676.39/291.54 WORST_CASE(Omega(n^1), O(n^10)) 676.39/291.56 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 676.39/291.56 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 676.39/291.56 676.39/291.56 676.39/291.56 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^10). 676.39/291.56 676.39/291.56 (0) CpxTRS 676.39/291.56 (1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms] 676.39/291.56 (2) CpxTRS 676.39/291.56 (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] 676.39/291.56 (4) CpxWeightedTrs 676.39/291.56 (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 676.39/291.56 (6) CpxTypedWeightedTrs 676.39/291.56 (7) CompletionProof [UPPER BOUND(ID), 0 ms] 676.39/291.56 (8) CpxTypedWeightedCompleteTrs 676.39/291.56 (9) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] 676.39/291.56 (10) CpxTypedWeightedCompleteTrs 676.39/291.56 (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] 676.39/291.56 (12) CpxRNTS 676.39/291.56 (13) InliningProof [UPPER BOUND(ID), 51 ms] 676.39/291.56 (14) CpxRNTS 676.39/291.56 (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] 676.39/291.56 (16) CpxRNTS 676.39/291.56 (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] 676.39/291.56 (18) CpxRNTS 676.39/291.56 (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 676.39/291.56 (20) CpxRNTS 676.39/291.56 (21) IntTrsBoundProof [UPPER BOUND(ID), 188 ms] 676.39/291.56 (22) CpxRNTS 676.39/291.56 (23) IntTrsBoundProof [UPPER BOUND(ID), 74 ms] 676.39/291.56 (24) CpxRNTS 676.39/291.56 (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 676.39/291.56 (26) CpxRNTS 676.39/291.56 (27) IntTrsBoundProof [UPPER BOUND(ID), 66 ms] 676.39/291.56 (28) CpxRNTS 676.39/291.56 (29) IntTrsBoundProof [UPPER BOUND(ID), 2 ms] 676.39/291.56 (30) CpxRNTS 676.39/291.56 (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 676.39/291.56 (32) CpxRNTS 676.39/291.56 (33) IntTrsBoundProof [UPPER BOUND(ID), 68 ms] 676.39/291.56 (34) CpxRNTS 676.39/291.56 (35) IntTrsBoundProof [UPPER BOUND(ID), 22 ms] 676.39/291.56 (36) CpxRNTS 676.39/291.56 (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 676.39/291.56 (38) CpxRNTS 676.39/291.56 (39) IntTrsBoundProof [UPPER BOUND(ID), 187 ms] 676.39/291.56 (40) CpxRNTS 676.39/291.56 (41) IntTrsBoundProof [UPPER BOUND(ID), 93 ms] 676.39/291.56 (42) CpxRNTS 676.39/291.56 (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 676.39/291.56 (44) CpxRNTS 676.39/291.56 (45) IntTrsBoundProof [UPPER BOUND(ID), 46 ms] 676.39/291.56 (46) CpxRNTS 676.39/291.56 (47) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] 676.39/291.56 (48) CpxRNTS 676.39/291.56 (49) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 676.39/291.56 (50) CpxRNTS 676.39/291.56 (51) IntTrsBoundProof [UPPER BOUND(ID), 209 ms] 676.39/291.56 (52) CpxRNTS 676.39/291.56 (53) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] 676.39/291.56 (54) CpxRNTS 676.39/291.56 (55) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 676.39/291.56 (56) CpxRNTS 676.39/291.56 (57) IntTrsBoundProof [UPPER BOUND(ID), 136 ms] 676.39/291.56 (58) CpxRNTS 676.39/291.56 (59) IntTrsBoundProof [UPPER BOUND(ID), 2 ms] 676.39/291.56 (60) CpxRNTS 676.39/291.56 (61) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 676.39/291.56 (62) CpxRNTS 676.39/291.56 (63) IntTrsBoundProof [UPPER BOUND(ID), 199 ms] 676.39/291.56 (64) CpxRNTS 676.39/291.56 (65) IntTrsBoundProof [UPPER BOUND(ID), 32 ms] 676.39/291.56 (66) CpxRNTS 676.39/291.56 (67) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 676.39/291.56 (68) CpxRNTS 676.39/291.56 (69) IntTrsBoundProof [UPPER BOUND(ID), 104 ms] 676.39/291.56 (70) CpxRNTS 676.39/291.56 (71) IntTrsBoundProof [UPPER BOUND(ID), 2 ms] 676.39/291.56 (72) CpxRNTS 676.39/291.56 (73) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 676.39/291.56 (74) CpxRNTS 676.39/291.56 (75) IntTrsBoundProof [UPPER BOUND(ID), 196 ms] 676.39/291.56 (76) CpxRNTS 676.39/291.56 (77) IntTrsBoundProof [UPPER BOUND(ID), 2 ms] 676.39/291.56 (78) CpxRNTS 676.39/291.56 (79) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 676.39/291.56 (80) CpxRNTS 676.39/291.56 (81) IntTrsBoundProof [UPPER BOUND(ID), 57 ms] 676.39/291.56 (82) CpxRNTS 676.39/291.56 (83) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] 676.39/291.56 (84) CpxRNTS 676.39/291.56 (85) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 676.39/291.56 (86) CpxRNTS 676.39/291.56 (87) IntTrsBoundProof [UPPER BOUND(ID), 107 ms] 676.39/291.56 (88) CpxRNTS 676.39/291.56 (89) IntTrsBoundProof [UPPER BOUND(ID), 2 ms] 676.39/291.56 (90) CpxRNTS 676.39/291.56 (91) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 676.39/291.56 (92) CpxRNTS 676.39/291.56 (93) IntTrsBoundProof [UPPER BOUND(ID), 104 ms] 676.39/291.56 (94) CpxRNTS 676.39/291.56 (95) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] 676.39/291.56 (96) CpxRNTS 676.39/291.56 (97) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 676.39/291.56 (98) CpxRNTS 676.39/291.56 (99) IntTrsBoundProof [UPPER BOUND(ID), 118 ms] 676.39/291.56 (100) CpxRNTS 676.39/291.56 (101) IntTrsBoundProof [UPPER BOUND(ID), 22 ms] 676.39/291.56 (102) CpxRNTS 676.39/291.56 (103) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 676.39/291.56 (104) CpxRNTS 676.39/291.56 (105) IntTrsBoundProof [UPPER BOUND(ID), 155 ms] 676.39/291.56 (106) CpxRNTS 676.39/291.56 (107) IntTrsBoundProof [UPPER BOUND(ID), 3 ms] 676.39/291.56 (108) CpxRNTS 676.39/291.56 (109) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 676.39/291.56 (110) CpxRNTS 676.39/291.56 (111) IntTrsBoundProof [UPPER BOUND(ID), 206 ms] 676.39/291.56 (112) CpxRNTS 676.39/291.56 (113) IntTrsBoundProof [UPPER BOUND(ID), 22 ms] 676.39/291.56 (114) CpxRNTS 676.39/291.56 (115) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 676.39/291.56 (116) CpxRNTS 676.39/291.56 (117) IntTrsBoundProof [UPPER BOUND(ID), 116 ms] 676.39/291.56 (118) CpxRNTS 676.39/291.56 (119) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] 676.39/291.56 (120) CpxRNTS 676.39/291.56 (121) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 676.39/291.56 (122) CpxRNTS 676.39/291.56 (123) IntTrsBoundProof [UPPER BOUND(ID), 178 ms] 676.39/291.56 (124) CpxRNTS 676.39/291.56 (125) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] 676.39/291.56 (126) CpxRNTS 676.39/291.56 (127) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 676.39/291.56 (128) CpxRNTS 676.39/291.56 (129) IntTrsBoundProof [UPPER BOUND(ID), 96 ms] 676.39/291.56 (130) CpxRNTS 676.39/291.56 (131) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] 676.39/291.56 (132) CpxRNTS 676.39/291.56 (133) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 676.39/291.56 (134) CpxRNTS 676.39/291.56 (135) IntTrsBoundProof [UPPER BOUND(ID), 208 ms] 676.39/291.56 (136) CpxRNTS 676.39/291.56 (137) IntTrsBoundProof [UPPER BOUND(ID), 42 ms] 676.39/291.56 (138) CpxRNTS 676.39/291.56 (139) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 676.39/291.56 (140) CpxRNTS 676.39/291.56 (141) IntTrsBoundProof [UPPER BOUND(ID), 55 ms] 676.39/291.56 (142) CpxRNTS 676.39/291.56 (143) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] 676.39/291.56 (144) CpxRNTS 676.39/291.56 (145) FinalProof [FINISHED, 0 ms] 676.39/291.56 (146) BOUNDS(1, n^10) 676.39/291.56 (147) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 676.39/291.56 (148) CpxTRS 676.39/291.56 (149) SlicingProof [LOWER BOUND(ID), 0 ms] 676.39/291.56 (150) CpxTRS 676.39/291.56 (151) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 676.39/291.56 (152) typed CpxTrs 676.39/291.56 (153) OrderProof [LOWER BOUND(ID), 0 ms] 676.39/291.56 (154) typed CpxTrs 676.39/291.56 (155) RewriteLemmaProof [LOWER BOUND(ID), 420 ms] 676.39/291.56 (156) BEST 676.39/291.56 (157) proven lower bound 676.39/291.56 (158) LowerBoundPropagationProof [FINISHED, 0 ms] 676.39/291.56 (159) BOUNDS(n^1, INF) 676.39/291.56 (160) typed CpxTrs 676.39/291.56 676.39/291.56 676.39/291.56 ---------------------------------------- 676.39/291.56 676.39/291.56 (0) 676.39/291.56 Obligation: 676.39/291.56 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^10). 676.39/291.56 676.39/291.56 676.39/291.56 The TRS R consists of the following rules: 676.39/291.56 676.39/291.56 f_0(x) -> a 676.39/291.56 f_1(x) -> g_1(x, x) 676.39/291.56 g_1(s(x), y) -> b(f_0(y), g_1(x, y)) 676.39/291.56 f_2(x) -> g_2(x, x) 676.39/291.56 g_2(s(x), y) -> b(f_1(y), g_2(x, y)) 676.39/291.56 f_3(x) -> g_3(x, x) 676.39/291.56 g_3(s(x), y) -> b(f_2(y), g_3(x, y)) 676.39/291.56 f_4(x) -> g_4(x, x) 676.39/291.56 g_4(s(x), y) -> b(f_3(y), g_4(x, y)) 676.39/291.56 f_5(x) -> g_5(x, x) 676.39/291.56 g_5(s(x), y) -> b(f_4(y), g_5(x, y)) 676.39/291.56 f_6(x) -> g_6(x, x) 676.39/291.56 g_6(s(x), y) -> b(f_5(y), g_6(x, y)) 676.39/291.56 f_7(x) -> g_7(x, x) 676.39/291.56 g_7(s(x), y) -> b(f_6(y), g_7(x, y)) 676.39/291.56 f_8(x) -> g_8(x, x) 676.39/291.56 g_8(s(x), y) -> b(f_7(y), g_8(x, y)) 676.39/291.56 f_9(x) -> g_9(x, x) 676.39/291.56 g_9(s(x), y) -> b(f_8(y), g_9(x, y)) 676.39/291.56 f_10(x) -> g_10(x, x) 676.39/291.56 g_10(s(x), y) -> b(f_9(y), g_10(x, y)) 676.39/291.56 676.39/291.56 S is empty. 676.39/291.56 Rewrite Strategy: FULL 676.39/291.56 ---------------------------------------- 676.39/291.56 676.39/291.56 (1) RcToIrcProof (BOTH BOUNDS(ID, ID)) 676.39/291.56 Converted rc-obligation to irc-obligation. 676.39/291.56 676.39/291.56 As the TRS does not nest defined symbols, we have rc = irc. 676.39/291.56 ---------------------------------------- 676.39/291.56 676.39/291.56 (2) 676.39/291.56 Obligation: 676.39/291.56 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^10). 676.39/291.56 676.39/291.56 676.39/291.56 The TRS R consists of the following rules: 676.39/291.56 676.39/291.56 f_0(x) -> a 676.39/291.56 f_1(x) -> g_1(x, x) 676.39/291.56 g_1(s(x), y) -> b(f_0(y), g_1(x, y)) 676.39/291.56 f_2(x) -> g_2(x, x) 676.39/291.56 g_2(s(x), y) -> b(f_1(y), g_2(x, y)) 676.39/291.56 f_3(x) -> g_3(x, x) 676.39/291.56 g_3(s(x), y) -> b(f_2(y), g_3(x, y)) 676.39/291.56 f_4(x) -> g_4(x, x) 676.39/291.56 g_4(s(x), y) -> b(f_3(y), g_4(x, y)) 676.39/291.56 f_5(x) -> g_5(x, x) 676.39/291.56 g_5(s(x), y) -> b(f_4(y), g_5(x, y)) 676.39/291.56 f_6(x) -> g_6(x, x) 676.39/291.56 g_6(s(x), y) -> b(f_5(y), g_6(x, y)) 676.39/291.56 f_7(x) -> g_7(x, x) 676.39/291.56 g_7(s(x), y) -> b(f_6(y), g_7(x, y)) 676.39/291.56 f_8(x) -> g_8(x, x) 676.39/291.56 g_8(s(x), y) -> b(f_7(y), g_8(x, y)) 676.39/291.56 f_9(x) -> g_9(x, x) 676.39/291.56 g_9(s(x), y) -> b(f_8(y), g_9(x, y)) 676.39/291.56 f_10(x) -> g_10(x, x) 676.39/291.56 g_10(s(x), y) -> b(f_9(y), g_10(x, y)) 676.39/291.56 676.39/291.56 S is empty. 676.39/291.56 Rewrite Strategy: INNERMOST 676.39/291.56 ---------------------------------------- 676.39/291.56 676.39/291.56 (3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) 676.39/291.56 Transformed relative TRS to weighted TRS 676.39/291.56 ---------------------------------------- 676.39/291.56 676.39/291.56 (4) 676.39/291.56 Obligation: 676.39/291.56 The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^10). 676.39/291.59 676.39/291.59 676.39/291.59 The TRS R consists of the following rules: 676.39/291.59 676.39/291.59 f_0(x) -> a [1] 676.39/291.59 f_1(x) -> g_1(x, x) [1] 676.39/291.59 g_1(s(x), y) -> b(f_0(y), g_1(x, y)) [1] 676.39/291.59 f_2(x) -> g_2(x, x) [1] 676.39/291.59 g_2(s(x), y) -> b(f_1(y), g_2(x, y)) [1] 676.39/291.59 f_3(x) -> g_3(x, x) [1] 676.39/291.59 g_3(s(x), y) -> b(f_2(y), g_3(x, y)) [1] 676.39/291.59 f_4(x) -> g_4(x, x) [1] 676.39/291.59 g_4(s(x), y) -> b(f_3(y), g_4(x, y)) [1] 676.39/291.59 f_5(x) -> g_5(x, x) [1] 676.39/291.59 g_5(s(x), y) -> b(f_4(y), g_5(x, y)) [1] 676.39/291.59 f_6(x) -> g_6(x, x) [1] 676.39/291.59 g_6(s(x), y) -> b(f_5(y), g_6(x, y)) [1] 676.39/291.59 f_7(x) -> g_7(x, x) [1] 676.39/291.59 g_7(s(x), y) -> b(f_6(y), g_7(x, y)) [1] 676.39/291.59 f_8(x) -> g_8(x, x) [1] 676.39/291.59 g_8(s(x), y) -> b(f_7(y), g_8(x, y)) [1] 676.39/291.59 f_9(x) -> g_9(x, x) [1] 676.39/291.59 g_9(s(x), y) -> b(f_8(y), g_9(x, y)) [1] 676.39/291.59 f_10(x) -> g_10(x, x) [1] 676.39/291.59 g_10(s(x), y) -> b(f_9(y), g_10(x, y)) [1] 676.39/291.59 676.39/291.59 Rewrite Strategy: INNERMOST 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (5) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 676.39/291.59 Infered types. 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (6) 676.39/291.59 Obligation: 676.39/291.59 Runtime Complexity Weighted TRS with Types. 676.39/291.59 The TRS R consists of the following rules: 676.39/291.59 676.39/291.59 f_0(x) -> a [1] 676.39/291.59 f_1(x) -> g_1(x, x) [1] 676.39/291.59 g_1(s(x), y) -> b(f_0(y), g_1(x, y)) [1] 676.39/291.59 f_2(x) -> g_2(x, x) [1] 676.39/291.59 g_2(s(x), y) -> b(f_1(y), g_2(x, y)) [1] 676.39/291.59 f_3(x) -> g_3(x, x) [1] 676.39/291.59 g_3(s(x), y) -> b(f_2(y), g_3(x, y)) [1] 676.39/291.59 f_4(x) -> g_4(x, x) [1] 676.39/291.59 g_4(s(x), y) -> b(f_3(y), g_4(x, y)) [1] 676.39/291.59 f_5(x) -> g_5(x, x) [1] 676.39/291.59 g_5(s(x), y) -> b(f_4(y), g_5(x, y)) [1] 676.39/291.59 f_6(x) -> g_6(x, x) [1] 676.39/291.59 g_6(s(x), y) -> b(f_5(y), g_6(x, y)) [1] 676.39/291.59 f_7(x) -> g_7(x, x) [1] 676.39/291.59 g_7(s(x), y) -> b(f_6(y), g_7(x, y)) [1] 676.39/291.59 f_8(x) -> g_8(x, x) [1] 676.39/291.59 g_8(s(x), y) -> b(f_7(y), g_8(x, y)) [1] 676.39/291.59 f_9(x) -> g_9(x, x) [1] 676.39/291.59 g_9(s(x), y) -> b(f_8(y), g_9(x, y)) [1] 676.39/291.59 f_10(x) -> g_10(x, x) [1] 676.39/291.59 g_10(s(x), y) -> b(f_9(y), g_10(x, y)) [1] 676.39/291.59 676.39/291.59 The TRS has the following type information: 676.39/291.59 f_0 :: s -> a:b 676.39/291.59 a :: a:b 676.39/291.59 f_1 :: s -> a:b 676.39/291.59 g_1 :: s -> s -> a:b 676.39/291.59 s :: s -> s 676.39/291.59 b :: a:b -> a:b -> a:b 676.39/291.59 f_2 :: s -> a:b 676.39/291.59 g_2 :: s -> s -> a:b 676.39/291.59 f_3 :: s -> a:b 676.39/291.59 g_3 :: s -> s -> a:b 676.39/291.59 f_4 :: s -> a:b 676.39/291.59 g_4 :: s -> s -> a:b 676.39/291.59 f_5 :: s -> a:b 676.39/291.59 g_5 :: s -> s -> a:b 676.39/291.59 f_6 :: s -> a:b 676.39/291.59 g_6 :: s -> s -> a:b 676.39/291.59 f_7 :: s -> a:b 676.39/291.59 g_7 :: s -> s -> a:b 676.39/291.59 f_8 :: s -> a:b 676.39/291.59 g_8 :: s -> s -> a:b 676.39/291.59 f_9 :: s -> a:b 676.39/291.59 g_9 :: s -> s -> a:b 676.39/291.59 f_10 :: s -> a:b 676.39/291.59 g_10 :: s -> s -> a:b 676.39/291.59 676.39/291.59 Rewrite Strategy: INNERMOST 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (7) CompletionProof (UPPER BOUND(ID)) 676.39/291.59 The transformation into a RNTS is sound, since: 676.39/291.59 676.39/291.59 (a) The obligation is a constructor system where every type has a constant constructor, 676.39/291.59 676.39/291.59 (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: 676.39/291.59 676.39/291.59 f_0_1 676.39/291.59 f_1_1 676.39/291.59 g_1_2 676.39/291.59 f_2_1 676.39/291.59 g_2_2 676.39/291.59 f_3_1 676.39/291.59 g_3_2 676.39/291.59 f_4_1 676.39/291.59 g_4_2 676.39/291.59 f_5_1 676.39/291.59 g_5_2 676.39/291.59 f_6_1 676.39/291.59 g_6_2 676.39/291.59 f_7_1 676.39/291.59 g_7_2 676.39/291.59 f_8_1 676.39/291.59 g_8_2 676.39/291.59 f_9_1 676.39/291.59 g_9_2 676.39/291.59 f_10_1 676.39/291.59 g_10_2 676.39/291.59 676.39/291.59 (c) The following functions are completely defined: 676.39/291.59 none 676.39/291.59 676.39/291.59 Due to the following rules being added: 676.39/291.59 none 676.39/291.59 676.39/291.59 And the following fresh constants: const 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (8) 676.39/291.59 Obligation: 676.39/291.59 Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: 676.39/291.59 676.39/291.59 Runtime Complexity Weighted TRS with Types. 676.39/291.59 The TRS R consists of the following rules: 676.39/291.59 676.39/291.59 f_0(x) -> a [1] 676.39/291.59 f_1(x) -> g_1(x, x) [1] 676.39/291.59 g_1(s(x), y) -> b(f_0(y), g_1(x, y)) [1] 676.39/291.59 f_2(x) -> g_2(x, x) [1] 676.39/291.59 g_2(s(x), y) -> b(f_1(y), g_2(x, y)) [1] 676.39/291.59 f_3(x) -> g_3(x, x) [1] 676.39/291.59 g_3(s(x), y) -> b(f_2(y), g_3(x, y)) [1] 676.39/291.59 f_4(x) -> g_4(x, x) [1] 676.39/291.59 g_4(s(x), y) -> b(f_3(y), g_4(x, y)) [1] 676.39/291.59 f_5(x) -> g_5(x, x) [1] 676.39/291.59 g_5(s(x), y) -> b(f_4(y), g_5(x, y)) [1] 676.39/291.59 f_6(x) -> g_6(x, x) [1] 676.39/291.59 g_6(s(x), y) -> b(f_5(y), g_6(x, y)) [1] 676.39/291.59 f_7(x) -> g_7(x, x) [1] 676.39/291.59 g_7(s(x), y) -> b(f_6(y), g_7(x, y)) [1] 676.39/291.59 f_8(x) -> g_8(x, x) [1] 676.39/291.59 g_8(s(x), y) -> b(f_7(y), g_8(x, y)) [1] 676.39/291.59 f_9(x) -> g_9(x, x) [1] 676.39/291.59 g_9(s(x), y) -> b(f_8(y), g_9(x, y)) [1] 676.39/291.59 f_10(x) -> g_10(x, x) [1] 676.39/291.59 g_10(s(x), y) -> b(f_9(y), g_10(x, y)) [1] 676.39/291.59 676.39/291.59 The TRS has the following type information: 676.39/291.59 f_0 :: s -> a:b 676.39/291.59 a :: a:b 676.39/291.59 f_1 :: s -> a:b 676.39/291.59 g_1 :: s -> s -> a:b 676.39/291.59 s :: s -> s 676.39/291.59 b :: a:b -> a:b -> a:b 676.39/291.59 f_2 :: s -> a:b 676.39/291.59 g_2 :: s -> s -> a:b 676.39/291.59 f_3 :: s -> a:b 676.39/291.59 g_3 :: s -> s -> a:b 676.39/291.59 f_4 :: s -> a:b 676.39/291.59 g_4 :: s -> s -> a:b 676.39/291.59 f_5 :: s -> a:b 676.39/291.59 g_5 :: s -> s -> a:b 676.39/291.59 f_6 :: s -> a:b 676.39/291.59 g_6 :: s -> s -> a:b 676.39/291.59 f_7 :: s -> a:b 676.39/291.59 g_7 :: s -> s -> a:b 676.39/291.59 f_8 :: s -> a:b 676.39/291.59 g_8 :: s -> s -> a:b 676.39/291.59 f_9 :: s -> a:b 676.39/291.59 g_9 :: s -> s -> a:b 676.39/291.59 f_10 :: s -> a:b 676.39/291.59 g_10 :: s -> s -> a:b 676.39/291.59 const :: s 676.39/291.59 676.39/291.59 Rewrite Strategy: INNERMOST 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (9) NarrowingProof (BOTH BOUNDS(ID, ID)) 676.39/291.59 Narrowed the inner basic terms of all right-hand sides by a single narrowing step. 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (10) 676.39/291.59 Obligation: 676.39/291.59 Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: 676.39/291.59 676.39/291.59 Runtime Complexity Weighted TRS with Types. 676.39/291.59 The TRS R consists of the following rules: 676.39/291.59 676.39/291.59 f_0(x) -> a [1] 676.39/291.59 f_1(x) -> g_1(x, x) [1] 676.39/291.59 g_1(s(x), y) -> b(f_0(y), g_1(x, y)) [1] 676.39/291.59 f_2(x) -> g_2(x, x) [1] 676.39/291.59 g_2(s(x), y) -> b(f_1(y), g_2(x, y)) [1] 676.39/291.59 f_3(x) -> g_3(x, x) [1] 676.39/291.59 g_3(s(x), y) -> b(f_2(y), g_3(x, y)) [1] 676.39/291.59 f_4(x) -> g_4(x, x) [1] 676.39/291.59 g_4(s(x), y) -> b(f_3(y), g_4(x, y)) [1] 676.39/291.59 f_5(x) -> g_5(x, x) [1] 676.39/291.59 g_5(s(x), y) -> b(f_4(y), g_5(x, y)) [1] 676.39/291.59 f_6(x) -> g_6(x, x) [1] 676.39/291.59 g_6(s(x), y) -> b(f_5(y), g_6(x, y)) [1] 676.39/291.59 f_7(x) -> g_7(x, x) [1] 676.39/291.59 g_7(s(x), y) -> b(f_6(y), g_7(x, y)) [1] 676.39/291.59 f_8(x) -> g_8(x, x) [1] 676.39/291.59 g_8(s(x), y) -> b(f_7(y), g_8(x, y)) [1] 676.39/291.59 f_9(x) -> g_9(x, x) [1] 676.39/291.59 g_9(s(x), y) -> b(f_8(y), g_9(x, y)) [1] 676.39/291.59 f_10(x) -> g_10(x, x) [1] 676.39/291.59 g_10(s(x), y) -> b(f_9(y), g_10(x, y)) [1] 676.39/291.59 676.39/291.59 The TRS has the following type information: 676.39/291.59 f_0 :: s -> a:b 676.39/291.59 a :: a:b 676.39/291.59 f_1 :: s -> a:b 676.39/291.59 g_1 :: s -> s -> a:b 676.39/291.59 s :: s -> s 676.39/291.59 b :: a:b -> a:b -> a:b 676.39/291.59 f_2 :: s -> a:b 676.39/291.59 g_2 :: s -> s -> a:b 676.39/291.59 f_3 :: s -> a:b 676.39/291.59 g_3 :: s -> s -> a:b 676.39/291.59 f_4 :: s -> a:b 676.39/291.59 g_4 :: s -> s -> a:b 676.39/291.59 f_5 :: s -> a:b 676.39/291.59 g_5 :: s -> s -> a:b 676.39/291.59 f_6 :: s -> a:b 676.39/291.59 g_6 :: s -> s -> a:b 676.39/291.59 f_7 :: s -> a:b 676.39/291.59 g_7 :: s -> s -> a:b 676.39/291.59 f_8 :: s -> a:b 676.39/291.59 g_8 :: s -> s -> a:b 676.39/291.59 f_9 :: s -> a:b 676.39/291.59 g_9 :: s -> s -> a:b 676.39/291.59 f_10 :: s -> a:b 676.39/291.59 g_10 :: s -> s -> a:b 676.39/291.59 const :: s 676.39/291.59 676.39/291.59 Rewrite Strategy: INNERMOST 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (11) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) 676.39/291.59 Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. 676.39/291.59 The constant constructors are abstracted as follows: 676.39/291.59 676.39/291.59 a => 0 676.39/291.59 const => 0 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (12) 676.39/291.59 Obligation: 676.39/291.59 Complexity RNTS consisting of the following rules: 676.39/291.59 676.39/291.59 f_0(z) -{ 1 }-> 0 :|: x >= 0, z = x 676.39/291.59 f_1(z) -{ 1 }-> g_1(x, x) :|: x >= 0, z = x 676.39/291.59 f_10(z) -{ 1 }-> g_10(x, x) :|: x >= 0, z = x 676.39/291.59 f_2(z) -{ 1 }-> g_2(x, x) :|: x >= 0, z = x 676.39/291.59 f_3(z) -{ 1 }-> g_3(x, x) :|: x >= 0, z = x 676.39/291.59 f_4(z) -{ 1 }-> g_4(x, x) :|: x >= 0, z = x 676.39/291.59 f_5(z) -{ 1 }-> g_5(x, x) :|: x >= 0, z = x 676.39/291.59 f_6(z) -{ 1 }-> g_6(x, x) :|: x >= 0, z = x 676.39/291.59 f_7(z) -{ 1 }-> g_7(x, x) :|: x >= 0, z = x 676.39/291.59 f_8(z) -{ 1 }-> g_8(x, x) :|: x >= 0, z = x 676.39/291.59 f_9(z) -{ 1 }-> g_9(x, x) :|: x >= 0, z = x 676.39/291.59 g_1(z, z') -{ 1 }-> 1 + f_0(y) + g_1(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y 676.39/291.59 g_10(z, z') -{ 1 }-> 1 + f_9(y) + g_10(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y 676.39/291.59 g_2(z, z') -{ 1 }-> 1 + f_1(y) + g_2(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y 676.39/291.59 g_3(z, z') -{ 1 }-> 1 + f_2(y) + g_3(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y 676.39/291.59 g_4(z, z') -{ 1 }-> 1 + f_3(y) + g_4(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y 676.39/291.59 g_5(z, z') -{ 1 }-> 1 + f_4(y) + g_5(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y 676.39/291.59 g_6(z, z') -{ 1 }-> 1 + f_5(y) + g_6(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y 676.39/291.59 g_7(z, z') -{ 1 }-> 1 + f_6(y) + g_7(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y 676.39/291.59 g_8(z, z') -{ 1 }-> 1 + f_7(y) + g_8(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y 676.39/291.59 g_9(z, z') -{ 1 }-> 1 + f_8(y) + g_9(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y 676.39/291.59 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (13) InliningProof (UPPER BOUND(ID)) 676.39/291.59 Inlined the following terminating rules on right-hand sides where appropriate: 676.39/291.59 676.39/291.59 f_0(z) -{ 1 }-> 0 :|: x >= 0, z = x 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (14) 676.39/291.59 Obligation: 676.39/291.59 Complexity RNTS consisting of the following rules: 676.39/291.59 676.39/291.59 f_0(z) -{ 1 }-> 0 :|: x >= 0, z = x 676.39/291.59 f_1(z) -{ 1 }-> g_1(x, x) :|: x >= 0, z = x 676.39/291.59 f_10(z) -{ 1 }-> g_10(x, x) :|: x >= 0, z = x 676.39/291.59 f_2(z) -{ 1 }-> g_2(x, x) :|: x >= 0, z = x 676.39/291.59 f_3(z) -{ 1 }-> g_3(x, x) :|: x >= 0, z = x 676.39/291.59 f_4(z) -{ 1 }-> g_4(x, x) :|: x >= 0, z = x 676.39/291.59 f_5(z) -{ 1 }-> g_5(x, x) :|: x >= 0, z = x 676.39/291.59 f_6(z) -{ 1 }-> g_6(x, x) :|: x >= 0, z = x 676.39/291.59 f_7(z) -{ 1 }-> g_7(x, x) :|: x >= 0, z = x 676.39/291.59 f_8(z) -{ 1 }-> g_8(x, x) :|: x >= 0, z = x 676.39/291.59 f_9(z) -{ 1 }-> g_9(x, x) :|: x >= 0, z = x 676.39/291.59 g_1(z, z') -{ 2 }-> 1 + 0 + g_1(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y, x' >= 0, y = x' 676.39/291.59 g_10(z, z') -{ 1 }-> 1 + f_9(y) + g_10(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y 676.39/291.59 g_2(z, z') -{ 1 }-> 1 + f_1(y) + g_2(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y 676.39/291.59 g_3(z, z') -{ 1 }-> 1 + f_2(y) + g_3(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y 676.39/291.59 g_4(z, z') -{ 1 }-> 1 + f_3(y) + g_4(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y 676.39/291.59 g_5(z, z') -{ 1 }-> 1 + f_4(y) + g_5(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y 676.39/291.59 g_6(z, z') -{ 1 }-> 1 + f_5(y) + g_6(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y 676.39/291.59 g_7(z, z') -{ 1 }-> 1 + f_6(y) + g_7(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y 676.39/291.59 g_8(z, z') -{ 1 }-> 1 + f_7(y) + g_8(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y 676.39/291.59 g_9(z, z') -{ 1 }-> 1 + f_8(y) + g_9(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y 676.39/291.59 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (15) SimplificationProof (BOTH BOUNDS(ID, ID)) 676.39/291.59 Simplified the RNTS by moving equalities from the constraints into the right-hand sides. 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (16) 676.39/291.59 Obligation: 676.39/291.59 Complexity RNTS consisting of the following rules: 676.39/291.59 676.39/291.59 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.59 f_1(z) -{ 1 }-> g_1(z, z) :|: z >= 0 676.39/291.59 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.59 f_2(z) -{ 1 }-> g_2(z, z) :|: z >= 0 676.39/291.59 f_3(z) -{ 1 }-> g_3(z, z) :|: z >= 0 676.39/291.59 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.39/291.59 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.39/291.59 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.39/291.59 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.39/291.59 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.59 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.59 g_1(z, z') -{ 2 }-> 1 + 0 + g_1(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_2(z, z') -{ 1 }-> 1 + f_1(z') + g_2(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_3(z, z') -{ 1 }-> 1 + f_2(z') + g_3(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_4(z, z') -{ 1 }-> 1 + f_3(z') + g_4(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) 676.39/291.59 Found the following analysis order by SCC decomposition: 676.39/291.59 676.39/291.59 { g_1 } 676.39/291.59 { f_0 } 676.39/291.59 { f_1 } 676.39/291.59 { g_2 } 676.39/291.59 { f_2 } 676.39/291.59 { g_3 } 676.39/291.59 { f_3 } 676.39/291.59 { g_4 } 676.39/291.59 { f_4 } 676.39/291.59 { g_5 } 676.39/291.59 { f_5 } 676.39/291.59 { g_6 } 676.39/291.59 { f_6 } 676.39/291.59 { g_7 } 676.39/291.59 { f_7 } 676.39/291.59 { g_8 } 676.39/291.59 { f_8 } 676.39/291.59 { g_9 } 676.39/291.59 { f_9 } 676.39/291.59 { g_10 } 676.39/291.59 { f_10 } 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (18) 676.39/291.59 Obligation: 676.39/291.59 Complexity RNTS consisting of the following rules: 676.39/291.59 676.39/291.59 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.59 f_1(z) -{ 1 }-> g_1(z, z) :|: z >= 0 676.39/291.59 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.59 f_2(z) -{ 1 }-> g_2(z, z) :|: z >= 0 676.39/291.59 f_3(z) -{ 1 }-> g_3(z, z) :|: z >= 0 676.39/291.59 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.39/291.59 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.39/291.59 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.39/291.59 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.39/291.59 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.59 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.59 g_1(z, z') -{ 2 }-> 1 + 0 + g_1(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_2(z, z') -{ 1 }-> 1 + f_1(z') + g_2(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_3(z, z') -{ 1 }-> 1 + f_2(z') + g_3(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_4(z, z') -{ 1 }-> 1 + f_3(z') + g_4(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 676.39/291.59 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} 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (19) ResultPropagationProof (UPPER BOUND(ID)) 676.39/291.59 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (20) 676.39/291.59 Obligation: 676.39/291.59 Complexity RNTS consisting of the following rules: 676.39/291.59 676.39/291.59 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.59 f_1(z) -{ 1 }-> g_1(z, z) :|: z >= 0 676.39/291.59 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.59 f_2(z) -{ 1 }-> g_2(z, z) :|: z >= 0 676.39/291.59 f_3(z) -{ 1 }-> g_3(z, z) :|: z >= 0 676.39/291.59 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.39/291.59 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.39/291.59 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.39/291.59 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.39/291.59 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.59 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.59 g_1(z, z') -{ 2 }-> 1 + 0 + g_1(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_2(z, z') -{ 1 }-> 1 + f_1(z') + g_2(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_3(z, z') -{ 1 }-> 1 + f_2(z') + g_3(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_4(z, z') -{ 1 }-> 1 + f_3(z') + g_4(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 676.39/291.59 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} 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (21) IntTrsBoundProof (UPPER BOUND(ID)) 676.39/291.59 676.39/291.59 Computed SIZE bound using CoFloCo for: g_1 676.39/291.59 after applying outer abstraction to obtain an ITS, 676.39/291.59 resulting in: O(1) with polynomial bound: 0 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (22) 676.39/291.59 Obligation: 676.39/291.59 Complexity RNTS consisting of the following rules: 676.39/291.59 676.39/291.59 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.59 f_1(z) -{ 1 }-> g_1(z, z) :|: z >= 0 676.39/291.59 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.59 f_2(z) -{ 1 }-> g_2(z, z) :|: z >= 0 676.39/291.59 f_3(z) -{ 1 }-> g_3(z, z) :|: z >= 0 676.39/291.59 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.39/291.59 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.39/291.59 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.39/291.59 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.39/291.59 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.59 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.59 g_1(z, z') -{ 2 }-> 1 + 0 + g_1(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_2(z, z') -{ 1 }-> 1 + f_1(z') + g_2(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_3(z, z') -{ 1 }-> 1 + f_2(z') + g_3(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_4(z, z') -{ 1 }-> 1 + f_3(z') + g_4(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 676.39/291.59 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} 676.39/291.59 Previous analysis results are: 676.39/291.59 g_1: runtime: ?, size: O(1) [0] 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (23) IntTrsBoundProof (UPPER BOUND(ID)) 676.39/291.59 676.39/291.59 Computed RUNTIME bound using CoFloCo for: g_1 676.39/291.59 after applying outer abstraction to obtain an ITS, 676.39/291.59 resulting in: O(n^1) with polynomial bound: 2*z 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (24) 676.39/291.59 Obligation: 676.39/291.59 Complexity RNTS consisting of the following rules: 676.39/291.59 676.39/291.59 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.59 f_1(z) -{ 1 }-> g_1(z, z) :|: z >= 0 676.39/291.59 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.59 f_2(z) -{ 1 }-> g_2(z, z) :|: z >= 0 676.39/291.59 f_3(z) -{ 1 }-> g_3(z, z) :|: z >= 0 676.39/291.59 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.39/291.59 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.39/291.59 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.39/291.59 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.39/291.59 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.59 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.59 g_1(z, z') -{ 2 }-> 1 + 0 + g_1(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_2(z, z') -{ 1 }-> 1 + f_1(z') + g_2(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_3(z, z') -{ 1 }-> 1 + f_2(z') + g_3(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_4(z, z') -{ 1 }-> 1 + f_3(z') + g_4(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 676.39/291.59 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} 676.39/291.59 Previous analysis results are: 676.39/291.59 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (25) ResultPropagationProof (UPPER BOUND(ID)) 676.39/291.59 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (26) 676.39/291.59 Obligation: 676.39/291.59 Complexity RNTS consisting of the following rules: 676.39/291.59 676.39/291.59 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.59 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.59 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.59 f_2(z) -{ 1 }-> g_2(z, z) :|: z >= 0 676.39/291.59 f_3(z) -{ 1 }-> g_3(z, z) :|: z >= 0 676.39/291.59 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.39/291.59 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.39/291.59 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.39/291.59 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.39/291.59 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.59 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.59 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_2(z, z') -{ 1 }-> 1 + f_1(z') + g_2(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_3(z, z') -{ 1 }-> 1 + f_2(z') + g_3(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_4(z, z') -{ 1 }-> 1 + f_3(z') + g_4(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 676.39/291.59 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} 676.39/291.59 Previous analysis results are: 676.39/291.59 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (27) IntTrsBoundProof (UPPER BOUND(ID)) 676.39/291.59 676.39/291.59 Computed SIZE bound using CoFloCo for: f_0 676.39/291.59 after applying outer abstraction to obtain an ITS, 676.39/291.59 resulting in: O(1) with polynomial bound: 0 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (28) 676.39/291.59 Obligation: 676.39/291.59 Complexity RNTS consisting of the following rules: 676.39/291.59 676.39/291.59 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.59 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.59 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.59 f_2(z) -{ 1 }-> g_2(z, z) :|: z >= 0 676.39/291.59 f_3(z) -{ 1 }-> g_3(z, z) :|: z >= 0 676.39/291.59 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.39/291.59 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.39/291.59 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.39/291.59 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.39/291.59 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.59 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.59 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_2(z, z') -{ 1 }-> 1 + f_1(z') + g_2(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_3(z, z') -{ 1 }-> 1 + f_2(z') + g_3(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_4(z, z') -{ 1 }-> 1 + f_3(z') + g_4(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 676.39/291.59 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} 676.39/291.59 Previous analysis results are: 676.39/291.59 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.59 f_0: runtime: ?, size: O(1) [0] 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (29) IntTrsBoundProof (UPPER BOUND(ID)) 676.39/291.59 676.39/291.59 Computed RUNTIME bound using CoFloCo for: f_0 676.39/291.59 after applying outer abstraction to obtain an ITS, 676.39/291.59 resulting in: O(1) with polynomial bound: 1 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (30) 676.39/291.59 Obligation: 676.39/291.59 Complexity RNTS consisting of the following rules: 676.39/291.59 676.39/291.59 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.59 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.59 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.59 f_2(z) -{ 1 }-> g_2(z, z) :|: z >= 0 676.39/291.59 f_3(z) -{ 1 }-> g_3(z, z) :|: z >= 0 676.39/291.59 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.39/291.59 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.39/291.59 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.39/291.59 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.39/291.59 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.59 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.59 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_2(z, z') -{ 1 }-> 1 + f_1(z') + g_2(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_3(z, z') -{ 1 }-> 1 + f_2(z') + g_3(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_4(z, z') -{ 1 }-> 1 + f_3(z') + g_4(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 676.39/291.59 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} 676.39/291.59 Previous analysis results are: 676.39/291.59 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.59 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (31) ResultPropagationProof (UPPER BOUND(ID)) 676.39/291.59 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (32) 676.39/291.59 Obligation: 676.39/291.59 Complexity RNTS consisting of the following rules: 676.39/291.59 676.39/291.59 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.59 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.59 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.59 f_2(z) -{ 1 }-> g_2(z, z) :|: z >= 0 676.39/291.59 f_3(z) -{ 1 }-> g_3(z, z) :|: z >= 0 676.39/291.59 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.39/291.59 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.39/291.59 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.39/291.59 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.39/291.59 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.59 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.59 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_2(z, z') -{ 1 }-> 1 + f_1(z') + g_2(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_3(z, z') -{ 1 }-> 1 + f_2(z') + g_3(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_4(z, z') -{ 1 }-> 1 + f_3(z') + g_4(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 676.39/291.59 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} 676.39/291.59 Previous analysis results are: 676.39/291.59 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.59 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (33) IntTrsBoundProof (UPPER BOUND(ID)) 676.39/291.59 676.39/291.59 Computed SIZE bound using CoFloCo for: f_1 676.39/291.59 after applying outer abstraction to obtain an ITS, 676.39/291.59 resulting in: O(1) with polynomial bound: 0 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (34) 676.39/291.59 Obligation: 676.39/291.59 Complexity RNTS consisting of the following rules: 676.39/291.59 676.39/291.59 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.59 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.59 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.59 f_2(z) -{ 1 }-> g_2(z, z) :|: z >= 0 676.39/291.59 f_3(z) -{ 1 }-> g_3(z, z) :|: z >= 0 676.39/291.59 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.39/291.59 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.39/291.59 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.39/291.59 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.39/291.59 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.59 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.59 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_2(z, z') -{ 1 }-> 1 + f_1(z') + g_2(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_3(z, z') -{ 1 }-> 1 + f_2(z') + g_3(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_4(z, z') -{ 1 }-> 1 + f_3(z') + g_4(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 676.39/291.59 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} 676.39/291.59 Previous analysis results are: 676.39/291.59 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.59 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.59 f_1: runtime: ?, size: O(1) [0] 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (35) IntTrsBoundProof (UPPER BOUND(ID)) 676.39/291.59 676.39/291.59 Computed RUNTIME bound using CoFloCo for: f_1 676.39/291.59 after applying outer abstraction to obtain an ITS, 676.39/291.59 resulting in: O(n^1) with polynomial bound: 1 + 2*z 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (36) 676.39/291.59 Obligation: 676.39/291.59 Complexity RNTS consisting of the following rules: 676.39/291.59 676.39/291.59 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.59 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.59 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.59 f_2(z) -{ 1 }-> g_2(z, z) :|: z >= 0 676.39/291.59 f_3(z) -{ 1 }-> g_3(z, z) :|: z >= 0 676.39/291.59 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.39/291.59 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.39/291.59 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.39/291.59 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.39/291.59 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.59 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.59 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_2(z, z') -{ 1 }-> 1 + f_1(z') + g_2(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_3(z, z') -{ 1 }-> 1 + f_2(z') + g_3(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_4(z, z') -{ 1 }-> 1 + f_3(z') + g_4(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 676.39/291.59 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} 676.39/291.59 Previous analysis results are: 676.39/291.59 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.59 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.59 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (37) ResultPropagationProof (UPPER BOUND(ID)) 676.39/291.59 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (38) 676.39/291.59 Obligation: 676.39/291.59 Complexity RNTS consisting of the following rules: 676.39/291.59 676.39/291.59 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.59 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.59 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.59 f_2(z) -{ 1 }-> g_2(z, z) :|: z >= 0 676.39/291.59 f_3(z) -{ 1 }-> g_3(z, z) :|: z >= 0 676.39/291.59 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.39/291.59 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.39/291.59 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.39/291.59 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.39/291.59 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.59 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.59 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_2(z, z') -{ 2 + 2*z' }-> 1 + s'' + g_2(z - 1, z') :|: s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 g_3(z, z') -{ 1 }-> 1 + f_2(z') + g_3(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_4(z, z') -{ 1 }-> 1 + f_3(z') + g_4(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 676.39/291.59 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} 676.39/291.59 Previous analysis results are: 676.39/291.59 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.59 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.59 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (39) IntTrsBoundProof (UPPER BOUND(ID)) 676.39/291.59 676.39/291.59 Computed SIZE bound using CoFloCo for: g_2 676.39/291.59 after applying outer abstraction to obtain an ITS, 676.39/291.59 resulting in: O(1) with polynomial bound: 0 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (40) 676.39/291.59 Obligation: 676.39/291.59 Complexity RNTS consisting of the following rules: 676.39/291.59 676.39/291.59 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.59 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.59 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.59 f_2(z) -{ 1 }-> g_2(z, z) :|: z >= 0 676.39/291.59 f_3(z) -{ 1 }-> g_3(z, z) :|: z >= 0 676.39/291.59 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.39/291.59 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.39/291.59 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.39/291.59 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.39/291.59 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.59 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.59 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_2(z, z') -{ 2 + 2*z' }-> 1 + s'' + g_2(z - 1, z') :|: s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 g_3(z, z') -{ 1 }-> 1 + f_2(z') + g_3(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_4(z, z') -{ 1 }-> 1 + f_3(z') + g_4(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 676.39/291.59 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} 676.39/291.59 Previous analysis results are: 676.39/291.59 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.59 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.59 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.59 g_2: runtime: ?, size: O(1) [0] 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (41) IntTrsBoundProof (UPPER BOUND(ID)) 676.39/291.59 676.39/291.59 Computed RUNTIME bound using CoFloCo for: g_2 676.39/291.59 after applying outer abstraction to obtain an ITS, 676.39/291.59 resulting in: O(n^2) with polynomial bound: 2*z + 2*z*z' 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (42) 676.39/291.59 Obligation: 676.39/291.59 Complexity RNTS consisting of the following rules: 676.39/291.59 676.39/291.59 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.59 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.59 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.59 f_2(z) -{ 1 }-> g_2(z, z) :|: z >= 0 676.39/291.59 f_3(z) -{ 1 }-> g_3(z, z) :|: z >= 0 676.39/291.59 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.39/291.59 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.39/291.59 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.39/291.59 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.39/291.59 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.59 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.59 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_2(z, z') -{ 2 + 2*z' }-> 1 + s'' + g_2(z - 1, z') :|: s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 g_3(z, z') -{ 1 }-> 1 + f_2(z') + g_3(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_4(z, z') -{ 1 }-> 1 + f_3(z') + g_4(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 676.39/291.59 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} 676.39/291.59 Previous analysis results are: 676.39/291.59 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.59 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.59 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.59 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (43) ResultPropagationProof (UPPER BOUND(ID)) 676.39/291.59 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (44) 676.39/291.59 Obligation: 676.39/291.59 Complexity RNTS consisting of the following rules: 676.39/291.59 676.39/291.59 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.59 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.59 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.59 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.59 f_3(z) -{ 1 }-> g_3(z, z) :|: z >= 0 676.39/291.59 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.39/291.59 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.39/291.59 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.39/291.59 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.39/291.59 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.59 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.59 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 g_3(z, z') -{ 1 }-> 1 + f_2(z') + g_3(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_4(z, z') -{ 1 }-> 1 + f_3(z') + g_4(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 676.39/291.59 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} 676.39/291.59 Previous analysis results are: 676.39/291.59 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.59 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.59 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.59 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (45) IntTrsBoundProof (UPPER BOUND(ID)) 676.39/291.59 676.39/291.59 Computed SIZE bound using CoFloCo for: f_2 676.39/291.59 after applying outer abstraction to obtain an ITS, 676.39/291.59 resulting in: O(1) with polynomial bound: 0 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (46) 676.39/291.59 Obligation: 676.39/291.59 Complexity RNTS consisting of the following rules: 676.39/291.59 676.39/291.59 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.59 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.59 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.59 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.59 f_3(z) -{ 1 }-> g_3(z, z) :|: z >= 0 676.39/291.59 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.39/291.59 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.39/291.59 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.39/291.59 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.39/291.59 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.59 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.59 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 g_3(z, z') -{ 1 }-> 1 + f_2(z') + g_3(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_4(z, z') -{ 1 }-> 1 + f_3(z') + g_4(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 676.39/291.59 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} 676.39/291.59 Previous analysis results are: 676.39/291.59 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.59 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.59 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.59 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.59 f_2: runtime: ?, size: O(1) [0] 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (47) IntTrsBoundProof (UPPER BOUND(ID)) 676.39/291.59 676.39/291.59 Computed RUNTIME bound using KoAT for: f_2 676.39/291.59 after applying outer abstraction to obtain an ITS, 676.39/291.59 resulting in: O(n^2) with polynomial bound: 1 + 2*z + 2*z^2 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (48) 676.39/291.59 Obligation: 676.39/291.59 Complexity RNTS consisting of the following rules: 676.39/291.59 676.39/291.59 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.59 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.59 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.59 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.59 f_3(z) -{ 1 }-> g_3(z, z) :|: z >= 0 676.39/291.59 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.39/291.59 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.39/291.59 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.39/291.59 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.39/291.59 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.59 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.59 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 g_3(z, z') -{ 1 }-> 1 + f_2(z') + g_3(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_4(z, z') -{ 1 }-> 1 + f_3(z') + g_4(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 676.39/291.59 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} 676.39/291.59 Previous analysis results are: 676.39/291.59 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.59 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.59 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.59 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.59 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (49) ResultPropagationProof (UPPER BOUND(ID)) 676.39/291.59 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (50) 676.39/291.59 Obligation: 676.39/291.59 Complexity RNTS consisting of the following rules: 676.39/291.59 676.39/291.59 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.59 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.59 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.59 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.59 f_3(z) -{ 1 }-> g_3(z, z) :|: z >= 0 676.39/291.59 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.39/291.59 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.39/291.59 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.39/291.59 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.39/291.59 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.59 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.59 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 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' >= 0 676.39/291.59 g_4(z, z') -{ 1 }-> 1 + f_3(z') + g_4(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 676.39/291.59 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} 676.39/291.59 Previous analysis results are: 676.39/291.59 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.59 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.59 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.59 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.59 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (51) IntTrsBoundProof (UPPER BOUND(ID)) 676.39/291.59 676.39/291.59 Computed SIZE bound using CoFloCo for: g_3 676.39/291.59 after applying outer abstraction to obtain an ITS, 676.39/291.59 resulting in: O(1) with polynomial bound: 0 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (52) 676.39/291.59 Obligation: 676.39/291.59 Complexity RNTS consisting of the following rules: 676.39/291.59 676.39/291.59 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.59 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.59 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.59 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.59 f_3(z) -{ 1 }-> g_3(z, z) :|: z >= 0 676.39/291.59 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.39/291.59 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.39/291.59 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.39/291.59 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.39/291.59 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.59 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.59 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 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' >= 0 676.39/291.59 g_4(z, z') -{ 1 }-> 1 + f_3(z') + g_4(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 676.39/291.59 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} 676.39/291.59 Previous analysis results are: 676.39/291.59 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.59 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.59 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.59 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.59 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.39/291.59 g_3: runtime: ?, size: O(1) [0] 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (53) IntTrsBoundProof (UPPER BOUND(ID)) 676.39/291.59 676.39/291.59 Computed RUNTIME bound using KoAT for: g_3 676.39/291.59 after applying outer abstraction to obtain an ITS, 676.39/291.59 resulting in: O(n^3) with polynomial bound: 2*z + 2*z*z' + 2*z*z'^2 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (54) 676.39/291.59 Obligation: 676.39/291.59 Complexity RNTS consisting of the following rules: 676.39/291.59 676.39/291.59 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.59 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.59 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.59 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.59 f_3(z) -{ 1 }-> g_3(z, z) :|: z >= 0 676.39/291.59 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.39/291.59 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.39/291.59 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.39/291.59 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.39/291.59 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.59 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.59 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 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' >= 0 676.39/291.59 g_4(z, z') -{ 1 }-> 1 + f_3(z') + g_4(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 676.39/291.59 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} 676.39/291.59 Previous analysis results are: 676.39/291.59 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.59 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.59 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.59 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.59 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.39/291.59 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (55) ResultPropagationProof (UPPER BOUND(ID)) 676.39/291.59 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (56) 676.39/291.59 Obligation: 676.39/291.59 Complexity RNTS consisting of the following rules: 676.39/291.59 676.39/291.59 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.59 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.59 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.59 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.59 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.39/291.59 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.39/291.59 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.39/291.59 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.39/291.59 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.39/291.59 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.59 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.59 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 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' >= 0 676.39/291.59 g_4(z, z') -{ 1 }-> 1 + f_3(z') + g_4(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 676.39/291.59 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} 676.39/291.59 Previous analysis results are: 676.39/291.59 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.59 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.59 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.59 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.59 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.39/291.59 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (57) IntTrsBoundProof (UPPER BOUND(ID)) 676.39/291.59 676.39/291.59 Computed SIZE bound using CoFloCo for: f_3 676.39/291.59 after applying outer abstraction to obtain an ITS, 676.39/291.59 resulting in: O(1) with polynomial bound: 0 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (58) 676.39/291.59 Obligation: 676.39/291.59 Complexity RNTS consisting of the following rules: 676.39/291.59 676.39/291.59 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.59 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.59 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.59 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.59 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.39/291.59 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.39/291.59 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.39/291.59 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.39/291.59 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.39/291.59 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.59 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.59 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 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' >= 0 676.39/291.59 g_4(z, z') -{ 1 }-> 1 + f_3(z') + g_4(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 676.39/291.59 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} 676.39/291.59 Previous analysis results are: 676.39/291.59 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.59 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.59 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.59 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.59 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.39/291.59 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.39/291.59 f_3: runtime: ?, size: O(1) [0] 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (59) IntTrsBoundProof (UPPER BOUND(ID)) 676.39/291.59 676.39/291.59 Computed RUNTIME bound using KoAT for: f_3 676.39/291.59 after applying outer abstraction to obtain an ITS, 676.39/291.59 resulting in: O(n^3) with polynomial bound: 1 + 2*z + 2*z^2 + 2*z^3 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (60) 676.39/291.59 Obligation: 676.39/291.59 Complexity RNTS consisting of the following rules: 676.39/291.59 676.39/291.59 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.59 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.59 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.59 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.59 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.39/291.59 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.39/291.59 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.39/291.59 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.39/291.59 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.39/291.59 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.59 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.59 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 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' >= 0 676.39/291.59 g_4(z, z') -{ 1 }-> 1 + f_3(z') + g_4(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 676.39/291.59 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} 676.39/291.59 Previous analysis results are: 676.39/291.59 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.59 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.59 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.59 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.59 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.39/291.59 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.39/291.59 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (61) ResultPropagationProof (UPPER BOUND(ID)) 676.39/291.59 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (62) 676.39/291.59 Obligation: 676.39/291.59 Complexity RNTS consisting of the following rules: 676.39/291.59 676.39/291.59 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.59 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.59 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.59 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.59 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.39/291.59 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.39/291.59 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.39/291.59 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.39/291.59 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.39/291.59 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.59 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.59 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 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' >= 0 676.39/291.59 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' >= 0 676.39/291.59 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 676.39/291.59 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} 676.39/291.59 Previous analysis results are: 676.39/291.59 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.59 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.59 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.59 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.59 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.39/291.59 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.39/291.59 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (63) IntTrsBoundProof (UPPER BOUND(ID)) 676.39/291.59 676.39/291.59 Computed SIZE bound using CoFloCo for: g_4 676.39/291.59 after applying outer abstraction to obtain an ITS, 676.39/291.59 resulting in: O(1) with polynomial bound: 0 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (64) 676.39/291.59 Obligation: 676.39/291.59 Complexity RNTS consisting of the following rules: 676.39/291.59 676.39/291.59 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.59 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.59 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.59 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.59 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.39/291.59 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.39/291.59 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.39/291.59 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.39/291.59 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.39/291.59 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.59 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.59 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 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' >= 0 676.39/291.59 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' >= 0 676.39/291.59 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 676.39/291.59 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} 676.39/291.59 Previous analysis results are: 676.39/291.59 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.59 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.59 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.59 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.59 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.39/291.59 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.39/291.59 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.39/291.59 g_4: runtime: ?, size: O(1) [0] 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (65) IntTrsBoundProof (UPPER BOUND(ID)) 676.39/291.59 676.39/291.59 Computed RUNTIME bound using KoAT for: g_4 676.39/291.59 after applying outer abstraction to obtain an ITS, 676.39/291.59 resulting in: O(n^4) with polynomial bound: 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (66) 676.39/291.59 Obligation: 676.39/291.59 Complexity RNTS consisting of the following rules: 676.39/291.59 676.39/291.59 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.59 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.59 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.59 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.59 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.39/291.59 f_4(z) -{ 1 }-> g_4(z, z) :|: z >= 0 676.39/291.59 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.39/291.59 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.39/291.59 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.39/291.59 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.59 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.59 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 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' >= 0 676.39/291.59 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' >= 0 676.39/291.59 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 676.39/291.59 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} 676.39/291.59 Previous analysis results are: 676.39/291.59 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.59 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.59 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.59 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.59 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.39/291.59 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.39/291.59 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.39/291.59 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (67) ResultPropagationProof (UPPER BOUND(ID)) 676.39/291.59 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (68) 676.39/291.59 Obligation: 676.39/291.59 Complexity RNTS consisting of the following rules: 676.39/291.59 676.39/291.59 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.59 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.59 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.59 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.59 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.39/291.59 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.39/291.59 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.39/291.59 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.39/291.59 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.39/291.59 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.59 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.59 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 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' >= 0 676.39/291.59 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' >= 0 676.39/291.59 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 676.39/291.59 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} 676.39/291.59 Previous analysis results are: 676.39/291.59 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.59 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.59 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.59 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.59 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.39/291.59 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.39/291.59 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.39/291.59 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (69) IntTrsBoundProof (UPPER BOUND(ID)) 676.39/291.59 676.39/291.59 Computed SIZE bound using CoFloCo for: f_4 676.39/291.59 after applying outer abstraction to obtain an ITS, 676.39/291.59 resulting in: O(1) with polynomial bound: 0 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (70) 676.39/291.59 Obligation: 676.39/291.59 Complexity RNTS consisting of the following rules: 676.39/291.59 676.39/291.59 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.59 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.59 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.59 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.59 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.39/291.59 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.39/291.59 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.39/291.59 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.39/291.59 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.39/291.59 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.59 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.59 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 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' >= 0 676.39/291.59 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' >= 0 676.39/291.59 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 676.39/291.59 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} 676.39/291.59 Previous analysis results are: 676.39/291.59 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.59 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.59 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.59 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.59 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.39/291.59 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.39/291.59 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.39/291.59 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.39/291.59 f_4: runtime: ?, size: O(1) [0] 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (71) IntTrsBoundProof (UPPER BOUND(ID)) 676.39/291.59 676.39/291.59 Computed RUNTIME bound using KoAT for: f_4 676.39/291.59 after applying outer abstraction to obtain an ITS, 676.39/291.59 resulting in: O(n^4) with polynomial bound: 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (72) 676.39/291.59 Obligation: 676.39/291.59 Complexity RNTS consisting of the following rules: 676.39/291.59 676.39/291.59 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.59 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.59 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.59 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.59 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.39/291.59 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.39/291.59 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.39/291.59 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.39/291.59 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.39/291.59 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.59 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.59 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 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' >= 0 676.39/291.59 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' >= 0 676.39/291.59 g_5(z, z') -{ 1 }-> 1 + f_4(z') + g_5(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 676.39/291.59 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} 676.39/291.59 Previous analysis results are: 676.39/291.59 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.59 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.59 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.59 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.59 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.39/291.59 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.39/291.59 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.39/291.59 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.39/291.59 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (73) ResultPropagationProof (UPPER BOUND(ID)) 676.39/291.59 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (74) 676.39/291.59 Obligation: 676.39/291.59 Complexity RNTS consisting of the following rules: 676.39/291.59 676.39/291.59 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.59 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.59 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.59 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.59 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.39/291.59 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.39/291.59 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.39/291.59 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.39/291.59 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.39/291.59 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.59 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.59 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 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' >= 0 676.39/291.59 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' >= 0 676.39/291.59 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' >= 0 676.39/291.59 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 676.39/291.59 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} 676.39/291.59 Previous analysis results are: 676.39/291.59 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.59 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.59 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.59 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.59 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.39/291.59 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.39/291.59 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.39/291.59 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.39/291.59 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (75) IntTrsBoundProof (UPPER BOUND(ID)) 676.39/291.59 676.39/291.59 Computed SIZE bound using CoFloCo for: g_5 676.39/291.59 after applying outer abstraction to obtain an ITS, 676.39/291.59 resulting in: O(1) with polynomial bound: 0 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (76) 676.39/291.59 Obligation: 676.39/291.59 Complexity RNTS consisting of the following rules: 676.39/291.59 676.39/291.59 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.59 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.59 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.59 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.59 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.39/291.59 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.39/291.59 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.39/291.59 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.39/291.59 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.39/291.59 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.59 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.59 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 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' >= 0 676.39/291.59 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' >= 0 676.39/291.59 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' >= 0 676.39/291.59 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 676.39/291.59 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} 676.39/291.59 Previous analysis results are: 676.39/291.59 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.59 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.59 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.59 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.59 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.39/291.59 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.39/291.59 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.39/291.59 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.39/291.59 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.39/291.59 g_5: runtime: ?, size: O(1) [0] 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (77) IntTrsBoundProof (UPPER BOUND(ID)) 676.39/291.59 676.39/291.59 Computed RUNTIME bound using KoAT for: g_5 676.39/291.59 after applying outer abstraction to obtain an ITS, 676.39/291.59 resulting in: O(n^5) with polynomial bound: 2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3 + 2*z*z'^4 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (78) 676.39/291.59 Obligation: 676.39/291.59 Complexity RNTS consisting of the following rules: 676.39/291.59 676.39/291.59 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.59 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.59 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.59 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.59 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.39/291.59 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.39/291.59 f_5(z) -{ 1 }-> g_5(z, z) :|: z >= 0 676.39/291.59 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.39/291.59 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.39/291.59 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.59 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.59 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 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' >= 0 676.39/291.59 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' >= 0 676.39/291.59 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' >= 0 676.39/291.59 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 676.39/291.59 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} 676.39/291.59 Previous analysis results are: 676.39/291.59 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.59 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.59 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.59 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.59 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.39/291.59 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.39/291.59 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.39/291.59 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.39/291.59 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.39/291.59 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] 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (79) ResultPropagationProof (UPPER BOUND(ID)) 676.39/291.59 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (80) 676.39/291.59 Obligation: 676.39/291.59 Complexity RNTS consisting of the following rules: 676.39/291.59 676.39/291.59 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.59 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.59 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.59 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.59 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.39/291.59 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.39/291.59 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.39/291.59 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.39/291.59 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.39/291.59 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.59 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.59 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 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' >= 0 676.39/291.59 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' >= 0 676.39/291.59 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' >= 0 676.39/291.59 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 676.39/291.59 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} 676.39/291.59 Previous analysis results are: 676.39/291.59 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.59 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.59 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.59 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.59 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.39/291.59 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.39/291.59 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.39/291.59 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.39/291.59 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.39/291.59 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] 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (81) IntTrsBoundProof (UPPER BOUND(ID)) 676.39/291.59 676.39/291.59 Computed SIZE bound using CoFloCo for: f_5 676.39/291.59 after applying outer abstraction to obtain an ITS, 676.39/291.59 resulting in: O(1) with polynomial bound: 0 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (82) 676.39/291.59 Obligation: 676.39/291.59 Complexity RNTS consisting of the following rules: 676.39/291.59 676.39/291.59 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.59 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.59 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.59 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.59 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.39/291.59 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.39/291.59 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.39/291.59 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.39/291.59 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.39/291.59 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.59 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.59 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 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' >= 0 676.39/291.59 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' >= 0 676.39/291.59 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' >= 0 676.39/291.59 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 676.39/291.59 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} 676.39/291.59 Previous analysis results are: 676.39/291.59 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.59 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.59 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.59 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.59 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.39/291.59 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.39/291.59 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.39/291.59 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.39/291.59 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.39/291.59 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] 676.39/291.59 f_5: runtime: ?, size: O(1) [0] 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (83) IntTrsBoundProof (UPPER BOUND(ID)) 676.39/291.59 676.39/291.59 Computed RUNTIME bound using KoAT for: f_5 676.39/291.59 after applying outer abstraction to obtain an ITS, 676.39/291.59 resulting in: O(n^5) with polynomial bound: 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (84) 676.39/291.59 Obligation: 676.39/291.59 Complexity RNTS consisting of the following rules: 676.39/291.59 676.39/291.59 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.59 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.59 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.59 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.59 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.39/291.59 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.39/291.59 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.39/291.59 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.39/291.59 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.39/291.59 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.59 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.59 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 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' >= 0 676.39/291.59 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' >= 0 676.39/291.59 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' >= 0 676.39/291.59 g_6(z, z') -{ 1 }-> 1 + f_5(z') + g_6(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 676.39/291.59 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} 676.39/291.59 Previous analysis results are: 676.39/291.59 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.59 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.59 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.59 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.59 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.39/291.59 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.39/291.59 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.39/291.59 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.39/291.59 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.39/291.59 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] 676.39/291.59 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] 676.39/291.59 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (85) ResultPropagationProof (UPPER BOUND(ID)) 676.39/291.59 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 676.39/291.59 ---------------------------------------- 676.39/291.59 676.39/291.59 (86) 676.39/291.59 Obligation: 676.39/291.59 Complexity RNTS consisting of the following rules: 676.39/291.59 676.39/291.59 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.59 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.59 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.59 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.59 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.39/291.59 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.39/291.59 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.39/291.59 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.39/291.59 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.39/291.59 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.59 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.59 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.59 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' >= 0 676.39/291.59 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' >= 0 676.39/291.59 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' >= 0 676.39/291.59 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' >= 0 676.39/291.59 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.59 676.39/291.59 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} 676.39/291.59 Previous analysis results are: 676.39/291.59 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.59 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.59 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.59 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.59 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.39/291.59 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.39/291.59 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.39/291.59 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.39/291.59 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.39/291.59 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] 676.39/291.59 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] 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (87) IntTrsBoundProof (UPPER BOUND(ID)) 676.39/291.60 676.39/291.60 Computed SIZE bound using CoFloCo for: g_6 676.39/291.60 after applying outer abstraction to obtain an ITS, 676.39/291.60 resulting in: O(1) with polynomial bound: 0 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (88) 676.39/291.60 Obligation: 676.39/291.60 Complexity RNTS consisting of the following rules: 676.39/291.60 676.39/291.60 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.60 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.60 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.60 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.60 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.39/291.60 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.39/291.60 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.39/291.60 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.39/291.60 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.39/291.60 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.60 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.60 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 676.39/291.60 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} 676.39/291.60 Previous analysis results are: 676.39/291.60 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.60 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.60 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.60 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.60 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.39/291.60 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.39/291.60 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.39/291.60 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.39/291.60 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 g_6: runtime: ?, size: O(1) [0] 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (89) IntTrsBoundProof (UPPER BOUND(ID)) 676.39/291.60 676.39/291.60 Computed RUNTIME bound using KoAT for: g_6 676.39/291.60 after applying outer abstraction to obtain an ITS, 676.39/291.60 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'^5 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (90) 676.39/291.60 Obligation: 676.39/291.60 Complexity RNTS consisting of the following rules: 676.39/291.60 676.39/291.60 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.60 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.60 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.60 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.60 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.39/291.60 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.39/291.60 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.39/291.60 f_6(z) -{ 1 }-> g_6(z, z) :|: z >= 0 676.39/291.60 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.39/291.60 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.60 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.60 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 676.39/291.60 Function symbols to be analyzed: {f_6}, {g_7}, {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.39/291.60 Previous analysis results are: 676.39/291.60 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.60 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.60 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.60 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.60 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.39/291.60 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.39/291.60 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.39/291.60 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.39/291.60 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (91) ResultPropagationProof (UPPER BOUND(ID)) 676.39/291.60 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (92) 676.39/291.60 Obligation: 676.39/291.60 Complexity RNTS consisting of the following rules: 676.39/291.60 676.39/291.60 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.60 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.60 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.60 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.60 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.39/291.60 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.39/291.60 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.39/291.60 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 >= 0 676.39/291.60 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.39/291.60 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.60 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.60 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 676.39/291.60 Function symbols to be analyzed: {f_6}, {g_7}, {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.39/291.60 Previous analysis results are: 676.39/291.60 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.60 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.60 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.60 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.60 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.39/291.60 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.39/291.60 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.39/291.60 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.39/291.60 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (93) IntTrsBoundProof (UPPER BOUND(ID)) 676.39/291.60 676.39/291.60 Computed SIZE bound using CoFloCo for: f_6 676.39/291.60 after applying outer abstraction to obtain an ITS, 676.39/291.60 resulting in: O(1) with polynomial bound: 0 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (94) 676.39/291.60 Obligation: 676.39/291.60 Complexity RNTS consisting of the following rules: 676.39/291.60 676.39/291.60 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.60 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.60 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.60 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.60 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.39/291.60 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.39/291.60 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.39/291.60 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 >= 0 676.39/291.60 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.39/291.60 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.60 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.60 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 676.39/291.60 Function symbols to be analyzed: {f_6}, {g_7}, {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.39/291.60 Previous analysis results are: 676.39/291.60 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.60 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.60 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.60 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.60 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.39/291.60 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.39/291.60 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.39/291.60 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.39/291.60 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 f_6: runtime: ?, size: O(1) [0] 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (95) IntTrsBoundProof (UPPER BOUND(ID)) 676.39/291.60 676.39/291.60 Computed RUNTIME bound using KoAT for: f_6 676.39/291.60 after applying outer abstraction to obtain an ITS, 676.39/291.60 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^6 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (96) 676.39/291.60 Obligation: 676.39/291.60 Complexity RNTS consisting of the following rules: 676.39/291.60 676.39/291.60 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.60 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.60 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.60 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.60 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.39/291.60 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.39/291.60 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.39/291.60 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 >= 0 676.39/291.60 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.39/291.60 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.60 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.60 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 g_7(z, z') -{ 1 }-> 1 + f_6(z') + g_7(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 676.39/291.60 Function symbols to be analyzed: {g_7}, {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.39/291.60 Previous analysis results are: 676.39/291.60 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.60 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.60 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.60 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.60 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.39/291.60 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.39/291.60 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.39/291.60 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.39/291.60 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (97) ResultPropagationProof (UPPER BOUND(ID)) 676.39/291.60 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (98) 676.39/291.60 Obligation: 676.39/291.60 Complexity RNTS consisting of the following rules: 676.39/291.60 676.39/291.60 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.60 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.60 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.60 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.60 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.39/291.60 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.39/291.60 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.39/291.60 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 >= 0 676.39/291.60 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.39/291.60 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.60 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.60 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 676.39/291.60 Function symbols to be analyzed: {g_7}, {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.39/291.60 Previous analysis results are: 676.39/291.60 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.60 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.60 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.60 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.60 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.39/291.60 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.39/291.60 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.39/291.60 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.39/291.60 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (99) IntTrsBoundProof (UPPER BOUND(ID)) 676.39/291.60 676.39/291.60 Computed SIZE bound using CoFloCo for: g_7 676.39/291.60 after applying outer abstraction to obtain an ITS, 676.39/291.60 resulting in: O(1) with polynomial bound: 0 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (100) 676.39/291.60 Obligation: 676.39/291.60 Complexity RNTS consisting of the following rules: 676.39/291.60 676.39/291.60 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.60 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.60 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.60 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.60 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.39/291.60 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.39/291.60 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.39/291.60 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 >= 0 676.39/291.60 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.39/291.60 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.60 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.60 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 676.39/291.60 Function symbols to be analyzed: {g_7}, {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.39/291.60 Previous analysis results are: 676.39/291.60 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.60 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.60 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.60 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.60 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.39/291.60 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.39/291.60 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.39/291.60 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.39/291.60 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 g_7: runtime: ?, size: O(1) [0] 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (101) IntTrsBoundProof (UPPER BOUND(ID)) 676.39/291.60 676.39/291.60 Computed RUNTIME bound using KoAT for: g_7 676.39/291.60 after applying outer abstraction to obtain an ITS, 676.39/291.60 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'^6 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (102) 676.39/291.60 Obligation: 676.39/291.60 Complexity RNTS consisting of the following rules: 676.39/291.60 676.39/291.60 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.60 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.60 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.60 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.60 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.39/291.60 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.39/291.60 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.39/291.60 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 >= 0 676.39/291.60 f_7(z) -{ 1 }-> g_7(z, z) :|: z >= 0 676.39/291.60 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.60 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.60 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 676.39/291.60 Function symbols to be analyzed: {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.39/291.60 Previous analysis results are: 676.39/291.60 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.60 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.60 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.60 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.60 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.39/291.60 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.39/291.60 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.39/291.60 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.39/291.60 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (103) ResultPropagationProof (UPPER BOUND(ID)) 676.39/291.60 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (104) 676.39/291.60 Obligation: 676.39/291.60 Complexity RNTS consisting of the following rules: 676.39/291.60 676.39/291.60 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.60 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.60 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.60 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.60 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.39/291.60 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.39/291.60 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.39/291.60 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 >= 0 676.39/291.60 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 >= 0 676.39/291.60 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.60 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.60 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 676.39/291.60 Function symbols to be analyzed: {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.39/291.60 Previous analysis results are: 676.39/291.60 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.60 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.60 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.60 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.60 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.39/291.60 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.39/291.60 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.39/291.60 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.39/291.60 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (105) IntTrsBoundProof (UPPER BOUND(ID)) 676.39/291.60 676.39/291.60 Computed SIZE bound using CoFloCo for: f_7 676.39/291.60 after applying outer abstraction to obtain an ITS, 676.39/291.60 resulting in: O(1) with polynomial bound: 0 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (106) 676.39/291.60 Obligation: 676.39/291.60 Complexity RNTS consisting of the following rules: 676.39/291.60 676.39/291.60 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.60 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.60 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.60 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.60 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.39/291.60 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.39/291.60 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.39/291.60 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 >= 0 676.39/291.60 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 >= 0 676.39/291.60 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.60 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.60 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 676.39/291.60 Function symbols to be analyzed: {f_7}, {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.39/291.60 Previous analysis results are: 676.39/291.60 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.60 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.60 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.60 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.60 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.39/291.60 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.39/291.60 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.39/291.60 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.39/291.60 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 f_7: runtime: ?, size: O(1) [0] 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (107) IntTrsBoundProof (UPPER BOUND(ID)) 676.39/291.60 676.39/291.60 Computed RUNTIME bound using KoAT for: f_7 676.39/291.60 after applying outer abstraction to obtain an ITS, 676.39/291.60 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^7 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (108) 676.39/291.60 Obligation: 676.39/291.60 Complexity RNTS consisting of the following rules: 676.39/291.60 676.39/291.60 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.60 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.60 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.60 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.60 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.39/291.60 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.39/291.60 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.39/291.60 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 >= 0 676.39/291.60 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 >= 0 676.39/291.60 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.60 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.60 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 g_8(z, z') -{ 1 }-> 1 + f_7(z') + g_8(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 676.39/291.60 Function symbols to be analyzed: {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.39/291.60 Previous analysis results are: 676.39/291.60 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.60 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.60 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.60 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.60 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.39/291.60 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.39/291.60 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.39/291.60 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.39/291.60 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (109) ResultPropagationProof (UPPER BOUND(ID)) 676.39/291.60 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (110) 676.39/291.60 Obligation: 676.39/291.60 Complexity RNTS consisting of the following rules: 676.39/291.60 676.39/291.60 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.60 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.60 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.60 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.60 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.39/291.60 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.39/291.60 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.39/291.60 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 >= 0 676.39/291.60 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 >= 0 676.39/291.60 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.60 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.60 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 676.39/291.60 Function symbols to be analyzed: {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.39/291.60 Previous analysis results are: 676.39/291.60 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.60 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.60 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.60 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.60 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.39/291.60 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.39/291.60 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.39/291.60 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.39/291.60 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (111) IntTrsBoundProof (UPPER BOUND(ID)) 676.39/291.60 676.39/291.60 Computed SIZE bound using CoFloCo for: g_8 676.39/291.60 after applying outer abstraction to obtain an ITS, 676.39/291.60 resulting in: O(1) with polynomial bound: 0 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (112) 676.39/291.60 Obligation: 676.39/291.60 Complexity RNTS consisting of the following rules: 676.39/291.60 676.39/291.60 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.60 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.60 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.60 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.60 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.39/291.60 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.39/291.60 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.39/291.60 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 >= 0 676.39/291.60 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 >= 0 676.39/291.60 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.60 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.60 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 676.39/291.60 Function symbols to be analyzed: {g_8}, {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.39/291.60 Previous analysis results are: 676.39/291.60 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.60 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.60 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.60 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.60 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.39/291.60 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.39/291.60 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.39/291.60 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.39/291.60 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 g_8: runtime: ?, size: O(1) [0] 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (113) IntTrsBoundProof (UPPER BOUND(ID)) 676.39/291.60 676.39/291.60 Computed RUNTIME bound using KoAT for: g_8 676.39/291.60 after applying outer abstraction to obtain an ITS, 676.39/291.60 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'^7 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (114) 676.39/291.60 Obligation: 676.39/291.60 Complexity RNTS consisting of the following rules: 676.39/291.60 676.39/291.60 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.60 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.60 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.60 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.60 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.39/291.60 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.39/291.60 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.39/291.60 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 >= 0 676.39/291.60 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 >= 0 676.39/291.60 f_8(z) -{ 1 }-> g_8(z, z) :|: z >= 0 676.39/291.60 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.60 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 676.39/291.60 Function symbols to be analyzed: {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.39/291.60 Previous analysis results are: 676.39/291.60 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.60 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.60 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.60 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.60 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.39/291.60 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.39/291.60 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.39/291.60 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.39/291.60 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (115) ResultPropagationProof (UPPER BOUND(ID)) 676.39/291.60 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (116) 676.39/291.60 Obligation: 676.39/291.60 Complexity RNTS consisting of the following rules: 676.39/291.60 676.39/291.60 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.60 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.60 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.60 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.60 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.39/291.60 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.39/291.60 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.39/291.60 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 >= 0 676.39/291.60 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 >= 0 676.39/291.60 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 >= 0 676.39/291.60 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.60 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 676.39/291.60 Function symbols to be analyzed: {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.39/291.60 Previous analysis results are: 676.39/291.60 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.60 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.60 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.60 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.60 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.39/291.60 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.39/291.60 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.39/291.60 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.39/291.60 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (117) IntTrsBoundProof (UPPER BOUND(ID)) 676.39/291.60 676.39/291.60 Computed SIZE bound using CoFloCo for: f_8 676.39/291.60 after applying outer abstraction to obtain an ITS, 676.39/291.60 resulting in: O(1) with polynomial bound: 0 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (118) 676.39/291.60 Obligation: 676.39/291.60 Complexity RNTS consisting of the following rules: 676.39/291.60 676.39/291.60 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.60 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.60 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.60 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.60 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.39/291.60 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.39/291.60 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.39/291.60 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 >= 0 676.39/291.60 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 >= 0 676.39/291.60 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 >= 0 676.39/291.60 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.60 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 676.39/291.60 Function symbols to be analyzed: {f_8}, {g_9}, {f_9}, {g_10}, {f_10} 676.39/291.60 Previous analysis results are: 676.39/291.60 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.60 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.60 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.60 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.60 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.39/291.60 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.39/291.60 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.39/291.60 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.39/291.60 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 f_8: runtime: ?, size: O(1) [0] 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (119) IntTrsBoundProof (UPPER BOUND(ID)) 676.39/291.60 676.39/291.60 Computed RUNTIME bound using KoAT for: f_8 676.39/291.60 after applying outer abstraction to obtain an ITS, 676.39/291.60 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^8 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (120) 676.39/291.60 Obligation: 676.39/291.60 Complexity RNTS consisting of the following rules: 676.39/291.60 676.39/291.60 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.60 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.60 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.60 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.60 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.39/291.60 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.39/291.60 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.39/291.60 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 >= 0 676.39/291.60 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 >= 0 676.39/291.60 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 >= 0 676.39/291.60 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.60 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 g_9(z, z') -{ 1 }-> 1 + f_8(z') + g_9(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 676.39/291.60 Function symbols to be analyzed: {g_9}, {f_9}, {g_10}, {f_10} 676.39/291.60 Previous analysis results are: 676.39/291.60 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.60 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.60 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.60 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.60 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.39/291.60 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.39/291.60 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.39/291.60 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.39/291.60 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (121) ResultPropagationProof (UPPER BOUND(ID)) 676.39/291.60 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (122) 676.39/291.60 Obligation: 676.39/291.60 Complexity RNTS consisting of the following rules: 676.39/291.60 676.39/291.60 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.60 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.60 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.60 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.60 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.39/291.60 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.39/291.60 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.39/291.60 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 >= 0 676.39/291.60 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 >= 0 676.39/291.60 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 >= 0 676.39/291.60 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.60 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 676.39/291.60 Function symbols to be analyzed: {g_9}, {f_9}, {g_10}, {f_10} 676.39/291.60 Previous analysis results are: 676.39/291.60 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.60 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.60 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.60 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.60 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.39/291.60 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.39/291.60 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.39/291.60 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.39/291.60 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (123) IntTrsBoundProof (UPPER BOUND(ID)) 676.39/291.60 676.39/291.60 Computed SIZE bound using CoFloCo for: g_9 676.39/291.60 after applying outer abstraction to obtain an ITS, 676.39/291.60 resulting in: O(1) with polynomial bound: 0 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (124) 676.39/291.60 Obligation: 676.39/291.60 Complexity RNTS consisting of the following rules: 676.39/291.60 676.39/291.60 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.60 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.60 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.60 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.60 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.39/291.60 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.39/291.60 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.39/291.60 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 >= 0 676.39/291.60 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 >= 0 676.39/291.60 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 >= 0 676.39/291.60 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.60 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 676.39/291.60 Function symbols to be analyzed: {g_9}, {f_9}, {g_10}, {f_10} 676.39/291.60 Previous analysis results are: 676.39/291.60 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.60 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.60 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.60 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.60 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.39/291.60 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.39/291.60 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.39/291.60 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.39/291.60 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 g_9: runtime: ?, size: O(1) [0] 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (125) IntTrsBoundProof (UPPER BOUND(ID)) 676.39/291.60 676.39/291.60 Computed RUNTIME bound using KoAT for: g_9 676.39/291.60 after applying outer abstraction to obtain an ITS, 676.39/291.60 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'^8 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (126) 676.39/291.60 Obligation: 676.39/291.60 Complexity RNTS consisting of the following rules: 676.39/291.60 676.39/291.60 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.60 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.60 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.60 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.60 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.39/291.60 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.39/291.60 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.39/291.60 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 >= 0 676.39/291.60 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 >= 0 676.39/291.60 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 >= 0 676.39/291.60 f_9(z) -{ 1 }-> g_9(z, z) :|: z >= 0 676.39/291.60 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 676.39/291.60 Function symbols to be analyzed: {f_9}, {g_10}, {f_10} 676.39/291.60 Previous analysis results are: 676.39/291.60 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.60 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.60 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.60 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.60 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.39/291.60 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.39/291.60 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.39/291.60 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.39/291.60 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (127) ResultPropagationProof (UPPER BOUND(ID)) 676.39/291.60 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (128) 676.39/291.60 Obligation: 676.39/291.60 Complexity RNTS consisting of the following rules: 676.39/291.60 676.39/291.60 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.60 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.60 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.60 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.60 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.39/291.60 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.39/291.60 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.39/291.60 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 >= 0 676.39/291.60 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 >= 0 676.39/291.60 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 >= 0 676.39/291.60 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 >= 0 676.39/291.60 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 676.39/291.60 Function symbols to be analyzed: {f_9}, {g_10}, {f_10} 676.39/291.60 Previous analysis results are: 676.39/291.60 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.60 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.60 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.60 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.60 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.39/291.60 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.39/291.60 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.39/291.60 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.39/291.60 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (129) IntTrsBoundProof (UPPER BOUND(ID)) 676.39/291.60 676.39/291.60 Computed SIZE bound using CoFloCo for: f_9 676.39/291.60 after applying outer abstraction to obtain an ITS, 676.39/291.60 resulting in: O(1) with polynomial bound: 0 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (130) 676.39/291.60 Obligation: 676.39/291.60 Complexity RNTS consisting of the following rules: 676.39/291.60 676.39/291.60 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.60 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.60 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.60 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.60 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.39/291.60 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.39/291.60 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.39/291.60 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 >= 0 676.39/291.60 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 >= 0 676.39/291.60 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 >= 0 676.39/291.60 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 >= 0 676.39/291.60 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 676.39/291.60 Function symbols to be analyzed: {f_9}, {g_10}, {f_10} 676.39/291.60 Previous analysis results are: 676.39/291.60 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.60 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.60 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.60 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.60 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.39/291.60 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.39/291.60 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.39/291.60 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.39/291.60 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 f_9: runtime: ?, size: O(1) [0] 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (131) IntTrsBoundProof (UPPER BOUND(ID)) 676.39/291.60 676.39/291.60 Computed RUNTIME bound using KoAT for: f_9 676.39/291.60 after applying outer abstraction to obtain an ITS, 676.39/291.60 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^9 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (132) 676.39/291.60 Obligation: 676.39/291.60 Complexity RNTS consisting of the following rules: 676.39/291.60 676.39/291.60 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.60 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.60 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.60 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.60 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.39/291.60 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.39/291.60 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.39/291.60 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 >= 0 676.39/291.60 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 >= 0 676.39/291.60 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 >= 0 676.39/291.60 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 >= 0 676.39/291.60 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 g_10(z, z') -{ 1 }-> 1 + f_9(z') + g_10(z - 1, z') :|: z - 1 >= 0, z' >= 0 676.39/291.60 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 676.39/291.60 Function symbols to be analyzed: {g_10}, {f_10} 676.39/291.60 Previous analysis results are: 676.39/291.60 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.60 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.60 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.60 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.60 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.39/291.60 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.39/291.60 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.39/291.60 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.39/291.60 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (133) ResultPropagationProof (UPPER BOUND(ID)) 676.39/291.60 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (134) 676.39/291.60 Obligation: 676.39/291.60 Complexity RNTS consisting of the following rules: 676.39/291.60 676.39/291.60 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.60 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.60 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.60 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.60 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.39/291.60 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.39/291.60 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.39/291.60 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 >= 0 676.39/291.60 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 >= 0 676.39/291.60 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 >= 0 676.39/291.60 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 >= 0 676.39/291.60 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 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' >= 0 676.39/291.60 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 676.39/291.60 Function symbols to be analyzed: {g_10}, {f_10} 676.39/291.60 Previous analysis results are: 676.39/291.60 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.60 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.60 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.60 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.60 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.39/291.60 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.39/291.60 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.39/291.60 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.39/291.60 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (135) IntTrsBoundProof (UPPER BOUND(ID)) 676.39/291.60 676.39/291.60 Computed SIZE bound using CoFloCo for: g_10 676.39/291.60 after applying outer abstraction to obtain an ITS, 676.39/291.60 resulting in: O(1) with polynomial bound: 0 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (136) 676.39/291.60 Obligation: 676.39/291.60 Complexity RNTS consisting of the following rules: 676.39/291.60 676.39/291.60 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.60 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.60 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.60 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.60 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.39/291.60 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.39/291.60 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.39/291.60 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 >= 0 676.39/291.60 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 >= 0 676.39/291.60 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 >= 0 676.39/291.60 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 >= 0 676.39/291.60 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 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' >= 0 676.39/291.60 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 676.39/291.60 Function symbols to be analyzed: {g_10}, {f_10} 676.39/291.60 Previous analysis results are: 676.39/291.60 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.60 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.60 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.60 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.60 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.39/291.60 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.39/291.60 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.39/291.60 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.39/291.60 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 g_10: runtime: ?, size: O(1) [0] 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (137) IntTrsBoundProof (UPPER BOUND(ID)) 676.39/291.60 676.39/291.60 Computed RUNTIME bound using KoAT for: g_10 676.39/291.60 after applying outer abstraction to obtain an ITS, 676.39/291.60 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'^9 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (138) 676.39/291.60 Obligation: 676.39/291.60 Complexity RNTS consisting of the following rules: 676.39/291.60 676.39/291.60 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.60 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.60 f_10(z) -{ 1 }-> g_10(z, z) :|: z >= 0 676.39/291.60 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.60 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.39/291.60 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.39/291.60 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.39/291.60 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 >= 0 676.39/291.60 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 >= 0 676.39/291.60 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 >= 0 676.39/291.60 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 >= 0 676.39/291.60 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 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' >= 0 676.39/291.60 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 676.39/291.60 Function symbols to be analyzed: {f_10} 676.39/291.60 Previous analysis results are: 676.39/291.60 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.60 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.60 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.60 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.60 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.39/291.60 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.39/291.60 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.39/291.60 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.39/291.60 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (139) ResultPropagationProof (UPPER BOUND(ID)) 676.39/291.60 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (140) 676.39/291.60 Obligation: 676.39/291.60 Complexity RNTS consisting of the following rules: 676.39/291.60 676.39/291.60 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.60 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.60 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 >= 0 676.39/291.60 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.60 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.39/291.60 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.39/291.60 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.39/291.60 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 >= 0 676.39/291.60 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 >= 0 676.39/291.60 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 >= 0 676.39/291.60 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 >= 0 676.39/291.60 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 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' >= 0 676.39/291.60 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 676.39/291.60 Function symbols to be analyzed: {f_10} 676.39/291.60 Previous analysis results are: 676.39/291.60 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.60 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.60 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.60 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.60 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.39/291.60 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.39/291.60 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.39/291.60 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.39/291.60 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (141) IntTrsBoundProof (UPPER BOUND(ID)) 676.39/291.60 676.39/291.60 Computed SIZE bound using CoFloCo for: f_10 676.39/291.60 after applying outer abstraction to obtain an ITS, 676.39/291.60 resulting in: O(1) with polynomial bound: 0 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (142) 676.39/291.60 Obligation: 676.39/291.60 Complexity RNTS consisting of the following rules: 676.39/291.60 676.39/291.60 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.60 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.60 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 >= 0 676.39/291.60 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.60 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.39/291.60 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.39/291.60 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.39/291.60 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 >= 0 676.39/291.60 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 >= 0 676.39/291.60 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 >= 0 676.39/291.60 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 >= 0 676.39/291.60 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 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' >= 0 676.39/291.60 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 676.39/291.60 Function symbols to be analyzed: {f_10} 676.39/291.60 Previous analysis results are: 676.39/291.60 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.60 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.60 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.60 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.60 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.39/291.60 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.39/291.60 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.39/291.60 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.39/291.60 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 f_10: runtime: ?, size: O(1) [0] 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (143) IntTrsBoundProof (UPPER BOUND(ID)) 676.39/291.60 676.39/291.60 Computed RUNTIME bound using KoAT for: f_10 676.39/291.60 after applying outer abstraction to obtain an ITS, 676.39/291.60 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^10 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (144) 676.39/291.60 Obligation: 676.39/291.60 Complexity RNTS consisting of the following rules: 676.39/291.60 676.39/291.60 f_0(z) -{ 1 }-> 0 :|: z >= 0 676.39/291.60 f_1(z) -{ 1 + 2*z }-> s :|: s >= 0, s <= 0, z >= 0 676.39/291.60 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 >= 0 676.39/291.60 f_2(z) -{ 1 + 2*z + 2*z^2 }-> s1 :|: s1 >= 0, s1 <= 0, z >= 0 676.39/291.60 f_3(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 }-> s4 :|: s4 >= 0, s4 <= 0, z >= 0 676.39/291.60 f_4(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 }-> s7 :|: s7 >= 0, s7 <= 0, z >= 0 676.39/291.60 f_5(z) -{ 1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4 + 2*z^5 }-> s10 :|: s10 >= 0, s10 <= 0, z >= 0 676.39/291.60 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 >= 0 676.39/291.60 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 >= 0 676.39/291.60 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 >= 0 676.39/291.60 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 >= 0 676.39/291.60 g_1(z, z') -{ 2*z }-> 1 + 0 + s' :|: s' >= 0, s' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 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' >= 0 676.39/291.60 g_2(z, z') -{ 2*z + 2*z*z' }-> 1 + s'' + s2 :|: s2 >= 0, s2 <= 0, s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 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' >= 0 676.39/291.60 676.39/291.60 Function symbols to be analyzed: 676.39/291.60 Previous analysis results are: 676.39/291.60 g_1: runtime: O(n^1) [2*z], size: O(1) [0] 676.39/291.60 f_0: runtime: O(1) [1], size: O(1) [0] 676.39/291.60 f_1: runtime: O(n^1) [1 + 2*z], size: O(1) [0] 676.39/291.60 g_2: runtime: O(n^2) [2*z + 2*z*z'], size: O(1) [0] 676.39/291.60 f_2: runtime: O(n^2) [1 + 2*z + 2*z^2], size: O(1) [0] 676.39/291.60 g_3: runtime: O(n^3) [2*z + 2*z*z' + 2*z*z'^2], size: O(1) [0] 676.39/291.60 f_3: runtime: O(n^3) [1 + 2*z + 2*z^2 + 2*z^3], size: O(1) [0] 676.39/291.60 g_4: runtime: O(n^4) [2*z + 2*z*z' + 2*z*z'^2 + 2*z*z'^3], size: O(1) [0] 676.39/291.60 f_4: runtime: O(n^4) [1 + 2*z + 2*z^2 + 2*z^3 + 2*z^4], size: O(1) [0] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 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] 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (145) FinalProof (FINISHED) 676.39/291.60 Computed overall runtime complexity 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (146) 676.39/291.60 BOUNDS(1, n^10) 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (147) RenamingProof (BOTH BOUNDS(ID, ID)) 676.39/291.60 Renamed function symbols to avoid clashes with predefined symbol. 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (148) 676.39/291.60 Obligation: 676.39/291.60 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 676.39/291.60 676.39/291.60 676.39/291.60 The TRS R consists of the following rules: 676.39/291.60 676.39/291.60 f_0(x) -> a 676.39/291.60 f_1(x) -> g_1(x, x) 676.39/291.60 g_1(s(x), y) -> b(f_0(y), g_1(x, y)) 676.39/291.60 f_2(x) -> g_2(x, x) 676.39/291.60 g_2(s(x), y) -> b(f_1(y), g_2(x, y)) 676.39/291.60 f_3(x) -> g_3(x, x) 676.39/291.60 g_3(s(x), y) -> b(f_2(y), g_3(x, y)) 676.39/291.60 f_4(x) -> g_4(x, x) 676.39/291.60 g_4(s(x), y) -> b(f_3(y), g_4(x, y)) 676.39/291.60 f_5(x) -> g_5(x, x) 676.39/291.60 g_5(s(x), y) -> b(f_4(y), g_5(x, y)) 676.39/291.60 f_6(x) -> g_6(x, x) 676.39/291.60 g_6(s(x), y) -> b(f_5(y), g_6(x, y)) 676.39/291.60 f_7(x) -> g_7(x, x) 676.39/291.60 g_7(s(x), y) -> b(f_6(y), g_7(x, y)) 676.39/291.60 f_8(x) -> g_8(x, x) 676.39/291.60 g_8(s(x), y) -> b(f_7(y), g_8(x, y)) 676.39/291.60 f_9(x) -> g_9(x, x) 676.39/291.60 g_9(s(x), y) -> b(f_8(y), g_9(x, y)) 676.39/291.60 f_10(x) -> g_10(x, x) 676.39/291.60 g_10(s(x), y) -> b(f_9(y), g_10(x, y)) 676.39/291.60 676.39/291.60 S is empty. 676.39/291.60 Rewrite Strategy: FULL 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (149) SlicingProof (LOWER BOUND(ID)) 676.39/291.60 Sliced the following arguments: 676.39/291.60 f_0/0 676.39/291.60 g_1/1 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (150) 676.39/291.60 Obligation: 676.39/291.60 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 676.39/291.60 676.39/291.60 676.39/291.60 The TRS R consists of the following rules: 676.39/291.60 676.39/291.60 f_0 -> a 676.39/291.60 f_1(x) -> g_1(x) 676.39/291.60 g_1(s(x)) -> b(f_0, g_1(x)) 676.39/291.60 f_2(x) -> g_2(x, x) 676.39/291.60 g_2(s(x), y) -> b(f_1(y), g_2(x, y)) 676.39/291.60 f_3(x) -> g_3(x, x) 676.39/291.60 g_3(s(x), y) -> b(f_2(y), g_3(x, y)) 676.39/291.60 f_4(x) -> g_4(x, x) 676.39/291.60 g_4(s(x), y) -> b(f_3(y), g_4(x, y)) 676.39/291.60 f_5(x) -> g_5(x, x) 676.39/291.60 g_5(s(x), y) -> b(f_4(y), g_5(x, y)) 676.39/291.60 f_6(x) -> g_6(x, x) 676.39/291.60 g_6(s(x), y) -> b(f_5(y), g_6(x, y)) 676.39/291.60 f_7(x) -> g_7(x, x) 676.39/291.60 g_7(s(x), y) -> b(f_6(y), g_7(x, y)) 676.39/291.60 f_8(x) -> g_8(x, x) 676.39/291.60 g_8(s(x), y) -> b(f_7(y), g_8(x, y)) 676.39/291.60 f_9(x) -> g_9(x, x) 676.39/291.60 g_9(s(x), y) -> b(f_8(y), g_9(x, y)) 676.39/291.60 f_10(x) -> g_10(x, x) 676.39/291.60 g_10(s(x), y) -> b(f_9(y), g_10(x, y)) 676.39/291.60 676.39/291.60 S is empty. 676.39/291.60 Rewrite Strategy: FULL 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (151) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 676.39/291.60 Infered types. 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (152) 676.39/291.60 Obligation: 676.39/291.60 TRS: 676.39/291.60 Rules: 676.39/291.60 f_0 -> a 676.39/291.60 f_1(x) -> g_1(x) 676.39/291.60 g_1(s(x)) -> b(f_0, g_1(x)) 676.39/291.60 f_2(x) -> g_2(x, x) 676.39/291.60 g_2(s(x), y) -> b(f_1(y), g_2(x, y)) 676.39/291.60 f_3(x) -> g_3(x, x) 676.39/291.60 g_3(s(x), y) -> b(f_2(y), g_3(x, y)) 676.39/291.60 f_4(x) -> g_4(x, x) 676.39/291.60 g_4(s(x), y) -> b(f_3(y), g_4(x, y)) 676.39/291.60 f_5(x) -> g_5(x, x) 676.39/291.60 g_5(s(x), y) -> b(f_4(y), g_5(x, y)) 676.39/291.60 f_6(x) -> g_6(x, x) 676.39/291.60 g_6(s(x), y) -> b(f_5(y), g_6(x, y)) 676.39/291.60 f_7(x) -> g_7(x, x) 676.39/291.60 g_7(s(x), y) -> b(f_6(y), g_7(x, y)) 676.39/291.60 f_8(x) -> g_8(x, x) 676.39/291.60 g_8(s(x), y) -> b(f_7(y), g_8(x, y)) 676.39/291.60 f_9(x) -> g_9(x, x) 676.39/291.60 g_9(s(x), y) -> b(f_8(y), g_9(x, y)) 676.39/291.60 f_10(x) -> g_10(x, x) 676.39/291.60 g_10(s(x), y) -> b(f_9(y), g_10(x, y)) 676.39/291.60 676.39/291.60 Types: 676.39/291.60 f_0 :: a:b 676.39/291.60 a :: a:b 676.39/291.60 f_1 :: s -> a:b 676.39/291.60 g_1 :: s -> a:b 676.39/291.60 s :: s -> s 676.39/291.60 b :: a:b -> a:b -> a:b 676.39/291.60 f_2 :: s -> a:b 676.39/291.60 g_2 :: s -> s -> a:b 676.39/291.60 f_3 :: s -> a:b 676.39/291.60 g_3 :: s -> s -> a:b 676.39/291.60 f_4 :: s -> a:b 676.39/291.60 g_4 :: s -> s -> a:b 676.39/291.60 f_5 :: s -> a:b 676.39/291.60 g_5 :: s -> s -> a:b 676.39/291.60 f_6 :: s -> a:b 676.39/291.60 g_6 :: s -> s -> a:b 676.39/291.60 f_7 :: s -> a:b 676.39/291.60 g_7 :: s -> s -> a:b 676.39/291.60 f_8 :: s -> a:b 676.39/291.60 g_8 :: s -> s -> a:b 676.39/291.60 f_9 :: s -> a:b 676.39/291.60 g_9 :: s -> s -> a:b 676.39/291.60 f_10 :: s -> a:b 676.39/291.60 g_10 :: s -> s -> a:b 676.39/291.60 hole_a:b1_11 :: a:b 676.39/291.60 hole_s2_11 :: s 676.39/291.60 gen_a:b3_11 :: Nat -> a:b 676.39/291.60 gen_s4_11 :: Nat -> s 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (153) OrderProof (LOWER BOUND(ID)) 676.39/291.60 Heuristically decided to analyse the following defined symbols: 676.39/291.60 g_1, g_2, g_3, g_4, g_5, g_6, g_7, g_8, g_9, g_10 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (154) 676.39/291.60 Obligation: 676.39/291.60 TRS: 676.39/291.60 Rules: 676.39/291.60 f_0 -> a 676.39/291.60 f_1(x) -> g_1(x) 676.39/291.60 g_1(s(x)) -> b(f_0, g_1(x)) 676.39/291.60 f_2(x) -> g_2(x, x) 676.39/291.60 g_2(s(x), y) -> b(f_1(y), g_2(x, y)) 676.39/291.60 f_3(x) -> g_3(x, x) 676.39/291.60 g_3(s(x), y) -> b(f_2(y), g_3(x, y)) 676.39/291.60 f_4(x) -> g_4(x, x) 676.39/291.60 g_4(s(x), y) -> b(f_3(y), g_4(x, y)) 676.39/291.60 f_5(x) -> g_5(x, x) 676.39/291.60 g_5(s(x), y) -> b(f_4(y), g_5(x, y)) 676.39/291.60 f_6(x) -> g_6(x, x) 676.39/291.60 g_6(s(x), y) -> b(f_5(y), g_6(x, y)) 676.39/291.60 f_7(x) -> g_7(x, x) 676.39/291.60 g_7(s(x), y) -> b(f_6(y), g_7(x, y)) 676.39/291.60 f_8(x) -> g_8(x, x) 676.39/291.60 g_8(s(x), y) -> b(f_7(y), g_8(x, y)) 676.39/291.60 f_9(x) -> g_9(x, x) 676.39/291.60 g_9(s(x), y) -> b(f_8(y), g_9(x, y)) 676.39/291.60 f_10(x) -> g_10(x, x) 676.39/291.60 g_10(s(x), y) -> b(f_9(y), g_10(x, y)) 676.39/291.60 676.39/291.60 Types: 676.39/291.60 f_0 :: a:b 676.39/291.60 a :: a:b 676.39/291.60 f_1 :: s -> a:b 676.39/291.60 g_1 :: s -> a:b 676.39/291.60 s :: s -> s 676.39/291.60 b :: a:b -> a:b -> a:b 676.39/291.60 f_2 :: s -> a:b 676.39/291.60 g_2 :: s -> s -> a:b 676.39/291.60 f_3 :: s -> a:b 676.39/291.60 g_3 :: s -> s -> a:b 676.39/291.60 f_4 :: s -> a:b 676.39/291.60 g_4 :: s -> s -> a:b 676.39/291.60 f_5 :: s -> a:b 676.39/291.60 g_5 :: s -> s -> a:b 676.39/291.60 f_6 :: s -> a:b 676.39/291.60 g_6 :: s -> s -> a:b 676.39/291.60 f_7 :: s -> a:b 676.39/291.60 g_7 :: s -> s -> a:b 676.39/291.60 f_8 :: s -> a:b 676.39/291.60 g_8 :: s -> s -> a:b 676.39/291.60 f_9 :: s -> a:b 676.39/291.60 g_9 :: s -> s -> a:b 676.39/291.60 f_10 :: s -> a:b 676.39/291.60 g_10 :: s -> s -> a:b 676.39/291.60 hole_a:b1_11 :: a:b 676.39/291.60 hole_s2_11 :: s 676.39/291.60 gen_a:b3_11 :: Nat -> a:b 676.39/291.60 gen_s4_11 :: Nat -> s 676.39/291.60 676.39/291.60 676.39/291.60 Generator Equations: 676.39/291.60 gen_a:b3_11(0) <=> a 676.39/291.60 gen_a:b3_11(+(x, 1)) <=> b(a, gen_a:b3_11(x)) 676.39/291.60 gen_s4_11(0) <=> hole_s2_11 676.39/291.60 gen_s4_11(+(x, 1)) <=> s(gen_s4_11(x)) 676.39/291.60 676.39/291.60 676.39/291.60 The following defined symbols remain to be analysed: 676.39/291.60 g_1, g_2, g_3, g_4, g_5, g_6, g_7, g_8, g_9, g_10 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (155) RewriteLemmaProof (LOWER BOUND(ID)) 676.39/291.60 Proved the following rewrite lemma: 676.39/291.60 g_1(gen_s4_11(+(1, n6_11))) -> *5_11, rt in Omega(n6_11) 676.39/291.60 676.39/291.60 Induction Base: 676.39/291.60 g_1(gen_s4_11(+(1, 0))) 676.39/291.60 676.39/291.60 Induction Step: 676.39/291.60 g_1(gen_s4_11(+(1, +(n6_11, 1)))) ->_R^Omega(1) 676.39/291.60 b(f_0, g_1(gen_s4_11(+(1, n6_11)))) ->_R^Omega(1) 676.39/291.60 b(a, g_1(gen_s4_11(+(1, n6_11)))) ->_IH 676.39/291.60 b(a, *5_11) 676.39/291.60 676.39/291.60 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (156) 676.39/291.60 Complex Obligation (BEST) 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (157) 676.39/291.60 Obligation: 676.39/291.60 Proved the lower bound n^1 for the following obligation: 676.39/291.60 676.39/291.60 TRS: 676.39/291.60 Rules: 676.39/291.60 f_0 -> a 676.39/291.60 f_1(x) -> g_1(x) 676.39/291.60 g_1(s(x)) -> b(f_0, g_1(x)) 676.39/291.60 f_2(x) -> g_2(x, x) 676.39/291.60 g_2(s(x), y) -> b(f_1(y), g_2(x, y)) 676.39/291.60 f_3(x) -> g_3(x, x) 676.39/291.60 g_3(s(x), y) -> b(f_2(y), g_3(x, y)) 676.39/291.60 f_4(x) -> g_4(x, x) 676.39/291.60 g_4(s(x), y) -> b(f_3(y), g_4(x, y)) 676.39/291.60 f_5(x) -> g_5(x, x) 676.39/291.60 g_5(s(x), y) -> b(f_4(y), g_5(x, y)) 676.39/291.60 f_6(x) -> g_6(x, x) 676.39/291.60 g_6(s(x), y) -> b(f_5(y), g_6(x, y)) 676.39/291.60 f_7(x) -> g_7(x, x) 676.39/291.60 g_7(s(x), y) -> b(f_6(y), g_7(x, y)) 676.39/291.60 f_8(x) -> g_8(x, x) 676.39/291.60 g_8(s(x), y) -> b(f_7(y), g_8(x, y)) 676.39/291.60 f_9(x) -> g_9(x, x) 676.39/291.60 g_9(s(x), y) -> b(f_8(y), g_9(x, y)) 676.39/291.60 f_10(x) -> g_10(x, x) 676.39/291.60 g_10(s(x), y) -> b(f_9(y), g_10(x, y)) 676.39/291.60 676.39/291.60 Types: 676.39/291.60 f_0 :: a:b 676.39/291.60 a :: a:b 676.39/291.60 f_1 :: s -> a:b 676.39/291.60 g_1 :: s -> a:b 676.39/291.60 s :: s -> s 676.39/291.60 b :: a:b -> a:b -> a:b 676.39/291.60 f_2 :: s -> a:b 676.39/291.60 g_2 :: s -> s -> a:b 676.39/291.60 f_3 :: s -> a:b 676.39/291.60 g_3 :: s -> s -> a:b 676.39/291.60 f_4 :: s -> a:b 676.39/291.60 g_4 :: s -> s -> a:b 676.39/291.60 f_5 :: s -> a:b 676.39/291.60 g_5 :: s -> s -> a:b 676.39/291.60 f_6 :: s -> a:b 676.39/291.60 g_6 :: s -> s -> a:b 676.39/291.60 f_7 :: s -> a:b 676.39/291.60 g_7 :: s -> s -> a:b 676.39/291.60 f_8 :: s -> a:b 676.39/291.60 g_8 :: s -> s -> a:b 676.39/291.60 f_9 :: s -> a:b 676.39/291.60 g_9 :: s -> s -> a:b 676.39/291.60 f_10 :: s -> a:b 676.39/291.60 g_10 :: s -> s -> a:b 676.39/291.60 hole_a:b1_11 :: a:b 676.39/291.60 hole_s2_11 :: s 676.39/291.60 gen_a:b3_11 :: Nat -> a:b 676.39/291.60 gen_s4_11 :: Nat -> s 676.39/291.60 676.39/291.60 676.39/291.60 Generator Equations: 676.39/291.60 gen_a:b3_11(0) <=> a 676.39/291.60 gen_a:b3_11(+(x, 1)) <=> b(a, gen_a:b3_11(x)) 676.39/291.60 gen_s4_11(0) <=> hole_s2_11 676.39/291.60 gen_s4_11(+(x, 1)) <=> s(gen_s4_11(x)) 676.39/291.60 676.39/291.60 676.39/291.60 The following defined symbols remain to be analysed: 676.39/291.60 g_1, g_2, g_3, g_4, g_5, g_6, g_7, g_8, g_9, g_10 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (158) LowerBoundPropagationProof (FINISHED) 676.39/291.60 Propagated lower bound. 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (159) 676.39/291.60 BOUNDS(n^1, INF) 676.39/291.60 676.39/291.60 ---------------------------------------- 676.39/291.60 676.39/291.60 (160) 676.39/291.60 Obligation: 676.39/291.60 TRS: 676.39/291.60 Rules: 676.39/291.60 f_0 -> a 676.39/291.60 f_1(x) -> g_1(x) 676.39/291.60 g_1(s(x)) -> b(f_0, g_1(x)) 676.39/291.60 f_2(x) -> g_2(x, x) 676.39/291.60 g_2(s(x), y) -> b(f_1(y), g_2(x, y)) 676.39/291.60 f_3(x) -> g_3(x, x) 676.39/291.60 g_3(s(x), y) -> b(f_2(y), g_3(x, y)) 676.39/291.60 f_4(x) -> g_4(x, x) 676.39/291.60 g_4(s(x), y) -> b(f_3(y), g_4(x, y)) 676.39/291.60 f_5(x) -> g_5(x, x) 676.39/291.60 g_5(s(x), y) -> b(f_4(y), g_5(x, y)) 676.39/291.60 f_6(x) -> g_6(x, x) 676.39/291.60 g_6(s(x), y) -> b(f_5(y), g_6(x, y)) 676.39/291.60 f_7(x) -> g_7(x, x) 676.39/291.60 g_7(s(x), y) -> b(f_6(y), g_7(x, y)) 676.39/291.60 f_8(x) -> g_8(x, x) 676.39/291.60 g_8(s(x), y) -> b(f_7(y), g_8(x, y)) 676.39/291.60 f_9(x) -> g_9(x, x) 676.39/291.60 g_9(s(x), y) -> b(f_8(y), g_9(x, y)) 676.39/291.60 f_10(x) -> g_10(x, x) 676.39/291.60 g_10(s(x), y) -> b(f_9(y), g_10(x, y)) 676.39/291.60 676.39/291.60 Types: 676.39/291.60 f_0 :: a:b 676.39/291.60 a :: a:b 676.39/291.60 f_1 :: s -> a:b 676.39/291.60 g_1 :: s -> a:b 676.39/291.60 s :: s -> s 676.39/291.60 b :: a:b -> a:b -> a:b 676.39/291.60 f_2 :: s -> a:b 676.39/291.60 g_2 :: s -> s -> a:b 676.39/291.60 f_3 :: s -> a:b 676.39/291.60 g_3 :: s -> s -> a:b 676.39/291.60 f_4 :: s -> a:b 676.39/291.60 g_4 :: s -> s -> a:b 676.39/291.60 f_5 :: s -> a:b 676.39/291.60 g_5 :: s -> s -> a:b 676.39/291.60 f_6 :: s -> a:b 676.39/291.60 g_6 :: s -> s -> a:b 676.39/291.60 f_7 :: s -> a:b 676.39/291.60 g_7 :: s -> s -> a:b 676.39/291.60 f_8 :: s -> a:b 676.39/291.60 g_8 :: s -> s -> a:b 676.39/291.60 f_9 :: s -> a:b 676.39/291.60 g_9 :: s -> s -> a:b 676.39/291.60 f_10 :: s -> a:b 676.39/291.60 g_10 :: s -> s -> a:b 676.39/291.60 hole_a:b1_11 :: a:b 676.39/291.60 hole_s2_11 :: s 676.39/291.60 gen_a:b3_11 :: Nat -> a:b 676.39/291.60 gen_s4_11 :: Nat -> s 676.39/291.60 676.39/291.60 676.39/291.60 Lemmas: 676.39/291.60 g_1(gen_s4_11(+(1, n6_11))) -> *5_11, rt in Omega(n6_11) 676.39/291.60 676.39/291.60 676.39/291.60 Generator Equations: 676.39/291.60 gen_a:b3_11(0) <=> a 676.39/291.60 gen_a:b3_11(+(x, 1)) <=> b(a, gen_a:b3_11(x)) 676.39/291.61 gen_s4_11(0) <=> hole_s2_11 676.39/291.61 gen_s4_11(+(x, 1)) <=> s(gen_s4_11(x)) 676.39/291.61 676.39/291.61 676.39/291.61 The following defined symbols remain to be analysed: 676.39/291.61 g_2, g_3, g_4, g_5, g_6, g_7, g_8, g_9, g_10 676.64/291.65 EOF