_9(y), g_10(x, y)) [1] 676.38/291.62 676.38/291.62 The TRS has the following type information: 676.38/291.62 f_0 :: s -> a:b 676.38/291.62 a :: a:b 676.38/291.62 f_1 :: s -> a:b 676.38/291.62 g_1 :: s -> s -> a:b 676.38/291.62 s :: s -> s 676.38/291.62 b :: a:b -> a:b -> a:b 676.38/291.62 f_2 :: s -> a:b 676.38/291.62 g_2 :: s -> s -> a:b 676.38/291.62 f_3 :: s -> a:b 676.38/291.62 g_3 :: s -> s -> a:b 676.38/291.62 f_4 :: s -> a:b 676.38/291.62 g_4 :: s -> s -> a:b 676.38/291.62 f_5 :: s -> a:b 676.38/291.62 g_5 :: s -> s -> a:b 676.38/291.62 f_6 :: s -> a:b 676.38/291.62 g_6 :: s -> s -> a:b 676.38/291.62 f_7 :: s -> a:b 676.38/291.62 g_7 :: s -> s -> a:b 676.38/291.62 f_8 :: s -> a:b 676.38/291.62 g_8 :: s -> s -> a:b 676.38/291.62 f_9 :: s -> a:b 676.38/291.62 g_9 :: s -> s -> a:b 676.38/291.62 f_10 :: s -> a:b 676.38/291.62 g_10 :: s -> s -> a:b 676.38/291.62 const :: s 676.38/291.62 676.38/291.62 Rewrite Strategy: INNERMOST 676.38/291.62 ---------------------------------------- 676.38/291.62 676.38/291.62 (7) NarrowingProof (BOTH BOUNDS(ID, ID)) 676.38/291.62 Narrowed the inner basic terms of all right-hand sides by a single narrowing step. 676.38/291.62 ---------------------------------------- 676.38/291.62 676.38/291.62 (8) 676.38/291.62 Obligation: 676.38/291.62 Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: 676.38/291.62 676.38/291.62 Runtime Complexity Weighted TRS with Types. 676.38/291.62 The TRS R consists of the following rules: 676.38/291.62 676.38/291.62 f_0(x) -> a [1] 676.38/291.62 f_1(x) -> g_1(x, x) [1] 676.38/291.62 g_1(s(x), y) -> b(f_0(y), g_1(x, y)) [1] 676.38/291.62 f_2(x) -> g_2(x, x) [1] 676.38/291.62 g_2(s(x), y) -> b(f_1(y), g_2(x, y)) [1] 676.38/291.62 f_3(x) -> g_3(x, x) [1] 676.38/291.62 g_3(s(x), y) -> b(f_2(y), g_3(x, y)) [1] 676.38/291.62 f_4(x) -> g_4(x, x) [1] 676.38/291.62 g_4(s(x), y) -> b(f_3(y), g_4(x, y)) [1] 676.38/291.62 f_5(x) -> g_5(x, x) [1] 676.38/291.62 g_5(s(x), y) -> b(f_4(y), g_5(x, y)) [1] 676.38/291.62 f_6(x) -> g_6(x, x) [1] 676.38/291.62 g_6(s(x), y) -> b(f_5(y), g_6(x, y)) [1] 676.38/291.62 f_7(x) -> g_7(x, x) [1] 676.38/291.62 g_7(s(x), y) -> b(f_6(y), g_7(x, y)) [1] 676.38/291.62 f_8(x) -> g_8(x, x) [1] 676.38/291.62 g_8(s(x), y) -> b(f_7(y), g_8(x, y)) [1] 676.38/291.62 f_9(x) -> g_9(x, x) [1] 676.38/291.62 g_9(s(x), y) -> b(f_8(y), g_9(x, y)) [1] 676.38/291.62 f_10(x) -> g_10(x, x) [1] 676.38/291.62 g_10(s(x), y) -> b(f_9(y), g_10(x, y)) [1] 676.38/291.62 676.38/291.62 The TRS has the following type information: 676.38/291.62 f_0 :: s -> a:b 676.38/291.62 a :: a:b 676.38/291.62 f_1 :: s -> a:b 676.38/291.62 g_1 :: s -> s -> a:b 676.38/291.62 s :: s -> s 676.38/291.62 b :: a:b -> a:b -> a:b 676.38/291.62 f_2 :: s -> a:b 676.38/291.62 g_2 :: s -> s -> a:b 676.38/291.62 f_3 :: s -> a:b 676.38/291.62 g_3 :: s -> s -> a:b 676.38/291.62 f_4 :: s -> a:b 676.38/291.62 g_4 :: s -> s -> a:b 676.38/291.62 f_5 :: s -> a:b 676.38/291.62 g_5 :: s -> s -> a:b 676.38/291.62 f_6 :: s -> a:b 676.38/291.62 g_6 :: s -> s -> a:b 676.38/291.62 f_7 :: s -> a:b 676.38/291.62 g_7 :: s -> s -> a:b 676.38/291.62 f_8 :: s -> a:b 676.38/291.62 g_8 :: s -> s -> a:b 676.38/291.62 f_9 :: s -> a:b 676.38/291.62 g_9 :: s -> s -> a:b 676.38/291.62 f_10 :: s -> a:b 676.38/291.62 g_10 :: s -> s -> a:b 676.38/291.62 const :: s 676.38/291.62 676.38/291.62 Rewrite Strategy: INNERMOST 676.38/291.62 ---------------------------------------- 676.38/291.62 676.38/291.62 (9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) 676.38/291.62 Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. 676.38/291.62 The constant constructors are abstracted as follows: 676.38/291.62 676.38/291.62 a => 0 676.38/291.62 const => 0 676.38/291.62 676.38/291.62 ---------------------------------------- 676.38/291.62 676.38/291.62 (10) 676.38/291.62 Obligation: 676.38/291.62 Complexity RNTS consisting of the following rules: 676.38/291.62 676.38/291.62 f_0(z) -{ 1 }-> 0 :|: x >= 0, z = x 676.38/291.62 f_1(z) -{ 1 }-> g_1(x, x) :|: x >= 0, z = x 676.38/291.62 f_10(z) -{ 1 }-> g_10(x, x) :|: x >= 0, z = x 676.38/291.62 f_2(z) -{ 1 }-> g_2(x, x) :|: x >= 0, z = x 676.38/291.62 f_3(z) -{ 1 }-> g_3(x, x) :|: x >= 0, z = x 676.38/291.62 f_4(z) -{ 1 }-> g_4(x, x) :|: x >= 0, z = x 676.38/291.62 f_5(z) -{ 1 }-> g_5(x, x) :|: x >= 0, z = x 676.38/291.62 f_6(z) -{ 1 }-> g_6(x, x) :|: x >= 0, z = x 676.38/291.62 f_7(z) -{ 1 }-> g_7(x, x) :|: x >= 0, z = x 676.38/291.62 f_8(z) -{ 1 }-> g_8(x, x) :|: x >= 0, z = x 676.38/291.62 f_9(z) -{ 1 }-> g_9(x, x) :|: x >= 0, z = x 676.38/291.62 g_1(z, z') -{ 1 }-> 1 + f_0(y) + g_1(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y 676.38/291.62 g_10(z, z') -{ 1 }-> 1 + f_9(y) + g_10(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y 676.38/291.62 g_2(z, z') -{ 1 }-> 1 + f_1(y) + g_2(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y 676.38/291.62 g_3(z, z') -{ 1 }-> 1 + f_2(y) + g_3(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y 676.38/291.62 g_4(z, z') -{ 1 }-> 1 + f_3(y) + g_4(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y 676.38/291.62 g_5(z, z') -{ 1 }-> 1 + f_4(y) + g_5(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y 676.38/291.62 g_6(z, z') -{ 1 }-> 1 + f_5(y) + g_6(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y 676.38/291.62 g_7(z, z') -{ 1 }-> 1 + f_6(y) + g_7(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y 676.38/291.62 g_8(z, z') -{ 1 }-> 1 + f_7(y) + g_8(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y 676.38/291.62 g_9(z, z') -{ 1 }-> 1 + f_8(y) + g_9(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y 676.38/291.62 676.38/291.62 676.38/291.62 ---------------------------------------- 676.38/291.62 676.38/291.62 (11) InliningProof (UPPER BOUND(ID)) 676.38/291.62 Inlined the following terminating rules on right-hand sides where appropriate: 676.38/291.62 676.38/291.62 f_0(z) -{ 1 }-> 0 :|: x >= 0, z = x 676.38/291.62 676.38/291.62 ---------------------------------------- 676.38/291.62 676.38/291.62 (12) 676.38/291.62 Obligation: 676.38/291.62 Complexity RNTS consisting of the following rules: 676.38/291.62 676.38/291.62 f_0(z) -{ 1 }-> 0 :|: x >= 0, z = x 676.38/291.62 f_1(z) -{ 1 }-> g_1(x, x) :|: x >= 0, z = x 676.38/291.62 f_10(z) -{ 1 }-> g_10(x, x) :|: x >= 0, z = x 676.38/291.62 f_2(z) -{ 1 }-> g_2(x, x) :|: x >= 0, z = x 676.38/291.62 f_3(z) -{ 1 }-> g_3(x, x) :|: x >= 0, z = x 676.38/291.62 f_4(z) -{ 1 }-> g_4(x, x) :|: x >= 0, z = x 676.38/291.62 f_5(z) -{ 1 }-> g_5(x, x) :|: x >= 0, z = x 676.38/291.62 f_6(z) -{ 1 }-> g_6(x, x) :|: x >= 0, z = x 676.38/291.62 f_7(z) -{ 1 }-> g_7(x, x) :|: x >= 0, z = x 676.38/291.62 f_8(z) -{ 1 }-> g_8(x, x) :|: x >= 0, z = x 676.38/291.62 f_9(z) -{ 1 }-> g_9(x, x) :|: x >= 0, z = x 676.38/291.62 g_1(z, z') -{ 2 }-> 1 + 0 + g_1(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y, x' >= 0, y = x' 676.38/291.62 g_10(z, z') -{ 1 }-> 1 + f_9(y) + g_10(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y 676.38/291.62 g_2(z, z') -{ 1 }-> 1 + f_1(y) + g_2(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y 676.38/291.62 g_3(z, z') -{ 1 }-> 1 + f_2(y) + g_3(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y 676.38/291.62 g_4(z, z') -{ 1 }-> 1 + f_3(y) + g_4(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y 676.38/291.62 g_5(z, z') -{ 1 }-> 1 + f_4(y) + g_5(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y 676.38/291.62 g_6(z, z') -{ 1 }-> 1 + f_5(y) + g_6(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y 676.38/291.62 g_7(z, z') -{ 1 }-> 1 + f_6(y) + g_7(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y 676.38/291.62 g_8(z, z') -{ 1 }-> 1 + f_7(y) + g_8(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y 676.38/291.62 g_9(z, z') -{ 1 }-> 1 + f_8(y) + g_9(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y 676.38/291.62 676.38/291.62 676.38/291.62 ---------------------------------------- 676.38/291.62 676.38/291.62 (13) SimplificationProof (BOTH BOUNDS(ID, ID)) 676.38/291.62 Simplified the RNTS by moving equalities from the constraints into the right-hand sides. 676.38/291.62 ---------------------------------------- 676.38/291.62 676.38/291.62 (14) 676.38/291.62 Obligation: 676.38/291.62 Complexity RNTS consisting of the following rules: 676.38/291.62 676.38/291.62 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.62 f_1(z) -{ 1 }-> g_1(z, z) :|: z >= 0 676.38/291.62 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.62 f_2(z) -{ 1 }-> g_2(z, z) :|: z >= 0 676.38/291.62 f_3(z) -{ 1 }-> g_3(z, z) :|: z >= 0 676.38/291.62 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.38/291.62 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.38/291.62 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.38/291.62 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.38/291.62 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.62 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.62 g_1(z, z') -{ 2 }-> 1 + 0 + g_1(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_2(z, z') -{ 1 }-> 1 + f_1(z') + g_2(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_3(z, z') -{ 1 }-> 1 + f_2(z') + g_3(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_4(z, z') -{ 1 }-> 1 + f_3(z') + g_4(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 676.38/291.62 676.38/291.62 ---------------------------------------- 676.38/291.62 676.38/291.62 (15) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) 676.38/291.62 Found the following analysis order by SCC decomposition: 676.38/291.62 676.38/291.62 { g_1 } 676.38/291.62 { f_0 } 676.38/291.62 { f_1 } 676.38/291.62 { g_2 } 676.38/291.62 { f_2 } 676.38/291.62 { g_3 } 676.38/291.62 { f_3 } 676.38/291.62 { g_4 } 676.38/291.62 { f_4 } 676.38/291.62 { g_5 } 676.38/291.62 { f_5 } 676.38/291.62 { g_6 } 676.38/291.62 { f_6 } 676.38/291.62 { g_7 } 676.38/291.62 { f_7 } 676.38/291.62 { g_8 } 676.38/291.62 { f_8 } 676.38/291.62 { g_9 } 676.38/291.62 { f_9 } 676.38/291.62 { g_10 } 676.38/291.62 { f_10 } 676.38/291.62 676.38/291.62 ---------------------------------------- 676.38/291.62 676.38/291.62 (16) 676.38/291.62 Obligation: 676.38/291.62 Complexity RNTS consisting of the following rules: 676.38/291.62 676.38/291.62 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.62 f_1(z) -{ 1 }-> g_1(z, z) :|: z >= 0 676.38/291.62 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.62 f_2(z) -{ 1 }-> g_2(z, z) :|: z >= 0 676.38/291.62 f_3(z) -{ 1 }-> g_3(z, z) :|: z >= 0 676.38/291.62 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.38/291.62 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.38/291.62 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.38/291.62 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.38/291.62 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.62 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.62 g_1(z, z') -{ 2 }-> 1 + 0 + g_1(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_2(z, z') -{ 1 }-> 1 + f_1(z') + g_2(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_3(z, z') -{ 1 }-> 1 + f_2(z') + g_3(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_4(z, z') -{ 1 }-> 1 + f_3(z') + g_4(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 676.38/291.62 Function symbols to be analyzed: {g_1}, {f_0}, {f_1}, {g_2}, {f_2}, {g_3}, {f_3}, {g_4}, {f_4}, {g_5}, {f_5}, {g_6}, {f_6}, {g_7}, {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.62 676.38/291.62 ---------------------------------------- 676.38/291.62 676.38/291.62 (17) ResultPropagationProof (UPPER BOUND(ID)) 676.38/291.62 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 676.38/291.62 ---------------------------------------- 676.38/291.62 676.38/291.62 (18) 676.38/291.62 Obligation: 676.38/291.62 Complexity RNTS consisting of the following rules: 676.38/291.62 676.38/291.62 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.62 f_1(z) -{ 1 }-> g_1(z, z) :|: z >= 0 676.38/291.62 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.62 f_2(z) -{ 1 }-> g_2(z, z) :|: z >= 0 676.38/291.62 f_3(z) -{ 1 }-> g_3(z, z) :|: z >= 0 676.38/291.62 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.38/291.62 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.38/291.62 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.38/291.62 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.38/291.62 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.62 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.62 g_1(z, z') -{ 2 }-> 1 + 0 + g_1(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_2(z, z') -{ 1 }-> 1 + f_1(z') + g_2(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_3(z, z') -{ 1 }-> 1 + f_2(z') + g_3(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_4(z, z') -{ 1 }-> 1 + f_3(z') + g_4(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 676.38/291.62 Function symbols to be analyzed: {g_1}, {f_0}, {f_1}, {g_2}, {f_2}, {g_3}, {f_3}, {g_4}, {f_4}, {g_5}, {f_5}, {g_6}, {f_6}, {g_7}, {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.62 676.38/291.62 ---------------------------------------- 676.38/291.62 676.38/291.62 (19) IntTrsBoundProof (UPPER BOUND(ID)) 676.38/291.62 676.38/291.62 Computed SIZE bound using CoFloCo for: g_1 676.38/291.62 after applying outer abstraction to obtain an ITS, 676.38/291.62 resulting in: O(1) with polynomial bound: 0 676.38/291.62 676.38/291.62 ---------------------------------------- 676.38/291.62 676.38/291.62 (20) 676.38/291.62 Obligation: 676.38/291.62 Complexity RNTS consisting of the following rules: 676.38/291.62 676.38/291.62 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.62 f_1(z) -{ 1 }-> g_1(z, z) :|: z >= 0 676.38/291.62 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.62 f_2(z) -{ 1 }-> g_2(z, z) :|: z >= 0 676.38/291.62 f_3(z) -{ 1 }-> g_3(z, z) :|: z >= 0 676.38/291.62 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.38/291.62 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.38/291.62 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.38/291.62 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.38/291.62 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.62 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.62 g_1(z, z') -{ 2 }-> 1 + 0 + g_1(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_2(z, z') -{ 1 }-> 1 + f_1(z') + g_2(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_3(z, z') -{ 1 }-> 1 + f_2(z') + g_3(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_4(z, z') -{ 1 }-> 1 + f_3(z') + g_4(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 676.38/291.62 Function symbols to be analyzed: {g_1}, {f_0}, {f_1}, {g_2}, {f_2}, {g_3}, {f_3}, {g_4}, {f_4}, {g_5}, {f_5}, {g_6}, {f_6}, {g_7}, {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.62 Previous analysis results are: 676.38/291.62 g_1: runtime: ?, size: O(1) [0] 676.38/291.62 676.38/291.62 ---------------------------------------- 676.38/291.62 676.38/291.62 (21) IntTrsBoundProof (UPPER BOUND(ID)) 676.38/291.62 676.38/291.62 Computed RUNTIME bound using CoFloCo for: g_1 676.38/291.62 after applying outer abstraction to obtain an ITS, 676.38/291.62 resulting in: O(n^1) with polynomial bound: 2*z 676.38/291.62 676.38/291.62 ---------------------------------------- 676.38/291.62 676.38/291.62 (22) 676.38/291.62 Obligation: 676.38/291.62 Complexity RNTS consisting of the following rules: 676.38/291.62 676.38/291.62 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.62 f_1(z) -{ 1 }-> g_1(z, z) :|: z >= 0 676.38/291.62 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.62 f_2(z) -{ 1 }-> g_2(z, z) :|: z >= 0 676.38/291.62 f_3(z) -{ 1 }-> g_3(z, z) :|: z >= 0 676.38/291.62 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.38/291.62 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.38/291.62 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.38/291.62 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.38/291.62 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.62 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.62 g_1(z, z') -{ 2 }-> 1 + 0 + g_1(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_2(z, z') -{ 1 }-> 1 + f_1(z') + g_2(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_3(z, z') -{ 1 }-> 1 + f_2(z') + g_3(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_4(z, z') -{ 1 }-> 1 + f_3(z') + g_4(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 676.38/291.62 Function symbols to be analyzed: {f_0}, {f_1}, {g_2}, {f_2}, {g_3}, {f_3}, {g_4}, {f_4}, {g_5}, {f_5}, {g_6}, {f_6}, {g_7}, {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.62 Previous analysis results are: 676.38/291.62 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.62 676.38/291.62 ---------------------------------------- 676.38/291.62 676.38/291.62 (23) ResultPropagationProof (UPPER BOUND(ID)) 676.38/291.62 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 676.38/291.62 ---------------------------------------- 676.38/291.62 676.38/291.62 (24) 676.38/291.62 Obligation: 676.38/291.62 Complexity RNTS consisting of the following rules: 676.38/291.62 676.38/291.62 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.62 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.62 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.62 f_2(z) -{ 1 }-> g_2(z, z) :|: z >= 0 676.38/291.62 f_3(z) -{ 1 }-> g_3(z, z) :|: z >= 0 676.38/291.62 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.38/291.62 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.38/291.62 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.38/291.62 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.38/291.62 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.62 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.62 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.62 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_2(z, z') -{ 1 }-> 1 + f_1(z') + g_2(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_3(z, z') -{ 1 }-> 1 + f_2(z') + g_3(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_4(z, z') -{ 1 }-> 1 + f_3(z') + g_4(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 676.38/291.62 Function symbols to be analyzed: {f_0}, {f_1}, {g_2}, {f_2}, {g_3}, {f_3}, {g_4}, {f_4}, {g_5}, {f_5}, {g_6}, {f_6}, {g_7}, {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.62 Previous analysis results are: 676.38/291.62 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.62 676.38/291.62 ---------------------------------------- 676.38/291.62 676.38/291.62 (25) IntTrsBoundProof (UPPER BOUND(ID)) 676.38/291.62 676.38/291.62 Computed SIZE bound using CoFloCo for: f_0 676.38/291.62 after applying outer abstraction to obtain an ITS, 676.38/291.62 resulting in: O(1) with polynomial bound: 0 676.38/291.62 676.38/291.62 ---------------------------------------- 676.38/291.62 676.38/291.62 (26) 676.38/291.62 Obligation: 676.38/291.62 Complexity RNTS consisting of the following rules: 676.38/291.62 676.38/291.62 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.62 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.62 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.62 f_2(z) -{ 1 }-> g_2(z, z) :|: z >= 0 676.38/291.62 f_3(z) -{ 1 }-> g_3(z, z) :|: z >= 0 676.38/291.62 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.38/291.62 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.38/291.62 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.38/291.62 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.38/291.62 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.62 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.62 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.62 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_2(z, z') -{ 1 }-> 1 + f_1(z') + g_2(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_3(z, z') -{ 1 }-> 1 + f_2(z') + g_3(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_4(z, z') -{ 1 }-> 1 + f_3(z') + g_4(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 676.38/291.62 Function symbols to be analyzed: {f_0}, {f_1}, {g_2}, {f_2}, {g_3}, {f_3}, {g_4}, {f_4}, {g_5}, {f_5}, {g_6}, {f_6}, {g_7}, {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.62 Previous analysis results are: 676.38/291.62 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.62 f_0: runtime: ?, size: O(1) [0] 676.38/291.62 676.38/291.62 ---------------------------------------- 676.38/291.62 676.38/291.62 (27) IntTrsBoundProof (UPPER BOUND(ID)) 676.38/291.62 676.38/291.62 Computed RUNTIME bound using CoFloCo for: f_0 676.38/291.62 after applying outer abstraction to obtain an ITS, 676.38/291.62 resulting in: O(1) with polynomial bound: 1 676.38/291.62 676.38/291.62 ---------------------------------------- 676.38/291.62 676.38/291.62 (28) 676.38/291.62 Obligation: 676.38/291.62 Complexity RNTS consisting of the following rules: 676.38/291.62 676.38/291.62 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.62 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.62 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.62 f_2(z) -{ 1 }-> g_2(z, z) :|: z >= 0 676.38/291.62 f_3(z) -{ 1 }-> g_3(z, z) :|: z >= 0 676.38/291.62 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.38/291.62 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.38/291.62 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.38/291.62 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.38/291.62 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.62 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.62 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.62 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_2(z, z') -{ 1 }-> 1 + f_1(z') + g_2(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_3(z, z') -{ 1 }-> 1 + f_2(z') + g_3(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_4(z, z') -{ 1 }-> 1 + f_3(z') + g_4(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 676.38/291.62 Function symbols to be analyzed: {f_1}, {g_2}, {f_2}, {g_3}, {f_3}, {g_4}, {f_4}, {g_5}, {f_5}, {g_6}, {f_6}, {g_7}, {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.62 Previous analysis results are: 676.38/291.62 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.62 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.62 676.38/291.62 ---------------------------------------- 676.38/291.62 676.38/291.62 (29) ResultPropagationProof (UPPER BOUND(ID)) 676.38/291.62 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 676.38/291.62 ---------------------------------------- 676.38/291.62 676.38/291.62 (30) 676.38/291.62 Obligation: 676.38/291.62 Complexity RNTS consisting of the following rules: 676.38/291.62 676.38/291.62 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.62 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.62 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.62 f_2(z) -{ 1 }-> g_2(z, z) :|: z >= 0 676.38/291.62 f_3(z) -{ 1 }-> g_3(z, z) :|: z >= 0 676.38/291.62 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.38/291.62 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.38/291.62 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.38/291.62 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.38/291.62 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.62 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.62 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.62 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_2(z, z') -{ 1 }-> 1 + f_1(z') + g_2(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_3(z, z') -{ 1 }-> 1 + f_2(z') + g_3(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_4(z, z') -{ 1 }-> 1 + f_3(z') + g_4(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.62 676.38/291.62 Function symbols to be analyzed: {f_1}, {g_2}, {f_2}, {g_3}, {f_3}, {g_4}, {f_4}, {g_5}, {f_5}, {g_6}, {f_6}, {g_7}, {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.62 Previous analysis results are: 676.38/291.62 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.62 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.62 676.38/291.62 ---------------------------------------- 676.38/291.62 676.38/291.62 (31) IntTrsBoundProof (UPPER BOUND(ID)) 676.38/291.62 676.38/291.62 Computed SIZE bound using CoFloCo for: f_1 676.38/291.62 after applying outer abstraction to obtain an ITS, 676.38/291.62 resulting in: O(1) with polynomial bound: 0 676.38/291.62 676.38/291.62 ---------------------------------------- 676.38/291.62 676.38/291.62 (32) 676.38/291.62 Obligation: 676.38/291.62 Complexity RNTS consisting of the following rules: 676.38/291.62 676.38/291.62 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.62 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.62 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.62 f_2(z) -{ 1 }-> g_2(z, z) :|: z >= 0 676.38/291.62 f_3(z) -{ 1 }-> g_3(z, z) :|: z >= 0 676.38/291.62 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.38/291.63 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.38/291.63 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.38/291.63 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.38/291.63 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.63 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.63 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.63 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_2(z, z') -{ 1 }-> 1 + f_1(z') + g_2(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_3(z, z') -{ 1 }-> 1 + f_2(z') + g_3(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_4(z, z') -{ 1 }-> 1 + f_3(z') + g_4(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 676.38/291.63 Function symbols to be analyzed: {f_1}, {g_2}, {f_2}, {g_3}, {f_3}, {g_4}, {f_4}, {g_5}, {f_5}, {g_6}, {f_6}, {g_7}, {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.63 Previous analysis results are: 676.38/291.63 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.63 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.63 f_1: runtime: ?, size: O(1) [0] 676.38/291.63 676.38/291.63 ---------------------------------------- 676.38/291.63 676.38/291.63 (33) IntTrsBoundProof (UPPER BOUND(ID)) 676.38/291.63 676.38/291.63 Computed RUNTIME bound using CoFloCo for: f_1 676.38/291.63 after applying outer abstraction to obtain an ITS, 676.38/291.63 resulting in: O(n^1) with polynomial bound: 1 + 2*z 676.38/291.63 676.38/291.63 ---------------------------------------- 676.38/291.63 676.38/291.63 (34) 676.38/291.63 Obligation: 676.38/291.63 Complexity RNTS consisting of the following rules: 676.38/291.63 676.38/291.63 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.63 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.63 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.63 f_2(z) -{ 1 }-> g_2(z, z) :|: z >= 0 676.38/291.63 f_3(z) -{ 1 }-> g_3(z, z) :|: z >= 0 676.38/291.63 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.38/291.63 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.38/291.63 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.38/291.63 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.38/291.63 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.63 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.63 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.63 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_2(z, z') -{ 1 }-> 1 + f_1(z') + g_2(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_3(z, z') -{ 1 }-> 1 + f_2(z') + g_3(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_4(z, z') -{ 1 }-> 1 + f_3(z') + g_4(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 676.38/291.63 Function symbols to be analyzed: {g_2}, {f_2}, {g_3}, {f_3}, {g_4}, {f_4}, {g_5}, {f_5}, {g_6}, {f_6}, {g_7}, {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.63 Previous analysis results are: 676.38/291.63 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.63 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.63 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.63 676.38/291.63 ---------------------------------------- 676.38/291.63 676.38/291.63 (35) ResultPropagationProof (UPPER BOUND(ID)) 676.38/291.63 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 676.38/291.63 ---------------------------------------- 676.38/291.63 676.38/291.63 (36) 676.38/291.63 Obligation: 676.38/291.63 Complexity RNTS consisting of the following rules: 676.38/291.63 676.38/291.63 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.63 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.63 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.63 f_2(z) -{ 1 }-> g_2(z, z) :|: z >= 0 676.38/291.63 f_3(z) -{ 1 }-> g_3(z, z) :|: z >= 0 676.38/291.63 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.38/291.63 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.38/291.63 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.38/291.63 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.38/291.63 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.63 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.63 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.63 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_2(z, z') -{ 2 + 2*z' }-> 1 + s'' + g_2(z - 1, z') :|: s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.63 g_3(z, z') -{ 1 }-> 1 + f_2(z') + g_3(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_4(z, z') -{ 1 }-> 1 + f_3(z') + g_4(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 676.38/291.63 Function symbols to be analyzed: {g_2}, {f_2}, {g_3}, {f_3}, {g_4}, {f_4}, {g_5}, {f_5}, {g_6}, {f_6}, {g_7}, {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.63 Previous analysis results are: 676.38/291.63 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.63 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.63 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.63 676.38/291.63 ---------------------------------------- 676.38/291.63 676.38/291.63 (37) IntTrsBoundProof (UPPER BOUND(ID)) 676.38/291.63 676.38/291.63 Computed SIZE bound using CoFloCo for: g_2 676.38/291.63 after applying outer abstraction to obtain an ITS, 676.38/291.63 resulting in: O(1) with polynomial bound: 0 676.38/291.63 676.38/291.63 ---------------------------------------- 676.38/291.63 676.38/291.63 (38) 676.38/291.63 Obligation: 676.38/291.63 Complexity RNTS consisting of the following rules: 676.38/291.63 676.38/291.63 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.63 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.63 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.63 f_2(z) -{ 1 }-> g_2(z, z) :|: z >= 0 676.38/291.63 f_3(z) -{ 1 }-> g_3(z, z) :|: z >= 0 676.38/291.63 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.38/291.63 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.38/291.63 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.38/291.63 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.38/291.63 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.63 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.63 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.63 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_2(z, z') -{ 2 + 2*z' }-> 1 + s'' + g_2(z - 1, z') :|: s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.63 g_3(z, z') -{ 1 }-> 1 + f_2(z') + g_3(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_4(z, z') -{ 1 }-> 1 + f_3(z') + g_4(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 676.38/291.63 Function symbols to be analyzed: {g_2}, {f_2}, {g_3}, {f_3}, {g_4}, {f_4}, {g_5}, {f_5}, {g_6}, {f_6}, {g_7}, {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.63 Previous analysis results are: 676.38/291.63 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.63 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.63 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.63 g_2: runtime: ?, size: O(1) [0] 676.38/291.63 676.38/291.63 ---------------------------------------- 676.38/291.63 676.38/291.63 (39) IntTrsBoundProof (UPPER BOUND(ID)) 676.38/291.63 676.38/291.63 Computed RUNTIME bound using CoFloCo for: g_2 676.38/291.63 after applying outer abstraction to obtain an ITS, 676.38/291.63 resulting in: O(n^2) with polynomial bound: 2*z + 2*z*z' 676.38/291.63 676.38/291.63 ---------------------------------------- 676.38/291.63 676.38/291.63 (40) 676.38/291.63 Obligation: 676.38/291.63 Complexity RNTS consisting of the following rules: 676.38/291.63 676.38/291.63 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.63 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.63 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.63 f_2(z) -{ 1 }-> g_2(z, z) :|: z >= 0 676.38/291.63 f_3(z) -{ 1 }-> g_3(z, z) :|: z >= 0 676.38/291.63 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.38/291.63 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.38/291.63 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.38/291.63 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.38/291.63 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.63 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.63 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.63 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_2(z, z') -{ 2 + 2*z' }-> 1 + s'' + g_2(z - 1, z') :|: s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.63 g_3(z, z') -{ 1 }-> 1 + f_2(z') + g_3(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_4(z, z') -{ 1 }-> 1 + f_3(z') + g_4(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 676.38/291.63 Function symbols to be analyzed: {f_2}, {g_3}, {f_3}, {g_4}, {f_4}, {g_5}, {f_5}, {g_6}, {f_6}, {g_7}, {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.63 Previous analysis results are: 676.38/291.63 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.63 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.63 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.63 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.63 676.38/291.63 ---------------------------------------- 676.38/291.63 676.38/291.63 (41) ResultPropagationProof (UPPER BOUND(ID)) 676.38/291.63 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 676.38/291.63 ---------------------------------------- 676.38/291.63 676.38/291.63 (42) 676.38/291.63 Obligation: 676.38/291.63 Complexity RNTS consisting of the following rules: 676.38/291.63 676.38/291.63 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.63 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.63 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.63 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.63 f_3(z) -{ 1 }-> g_3(z, z) :|: z >= 0 676.38/291.63 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.38/291.63 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.38/291.63 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.38/291.63 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.38/291.63 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.63 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.63 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.63 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.63 g_3(z, z') -{ 1 }-> 1 + f_2(z') + g_3(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_4(z, z') -{ 1 }-> 1 + f_3(z') + g_4(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 676.38/291.63 Function symbols to be analyzed: {f_2}, {g_3}, {f_3}, {g_4}, {f_4}, {g_5}, {f_5}, {g_6}, {f_6}, {g_7}, {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.63 Previous analysis results are: 676.38/291.63 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.63 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.63 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.63 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.63 676.38/291.63 ---------------------------------------- 676.38/291.63 676.38/291.63 (43) IntTrsBoundProof (UPPER BOUND(ID)) 676.38/291.63 676.38/291.63 Computed SIZE bound using CoFloCo for: f_2 676.38/291.63 after applying outer abstraction to obtain an ITS, 676.38/291.63 resulting in: O(1) with polynomial bound: 0 676.38/291.63 676.38/291.63 ---------------------------------------- 676.38/291.63 676.38/291.63 (44) 676.38/291.63 Obligation: 676.38/291.63 Complexity RNTS consisting of the following rules: 676.38/291.63 676.38/291.63 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.63 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.63 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.63 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.63 f_3(z) -{ 1 }-> g_3(z, z) :|: z >= 0 676.38/291.63 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.38/291.63 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.38/291.63 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.38/291.63 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.38/291.63 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.63 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.63 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.63 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.63 g_3(z, z') -{ 1 }-> 1 + f_2(z') + g_3(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_4(z, z') -{ 1 }-> 1 + f_3(z') + g_4(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 676.38/291.63 Function symbols to be analyzed: {f_2}, {g_3}, {f_3}, {g_4}, {f_4}, {g_5}, {f_5}, {g_6}, {f_6}, {g_7}, {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.63 Previous analysis results are: 676.38/291.63 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.63 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.63 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.63 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.63 f_2: runtime: ?, size: O(1) [0] 676.38/291.63 676.38/291.63 ---------------------------------------- 676.38/291.63 676.38/291.63 (45) IntTrsBoundProof (UPPER BOUND(ID)) 676.38/291.63 676.38/291.63 Computed RUNTIME bound using KoAT for: f_2 676.38/291.63 after applying outer abstraction to obtain an ITS, 676.38/291.63 resulting in: O(n^2) with polynomial bound: 1 + 2*z + 2*z^2 676.38/291.63 676.38/291.63 ---------------------------------------- 676.38/291.63 676.38/291.63 (46) 676.38/291.63 Obligation: 676.38/291.63 Complexity RNTS consisting of the following rules: 676.38/291.63 676.38/291.63 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.63 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.63 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.63 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.63 f_3(z) -{ 1 }-> g_3(z, z) :|: z >= 0 676.38/291.63 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.38/291.63 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.38/291.63 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.38/291.63 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.38/291.63 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.63 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.63 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.63 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.63 g_3(z, z') -{ 1 }-> 1 + f_2(z') + g_3(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_4(z, z') -{ 1 }-> 1 + f_3(z') + g_4(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 676.38/291.63 Function symbols to be analyzed: {g_3}, {f_3}, {g_4}, {f_4}, {g_5}, {f_5}, {g_6}, {f_6}, {g_7}, {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.63 Previous analysis results are: 676.38/291.63 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.63 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.63 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.63 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.63 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.38/291.63 676.38/291.63 ---------------------------------------- 676.38/291.63 676.38/291.63 (47) ResultPropagationProof (UPPER BOUND(ID)) 676.38/291.63 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 676.38/291.63 ---------------------------------------- 676.38/291.63 676.38/291.63 (48) 676.38/291.63 Obligation: 676.38/291.63 Complexity RNTS consisting of the following rules: 676.38/291.63 676.38/291.63 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.63 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.63 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.63 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.63 f_3(z) -{ 1 }-> g_3(z, z) :|: z >= 0 676.38/291.63 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.38/291.63 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.38/291.63 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.38/291.63 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.38/291.63 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.63 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.63 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.63 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.63 g_3(z, z') -{ 2 + 2*z' + 2*z'^2 }-> 1 + s3 + g_3(z - 1, z') :|: s3 >= 0, s3 <= 0, z - 1 >= 0, z' >= 0 676.38/291.63 g_4(z, z') -{ 1 }-> 1 + f_3(z') + g_4(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 676.38/291.63 Function symbols to be analyzed: {g_3}, {f_3}, {g_4}, {f_4}, {g_5}, {f_5}, {g_6}, {f_6}, {g_7}, {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.63 Previous analysis results are: 676.38/291.63 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.63 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.63 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.63 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.63 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.38/291.63 676.38/291.63 ---------------------------------------- 676.38/291.63 676.38/291.63 (49) IntTrsBoundProof (UPPER BOUND(ID)) 676.38/291.63 676.38/291.63 Computed SIZE bound using CoFloCo for: g_3 676.38/291.63 after applying outer abstraction to obtain an ITS, 676.38/291.63 resulting in: O(1) with polynomial bound: 0 676.38/291.63 676.38/291.63 ---------------------------------------- 676.38/291.63 676.38/291.63 (50) 676.38/291.63 Obligation: 676.38/291.63 Complexity RNTS consisting of the following rules: 676.38/291.63 676.38/291.63 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.63 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.63 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.63 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.63 f_3(z) -{ 1 }-> g_3(z, z) :|: z >= 0 676.38/291.63 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.38/291.63 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.38/291.63 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.38/291.63 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.38/291.63 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.63 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.63 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.63 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.63 g_3(z, z') -{ 2 + 2*z' + 2*z'^2 }-> 1 + s3 + g_3(z - 1, z') :|: s3 >= 0, s3 <= 0, z - 1 >= 0, z' >= 0 676.38/291.63 g_4(z, z') -{ 1 }-> 1 + f_3(z') + g_4(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 676.38/291.63 Function symbols to be analyzed: {g_3}, {f_3}, {g_4}, {f_4}, {g_5}, {f_5}, {g_6}, {f_6}, {g_7}, {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.63 Previous analysis results are: 676.38/291.63 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.63 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.63 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.63 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.63 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.38/291.63 g_3: runtime: ?, size: O(1) [0] 676.38/291.63 676.38/291.63 ---------------------------------------- 676.38/291.63 676.38/291.63 (51) IntTrsBoundProof (UPPER BOUND(ID)) 676.38/291.63 676.38/291.63 Computed RUNTIME bound using KoAT for: g_3 676.38/291.63 after applying outer abstraction to obtain an ITS, 676.38/291.63 resulting in: O(n^3) with polynomial bound: 2*z + 2*z*z' + 2*z*z'^2 676.38/291.63 676.38/291.63 ---------------------------------------- 676.38/291.63 676.38/291.63 (52) 676.38/291.63 Obligation: 676.38/291.63 Complexity RNTS consisting of the following rules: 676.38/291.63 676.38/291.63 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.63 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.63 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.63 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.63 f_3(z) -{ 1 }-> g_3(z, z) :|: z >= 0 676.38/291.63 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.38/291.63 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.38/291.63 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.38/291.63 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.38/291.63 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.63 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.63 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.63 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.63 g_3(z, z') -{ 2 + 2*z' + 2*z'^2 }-> 1 + s3 + g_3(z - 1, z') :|: s3 >= 0, s3 <= 0, z - 1 >= 0, z' >= 0 676.38/291.63 g_4(z, z') -{ 1 }-> 1 + f_3(z') + g_4(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 676.38/291.63 Function symbols to be analyzed: {f_3}, {g_4}, {f_4}, {g_5}, {f_5}, {g_6}, {f_6}, {g_7}, {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.63 Previous analysis results are: 676.38/291.63 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.63 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.63 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.63 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.63 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.38/291.63 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.38/291.63 676.38/291.63 ---------------------------------------- 676.38/291.63 676.38/291.63 (53) ResultPropagationProof (UPPER BOUND(ID)) 676.38/291.63 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 676.38/291.63 ---------------------------------------- 676.38/291.63 676.38/291.63 (54) 676.38/291.63 Obligation: 676.38/291.63 Complexity RNTS consisting of the following rules: 676.38/291.63 676.38/291.63 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.63 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.63 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.63 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.63 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.38/291.63 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.38/291.63 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.38/291.63 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.38/291.63 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.38/291.63 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.63 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.63 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.63 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.63 g_3(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 }-> 1 + s3 + s5 :|: s5 >= 0, s5 <= 0, s3 >= 0, s3 <= 0, z - 1 >= 0, z' >= 0 676.38/291.63 g_4(z, z') -{ 1 }-> 1 + f_3(z') + g_4(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 676.38/291.63 Function symbols to be analyzed: {f_3}, {g_4}, {f_4}, {g_5}, {f_5}, {g_6}, {f_6}, {g_7}, {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.63 Previous analysis results are: 676.38/291.63 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.63 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.63 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.63 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.63 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.38/291.63 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.38/291.63 676.38/291.63 ---------------------------------------- 676.38/291.63 676.38/291.63 (55) IntTrsBoundProof (UPPER BOUND(ID)) 676.38/291.63 676.38/291.63 Computed SIZE bound using CoFloCo for: f_3 676.38/291.63 after applying outer abstraction to obtain an ITS, 676.38/291.63 resulting in: O(1) with polynomial bound: 0 676.38/291.63 676.38/291.63 ---------------------------------------- 676.38/291.63 676.38/291.63 (56) 676.38/291.63 Obligation: 676.38/291.63 Complexity RNTS consisting of the following rules: 676.38/291.63 676.38/291.63 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.63 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.63 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.63 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.63 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.38/291.63 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.38/291.63 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.38/291.63 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.38/291.63 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.38/291.63 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.63 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.63 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.63 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.63 g_3(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 }-> 1 + s3 + s5 :|: s5 >= 0, s5 <= 0, s3 >= 0, s3 <= 0, z - 1 >= 0, z' >= 0 676.38/291.63 g_4(z, z') -{ 1 }-> 1 + f_3(z') + g_4(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 676.38/291.63 Function symbols to be analyzed: {f_3}, {g_4}, {f_4}, {g_5}, {f_5}, {g_6}, {f_6}, {g_7}, {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.63 Previous analysis results are: 676.38/291.63 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.63 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.63 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.63 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.63 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.38/291.63 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.38/291.63 f_3: runtime: ?, size: O(1) [0] 676.38/291.63 676.38/291.63 ---------------------------------------- 676.38/291.63 676.38/291.63 (57) IntTrsBoundProof (UPPER BOUND(ID)) 676.38/291.63 676.38/291.63 Computed RUNTIME bound using KoAT for: f_3 676.38/291.63 after applying outer abstraction to obtain an ITS, 676.38/291.63 resulting in: O(n^3) with polynomial bound: 1 + 2*z + 2*z^2 + 2*z^3 676.38/291.63 676.38/291.63 ---------------------------------------- 676.38/291.63 676.38/291.63 (58) 676.38/291.63 Obligation: 676.38/291.63 Complexity RNTS consisting of the following rules: 676.38/291.63 676.38/291.63 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.63 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.63 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.63 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.63 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.38/291.63 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.38/291.63 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.38/291.63 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.38/291.63 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.38/291.63 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.63 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.63 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.63 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.63 g_3(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 }-> 1 + s3 + s5 :|: s5 >= 0, s5 <= 0, s3 >= 0, s3 <= 0, z - 1 >= 0, z' >= 0 676.38/291.63 g_4(z, z') -{ 1 }-> 1 + f_3(z') + g_4(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.63 676.38/291.63 Function symbols to be analyzed: {g_4}, {f_4}, {g_5}, {f_5}, {g_6}, {f_6}, {g_7}, {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.63 Previous analysis results are: 676.38/291.63 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.63 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.63 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.63 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.63 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.38/291.63 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.38/291.64 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (59) ResultPropagationProof (UPPER BOUND(ID)) 676.38/291.64 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (60) 676.38/291.64 Obligation: 676.38/291.64 Complexity RNTS consisting of the following rules: 676.38/291.64 676.38/291.64 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.64 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.64 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.64 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.64 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.38/291.64 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.38/291.64 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.38/291.64 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.38/291.64 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.38/291.64 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.64 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.64 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_3(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 }-> 1 + s3 + s5 :|: s5 >= 0, s5 <= 0, s3 >= 0, s3 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_4(z, z') -{ 2 + 2*z' + 2*z'^2 + 2*z'^3 }-> 1 + s6 + g_4(z - 1, z') :|: s6 >= 0, s6 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 676.38/291.64 Function symbols to be analyzed: {g_4}, {f_4}, {g_5}, {f_5}, {g_6}, {f_6}, {g_7}, {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.64 Previous analysis results are: 676.38/291.64 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.64 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.64 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.64 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.64 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.38/291.64 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.38/291.64 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (61) IntTrsBoundProof (UPPER BOUND(ID)) 676.38/291.64 676.38/291.64 Computed SIZE bound using CoFloCo for: g_4 676.38/291.64 after applying outer abstraction to obtain an ITS, 676.38/291.64 resulting in: O(1) with polynomial bound: 0 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (62) 676.38/291.64 Obligation: 676.38/291.64 Complexity RNTS consisting of the following rules: 676.38/291.64 676.38/291.64 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.64 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.64 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.64 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.64 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.38/291.64 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.38/291.64 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.38/291.64 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.38/291.64 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.38/291.64 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.64 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.64 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_3(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 }-> 1 + s3 + s5 :|: s5 >= 0, s5 <= 0, s3 >= 0, s3 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_4(z, z') -{ 2 + 2*z' + 2*z'^2 + 2*z'^3 }-> 1 + s6 + g_4(z - 1, z') :|: s6 >= 0, s6 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 676.38/291.64 Function symbols to be analyzed: {g_4}, {f_4}, {g_5}, {f_5}, {g_6}, {f_6}, {g_7}, {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.64 Previous analysis results are: 676.38/291.64 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.64 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.64 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.64 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.64 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.38/291.64 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.38/291.64 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.38/291.64 g_4: runtime: ?, size: O(1) [0] 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (63) IntTrsBoundProof (UPPER BOUND(ID)) 676.38/291.64 676.38/291.64 Computed RUNTIME bound using KoAT for: g_4 676.38/291.64 after applying outer abstraction to obtain an ITS, 676.38/291.64 resulting in: O(n^4) with polynomial bound: 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (64) 676.38/291.64 Obligation: 676.38/291.64 Complexity RNTS consisting of the following rules: 676.38/291.64 676.38/291.64 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.64 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.64 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.64 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.64 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.38/291.64 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.38/291.64 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.38/291.64 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.38/291.64 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.38/291.64 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.64 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.64 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_3(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 }-> 1 + s3 + s5 :|: s5 >= 0, s5 <= 0, s3 >= 0, s3 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_4(z, z') -{ 2 + 2*z' + 2*z'^2 + 2*z'^3 }-> 1 + s6 + g_4(z - 1, z') :|: s6 >= 0, s6 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 676.38/291.64 Function symbols to be analyzed: {f_4}, {g_5}, {f_5}, {g_6}, {f_6}, {g_7}, {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.64 Previous analysis results are: 676.38/291.64 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.64 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.64 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.64 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.64 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.38/291.64 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.38/291.64 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.38/291.64 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (65) ResultPropagationProof (UPPER BOUND(ID)) 676.38/291.64 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (66) 676.38/291.64 Obligation: 676.38/291.64 Complexity RNTS consisting of the following rules: 676.38/291.64 676.38/291.64 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.64 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.64 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.64 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.64 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.38/291.64 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.38/291.64 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.38/291.64 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.38/291.64 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.38/291.64 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.64 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.64 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_3(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 }-> 1 + s3 + s5 :|: s5 >= 0, s5 <= 0, s3 >= 0, s3 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_4(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 }-> 1 + s6 + s8 :|: s8 >= 0, s8 <= 0, s6 >= 0, s6 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 676.38/291.64 Function symbols to be analyzed: {f_4}, {g_5}, {f_5}, {g_6}, {f_6}, {g_7}, {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.64 Previous analysis results are: 676.38/291.64 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.64 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.64 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.64 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.64 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.38/291.64 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.38/291.64 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.38/291.64 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (67) IntTrsBoundProof (UPPER BOUND(ID)) 676.38/291.64 676.38/291.64 Computed SIZE bound using CoFloCo for: f_4 676.38/291.64 after applying outer abstraction to obtain an ITS, 676.38/291.64 resulting in: O(1) with polynomial bound: 0 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (68) 676.38/291.64 Obligation: 676.38/291.64 Complexity RNTS consisting of the following rules: 676.38/291.64 676.38/291.64 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.64 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.64 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.64 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.64 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.38/291.64 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.38/291.64 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.38/291.64 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.38/291.64 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.38/291.64 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.64 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.64 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_3(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 }-> 1 + s3 + s5 :|: s5 >= 0, s5 <= 0, s3 >= 0, s3 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_4(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 }-> 1 + s6 + s8 :|: s8 >= 0, s8 <= 0, s6 >= 0, s6 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 676.38/291.64 Function symbols to be analyzed: {f_4}, {g_5}, {f_5}, {g_6}, {f_6}, {g_7}, {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.64 Previous analysis results are: 676.38/291.64 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.64 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.64 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.64 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.64 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.38/291.64 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.38/291.64 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.38/291.64 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.38/291.64 f_4: runtime: ?, size: O(1) [0] 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (69) IntTrsBoundProof (UPPER BOUND(ID)) 676.38/291.64 676.38/291.64 Computed RUNTIME bound using KoAT for: f_4 676.38/291.64 after applying outer abstraction to obtain an ITS, 676.38/291.64 resulting in: O(n^4) with polynomial bound: 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (70) 676.38/291.64 Obligation: 676.38/291.64 Complexity RNTS consisting of the following rules: 676.38/291.64 676.38/291.64 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.64 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.64 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.64 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.64 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.38/291.64 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.38/291.64 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.38/291.64 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.38/291.64 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.38/291.64 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.64 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.64 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_3(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 }-> 1 + s3 + s5 :|: s5 >= 0, s5 <= 0, s3 >= 0, s3 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_4(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 }-> 1 + s6 + s8 :|: s8 >= 0, s8 <= 0, s6 >= 0, s6 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 676.38/291.64 Function symbols to be analyzed: {g_5}, {f_5}, {g_6}, {f_6}, {g_7}, {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.64 Previous analysis results are: 676.38/291.64 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.64 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.64 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.64 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.64 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.38/291.64 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.38/291.64 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.38/291.64 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.38/291.64 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (71) ResultPropagationProof (UPPER BOUND(ID)) 676.38/291.64 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (72) 676.38/291.64 Obligation: 676.38/291.64 Complexity RNTS consisting of the following rules: 676.38/291.64 676.38/291.64 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.64 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.64 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.64 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.64 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.38/291.64 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.38/291.64 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.38/291.64 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.38/291.64 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.38/291.64 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.64 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.64 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_3(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 }-> 1 + s3 + s5 :|: s5 >= 0, s5 <= 0, s3 >= 0, s3 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_4(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 }-> 1 + s6 + s8 :|: s8 >= 0, s8 <= 0, s6 >= 0, s6 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_5(z, z') -{ 2 + 2*z' + 2*z'^2 + 2*z'^3 + 2*z'^4 }-> 1 + s9 + g_5(z - 1, z') :|: s9 >= 0, s9 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 676.38/291.64 Function symbols to be analyzed: {g_5}, {f_5}, {g_6}, {f_6}, {g_7}, {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.64 Previous analysis results are: 676.38/291.64 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.64 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.64 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.64 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.64 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.38/291.64 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.38/291.64 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.38/291.64 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.38/291.64 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (73) IntTrsBoundProof (UPPER BOUND(ID)) 676.38/291.64 676.38/291.64 Computed SIZE bound using CoFloCo for: g_5 676.38/291.64 after applying outer abstraction to obtain an ITS, 676.38/291.64 resulting in: O(1) with polynomial bound: 0 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (74) 676.38/291.64 Obligation: 676.38/291.64 Complexity RNTS consisting of the following rules: 676.38/291.64 676.38/291.64 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.64 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.64 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.64 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.64 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.38/291.64 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.38/291.64 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.38/291.64 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.38/291.64 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.38/291.64 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.64 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.64 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_3(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 }-> 1 + s3 + s5 :|: s5 >= 0, s5 <= 0, s3 >= 0, s3 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_4(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 }-> 1 + s6 + s8 :|: s8 >= 0, s8 <= 0, s6 >= 0, s6 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_5(z, z') -{ 2 + 2*z' + 2*z'^2 + 2*z'^3 + 2*z'^4 }-> 1 + s9 + g_5(z - 1, z') :|: s9 >= 0, s9 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 676.38/291.64 Function symbols to be analyzed: {g_5}, {f_5}, {g_6}, {f_6}, {g_7}, {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.64 Previous analysis results are: 676.38/291.64 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.64 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.64 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.64 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.64 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.38/291.64 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.38/291.64 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.38/291.64 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.38/291.64 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.38/291.64 g_5: runtime: ?, size: O(1) [0] 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (75) IntTrsBoundProof (UPPER BOUND(ID)) 676.38/291.64 676.38/291.64 Computed RUNTIME bound using KoAT for: g_5 676.38/291.64 after applying outer abstraction to obtain an ITS, 676.38/291.64 resulting in: O(n^5) with polynomial bound: 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (76) 676.38/291.64 Obligation: 676.38/291.64 Complexity RNTS consisting of the following rules: 676.38/291.64 676.38/291.64 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.64 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.64 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.64 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.64 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.38/291.64 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.38/291.64 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.38/291.64 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.38/291.64 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.38/291.64 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.64 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.64 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_3(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 }-> 1 + s3 + s5 :|: s5 >= 0, s5 <= 0, s3 >= 0, s3 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_4(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 }-> 1 + s6 + s8 :|: s8 >= 0, s8 <= 0, s6 >= 0, s6 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_5(z, z') -{ 2 + 2*z' + 2*z'^2 + 2*z'^3 + 2*z'^4 }-> 1 + s9 + g_5(z - 1, z') :|: s9 >= 0, s9 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 676.38/291.64 Function symbols to be analyzed: {f_5}, {g_6}, {f_6}, {g_7}, {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.64 Previous analysis results are: 676.38/291.64 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.64 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.64 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.64 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.64 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.38/291.64 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.38/291.64 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.38/291.64 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.38/291.64 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.38/291.64 g_5: runtime: O(n^5) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4], size: O(1) [0] 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (77) ResultPropagationProof (UPPER BOUND(ID)) 676.38/291.64 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (78) 676.38/291.64 Obligation: 676.38/291.64 Complexity RNTS consisting of the following rules: 676.38/291.64 676.38/291.64 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.64 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.64 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.64 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.64 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.38/291.64 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.38/291.64 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.38/291.64 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.38/291.64 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.38/291.64 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.64 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.64 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_3(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 }-> 1 + s3 + s5 :|: s5 >= 0, s5 <= 0, s3 >= 0, s3 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_4(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 }-> 1 + s6 + s8 :|: s8 >= 0, s8 <= 0, s6 >= 0, s6 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_5(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 }-> 1 + s9 + s11 :|: s11 >= 0, s11 <= 0, s9 >= 0, s9 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 676.38/291.64 Function symbols to be analyzed: {f_5}, {g_6}, {f_6}, {g_7}, {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.64 Previous analysis results are: 676.38/291.64 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.64 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.64 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.64 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.64 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.38/291.64 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.38/291.64 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.38/291.64 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.38/291.64 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.38/291.64 g_5: runtime: O(n^5) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4], size: O(1) [0] 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (79) IntTrsBoundProof (UPPER BOUND(ID)) 676.38/291.64 676.38/291.64 Computed SIZE bound using CoFloCo for: f_5 676.38/291.64 after applying outer abstraction to obtain an ITS, 676.38/291.64 resulting in: O(1) with polynomial bound: 0 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (80) 676.38/291.64 Obligation: 676.38/291.64 Complexity RNTS consisting of the following rules: 676.38/291.64 676.38/291.64 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.64 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.64 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.64 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.64 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.38/291.64 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.38/291.64 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.38/291.64 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.38/291.64 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.38/291.64 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.64 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.64 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_3(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 }-> 1 + s3 + s5 :|: s5 >= 0, s5 <= 0, s3 >= 0, s3 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_4(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 }-> 1 + s6 + s8 :|: s8 >= 0, s8 <= 0, s6 >= 0, s6 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_5(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 }-> 1 + s9 + s11 :|: s11 >= 0, s11 <= 0, s9 >= 0, s9 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 676.38/291.64 Function symbols to be analyzed: {f_5}, {g_6}, {f_6}, {g_7}, {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.64 Previous analysis results are: 676.38/291.64 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.64 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.64 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.64 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.64 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.38/291.64 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.38/291.64 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.38/291.64 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.38/291.64 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.38/291.64 g_5: runtime: O(n^5) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4], size: O(1) [0] 676.38/291.64 f_5: runtime: ?, size: O(1) [0] 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (81) IntTrsBoundProof (UPPER BOUND(ID)) 676.38/291.64 676.38/291.64 Computed RUNTIME bound using KoAT for: f_5 676.38/291.64 after applying outer abstraction to obtain an ITS, 676.38/291.64 resulting in: O(n^5) with polynomial bound: 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (82) 676.38/291.64 Obligation: 676.38/291.64 Complexity RNTS consisting of the following rules: 676.38/291.64 676.38/291.64 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.64 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.64 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.64 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.64 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.38/291.64 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.38/291.64 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.38/291.64 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.38/291.64 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.38/291.64 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.64 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.64 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_3(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 }-> 1 + s3 + s5 :|: s5 >= 0, s5 <= 0, s3 >= 0, s3 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_4(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 }-> 1 + s6 + s8 :|: s8 >= 0, s8 <= 0, s6 >= 0, s6 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_5(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 }-> 1 + s9 + s11 :|: s11 >= 0, s11 <= 0, s9 >= 0, s9 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 676.38/291.64 Function symbols to be analyzed: {g_6}, {f_6}, {g_7}, {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.64 Previous analysis results are: 676.38/291.64 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.64 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.64 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.64 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.64 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.38/291.64 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.38/291.64 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.38/291.64 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.38/291.64 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.38/291.64 g_5: runtime: O(n^5) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4], size: O(1) [0] 676.38/291.64 f_5: runtime: O(n^5) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5], size: O(1) [0] 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (83) ResultPropagationProof (UPPER BOUND(ID)) 676.38/291.64 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (84) 676.38/291.64 Obligation: 676.38/291.64 Complexity RNTS consisting of the following rules: 676.38/291.64 676.38/291.64 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.64 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.64 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.64 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.64 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.38/291.64 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.38/291.64 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.38/291.64 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.38/291.64 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.38/291.64 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.64 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.64 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_3(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 }-> 1 + s3 + s5 :|: s5 >= 0, s5 <= 0, s3 >= 0, s3 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_4(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 }-> 1 + s6 + s8 :|: s8 >= 0, s8 <= 0, s6 >= 0, s6 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_5(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 }-> 1 + s9 + s11 :|: s11 >= 0, s11 <= 0, s9 >= 0, s9 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_6(z, z') -{ 2 + 2*z' + 2*z'^2 + 2*z'^3 + 2*z'^4 + 2*z'^5 }-> 1 + s12 + g_6(z - 1, z') :|: s12 >= 0, s12 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 676.38/291.64 Function symbols to be analyzed: {g_6}, {f_6}, {g_7}, {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.64 Previous analysis results are: 676.38/291.64 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.64 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.64 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.64 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.64 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.38/291.64 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.38/291.64 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.38/291.64 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.38/291.64 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.38/291.64 g_5: runtime: O(n^5) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4], size: O(1) [0] 676.38/291.64 f_5: runtime: O(n^5) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5], size: O(1) [0] 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (85) IntTrsBoundProof (UPPER BOUND(ID)) 676.38/291.64 676.38/291.64 Computed SIZE bound using CoFloCo for: g_6 676.38/291.64 after applying outer abstraction to obtain an ITS, 676.38/291.64 resulting in: O(1) with polynomial bound: 0 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (86) 676.38/291.64 Obligation: 676.38/291.64 Complexity RNTS consisting of the following rules: 676.38/291.64 676.38/291.64 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.64 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.64 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.64 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.64 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.38/291.64 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.38/291.64 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.38/291.64 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.38/291.64 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.38/291.64 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.64 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.64 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_3(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 }-> 1 + s3 + s5 :|: s5 >= 0, s5 <= 0, s3 >= 0, s3 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_4(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 }-> 1 + s6 + s8 :|: s8 >= 0, s8 <= 0, s6 >= 0, s6 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_5(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 }-> 1 + s9 + s11 :|: s11 >= 0, s11 <= 0, s9 >= 0, s9 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_6(z, z') -{ 2 + 2*z' + 2*z'^2 + 2*z'^3 + 2*z'^4 + 2*z'^5 }-> 1 + s12 + g_6(z - 1, z') :|: s12 >= 0, s12 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 676.38/291.64 Function symbols to be analyzed: {g_6}, {f_6}, {g_7}, {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.64 Previous analysis results are: 676.38/291.64 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.64 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.64 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.64 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.64 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.38/291.64 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.38/291.64 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.38/291.64 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.38/291.64 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.38/291.64 g_5: runtime: O(n^5) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4], size: O(1) [0] 676.38/291.64 f_5: runtime: O(n^5) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5], size: O(1) [0] 676.38/291.64 g_6: runtime: ?, size: O(1) [0] 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (87) IntTrsBoundProof (UPPER BOUND(ID)) 676.38/291.64 676.38/291.64 Computed RUNTIME bound using KoAT for: g_6 676.38/291.64 after applying outer abstraction to obtain an ITS, 676.38/291.64 resulting in: O(n^6) with polynomial bound: 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (88) 676.38/291.64 Obligation: 676.38/291.64 Complexity RNTS consisting of the following rules: 676.38/291.64 676.38/291.64 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.64 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.64 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.64 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.64 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.38/291.64 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.38/291.64 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.38/291.64 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.38/291.64 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.38/291.64 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.64 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.64 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_3(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 }-> 1 + s3 + s5 :|: s5 >= 0, s5 <= 0, s3 >= 0, s3 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_4(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 }-> 1 + s6 + s8 :|: s8 >= 0, s8 <= 0, s6 >= 0, s6 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_5(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 }-> 1 + s9 + s11 :|: s11 >= 0, s11 <= 0, s9 >= 0, s9 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_6(z, z') -{ 2 + 2*z' + 2*z'^2 + 2*z'^3 + 2*z'^4 + 2*z'^5 }-> 1 + s12 + g_6(z - 1, z') :|: s12 >= 0, s12 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 676.38/291.64 Function symbols to be analyzed: {f_6}, {g_7}, {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.64 Previous analysis results are: 676.38/291.64 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.64 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.64 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.64 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.64 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.38/291.64 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.38/291.64 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.38/291.64 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.38/291.64 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.38/291.64 g_5: runtime: O(n^5) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4], size: O(1) [0] 676.38/291.64 f_5: runtime: O(n^5) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5], size: O(1) [0] 676.38/291.64 g_6: runtime: O(n^6) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5], size: O(1) [0] 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (89) ResultPropagationProof (UPPER BOUND(ID)) 676.38/291.64 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (90) 676.38/291.64 Obligation: 676.38/291.64 Complexity RNTS consisting of the following rules: 676.38/291.64 676.38/291.64 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.64 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.64 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.64 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.64 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.38/291.64 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.38/291.64 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.38/291.64 f_6(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 }-> s13 :|: s13 >= 0, s13 <= 0, z >= 0 676.38/291.64 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.38/291.64 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.64 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.64 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_3(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 }-> 1 + s3 + s5 :|: s5 >= 0, s5 <= 0, s3 >= 0, s3 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_4(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 }-> 1 + s6 + s8 :|: s8 >= 0, s8 <= 0, s6 >= 0, s6 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_5(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 }-> 1 + s9 + s11 :|: s11 >= 0, s11 <= 0, s9 >= 0, s9 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_6(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 }-> 1 + s12 + s14 :|: s14 >= 0, s14 <= 0, s12 >= 0, s12 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 676.38/291.64 Function symbols to be analyzed: {f_6}, {g_7}, {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.64 Previous analysis results are: 676.38/291.64 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.64 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.64 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.64 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.64 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.38/291.64 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.38/291.64 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.38/291.64 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.38/291.64 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.38/291.64 g_5: runtime: O(n^5) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4], size: O(1) [0] 676.38/291.64 f_5: runtime: O(n^5) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5], size: O(1) [0] 676.38/291.64 g_6: runtime: O(n^6) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5], size: O(1) [0] 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (91) IntTrsBoundProof (UPPER BOUND(ID)) 676.38/291.64 676.38/291.64 Computed SIZE bound using CoFloCo for: f_6 676.38/291.64 after applying outer abstraction to obtain an ITS, 676.38/291.64 resulting in: O(1) with polynomial bound: 0 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (92) 676.38/291.64 Obligation: 676.38/291.64 Complexity RNTS consisting of the following rules: 676.38/291.64 676.38/291.64 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.64 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.64 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.64 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.64 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.38/291.64 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.38/291.64 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.38/291.64 f_6(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 }-> s13 :|: s13 >= 0, s13 <= 0, z >= 0 676.38/291.64 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.38/291.64 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.64 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.64 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_3(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 }-> 1 + s3 + s5 :|: s5 >= 0, s5 <= 0, s3 >= 0, s3 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_4(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 }-> 1 + s6 + s8 :|: s8 >= 0, s8 <= 0, s6 >= 0, s6 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_5(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 }-> 1 + s9 + s11 :|: s11 >= 0, s11 <= 0, s9 >= 0, s9 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_6(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 }-> 1 + s12 + s14 :|: s14 >= 0, s14 <= 0, s12 >= 0, s12 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 676.38/291.64 Function symbols to be analyzed: {f_6}, {g_7}, {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.64 Previous analysis results are: 676.38/291.64 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.64 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.64 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.64 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.64 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.38/291.64 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.38/291.64 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.38/291.64 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.38/291.64 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.38/291.64 g_5: runtime: O(n^5) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4], size: O(1) [0] 676.38/291.64 f_5: runtime: O(n^5) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5], size: O(1) [0] 676.38/291.64 g_6: runtime: O(n^6) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5], size: O(1) [0] 676.38/291.64 f_6: runtime: ?, size: O(1) [0] 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (93) IntTrsBoundProof (UPPER BOUND(ID)) 676.38/291.64 676.38/291.64 Computed RUNTIME bound using KoAT for: f_6 676.38/291.64 after applying outer abstraction to obtain an ITS, 676.38/291.64 resulting in: O(n^6) with polynomial bound: 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (94) 676.38/291.64 Obligation: 676.38/291.64 Complexity RNTS consisting of the following rules: 676.38/291.64 676.38/291.64 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.64 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.64 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.64 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.64 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.38/291.64 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.38/291.64 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.38/291.64 f_6(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 }-> s13 :|: s13 >= 0, s13 <= 0, z >= 0 676.38/291.64 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.38/291.64 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.64 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.64 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_3(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 }-> 1 + s3 + s5 :|: s5 >= 0, s5 <= 0, s3 >= 0, s3 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_4(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 }-> 1 + s6 + s8 :|: s8 >= 0, s8 <= 0, s6 >= 0, s6 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_5(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 }-> 1 + s9 + s11 :|: s11 >= 0, s11 <= 0, s9 >= 0, s9 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_6(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 }-> 1 + s12 + s14 :|: s14 >= 0, s14 <= 0, s12 >= 0, s12 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 676.38/291.64 Function symbols to be analyzed: {g_7}, {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.64 Previous analysis results are: 676.38/291.64 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.64 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.64 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.64 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.64 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.38/291.64 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.38/291.64 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.38/291.64 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.38/291.64 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.38/291.64 g_5: runtime: O(n^5) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4], size: O(1) [0] 676.38/291.64 f_5: runtime: O(n^5) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5], size: O(1) [0] 676.38/291.64 g_6: runtime: O(n^6) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5], size: O(1) [0] 676.38/291.64 f_6: runtime: O(n^6) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6], size: O(1) [0] 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (95) ResultPropagationProof (UPPER BOUND(ID)) 676.38/291.64 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (96) 676.38/291.64 Obligation: 676.38/291.64 Complexity RNTS consisting of the following rules: 676.38/291.64 676.38/291.64 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.64 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.64 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.64 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.64 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.38/291.64 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.38/291.64 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.38/291.64 f_6(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 }-> s13 :|: s13 >= 0, s13 <= 0, z >= 0 676.38/291.64 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.38/291.64 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.64 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.64 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_3(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 }-> 1 + s3 + s5 :|: s5 >= 0, s5 <= 0, s3 >= 0, s3 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_4(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 }-> 1 + s6 + s8 :|: s8 >= 0, s8 <= 0, s6 >= 0, s6 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_5(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 }-> 1 + s9 + s11 :|: s11 >= 0, s11 <= 0, s9 >= 0, s9 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_6(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 }-> 1 + s12 + s14 :|: s14 >= 0, s14 <= 0, s12 >= 0, s12 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_7(z, z') -{ 2 + 2*z' + 2*z'^2 + 2*z'^3 + 2*z'^4 + 2*z'^5 + 2*z'^6 }-> 1 + s15 + g_7(z - 1, z') :|: s15 >= 0, s15 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 676.38/291.64 Function symbols to be analyzed: {g_7}, {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.64 Previous analysis results are: 676.38/291.64 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.64 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.64 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.64 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.64 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.38/291.64 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.38/291.64 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.38/291.64 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.38/291.64 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.38/291.64 g_5: runtime: O(n^5) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4], size: O(1) [0] 676.38/291.64 f_5: runtime: O(n^5) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5], size: O(1) [0] 676.38/291.64 g_6: runtime: O(n^6) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5], size: O(1) [0] 676.38/291.64 f_6: runtime: O(n^6) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6], size: O(1) [0] 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (97) IntTrsBoundProof (UPPER BOUND(ID)) 676.38/291.64 676.38/291.64 Computed SIZE bound using CoFloCo for: g_7 676.38/291.64 after applying outer abstraction to obtain an ITS, 676.38/291.64 resulting in: O(1) with polynomial bound: 0 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (98) 676.38/291.64 Obligation: 676.38/291.64 Complexity RNTS consisting of the following rules: 676.38/291.64 676.38/291.64 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.64 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.64 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.64 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.64 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.38/291.64 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.38/291.64 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.38/291.64 f_6(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 }-> s13 :|: s13 >= 0, s13 <= 0, z >= 0 676.38/291.64 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.38/291.64 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.64 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.64 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_3(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 }-> 1 + s3 + s5 :|: s5 >= 0, s5 <= 0, s3 >= 0, s3 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_4(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 }-> 1 + s6 + s8 :|: s8 >= 0, s8 <= 0, s6 >= 0, s6 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_5(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 }-> 1 + s9 + s11 :|: s11 >= 0, s11 <= 0, s9 >= 0, s9 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_6(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 }-> 1 + s12 + s14 :|: s14 >= 0, s14 <= 0, s12 >= 0, s12 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_7(z, z') -{ 2 + 2*z' + 2*z'^2 + 2*z'^3 + 2*z'^4 + 2*z'^5 + 2*z'^6 }-> 1 + s15 + g_7(z - 1, z') :|: s15 >= 0, s15 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 676.38/291.64 Function symbols to be analyzed: {g_7}, {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.64 Previous analysis results are: 676.38/291.64 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.64 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.64 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.64 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.64 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.38/291.64 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.38/291.64 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.38/291.64 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.38/291.64 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.38/291.64 g_5: runtime: O(n^5) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4], size: O(1) [0] 676.38/291.64 f_5: runtime: O(n^5) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5], size: O(1) [0] 676.38/291.64 g_6: runtime: O(n^6) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5], size: O(1) [0] 676.38/291.64 f_6: runtime: O(n^6) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6], size: O(1) [0] 676.38/291.64 g_7: runtime: ?, size: O(1) [0] 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (99) IntTrsBoundProof (UPPER BOUND(ID)) 676.38/291.64 676.38/291.64 Computed RUNTIME bound using KoAT for: g_7 676.38/291.64 after applying outer abstraction to obtain an ITS, 676.38/291.64 resulting in: O(n^7) with polynomial bound: 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (100) 676.38/291.64 Obligation: 676.38/291.64 Complexity RNTS consisting of the following rules: 676.38/291.64 676.38/291.64 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.64 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.64 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.64 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.64 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.38/291.64 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.38/291.64 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.38/291.64 f_6(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 }-> s13 :|: s13 >= 0, s13 <= 0, z >= 0 676.38/291.64 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.38/291.64 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.64 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.64 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_3(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 }-> 1 + s3 + s5 :|: s5 >= 0, s5 <= 0, s3 >= 0, s3 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_4(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 }-> 1 + s6 + s8 :|: s8 >= 0, s8 <= 0, s6 >= 0, s6 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_5(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 }-> 1 + s9 + s11 :|: s11 >= 0, s11 <= 0, s9 >= 0, s9 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_6(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 }-> 1 + s12 + s14 :|: s14 >= 0, s14 <= 0, s12 >= 0, s12 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_7(z, z') -{ 2 + 2*z' + 2*z'^2 + 2*z'^3 + 2*z'^4 + 2*z'^5 + 2*z'^6 }-> 1 + s15 + g_7(z - 1, z') :|: s15 >= 0, s15 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 676.38/291.64 Function symbols to be analyzed: {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.64 Previous analysis results are: 676.38/291.64 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.64 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.64 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.64 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.64 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.38/291.64 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.38/291.64 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.38/291.64 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.38/291.64 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.38/291.64 g_5: runtime: O(n^5) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4], size: O(1) [0] 676.38/291.64 f_5: runtime: O(n^5) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5], size: O(1) [0] 676.38/291.64 g_6: runtime: O(n^6) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5], size: O(1) [0] 676.38/291.64 f_6: runtime: O(n^6) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6], size: O(1) [0] 676.38/291.64 g_7: runtime: O(n^7) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6], size: O(1) [0] 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (101) ResultPropagationProof (UPPER BOUND(ID)) 676.38/291.64 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (102) 676.38/291.64 Obligation: 676.38/291.64 Complexity RNTS consisting of the following rules: 676.38/291.64 676.38/291.64 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.64 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.64 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.64 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.64 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.38/291.64 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.38/291.64 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.38/291.64 f_6(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 }-> s13 :|: s13 >= 0, s13 <= 0, z >= 0 676.38/291.64 f_7(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 }-> s16 :|: s16 >= 0, s16 <= 0, z >= 0 676.38/291.64 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.64 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.64 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_3(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 }-> 1 + s3 + s5 :|: s5 >= 0, s5 <= 0, s3 >= 0, s3 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_4(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 }-> 1 + s6 + s8 :|: s8 >= 0, s8 <= 0, s6 >= 0, s6 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_5(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 }-> 1 + s9 + s11 :|: s11 >= 0, s11 <= 0, s9 >= 0, s9 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_6(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 }-> 1 + s12 + s14 :|: s14 >= 0, s14 <= 0, s12 >= 0, s12 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_7(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 }-> 1 + s15 + s17 :|: s17 >= 0, s17 <= 0, s15 >= 0, s15 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 676.38/291.64 Function symbols to be analyzed: {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.64 Previous analysis results are: 676.38/291.64 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.64 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.64 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.64 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.64 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.38/291.64 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.38/291.64 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.38/291.64 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.38/291.64 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.38/291.64 g_5: runtime: O(n^5) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4], size: O(1) [0] 676.38/291.64 f_5: runtime: O(n^5) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5], size: O(1) [0] 676.38/291.64 g_6: runtime: O(n^6) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5], size: O(1) [0] 676.38/291.64 f_6: runtime: O(n^6) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6], size: O(1) [0] 676.38/291.64 g_7: runtime: O(n^7) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6], size: O(1) [0] 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (103) IntTrsBoundProof (UPPER BOUND(ID)) 676.38/291.64 676.38/291.64 Computed SIZE bound using CoFloCo for: f_7 676.38/291.64 after applying outer abstraction to obtain an ITS, 676.38/291.64 resulting in: O(1) with polynomial bound: 0 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (104) 676.38/291.64 Obligation: 676.38/291.64 Complexity RNTS consisting of the following rules: 676.38/291.64 676.38/291.64 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.64 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.64 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.64 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.64 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.38/291.64 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.38/291.64 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.38/291.64 f_6(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 }-> s13 :|: s13 >= 0, s13 <= 0, z >= 0 676.38/291.64 f_7(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 }-> s16 :|: s16 >= 0, s16 <= 0, z >= 0 676.38/291.64 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.64 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.64 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_3(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 }-> 1 + s3 + s5 :|: s5 >= 0, s5 <= 0, s3 >= 0, s3 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_4(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 }-> 1 + s6 + s8 :|: s8 >= 0, s8 <= 0, s6 >= 0, s6 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_5(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 }-> 1 + s9 + s11 :|: s11 >= 0, s11 <= 0, s9 >= 0, s9 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_6(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 }-> 1 + s12 + s14 :|: s14 >= 0, s14 <= 0, s12 >= 0, s12 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_7(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 }-> 1 + s15 + s17 :|: s17 >= 0, s17 <= 0, s15 >= 0, s15 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 676.38/291.64 Function symbols to be analyzed: {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.64 Previous analysis results are: 676.38/291.64 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.64 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.64 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.64 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.64 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.38/291.64 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.38/291.64 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.38/291.64 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.38/291.64 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.38/291.64 g_5: runtime: O(n^5) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4], size: O(1) [0] 676.38/291.64 f_5: runtime: O(n^5) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5], size: O(1) [0] 676.38/291.64 g_6: runtime: O(n^6) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5], size: O(1) [0] 676.38/291.64 f_6: runtime: O(n^6) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6], size: O(1) [0] 676.38/291.64 g_7: runtime: O(n^7) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6], size: O(1) [0] 676.38/291.64 f_7: runtime: ?, size: O(1) [0] 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (105) IntTrsBoundProof (UPPER BOUND(ID)) 676.38/291.64 676.38/291.64 Computed RUNTIME bound using KoAT for: f_7 676.38/291.64 after applying outer abstraction to obtain an ITS, 676.38/291.64 resulting in: O(n^7) with polynomial bound: 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (106) 676.38/291.64 Obligation: 676.38/291.64 Complexity RNTS consisting of the following rules: 676.38/291.64 676.38/291.64 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.64 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.64 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.64 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.64 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.38/291.64 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.38/291.64 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.38/291.64 f_6(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 }-> s13 :|: s13 >= 0, s13 <= 0, z >= 0 676.38/291.64 f_7(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 }-> s16 :|: s16 >= 0, s16 <= 0, z >= 0 676.38/291.64 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.64 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.64 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_3(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 }-> 1 + s3 + s5 :|: s5 >= 0, s5 <= 0, s3 >= 0, s3 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_4(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 }-> 1 + s6 + s8 :|: s8 >= 0, s8 <= 0, s6 >= 0, s6 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_5(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 }-> 1 + s9 + s11 :|: s11 >= 0, s11 <= 0, s9 >= 0, s9 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_6(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 }-> 1 + s12 + s14 :|: s14 >= 0, s14 <= 0, s12 >= 0, s12 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_7(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 }-> 1 + s15 + s17 :|: s17 >= 0, s17 <= 0, s15 >= 0, s15 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 676.38/291.64 Function symbols to be analyzed: {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.64 Previous analysis results are: 676.38/291.64 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.64 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.64 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.64 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.64 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.38/291.64 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.38/291.64 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.38/291.64 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.38/291.64 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.38/291.64 g_5: runtime: O(n^5) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4], size: O(1) [0] 676.38/291.64 f_5: runtime: O(n^5) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5], size: O(1) [0] 676.38/291.64 g_6: runtime: O(n^6) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5], size: O(1) [0] 676.38/291.64 f_6: runtime: O(n^6) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6], size: O(1) [0] 676.38/291.64 g_7: runtime: O(n^7) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6], size: O(1) [0] 676.38/291.64 f_7: runtime: O(n^7) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7], size: O(1) [0] 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (107) ResultPropagationProof (UPPER BOUND(ID)) 676.38/291.64 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (108) 676.38/291.64 Obligation: 676.38/291.64 Complexity RNTS consisting of the following rules: 676.38/291.64 676.38/291.64 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.64 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.64 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.64 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.64 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.38/291.64 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.38/291.64 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.38/291.64 f_6(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 }-> s13 :|: s13 >= 0, s13 <= 0, z >= 0 676.38/291.64 f_7(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 }-> s16 :|: s16 >= 0, s16 <= 0, z >= 0 676.38/291.64 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.64 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.64 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_3(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 }-> 1 + s3 + s5 :|: s5 >= 0, s5 <= 0, s3 >= 0, s3 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_4(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 }-> 1 + s6 + s8 :|: s8 >= 0, s8 <= 0, s6 >= 0, s6 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_5(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 }-> 1 + s9 + s11 :|: s11 >= 0, s11 <= 0, s9 >= 0, s9 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_6(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 }-> 1 + s12 + s14 :|: s14 >= 0, s14 <= 0, s12 >= 0, s12 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_7(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 }-> 1 + s15 + s17 :|: s17 >= 0, s17 <= 0, s15 >= 0, s15 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_8(z, z') -{ 2 + 2*z' + 2*z'^2 + 2*z'^3 + 2*z'^4 + 2*z'^5 + 2*z'^6 + 2*z'^7 }-> 1 + s18 + g_8(z - 1, z') :|: s18 >= 0, s18 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 676.38/291.64 Function symbols to be analyzed: {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.64 Previous analysis results are: 676.38/291.64 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.64 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.64 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.64 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.64 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.38/291.64 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.38/291.64 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.38/291.64 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.38/291.64 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.38/291.64 g_5: runtime: O(n^5) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4], size: O(1) [0] 676.38/291.64 f_5: runtime: O(n^5) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5], size: O(1) [0] 676.38/291.64 g_6: runtime: O(n^6) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5], size: O(1) [0] 676.38/291.64 f_6: runtime: O(n^6) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6], size: O(1) [0] 676.38/291.64 g_7: runtime: O(n^7) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6], size: O(1) [0] 676.38/291.64 f_7: runtime: O(n^7) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7], size: O(1) [0] 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (109) IntTrsBoundProof (UPPER BOUND(ID)) 676.38/291.64 676.38/291.64 Computed SIZE bound using CoFloCo for: g_8 676.38/291.64 after applying outer abstraction to obtain an ITS, 676.38/291.64 resulting in: O(1) with polynomial bound: 0 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (110) 676.38/291.64 Obligation: 676.38/291.64 Complexity RNTS consisting of the following rules: 676.38/291.64 676.38/291.64 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.64 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.64 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.64 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.64 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.38/291.64 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.38/291.64 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.38/291.64 f_6(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 }-> s13 :|: s13 >= 0, s13 <= 0, z >= 0 676.38/291.64 f_7(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 }-> s16 :|: s16 >= 0, s16 <= 0, z >= 0 676.38/291.64 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.64 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.64 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_3(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 }-> 1 + s3 + s5 :|: s5 >= 0, s5 <= 0, s3 >= 0, s3 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_4(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 }-> 1 + s6 + s8 :|: s8 >= 0, s8 <= 0, s6 >= 0, s6 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_5(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 }-> 1 + s9 + s11 :|: s11 >= 0, s11 <= 0, s9 >= 0, s9 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_6(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 }-> 1 + s12 + s14 :|: s14 >= 0, s14 <= 0, s12 >= 0, s12 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_7(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 }-> 1 + s15 + s17 :|: s17 >= 0, s17 <= 0, s15 >= 0, s15 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_8(z, z') -{ 2 + 2*z' + 2*z'^2 + 2*z'^3 + 2*z'^4 + 2*z'^5 + 2*z'^6 + 2*z'^7 }-> 1 + s18 + g_8(z - 1, z') :|: s18 >= 0, s18 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 676.38/291.64 Function symbols to be analyzed: {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.64 Previous analysis results are: 676.38/291.64 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.64 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.64 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.64 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.64 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.38/291.64 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.38/291.64 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.38/291.64 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.38/291.64 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.38/291.64 g_5: runtime: O(n^5) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4], size: O(1) [0] 676.38/291.64 f_5: runtime: O(n^5) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5], size: O(1) [0] 676.38/291.64 g_6: runtime: O(n^6) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5], size: O(1) [0] 676.38/291.64 f_6: runtime: O(n^6) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6], size: O(1) [0] 676.38/291.64 g_7: runtime: O(n^7) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6], size: O(1) [0] 676.38/291.64 f_7: runtime: O(n^7) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7], size: O(1) [0] 676.38/291.64 g_8: runtime: ?, size: O(1) [0] 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (111) IntTrsBoundProof (UPPER BOUND(ID)) 676.38/291.64 676.38/291.64 Computed RUNTIME bound using KoAT for: g_8 676.38/291.64 after applying outer abstraction to obtain an ITS, 676.38/291.64 resulting in: O(n^8) with polynomial bound: 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (112) 676.38/291.64 Obligation: 676.38/291.64 Complexity RNTS consisting of the following rules: 676.38/291.64 676.38/291.64 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.64 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.64 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.64 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.64 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.38/291.64 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.38/291.64 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.38/291.64 f_6(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 }-> s13 :|: s13 >= 0, s13 <= 0, z >= 0 676.38/291.64 f_7(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 }-> s16 :|: s16 >= 0, s16 <= 0, z >= 0 676.38/291.64 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.38/291.64 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.64 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_3(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 }-> 1 + s3 + s5 :|: s5 >= 0, s5 <= 0, s3 >= 0, s3 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_4(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 }-> 1 + s6 + s8 :|: s8 >= 0, s8 <= 0, s6 >= 0, s6 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_5(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 }-> 1 + s9 + s11 :|: s11 >= 0, s11 <= 0, s9 >= 0, s9 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_6(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 }-> 1 + s12 + s14 :|: s14 >= 0, s14 <= 0, s12 >= 0, s12 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_7(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 }-> 1 + s15 + s17 :|: s17 >= 0, s17 <= 0, s15 >= 0, s15 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_8(z, z') -{ 2 + 2*z' + 2*z'^2 + 2*z'^3 + 2*z'^4 + 2*z'^5 + 2*z'^6 + 2*z'^7 }-> 1 + s18 + g_8(z - 1, z') :|: s18 >= 0, s18 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 676.38/291.64 Function symbols to be analyzed: {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.64 Previous analysis results are: 676.38/291.64 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.64 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.64 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.64 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.64 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.38/291.64 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.38/291.64 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.38/291.64 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.38/291.64 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.38/291.64 g_5: runtime: O(n^5) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4], size: O(1) [0] 676.38/291.64 f_5: runtime: O(n^5) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5], size: O(1) [0] 676.38/291.64 g_6: runtime: O(n^6) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5], size: O(1) [0] 676.38/291.64 f_6: runtime: O(n^6) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6], size: O(1) [0] 676.38/291.64 g_7: runtime: O(n^7) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6], size: O(1) [0] 676.38/291.64 f_7: runtime: O(n^7) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7], size: O(1) [0] 676.38/291.64 g_8: runtime: O(n^8) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7], size: O(1) [0] 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (113) ResultPropagationProof (UPPER BOUND(ID)) 676.38/291.64 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (114) 676.38/291.64 Obligation: 676.38/291.64 Complexity RNTS consisting of the following rules: 676.38/291.64 676.38/291.64 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.64 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.64 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.64 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.64 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.38/291.64 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.38/291.64 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.38/291.64 f_6(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 }-> s13 :|: s13 >= 0, s13 <= 0, z >= 0 676.38/291.64 f_7(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 }-> s16 :|: s16 >= 0, s16 <= 0, z >= 0 676.38/291.64 f_8(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8 }-> s19 :|: s19 >= 0, s19 <= 0, z >= 0 676.38/291.64 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.64 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_3(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 }-> 1 + s3 + s5 :|: s5 >= 0, s5 <= 0, s3 >= 0, s3 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_4(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 }-> 1 + s6 + s8 :|: s8 >= 0, s8 <= 0, s6 >= 0, s6 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_5(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 }-> 1 + s9 + s11 :|: s11 >= 0, s11 <= 0, s9 >= 0, s9 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_6(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 }-> 1 + s12 + s14 :|: s14 >= 0, s14 <= 0, s12 >= 0, s12 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_7(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 }-> 1 + s15 + s17 :|: s17 >= 0, s17 <= 0, s15 >= 0, s15 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_8(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7 }-> 1 + s18 + s20 :|: s20 >= 0, s20 <= 0, s18 >= 0, s18 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 676.38/291.64 Function symbols to be analyzed: {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.64 Previous analysis results are: 676.38/291.64 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.64 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.64 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.64 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.64 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.38/291.64 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.38/291.64 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.38/291.64 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.38/291.64 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.38/291.64 g_5: runtime: O(n^5) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4], size: O(1) [0] 676.38/291.64 f_5: runtime: O(n^5) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5], size: O(1) [0] 676.38/291.64 g_6: runtime: O(n^6) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5], size: O(1) [0] 676.38/291.64 f_6: runtime: O(n^6) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6], size: O(1) [0] 676.38/291.64 g_7: runtime: O(n^7) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6], size: O(1) [0] 676.38/291.64 f_7: runtime: O(n^7) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7], size: O(1) [0] 676.38/291.64 g_8: runtime: O(n^8) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7], size: O(1) [0] 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (115) IntTrsBoundProof (UPPER BOUND(ID)) 676.38/291.64 676.38/291.64 Computed SIZE bound using CoFloCo for: f_8 676.38/291.64 after applying outer abstraction to obtain an ITS, 676.38/291.64 resulting in: O(1) with polynomial bound: 0 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (116) 676.38/291.64 Obligation: 676.38/291.64 Complexity RNTS consisting of the following rules: 676.38/291.64 676.38/291.64 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.64 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.64 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.64 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.64 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.38/291.64 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.38/291.64 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.38/291.64 f_6(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 }-> s13 :|: s13 >= 0, s13 <= 0, z >= 0 676.38/291.64 f_7(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 }-> s16 :|: s16 >= 0, s16 <= 0, z >= 0 676.38/291.64 f_8(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8 }-> s19 :|: s19 >= 0, s19 <= 0, z >= 0 676.38/291.64 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.64 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_3(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 }-> 1 + s3 + s5 :|: s5 >= 0, s5 <= 0, s3 >= 0, s3 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_4(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 }-> 1 + s6 + s8 :|: s8 >= 0, s8 <= 0, s6 >= 0, s6 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_5(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 }-> 1 + s9 + s11 :|: s11 >= 0, s11 <= 0, s9 >= 0, s9 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_6(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 }-> 1 + s12 + s14 :|: s14 >= 0, s14 <= 0, s12 >= 0, s12 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_7(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 }-> 1 + s15 + s17 :|: s17 >= 0, s17 <= 0, s15 >= 0, s15 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_8(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7 }-> 1 + s18 + s20 :|: s20 >= 0, s20 <= 0, s18 >= 0, s18 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 676.38/291.64 Function symbols to be analyzed: {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.64 Previous analysis results are: 676.38/291.64 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.64 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.64 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.64 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.64 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.38/291.64 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.38/291.64 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.38/291.64 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.38/291.64 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.38/291.64 g_5: runtime: O(n^5) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4], size: O(1) [0] 676.38/291.64 f_5: runtime: O(n^5) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5], size: O(1) [0] 676.38/291.64 g_6: runtime: O(n^6) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5], size: O(1) [0] 676.38/291.64 f_6: runtime: O(n^6) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6], size: O(1) [0] 676.38/291.64 g_7: runtime: O(n^7) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6], size: O(1) [0] 676.38/291.64 f_7: runtime: O(n^7) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7], size: O(1) [0] 676.38/291.64 g_8: runtime: O(n^8) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7], size: O(1) [0] 676.38/291.64 f_8: runtime: ?, size: O(1) [0] 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (117) IntTrsBoundProof (UPPER BOUND(ID)) 676.38/291.64 676.38/291.64 Computed RUNTIME bound using KoAT for: f_8 676.38/291.64 after applying outer abstraction to obtain an ITS, 676.38/291.64 resulting in: O(n^8) with polynomial bound: 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (118) 676.38/291.64 Obligation: 676.38/291.64 Complexity RNTS consisting of the following rules: 676.38/291.64 676.38/291.64 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.64 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.64 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.64 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.64 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.38/291.64 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.38/291.64 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.38/291.64 f_6(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 }-> s13 :|: s13 >= 0, s13 <= 0, z >= 0 676.38/291.64 f_7(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 }-> s16 :|: s16 >= 0, s16 <= 0, z >= 0 676.38/291.64 f_8(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8 }-> s19 :|: s19 >= 0, s19 <= 0, z >= 0 676.38/291.64 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.64 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_3(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 }-> 1 + s3 + s5 :|: s5 >= 0, s5 <= 0, s3 >= 0, s3 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_4(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 }-> 1 + s6 + s8 :|: s8 >= 0, s8 <= 0, s6 >= 0, s6 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_5(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 }-> 1 + s9 + s11 :|: s11 >= 0, s11 <= 0, s9 >= 0, s9 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_6(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 }-> 1 + s12 + s14 :|: s14 >= 0, s14 <= 0, s12 >= 0, s12 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_7(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 }-> 1 + s15 + s17 :|: s17 >= 0, s17 <= 0, s15 >= 0, s15 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_8(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7 }-> 1 + s18 + s20 :|: s20 >= 0, s20 <= 0, s18 >= 0, s18 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 676.38/291.64 Function symbols to be analyzed: {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.64 Previous analysis results are: 676.38/291.64 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.64 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.64 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.64 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.64 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.38/291.64 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.38/291.64 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.38/291.64 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.38/291.64 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.38/291.64 g_5: runtime: O(n^5) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4], size: O(1) [0] 676.38/291.64 f_5: runtime: O(n^5) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5], size: O(1) [0] 676.38/291.64 g_6: runtime: O(n^6) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5], size: O(1) [0] 676.38/291.64 f_6: runtime: O(n^6) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6], size: O(1) [0] 676.38/291.64 g_7: runtime: O(n^7) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6], size: O(1) [0] 676.38/291.64 f_7: runtime: O(n^7) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7], size: O(1) [0] 676.38/291.64 g_8: runtime: O(n^8) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7], size: O(1) [0] 676.38/291.64 f_8: runtime: O(n^8) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8], size: O(1) [0] 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (119) ResultPropagationProof (UPPER BOUND(ID)) 676.38/291.64 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (120) 676.38/291.64 Obligation: 676.38/291.64 Complexity RNTS consisting of the following rules: 676.38/291.64 676.38/291.64 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.64 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.64 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.64 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.64 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.38/291.64 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.38/291.64 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.38/291.64 f_6(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 }-> s13 :|: s13 >= 0, s13 <= 0, z >= 0 676.38/291.64 f_7(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 }-> s16 :|: s16 >= 0, s16 <= 0, z >= 0 676.38/291.64 f_8(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8 }-> s19 :|: s19 >= 0, s19 <= 0, z >= 0 676.38/291.64 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.64 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_3(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 }-> 1 + s3 + s5 :|: s5 >= 0, s5 <= 0, s3 >= 0, s3 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_4(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 }-> 1 + s6 + s8 :|: s8 >= 0, s8 <= 0, s6 >= 0, s6 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_5(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 }-> 1 + s9 + s11 :|: s11 >= 0, s11 <= 0, s9 >= 0, s9 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_6(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 }-> 1 + s12 + s14 :|: s14 >= 0, s14 <= 0, s12 >= 0, s12 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_7(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 }-> 1 + s15 + s17 :|: s17 >= 0, s17 <= 0, s15 >= 0, s15 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_8(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7 }-> 1 + s18 + s20 :|: s20 >= 0, s20 <= 0, s18 >= 0, s18 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_9(z, z') -{ 2 + 2*z' + 2*z'^2 + 2*z'^3 + 2*z'^4 + 2*z'^5 + 2*z'^6 + 2*z'^7 + 2*z'^8 }-> 1 + s21 + g_9(z - 1, z') :|: s21 >= 0, s21 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 676.38/291.64 Function symbols to be analyzed: {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.64 Previous analysis results are: 676.38/291.64 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.64 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.64 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.64 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.64 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.38/291.64 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.38/291.64 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.38/291.64 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.38/291.64 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.38/291.64 g_5: runtime: O(n^5) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4], size: O(1) [0] 676.38/291.64 f_5: runtime: O(n^5) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5], size: O(1) [0] 676.38/291.64 g_6: runtime: O(n^6) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5], size: O(1) [0] 676.38/291.64 f_6: runtime: O(n^6) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6], size: O(1) [0] 676.38/291.64 g_7: runtime: O(n^7) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6], size: O(1) [0] 676.38/291.64 f_7: runtime: O(n^7) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7], size: O(1) [0] 676.38/291.64 g_8: runtime: O(n^8) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7], size: O(1) [0] 676.38/291.64 f_8: runtime: O(n^8) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8], size: O(1) [0] 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (121) IntTrsBoundProof (UPPER BOUND(ID)) 676.38/291.64 676.38/291.64 Computed SIZE bound using CoFloCo for: g_9 676.38/291.64 after applying outer abstraction to obtain an ITS, 676.38/291.64 resulting in: O(1) with polynomial bound: 0 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (122) 676.38/291.64 Obligation: 676.38/291.64 Complexity RNTS consisting of the following rules: 676.38/291.64 676.38/291.64 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.64 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.64 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.64 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.64 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.38/291.64 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.38/291.64 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.38/291.64 f_6(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 }-> s13 :|: s13 >= 0, s13 <= 0, z >= 0 676.38/291.64 f_7(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 }-> s16 :|: s16 >= 0, s16 <= 0, z >= 0 676.38/291.64 f_8(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8 }-> s19 :|: s19 >= 0, s19 <= 0, z >= 0 676.38/291.64 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.64 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_3(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 }-> 1 + s3 + s5 :|: s5 >= 0, s5 <= 0, s3 >= 0, s3 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_4(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 }-> 1 + s6 + s8 :|: s8 >= 0, s8 <= 0, s6 >= 0, s6 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_5(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 }-> 1 + s9 + s11 :|: s11 >= 0, s11 <= 0, s9 >= 0, s9 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_6(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 }-> 1 + s12 + s14 :|: s14 >= 0, s14 <= 0, s12 >= 0, s12 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_7(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 }-> 1 + s15 + s17 :|: s17 >= 0, s17 <= 0, s15 >= 0, s15 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_8(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7 }-> 1 + s18 + s20 :|: s20 >= 0, s20 <= 0, s18 >= 0, s18 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_9(z, z') -{ 2 + 2*z' + 2*z'^2 + 2*z'^3 + 2*z'^4 + 2*z'^5 + 2*z'^6 + 2*z'^7 + 2*z'^8 }-> 1 + s21 + g_9(z - 1, z') :|: s21 >= 0, s21 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 676.38/291.64 Function symbols to be analyzed: {g_9}, {f_9}, {g_10}, {f_10} 676.38/291.64 Previous analysis results are: 676.38/291.64 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.64 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.64 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.64 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.64 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.38/291.64 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.38/291.64 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.38/291.64 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.38/291.64 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.38/291.64 g_5: runtime: O(n^5) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4], size: O(1) [0] 676.38/291.64 f_5: runtime: O(n^5) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5], size: O(1) [0] 676.38/291.64 g_6: runtime: O(n^6) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5], size: O(1) [0] 676.38/291.64 f_6: runtime: O(n^6) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6], size: O(1) [0] 676.38/291.64 g_7: runtime: O(n^7) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6], size: O(1) [0] 676.38/291.64 f_7: runtime: O(n^7) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7], size: O(1) [0] 676.38/291.64 g_8: runtime: O(n^8) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7], size: O(1) [0] 676.38/291.64 f_8: runtime: O(n^8) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8], size: O(1) [0] 676.38/291.64 g_9: runtime: ?, size: O(1) [0] 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (123) IntTrsBoundProof (UPPER BOUND(ID)) 676.38/291.64 676.38/291.64 Computed RUNTIME bound using KoAT for: g_9 676.38/291.64 after applying outer abstraction to obtain an ITS, 676.38/291.64 resulting in: O(n^9) with polynomial bound: 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7 + 2*z*z'^8 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (124) 676.38/291.64 Obligation: 676.38/291.64 Complexity RNTS consisting of the following rules: 676.38/291.64 676.38/291.64 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.64 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.64 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.64 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.64 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.38/291.64 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.38/291.64 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.38/291.64 f_6(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 }-> s13 :|: s13 >= 0, s13 <= 0, z >= 0 676.38/291.64 f_7(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 }-> s16 :|: s16 >= 0, s16 <= 0, z >= 0 676.38/291.64 f_8(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8 }-> s19 :|: s19 >= 0, s19 <= 0, z >= 0 676.38/291.64 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.38/291.64 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_3(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 }-> 1 + s3 + s5 :|: s5 >= 0, s5 <= 0, s3 >= 0, s3 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_4(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 }-> 1 + s6 + s8 :|: s8 >= 0, s8 <= 0, s6 >= 0, s6 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_5(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 }-> 1 + s9 + s11 :|: s11 >= 0, s11 <= 0, s9 >= 0, s9 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_6(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 }-> 1 + s12 + s14 :|: s14 >= 0, s14 <= 0, s12 >= 0, s12 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_7(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 }-> 1 + s15 + s17 :|: s17 >= 0, s17 <= 0, s15 >= 0, s15 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_8(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7 }-> 1 + s18 + s20 :|: s20 >= 0, s20 <= 0, s18 >= 0, s18 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_9(z, z') -{ 2 + 2*z' + 2*z'^2 + 2*z'^3 + 2*z'^4 + 2*z'^5 + 2*z'^6 + 2*z'^7 + 2*z'^8 }-> 1 + s21 + g_9(z - 1, z') :|: s21 >= 0, s21 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 676.38/291.64 Function symbols to be analyzed: {f_9}, {g_10}, {f_10} 676.38/291.64 Previous analysis results are: 676.38/291.64 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.64 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.64 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.64 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.64 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.38/291.64 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.38/291.64 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.38/291.64 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.38/291.64 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.38/291.64 g_5: runtime: O(n^5) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4], size: O(1) [0] 676.38/291.64 f_5: runtime: O(n^5) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5], size: O(1) [0] 676.38/291.64 g_6: runtime: O(n^6) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5], size: O(1) [0] 676.38/291.64 f_6: runtime: O(n^6) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6], size: O(1) [0] 676.38/291.64 g_7: runtime: O(n^7) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6], size: O(1) [0] 676.38/291.64 f_7: runtime: O(n^7) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7], size: O(1) [0] 676.38/291.64 g_8: runtime: O(n^8) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7], size: O(1) [0] 676.38/291.64 f_8: runtime: O(n^8) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8], size: O(1) [0] 676.38/291.64 g_9: runtime: O(n^9) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7 + 2*z*z'^8], size: O(1) [0] 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (125) ResultPropagationProof (UPPER BOUND(ID)) 676.38/291.64 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (126) 676.38/291.64 Obligation: 676.38/291.64 Complexity RNTS consisting of the following rules: 676.38/291.64 676.38/291.64 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.64 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.64 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.64 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.64 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.38/291.64 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.38/291.64 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.38/291.64 f_6(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 }-> s13 :|: s13 >= 0, s13 <= 0, z >= 0 676.38/291.64 f_7(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 }-> s16 :|: s16 >= 0, s16 <= 0, z >= 0 676.38/291.64 f_8(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8 }-> s19 :|: s19 >= 0, s19 <= 0, z >= 0 676.38/291.64 f_9(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8 + 2*z^9 }-> s22 :|: s22 >= 0, s22 <= 0, z >= 0 676.38/291.64 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_3(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 }-> 1 + s3 + s5 :|: s5 >= 0, s5 <= 0, s3 >= 0, s3 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_4(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 }-> 1 + s6 + s8 :|: s8 >= 0, s8 <= 0, s6 >= 0, s6 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_5(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 }-> 1 + s9 + s11 :|: s11 >= 0, s11 <= 0, s9 >= 0, s9 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_6(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 }-> 1 + s12 + s14 :|: s14 >= 0, s14 <= 0, s12 >= 0, s12 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_7(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 }-> 1 + s15 + s17 :|: s17 >= 0, s17 <= 0, s15 >= 0, s15 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_8(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7 }-> 1 + s18 + s20 :|: s20 >= 0, s20 <= 0, s18 >= 0, s18 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_9(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7 + 2*z*z'^8 }-> 1 + s21 + s23 :|: s23 >= 0, s23 <= 0, s21 >= 0, s21 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 676.38/291.64 Function symbols to be analyzed: {f_9}, {g_10}, {f_10} 676.38/291.64 Previous analysis results are: 676.38/291.64 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.64 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.64 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.64 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.64 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.38/291.64 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.38/291.64 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.38/291.64 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.38/291.64 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.38/291.64 g_5: runtime: O(n^5) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4], size: O(1) [0] 676.38/291.64 f_5: runtime: O(n^5) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5], size: O(1) [0] 676.38/291.64 g_6: runtime: O(n^6) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5], size: O(1) [0] 676.38/291.64 f_6: runtime: O(n^6) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6], size: O(1) [0] 676.38/291.64 g_7: runtime: O(n^7) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6], size: O(1) [0] 676.38/291.64 f_7: runtime: O(n^7) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7], size: O(1) [0] 676.38/291.64 g_8: runtime: O(n^8) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7], size: O(1) [0] 676.38/291.64 f_8: runtime: O(n^8) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8], size: O(1) [0] 676.38/291.64 g_9: runtime: O(n^9) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7 + 2*z*z'^8], size: O(1) [0] 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (127) IntTrsBoundProof (UPPER BOUND(ID)) 676.38/291.64 676.38/291.64 Computed SIZE bound using CoFloCo for: f_9 676.38/291.64 after applying outer abstraction to obtain an ITS, 676.38/291.64 resulting in: O(1) with polynomial bound: 0 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (128) 676.38/291.64 Obligation: 676.38/291.64 Complexity RNTS consisting of the following rules: 676.38/291.64 676.38/291.64 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.64 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.64 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.64 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.64 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.38/291.64 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.38/291.64 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.38/291.64 f_6(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 }-> s13 :|: s13 >= 0, s13 <= 0, z >= 0 676.38/291.64 f_7(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 }-> s16 :|: s16 >= 0, s16 <= 0, z >= 0 676.38/291.64 f_8(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8 }-> s19 :|: s19 >= 0, s19 <= 0, z >= 0 676.38/291.64 f_9(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8 + 2*z^9 }-> s22 :|: s22 >= 0, s22 <= 0, z >= 0 676.38/291.64 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_3(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 }-> 1 + s3 + s5 :|: s5 >= 0, s5 <= 0, s3 >= 0, s3 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_4(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 }-> 1 + s6 + s8 :|: s8 >= 0, s8 <= 0, s6 >= 0, s6 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_5(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 }-> 1 + s9 + s11 :|: s11 >= 0, s11 <= 0, s9 >= 0, s9 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_6(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 }-> 1 + s12 + s14 :|: s14 >= 0, s14 <= 0, s12 >= 0, s12 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_7(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 }-> 1 + s15 + s17 :|: s17 >= 0, s17 <= 0, s15 >= 0, s15 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_8(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7 }-> 1 + s18 + s20 :|: s20 >= 0, s20 <= 0, s18 >= 0, s18 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_9(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7 + 2*z*z'^8 }-> 1 + s21 + s23 :|: s23 >= 0, s23 <= 0, s21 >= 0, s21 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 676.38/291.64 Function symbols to be analyzed: {f_9}, {g_10}, {f_10} 676.38/291.64 Previous analysis results are: 676.38/291.64 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.64 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.64 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.64 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.64 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.38/291.64 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.38/291.64 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.38/291.64 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.38/291.64 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.38/291.64 g_5: runtime: O(n^5) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4], size: O(1) [0] 676.38/291.64 f_5: runtime: O(n^5) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5], size: O(1) [0] 676.38/291.64 g_6: runtime: O(n^6) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5], size: O(1) [0] 676.38/291.64 f_6: runtime: O(n^6) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6], size: O(1) [0] 676.38/291.64 g_7: runtime: O(n^7) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6], size: O(1) [0] 676.38/291.64 f_7: runtime: O(n^7) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7], size: O(1) [0] 676.38/291.64 g_8: runtime: O(n^8) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7], size: O(1) [0] 676.38/291.64 f_8: runtime: O(n^8) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8], size: O(1) [0] 676.38/291.64 g_9: runtime: O(n^9) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7 + 2*z*z'^8], size: O(1) [0] 676.38/291.64 f_9: runtime: ?, size: O(1) [0] 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (129) IntTrsBoundProof (UPPER BOUND(ID)) 676.38/291.64 676.38/291.64 Computed RUNTIME bound using KoAT for: f_9 676.38/291.64 after applying outer abstraction to obtain an ITS, 676.38/291.64 resulting in: O(n^9) with polynomial bound: 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8 + 2*z^9 676.38/291.64 676.38/291.64 ---------------------------------------- 676.38/291.64 676.38/291.64 (130) 676.38/291.64 Obligation: 676.38/291.64 Complexity RNTS consisting of the following rules: 676.38/291.64 676.38/291.64 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.64 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.64 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.64 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.64 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.38/291.64 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.38/291.64 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.38/291.64 f_6(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 }-> s13 :|: s13 >= 0, s13 <= 0, z >= 0 676.38/291.64 f_7(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 }-> s16 :|: s16 >= 0, s16 <= 0, z >= 0 676.38/291.64 f_8(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8 }-> s19 :|: s19 >= 0, s19 <= 0, z >= 0 676.38/291.64 f_9(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8 + 2*z^9 }-> s22 :|: s22 >= 0, s22 <= 0, z >= 0 676.38/291.64 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.38/291.64 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_3(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 }-> 1 + s3 + s5 :|: s5 >= 0, s5 <= 0, s3 >= 0, s3 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_4(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 }-> 1 + s6 + s8 :|: s8 >= 0, s8 <= 0, s6 >= 0, s6 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_5(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 }-> 1 + s9 + s11 :|: s11 >= 0, s11 <= 0, s9 >= 0, s9 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_6(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 }-> 1 + s12 + s14 :|: s14 >= 0, s14 <= 0, s12 >= 0, s12 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_7(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 }-> 1 + s15 + s17 :|: s17 >= 0, s17 <= 0, s15 >= 0, s15 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_8(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7 }-> 1 + s18 + s20 :|: s20 >= 0, s20 <= 0, s18 >= 0, s18 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 g_9(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7 + 2*z*z'^8 }-> 1 + s21 + s23 :|: s23 >= 0, s23 <= 0, s21 >= 0, s21 <= 0, z - 1 >= 0, z' >= 0 676.38/291.64 676.38/291.64 Function symbols to be analyzed: {g_10}, {f_10} 676.38/291.64 Previous analysis results are: 676.38/291.64 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.64 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.64 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.64 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.64 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.38/291.64 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.38/291.64 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.38/291.64 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.38/291.64 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.38/291.65 g_5: runtime: O(n^5) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4], size: O(1) [0] 676.38/291.65 f_5: runtime: O(n^5) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5], size: O(1) [0] 676.38/291.65 g_6: runtime: O(n^6) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5], size: O(1) [0] 676.38/291.65 f_6: runtime: O(n^6) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6], size: O(1) [0] 676.38/291.65 g_7: runtime: O(n^7) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6], size: O(1) [0] 676.38/291.65 f_7: runtime: O(n^7) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7], size: O(1) [0] 676.38/291.65 g_8: runtime: O(n^8) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7], size: O(1) [0] 676.38/291.65 f_8: runtime: O(n^8) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8], size: O(1) [0] 676.38/291.65 g_9: runtime: O(n^9) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7 + 2*z*z'^8], size: O(1) [0] 676.38/291.65 f_9: runtime: O(n^9) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8 + 2*z^9], size: O(1) [0] 676.38/291.65 676.38/291.65 ---------------------------------------- 676.38/291.65 676.38/291.65 (131) ResultPropagationProof (UPPER BOUND(ID)) 676.38/291.65 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 676.38/291.65 ---------------------------------------- 676.38/291.65 676.38/291.65 (132) 676.38/291.65 Obligation: 676.38/291.65 Complexity RNTS consisting of the following rules: 676.38/291.65 676.38/291.65 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.65 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.65 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.65 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.65 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.38/291.65 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.38/291.65 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.38/291.65 f_6(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 }-> s13 :|: s13 >= 0, s13 <= 0, z >= 0 676.38/291.65 f_7(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 }-> s16 :|: s16 >= 0, s16 <= 0, z >= 0 676.38/291.65 f_8(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8 }-> s19 :|: s19 >= 0, s19 <= 0, z >= 0 676.38/291.65 f_9(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8 + 2*z^9 }-> s22 :|: s22 >= 0, s22 <= 0, z >= 0 676.38/291.65 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_10(z, z') -{ 2 + 2*z' + 2*z'^2 + 2*z'^3 + 2*z'^4 + 2*z'^5 + 2*z'^6 + 2*z'^7 + 2*z'^8 + 2*z'^9 }-> 1 + s24 + g_10(z - 1, z') :|: s24 >= 0, s24 <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_3(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 }-> 1 + s3 + s5 :|: s5 >= 0, s5 <= 0, s3 >= 0, s3 <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_4(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 }-> 1 + s6 + s8 :|: s8 >= 0, s8 <= 0, s6 >= 0, s6 <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_5(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 }-> 1 + s9 + s11 :|: s11 >= 0, s11 <= 0, s9 >= 0, s9 <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_6(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 }-> 1 + s12 + s14 :|: s14 >= 0, s14 <= 0, s12 >= 0, s12 <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_7(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 }-> 1 + s15 + s17 :|: s17 >= 0, s17 <= 0, s15 >= 0, s15 <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_8(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7 }-> 1 + s18 + s20 :|: s20 >= 0, s20 <= 0, s18 >= 0, s18 <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_9(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7 + 2*z*z'^8 }-> 1 + s21 + s23 :|: s23 >= 0, s23 <= 0, s21 >= 0, s21 <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 676.38/291.65 Function symbols to be analyzed: {g_10}, {f_10} 676.38/291.65 Previous analysis results are: 676.38/291.65 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.65 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.65 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.65 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.65 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.38/291.65 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.38/291.65 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.38/291.65 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.38/291.65 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.38/291.65 g_5: runtime: O(n^5) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4], size: O(1) [0] 676.38/291.65 f_5: runtime: O(n^5) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5], size: O(1) [0] 676.38/291.65 g_6: runtime: O(n^6) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5], size: O(1) [0] 676.38/291.65 f_6: runtime: O(n^6) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6], size: O(1) [0] 676.38/291.65 g_7: runtime: O(n^7) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6], size: O(1) [0] 676.38/291.65 f_7: runtime: O(n^7) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7], size: O(1) [0] 676.38/291.65 g_8: runtime: O(n^8) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7], size: O(1) [0] 676.38/291.65 f_8: runtime: O(n^8) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8], size: O(1) [0] 676.38/291.65 g_9: runtime: O(n^9) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7 + 2*z*z'^8], size: O(1) [0] 676.38/291.65 f_9: runtime: O(n^9) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8 + 2*z^9], size: O(1) [0] 676.38/291.65 676.38/291.65 ---------------------------------------- 676.38/291.65 676.38/291.65 (133) IntTrsBoundProof (UPPER BOUND(ID)) 676.38/291.65 676.38/291.65 Computed SIZE bound using CoFloCo for: g_10 676.38/291.65 after applying outer abstraction to obtain an ITS, 676.38/291.65 resulting in: O(1) with polynomial bound: 0 676.38/291.65 676.38/291.65 ---------------------------------------- 676.38/291.65 676.38/291.65 (134) 676.38/291.65 Obligation: 676.38/291.65 Complexity RNTS consisting of the following rules: 676.38/291.65 676.38/291.65 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.65 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.65 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.65 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.65 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.38/291.65 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.38/291.65 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.38/291.65 f_6(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 }-> s13 :|: s13 >= 0, s13 <= 0, z >= 0 676.38/291.65 f_7(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 }-> s16 :|: s16 >= 0, s16 <= 0, z >= 0 676.38/291.65 f_8(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8 }-> s19 :|: s19 >= 0, s19 <= 0, z >= 0 676.38/291.65 f_9(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8 + 2*z^9 }-> s22 :|: s22 >= 0, s22 <= 0, z >= 0 676.38/291.65 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_10(z, z') -{ 2 + 2*z' + 2*z'^2 + 2*z'^3 + 2*z'^4 + 2*z'^5 + 2*z'^6 + 2*z'^7 + 2*z'^8 + 2*z'^9 }-> 1 + s24 + g_10(z - 1, z') :|: s24 >= 0, s24 <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_3(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 }-> 1 + s3 + s5 :|: s5 >= 0, s5 <= 0, s3 >= 0, s3 <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_4(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 }-> 1 + s6 + s8 :|: s8 >= 0, s8 <= 0, s6 >= 0, s6 <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_5(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 }-> 1 + s9 + s11 :|: s11 >= 0, s11 <= 0, s9 >= 0, s9 <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_6(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 }-> 1 + s12 + s14 :|: s14 >= 0, s14 <= 0, s12 >= 0, s12 <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_7(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 }-> 1 + s15 + s17 :|: s17 >= 0, s17 <= 0, s15 >= 0, s15 <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_8(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7 }-> 1 + s18 + s20 :|: s20 >= 0, s20 <= 0, s18 >= 0, s18 <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_9(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7 + 2*z*z'^8 }-> 1 + s21 + s23 :|: s23 >= 0, s23 <= 0, s21 >= 0, s21 <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 676.38/291.65 Function symbols to be analyzed: {g_10}, {f_10} 676.38/291.65 Previous analysis results are: 676.38/291.65 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.65 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.65 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.65 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.65 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.38/291.65 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.38/291.65 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.38/291.65 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.38/291.65 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.38/291.65 g_5: runtime: O(n^5) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4], size: O(1) [0] 676.38/291.65 f_5: runtime: O(n^5) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5], size: O(1) [0] 676.38/291.65 g_6: runtime: O(n^6) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5], size: O(1) [0] 676.38/291.65 f_6: runtime: O(n^6) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6], size: O(1) [0] 676.38/291.65 g_7: runtime: O(n^7) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6], size: O(1) [0] 676.38/291.65 f_7: runtime: O(n^7) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7], size: O(1) [0] 676.38/291.65 g_8: runtime: O(n^8) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7], size: O(1) [0] 676.38/291.65 f_8: runtime: O(n^8) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8], size: O(1) [0] 676.38/291.65 g_9: runtime: O(n^9) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7 + 2*z*z'^8], size: O(1) [0] 676.38/291.65 f_9: runtime: O(n^9) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8 + 2*z^9], size: O(1) [0] 676.38/291.65 g_10: runtime: ?, size: O(1) [0] 676.38/291.65 676.38/291.65 ---------------------------------------- 676.38/291.65 676.38/291.65 (135) IntTrsBoundProof (UPPER BOUND(ID)) 676.38/291.65 676.38/291.65 Computed RUNTIME bound using KoAT for: g_10 676.38/291.65 after applying outer abstraction to obtain an ITS, 676.38/291.65 resulting in: O(n^10) with polynomial bound: 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7 + 2*z*z'^8 + 2*z*z'^9 676.38/291.65 676.38/291.65 ---------------------------------------- 676.38/291.65 676.38/291.65 (136) 676.38/291.65 Obligation: 676.38/291.65 Complexity RNTS consisting of the following rules: 676.38/291.65 676.38/291.65 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.65 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.65 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.38/291.65 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.65 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.38/291.65 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.38/291.65 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.38/291.65 f_6(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 }-> s13 :|: s13 >= 0, s13 <= 0, z >= 0 676.38/291.65 f_7(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 }-> s16 :|: s16 >= 0, s16 <= 0, z >= 0 676.38/291.65 f_8(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8 }-> s19 :|: s19 >= 0, s19 <= 0, z >= 0 676.38/291.65 f_9(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8 + 2*z^9 }-> s22 :|: s22 >= 0, s22 <= 0, z >= 0 676.38/291.65 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_10(z, z') -{ 2 + 2*z' + 2*z'^2 + 2*z'^3 + 2*z'^4 + 2*z'^5 + 2*z'^6 + 2*z'^7 + 2*z'^8 + 2*z'^9 }-> 1 + s24 + g_10(z - 1, z') :|: s24 >= 0, s24 <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_3(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 }-> 1 + s3 + s5 :|: s5 >= 0, s5 <= 0, s3 >= 0, s3 <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_4(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 }-> 1 + s6 + s8 :|: s8 >= 0, s8 <= 0, s6 >= 0, s6 <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_5(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 }-> 1 + s9 + s11 :|: s11 >= 0, s11 <= 0, s9 >= 0, s9 <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_6(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 }-> 1 + s12 + s14 :|: s14 >= 0, s14 <= 0, s12 >= 0, s12 <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_7(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 }-> 1 + s15 + s17 :|: s17 >= 0, s17 <= 0, s15 >= 0, s15 <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_8(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7 }-> 1 + s18 + s20 :|: s20 >= 0, s20 <= 0, s18 >= 0, s18 <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_9(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7 + 2*z*z'^8 }-> 1 + s21 + s23 :|: s23 >= 0, s23 <= 0, s21 >= 0, s21 <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 676.38/291.65 Function symbols to be analyzed: {f_10} 676.38/291.65 Previous analysis results are: 676.38/291.65 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.65 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.65 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.65 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.65 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.38/291.65 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.38/291.65 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.38/291.65 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.38/291.65 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.38/291.65 g_5: runtime: O(n^5) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4], size: O(1) [0] 676.38/291.65 f_5: runtime: O(n^5) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5], size: O(1) [0] 676.38/291.65 g_6: runtime: O(n^6) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5], size: O(1) [0] 676.38/291.65 f_6: runtime: O(n^6) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6], size: O(1) [0] 676.38/291.65 g_7: runtime: O(n^7) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6], size: O(1) [0] 676.38/291.65 f_7: runtime: O(n^7) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7], size: O(1) [0] 676.38/291.65 g_8: runtime: O(n^8) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7], size: O(1) [0] 676.38/291.65 f_8: runtime: O(n^8) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8], size: O(1) [0] 676.38/291.65 g_9: runtime: O(n^9) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7 + 2*z*z'^8], size: O(1) [0] 676.38/291.65 f_9: runtime: O(n^9) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8 + 2*z^9], size: O(1) [0] 676.38/291.65 g_10: runtime: O(n^10) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7 + 2*z*z'^8 + 2*z*z'^9], size: O(1) [0] 676.38/291.65 676.38/291.65 ---------------------------------------- 676.38/291.65 676.38/291.65 (137) ResultPropagationProof (UPPER BOUND(ID)) 676.38/291.65 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 676.38/291.65 ---------------------------------------- 676.38/291.65 676.38/291.65 (138) 676.38/291.65 Obligation: 676.38/291.65 Complexity RNTS consisting of the following rules: 676.38/291.65 676.38/291.65 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.65 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.65 f_10(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8 + 2*z^9 + 2*z^10 }-> s25 :|: s25 >= 0, s25 <= 0, z >= 0 676.38/291.65 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.65 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.38/291.65 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.38/291.65 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.38/291.65 f_6(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 }-> s13 :|: s13 >= 0, s13 <= 0, z >= 0 676.38/291.65 f_7(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 }-> s16 :|: s16 >= 0, s16 <= 0, z >= 0 676.38/291.65 f_8(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8 }-> s19 :|: s19 >= 0, s19 <= 0, z >= 0 676.38/291.65 f_9(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8 + 2*z^9 }-> s22 :|: s22 >= 0, s22 <= 0, z >= 0 676.38/291.65 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_10(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7 + 2*z*z'^8 + 2*z*z'^9 }-> 1 + s24 + s26 :|: s26 >= 0, s26 <= 0, s24 >= 0, s24 <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_3(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 }-> 1 + s3 + s5 :|: s5 >= 0, s5 <= 0, s3 >= 0, s3 <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_4(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 }-> 1 + s6 + s8 :|: s8 >= 0, s8 <= 0, s6 >= 0, s6 <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_5(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 }-> 1 + s9 + s11 :|: s11 >= 0, s11 <= 0, s9 >= 0, s9 <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_6(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 }-> 1 + s12 + s14 :|: s14 >= 0, s14 <= 0, s12 >= 0, s12 <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_7(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 }-> 1 + s15 + s17 :|: s17 >= 0, s17 <= 0, s15 >= 0, s15 <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_8(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7 }-> 1 + s18 + s20 :|: s20 >= 0, s20 <= 0, s18 >= 0, s18 <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_9(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7 + 2*z*z'^8 }-> 1 + s21 + s23 :|: s23 >= 0, s23 <= 0, s21 >= 0, s21 <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 676.38/291.65 Function symbols to be analyzed: {f_10} 676.38/291.65 Previous analysis results are: 676.38/291.65 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.65 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.65 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.65 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.65 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.38/291.65 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.38/291.65 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.38/291.65 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.38/291.65 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.38/291.65 g_5: runtime: O(n^5) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4], size: O(1) [0] 676.38/291.65 f_5: runtime: O(n^5) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5], size: O(1) [0] 676.38/291.65 g_6: runtime: O(n^6) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5], size: O(1) [0] 676.38/291.65 f_6: runtime: O(n^6) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6], size: O(1) [0] 676.38/291.65 g_7: runtime: O(n^7) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6], size: O(1) [0] 676.38/291.65 f_7: runtime: O(n^7) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7], size: O(1) [0] 676.38/291.65 g_8: runtime: O(n^8) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7], size: O(1) [0] 676.38/291.65 f_8: runtime: O(n^8) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8], size: O(1) [0] 676.38/291.65 g_9: runtime: O(n^9) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7 + 2*z*z'^8], size: O(1) [0] 676.38/291.65 f_9: runtime: O(n^9) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8 + 2*z^9], size: O(1) [0] 676.38/291.65 g_10: runtime: O(n^10) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7 + 2*z*z'^8 + 2*z*z'^9], size: O(1) [0] 676.38/291.65 676.38/291.65 ---------------------------------------- 676.38/291.65 676.38/291.65 (139) IntTrsBoundProof (UPPER BOUND(ID)) 676.38/291.65 676.38/291.65 Computed SIZE bound using CoFloCo for: f_10 676.38/291.65 after applying outer abstraction to obtain an ITS, 676.38/291.65 resulting in: O(1) with polynomial bound: 0 676.38/291.65 676.38/291.65 ---------------------------------------- 676.38/291.65 676.38/291.65 (140) 676.38/291.65 Obligation: 676.38/291.65 Complexity RNTS consisting of the following rules: 676.38/291.65 676.38/291.65 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.65 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.65 f_10(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8 + 2*z^9 + 2*z^10 }-> s25 :|: s25 >= 0, s25 <= 0, z >= 0 676.38/291.65 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.65 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.38/291.65 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.38/291.65 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.38/291.65 f_6(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 }-> s13 :|: s13 >= 0, s13 <= 0, z >= 0 676.38/291.65 f_7(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 }-> s16 :|: s16 >= 0, s16 <= 0, z >= 0 676.38/291.65 f_8(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8 }-> s19 :|: s19 >= 0, s19 <= 0, z >= 0 676.38/291.65 f_9(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8 + 2*z^9 }-> s22 :|: s22 >= 0, s22 <= 0, z >= 0 676.38/291.65 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_10(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7 + 2*z*z'^8 + 2*z*z'^9 }-> 1 + s24 + s26 :|: s26 >= 0, s26 <= 0, s24 >= 0, s24 <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_3(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 }-> 1 + s3 + s5 :|: s5 >= 0, s5 <= 0, s3 >= 0, s3 <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_4(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 }-> 1 + s6 + s8 :|: s8 >= 0, s8 <= 0, s6 >= 0, s6 <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_5(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 }-> 1 + s9 + s11 :|: s11 >= 0, s11 <= 0, s9 >= 0, s9 <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_6(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 }-> 1 + s12 + s14 :|: s14 >= 0, s14 <= 0, s12 >= 0, s12 <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_7(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 }-> 1 + s15 + s17 :|: s17 >= 0, s17 <= 0, s15 >= 0, s15 <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_8(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7 }-> 1 + s18 + s20 :|: s20 >= 0, s20 <= 0, s18 >= 0, s18 <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_9(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7 + 2*z*z'^8 }-> 1 + s21 + s23 :|: s23 >= 0, s23 <= 0, s21 >= 0, s21 <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 676.38/291.65 Function symbols to be analyzed: {f_10} 676.38/291.65 Previous analysis results are: 676.38/291.65 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.65 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.65 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.65 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.65 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.38/291.65 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.38/291.65 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.38/291.65 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.38/291.65 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.38/291.65 g_5: runtime: O(n^5) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4], size: O(1) [0] 676.38/291.65 f_5: runtime: O(n^5) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5], size: O(1) [0] 676.38/291.65 g_6: runtime: O(n^6) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5], size: O(1) [0] 676.38/291.65 f_6: runtime: O(n^6) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6], size: O(1) [0] 676.38/291.65 g_7: runtime: O(n^7) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6], size: O(1) [0] 676.38/291.65 f_7: runtime: O(n^7) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7], size: O(1) [0] 676.38/291.65 g_8: runtime: O(n^8) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7], size: O(1) [0] 676.38/291.65 f_8: runtime: O(n^8) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8], size: O(1) [0] 676.38/291.65 g_9: runtime: O(n^9) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7 + 2*z*z'^8], size: O(1) [0] 676.38/291.65 f_9: runtime: O(n^9) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8 + 2*z^9], size: O(1) [0] 676.38/291.65 g_10: runtime: O(n^10) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7 + 2*z*z'^8 + 2*z*z'^9], size: O(1) [0] 676.38/291.65 f_10: runtime: ?, size: O(1) [0] 676.38/291.65 676.38/291.65 ---------------------------------------- 676.38/291.65 676.38/291.65 (141) IntTrsBoundProof (UPPER BOUND(ID)) 676.38/291.65 676.38/291.65 Computed RUNTIME bound using KoAT for: f_10 676.38/291.65 after applying outer abstraction to obtain an ITS, 676.38/291.65 resulting in: O(n^10) with polynomial bound: 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8 + 2*z^9 + 2*z^10 676.38/291.65 676.38/291.65 ---------------------------------------- 676.38/291.65 676.38/291.65 (142) 676.38/291.65 Obligation: 676.38/291.65 Complexity RNTS consisting of the following rules: 676.38/291.65 676.38/291.65 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.38/291.65 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.38/291.65 f_10(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8 + 2*z^9 + 2*z^10 }-> s25 :|: s25 >= 0, s25 <= 0, z >= 0 676.38/291.65 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.38/291.65 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.38/291.65 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.38/291.65 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.38/291.65 f_6(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 }-> s13 :|: s13 >= 0, s13 <= 0, z >= 0 676.38/291.65 f_7(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 }-> s16 :|: s16 >= 0, s16 <= 0, z >= 0 676.38/291.65 f_8(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8 }-> s19 :|: s19 >= 0, s19 <= 0, z >= 0 676.38/291.65 f_9(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8 + 2*z^9 }-> s22 :|: s22 >= 0, s22 <= 0, z >= 0 676.38/291.65 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_10(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7 + 2*z*z'^8 + 2*z*z'^9 }-> 1 + s24 + s26 :|: s26 >= 0, s26 <= 0, s24 >= 0, s24 <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_3(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 }-> 1 + s3 + s5 :|: s5 >= 0, s5 <= 0, s3 >= 0, s3 <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_4(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 }-> 1 + s6 + s8 :|: s8 >= 0, s8 <= 0, s6 >= 0, s6 <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_5(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 }-> 1 + s9 + s11 :|: s11 >= 0, s11 <= 0, s9 >= 0, s9 <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_6(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 }-> 1 + s12 + s14 :|: s14 >= 0, s14 <= 0, s12 >= 0, s12 <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_7(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 }-> 1 + s15 + s17 :|: s17 >= 0, s17 <= 0, s15 >= 0, s15 <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_8(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7 }-> 1 + s18 + s20 :|: s20 >= 0, s20 <= 0, s18 >= 0, s18 <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 g_9(z, z') -{ 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7 + 2*z*z'^8 }-> 1 + s21 + s23 :|: s23 >= 0, s23 <= 0, s21 >= 0, s21 <= 0, z - 1 >= 0, z' >= 0 676.38/291.65 676.38/291.65 Function symbols to be analyzed: 676.38/291.65 Previous analysis results are: 676.38/291.65 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.38/291.65 f_0: runtime: O(1) [1], size: O(1) [0] 676.38/291.65 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.38/291.65 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.38/291.65 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.38/291.65 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.38/291.65 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.38/291.65 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.38/291.65 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.38/291.65 g_5: runtime: O(n^5) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4], size: O(1) [0] 676.38/291.65 f_5: runtime: O(n^5) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5], size: O(1) [0] 676.38/291.65 g_6: runtime: O(n^6) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5], size: O(1) [0] 676.38/291.65 f_6: runtime: O(n^6) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6], size: O(1) [0] 676.38/291.65 g_7: runtime: O(n^7) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6], size: O(1) [0] 676.38/291.65 f_7: runtime: O(n^7) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7], size: O(1) [0] 676.38/291.65 g_8: runtime: O(n^8) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7], size: O(1) [0] 676.38/291.65 f_8: runtime: O(n^8) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8], size: O(1) [0] 676.38/291.65 g_9: runtime: O(n^9) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7 + 2*z*z'^8], size: O(1) [0] 676.38/291.65 f_9: runtime: O(n^9) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8 + 2*z^9], size: O(1) [0] 676.38/291.65 g_10: runtime: O(n^10) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 + 2*z*z'^5 + 2*z*z'^6 + 2*z*z'^7 + 2*z*z'^8 + 2*z*z'^9], size: O(1) [0] 676.38/291.65 f_10: runtime: O(n^10) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 + 2*z^6 + 2*z^7 + 2*z^8 + 2*z^9 + 2*z^10], size: O(1) [0] 676.38/291.65 676.38/291.65 ---------------------------------------- 676.38/291.65 676.38/291.65 (143) FinalProof (FINISHED) 676.38/291.65 Computed overall runtime complexity 676.38/291.65 ---------------------------------------- 676.38/291.65 676.38/291.65 (144) 676.38/291.65 BOUNDS(1, n^10) 676.38/291.65 676.38/291.65 ---------------------------------------- 676.38/291.65 676.38/291.65 (145) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 676.38/291.65 Transformed a relative TRS into a decreasing-loop problem. 676.38/291.65 ---------------------------------------- 676.38/291.65 676.38/291.65 (146) 676.38/291.65 Obligation: 676.38/291.65 Analyzing the following TRS for decreasing loops: 676.38/291.65 676.38/291.65 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^10). 676.38/291.65 676.38/291.65 676.38/291.65 The TRS R consists of the following rules: 676.38/291.65 676.38/291.65 f_0(x) -> a 676.38/291.65 f_1(x) -> g_1(x, x) 676.38/291.65 g_1(s(x), y) -> b(f_0(y), g_1(x, y)) 676.38/291.65 f_2(x) -> g_2(x, x) 676.38/291.65 g_2(s(x), y) -> b(f_1(y), g_2(x, y)) 676.38/291.65 f_3(x) -> g_3(x, x) 676.38/291.65 g_3(s(x), y) -> b(f_2(y), g_3(x, y)) 676.38/291.65 f_4(x) -> g_4(x, x) 676.38/291.65 g_4(s(x), y) -> b(f_3(y), g_4(x, y)) 676.38/291.65 f_5(x) -> g_5(x, x) 676.38/291.65 g_5(s(x), y) -> b(f_4(y), g_5(x, y)) 676.38/291.65 f_6(x) -> g_6(x, x) 676.38/291.65 g_6(s(x), y) -> b(f_5(y), g_6(x, y)) 676.38/291.65 f_7(x) -> g_7(x, x) 676.38/291.65 g_7(s(x), y) -> b(f_6(y), g_7(x, y)) 676.38/291.65 f_8(x) -> g_8(x, x) 676.38/291.65 g_8(s(x), y) -> b(f_7(y), g_8(x, y)) 676.38/291.65 f_9(x) -> g_9(x, x) 676.38/291.65 g_9(s(x), y) -> b(f_8(y), g_9(x, y)) 676.38/291.65 f_10(x) -> g_10(x, x) 676.38/291.65 g_10(s(x), y) -> b(f_9(y), g_10(x, y)) 676.38/291.65 676.38/291.65 S is empty. 676.38/291.65 Rewrite Strategy: INNERMOST 676.38/291.65 ---------------------------------------- 676.38/291.65 676.38/291.65 (147) DecreasingLoopProof (LOWER BOUND(ID)) 676.38/291.65 The following loop(s) give(s) rise to the lower bound Omega(n^1): 676.38/291.65 676.38/291.65 The rewrite sequence 676.38/291.65 676.38/291.65 g_2(s(x), y) ->^+ b(f_1(y), g_2(x, y)) 676.38/291.65 676.38/291.65 gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. 676.38/291.65 676.38/291.65 The pumping substitution is [x / s(x)]. 676.38/291.65 676.38/291.65 The result substitution is [ ]. 676.38/291.65 676.38/291.65 676.38/291.65 676.38/291.65 676.38/291.65 ---------------------------------------- 676.38/291.65 676.38/291.65 (148) 676.38/291.65 Complex Obligation (BEST) 676.38/291.65 676.38/291.65 ---------------------------------------- 676.38/291.65 676.38/291.65 (149) 676.38/291.65 Obligation: 676.38/291.65 Proved the lower bound n^1 for the following obligation: 676.38/291.65 676.38/291.65 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^10). 676.38/291.65 676.38/291.65 676.38/291.65 The TRS R consists of the following rules: 676.38/291.65 676.38/291.65 f_0(x) -> a 676.38/291.65 f_1(x) -> g_1(x, x) 676.38/291.65 g_1(s(x), y) -> b(f_0(y), g_1(x, y)) 676.38/291.65 f_2(x) -> g_2(x, x) 676.38/291.65 g_2(s(x), y) -> b(f_1(y), g_2(x, y)) 676.38/291.65 f_3(x) -> g_3(x, x) 676.38/291.65 g_3(s(x), y) -> b(f_2(y), g_3(x, y)) 676.38/291.65 f_4(x) -> g_4(x, x) 676.38/291.65 g_4(s(x), y) -> b(f_3(y), g_4(x, y)) 676.38/291.65 f_5(x) -> g_5(x, x) 676.38/291.65 g_5(s(x), y) -> b(f_4(y), g_5(x, y)) 676.38/291.65 f_6(x) -> g_6(x, x) 676.38/291.65 g_6(s(x), y) -> b(f_5(y), g_6(x, y)) 676.38/291.65 f_7(x) -> g_7(x, x) 676.38/291.65 g_7(s(x), y) -> b(f_6(y), g_7(x, y)) 676.38/291.65 f_8(x) -> g_8(x, x) 676.38/291.65 g_8(s(x), y) -> b(f_7(y), g_8(x, y)) 676.38/291.65 f_9(x) -> g_9(x, x) 676.38/291.65 g_9(s(x), y) -> b(f_8(y), g_9(x, y)) 676.38/291.65 f_10(x) -> g_10(x, x) 676.38/291.65 g_10(s(x), y) -> b(f_9(y), g_10(x, y)) 676.38/291.65 676.38/291.65 S is empty. 676.38/291.65 Rewrite Strategy: INNERMOST 676.38/291.65 ---------------------------------------- 676.38/291.65 676.38/291.65 (150) LowerBoundPropagationProof (FINISHED) 676.38/291.65 Propagated lower bound. 676.38/291.65 ---------------------------------------- 676.38/291.65 676.38/291.65 (151) 676.38/291.65 BOUNDS(n^1, INF) 676.38/291.65 676.38/291.65 ---------------------------------------- 676.38/291.65 676.38/291.65 (152) 676.38/291.65 Obligation: 676.38/291.65 Analyzing the following TRS for decreasing loops: 676.38/291.65 676.38/291.65 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^10). 676.38/291.65 676.38/291.65 676.38/291.65 The TRS R consists of the following rules: 676.38/291.65 676.38/291.65 f_0(x) -> a 676.38/291.65 f_1(x) -> g_1(x, x) 676.38/291.65 g_1(s(x), y) -> b(f_0(y), g_1(x, y)) 676.38/291.65 f_2(x) -> g_2(x, x) 676.38/291.65 g_2(s(x), y) -> b(f_1(y), g_2(x, y)) 676.38/291.65 f_3(x) -> g_3(x, x) 676.38/291.65 g_3(s(x), y) -> b(f_2(y), g_3(x, y)) 676.38/291.65 f_4(x) -> g_4(x, x) 676.38/291.65 g_4(s(x), y) -> b(f_3(y), g_4(x, y)) 676.38/291.65 f_5(x) -> g_5(x, x) 676.38/291.65 g_5(s(x), y) -> b(f_4(y), g_5(x, y)) 676.38/291.65 f_6(x) -> g_6(x, x) 676.38/291.65 g_6(s(x), y) -> b(f_5(y), g_6(x, y)) 676.38/291.65 f_7(x) -> g_7(x, x) 676.38/291.65 g_7(s(x), y) -> b(f_6(y), g_7(x, y)) 676.38/291.65 f_8(x) -> g_8(x, x) 676.38/291.65 g_8(s(x), y) -> b(f_7(y), g_8(x, y)) 676.38/291.65 f_9(x) -> g_9(x, x) 676.38/291.65 g_9(s(x), y) -> b(f_8(y), g_9(x, y)) 676.38/291.65 f_10(x) -> g_10(x, x) 676.38/291.65 g_10(s(x), y) -> b(f_9(y), g_10(x, y)) 676.38/291.65 676.38/291.65 S is empty. 676.38/291.65 Rewrite Strategy: INNERMOST 676.38/291.70 EOF