WORST_CASE(Omega(n^1),O(n^2)) proof of /export/starexec/sandbox/benchmark/theBenchmark.trs # AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). (0) CpxRelTRS (1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 2141 ms] (2) CpxRelTRS (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (4) CpxWeightedTrs (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CpxTypedWeightedTrs (7) CompletionProof [UPPER BOUND(ID), 0 ms] (8) CpxTypedWeightedCompleteTrs (9) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (12) CpxRNTS (13) SimplificationProof [BOTH BOUNDS(ID, ID), 2 ms] (14) CpxRNTS (15) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 2 ms] (16) CpxRNTS (17) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (18) CpxRNTS (19) IntTrsBoundProof [UPPER BOUND(ID), 399 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 58 ms] (22) CpxRNTS (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 293 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 4 ms] (28) CpxRNTS (29) ResultPropagationProof [UPPER BOUND(ID), 3 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 291 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 116 ms] (34) CpxRNTS (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 231 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 73 ms] (40) CpxRNTS (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (42) CpxRNTS (43) IntTrsBoundProof [UPPER BOUND(ID), 341 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 15 ms] (46) CpxRNTS (47) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (48) CpxRNTS (49) IntTrsBoundProof [UPPER BOUND(ID), 291 ms] (50) CpxRNTS (51) IntTrsBoundProof [UPPER BOUND(ID), 135 ms] (52) CpxRNTS (53) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (54) CpxRNTS (55) IntTrsBoundProof [UPPER BOUND(ID), 1946 ms] (56) CpxRNTS (57) IntTrsBoundProof [UPPER BOUND(ID), 500 ms] (58) CpxRNTS (59) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (60) CpxRNTS (61) IntTrsBoundProof [UPPER BOUND(ID), 11.5 s] (62) CpxRNTS (63) IntTrsBoundProof [UPPER BOUND(ID), 2577 ms] (64) CpxRNTS (65) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (66) CpxRNTS (67) IntTrsBoundProof [UPPER BOUND(ID), 138 ms] (68) CpxRNTS (69) IntTrsBoundProof [UPPER BOUND(ID), 3 ms] (70) CpxRNTS (71) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (72) CpxRNTS (73) IntTrsBoundProof [UPPER BOUND(ID), 558 ms] (74) CpxRNTS (75) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] (76) CpxRNTS (77) FinalProof [FINISHED, 0 ms] (78) BOUNDS(1, n^2) (79) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CpxRelTRS (81) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (82) typed CpxTrs (83) OrderProof [LOWER BOUND(ID), 0 ms] (84) typed CpxTrs (85) RewriteLemmaProof [LOWER BOUND(ID), 241 ms] (86) typed CpxTrs (87) RewriteLemmaProof [LOWER BOUND(ID), 54 ms] (88) typed CpxTrs (89) RewriteLemmaProof [LOWER BOUND(ID), 62 ms] (90) BEST (91) proven lower bound (92) LowerBoundPropagationProof [FINISHED, 0 ms] (93) BOUNDS(n^1, INF) (94) typed CpxTrs (95) RewriteLemmaProof [LOWER BOUND(ID), 7 ms] (96) typed CpxTrs (97) RewriteLemmaProof [LOWER BOUND(ID), 51 ms] (98) typed CpxTrs (99) RewriteLemmaProof [LOWER BOUND(ID), 69 ms] (100) typed CpxTrs ---------------------------------------- (0) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: DIVISION(z0, z1) -> c(DIV(z0, z1, 0)) DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1)) DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2)) IF(true, z0, z1, z2) -> c3 IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1))) MINUS(z0, 0) -> c5 MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1)) LT(z0, 0) -> c7 LT(0, s(z0)) -> c8 LT(s(z0), s(z1)) -> c9(LT(z0, z1)) INC(0) -> c10 INC(s(z0)) -> c11(INC(z0)) The (relative) TRS S consists of the following rules: division(z0, z1) -> div(z0, z1, 0) div(z0, z1, z2) -> if(lt(z0, z1), z0, z1, inc(z2)) if(true, z0, z1, z2) -> z2 if(false, z0, s(z1), z2) -> div(minus(z0, s(z1)), s(z1), z2) minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) lt(z0, 0) -> false lt(0, s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) inc(0) -> s(0) inc(s(z0)) -> s(inc(z0)) Rewrite Strategy: INNERMOST ---------------------------------------- (1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: DIVISION(z0, z1) -> c(DIV(z0, z1, 0)) DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1)) DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2)) IF(true, z0, z1, z2) -> c3 IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1))) MINUS(z0, 0) -> c5 MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1)) LT(z0, 0) -> c7 LT(0, s(z0)) -> c8 LT(s(z0), s(z1)) -> c9(LT(z0, z1)) INC(0) -> c10 INC(s(z0)) -> c11(INC(z0)) The (relative) TRS S consists of the following rules: division(z0, z1) -> div(z0, z1, 0) div(z0, z1, z2) -> if(lt(z0, z1), z0, z1, inc(z2)) if(true, z0, z1, z2) -> z2 if(false, z0, s(z1), z2) -> div(minus(z0, s(z1)), s(z1), z2) minus(z0, 0) -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) lt(z0, 0) -> false lt(0, s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) inc(0) -> s(0) inc(s(z0)) -> s(inc(z0)) Rewrite Strategy: INNERMOST ---------------------------------------- (3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2). The TRS R consists of the following rules: DIVISION(z0, z1) -> c(DIV(z0, z1, 0)) [1] DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1)) [1] DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2)) [1] IF(true, z0, z1, z2) -> c3 [1] IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1))) [1] MINUS(z0, 0) -> c5 [1] MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1)) [1] LT(z0, 0) -> c7 [1] LT(0, s(z0)) -> c8 [1] LT(s(z0), s(z1)) -> c9(LT(z0, z1)) [1] INC(0) -> c10 [1] INC(s(z0)) -> c11(INC(z0)) [1] division(z0, z1) -> div(z0, z1, 0) [0] div(z0, z1, z2) -> if(lt(z0, z1), z0, z1, inc(z2)) [0] if(true, z0, z1, z2) -> z2 [0] if(false, z0, s(z1), z2) -> div(minus(z0, s(z1)), s(z1), z2) [0] minus(z0, 0) -> z0 [0] minus(s(z0), s(z1)) -> minus(z0, z1) [0] lt(z0, 0) -> false [0] lt(0, s(z0)) -> true [0] lt(s(z0), s(z1)) -> lt(z0, z1) [0] inc(0) -> s(0) [0] inc(s(z0)) -> s(inc(z0)) [0] Rewrite Strategy: INNERMOST ---------------------------------------- (5) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (6) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: DIVISION(z0, z1) -> c(DIV(z0, z1, 0)) [1] DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1)) [1] DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2)) [1] IF(true, z0, z1, z2) -> c3 [1] IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1))) [1] MINUS(z0, 0) -> c5 [1] MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1)) [1] LT(z0, 0) -> c7 [1] LT(0, s(z0)) -> c8 [1] LT(s(z0), s(z1)) -> c9(LT(z0, z1)) [1] INC(0) -> c10 [1] INC(s(z0)) -> c11(INC(z0)) [1] division(z0, z1) -> div(z0, z1, 0) [0] div(z0, z1, z2) -> if(lt(z0, z1), z0, z1, inc(z2)) [0] if(true, z0, z1, z2) -> z2 [0] if(false, z0, s(z1), z2) -> div(minus(z0, s(z1)), s(z1), z2) [0] minus(z0, 0) -> z0 [0] minus(s(z0), s(z1)) -> minus(z0, z1) [0] lt(z0, 0) -> false [0] lt(0, s(z0)) -> true [0] lt(s(z0), s(z1)) -> lt(z0, z1) [0] inc(0) -> s(0) [0] inc(s(z0)) -> s(inc(z0)) [0] The TRS has the following type information: DIVISION :: 0:s -> 0:s -> c c :: c1:c2 -> c DIV :: 0:s -> 0:s -> 0:s -> c1:c2 0 :: 0:s c1 :: c3:c4 -> c7:c8:c9 -> c1:c2 IF :: true:false -> 0:s -> 0:s -> 0:s -> c3:c4 lt :: 0:s -> 0:s -> true:false inc :: 0:s -> 0:s LT :: 0:s -> 0:s -> c7:c8:c9 c2 :: c3:c4 -> c10:c11 -> c1:c2 INC :: 0:s -> c10:c11 true :: true:false c3 :: c3:c4 false :: true:false s :: 0:s -> 0:s c4 :: c1:c2 -> c5:c6 -> c3:c4 minus :: 0:s -> 0:s -> 0:s MINUS :: 0:s -> 0:s -> c5:c6 c5 :: c5:c6 c6 :: c5:c6 -> c5:c6 c7 :: c7:c8:c9 c8 :: c7:c8:c9 c9 :: c7:c8:c9 -> c7:c8:c9 c10 :: c10:c11 c11 :: c10:c11 -> c10:c11 division :: 0:s -> 0:s -> 0:s div :: 0:s -> 0:s -> 0:s -> 0:s if :: true:false -> 0:s -> 0:s -> 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (7) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: DIVISION_2 DIV_3 IF_4 MINUS_2 LT_2 INC_1 (c) The following functions are completely defined: division_2 div_3 if_4 minus_2 lt_2 inc_1 Due to the following rules being added: division(v0, v1) -> 0 [0] div(v0, v1, v2) -> 0 [0] if(v0, v1, v2, v3) -> 0 [0] minus(v0, v1) -> 0 [0] lt(v0, v1) -> null_lt [0] inc(v0) -> 0 [0] And the following fresh constants: null_lt, const, const1 ---------------------------------------- (8) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: DIVISION(z0, z1) -> c(DIV(z0, z1, 0)) [1] DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1)) [1] DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2)) [1] IF(true, z0, z1, z2) -> c3 [1] IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1))) [1] MINUS(z0, 0) -> c5 [1] MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1)) [1] LT(z0, 0) -> c7 [1] LT(0, s(z0)) -> c8 [1] LT(s(z0), s(z1)) -> c9(LT(z0, z1)) [1] INC(0) -> c10 [1] INC(s(z0)) -> c11(INC(z0)) [1] division(z0, z1) -> div(z0, z1, 0) [0] div(z0, z1, z2) -> if(lt(z0, z1), z0, z1, inc(z2)) [0] if(true, z0, z1, z2) -> z2 [0] if(false, z0, s(z1), z2) -> div(minus(z0, s(z1)), s(z1), z2) [0] minus(z0, 0) -> z0 [0] minus(s(z0), s(z1)) -> minus(z0, z1) [0] lt(z0, 0) -> false [0] lt(0, s(z0)) -> true [0] lt(s(z0), s(z1)) -> lt(z0, z1) [0] inc(0) -> s(0) [0] inc(s(z0)) -> s(inc(z0)) [0] division(v0, v1) -> 0 [0] div(v0, v1, v2) -> 0 [0] if(v0, v1, v2, v3) -> 0 [0] minus(v0, v1) -> 0 [0] lt(v0, v1) -> null_lt [0] inc(v0) -> 0 [0] The TRS has the following type information: DIVISION :: 0:s -> 0:s -> c c :: c1:c2 -> c DIV :: 0:s -> 0:s -> 0:s -> c1:c2 0 :: 0:s c1 :: c3:c4 -> c7:c8:c9 -> c1:c2 IF :: true:false:null_lt -> 0:s -> 0:s -> 0:s -> c3:c4 lt :: 0:s -> 0:s -> true:false:null_lt inc :: 0:s -> 0:s LT :: 0:s -> 0:s -> c7:c8:c9 c2 :: c3:c4 -> c10:c11 -> c1:c2 INC :: 0:s -> c10:c11 true :: true:false:null_lt c3 :: c3:c4 false :: true:false:null_lt s :: 0:s -> 0:s c4 :: c1:c2 -> c5:c6 -> c3:c4 minus :: 0:s -> 0:s -> 0:s MINUS :: 0:s -> 0:s -> c5:c6 c5 :: c5:c6 c6 :: c5:c6 -> c5:c6 c7 :: c7:c8:c9 c8 :: c7:c8:c9 c9 :: c7:c8:c9 -> c7:c8:c9 c10 :: c10:c11 c11 :: c10:c11 -> c10:c11 division :: 0:s -> 0:s -> 0:s div :: 0:s -> 0:s -> 0:s -> 0:s if :: true:false:null_lt -> 0:s -> 0:s -> 0:s -> 0:s null_lt :: true:false:null_lt const :: c const1 :: c1:c2 Rewrite Strategy: INNERMOST ---------------------------------------- (9) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: DIVISION(z0, z1) -> c(DIV(z0, z1, 0)) [1] DIV(z0, 0, 0) -> c1(IF(false, z0, 0, s(0)), LT(z0, 0)) [1] DIV(z0, 0, s(z01)) -> c1(IF(false, z0, 0, s(inc(z01))), LT(z0, 0)) [1] DIV(z0, 0, z2) -> c1(IF(false, z0, 0, 0), LT(z0, 0)) [1] DIV(0, s(z0'), 0) -> c1(IF(true, 0, s(z0'), s(0)), LT(0, s(z0'))) [1] DIV(0, s(z0'), s(z02)) -> c1(IF(true, 0, s(z0'), s(inc(z02))), LT(0, s(z0'))) [1] DIV(0, s(z0'), z2) -> c1(IF(true, 0, s(z0'), 0), LT(0, s(z0'))) [1] DIV(s(z0''), s(z1'), 0) -> c1(IF(lt(z0'', z1'), s(z0''), s(z1'), s(0)), LT(s(z0''), s(z1'))) [1] DIV(s(z0''), s(z1'), s(z03)) -> c1(IF(lt(z0'', z1'), s(z0''), s(z1'), s(inc(z03))), LT(s(z0''), s(z1'))) [1] DIV(s(z0''), s(z1'), z2) -> c1(IF(lt(z0'', z1'), s(z0''), s(z1'), 0), LT(s(z0''), s(z1'))) [1] DIV(z0, z1, 0) -> c1(IF(null_lt, z0, z1, s(0)), LT(z0, z1)) [1] DIV(z0, z1, s(z04)) -> c1(IF(null_lt, z0, z1, s(inc(z04))), LT(z0, z1)) [1] DIV(z0, z1, z2) -> c1(IF(null_lt, z0, z1, 0), LT(z0, z1)) [1] DIV(z0, 0, 0) -> c2(IF(false, z0, 0, s(0)), INC(0)) [1] DIV(z0, 0, s(z07)) -> c2(IF(false, z0, 0, s(inc(z07))), INC(s(z07))) [1] DIV(z0, 0, z2) -> c2(IF(false, z0, 0, 0), INC(z2)) [1] DIV(0, s(z05), 0) -> c2(IF(true, 0, s(z05), s(0)), INC(0)) [1] DIV(0, s(z05), s(z08)) -> c2(IF(true, 0, s(z05), s(inc(z08))), INC(s(z08))) [1] DIV(0, s(z05), z2) -> c2(IF(true, 0, s(z05), 0), INC(z2)) [1] DIV(s(z06), s(z1''), 0) -> c2(IF(lt(z06, z1''), s(z06), s(z1''), s(0)), INC(0)) [1] DIV(s(z06), s(z1''), s(z09)) -> c2(IF(lt(z06, z1''), s(z06), s(z1''), s(inc(z09))), INC(s(z09))) [1] DIV(s(z06), s(z1''), z2) -> c2(IF(lt(z06, z1''), s(z06), s(z1''), 0), INC(z2)) [1] DIV(z0, z1, 0) -> c2(IF(null_lt, z0, z1, s(0)), INC(0)) [1] DIV(z0, z1, s(z010)) -> c2(IF(null_lt, z0, z1, s(inc(z010))), INC(s(z010))) [1] DIV(z0, z1, z2) -> c2(IF(null_lt, z0, z1, 0), INC(z2)) [1] IF(true, z0, z1, z2) -> c3 [1] IF(false, s(z011), s(z1), z2) -> c4(DIV(minus(z011, z1), s(z1), z2), MINUS(s(z011), s(z1))) [1] IF(false, z0, s(z1), z2) -> c4(DIV(0, s(z1), z2), MINUS(z0, s(z1))) [1] MINUS(z0, 0) -> c5 [1] MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1)) [1] LT(z0, 0) -> c7 [1] LT(0, s(z0)) -> c8 [1] LT(s(z0), s(z1)) -> c9(LT(z0, z1)) [1] INC(0) -> c10 [1] INC(s(z0)) -> c11(INC(z0)) [1] division(z0, z1) -> div(z0, z1, 0) [0] div(z0, 0, 0) -> if(false, z0, 0, s(0)) [0] div(z0, 0, s(z014)) -> if(false, z0, 0, s(inc(z014))) [0] div(z0, 0, z2) -> if(false, z0, 0, 0) [0] div(0, s(z012), 0) -> if(true, 0, s(z012), s(0)) [0] div(0, s(z012), s(z015)) -> if(true, 0, s(z012), s(inc(z015))) [0] div(0, s(z012), z2) -> if(true, 0, s(z012), 0) [0] div(s(z013), s(z11), 0) -> if(lt(z013, z11), s(z013), s(z11), s(0)) [0] div(s(z013), s(z11), s(z016)) -> if(lt(z013, z11), s(z013), s(z11), s(inc(z016))) [0] div(s(z013), s(z11), z2) -> if(lt(z013, z11), s(z013), s(z11), 0) [0] div(z0, z1, 0) -> if(null_lt, z0, z1, s(0)) [0] div(z0, z1, s(z017)) -> if(null_lt, z0, z1, s(inc(z017))) [0] div(z0, z1, z2) -> if(null_lt, z0, z1, 0) [0] if(true, z0, z1, z2) -> z2 [0] if(false, s(z018), s(z1), z2) -> div(minus(z018, z1), s(z1), z2) [0] if(false, z0, s(z1), z2) -> div(0, s(z1), z2) [0] minus(z0, 0) -> z0 [0] minus(s(z0), s(z1)) -> minus(z0, z1) [0] lt(z0, 0) -> false [0] lt(0, s(z0)) -> true [0] lt(s(z0), s(z1)) -> lt(z0, z1) [0] inc(0) -> s(0) [0] inc(s(z0)) -> s(inc(z0)) [0] division(v0, v1) -> 0 [0] div(v0, v1, v2) -> 0 [0] if(v0, v1, v2, v3) -> 0 [0] minus(v0, v1) -> 0 [0] lt(v0, v1) -> null_lt [0] inc(v0) -> 0 [0] The TRS has the following type information: DIVISION :: 0:s -> 0:s -> c c :: c1:c2 -> c DIV :: 0:s -> 0:s -> 0:s -> c1:c2 0 :: 0:s c1 :: c3:c4 -> c7:c8:c9 -> c1:c2 IF :: true:false:null_lt -> 0:s -> 0:s -> 0:s -> c3:c4 lt :: 0:s -> 0:s -> true:false:null_lt inc :: 0:s -> 0:s LT :: 0:s -> 0:s -> c7:c8:c9 c2 :: c3:c4 -> c10:c11 -> c1:c2 INC :: 0:s -> c10:c11 true :: true:false:null_lt c3 :: c3:c4 false :: true:false:null_lt s :: 0:s -> 0:s c4 :: c1:c2 -> c5:c6 -> c3:c4 minus :: 0:s -> 0:s -> 0:s MINUS :: 0:s -> 0:s -> c5:c6 c5 :: c5:c6 c6 :: c5:c6 -> c5:c6 c7 :: c7:c8:c9 c8 :: c7:c8:c9 c9 :: c7:c8:c9 -> c7:c8:c9 c10 :: c10:c11 c11 :: c10:c11 -> c10:c11 division :: 0:s -> 0:s -> 0:s div :: 0:s -> 0:s -> 0:s -> 0:s if :: true:false:null_lt -> 0:s -> 0:s -> 0:s -> 0:s null_lt :: true:false:null_lt const :: c const1 :: c1:c2 Rewrite Strategy: INNERMOST ---------------------------------------- (11) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 true => 2 c3 => 0 false => 1 c5 => 0 c7 => 0 c8 => 1 c10 => 0 null_lt => 0 const => 0 const1 => 0 ---------------------------------------- (12) Obligation: Complexity RNTS consisting of the following rules: DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z0'', z1'), 1 + z0'', 1 + z1', 0) + LT(1 + z0'', 1 + z1') :|: z'' = z2, z' = 1 + z1', z1' >= 0, z0'' >= 0, z2 >= 0, z = 1 + z0'' DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z0'', z1'), 1 + z0'', 1 + z1', 1 + inc(z03)) + LT(1 + z0'', 1 + z1') :|: z'' = 1 + z03, z' = 1 + z1', z1' >= 0, z0'' >= 0, z03 >= 0, z = 1 + z0'' DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z0'', z1'), 1 + z0'', 1 + z1', 1 + 0) + LT(1 + z0'', 1 + z1') :|: z'' = 0, z' = 1 + z1', z1' >= 0, z0'' >= 0, z = 1 + z0'' DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z06, z1''), 1 + z06, 1 + z1'', 0) + INC(z2) :|: z'' = z2, z' = 1 + z1'', z06 >= 0, z2 >= 0, z = 1 + z06, z1'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z06, z1''), 1 + z06, 1 + z1'', 1 + inc(z09)) + INC(1 + z09) :|: z' = 1 + z1'', z'' = 1 + z09, z06 >= 0, z09 >= 0, z = 1 + z06, z1'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z06, z1''), 1 + z06, 1 + z1'', 1 + 0) + INC(0) :|: z'' = 0, z' = 1 + z1'', z06 >= 0, z = 1 + z06, z1'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + z0', 0) + LT(0, 1 + z0') :|: z'' = z2, z0' >= 0, z = 0, z' = 1 + z0', z2 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + z0', 1 + inc(z02)) + LT(0, 1 + z0') :|: z0' >= 0, z02 >= 0, z = 0, z' = 1 + z0', z'' = 1 + z02 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + z0', 1 + 0) + LT(0, 1 + z0') :|: z'' = 0, z0' >= 0, z = 0, z' = 1 + z0' DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + z05, 0) + INC(z2) :|: z'' = z2, z05 >= 0, z = 0, z' = 1 + z05, z2 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + z05, 1 + inc(z08)) + INC(1 + z08) :|: z08 >= 0, z'' = 1 + z08, z05 >= 0, z = 0, z' = 1 + z05 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + z05, 1 + 0) + INC(0) :|: z'' = 0, z05 >= 0, z = 0, z' = 1 + z05 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z0, 0, 0) + LT(z0, 0) :|: z'' = z2, z = z0, z0 >= 0, z2 >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z0, 0, 0) + INC(z2) :|: z'' = z2, z = z0, z0 >= 0, z2 >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z0, 0, 1 + inc(z01)) + LT(z0, 0) :|: z = z0, z01 >= 0, z0 >= 0, z' = 0, z'' = 1 + z01 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z0, 0, 1 + inc(z07)) + INC(1 + z07) :|: z = z0, z07 >= 0, z'' = 1 + z07, z0 >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z0, 0, 1 + 0) + LT(z0, 0) :|: z'' = 0, z = z0, z0 >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z0, 0, 1 + 0) + INC(0) :|: z'' = 0, z = z0, z0 >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z0, z1, 0) + LT(z0, z1) :|: z'' = z2, z = z0, z1 >= 0, z' = z1, z0 >= 0, z2 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z0, z1, 0) + INC(z2) :|: z'' = z2, z = z0, z1 >= 0, z' = z1, z0 >= 0, z2 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z0, z1, 1 + inc(z010)) + INC(1 + z010) :|: z = z0, z1 >= 0, z'' = 1 + z010, z' = z1, z0 >= 0, z010 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z0, z1, 1 + inc(z04)) + LT(z0, z1) :|: z'' = 1 + z04, z04 >= 0, z = z0, z1 >= 0, z' = z1, z0 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z0, z1, 1 + 0) + LT(z0, z1) :|: z'' = 0, z = z0, z1 >= 0, z' = z1, z0 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z0, z1, 1 + 0) + INC(0) :|: z'' = 0, z = z0, z1 >= 0, z' = z1, z0 >= 0 DIVISION(z, z') -{ 1 }-> 1 + DIV(z0, z1, 0) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z1 >= 0, z0 >= 0, z3 = z2, z' = z0, z2 >= 0, z'' = z1 IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(minus(z011, z1), 1 + z1, z2) + MINUS(1 + z011, 1 + z1) :|: z011 >= 0, z1 >= 0, z = 1, z' = 1 + z011, z'' = 1 + z1, z3 = z2, z2 >= 0 IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(0, 1 + z1, z2) + MINUS(z0, 1 + z1) :|: z1 >= 0, z = 1, z0 >= 0, z'' = 1 + z1, z3 = z2, z' = z0, z2 >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 }-> 1 + INC(z0) :|: z = 1 + z0, z0 >= 0 LT(z, z') -{ 1 }-> 1 :|: z0 >= 0, z' = 1 + z0, z = 0 LT(z, z') -{ 1 }-> 0 :|: z = z0, z0 >= 0, z' = 0 LT(z, z') -{ 1 }-> 1 + LT(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 MINUS(z, z') -{ 1 }-> 0 :|: z = z0, z0 >= 0, z' = 0 MINUS(z, z') -{ 1 }-> 1 + MINUS(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 div(z, z', z'') -{ 0 }-> if(lt(z013, z11), 1 + z013, 1 + z11, 0) :|: z' = 1 + z11, z'' = z2, z11 >= 0, z = 1 + z013, z013 >= 0, z2 >= 0 div(z, z', z'') -{ 0 }-> if(lt(z013, z11), 1 + z013, 1 + z11, 1 + inc(z016)) :|: z' = 1 + z11, z11 >= 0, z = 1 + z013, z'' = 1 + z016, z013 >= 0, z016 >= 0 div(z, z', z'') -{ 0 }-> if(lt(z013, z11), 1 + z013, 1 + z11, 1 + 0) :|: z'' = 0, z' = 1 + z11, z11 >= 0, z = 1 + z013, z013 >= 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + z012, 0) :|: z'' = z2, z' = 1 + z012, z012 >= 0, z = 0, z2 >= 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + z012, 1 + inc(z015)) :|: z'' = 1 + z015, z015 >= 0, z' = 1 + z012, z012 >= 0, z = 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + z012, 1 + 0) :|: z'' = 0, z' = 1 + z012, z012 >= 0, z = 0 div(z, z', z'') -{ 0 }-> if(1, z0, 0, 0) :|: z'' = z2, z = z0, z0 >= 0, z2 >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(1, z0, 0, 1 + inc(z014)) :|: z'' = 1 + z014, z = z0, z014 >= 0, z0 >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(1, z0, 0, 1 + 0) :|: z'' = 0, z = z0, z0 >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(0, z0, z1, 0) :|: z'' = z2, z = z0, z1 >= 0, z' = z1, z0 >= 0, z2 >= 0 div(z, z', z'') -{ 0 }-> if(0, z0, z1, 1 + inc(z017)) :|: z'' = 1 + z017, z = z0, z1 >= 0, z' = z1, z0 >= 0, z017 >= 0 div(z, z', z'') -{ 0 }-> if(0, z0, z1, 1 + 0) :|: z'' = 0, z = z0, z1 >= 0, z' = z1, z0 >= 0 div(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 division(z, z') -{ 0 }-> div(z0, z1, 0) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 division(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 if(z, z', z'', z3) -{ 0 }-> z2 :|: z = 2, z1 >= 0, z0 >= 0, z3 = z2, z' = z0, z2 >= 0, z'' = z1 if(z, z', z'', z3) -{ 0 }-> div(minus(z018, z1), 1 + z1, z2) :|: z1 >= 0, z = 1, z' = 1 + z018, z'' = 1 + z1, z3 = z2, z018 >= 0, z2 >= 0 if(z, z', z'', z3) -{ 0 }-> div(0, 1 + z1, z2) :|: z1 >= 0, z = 1, z0 >= 0, z'' = 1 + z1, z3 = z2, z' = z0, z2 >= 0 if(z, z', z'', z3) -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, z3 = v3, v2 >= 0, v3 >= 0 inc(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 inc(z) -{ 0 }-> 1 + inc(z0) :|: z = 1 + z0, z0 >= 0 inc(z) -{ 0 }-> 1 + 0 :|: z = 0 lt(z, z') -{ 0 }-> lt(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 lt(z, z') -{ 0 }-> 2 :|: z0 >= 0, z' = 1 + z0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z = z0, z0 >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 minus(z, z') -{ 0 }-> z0 :|: z = z0, z0 >= 0, z' = 0 minus(z, z') -{ 0 }-> minus(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 ---------------------------------------- (13) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + INC(z'') :|: z - 1 >= 0, z'' >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + inc(z'' - 1)) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + LT(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + INC(z'') :|: z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + inc(z'' - 1)) + LT(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + LT(0, 1 + (z' - 1)) :|: z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + LT(z, 0) :|: z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + INC(z'') :|: z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + inc(z'' - 1)) + LT(z, 0) :|: z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + LT(z, 0) :|: z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + INC(0) :|: z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + LT(z, z') :|: z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + INC(z'') :|: z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + inc(z'' - 1)) + LT(z, z') :|: z'' - 1 >= 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z' >= 0, z >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + LT(z, z') :|: z'' = 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + INC(0) :|: z'' = 0, z' >= 0, z >= 0 DIVISION(z, z') -{ 1 }-> 1 + DIV(z, z', 0) :|: z' >= 0, z >= 0 IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(minus(z' - 1, z'' - 1), 1 + (z'' - 1), z3) + MINUS(1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 1, z3 >= 0 IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 }-> 1 + INC(z - 1) :|: z - 1 >= 0 LT(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 LT(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 LT(z, z') -{ 1 }-> 1 + LT(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + inc(z'' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 0) :|: z' - 1 >= 0, z = 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 1 + inc(z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 1 + 0) :|: z'' = 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 0) :|: z >= 0, z'' >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 1 + inc(z'' - 1)) :|: z'' - 1 >= 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 1 + 0) :|: z'' = 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 1 + inc(z'' - 1)) :|: z' >= 0, z >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0 div(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 division(z, z') -{ 0 }-> div(z, z', 0) :|: z' >= 0, z >= 0 division(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> div(minus(z' - 1, z'' - 1), 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> div(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + inc(z - 1) :|: z - 1 >= 0 inc(z) -{ 0 }-> 1 + 0 :|: z = 0 lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 ---------------------------------------- (15) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { minus } { inc } { LT } { INC } { lt } { MINUS } { div, if } { IF, DIV } { division } { DIVISION } ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + INC(z'') :|: z - 1 >= 0, z'' >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + inc(z'' - 1)) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + LT(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + INC(z'') :|: z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + inc(z'' - 1)) + LT(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + LT(0, 1 + (z' - 1)) :|: z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + LT(z, 0) :|: z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + INC(z'') :|: z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + inc(z'' - 1)) + LT(z, 0) :|: z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + LT(z, 0) :|: z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + INC(0) :|: z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + LT(z, z') :|: z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + INC(z'') :|: z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + inc(z'' - 1)) + LT(z, z') :|: z'' - 1 >= 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z' >= 0, z >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + LT(z, z') :|: z'' = 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + INC(0) :|: z'' = 0, z' >= 0, z >= 0 DIVISION(z, z') -{ 1 }-> 1 + DIV(z, z', 0) :|: z' >= 0, z >= 0 IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(minus(z' - 1, z'' - 1), 1 + (z'' - 1), z3) + MINUS(1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 1, z3 >= 0 IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 }-> 1 + INC(z - 1) :|: z - 1 >= 0 LT(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 LT(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 LT(z, z') -{ 1 }-> 1 + LT(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + inc(z'' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 0) :|: z' - 1 >= 0, z = 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 1 + inc(z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 1 + 0) :|: z'' = 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 0) :|: z >= 0, z'' >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 1 + inc(z'' - 1)) :|: z'' - 1 >= 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 1 + 0) :|: z'' = 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 1 + inc(z'' - 1)) :|: z' >= 0, z >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0 div(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 division(z, z') -{ 0 }-> div(z, z', 0) :|: z' >= 0, z >= 0 division(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> div(minus(z' - 1, z'' - 1), 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> div(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + inc(z - 1) :|: z - 1 >= 0 inc(z) -{ 0 }-> 1 + 0 :|: z = 0 lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {minus}, {inc}, {LT}, {INC}, {lt}, {MINUS}, {div,if}, {IF,DIV}, {division}, {DIVISION} ---------------------------------------- (17) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + INC(z'') :|: z - 1 >= 0, z'' >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + inc(z'' - 1)) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + LT(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + INC(z'') :|: z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + inc(z'' - 1)) + LT(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + LT(0, 1 + (z' - 1)) :|: z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + LT(z, 0) :|: z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + INC(z'') :|: z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + inc(z'' - 1)) + LT(z, 0) :|: z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + LT(z, 0) :|: z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + INC(0) :|: z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + LT(z, z') :|: z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + INC(z'') :|: z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + inc(z'' - 1)) + LT(z, z') :|: z'' - 1 >= 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z' >= 0, z >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + LT(z, z') :|: z'' = 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + INC(0) :|: z'' = 0, z' >= 0, z >= 0 DIVISION(z, z') -{ 1 }-> 1 + DIV(z, z', 0) :|: z' >= 0, z >= 0 IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(minus(z' - 1, z'' - 1), 1 + (z'' - 1), z3) + MINUS(1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 1, z3 >= 0 IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 }-> 1 + INC(z - 1) :|: z - 1 >= 0 LT(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 LT(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 LT(z, z') -{ 1 }-> 1 + LT(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + inc(z'' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 0) :|: z' - 1 >= 0, z = 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 1 + inc(z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 1 + 0) :|: z'' = 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 0) :|: z >= 0, z'' >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 1 + inc(z'' - 1)) :|: z'' - 1 >= 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 1 + 0) :|: z'' = 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 1 + inc(z'' - 1)) :|: z' >= 0, z >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0 div(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 division(z, z') -{ 0 }-> div(z, z', 0) :|: z' >= 0, z >= 0 division(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> div(minus(z' - 1, z'' - 1), 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> div(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + inc(z - 1) :|: z - 1 >= 0 inc(z) -{ 0 }-> 1 + 0 :|: z = 0 lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {minus}, {inc}, {LT}, {INC}, {lt}, {MINUS}, {div,if}, {IF,DIV}, {division}, {DIVISION} ---------------------------------------- (19) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: minus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + INC(z'') :|: z - 1 >= 0, z'' >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + inc(z'' - 1)) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + LT(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + INC(z'') :|: z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + inc(z'' - 1)) + LT(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + LT(0, 1 + (z' - 1)) :|: z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + LT(z, 0) :|: z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + INC(z'') :|: z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + inc(z'' - 1)) + LT(z, 0) :|: z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + LT(z, 0) :|: z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + INC(0) :|: z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + LT(z, z') :|: z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + INC(z'') :|: z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + inc(z'' - 1)) + LT(z, z') :|: z'' - 1 >= 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z' >= 0, z >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + LT(z, z') :|: z'' = 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + INC(0) :|: z'' = 0, z' >= 0, z >= 0 DIVISION(z, z') -{ 1 }-> 1 + DIV(z, z', 0) :|: z' >= 0, z >= 0 IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(minus(z' - 1, z'' - 1), 1 + (z'' - 1), z3) + MINUS(1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 1, z3 >= 0 IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 }-> 1 + INC(z - 1) :|: z - 1 >= 0 LT(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 LT(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 LT(z, z') -{ 1 }-> 1 + LT(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + inc(z'' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 0) :|: z' - 1 >= 0, z = 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 1 + inc(z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 1 + 0) :|: z'' = 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 0) :|: z >= 0, z'' >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 1 + inc(z'' - 1)) :|: z'' - 1 >= 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 1 + 0) :|: z'' = 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 1 + inc(z'' - 1)) :|: z' >= 0, z >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0 div(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 division(z, z') -{ 0 }-> div(z, z', 0) :|: z' >= 0, z >= 0 division(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> div(minus(z' - 1, z'' - 1), 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> div(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + inc(z - 1) :|: z - 1 >= 0 inc(z) -{ 0 }-> 1 + 0 :|: z = 0 lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {minus}, {inc}, {LT}, {INC}, {lt}, {MINUS}, {div,if}, {IF,DIV}, {division}, {DIVISION} Previous analysis results are: minus: runtime: ?, size: O(n^1) [z] ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: minus after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + INC(z'') :|: z - 1 >= 0, z'' >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + inc(z'' - 1)) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + LT(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + INC(z'') :|: z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + inc(z'' - 1)) + LT(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + LT(0, 1 + (z' - 1)) :|: z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + LT(z, 0) :|: z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + INC(z'') :|: z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + inc(z'' - 1)) + LT(z, 0) :|: z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + LT(z, 0) :|: z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + INC(0) :|: z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + LT(z, z') :|: z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + INC(z'') :|: z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + inc(z'' - 1)) + LT(z, z') :|: z'' - 1 >= 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z' >= 0, z >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + LT(z, z') :|: z'' = 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + INC(0) :|: z'' = 0, z' >= 0, z >= 0 DIVISION(z, z') -{ 1 }-> 1 + DIV(z, z', 0) :|: z' >= 0, z >= 0 IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(minus(z' - 1, z'' - 1), 1 + (z'' - 1), z3) + MINUS(1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 1, z3 >= 0 IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 }-> 1 + INC(z - 1) :|: z - 1 >= 0 LT(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 LT(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 LT(z, z') -{ 1 }-> 1 + LT(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + inc(z'' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 0) :|: z' - 1 >= 0, z = 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 1 + inc(z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 1 + 0) :|: z'' = 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 0) :|: z >= 0, z'' >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 1 + inc(z'' - 1)) :|: z'' - 1 >= 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 1 + 0) :|: z'' = 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 1 + inc(z'' - 1)) :|: z' >= 0, z >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0 div(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 division(z, z') -{ 0 }-> div(z, z', 0) :|: z' >= 0, z >= 0 division(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> div(minus(z' - 1, z'' - 1), 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> div(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + inc(z - 1) :|: z - 1 >= 0 inc(z) -{ 0 }-> 1 + 0 :|: z = 0 lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {inc}, {LT}, {INC}, {lt}, {MINUS}, {div,if}, {IF,DIV}, {division}, {DIVISION} Previous analysis results are: minus: runtime: O(1) [0], size: O(n^1) [z] ---------------------------------------- (23) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + INC(z'') :|: z - 1 >= 0, z'' >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + inc(z'' - 1)) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + LT(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + INC(z'') :|: z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + inc(z'' - 1)) + LT(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + LT(0, 1 + (z' - 1)) :|: z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + LT(z, 0) :|: z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + INC(z'') :|: z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + inc(z'' - 1)) + LT(z, 0) :|: z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + LT(z, 0) :|: z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + INC(0) :|: z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + LT(z, z') :|: z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + INC(z'') :|: z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + inc(z'' - 1)) + LT(z, z') :|: z'' - 1 >= 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z' >= 0, z >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + LT(z, z') :|: z'' = 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + INC(0) :|: z'' = 0, z' >= 0, z >= 0 DIVISION(z, z') -{ 1 }-> 1 + DIV(z, z', 0) :|: z' >= 0, z >= 0 IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(s, 1 + (z'' - 1), z3) + MINUS(1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= z' - 1, z' - 1 >= 0, z'' - 1 >= 0, z = 1, z3 >= 0 IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 }-> 1 + INC(z - 1) :|: z - 1 >= 0 LT(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 LT(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 LT(z, z') -{ 1 }-> 1 + LT(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + inc(z'' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 0) :|: z' - 1 >= 0, z = 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 1 + inc(z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 1 + 0) :|: z'' = 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 0) :|: z >= 0, z'' >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 1 + inc(z'' - 1)) :|: z'' - 1 >= 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 1 + 0) :|: z'' = 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 1 + inc(z'' - 1)) :|: z' >= 0, z >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0 div(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 division(z, z') -{ 0 }-> div(z, z', 0) :|: z' >= 0, z >= 0 division(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> div(s', 1 + (z'' - 1), z3) :|: s' >= 0, s' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> div(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + inc(z - 1) :|: z - 1 >= 0 inc(z) -{ 0 }-> 1 + 0 :|: z = 0 lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {inc}, {LT}, {INC}, {lt}, {MINUS}, {div,if}, {IF,DIV}, {division}, {DIVISION} Previous analysis results are: minus: runtime: O(1) [0], size: O(n^1) [z] ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: inc after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + INC(z'') :|: z - 1 >= 0, z'' >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + inc(z'' - 1)) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + LT(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + INC(z'') :|: z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + inc(z'' - 1)) + LT(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + LT(0, 1 + (z' - 1)) :|: z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + LT(z, 0) :|: z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + INC(z'') :|: z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + inc(z'' - 1)) + LT(z, 0) :|: z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + LT(z, 0) :|: z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + INC(0) :|: z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + LT(z, z') :|: z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + INC(z'') :|: z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + inc(z'' - 1)) + LT(z, z') :|: z'' - 1 >= 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z' >= 0, z >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + LT(z, z') :|: z'' = 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + INC(0) :|: z'' = 0, z' >= 0, z >= 0 DIVISION(z, z') -{ 1 }-> 1 + DIV(z, z', 0) :|: z' >= 0, z >= 0 IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(s, 1 + (z'' - 1), z3) + MINUS(1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= z' - 1, z' - 1 >= 0, z'' - 1 >= 0, z = 1, z3 >= 0 IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 }-> 1 + INC(z - 1) :|: z - 1 >= 0 LT(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 LT(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 LT(z, z') -{ 1 }-> 1 + LT(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + inc(z'' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 0) :|: z' - 1 >= 0, z = 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 1 + inc(z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 1 + 0) :|: z'' = 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 0) :|: z >= 0, z'' >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 1 + inc(z'' - 1)) :|: z'' - 1 >= 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 1 + 0) :|: z'' = 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 1 + inc(z'' - 1)) :|: z' >= 0, z >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0 div(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 division(z, z') -{ 0 }-> div(z, z', 0) :|: z' >= 0, z >= 0 division(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> div(s', 1 + (z'' - 1), z3) :|: s' >= 0, s' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> div(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + inc(z - 1) :|: z - 1 >= 0 inc(z) -{ 0 }-> 1 + 0 :|: z = 0 lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {inc}, {LT}, {INC}, {lt}, {MINUS}, {div,if}, {IF,DIV}, {division}, {DIVISION} Previous analysis results are: minus: runtime: O(1) [0], size: O(n^1) [z] inc: runtime: ?, size: O(n^1) [1 + z] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: inc after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + INC(z'') :|: z - 1 >= 0, z'' >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + inc(z'' - 1)) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + LT(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + INC(z'') :|: z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + inc(z'' - 1)) + LT(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + LT(0, 1 + (z' - 1)) :|: z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + LT(z, 0) :|: z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + INC(z'') :|: z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + inc(z'' - 1)) + LT(z, 0) :|: z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + LT(z, 0) :|: z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + INC(0) :|: z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + LT(z, z') :|: z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + INC(z'') :|: z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + inc(z'' - 1)) + LT(z, z') :|: z'' - 1 >= 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + inc(z'' - 1)) + INC(1 + (z'' - 1)) :|: z' >= 0, z >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + LT(z, z') :|: z'' = 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + INC(0) :|: z'' = 0, z' >= 0, z >= 0 DIVISION(z, z') -{ 1 }-> 1 + DIV(z, z', 0) :|: z' >= 0, z >= 0 IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(s, 1 + (z'' - 1), z3) + MINUS(1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= z' - 1, z' - 1 >= 0, z'' - 1 >= 0, z = 1, z3 >= 0 IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 }-> 1 + INC(z - 1) :|: z - 1 >= 0 LT(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 LT(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 LT(z, z') -{ 1 }-> 1 + LT(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + inc(z'' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 0) :|: z' - 1 >= 0, z = 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 1 + inc(z'' - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 1 + 0) :|: z'' = 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 0) :|: z >= 0, z'' >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 1 + inc(z'' - 1)) :|: z'' - 1 >= 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 1 + 0) :|: z'' = 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 1 + inc(z'' - 1)) :|: z' >= 0, z >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0 div(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 division(z, z') -{ 0 }-> div(z, z', 0) :|: z' >= 0, z >= 0 division(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> div(s', 1 + (z'' - 1), z3) :|: s' >= 0, s' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> div(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + inc(z - 1) :|: z - 1 >= 0 inc(z) -{ 0 }-> 1 + 0 :|: z = 0 lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {LT}, {INC}, {lt}, {MINUS}, {div,if}, {IF,DIV}, {division}, {DIVISION} Previous analysis results are: minus: runtime: O(1) [0], size: O(n^1) [z] inc: runtime: O(1) [0], size: O(n^1) [1 + z] ---------------------------------------- (29) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + INC(z'') :|: z - 1 >= 0, z'' >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + s3) + LT(1 + (z - 1), 1 + (z' - 1)) :|: s3 >= 0, s3 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + s7) + INC(1 + (z'' - 1)) :|: s7 >= 0, s7 <= z'' - 1 + 1, z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + LT(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + INC(z'') :|: z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s2) + LT(0, 1 + (z' - 1)) :|: s2 >= 0, s2 <= z'' - 1 + 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s6) + INC(1 + (z'' - 1)) :|: s6 >= 0, s6 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + LT(0, 1 + (z' - 1)) :|: z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + LT(z, 0) :|: z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + INC(z'') :|: z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + s1) + LT(z, 0) :|: s1 >= 0, s1 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + s5) + INC(1 + (z'' - 1)) :|: s5 >= 0, s5 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + LT(z, 0) :|: z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + INC(0) :|: z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + LT(z, z') :|: z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + INC(z'') :|: z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + s4) + LT(z, z') :|: s4 >= 0, s4 <= z'' - 1 + 1, z'' - 1 >= 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + s8) + INC(1 + (z'' - 1)) :|: s8 >= 0, s8 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + LT(z, z') :|: z'' = 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + INC(0) :|: z'' = 0, z' >= 0, z >= 0 DIVISION(z, z') -{ 1 }-> 1 + DIV(z, z', 0) :|: z' >= 0, z >= 0 IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(s, 1 + (z'' - 1), z3) + MINUS(1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= z' - 1, z' - 1 >= 0, z'' - 1 >= 0, z = 1, z3 >= 0 IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 }-> 1 + INC(z - 1) :|: z - 1 >= 0 LT(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 LT(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 LT(z, z') -{ 1 }-> 1 + LT(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + s11) :|: s11 >= 0, s11 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 0) :|: z' - 1 >= 0, z = 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 1 + s10) :|: s10 >= 0, s10 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 1 + 0) :|: z'' = 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 0) :|: z >= 0, z'' >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 1 + s9) :|: s9 >= 0, s9 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 1 + 0) :|: z'' = 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 1 + s12) :|: s12 >= 0, s12 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0 div(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 division(z, z') -{ 0 }-> div(z, z', 0) :|: z' >= 0, z >= 0 division(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> div(s', 1 + (z'' - 1), z3) :|: s' >= 0, s' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> div(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s13 :|: s13 >= 0, s13 <= z - 1 + 1, z - 1 >= 0 inc(z) -{ 0 }-> 1 + 0 :|: z = 0 lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {LT}, {INC}, {lt}, {MINUS}, {div,if}, {IF,DIV}, {division}, {DIVISION} Previous analysis results are: minus: runtime: O(1) [0], size: O(n^1) [z] inc: runtime: O(1) [0], size: O(n^1) [1 + z] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: LT after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + INC(z'') :|: z - 1 >= 0, z'' >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + s3) + LT(1 + (z - 1), 1 + (z' - 1)) :|: s3 >= 0, s3 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + s7) + INC(1 + (z'' - 1)) :|: s7 >= 0, s7 <= z'' - 1 + 1, z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + LT(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + INC(z'') :|: z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s2) + LT(0, 1 + (z' - 1)) :|: s2 >= 0, s2 <= z'' - 1 + 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s6) + INC(1 + (z'' - 1)) :|: s6 >= 0, s6 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + LT(0, 1 + (z' - 1)) :|: z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + LT(z, 0) :|: z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + INC(z'') :|: z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + s1) + LT(z, 0) :|: s1 >= 0, s1 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + s5) + INC(1 + (z'' - 1)) :|: s5 >= 0, s5 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + LT(z, 0) :|: z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + INC(0) :|: z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + LT(z, z') :|: z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + INC(z'') :|: z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + s4) + LT(z, z') :|: s4 >= 0, s4 <= z'' - 1 + 1, z'' - 1 >= 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + s8) + INC(1 + (z'' - 1)) :|: s8 >= 0, s8 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + LT(z, z') :|: z'' = 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + INC(0) :|: z'' = 0, z' >= 0, z >= 0 DIVISION(z, z') -{ 1 }-> 1 + DIV(z, z', 0) :|: z' >= 0, z >= 0 IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(s, 1 + (z'' - 1), z3) + MINUS(1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= z' - 1, z' - 1 >= 0, z'' - 1 >= 0, z = 1, z3 >= 0 IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 }-> 1 + INC(z - 1) :|: z - 1 >= 0 LT(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 LT(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 LT(z, z') -{ 1 }-> 1 + LT(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + s11) :|: s11 >= 0, s11 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 0) :|: z' - 1 >= 0, z = 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 1 + s10) :|: s10 >= 0, s10 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 1 + 0) :|: z'' = 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 0) :|: z >= 0, z'' >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 1 + s9) :|: s9 >= 0, s9 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 1 + 0) :|: z'' = 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 1 + s12) :|: s12 >= 0, s12 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0 div(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 division(z, z') -{ 0 }-> div(z, z', 0) :|: z' >= 0, z >= 0 division(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> div(s', 1 + (z'' - 1), z3) :|: s' >= 0, s' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> div(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s13 :|: s13 >= 0, s13 <= z - 1 + 1, z - 1 >= 0 inc(z) -{ 0 }-> 1 + 0 :|: z = 0 lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {LT}, {INC}, {lt}, {MINUS}, {div,if}, {IF,DIV}, {division}, {DIVISION} Previous analysis results are: minus: runtime: O(1) [0], size: O(n^1) [z] inc: runtime: O(1) [0], size: O(n^1) [1 + z] LT: runtime: ?, size: O(n^1) [z'] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: LT after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z' ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + INC(z'') :|: z - 1 >= 0, z'' >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + s3) + LT(1 + (z - 1), 1 + (z' - 1)) :|: s3 >= 0, s3 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + s7) + INC(1 + (z'' - 1)) :|: s7 >= 0, s7 <= z'' - 1 + 1, z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + LT(1 + (z - 1), 1 + (z' - 1)) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + LT(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + INC(z'') :|: z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s2) + LT(0, 1 + (z' - 1)) :|: s2 >= 0, s2 <= z'' - 1 + 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s6) + INC(1 + (z'' - 1)) :|: s6 >= 0, s6 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + LT(0, 1 + (z' - 1)) :|: z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + LT(z, 0) :|: z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + INC(z'') :|: z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + s1) + LT(z, 0) :|: s1 >= 0, s1 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + s5) + INC(1 + (z'' - 1)) :|: s5 >= 0, s5 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + LT(z, 0) :|: z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + INC(0) :|: z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + LT(z, z') :|: z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + INC(z'') :|: z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + s4) + LT(z, z') :|: s4 >= 0, s4 <= z'' - 1 + 1, z'' - 1 >= 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + s8) + INC(1 + (z'' - 1)) :|: s8 >= 0, s8 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + LT(z, z') :|: z'' = 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + INC(0) :|: z'' = 0, z' >= 0, z >= 0 DIVISION(z, z') -{ 1 }-> 1 + DIV(z, z', 0) :|: z' >= 0, z >= 0 IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(s, 1 + (z'' - 1), z3) + MINUS(1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= z' - 1, z' - 1 >= 0, z'' - 1 >= 0, z = 1, z3 >= 0 IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 }-> 1 + INC(z - 1) :|: z - 1 >= 0 LT(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 LT(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 LT(z, z') -{ 1 }-> 1 + LT(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + s11) :|: s11 >= 0, s11 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 0) :|: z' - 1 >= 0, z = 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 1 + s10) :|: s10 >= 0, s10 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 1 + 0) :|: z'' = 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 0) :|: z >= 0, z'' >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 1 + s9) :|: s9 >= 0, s9 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 1 + 0) :|: z'' = 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 1 + s12) :|: s12 >= 0, s12 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0 div(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 division(z, z') -{ 0 }-> div(z, z', 0) :|: z' >= 0, z >= 0 division(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> div(s', 1 + (z'' - 1), z3) :|: s' >= 0, s' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> div(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s13 :|: s13 >= 0, s13 <= z - 1 + 1, z - 1 >= 0 inc(z) -{ 0 }-> 1 + 0 :|: z = 0 lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {INC}, {lt}, {MINUS}, {div,if}, {IF,DIV}, {division}, {DIVISION} Previous analysis results are: minus: runtime: O(1) [0], size: O(n^1) [z] inc: runtime: O(1) [0], size: O(n^1) [1 + z] LT: runtime: O(n^1) [2 + z'], size: O(n^1) [z'] ---------------------------------------- (35) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + s22 :|: s22 >= 0, s22 <= 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + INC(z'') :|: z - 1 >= 0, z'' >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + s3) + s21 :|: s21 >= 0, s21 <= 1 + (z' - 1), s3 >= 0, s3 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + s7) + INC(1 + (z'' - 1)) :|: s7 >= 0, s7 <= z'' - 1 + 1, z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + s20 :|: s20 >= 0, s20 <= 1 + (z' - 1), z'' = 0, z' - 1 >= 0, z - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + s19 :|: s19 >= 0, s19 <= 1 + (z' - 1), z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + INC(z'') :|: z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s2) + s18 :|: s18 >= 0, s18 <= 1 + (z' - 1), s2 >= 0, s2 <= z'' - 1 + 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s6) + INC(1 + (z'' - 1)) :|: s6 >= 0, s6 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + s17 :|: s17 >= 0, s17 <= 1 + (z' - 1), z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 3 }-> 1 + IF(1, z, 0, 0) + s16 :|: s16 >= 0, s16 <= 0, z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + INC(z'') :|: z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 3 }-> 1 + IF(1, z, 0, 1 + s1) + s15 :|: s15 >= 0, s15 <= 0, s1 >= 0, s1 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + s5) + INC(1 + (z'' - 1)) :|: s5 >= 0, s5 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 3 }-> 1 + IF(1, z, 0, 1 + 0) + s14 :|: s14 >= 0, s14 <= 0, z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + INC(0) :|: z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(0, z, z', 0) + s25 :|: s25 >= 0, s25 <= z', z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + INC(z'') :|: z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(0, z, z', 1 + s4) + s24 :|: s24 >= 0, s24 <= z', s4 >= 0, s4 <= z'' - 1 + 1, z'' - 1 >= 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + s8) + INC(1 + (z'' - 1)) :|: s8 >= 0, s8 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(0, z, z', 1 + 0) + s23 :|: s23 >= 0, s23 <= z', z'' = 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + INC(0) :|: z'' = 0, z' >= 0, z >= 0 DIVISION(z, z') -{ 1 }-> 1 + DIV(z, z', 0) :|: z' >= 0, z >= 0 IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(s, 1 + (z'' - 1), z3) + MINUS(1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= z' - 1, z' - 1 >= 0, z'' - 1 >= 0, z = 1, z3 >= 0 IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 }-> 1 + INC(z - 1) :|: z - 1 >= 0 LT(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 LT(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 LT(z, z') -{ 2 + z' }-> 1 + s26 :|: s26 >= 0, s26 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + s11) :|: s11 >= 0, s11 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 0) :|: z' - 1 >= 0, z = 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 1 + s10) :|: s10 >= 0, s10 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 1 + 0) :|: z'' = 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 0) :|: z >= 0, z'' >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 1 + s9) :|: s9 >= 0, s9 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 1 + 0) :|: z'' = 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 1 + s12) :|: s12 >= 0, s12 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0 div(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 division(z, z') -{ 0 }-> div(z, z', 0) :|: z' >= 0, z >= 0 division(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> div(s', 1 + (z'' - 1), z3) :|: s' >= 0, s' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> div(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s13 :|: s13 >= 0, s13 <= z - 1 + 1, z - 1 >= 0 inc(z) -{ 0 }-> 1 + 0 :|: z = 0 lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {INC}, {lt}, {MINUS}, {div,if}, {IF,DIV}, {division}, {DIVISION} Previous analysis results are: minus: runtime: O(1) [0], size: O(n^1) [z] inc: runtime: O(1) [0], size: O(n^1) [1 + z] LT: runtime: O(n^1) [2 + z'], size: O(n^1) [z'] ---------------------------------------- (37) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: INC after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + s22 :|: s22 >= 0, s22 <= 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + INC(z'') :|: z - 1 >= 0, z'' >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + s3) + s21 :|: s21 >= 0, s21 <= 1 + (z' - 1), s3 >= 0, s3 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + s7) + INC(1 + (z'' - 1)) :|: s7 >= 0, s7 <= z'' - 1 + 1, z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + s20 :|: s20 >= 0, s20 <= 1 + (z' - 1), z'' = 0, z' - 1 >= 0, z - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + s19 :|: s19 >= 0, s19 <= 1 + (z' - 1), z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + INC(z'') :|: z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s2) + s18 :|: s18 >= 0, s18 <= 1 + (z' - 1), s2 >= 0, s2 <= z'' - 1 + 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s6) + INC(1 + (z'' - 1)) :|: s6 >= 0, s6 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + s17 :|: s17 >= 0, s17 <= 1 + (z' - 1), z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 3 }-> 1 + IF(1, z, 0, 0) + s16 :|: s16 >= 0, s16 <= 0, z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + INC(z'') :|: z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 3 }-> 1 + IF(1, z, 0, 1 + s1) + s15 :|: s15 >= 0, s15 <= 0, s1 >= 0, s1 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + s5) + INC(1 + (z'' - 1)) :|: s5 >= 0, s5 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 3 }-> 1 + IF(1, z, 0, 1 + 0) + s14 :|: s14 >= 0, s14 <= 0, z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + INC(0) :|: z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(0, z, z', 0) + s25 :|: s25 >= 0, s25 <= z', z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + INC(z'') :|: z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(0, z, z', 1 + s4) + s24 :|: s24 >= 0, s24 <= z', s4 >= 0, s4 <= z'' - 1 + 1, z'' - 1 >= 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + s8) + INC(1 + (z'' - 1)) :|: s8 >= 0, s8 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(0, z, z', 1 + 0) + s23 :|: s23 >= 0, s23 <= z', z'' = 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + INC(0) :|: z'' = 0, z' >= 0, z >= 0 DIVISION(z, z') -{ 1 }-> 1 + DIV(z, z', 0) :|: z' >= 0, z >= 0 IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(s, 1 + (z'' - 1), z3) + MINUS(1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= z' - 1, z' - 1 >= 0, z'' - 1 >= 0, z = 1, z3 >= 0 IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 }-> 1 + INC(z - 1) :|: z - 1 >= 0 LT(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 LT(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 LT(z, z') -{ 2 + z' }-> 1 + s26 :|: s26 >= 0, s26 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + s11) :|: s11 >= 0, s11 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 0) :|: z' - 1 >= 0, z = 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 1 + s10) :|: s10 >= 0, s10 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 1 + 0) :|: z'' = 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 0) :|: z >= 0, z'' >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 1 + s9) :|: s9 >= 0, s9 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 1 + 0) :|: z'' = 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 1 + s12) :|: s12 >= 0, s12 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0 div(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 division(z, z') -{ 0 }-> div(z, z', 0) :|: z' >= 0, z >= 0 division(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> div(s', 1 + (z'' - 1), z3) :|: s' >= 0, s' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> div(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s13 :|: s13 >= 0, s13 <= z - 1 + 1, z - 1 >= 0 inc(z) -{ 0 }-> 1 + 0 :|: z = 0 lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {INC}, {lt}, {MINUS}, {div,if}, {IF,DIV}, {division}, {DIVISION} Previous analysis results are: minus: runtime: O(1) [0], size: O(n^1) [z] inc: runtime: O(1) [0], size: O(n^1) [1 + z] LT: runtime: O(n^1) [2 + z'], size: O(n^1) [z'] INC: runtime: ?, size: O(n^1) [z] ---------------------------------------- (39) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: INC after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + s22 :|: s22 >= 0, s22 <= 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + INC(z'') :|: z - 1 >= 0, z'' >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + s3) + s21 :|: s21 >= 0, s21 <= 1 + (z' - 1), s3 >= 0, s3 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + s7) + INC(1 + (z'' - 1)) :|: s7 >= 0, s7 <= z'' - 1 + 1, z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + s20 :|: s20 >= 0, s20 <= 1 + (z' - 1), z'' = 0, z' - 1 >= 0, z - 1 >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + s19 :|: s19 >= 0, s19 <= 1 + (z' - 1), z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + INC(z'') :|: z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s2) + s18 :|: s18 >= 0, s18 <= 1 + (z' - 1), s2 >= 0, s2 <= z'' - 1 + 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s6) + INC(1 + (z'' - 1)) :|: s6 >= 0, s6 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + s17 :|: s17 >= 0, s17 <= 1 + (z' - 1), z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + INC(0) :|: z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 3 }-> 1 + IF(1, z, 0, 0) + s16 :|: s16 >= 0, s16 <= 0, z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 0) + INC(z'') :|: z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 3 }-> 1 + IF(1, z, 0, 1 + s1) + s15 :|: s15 >= 0, s15 <= 0, s1 >= 0, s1 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + s5) + INC(1 + (z'' - 1)) :|: s5 >= 0, s5 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 3 }-> 1 + IF(1, z, 0, 1 + 0) + s14 :|: s14 >= 0, s14 <= 0, z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(1, z, 0, 1 + 0) + INC(0) :|: z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(0, z, z', 0) + s25 :|: s25 >= 0, s25 <= z', z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 0) + INC(z'') :|: z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(0, z, z', 1 + s4) + s24 :|: s24 >= 0, s24 <= z', s4 >= 0, s4 <= z'' - 1 + 1, z'' - 1 >= 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + s8) + INC(1 + (z'' - 1)) :|: s8 >= 0, s8 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(0, z, z', 1 + 0) + s23 :|: s23 >= 0, s23 <= z', z'' = 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 1 }-> 1 + IF(0, z, z', 1 + 0) + INC(0) :|: z'' = 0, z' >= 0, z >= 0 DIVISION(z, z') -{ 1 }-> 1 + DIV(z, z', 0) :|: z' >= 0, z >= 0 IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(s, 1 + (z'' - 1), z3) + MINUS(1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= z' - 1, z' - 1 >= 0, z'' - 1 >= 0, z = 1, z3 >= 0 IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 }-> 1 + INC(z - 1) :|: z - 1 >= 0 LT(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 LT(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 LT(z, z') -{ 2 + z' }-> 1 + s26 :|: s26 >= 0, s26 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + s11) :|: s11 >= 0, s11 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 0) :|: z' - 1 >= 0, z = 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 1 + s10) :|: s10 >= 0, s10 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 1 + 0) :|: z'' = 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 0) :|: z >= 0, z'' >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 1 + s9) :|: s9 >= 0, s9 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 1 + 0) :|: z'' = 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 1 + s12) :|: s12 >= 0, s12 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0 div(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 division(z, z') -{ 0 }-> div(z, z', 0) :|: z' >= 0, z >= 0 division(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> div(s', 1 + (z'' - 1), z3) :|: s' >= 0, s' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> div(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s13 :|: s13 >= 0, s13 <= z - 1 + 1, z - 1 >= 0 inc(z) -{ 0 }-> 1 + 0 :|: z = 0 lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {lt}, {MINUS}, {div,if}, {IF,DIV}, {division}, {DIVISION} Previous analysis results are: minus: runtime: O(1) [0], size: O(n^1) [z] inc: runtime: O(1) [0], size: O(n^1) [1 + z] LT: runtime: O(n^1) [2 + z'], size: O(n^1) [z'] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] ---------------------------------------- (41) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + s22 :|: s22 >= 0, s22 <= 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0, z'' >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + s35 :|: s35 >= 0, s35 <= z'', z - 1 >= 0, z'' >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + s3) + s21 :|: s21 >= 0, s21 <= 1 + (z' - 1), s3 >= 0, s3 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + s7) + s34 :|: s34 >= 0, s34 <= 1 + (z'' - 1), s7 >= 0, s7 <= z'' - 1 + 1, z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + s20 :|: s20 >= 0, s20 <= 1 + (z' - 1), z'' = 0, z' - 1 >= 0, z - 1 >= 0 DIV(z, z', z'') -{ 2 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + s33 :|: s33 >= 0, s33 <= 0, z'' = 0, z - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + s19 :|: s19 >= 0, s19 <= 1 + (z' - 1), z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + s32 :|: s32 >= 0, s32 <= z'', z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s2) + s18 :|: s18 >= 0, s18 <= 1 + (z' - 1), s2 >= 0, s2 <= z'' - 1 + 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s6) + s31 :|: s31 >= 0, s31 <= 1 + (z'' - 1), s6 >= 0, s6 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + s17 :|: s17 >= 0, s17 <= 1 + (z' - 1), z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 2 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + s30 :|: s30 >= 0, s30 <= 0, z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 3 }-> 1 + IF(1, z, 0, 0) + s16 :|: s16 >= 0, s16 <= 0, z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, z, 0, 0) + s29 :|: s29 >= 0, s29 <= z'', z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 3 }-> 1 + IF(1, z, 0, 1 + s1) + s15 :|: s15 >= 0, s15 <= 0, s1 >= 0, s1 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, z, 0, 1 + s5) + s28 :|: s28 >= 0, s28 <= 1 + (z'' - 1), s5 >= 0, s5 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 3 }-> 1 + IF(1, z, 0, 1 + 0) + s14 :|: s14 >= 0, s14 <= 0, z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 2 }-> 1 + IF(1, z, 0, 1 + 0) + s27 :|: s27 >= 0, s27 <= 0, z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(0, z, z', 0) + s25 :|: s25 >= 0, s25 <= z', z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', 0) + s38 :|: s38 >= 0, s38 <= z'', z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(0, z, z', 1 + s4) + s24 :|: s24 >= 0, s24 <= z', s4 >= 0, s4 <= z'' - 1 + 1, z'' - 1 >= 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', 1 + s8) + s37 :|: s37 >= 0, s37 <= 1 + (z'' - 1), s8 >= 0, s8 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(0, z, z', 1 + 0) + s23 :|: s23 >= 0, s23 <= z', z'' = 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + s36 :|: s36 >= 0, s36 <= 0, z'' = 0, z' >= 0, z >= 0 DIVISION(z, z') -{ 1 }-> 1 + DIV(z, z', 0) :|: z' >= 0, z >= 0 IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(s, 1 + (z'' - 1), z3) + MINUS(1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= z' - 1, z' - 1 >= 0, z'' - 1 >= 0, z = 1, z3 >= 0 IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 + z }-> 1 + s39 :|: s39 >= 0, s39 <= z - 1, z - 1 >= 0 LT(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 LT(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 LT(z, z') -{ 2 + z' }-> 1 + s26 :|: s26 >= 0, s26 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + s11) :|: s11 >= 0, s11 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 0) :|: z' - 1 >= 0, z = 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 1 + s10) :|: s10 >= 0, s10 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 1 + 0) :|: z'' = 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 0) :|: z >= 0, z'' >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 1 + s9) :|: s9 >= 0, s9 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 1 + 0) :|: z'' = 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 1 + s12) :|: s12 >= 0, s12 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0 div(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 division(z, z') -{ 0 }-> div(z, z', 0) :|: z' >= 0, z >= 0 division(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> div(s', 1 + (z'' - 1), z3) :|: s' >= 0, s' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> div(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s13 :|: s13 >= 0, s13 <= z - 1 + 1, z - 1 >= 0 inc(z) -{ 0 }-> 1 + 0 :|: z = 0 lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {lt}, {MINUS}, {div,if}, {IF,DIV}, {division}, {DIVISION} Previous analysis results are: minus: runtime: O(1) [0], size: O(n^1) [z] inc: runtime: O(1) [0], size: O(n^1) [1 + z] LT: runtime: O(n^1) [2 + z'], size: O(n^1) [z'] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] ---------------------------------------- (43) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: lt after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + s22 :|: s22 >= 0, s22 <= 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0, z'' >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + s35 :|: s35 >= 0, s35 <= z'', z - 1 >= 0, z'' >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + s3) + s21 :|: s21 >= 0, s21 <= 1 + (z' - 1), s3 >= 0, s3 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + s7) + s34 :|: s34 >= 0, s34 <= 1 + (z'' - 1), s7 >= 0, s7 <= z'' - 1 + 1, z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + s20 :|: s20 >= 0, s20 <= 1 + (z' - 1), z'' = 0, z' - 1 >= 0, z - 1 >= 0 DIV(z, z', z'') -{ 2 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + s33 :|: s33 >= 0, s33 <= 0, z'' = 0, z - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + s19 :|: s19 >= 0, s19 <= 1 + (z' - 1), z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + s32 :|: s32 >= 0, s32 <= z'', z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s2) + s18 :|: s18 >= 0, s18 <= 1 + (z' - 1), s2 >= 0, s2 <= z'' - 1 + 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s6) + s31 :|: s31 >= 0, s31 <= 1 + (z'' - 1), s6 >= 0, s6 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + s17 :|: s17 >= 0, s17 <= 1 + (z' - 1), z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 2 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + s30 :|: s30 >= 0, s30 <= 0, z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 3 }-> 1 + IF(1, z, 0, 0) + s16 :|: s16 >= 0, s16 <= 0, z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, z, 0, 0) + s29 :|: s29 >= 0, s29 <= z'', z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 3 }-> 1 + IF(1, z, 0, 1 + s1) + s15 :|: s15 >= 0, s15 <= 0, s1 >= 0, s1 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, z, 0, 1 + s5) + s28 :|: s28 >= 0, s28 <= 1 + (z'' - 1), s5 >= 0, s5 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 3 }-> 1 + IF(1, z, 0, 1 + 0) + s14 :|: s14 >= 0, s14 <= 0, z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 2 }-> 1 + IF(1, z, 0, 1 + 0) + s27 :|: s27 >= 0, s27 <= 0, z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(0, z, z', 0) + s25 :|: s25 >= 0, s25 <= z', z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', 0) + s38 :|: s38 >= 0, s38 <= z'', z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(0, z, z', 1 + s4) + s24 :|: s24 >= 0, s24 <= z', s4 >= 0, s4 <= z'' - 1 + 1, z'' - 1 >= 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', 1 + s8) + s37 :|: s37 >= 0, s37 <= 1 + (z'' - 1), s8 >= 0, s8 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(0, z, z', 1 + 0) + s23 :|: s23 >= 0, s23 <= z', z'' = 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + s36 :|: s36 >= 0, s36 <= 0, z'' = 0, z' >= 0, z >= 0 DIVISION(z, z') -{ 1 }-> 1 + DIV(z, z', 0) :|: z' >= 0, z >= 0 IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(s, 1 + (z'' - 1), z3) + MINUS(1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= z' - 1, z' - 1 >= 0, z'' - 1 >= 0, z = 1, z3 >= 0 IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 + z }-> 1 + s39 :|: s39 >= 0, s39 <= z - 1, z - 1 >= 0 LT(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 LT(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 LT(z, z') -{ 2 + z' }-> 1 + s26 :|: s26 >= 0, s26 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + s11) :|: s11 >= 0, s11 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 0) :|: z' - 1 >= 0, z = 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 1 + s10) :|: s10 >= 0, s10 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 1 + 0) :|: z'' = 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 0) :|: z >= 0, z'' >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 1 + s9) :|: s9 >= 0, s9 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 1 + 0) :|: z'' = 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 1 + s12) :|: s12 >= 0, s12 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0 div(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 division(z, z') -{ 0 }-> div(z, z', 0) :|: z' >= 0, z >= 0 division(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> div(s', 1 + (z'' - 1), z3) :|: s' >= 0, s' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> div(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s13 :|: s13 >= 0, s13 <= z - 1 + 1, z - 1 >= 0 inc(z) -{ 0 }-> 1 + 0 :|: z = 0 lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {lt}, {MINUS}, {div,if}, {IF,DIV}, {division}, {DIVISION} Previous analysis results are: minus: runtime: O(1) [0], size: O(n^1) [z] inc: runtime: O(1) [0], size: O(n^1) [1 + z] LT: runtime: O(n^1) [2 + z'], size: O(n^1) [z'] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] lt: runtime: ?, size: O(1) [2] ---------------------------------------- (45) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: lt after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + s22 :|: s22 >= 0, s22 <= 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0, z'' >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) + s35 :|: s35 >= 0, s35 <= z'', z - 1 >= 0, z'' >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + s3) + s21 :|: s21 >= 0, s21 <= 1 + (z' - 1), s3 >= 0, s3 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + s7) + s34 :|: s34 >= 0, s34 <= 1 + (z'' - 1), s7 >= 0, s7 <= z'' - 1 + 1, z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + s20 :|: s20 >= 0, s20 <= 1 + (z' - 1), z'' = 0, z' - 1 >= 0, z - 1 >= 0 DIV(z, z', z'') -{ 2 }-> 1 + IF(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) + s33 :|: s33 >= 0, s33 <= 0, z'' = 0, z - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + s19 :|: s19 >= 0, s19 <= 1 + (z' - 1), z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + s32 :|: s32 >= 0, s32 <= z'', z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s2) + s18 :|: s18 >= 0, s18 <= 1 + (z' - 1), s2 >= 0, s2 <= z'' - 1 + 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s6) + s31 :|: s31 >= 0, s31 <= 1 + (z'' - 1), s6 >= 0, s6 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + s17 :|: s17 >= 0, s17 <= 1 + (z' - 1), z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 2 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + s30 :|: s30 >= 0, s30 <= 0, z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 3 }-> 1 + IF(1, z, 0, 0) + s16 :|: s16 >= 0, s16 <= 0, z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, z, 0, 0) + s29 :|: s29 >= 0, s29 <= z'', z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 3 }-> 1 + IF(1, z, 0, 1 + s1) + s15 :|: s15 >= 0, s15 <= 0, s1 >= 0, s1 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, z, 0, 1 + s5) + s28 :|: s28 >= 0, s28 <= 1 + (z'' - 1), s5 >= 0, s5 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 3 }-> 1 + IF(1, z, 0, 1 + 0) + s14 :|: s14 >= 0, s14 <= 0, z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 2 }-> 1 + IF(1, z, 0, 1 + 0) + s27 :|: s27 >= 0, s27 <= 0, z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(0, z, z', 0) + s25 :|: s25 >= 0, s25 <= z', z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', 0) + s38 :|: s38 >= 0, s38 <= z'', z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(0, z, z', 1 + s4) + s24 :|: s24 >= 0, s24 <= z', s4 >= 0, s4 <= z'' - 1 + 1, z'' - 1 >= 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', 1 + s8) + s37 :|: s37 >= 0, s37 <= 1 + (z'' - 1), s8 >= 0, s8 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(0, z, z', 1 + 0) + s23 :|: s23 >= 0, s23 <= z', z'' = 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + s36 :|: s36 >= 0, s36 <= 0, z'' = 0, z' >= 0, z >= 0 DIVISION(z, z') -{ 1 }-> 1 + DIV(z, z', 0) :|: z' >= 0, z >= 0 IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(s, 1 + (z'' - 1), z3) + MINUS(1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= z' - 1, z' - 1 >= 0, z'' - 1 >= 0, z = 1, z3 >= 0 IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 + z }-> 1 + s39 :|: s39 >= 0, s39 <= z - 1, z - 1 >= 0 LT(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 LT(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 LT(z, z') -{ 2 + z' }-> 1 + s26 :|: s26 >= 0, s26 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 0) :|: z' - 1 >= 0, z - 1 >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + s11) :|: s11 >= 0, s11 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), 1 + 0) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 0) :|: z' - 1 >= 0, z = 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 1 + s10) :|: s10 >= 0, s10 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 1 + 0) :|: z'' = 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 0) :|: z >= 0, z'' >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 1 + s9) :|: s9 >= 0, s9 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 1 + 0) :|: z'' = 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 1 + s12) :|: s12 >= 0, s12 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0 div(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 division(z, z') -{ 0 }-> div(z, z', 0) :|: z' >= 0, z >= 0 division(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> div(s', 1 + (z'' - 1), z3) :|: s' >= 0, s' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> div(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s13 :|: s13 >= 0, s13 <= z - 1 + 1, z - 1 >= 0 inc(z) -{ 0 }-> 1 + 0 :|: z = 0 lt(z, z') -{ 0 }-> lt(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {MINUS}, {div,if}, {IF,DIV}, {division}, {DIVISION} Previous analysis results are: minus: runtime: O(1) [0], size: O(n^1) [z] inc: runtime: O(1) [0], size: O(n^1) [1 + z] LT: runtime: O(n^1) [2 + z'], size: O(n^1) [z'] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] lt: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (47) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(s40, 1 + (z - 1), 1 + (z' - 1), 1 + 0) + s20 :|: s40 >= 0, s40 <= 2, s20 >= 0, s20 <= 1 + (z' - 1), z'' = 0, z' - 1 >= 0, z - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(s41, 1 + (z - 1), 1 + (z' - 1), 1 + s3) + s21 :|: s41 >= 0, s41 <= 2, s21 >= 0, s21 <= 1 + (z' - 1), s3 >= 0, s3 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(s42, 1 + (z - 1), 1 + (z' - 1), 0) + s22 :|: s42 >= 0, s42 <= 2, s22 >= 0, s22 <= 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0, z'' >= 0 DIV(z, z', z'') -{ 2 }-> 1 + IF(s43, 1 + (z - 1), 1 + (z' - 1), 1 + 0) + s33 :|: s43 >= 0, s43 <= 2, s33 >= 0, s33 <= 0, z'' = 0, z - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(s44, 1 + (z - 1), 1 + (z' - 1), 1 + s7) + s34 :|: s44 >= 0, s44 <= 2, s34 >= 0, s34 <= 1 + (z'' - 1), s7 >= 0, s7 <= z'' - 1 + 1, z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(s45, 1 + (z - 1), 1 + (z' - 1), 0) + s35 :|: s45 >= 0, s45 <= 2, s35 >= 0, s35 <= z'', z - 1 >= 0, z'' >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + s19 :|: s19 >= 0, s19 <= 1 + (z' - 1), z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + s32 :|: s32 >= 0, s32 <= z'', z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s2) + s18 :|: s18 >= 0, s18 <= 1 + (z' - 1), s2 >= 0, s2 <= z'' - 1 + 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s6) + s31 :|: s31 >= 0, s31 <= 1 + (z'' - 1), s6 >= 0, s6 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + s17 :|: s17 >= 0, s17 <= 1 + (z' - 1), z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 2 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + s30 :|: s30 >= 0, s30 <= 0, z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 3 }-> 1 + IF(1, z, 0, 0) + s16 :|: s16 >= 0, s16 <= 0, z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, z, 0, 0) + s29 :|: s29 >= 0, s29 <= z'', z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 3 }-> 1 + IF(1, z, 0, 1 + s1) + s15 :|: s15 >= 0, s15 <= 0, s1 >= 0, s1 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, z, 0, 1 + s5) + s28 :|: s28 >= 0, s28 <= 1 + (z'' - 1), s5 >= 0, s5 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 3 }-> 1 + IF(1, z, 0, 1 + 0) + s14 :|: s14 >= 0, s14 <= 0, z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 2 }-> 1 + IF(1, z, 0, 1 + 0) + s27 :|: s27 >= 0, s27 <= 0, z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(0, z, z', 0) + s25 :|: s25 >= 0, s25 <= z', z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', 0) + s38 :|: s38 >= 0, s38 <= z'', z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(0, z, z', 1 + s4) + s24 :|: s24 >= 0, s24 <= z', s4 >= 0, s4 <= z'' - 1 + 1, z'' - 1 >= 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', 1 + s8) + s37 :|: s37 >= 0, s37 <= 1 + (z'' - 1), s8 >= 0, s8 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(0, z, z', 1 + 0) + s23 :|: s23 >= 0, s23 <= z', z'' = 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + s36 :|: s36 >= 0, s36 <= 0, z'' = 0, z' >= 0, z >= 0 DIVISION(z, z') -{ 1 }-> 1 + DIV(z, z', 0) :|: z' >= 0, z >= 0 IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(s, 1 + (z'' - 1), z3) + MINUS(1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= z' - 1, z' - 1 >= 0, z'' - 1 >= 0, z = 1, z3 >= 0 IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 + z }-> 1 + s39 :|: s39 >= 0, s39 <= z - 1, z - 1 >= 0 LT(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 LT(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 LT(z, z') -{ 2 + z' }-> 1 + s26 :|: s26 >= 0, s26 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> if(s46, 1 + (z - 1), 1 + (z' - 1), 1 + 0) :|: s46 >= 0, s46 <= 2, z'' = 0, z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> if(s47, 1 + (z - 1), 1 + (z' - 1), 1 + s11) :|: s47 >= 0, s47 <= 2, s11 >= 0, s11 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> if(s48, 1 + (z - 1), 1 + (z' - 1), 0) :|: s48 >= 0, s48 <= 2, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 0) :|: z' - 1 >= 0, z = 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 1 + s10) :|: s10 >= 0, s10 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 1 + 0) :|: z'' = 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 0) :|: z >= 0, z'' >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 1 + s9) :|: s9 >= 0, s9 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 1 + 0) :|: z'' = 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 1 + s12) :|: s12 >= 0, s12 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0 div(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 division(z, z') -{ 0 }-> div(z, z', 0) :|: z' >= 0, z >= 0 division(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> div(s', 1 + (z'' - 1), z3) :|: s' >= 0, s' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> div(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s13 :|: s13 >= 0, s13 <= z - 1 + 1, z - 1 >= 0 inc(z) -{ 0 }-> 1 + 0 :|: z = 0 lt(z, z') -{ 0 }-> s49 :|: s49 >= 0, s49 <= 2, z' - 1 >= 0, z - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {MINUS}, {div,if}, {IF,DIV}, {division}, {DIVISION} Previous analysis results are: minus: runtime: O(1) [0], size: O(n^1) [z] inc: runtime: O(1) [0], size: O(n^1) [1 + z] LT: runtime: O(n^1) [2 + z'], size: O(n^1) [z'] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] lt: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (49) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: MINUS after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(s40, 1 + (z - 1), 1 + (z' - 1), 1 + 0) + s20 :|: s40 >= 0, s40 <= 2, s20 >= 0, s20 <= 1 + (z' - 1), z'' = 0, z' - 1 >= 0, z - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(s41, 1 + (z - 1), 1 + (z' - 1), 1 + s3) + s21 :|: s41 >= 0, s41 <= 2, s21 >= 0, s21 <= 1 + (z' - 1), s3 >= 0, s3 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(s42, 1 + (z - 1), 1 + (z' - 1), 0) + s22 :|: s42 >= 0, s42 <= 2, s22 >= 0, s22 <= 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0, z'' >= 0 DIV(z, z', z'') -{ 2 }-> 1 + IF(s43, 1 + (z - 1), 1 + (z' - 1), 1 + 0) + s33 :|: s43 >= 0, s43 <= 2, s33 >= 0, s33 <= 0, z'' = 0, z - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(s44, 1 + (z - 1), 1 + (z' - 1), 1 + s7) + s34 :|: s44 >= 0, s44 <= 2, s34 >= 0, s34 <= 1 + (z'' - 1), s7 >= 0, s7 <= z'' - 1 + 1, z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(s45, 1 + (z - 1), 1 + (z' - 1), 0) + s35 :|: s45 >= 0, s45 <= 2, s35 >= 0, s35 <= z'', z - 1 >= 0, z'' >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + s19 :|: s19 >= 0, s19 <= 1 + (z' - 1), z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + s32 :|: s32 >= 0, s32 <= z'', z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s2) + s18 :|: s18 >= 0, s18 <= 1 + (z' - 1), s2 >= 0, s2 <= z'' - 1 + 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s6) + s31 :|: s31 >= 0, s31 <= 1 + (z'' - 1), s6 >= 0, s6 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + s17 :|: s17 >= 0, s17 <= 1 + (z' - 1), z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 2 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + s30 :|: s30 >= 0, s30 <= 0, z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 3 }-> 1 + IF(1, z, 0, 0) + s16 :|: s16 >= 0, s16 <= 0, z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, z, 0, 0) + s29 :|: s29 >= 0, s29 <= z'', z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 3 }-> 1 + IF(1, z, 0, 1 + s1) + s15 :|: s15 >= 0, s15 <= 0, s1 >= 0, s1 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, z, 0, 1 + s5) + s28 :|: s28 >= 0, s28 <= 1 + (z'' - 1), s5 >= 0, s5 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 3 }-> 1 + IF(1, z, 0, 1 + 0) + s14 :|: s14 >= 0, s14 <= 0, z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 2 }-> 1 + IF(1, z, 0, 1 + 0) + s27 :|: s27 >= 0, s27 <= 0, z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(0, z, z', 0) + s25 :|: s25 >= 0, s25 <= z', z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', 0) + s38 :|: s38 >= 0, s38 <= z'', z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(0, z, z', 1 + s4) + s24 :|: s24 >= 0, s24 <= z', s4 >= 0, s4 <= z'' - 1 + 1, z'' - 1 >= 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', 1 + s8) + s37 :|: s37 >= 0, s37 <= 1 + (z'' - 1), s8 >= 0, s8 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(0, z, z', 1 + 0) + s23 :|: s23 >= 0, s23 <= z', z'' = 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + s36 :|: s36 >= 0, s36 <= 0, z'' = 0, z' >= 0, z >= 0 DIVISION(z, z') -{ 1 }-> 1 + DIV(z, z', 0) :|: z' >= 0, z >= 0 IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(s, 1 + (z'' - 1), z3) + MINUS(1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= z' - 1, z' - 1 >= 0, z'' - 1 >= 0, z = 1, z3 >= 0 IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 + z }-> 1 + s39 :|: s39 >= 0, s39 <= z - 1, z - 1 >= 0 LT(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 LT(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 LT(z, z') -{ 2 + z' }-> 1 + s26 :|: s26 >= 0, s26 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> if(s46, 1 + (z - 1), 1 + (z' - 1), 1 + 0) :|: s46 >= 0, s46 <= 2, z'' = 0, z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> if(s47, 1 + (z - 1), 1 + (z' - 1), 1 + s11) :|: s47 >= 0, s47 <= 2, s11 >= 0, s11 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> if(s48, 1 + (z - 1), 1 + (z' - 1), 0) :|: s48 >= 0, s48 <= 2, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 0) :|: z' - 1 >= 0, z = 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 1 + s10) :|: s10 >= 0, s10 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 1 + 0) :|: z'' = 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 0) :|: z >= 0, z'' >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 1 + s9) :|: s9 >= 0, s9 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 1 + 0) :|: z'' = 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 1 + s12) :|: s12 >= 0, s12 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0 div(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 division(z, z') -{ 0 }-> div(z, z', 0) :|: z' >= 0, z >= 0 division(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> div(s', 1 + (z'' - 1), z3) :|: s' >= 0, s' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> div(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s13 :|: s13 >= 0, s13 <= z - 1 + 1, z - 1 >= 0 inc(z) -{ 0 }-> 1 + 0 :|: z = 0 lt(z, z') -{ 0 }-> s49 :|: s49 >= 0, s49 <= 2, z' - 1 >= 0, z - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {MINUS}, {div,if}, {IF,DIV}, {division}, {DIVISION} Previous analysis results are: minus: runtime: O(1) [0], size: O(n^1) [z] inc: runtime: O(1) [0], size: O(n^1) [1 + z] LT: runtime: O(n^1) [2 + z'], size: O(n^1) [z'] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] lt: runtime: O(1) [0], size: O(1) [2] MINUS: runtime: ?, size: O(n^1) [z'] ---------------------------------------- (51) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: MINUS after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z' ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(s40, 1 + (z - 1), 1 + (z' - 1), 1 + 0) + s20 :|: s40 >= 0, s40 <= 2, s20 >= 0, s20 <= 1 + (z' - 1), z'' = 0, z' - 1 >= 0, z - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(s41, 1 + (z - 1), 1 + (z' - 1), 1 + s3) + s21 :|: s41 >= 0, s41 <= 2, s21 >= 0, s21 <= 1 + (z' - 1), s3 >= 0, s3 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(s42, 1 + (z - 1), 1 + (z' - 1), 0) + s22 :|: s42 >= 0, s42 <= 2, s22 >= 0, s22 <= 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0, z'' >= 0 DIV(z, z', z'') -{ 2 }-> 1 + IF(s43, 1 + (z - 1), 1 + (z' - 1), 1 + 0) + s33 :|: s43 >= 0, s43 <= 2, s33 >= 0, s33 <= 0, z'' = 0, z - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(s44, 1 + (z - 1), 1 + (z' - 1), 1 + s7) + s34 :|: s44 >= 0, s44 <= 2, s34 >= 0, s34 <= 1 + (z'' - 1), s7 >= 0, s7 <= z'' - 1 + 1, z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(s45, 1 + (z - 1), 1 + (z' - 1), 0) + s35 :|: s45 >= 0, s45 <= 2, s35 >= 0, s35 <= z'', z - 1 >= 0, z'' >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + s19 :|: s19 >= 0, s19 <= 1 + (z' - 1), z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + s32 :|: s32 >= 0, s32 <= z'', z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s2) + s18 :|: s18 >= 0, s18 <= 1 + (z' - 1), s2 >= 0, s2 <= z'' - 1 + 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s6) + s31 :|: s31 >= 0, s31 <= 1 + (z'' - 1), s6 >= 0, s6 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + s17 :|: s17 >= 0, s17 <= 1 + (z' - 1), z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 2 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + s30 :|: s30 >= 0, s30 <= 0, z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 3 }-> 1 + IF(1, z, 0, 0) + s16 :|: s16 >= 0, s16 <= 0, z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, z, 0, 0) + s29 :|: s29 >= 0, s29 <= z'', z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 3 }-> 1 + IF(1, z, 0, 1 + s1) + s15 :|: s15 >= 0, s15 <= 0, s1 >= 0, s1 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, z, 0, 1 + s5) + s28 :|: s28 >= 0, s28 <= 1 + (z'' - 1), s5 >= 0, s5 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 3 }-> 1 + IF(1, z, 0, 1 + 0) + s14 :|: s14 >= 0, s14 <= 0, z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 2 }-> 1 + IF(1, z, 0, 1 + 0) + s27 :|: s27 >= 0, s27 <= 0, z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(0, z, z', 0) + s25 :|: s25 >= 0, s25 <= z', z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', 0) + s38 :|: s38 >= 0, s38 <= z'', z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(0, z, z', 1 + s4) + s24 :|: s24 >= 0, s24 <= z', s4 >= 0, s4 <= z'' - 1 + 1, z'' - 1 >= 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', 1 + s8) + s37 :|: s37 >= 0, s37 <= 1 + (z'' - 1), s8 >= 0, s8 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(0, z, z', 1 + 0) + s23 :|: s23 >= 0, s23 <= z', z'' = 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + s36 :|: s36 >= 0, s36 <= 0, z'' = 0, z' >= 0, z >= 0 DIVISION(z, z') -{ 1 }-> 1 + DIV(z, z', 0) :|: z' >= 0, z >= 0 IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(s, 1 + (z'' - 1), z3) + MINUS(1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= z' - 1, z' - 1 >= 0, z'' - 1 >= 0, z = 1, z3 >= 0 IF(z, z', z'', z3) -{ 1 }-> 1 + DIV(0, 1 + (z'' - 1), z3) + MINUS(z', 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 + z }-> 1 + s39 :|: s39 >= 0, s39 <= z - 1, z - 1 >= 0 LT(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 LT(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 LT(z, z') -{ 2 + z' }-> 1 + s26 :|: s26 >= 0, s26 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 MINUS(z, z') -{ 1 }-> 1 + MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> if(s46, 1 + (z - 1), 1 + (z' - 1), 1 + 0) :|: s46 >= 0, s46 <= 2, z'' = 0, z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> if(s47, 1 + (z - 1), 1 + (z' - 1), 1 + s11) :|: s47 >= 0, s47 <= 2, s11 >= 0, s11 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> if(s48, 1 + (z - 1), 1 + (z' - 1), 0) :|: s48 >= 0, s48 <= 2, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 0) :|: z' - 1 >= 0, z = 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 1 + s10) :|: s10 >= 0, s10 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 1 + 0) :|: z'' = 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 0) :|: z >= 0, z'' >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 1 + s9) :|: s9 >= 0, s9 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 1 + 0) :|: z'' = 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 1 + s12) :|: s12 >= 0, s12 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0 div(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 division(z, z') -{ 0 }-> div(z, z', 0) :|: z' >= 0, z >= 0 division(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> div(s', 1 + (z'' - 1), z3) :|: s' >= 0, s' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> div(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s13 :|: s13 >= 0, s13 <= z - 1 + 1, z - 1 >= 0 inc(z) -{ 0 }-> 1 + 0 :|: z = 0 lt(z, z') -{ 0 }-> s49 :|: s49 >= 0, s49 <= 2, z' - 1 >= 0, z - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {div,if}, {IF,DIV}, {division}, {DIVISION} Previous analysis results are: minus: runtime: O(1) [0], size: O(n^1) [z] inc: runtime: O(1) [0], size: O(n^1) [1 + z] LT: runtime: O(n^1) [2 + z'], size: O(n^1) [z'] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] lt: runtime: O(1) [0], size: O(1) [2] MINUS: runtime: O(n^1) [1 + z'], size: O(n^1) [z'] ---------------------------------------- (53) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (54) Obligation: Complexity RNTS consisting of the following rules: DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(s40, 1 + (z - 1), 1 + (z' - 1), 1 + 0) + s20 :|: s40 >= 0, s40 <= 2, s20 >= 0, s20 <= 1 + (z' - 1), z'' = 0, z' - 1 >= 0, z - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(s41, 1 + (z - 1), 1 + (z' - 1), 1 + s3) + s21 :|: s41 >= 0, s41 <= 2, s21 >= 0, s21 <= 1 + (z' - 1), s3 >= 0, s3 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(s42, 1 + (z - 1), 1 + (z' - 1), 0) + s22 :|: s42 >= 0, s42 <= 2, s22 >= 0, s22 <= 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0, z'' >= 0 DIV(z, z', z'') -{ 2 }-> 1 + IF(s43, 1 + (z - 1), 1 + (z' - 1), 1 + 0) + s33 :|: s43 >= 0, s43 <= 2, s33 >= 0, s33 <= 0, z'' = 0, z - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(s44, 1 + (z - 1), 1 + (z' - 1), 1 + s7) + s34 :|: s44 >= 0, s44 <= 2, s34 >= 0, s34 <= 1 + (z'' - 1), s7 >= 0, s7 <= z'' - 1 + 1, z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(s45, 1 + (z - 1), 1 + (z' - 1), 0) + s35 :|: s45 >= 0, s45 <= 2, s35 >= 0, s35 <= z'', z - 1 >= 0, z'' >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + s19 :|: s19 >= 0, s19 <= 1 + (z' - 1), z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + s32 :|: s32 >= 0, s32 <= z'', z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s2) + s18 :|: s18 >= 0, s18 <= 1 + (z' - 1), s2 >= 0, s2 <= z'' - 1 + 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s6) + s31 :|: s31 >= 0, s31 <= 1 + (z'' - 1), s6 >= 0, s6 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + s17 :|: s17 >= 0, s17 <= 1 + (z' - 1), z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 2 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + s30 :|: s30 >= 0, s30 <= 0, z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 3 }-> 1 + IF(1, z, 0, 0) + s16 :|: s16 >= 0, s16 <= 0, z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, z, 0, 0) + s29 :|: s29 >= 0, s29 <= z'', z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 3 }-> 1 + IF(1, z, 0, 1 + s1) + s15 :|: s15 >= 0, s15 <= 0, s1 >= 0, s1 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, z, 0, 1 + s5) + s28 :|: s28 >= 0, s28 <= 1 + (z'' - 1), s5 >= 0, s5 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 3 }-> 1 + IF(1, z, 0, 1 + 0) + s14 :|: s14 >= 0, s14 <= 0, z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 2 }-> 1 + IF(1, z, 0, 1 + 0) + s27 :|: s27 >= 0, s27 <= 0, z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(0, z, z', 0) + s25 :|: s25 >= 0, s25 <= z', z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', 0) + s38 :|: s38 >= 0, s38 <= z'', z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(0, z, z', 1 + s4) + s24 :|: s24 >= 0, s24 <= z', s4 >= 0, s4 <= z'' - 1 + 1, z'' - 1 >= 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', 1 + s8) + s37 :|: s37 >= 0, s37 <= 1 + (z'' - 1), s8 >= 0, s8 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(0, z, z', 1 + 0) + s23 :|: s23 >= 0, s23 <= z', z'' = 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + s36 :|: s36 >= 0, s36 <= 0, z'' = 0, z' >= 0, z >= 0 DIVISION(z, z') -{ 1 }-> 1 + DIV(z, z', 0) :|: z' >= 0, z >= 0 IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 IF(z, z', z'', z3) -{ 2 + z'' }-> 1 + DIV(s, 1 + (z'' - 1), z3) + s50 :|: s50 >= 0, s50 <= 1 + (z'' - 1), s >= 0, s <= z' - 1, z' - 1 >= 0, z'' - 1 >= 0, z = 1, z3 >= 0 IF(z, z', z'', z3) -{ 2 + z'' }-> 1 + DIV(0, 1 + (z'' - 1), z3) + s51 :|: s51 >= 0, s51 <= 1 + (z'' - 1), z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 + z }-> 1 + s39 :|: s39 >= 0, s39 <= z - 1, z - 1 >= 0 LT(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 LT(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 LT(z, z') -{ 2 + z' }-> 1 + s26 :|: s26 >= 0, s26 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 MINUS(z, z') -{ 1 + z' }-> 1 + s52 :|: s52 >= 0, s52 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> if(s46, 1 + (z - 1), 1 + (z' - 1), 1 + 0) :|: s46 >= 0, s46 <= 2, z'' = 0, z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> if(s47, 1 + (z - 1), 1 + (z' - 1), 1 + s11) :|: s47 >= 0, s47 <= 2, s11 >= 0, s11 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> if(s48, 1 + (z - 1), 1 + (z' - 1), 0) :|: s48 >= 0, s48 <= 2, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 0) :|: z' - 1 >= 0, z = 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 1 + s10) :|: s10 >= 0, s10 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 1 + 0) :|: z'' = 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 0) :|: z >= 0, z'' >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 1 + s9) :|: s9 >= 0, s9 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 1 + 0) :|: z'' = 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 1 + s12) :|: s12 >= 0, s12 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0 div(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 division(z, z') -{ 0 }-> div(z, z', 0) :|: z' >= 0, z >= 0 division(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> div(s', 1 + (z'' - 1), z3) :|: s' >= 0, s' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> div(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s13 :|: s13 >= 0, s13 <= z - 1 + 1, z - 1 >= 0 inc(z) -{ 0 }-> 1 + 0 :|: z = 0 lt(z, z') -{ 0 }-> s49 :|: s49 >= 0, s49 <= 2, z' - 1 >= 0, z - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {div,if}, {IF,DIV}, {division}, {DIVISION} Previous analysis results are: minus: runtime: O(1) [0], size: O(n^1) [z] inc: runtime: O(1) [0], size: O(n^1) [1 + z] LT: runtime: O(n^1) [2 + z'], size: O(n^1) [z'] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] lt: runtime: O(1) [0], size: O(1) [2] MINUS: runtime: O(n^1) [1 + z'], size: O(n^1) [z'] ---------------------------------------- (55) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: div after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z + z'' Computed SIZE bound using CoFloCo for: if after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z' + z3 ---------------------------------------- (56) Obligation: Complexity RNTS consisting of the following rules: DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(s40, 1 + (z - 1), 1 + (z' - 1), 1 + 0) + s20 :|: s40 >= 0, s40 <= 2, s20 >= 0, s20 <= 1 + (z' - 1), z'' = 0, z' - 1 >= 0, z - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(s41, 1 + (z - 1), 1 + (z' - 1), 1 + s3) + s21 :|: s41 >= 0, s41 <= 2, s21 >= 0, s21 <= 1 + (z' - 1), s3 >= 0, s3 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(s42, 1 + (z - 1), 1 + (z' - 1), 0) + s22 :|: s42 >= 0, s42 <= 2, s22 >= 0, s22 <= 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0, z'' >= 0 DIV(z, z', z'') -{ 2 }-> 1 + IF(s43, 1 + (z - 1), 1 + (z' - 1), 1 + 0) + s33 :|: s43 >= 0, s43 <= 2, s33 >= 0, s33 <= 0, z'' = 0, z - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(s44, 1 + (z - 1), 1 + (z' - 1), 1 + s7) + s34 :|: s44 >= 0, s44 <= 2, s34 >= 0, s34 <= 1 + (z'' - 1), s7 >= 0, s7 <= z'' - 1 + 1, z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(s45, 1 + (z - 1), 1 + (z' - 1), 0) + s35 :|: s45 >= 0, s45 <= 2, s35 >= 0, s35 <= z'', z - 1 >= 0, z'' >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + s19 :|: s19 >= 0, s19 <= 1 + (z' - 1), z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + s32 :|: s32 >= 0, s32 <= z'', z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s2) + s18 :|: s18 >= 0, s18 <= 1 + (z' - 1), s2 >= 0, s2 <= z'' - 1 + 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s6) + s31 :|: s31 >= 0, s31 <= 1 + (z'' - 1), s6 >= 0, s6 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + s17 :|: s17 >= 0, s17 <= 1 + (z' - 1), z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 2 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + s30 :|: s30 >= 0, s30 <= 0, z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 3 }-> 1 + IF(1, z, 0, 0) + s16 :|: s16 >= 0, s16 <= 0, z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, z, 0, 0) + s29 :|: s29 >= 0, s29 <= z'', z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 3 }-> 1 + IF(1, z, 0, 1 + s1) + s15 :|: s15 >= 0, s15 <= 0, s1 >= 0, s1 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, z, 0, 1 + s5) + s28 :|: s28 >= 0, s28 <= 1 + (z'' - 1), s5 >= 0, s5 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 3 }-> 1 + IF(1, z, 0, 1 + 0) + s14 :|: s14 >= 0, s14 <= 0, z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 2 }-> 1 + IF(1, z, 0, 1 + 0) + s27 :|: s27 >= 0, s27 <= 0, z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(0, z, z', 0) + s25 :|: s25 >= 0, s25 <= z', z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', 0) + s38 :|: s38 >= 0, s38 <= z'', z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(0, z, z', 1 + s4) + s24 :|: s24 >= 0, s24 <= z', s4 >= 0, s4 <= z'' - 1 + 1, z'' - 1 >= 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', 1 + s8) + s37 :|: s37 >= 0, s37 <= 1 + (z'' - 1), s8 >= 0, s8 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(0, z, z', 1 + 0) + s23 :|: s23 >= 0, s23 <= z', z'' = 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + s36 :|: s36 >= 0, s36 <= 0, z'' = 0, z' >= 0, z >= 0 DIVISION(z, z') -{ 1 }-> 1 + DIV(z, z', 0) :|: z' >= 0, z >= 0 IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 IF(z, z', z'', z3) -{ 2 + z'' }-> 1 + DIV(s, 1 + (z'' - 1), z3) + s50 :|: s50 >= 0, s50 <= 1 + (z'' - 1), s >= 0, s <= z' - 1, z' - 1 >= 0, z'' - 1 >= 0, z = 1, z3 >= 0 IF(z, z', z'', z3) -{ 2 + z'' }-> 1 + DIV(0, 1 + (z'' - 1), z3) + s51 :|: s51 >= 0, s51 <= 1 + (z'' - 1), z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 + z }-> 1 + s39 :|: s39 >= 0, s39 <= z - 1, z - 1 >= 0 LT(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 LT(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 LT(z, z') -{ 2 + z' }-> 1 + s26 :|: s26 >= 0, s26 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 MINUS(z, z') -{ 1 + z' }-> 1 + s52 :|: s52 >= 0, s52 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> if(s46, 1 + (z - 1), 1 + (z' - 1), 1 + 0) :|: s46 >= 0, s46 <= 2, z'' = 0, z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> if(s47, 1 + (z - 1), 1 + (z' - 1), 1 + s11) :|: s47 >= 0, s47 <= 2, s11 >= 0, s11 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> if(s48, 1 + (z - 1), 1 + (z' - 1), 0) :|: s48 >= 0, s48 <= 2, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 0) :|: z' - 1 >= 0, z = 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 1 + s10) :|: s10 >= 0, s10 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 1 + 0) :|: z'' = 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 0) :|: z >= 0, z'' >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 1 + s9) :|: s9 >= 0, s9 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 1 + 0) :|: z'' = 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 1 + s12) :|: s12 >= 0, s12 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0 div(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 division(z, z') -{ 0 }-> div(z, z', 0) :|: z' >= 0, z >= 0 division(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> div(s', 1 + (z'' - 1), z3) :|: s' >= 0, s' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> div(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s13 :|: s13 >= 0, s13 <= z - 1 + 1, z - 1 >= 0 inc(z) -{ 0 }-> 1 + 0 :|: z = 0 lt(z, z') -{ 0 }-> s49 :|: s49 >= 0, s49 <= 2, z' - 1 >= 0, z - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {div,if}, {IF,DIV}, {division}, {DIVISION} Previous analysis results are: minus: runtime: O(1) [0], size: O(n^1) [z] inc: runtime: O(1) [0], size: O(n^1) [1 + z] LT: runtime: O(n^1) [2 + z'], size: O(n^1) [z'] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] lt: runtime: O(1) [0], size: O(1) [2] MINUS: runtime: O(n^1) [1 + z'], size: O(n^1) [z'] div: runtime: ?, size: O(n^1) [1 + z + z''] if: runtime: ?, size: O(n^1) [1 + z' + z3] ---------------------------------------- (57) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: div after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 Computed RUNTIME bound using CoFloCo for: if after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (58) Obligation: Complexity RNTS consisting of the following rules: DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(s40, 1 + (z - 1), 1 + (z' - 1), 1 + 0) + s20 :|: s40 >= 0, s40 <= 2, s20 >= 0, s20 <= 1 + (z' - 1), z'' = 0, z' - 1 >= 0, z - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(s41, 1 + (z - 1), 1 + (z' - 1), 1 + s3) + s21 :|: s41 >= 0, s41 <= 2, s21 >= 0, s21 <= 1 + (z' - 1), s3 >= 0, s3 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(s42, 1 + (z - 1), 1 + (z' - 1), 0) + s22 :|: s42 >= 0, s42 <= 2, s22 >= 0, s22 <= 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0, z'' >= 0 DIV(z, z', z'') -{ 2 }-> 1 + IF(s43, 1 + (z - 1), 1 + (z' - 1), 1 + 0) + s33 :|: s43 >= 0, s43 <= 2, s33 >= 0, s33 <= 0, z'' = 0, z - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(s44, 1 + (z - 1), 1 + (z' - 1), 1 + s7) + s34 :|: s44 >= 0, s44 <= 2, s34 >= 0, s34 <= 1 + (z'' - 1), s7 >= 0, s7 <= z'' - 1 + 1, z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(s45, 1 + (z - 1), 1 + (z' - 1), 0) + s35 :|: s45 >= 0, s45 <= 2, s35 >= 0, s35 <= z'', z - 1 >= 0, z'' >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + s19 :|: s19 >= 0, s19 <= 1 + (z' - 1), z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + s32 :|: s32 >= 0, s32 <= z'', z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s2) + s18 :|: s18 >= 0, s18 <= 1 + (z' - 1), s2 >= 0, s2 <= z'' - 1 + 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s6) + s31 :|: s31 >= 0, s31 <= 1 + (z'' - 1), s6 >= 0, s6 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + s17 :|: s17 >= 0, s17 <= 1 + (z' - 1), z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 2 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + s30 :|: s30 >= 0, s30 <= 0, z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 3 }-> 1 + IF(1, z, 0, 0) + s16 :|: s16 >= 0, s16 <= 0, z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, z, 0, 0) + s29 :|: s29 >= 0, s29 <= z'', z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 3 }-> 1 + IF(1, z, 0, 1 + s1) + s15 :|: s15 >= 0, s15 <= 0, s1 >= 0, s1 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, z, 0, 1 + s5) + s28 :|: s28 >= 0, s28 <= 1 + (z'' - 1), s5 >= 0, s5 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 3 }-> 1 + IF(1, z, 0, 1 + 0) + s14 :|: s14 >= 0, s14 <= 0, z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 2 }-> 1 + IF(1, z, 0, 1 + 0) + s27 :|: s27 >= 0, s27 <= 0, z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(0, z, z', 0) + s25 :|: s25 >= 0, s25 <= z', z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', 0) + s38 :|: s38 >= 0, s38 <= z'', z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(0, z, z', 1 + s4) + s24 :|: s24 >= 0, s24 <= z', s4 >= 0, s4 <= z'' - 1 + 1, z'' - 1 >= 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', 1 + s8) + s37 :|: s37 >= 0, s37 <= 1 + (z'' - 1), s8 >= 0, s8 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(0, z, z', 1 + 0) + s23 :|: s23 >= 0, s23 <= z', z'' = 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + s36 :|: s36 >= 0, s36 <= 0, z'' = 0, z' >= 0, z >= 0 DIVISION(z, z') -{ 1 }-> 1 + DIV(z, z', 0) :|: z' >= 0, z >= 0 IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 IF(z, z', z'', z3) -{ 2 + z'' }-> 1 + DIV(s, 1 + (z'' - 1), z3) + s50 :|: s50 >= 0, s50 <= 1 + (z'' - 1), s >= 0, s <= z' - 1, z' - 1 >= 0, z'' - 1 >= 0, z = 1, z3 >= 0 IF(z, z', z'', z3) -{ 2 + z'' }-> 1 + DIV(0, 1 + (z'' - 1), z3) + s51 :|: s51 >= 0, s51 <= 1 + (z'' - 1), z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 + z }-> 1 + s39 :|: s39 >= 0, s39 <= z - 1, z - 1 >= 0 LT(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 LT(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 LT(z, z') -{ 2 + z' }-> 1 + s26 :|: s26 >= 0, s26 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 MINUS(z, z') -{ 1 + z' }-> 1 + s52 :|: s52 >= 0, s52 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> if(s46, 1 + (z - 1), 1 + (z' - 1), 1 + 0) :|: s46 >= 0, s46 <= 2, z'' = 0, z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> if(s47, 1 + (z - 1), 1 + (z' - 1), 1 + s11) :|: s47 >= 0, s47 <= 2, s11 >= 0, s11 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> if(s48, 1 + (z - 1), 1 + (z' - 1), 0) :|: s48 >= 0, s48 <= 2, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 0) :|: z' - 1 >= 0, z = 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 1 + s10) :|: s10 >= 0, s10 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> if(2, 0, 1 + (z' - 1), 1 + 0) :|: z'' = 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 0) :|: z >= 0, z'' >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 1 + s9) :|: s9 >= 0, s9 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(1, z, 0, 1 + 0) :|: z'' = 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 0) :|: z' >= 0, z >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 1 + s12) :|: s12 >= 0, s12 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> if(0, z, z', 1 + 0) :|: z'' = 0, z' >= 0, z >= 0 div(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 division(z, z') -{ 0 }-> div(z, z', 0) :|: z' >= 0, z >= 0 division(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> div(s', 1 + (z'' - 1), z3) :|: s' >= 0, s' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> div(0, 1 + (z'' - 1), z3) :|: z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s13 :|: s13 >= 0, s13 <= z - 1 + 1, z - 1 >= 0 inc(z) -{ 0 }-> 1 + 0 :|: z = 0 lt(z, z') -{ 0 }-> s49 :|: s49 >= 0, s49 <= 2, z' - 1 >= 0, z - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {IF,DIV}, {division}, {DIVISION} Previous analysis results are: minus: runtime: O(1) [0], size: O(n^1) [z] inc: runtime: O(1) [0], size: O(n^1) [1 + z] LT: runtime: O(n^1) [2 + z'], size: O(n^1) [z'] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] lt: runtime: O(1) [0], size: O(1) [2] MINUS: runtime: O(n^1) [1 + z'], size: O(n^1) [z'] div: runtime: O(1) [0], size: O(n^1) [1 + z + z''] if: runtime: O(1) [0], size: O(n^1) [1 + z' + z3] ---------------------------------------- (59) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (60) Obligation: Complexity RNTS consisting of the following rules: DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(s40, 1 + (z - 1), 1 + (z' - 1), 1 + 0) + s20 :|: s40 >= 0, s40 <= 2, s20 >= 0, s20 <= 1 + (z' - 1), z'' = 0, z' - 1 >= 0, z - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(s41, 1 + (z - 1), 1 + (z' - 1), 1 + s3) + s21 :|: s41 >= 0, s41 <= 2, s21 >= 0, s21 <= 1 + (z' - 1), s3 >= 0, s3 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(s42, 1 + (z - 1), 1 + (z' - 1), 0) + s22 :|: s42 >= 0, s42 <= 2, s22 >= 0, s22 <= 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0, z'' >= 0 DIV(z, z', z'') -{ 2 }-> 1 + IF(s43, 1 + (z - 1), 1 + (z' - 1), 1 + 0) + s33 :|: s43 >= 0, s43 <= 2, s33 >= 0, s33 <= 0, z'' = 0, z - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(s44, 1 + (z - 1), 1 + (z' - 1), 1 + s7) + s34 :|: s44 >= 0, s44 <= 2, s34 >= 0, s34 <= 1 + (z'' - 1), s7 >= 0, s7 <= z'' - 1 + 1, z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(s45, 1 + (z - 1), 1 + (z' - 1), 0) + s35 :|: s45 >= 0, s45 <= 2, s35 >= 0, s35 <= z'', z - 1 >= 0, z'' >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + s19 :|: s19 >= 0, s19 <= 1 + (z' - 1), z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + s32 :|: s32 >= 0, s32 <= z'', z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s2) + s18 :|: s18 >= 0, s18 <= 1 + (z' - 1), s2 >= 0, s2 <= z'' - 1 + 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s6) + s31 :|: s31 >= 0, s31 <= 1 + (z'' - 1), s6 >= 0, s6 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + s17 :|: s17 >= 0, s17 <= 1 + (z' - 1), z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 2 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + s30 :|: s30 >= 0, s30 <= 0, z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 3 }-> 1 + IF(1, z, 0, 0) + s16 :|: s16 >= 0, s16 <= 0, z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, z, 0, 0) + s29 :|: s29 >= 0, s29 <= z'', z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 3 }-> 1 + IF(1, z, 0, 1 + s1) + s15 :|: s15 >= 0, s15 <= 0, s1 >= 0, s1 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, z, 0, 1 + s5) + s28 :|: s28 >= 0, s28 <= 1 + (z'' - 1), s5 >= 0, s5 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 3 }-> 1 + IF(1, z, 0, 1 + 0) + s14 :|: s14 >= 0, s14 <= 0, z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 2 }-> 1 + IF(1, z, 0, 1 + 0) + s27 :|: s27 >= 0, s27 <= 0, z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(0, z, z', 0) + s25 :|: s25 >= 0, s25 <= z', z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', 0) + s38 :|: s38 >= 0, s38 <= z'', z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(0, z, z', 1 + s4) + s24 :|: s24 >= 0, s24 <= z', s4 >= 0, s4 <= z'' - 1 + 1, z'' - 1 >= 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', 1 + s8) + s37 :|: s37 >= 0, s37 <= 1 + (z'' - 1), s8 >= 0, s8 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(0, z, z', 1 + 0) + s23 :|: s23 >= 0, s23 <= z', z'' = 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + s36 :|: s36 >= 0, s36 <= 0, z'' = 0, z' >= 0, z >= 0 DIVISION(z, z') -{ 1 }-> 1 + DIV(z, z', 0) :|: z' >= 0, z >= 0 IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 IF(z, z', z'', z3) -{ 2 + z'' }-> 1 + DIV(s, 1 + (z'' - 1), z3) + s50 :|: s50 >= 0, s50 <= 1 + (z'' - 1), s >= 0, s <= z' - 1, z' - 1 >= 0, z'' - 1 >= 0, z = 1, z3 >= 0 IF(z, z', z'', z3) -{ 2 + z'' }-> 1 + DIV(0, 1 + (z'' - 1), z3) + s51 :|: s51 >= 0, s51 <= 1 + (z'' - 1), z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 + z }-> 1 + s39 :|: s39 >= 0, s39 <= z - 1, z - 1 >= 0 LT(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 LT(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 LT(z, z') -{ 2 + z' }-> 1 + s26 :|: s26 >= 0, s26 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 MINUS(z, z') -{ 1 + z' }-> 1 + s52 :|: s52 >= 0, s52 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> s54 :|: s54 >= 0, s54 <= 1 + 0 + 1 + z, z'' = 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> s55 :|: s55 >= 0, s55 <= 1 + s9 + 1 + z, s9 >= 0, s9 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> s56 :|: s56 >= 0, s56 <= 0 + 1 + z, z >= 0, z'' >= 0, z' = 0 div(z, z', z'') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 1 + 0 + 1 + 0, z'' = 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 1 + s10 + 1 + 0, s10 >= 0, s10 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> s59 :|: s59 >= 0, s59 <= 0 + 1 + 0, z' - 1 >= 0, z = 0, z'' >= 0 div(z, z', z'') -{ 0 }-> s60 :|: s60 >= 0, s60 <= 1 + 0 + 1 + (1 + (z - 1)), s46 >= 0, s46 <= 2, z'' = 0, z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> s61 :|: s61 >= 0, s61 <= 1 + s11 + 1 + (1 + (z - 1)), s47 >= 0, s47 <= 2, s11 >= 0, s11 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> s62 :|: s62 >= 0, s62 <= 0 + 1 + (1 + (z - 1)), s48 >= 0, s48 <= 2, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> s63 :|: s63 >= 0, s63 <= 1 + 0 + 1 + z, z'' = 0, z' >= 0, z >= 0 div(z, z', z'') -{ 0 }-> s64 :|: s64 >= 0, s64 <= 1 + s12 + 1 + z, s12 >= 0, s12 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> s65 :|: s65 >= 0, s65 <= 0 + 1 + z, z' >= 0, z >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 division(z, z') -{ 0 }-> s53 :|: s53 >= 0, s53 <= z + 0 + 1, z' >= 0, z >= 0 division(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'', z3) -{ 0 }-> s66 :|: s66 >= 0, s66 <= s' + z3 + 1, s' >= 0, s' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> s67 :|: s67 >= 0, s67 <= 0 + z3 + 1, z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s13 :|: s13 >= 0, s13 <= z - 1 + 1, z - 1 >= 0 inc(z) -{ 0 }-> 1 + 0 :|: z = 0 lt(z, z') -{ 0 }-> s49 :|: s49 >= 0, s49 <= 2, z' - 1 >= 0, z - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {IF,DIV}, {division}, {DIVISION} Previous analysis results are: minus: runtime: O(1) [0], size: O(n^1) [z] inc: runtime: O(1) [0], size: O(n^1) [1 + z] LT: runtime: O(n^1) [2 + z'], size: O(n^1) [z'] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] lt: runtime: O(1) [0], size: O(1) [2] MINUS: runtime: O(n^1) [1 + z'], size: O(n^1) [z'] div: runtime: O(1) [0], size: O(n^1) [1 + z + z''] if: runtime: O(1) [0], size: O(n^1) [1 + z' + z3] ---------------------------------------- (61) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: IF after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 35 + 28*z' + 4*z'*z'' + 6*z'*z3 + 6*z'^2 + 3*z'' + 4*z3 Computed SIZE bound using KoAT for: DIV after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 928 + 576*z + 48*z*z' + 36*z*z'' + 108*z^2 + 63*z' + 40*z'' ---------------------------------------- (62) Obligation: Complexity RNTS consisting of the following rules: DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(s40, 1 + (z - 1), 1 + (z' - 1), 1 + 0) + s20 :|: s40 >= 0, s40 <= 2, s20 >= 0, s20 <= 1 + (z' - 1), z'' = 0, z' - 1 >= 0, z - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(s41, 1 + (z - 1), 1 + (z' - 1), 1 + s3) + s21 :|: s41 >= 0, s41 <= 2, s21 >= 0, s21 <= 1 + (z' - 1), s3 >= 0, s3 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(s42, 1 + (z - 1), 1 + (z' - 1), 0) + s22 :|: s42 >= 0, s42 <= 2, s22 >= 0, s22 <= 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0, z'' >= 0 DIV(z, z', z'') -{ 2 }-> 1 + IF(s43, 1 + (z - 1), 1 + (z' - 1), 1 + 0) + s33 :|: s43 >= 0, s43 <= 2, s33 >= 0, s33 <= 0, z'' = 0, z - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(s44, 1 + (z - 1), 1 + (z' - 1), 1 + s7) + s34 :|: s44 >= 0, s44 <= 2, s34 >= 0, s34 <= 1 + (z'' - 1), s7 >= 0, s7 <= z'' - 1 + 1, z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(s45, 1 + (z - 1), 1 + (z' - 1), 0) + s35 :|: s45 >= 0, s45 <= 2, s35 >= 0, s35 <= z'', z - 1 >= 0, z'' >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + s19 :|: s19 >= 0, s19 <= 1 + (z' - 1), z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + s32 :|: s32 >= 0, s32 <= z'', z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s2) + s18 :|: s18 >= 0, s18 <= 1 + (z' - 1), s2 >= 0, s2 <= z'' - 1 + 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s6) + s31 :|: s31 >= 0, s31 <= 1 + (z'' - 1), s6 >= 0, s6 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + s17 :|: s17 >= 0, s17 <= 1 + (z' - 1), z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 2 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + s30 :|: s30 >= 0, s30 <= 0, z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 3 }-> 1 + IF(1, z, 0, 0) + s16 :|: s16 >= 0, s16 <= 0, z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, z, 0, 0) + s29 :|: s29 >= 0, s29 <= z'', z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 3 }-> 1 + IF(1, z, 0, 1 + s1) + s15 :|: s15 >= 0, s15 <= 0, s1 >= 0, s1 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, z, 0, 1 + s5) + s28 :|: s28 >= 0, s28 <= 1 + (z'' - 1), s5 >= 0, s5 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 3 }-> 1 + IF(1, z, 0, 1 + 0) + s14 :|: s14 >= 0, s14 <= 0, z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 2 }-> 1 + IF(1, z, 0, 1 + 0) + s27 :|: s27 >= 0, s27 <= 0, z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(0, z, z', 0) + s25 :|: s25 >= 0, s25 <= z', z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', 0) + s38 :|: s38 >= 0, s38 <= z'', z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(0, z, z', 1 + s4) + s24 :|: s24 >= 0, s24 <= z', s4 >= 0, s4 <= z'' - 1 + 1, z'' - 1 >= 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', 1 + s8) + s37 :|: s37 >= 0, s37 <= 1 + (z'' - 1), s8 >= 0, s8 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(0, z, z', 1 + 0) + s23 :|: s23 >= 0, s23 <= z', z'' = 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + s36 :|: s36 >= 0, s36 <= 0, z'' = 0, z' >= 0, z >= 0 DIVISION(z, z') -{ 1 }-> 1 + DIV(z, z', 0) :|: z' >= 0, z >= 0 IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 IF(z, z', z'', z3) -{ 2 + z'' }-> 1 + DIV(s, 1 + (z'' - 1), z3) + s50 :|: s50 >= 0, s50 <= 1 + (z'' - 1), s >= 0, s <= z' - 1, z' - 1 >= 0, z'' - 1 >= 0, z = 1, z3 >= 0 IF(z, z', z'', z3) -{ 2 + z'' }-> 1 + DIV(0, 1 + (z'' - 1), z3) + s51 :|: s51 >= 0, s51 <= 1 + (z'' - 1), z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 + z }-> 1 + s39 :|: s39 >= 0, s39 <= z - 1, z - 1 >= 0 LT(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 LT(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 LT(z, z') -{ 2 + z' }-> 1 + s26 :|: s26 >= 0, s26 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 MINUS(z, z') -{ 1 + z' }-> 1 + s52 :|: s52 >= 0, s52 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> s54 :|: s54 >= 0, s54 <= 1 + 0 + 1 + z, z'' = 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> s55 :|: s55 >= 0, s55 <= 1 + s9 + 1 + z, s9 >= 0, s9 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> s56 :|: s56 >= 0, s56 <= 0 + 1 + z, z >= 0, z'' >= 0, z' = 0 div(z, z', z'') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 1 + 0 + 1 + 0, z'' = 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 1 + s10 + 1 + 0, s10 >= 0, s10 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> s59 :|: s59 >= 0, s59 <= 0 + 1 + 0, z' - 1 >= 0, z = 0, z'' >= 0 div(z, z', z'') -{ 0 }-> s60 :|: s60 >= 0, s60 <= 1 + 0 + 1 + (1 + (z - 1)), s46 >= 0, s46 <= 2, z'' = 0, z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> s61 :|: s61 >= 0, s61 <= 1 + s11 + 1 + (1 + (z - 1)), s47 >= 0, s47 <= 2, s11 >= 0, s11 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> s62 :|: s62 >= 0, s62 <= 0 + 1 + (1 + (z - 1)), s48 >= 0, s48 <= 2, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> s63 :|: s63 >= 0, s63 <= 1 + 0 + 1 + z, z'' = 0, z' >= 0, z >= 0 div(z, z', z'') -{ 0 }-> s64 :|: s64 >= 0, s64 <= 1 + s12 + 1 + z, s12 >= 0, s12 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> s65 :|: s65 >= 0, s65 <= 0 + 1 + z, z' >= 0, z >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 division(z, z') -{ 0 }-> s53 :|: s53 >= 0, s53 <= z + 0 + 1, z' >= 0, z >= 0 division(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'', z3) -{ 0 }-> s66 :|: s66 >= 0, s66 <= s' + z3 + 1, s' >= 0, s' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> s67 :|: s67 >= 0, s67 <= 0 + z3 + 1, z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s13 :|: s13 >= 0, s13 <= z - 1 + 1, z - 1 >= 0 inc(z) -{ 0 }-> 1 + 0 :|: z = 0 lt(z, z') -{ 0 }-> s49 :|: s49 >= 0, s49 <= 2, z' - 1 >= 0, z - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {IF,DIV}, {division}, {DIVISION} Previous analysis results are: minus: runtime: O(1) [0], size: O(n^1) [z] inc: runtime: O(1) [0], size: O(n^1) [1 + z] LT: runtime: O(n^1) [2 + z'], size: O(n^1) [z'] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] lt: runtime: O(1) [0], size: O(1) [2] MINUS: runtime: O(n^1) [1 + z'], size: O(n^1) [z'] div: runtime: O(1) [0], size: O(n^1) [1 + z + z''] if: runtime: O(1) [0], size: O(n^1) [1 + z' + z3] IF: runtime: ?, size: O(n^2) [35 + 28*z' + 4*z'*z'' + 6*z'*z3 + 6*z'^2 + 3*z'' + 4*z3] DIV: runtime: ?, size: O(n^2) [928 + 576*z + 48*z*z' + 36*z*z'' + 108*z^2 + 63*z' + 40*z''] ---------------------------------------- (63) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: IF after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 153 + 1109*z' + 52*z'*z'' + 90*z'*z3 + 90*z'^2 + 7*z'' + 12*z3 Computed RUNTIME bound using KoAT for: DIV after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 3924 + 21042*z + 624*z*z' + 540*z*z'' + 1620*z^2 + 135*z' + 104*z'' ---------------------------------------- (64) Obligation: Complexity RNTS consisting of the following rules: DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(s40, 1 + (z - 1), 1 + (z' - 1), 1 + 0) + s20 :|: s40 >= 0, s40 <= 2, s20 >= 0, s20 <= 1 + (z' - 1), z'' = 0, z' - 1 >= 0, z - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(s41, 1 + (z - 1), 1 + (z' - 1), 1 + s3) + s21 :|: s41 >= 0, s41 <= 2, s21 >= 0, s21 <= 1 + (z' - 1), s3 >= 0, s3 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(s42, 1 + (z - 1), 1 + (z' - 1), 0) + s22 :|: s42 >= 0, s42 <= 2, s22 >= 0, s22 <= 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0, z'' >= 0 DIV(z, z', z'') -{ 2 }-> 1 + IF(s43, 1 + (z - 1), 1 + (z' - 1), 1 + 0) + s33 :|: s43 >= 0, s43 <= 2, s33 >= 0, s33 <= 0, z'' = 0, z - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(s44, 1 + (z - 1), 1 + (z' - 1), 1 + s7) + s34 :|: s44 >= 0, s44 <= 2, s34 >= 0, s34 <= 1 + (z'' - 1), s7 >= 0, s7 <= z'' - 1 + 1, z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(s45, 1 + (z - 1), 1 + (z' - 1), 0) + s35 :|: s45 >= 0, s45 <= 2, s35 >= 0, s35 <= z'', z - 1 >= 0, z'' >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + s19 :|: s19 >= 0, s19 <= 1 + (z' - 1), z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, 1 + (z' - 1), 0) + s32 :|: s32 >= 0, s32 <= z'', z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s2) + s18 :|: s18 >= 0, s18 <= 1 + (z' - 1), s2 >= 0, s2 <= z'' - 1 + 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + s6) + s31 :|: s31 >= 0, s31 <= 1 + (z'' - 1), s6 >= 0, s6 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + s17 :|: s17 >= 0, s17 <= 1 + (z' - 1), z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 2 }-> 1 + IF(2, 0, 1 + (z' - 1), 1 + 0) + s30 :|: s30 >= 0, s30 <= 0, z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 3 }-> 1 + IF(1, z, 0, 0) + s16 :|: s16 >= 0, s16 <= 0, z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, z, 0, 0) + s29 :|: s29 >= 0, s29 <= z'', z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 3 }-> 1 + IF(1, z, 0, 1 + s1) + s15 :|: s15 >= 0, s15 <= 0, s1 >= 0, s1 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(1, z, 0, 1 + s5) + s28 :|: s28 >= 0, s28 <= 1 + (z'' - 1), s5 >= 0, s5 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 3 }-> 1 + IF(1, z, 0, 1 + 0) + s14 :|: s14 >= 0, s14 <= 0, z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 2 }-> 1 + IF(1, z, 0, 1 + 0) + s27 :|: s27 >= 0, s27 <= 0, z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(0, z, z', 0) + s25 :|: s25 >= 0, s25 <= z', z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', 0) + s38 :|: s38 >= 0, s38 <= z'', z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(0, z, z', 1 + s4) + s24 :|: s24 >= 0, s24 <= z', s4 >= 0, s4 <= z'' - 1 + 1, z'' - 1 >= 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 2 + z'' }-> 1 + IF(0, z, z', 1 + s8) + s37 :|: s37 >= 0, s37 <= 1 + (z'' - 1), s8 >= 0, s8 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 3 + z' }-> 1 + IF(0, z, z', 1 + 0) + s23 :|: s23 >= 0, s23 <= z', z'' = 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 2 }-> 1 + IF(0, z, z', 1 + 0) + s36 :|: s36 >= 0, s36 <= 0, z'' = 0, z' >= 0, z >= 0 DIVISION(z, z') -{ 1 }-> 1 + DIV(z, z', 0) :|: z' >= 0, z >= 0 IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 IF(z, z', z'', z3) -{ 2 + z'' }-> 1 + DIV(s, 1 + (z'' - 1), z3) + s50 :|: s50 >= 0, s50 <= 1 + (z'' - 1), s >= 0, s <= z' - 1, z' - 1 >= 0, z'' - 1 >= 0, z = 1, z3 >= 0 IF(z, z', z'', z3) -{ 2 + z'' }-> 1 + DIV(0, 1 + (z'' - 1), z3) + s51 :|: s51 >= 0, s51 <= 1 + (z'' - 1), z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 + z }-> 1 + s39 :|: s39 >= 0, s39 <= z - 1, z - 1 >= 0 LT(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 LT(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 LT(z, z') -{ 2 + z' }-> 1 + s26 :|: s26 >= 0, s26 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 MINUS(z, z') -{ 1 + z' }-> 1 + s52 :|: s52 >= 0, s52 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> s54 :|: s54 >= 0, s54 <= 1 + 0 + 1 + z, z'' = 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> s55 :|: s55 >= 0, s55 <= 1 + s9 + 1 + z, s9 >= 0, s9 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> s56 :|: s56 >= 0, s56 <= 0 + 1 + z, z >= 0, z'' >= 0, z' = 0 div(z, z', z'') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 1 + 0 + 1 + 0, z'' = 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 1 + s10 + 1 + 0, s10 >= 0, s10 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> s59 :|: s59 >= 0, s59 <= 0 + 1 + 0, z' - 1 >= 0, z = 0, z'' >= 0 div(z, z', z'') -{ 0 }-> s60 :|: s60 >= 0, s60 <= 1 + 0 + 1 + (1 + (z - 1)), s46 >= 0, s46 <= 2, z'' = 0, z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> s61 :|: s61 >= 0, s61 <= 1 + s11 + 1 + (1 + (z - 1)), s47 >= 0, s47 <= 2, s11 >= 0, s11 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> s62 :|: s62 >= 0, s62 <= 0 + 1 + (1 + (z - 1)), s48 >= 0, s48 <= 2, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> s63 :|: s63 >= 0, s63 <= 1 + 0 + 1 + z, z'' = 0, z' >= 0, z >= 0 div(z, z', z'') -{ 0 }-> s64 :|: s64 >= 0, s64 <= 1 + s12 + 1 + z, s12 >= 0, s12 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> s65 :|: s65 >= 0, s65 <= 0 + 1 + z, z' >= 0, z >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 division(z, z') -{ 0 }-> s53 :|: s53 >= 0, s53 <= z + 0 + 1, z' >= 0, z >= 0 division(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'', z3) -{ 0 }-> s66 :|: s66 >= 0, s66 <= s' + z3 + 1, s' >= 0, s' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> s67 :|: s67 >= 0, s67 <= 0 + z3 + 1, z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s13 :|: s13 >= 0, s13 <= z - 1 + 1, z - 1 >= 0 inc(z) -{ 0 }-> 1 + 0 :|: z = 0 lt(z, z') -{ 0 }-> s49 :|: s49 >= 0, s49 <= 2, z' - 1 >= 0, z - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {division}, {DIVISION} Previous analysis results are: minus: runtime: O(1) [0], size: O(n^1) [z] inc: runtime: O(1) [0], size: O(n^1) [1 + z] LT: runtime: O(n^1) [2 + z'], size: O(n^1) [z'] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] lt: runtime: O(1) [0], size: O(1) [2] MINUS: runtime: O(n^1) [1 + z'], size: O(n^1) [z'] div: runtime: O(1) [0], size: O(n^1) [1 + z + z''] if: runtime: O(1) [0], size: O(n^1) [1 + z' + z3] IF: runtime: O(n^2) [153 + 1109*z' + 52*z'*z'' + 90*z'*z3 + 90*z'^2 + 7*z'' + 12*z3], size: O(n^2) [35 + 28*z' + 4*z'*z'' + 6*z'*z3 + 6*z'^2 + 3*z'' + 4*z3] DIV: runtime: O(n^2) [3924 + 21042*z + 624*z*z' + 540*z*z'' + 1620*z^2 + 135*z' + 104*z''], size: O(n^2) [928 + 576*z + 48*z*z' + 36*z*z'' + 108*z^2 + 63*z' + 40*z''] ---------------------------------------- (65) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (66) Obligation: Complexity RNTS consisting of the following rules: DIV(z, z', z'') -{ 168 + 1199*z + 90*z^2 }-> 1 + s69 + s14 :|: s69 >= 0, s69 <= 3 * 0 + 28 * z + 4 * (1 + 0) + 4 * (0 * z) + 6 * (z * z) + 6 * ((1 + 0) * z) + 35, s14 >= 0, s14 <= 0, z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 168 + 12*s1 + 90*s1*z + 1199*z + 90*z^2 }-> 1 + s70 + s15 :|: s70 >= 0, s70 <= 3 * 0 + 28 * z + 4 * (1 + s1) + 4 * (0 * z) + 6 * (z * z) + 6 * ((1 + s1) * z) + 35, s15 >= 0, s15 <= 0, s1 >= 0, s1 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 156 + 1109*z + 90*z^2 }-> 1 + s71 + s16 :|: s71 >= 0, s71 <= 3 * 0 + 28 * z + 4 * 0 + 4 * (0 * z) + 6 * (z * z) + 6 * (0 * z) + 35, s16 >= 0, s16 <= 0, z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 168 + 8*z' }-> 1 + s72 + s17 :|: s72 >= 0, s72 <= 3 * (1 + (z' - 1)) + 28 * 0 + 4 * (1 + 0) + 4 * ((1 + (z' - 1)) * 0) + 6 * (0 * 0) + 6 * ((1 + 0) * 0) + 35, s17 >= 0, s17 <= 1 + (z' - 1), z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 168 + 12*s2 + 8*z' }-> 1 + s73 + s18 :|: s73 >= 0, s73 <= 3 * (1 + (z' - 1)) + 28 * 0 + 4 * (1 + s2) + 4 * ((1 + (z' - 1)) * 0) + 6 * (0 * 0) + 6 * ((1 + s2) * 0) + 35, s18 >= 0, s18 <= 1 + (z' - 1), s2 >= 0, s2 <= z'' - 1 + 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 156 + 8*z' }-> 1 + s74 + s19 :|: s74 >= 0, s74 <= 3 * (1 + (z' - 1)) + 28 * 0 + 4 * 0 + 4 * ((1 + (z' - 1)) * 0) + 6 * (0 * 0) + 6 * (0 * 0) + 35, s19 >= 0, s19 <= 1 + (z' - 1), z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 168 + 1199*z + 52*z*z' + 90*z^2 + 8*z' }-> 1 + s75 + s20 :|: s75 >= 0, s75 <= 3 * (1 + (z' - 1)) + 28 * (1 + (z - 1)) + 4 * (1 + 0) + 4 * ((1 + (z' - 1)) * (1 + (z - 1))) + 6 * ((1 + (z - 1)) * (1 + (z - 1))) + 6 * ((1 + 0) * (1 + (z - 1))) + 35, s40 >= 0, s40 <= 2, s20 >= 0, s20 <= 1 + (z' - 1), z'' = 0, z' - 1 >= 0, z - 1 >= 0 DIV(z, z', z'') -{ 168 + 12*s3 + 90*s3*z + 1199*z + 52*z*z' + 90*z^2 + 8*z' }-> 1 + s76 + s21 :|: s76 >= 0, s76 <= 3 * (1 + (z' - 1)) + 28 * (1 + (z - 1)) + 4 * (1 + s3) + 4 * ((1 + (z' - 1)) * (1 + (z - 1))) + 6 * ((1 + (z - 1)) * (1 + (z - 1))) + 6 * ((1 + s3) * (1 + (z - 1))) + 35, s41 >= 0, s41 <= 2, s21 >= 0, s21 <= 1 + (z' - 1), s3 >= 0, s3 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 156 + 1109*z + 52*z*z' + 90*z^2 + 8*z' }-> 1 + s77 + s22 :|: s77 >= 0, s77 <= 3 * (1 + (z' - 1)) + 28 * (1 + (z - 1)) + 4 * 0 + 4 * ((1 + (z' - 1)) * (1 + (z - 1))) + 6 * ((1 + (z - 1)) * (1 + (z - 1))) + 6 * (0 * (1 + (z - 1))) + 35, s42 >= 0, s42 <= 2, s22 >= 0, s22 <= 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0, z'' >= 0 DIV(z, z', z'') -{ 168 + 1199*z + 52*z*z' + 90*z^2 + 8*z' }-> 1 + s78 + s23 :|: s78 >= 0, s78 <= 3 * z' + 28 * z + 4 * (1 + 0) + 4 * (z' * z) + 6 * (z * z) + 6 * ((1 + 0) * z) + 35, s23 >= 0, s23 <= z', z'' = 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 168 + 12*s4 + 90*s4*z + 1199*z + 52*z*z' + 90*z^2 + 8*z' }-> 1 + s79 + s24 :|: s79 >= 0, s79 <= 3 * z' + 28 * z + 4 * (1 + s4) + 4 * (z' * z) + 6 * (z * z) + 6 * ((1 + s4) * z) + 35, s24 >= 0, s24 <= z', s4 >= 0, s4 <= z'' - 1 + 1, z'' - 1 >= 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 156 + 1109*z + 52*z*z' + 90*z^2 + 8*z' }-> 1 + s80 + s25 :|: s80 >= 0, s80 <= 3 * z' + 28 * z + 4 * 0 + 4 * (z' * z) + 6 * (z * z) + 6 * (0 * z) + 35, s25 >= 0, s25 <= z', z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 167 + 1199*z + 90*z^2 }-> 1 + s81 + s27 :|: s81 >= 0, s81 <= 3 * 0 + 28 * z + 4 * (1 + 0) + 4 * (0 * z) + 6 * (z * z) + 6 * ((1 + 0) * z) + 35, s27 >= 0, s27 <= 0, z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 167 + 12*s5 + 90*s5*z + 1199*z + 90*z^2 + z'' }-> 1 + s82 + s28 :|: s82 >= 0, s82 <= 3 * 0 + 28 * z + 4 * (1 + s5) + 4 * (0 * z) + 6 * (z * z) + 6 * ((1 + s5) * z) + 35, s28 >= 0, s28 <= 1 + (z'' - 1), s5 >= 0, s5 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 155 + 1109*z + 90*z^2 + z'' }-> 1 + s83 + s29 :|: s83 >= 0, s83 <= 3 * 0 + 28 * z + 4 * 0 + 4 * (0 * z) + 6 * (z * z) + 6 * (0 * z) + 35, s29 >= 0, s29 <= z'', z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 167 + 7*z' }-> 1 + s84 + s30 :|: s84 >= 0, s84 <= 3 * (1 + (z' - 1)) + 28 * 0 + 4 * (1 + 0) + 4 * ((1 + (z' - 1)) * 0) + 6 * (0 * 0) + 6 * ((1 + 0) * 0) + 35, s30 >= 0, s30 <= 0, z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 167 + 12*s6 + 7*z' + z'' }-> 1 + s85 + s31 :|: s85 >= 0, s85 <= 3 * (1 + (z' - 1)) + 28 * 0 + 4 * (1 + s6) + 4 * ((1 + (z' - 1)) * 0) + 6 * (0 * 0) + 6 * ((1 + s6) * 0) + 35, s31 >= 0, s31 <= 1 + (z'' - 1), s6 >= 0, s6 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 155 + 7*z' + z'' }-> 1 + s86 + s32 :|: s86 >= 0, s86 <= 3 * (1 + (z' - 1)) + 28 * 0 + 4 * 0 + 4 * ((1 + (z' - 1)) * 0) + 6 * (0 * 0) + 6 * (0 * 0) + 35, s32 >= 0, s32 <= z'', z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 167 + 1199*z + 52*z*z' + 90*z^2 + 7*z' }-> 1 + s87 + s33 :|: s87 >= 0, s87 <= 3 * (1 + (z' - 1)) + 28 * (1 + (z - 1)) + 4 * (1 + 0) + 4 * ((1 + (z' - 1)) * (1 + (z - 1))) + 6 * ((1 + (z - 1)) * (1 + (z - 1))) + 6 * ((1 + 0) * (1 + (z - 1))) + 35, s43 >= 0, s43 <= 2, s33 >= 0, s33 <= 0, z'' = 0, z - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 167 + 12*s7 + 90*s7*z + 1199*z + 52*z*z' + 90*z^2 + 7*z' + z'' }-> 1 + s88 + s34 :|: s88 >= 0, s88 <= 3 * (1 + (z' - 1)) + 28 * (1 + (z - 1)) + 4 * (1 + s7) + 4 * ((1 + (z' - 1)) * (1 + (z - 1))) + 6 * ((1 + (z - 1)) * (1 + (z - 1))) + 6 * ((1 + s7) * (1 + (z - 1))) + 35, s44 >= 0, s44 <= 2, s34 >= 0, s34 <= 1 + (z'' - 1), s7 >= 0, s7 <= z'' - 1 + 1, z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 155 + 1109*z + 52*z*z' + 90*z^2 + 7*z' + z'' }-> 1 + s89 + s35 :|: s89 >= 0, s89 <= 3 * (1 + (z' - 1)) + 28 * (1 + (z - 1)) + 4 * 0 + 4 * ((1 + (z' - 1)) * (1 + (z - 1))) + 6 * ((1 + (z - 1)) * (1 + (z - 1))) + 6 * (0 * (1 + (z - 1))) + 35, s45 >= 0, s45 <= 2, s35 >= 0, s35 <= z'', z - 1 >= 0, z'' >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 167 + 1199*z + 52*z*z' + 90*z^2 + 7*z' }-> 1 + s90 + s36 :|: s90 >= 0, s90 <= 3 * z' + 28 * z + 4 * (1 + 0) + 4 * (z' * z) + 6 * (z * z) + 6 * ((1 + 0) * z) + 35, s36 >= 0, s36 <= 0, z'' = 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 167 + 12*s8 + 90*s8*z + 1199*z + 52*z*z' + 90*z^2 + 7*z' + z'' }-> 1 + s91 + s37 :|: s91 >= 0, s91 <= 3 * z' + 28 * z + 4 * (1 + s8) + 4 * (z' * z) + 6 * (z * z) + 6 * ((1 + s8) * z) + 35, s37 >= 0, s37 <= 1 + (z'' - 1), s8 >= 0, s8 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 155 + 1109*z + 52*z*z' + 90*z^2 + 7*z' + z'' }-> 1 + s92 + s38 :|: s92 >= 0, s92 <= 3 * z' + 28 * z + 4 * 0 + 4 * (z' * z) + 6 * (z * z) + 6 * (0 * z) + 35, s38 >= 0, s38 <= z'', z' >= 0, z >= 0, z'' >= 0 DIVISION(z, z') -{ 3925 + 21042*z + 624*z*z' + 1620*z^2 + 135*z' }-> 1 + s68 :|: s68 >= 0, s68 <= 576 * z + 108 * (z * z) + 40 * 0 + 36 * (0 * z) + 63 * z' + 48 * (z' * z) + 928, z' >= 0, z >= 0 IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 IF(z, z', z'', z3) -{ 3926 + 21042*s + 624*s*z'' + 540*s*z3 + 1620*s^2 + 136*z'' + 104*z3 }-> 1 + s93 + s50 :|: s93 >= 0, s93 <= 576 * s + 108 * (s * s) + 40 * z3 + 36 * (z3 * s) + 63 * (1 + (z'' - 1)) + 48 * ((1 + (z'' - 1)) * s) + 928, s50 >= 0, s50 <= 1 + (z'' - 1), s >= 0, s <= z' - 1, z' - 1 >= 0, z'' - 1 >= 0, z = 1, z3 >= 0 IF(z, z', z'', z3) -{ 3926 + 136*z'' + 104*z3 }-> 1 + s94 + s51 :|: s94 >= 0, s94 <= 576 * 0 + 108 * (0 * 0) + 40 * z3 + 36 * (z3 * 0) + 63 * (1 + (z'' - 1)) + 48 * ((1 + (z'' - 1)) * 0) + 928, s51 >= 0, s51 <= 1 + (z'' - 1), z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 + z }-> 1 + s39 :|: s39 >= 0, s39 <= z - 1, z - 1 >= 0 LT(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 LT(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 LT(z, z') -{ 2 + z' }-> 1 + s26 :|: s26 >= 0, s26 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 MINUS(z, z') -{ 1 + z' }-> 1 + s52 :|: s52 >= 0, s52 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> s54 :|: s54 >= 0, s54 <= 1 + 0 + 1 + z, z'' = 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> s55 :|: s55 >= 0, s55 <= 1 + s9 + 1 + z, s9 >= 0, s9 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> s56 :|: s56 >= 0, s56 <= 0 + 1 + z, z >= 0, z'' >= 0, z' = 0 div(z, z', z'') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 1 + 0 + 1 + 0, z'' = 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 1 + s10 + 1 + 0, s10 >= 0, s10 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> s59 :|: s59 >= 0, s59 <= 0 + 1 + 0, z' - 1 >= 0, z = 0, z'' >= 0 div(z, z', z'') -{ 0 }-> s60 :|: s60 >= 0, s60 <= 1 + 0 + 1 + (1 + (z - 1)), s46 >= 0, s46 <= 2, z'' = 0, z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> s61 :|: s61 >= 0, s61 <= 1 + s11 + 1 + (1 + (z - 1)), s47 >= 0, s47 <= 2, s11 >= 0, s11 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> s62 :|: s62 >= 0, s62 <= 0 + 1 + (1 + (z - 1)), s48 >= 0, s48 <= 2, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> s63 :|: s63 >= 0, s63 <= 1 + 0 + 1 + z, z'' = 0, z' >= 0, z >= 0 div(z, z', z'') -{ 0 }-> s64 :|: s64 >= 0, s64 <= 1 + s12 + 1 + z, s12 >= 0, s12 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> s65 :|: s65 >= 0, s65 <= 0 + 1 + z, z' >= 0, z >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 division(z, z') -{ 0 }-> s53 :|: s53 >= 0, s53 <= z + 0 + 1, z' >= 0, z >= 0 division(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'', z3) -{ 0 }-> s66 :|: s66 >= 0, s66 <= s' + z3 + 1, s' >= 0, s' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> s67 :|: s67 >= 0, s67 <= 0 + z3 + 1, z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s13 :|: s13 >= 0, s13 <= z - 1 + 1, z - 1 >= 0 inc(z) -{ 0 }-> 1 + 0 :|: z = 0 lt(z, z') -{ 0 }-> s49 :|: s49 >= 0, s49 <= 2, z' - 1 >= 0, z - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {division}, {DIVISION} Previous analysis results are: minus: runtime: O(1) [0], size: O(n^1) [z] inc: runtime: O(1) [0], size: O(n^1) [1 + z] LT: runtime: O(n^1) [2 + z'], size: O(n^1) [z'] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] lt: runtime: O(1) [0], size: O(1) [2] MINUS: runtime: O(n^1) [1 + z'], size: O(n^1) [z'] div: runtime: O(1) [0], size: O(n^1) [1 + z + z''] if: runtime: O(1) [0], size: O(n^1) [1 + z' + z3] IF: runtime: O(n^2) [153 + 1109*z' + 52*z'*z'' + 90*z'*z3 + 90*z'^2 + 7*z'' + 12*z3], size: O(n^2) [35 + 28*z' + 4*z'*z'' + 6*z'*z3 + 6*z'^2 + 3*z'' + 4*z3] DIV: runtime: O(n^2) [3924 + 21042*z + 624*z*z' + 540*z*z'' + 1620*z^2 + 135*z' + 104*z''], size: O(n^2) [928 + 576*z + 48*z*z' + 36*z*z'' + 108*z^2 + 63*z' + 40*z''] ---------------------------------------- (67) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: division after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (68) Obligation: Complexity RNTS consisting of the following rules: DIV(z, z', z'') -{ 168 + 1199*z + 90*z^2 }-> 1 + s69 + s14 :|: s69 >= 0, s69 <= 3 * 0 + 28 * z + 4 * (1 + 0) + 4 * (0 * z) + 6 * (z * z) + 6 * ((1 + 0) * z) + 35, s14 >= 0, s14 <= 0, z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 168 + 12*s1 + 90*s1*z + 1199*z + 90*z^2 }-> 1 + s70 + s15 :|: s70 >= 0, s70 <= 3 * 0 + 28 * z + 4 * (1 + s1) + 4 * (0 * z) + 6 * (z * z) + 6 * ((1 + s1) * z) + 35, s15 >= 0, s15 <= 0, s1 >= 0, s1 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 156 + 1109*z + 90*z^2 }-> 1 + s71 + s16 :|: s71 >= 0, s71 <= 3 * 0 + 28 * z + 4 * 0 + 4 * (0 * z) + 6 * (z * z) + 6 * (0 * z) + 35, s16 >= 0, s16 <= 0, z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 168 + 8*z' }-> 1 + s72 + s17 :|: s72 >= 0, s72 <= 3 * (1 + (z' - 1)) + 28 * 0 + 4 * (1 + 0) + 4 * ((1 + (z' - 1)) * 0) + 6 * (0 * 0) + 6 * ((1 + 0) * 0) + 35, s17 >= 0, s17 <= 1 + (z' - 1), z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 168 + 12*s2 + 8*z' }-> 1 + s73 + s18 :|: s73 >= 0, s73 <= 3 * (1 + (z' - 1)) + 28 * 0 + 4 * (1 + s2) + 4 * ((1 + (z' - 1)) * 0) + 6 * (0 * 0) + 6 * ((1 + s2) * 0) + 35, s18 >= 0, s18 <= 1 + (z' - 1), s2 >= 0, s2 <= z'' - 1 + 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 156 + 8*z' }-> 1 + s74 + s19 :|: s74 >= 0, s74 <= 3 * (1 + (z' - 1)) + 28 * 0 + 4 * 0 + 4 * ((1 + (z' - 1)) * 0) + 6 * (0 * 0) + 6 * (0 * 0) + 35, s19 >= 0, s19 <= 1 + (z' - 1), z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 168 + 1199*z + 52*z*z' + 90*z^2 + 8*z' }-> 1 + s75 + s20 :|: s75 >= 0, s75 <= 3 * (1 + (z' - 1)) + 28 * (1 + (z - 1)) + 4 * (1 + 0) + 4 * ((1 + (z' - 1)) * (1 + (z - 1))) + 6 * ((1 + (z - 1)) * (1 + (z - 1))) + 6 * ((1 + 0) * (1 + (z - 1))) + 35, s40 >= 0, s40 <= 2, s20 >= 0, s20 <= 1 + (z' - 1), z'' = 0, z' - 1 >= 0, z - 1 >= 0 DIV(z, z', z'') -{ 168 + 12*s3 + 90*s3*z + 1199*z + 52*z*z' + 90*z^2 + 8*z' }-> 1 + s76 + s21 :|: s76 >= 0, s76 <= 3 * (1 + (z' - 1)) + 28 * (1 + (z - 1)) + 4 * (1 + s3) + 4 * ((1 + (z' - 1)) * (1 + (z - 1))) + 6 * ((1 + (z - 1)) * (1 + (z - 1))) + 6 * ((1 + s3) * (1 + (z - 1))) + 35, s41 >= 0, s41 <= 2, s21 >= 0, s21 <= 1 + (z' - 1), s3 >= 0, s3 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 156 + 1109*z + 52*z*z' + 90*z^2 + 8*z' }-> 1 + s77 + s22 :|: s77 >= 0, s77 <= 3 * (1 + (z' - 1)) + 28 * (1 + (z - 1)) + 4 * 0 + 4 * ((1 + (z' - 1)) * (1 + (z - 1))) + 6 * ((1 + (z - 1)) * (1 + (z - 1))) + 6 * (0 * (1 + (z - 1))) + 35, s42 >= 0, s42 <= 2, s22 >= 0, s22 <= 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0, z'' >= 0 DIV(z, z', z'') -{ 168 + 1199*z + 52*z*z' + 90*z^2 + 8*z' }-> 1 + s78 + s23 :|: s78 >= 0, s78 <= 3 * z' + 28 * z + 4 * (1 + 0) + 4 * (z' * z) + 6 * (z * z) + 6 * ((1 + 0) * z) + 35, s23 >= 0, s23 <= z', z'' = 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 168 + 12*s4 + 90*s4*z + 1199*z + 52*z*z' + 90*z^2 + 8*z' }-> 1 + s79 + s24 :|: s79 >= 0, s79 <= 3 * z' + 28 * z + 4 * (1 + s4) + 4 * (z' * z) + 6 * (z * z) + 6 * ((1 + s4) * z) + 35, s24 >= 0, s24 <= z', s4 >= 0, s4 <= z'' - 1 + 1, z'' - 1 >= 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 156 + 1109*z + 52*z*z' + 90*z^2 + 8*z' }-> 1 + s80 + s25 :|: s80 >= 0, s80 <= 3 * z' + 28 * z + 4 * 0 + 4 * (z' * z) + 6 * (z * z) + 6 * (0 * z) + 35, s25 >= 0, s25 <= z', z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 167 + 1199*z + 90*z^2 }-> 1 + s81 + s27 :|: s81 >= 0, s81 <= 3 * 0 + 28 * z + 4 * (1 + 0) + 4 * (0 * z) + 6 * (z * z) + 6 * ((1 + 0) * z) + 35, s27 >= 0, s27 <= 0, z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 167 + 12*s5 + 90*s5*z + 1199*z + 90*z^2 + z'' }-> 1 + s82 + s28 :|: s82 >= 0, s82 <= 3 * 0 + 28 * z + 4 * (1 + s5) + 4 * (0 * z) + 6 * (z * z) + 6 * ((1 + s5) * z) + 35, s28 >= 0, s28 <= 1 + (z'' - 1), s5 >= 0, s5 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 155 + 1109*z + 90*z^2 + z'' }-> 1 + s83 + s29 :|: s83 >= 0, s83 <= 3 * 0 + 28 * z + 4 * 0 + 4 * (0 * z) + 6 * (z * z) + 6 * (0 * z) + 35, s29 >= 0, s29 <= z'', z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 167 + 7*z' }-> 1 + s84 + s30 :|: s84 >= 0, s84 <= 3 * (1 + (z' - 1)) + 28 * 0 + 4 * (1 + 0) + 4 * ((1 + (z' - 1)) * 0) + 6 * (0 * 0) + 6 * ((1 + 0) * 0) + 35, s30 >= 0, s30 <= 0, z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 167 + 12*s6 + 7*z' + z'' }-> 1 + s85 + s31 :|: s85 >= 0, s85 <= 3 * (1 + (z' - 1)) + 28 * 0 + 4 * (1 + s6) + 4 * ((1 + (z' - 1)) * 0) + 6 * (0 * 0) + 6 * ((1 + s6) * 0) + 35, s31 >= 0, s31 <= 1 + (z'' - 1), s6 >= 0, s6 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 155 + 7*z' + z'' }-> 1 + s86 + s32 :|: s86 >= 0, s86 <= 3 * (1 + (z' - 1)) + 28 * 0 + 4 * 0 + 4 * ((1 + (z' - 1)) * 0) + 6 * (0 * 0) + 6 * (0 * 0) + 35, s32 >= 0, s32 <= z'', z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 167 + 1199*z + 52*z*z' + 90*z^2 + 7*z' }-> 1 + s87 + s33 :|: s87 >= 0, s87 <= 3 * (1 + (z' - 1)) + 28 * (1 + (z - 1)) + 4 * (1 + 0) + 4 * ((1 + (z' - 1)) * (1 + (z - 1))) + 6 * ((1 + (z - 1)) * (1 + (z - 1))) + 6 * ((1 + 0) * (1 + (z - 1))) + 35, s43 >= 0, s43 <= 2, s33 >= 0, s33 <= 0, z'' = 0, z - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 167 + 12*s7 + 90*s7*z + 1199*z + 52*z*z' + 90*z^2 + 7*z' + z'' }-> 1 + s88 + s34 :|: s88 >= 0, s88 <= 3 * (1 + (z' - 1)) + 28 * (1 + (z - 1)) + 4 * (1 + s7) + 4 * ((1 + (z' - 1)) * (1 + (z - 1))) + 6 * ((1 + (z - 1)) * (1 + (z - 1))) + 6 * ((1 + s7) * (1 + (z - 1))) + 35, s44 >= 0, s44 <= 2, s34 >= 0, s34 <= 1 + (z'' - 1), s7 >= 0, s7 <= z'' - 1 + 1, z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 155 + 1109*z + 52*z*z' + 90*z^2 + 7*z' + z'' }-> 1 + s89 + s35 :|: s89 >= 0, s89 <= 3 * (1 + (z' - 1)) + 28 * (1 + (z - 1)) + 4 * 0 + 4 * ((1 + (z' - 1)) * (1 + (z - 1))) + 6 * ((1 + (z - 1)) * (1 + (z - 1))) + 6 * (0 * (1 + (z - 1))) + 35, s45 >= 0, s45 <= 2, s35 >= 0, s35 <= z'', z - 1 >= 0, z'' >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 167 + 1199*z + 52*z*z' + 90*z^2 + 7*z' }-> 1 + s90 + s36 :|: s90 >= 0, s90 <= 3 * z' + 28 * z + 4 * (1 + 0) + 4 * (z' * z) + 6 * (z * z) + 6 * ((1 + 0) * z) + 35, s36 >= 0, s36 <= 0, z'' = 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 167 + 12*s8 + 90*s8*z + 1199*z + 52*z*z' + 90*z^2 + 7*z' + z'' }-> 1 + s91 + s37 :|: s91 >= 0, s91 <= 3 * z' + 28 * z + 4 * (1 + s8) + 4 * (z' * z) + 6 * (z * z) + 6 * ((1 + s8) * z) + 35, s37 >= 0, s37 <= 1 + (z'' - 1), s8 >= 0, s8 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 155 + 1109*z + 52*z*z' + 90*z^2 + 7*z' + z'' }-> 1 + s92 + s38 :|: s92 >= 0, s92 <= 3 * z' + 28 * z + 4 * 0 + 4 * (z' * z) + 6 * (z * z) + 6 * (0 * z) + 35, s38 >= 0, s38 <= z'', z' >= 0, z >= 0, z'' >= 0 DIVISION(z, z') -{ 3925 + 21042*z + 624*z*z' + 1620*z^2 + 135*z' }-> 1 + s68 :|: s68 >= 0, s68 <= 576 * z + 108 * (z * z) + 40 * 0 + 36 * (0 * z) + 63 * z' + 48 * (z' * z) + 928, z' >= 0, z >= 0 IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 IF(z, z', z'', z3) -{ 3926 + 21042*s + 624*s*z'' + 540*s*z3 + 1620*s^2 + 136*z'' + 104*z3 }-> 1 + s93 + s50 :|: s93 >= 0, s93 <= 576 * s + 108 * (s * s) + 40 * z3 + 36 * (z3 * s) + 63 * (1 + (z'' - 1)) + 48 * ((1 + (z'' - 1)) * s) + 928, s50 >= 0, s50 <= 1 + (z'' - 1), s >= 0, s <= z' - 1, z' - 1 >= 0, z'' - 1 >= 0, z = 1, z3 >= 0 IF(z, z', z'', z3) -{ 3926 + 136*z'' + 104*z3 }-> 1 + s94 + s51 :|: s94 >= 0, s94 <= 576 * 0 + 108 * (0 * 0) + 40 * z3 + 36 * (z3 * 0) + 63 * (1 + (z'' - 1)) + 48 * ((1 + (z'' - 1)) * 0) + 928, s51 >= 0, s51 <= 1 + (z'' - 1), z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 + z }-> 1 + s39 :|: s39 >= 0, s39 <= z - 1, z - 1 >= 0 LT(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 LT(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 LT(z, z') -{ 2 + z' }-> 1 + s26 :|: s26 >= 0, s26 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 MINUS(z, z') -{ 1 + z' }-> 1 + s52 :|: s52 >= 0, s52 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> s54 :|: s54 >= 0, s54 <= 1 + 0 + 1 + z, z'' = 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> s55 :|: s55 >= 0, s55 <= 1 + s9 + 1 + z, s9 >= 0, s9 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> s56 :|: s56 >= 0, s56 <= 0 + 1 + z, z >= 0, z'' >= 0, z' = 0 div(z, z', z'') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 1 + 0 + 1 + 0, z'' = 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 1 + s10 + 1 + 0, s10 >= 0, s10 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> s59 :|: s59 >= 0, s59 <= 0 + 1 + 0, z' - 1 >= 0, z = 0, z'' >= 0 div(z, z', z'') -{ 0 }-> s60 :|: s60 >= 0, s60 <= 1 + 0 + 1 + (1 + (z - 1)), s46 >= 0, s46 <= 2, z'' = 0, z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> s61 :|: s61 >= 0, s61 <= 1 + s11 + 1 + (1 + (z - 1)), s47 >= 0, s47 <= 2, s11 >= 0, s11 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> s62 :|: s62 >= 0, s62 <= 0 + 1 + (1 + (z - 1)), s48 >= 0, s48 <= 2, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> s63 :|: s63 >= 0, s63 <= 1 + 0 + 1 + z, z'' = 0, z' >= 0, z >= 0 div(z, z', z'') -{ 0 }-> s64 :|: s64 >= 0, s64 <= 1 + s12 + 1 + z, s12 >= 0, s12 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> s65 :|: s65 >= 0, s65 <= 0 + 1 + z, z' >= 0, z >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 division(z, z') -{ 0 }-> s53 :|: s53 >= 0, s53 <= z + 0 + 1, z' >= 0, z >= 0 division(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'', z3) -{ 0 }-> s66 :|: s66 >= 0, s66 <= s' + z3 + 1, s' >= 0, s' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> s67 :|: s67 >= 0, s67 <= 0 + z3 + 1, z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s13 :|: s13 >= 0, s13 <= z - 1 + 1, z - 1 >= 0 inc(z) -{ 0 }-> 1 + 0 :|: z = 0 lt(z, z') -{ 0 }-> s49 :|: s49 >= 0, s49 <= 2, z' - 1 >= 0, z - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {division}, {DIVISION} Previous analysis results are: minus: runtime: O(1) [0], size: O(n^1) [z] inc: runtime: O(1) [0], size: O(n^1) [1 + z] LT: runtime: O(n^1) [2 + z'], size: O(n^1) [z'] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] lt: runtime: O(1) [0], size: O(1) [2] MINUS: runtime: O(n^1) [1 + z'], size: O(n^1) [z'] div: runtime: O(1) [0], size: O(n^1) [1 + z + z''] if: runtime: O(1) [0], size: O(n^1) [1 + z' + z3] IF: runtime: O(n^2) [153 + 1109*z' + 52*z'*z'' + 90*z'*z3 + 90*z'^2 + 7*z'' + 12*z3], size: O(n^2) [35 + 28*z' + 4*z'*z'' + 6*z'*z3 + 6*z'^2 + 3*z'' + 4*z3] DIV: runtime: O(n^2) [3924 + 21042*z + 624*z*z' + 540*z*z'' + 1620*z^2 + 135*z' + 104*z''], size: O(n^2) [928 + 576*z + 48*z*z' + 36*z*z'' + 108*z^2 + 63*z' + 40*z''] division: runtime: ?, size: O(n^1) [1 + z] ---------------------------------------- (69) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: division after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (70) Obligation: Complexity RNTS consisting of the following rules: DIV(z, z', z'') -{ 168 + 1199*z + 90*z^2 }-> 1 + s69 + s14 :|: s69 >= 0, s69 <= 3 * 0 + 28 * z + 4 * (1 + 0) + 4 * (0 * z) + 6 * (z * z) + 6 * ((1 + 0) * z) + 35, s14 >= 0, s14 <= 0, z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 168 + 12*s1 + 90*s1*z + 1199*z + 90*z^2 }-> 1 + s70 + s15 :|: s70 >= 0, s70 <= 3 * 0 + 28 * z + 4 * (1 + s1) + 4 * (0 * z) + 6 * (z * z) + 6 * ((1 + s1) * z) + 35, s15 >= 0, s15 <= 0, s1 >= 0, s1 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 156 + 1109*z + 90*z^2 }-> 1 + s71 + s16 :|: s71 >= 0, s71 <= 3 * 0 + 28 * z + 4 * 0 + 4 * (0 * z) + 6 * (z * z) + 6 * (0 * z) + 35, s16 >= 0, s16 <= 0, z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 168 + 8*z' }-> 1 + s72 + s17 :|: s72 >= 0, s72 <= 3 * (1 + (z' - 1)) + 28 * 0 + 4 * (1 + 0) + 4 * ((1 + (z' - 1)) * 0) + 6 * (0 * 0) + 6 * ((1 + 0) * 0) + 35, s17 >= 0, s17 <= 1 + (z' - 1), z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 168 + 12*s2 + 8*z' }-> 1 + s73 + s18 :|: s73 >= 0, s73 <= 3 * (1 + (z' - 1)) + 28 * 0 + 4 * (1 + s2) + 4 * ((1 + (z' - 1)) * 0) + 6 * (0 * 0) + 6 * ((1 + s2) * 0) + 35, s18 >= 0, s18 <= 1 + (z' - 1), s2 >= 0, s2 <= z'' - 1 + 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 156 + 8*z' }-> 1 + s74 + s19 :|: s74 >= 0, s74 <= 3 * (1 + (z' - 1)) + 28 * 0 + 4 * 0 + 4 * ((1 + (z' - 1)) * 0) + 6 * (0 * 0) + 6 * (0 * 0) + 35, s19 >= 0, s19 <= 1 + (z' - 1), z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 168 + 1199*z + 52*z*z' + 90*z^2 + 8*z' }-> 1 + s75 + s20 :|: s75 >= 0, s75 <= 3 * (1 + (z' - 1)) + 28 * (1 + (z - 1)) + 4 * (1 + 0) + 4 * ((1 + (z' - 1)) * (1 + (z - 1))) + 6 * ((1 + (z - 1)) * (1 + (z - 1))) + 6 * ((1 + 0) * (1 + (z - 1))) + 35, s40 >= 0, s40 <= 2, s20 >= 0, s20 <= 1 + (z' - 1), z'' = 0, z' - 1 >= 0, z - 1 >= 0 DIV(z, z', z'') -{ 168 + 12*s3 + 90*s3*z + 1199*z + 52*z*z' + 90*z^2 + 8*z' }-> 1 + s76 + s21 :|: s76 >= 0, s76 <= 3 * (1 + (z' - 1)) + 28 * (1 + (z - 1)) + 4 * (1 + s3) + 4 * ((1 + (z' - 1)) * (1 + (z - 1))) + 6 * ((1 + (z - 1)) * (1 + (z - 1))) + 6 * ((1 + s3) * (1 + (z - 1))) + 35, s41 >= 0, s41 <= 2, s21 >= 0, s21 <= 1 + (z' - 1), s3 >= 0, s3 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 156 + 1109*z + 52*z*z' + 90*z^2 + 8*z' }-> 1 + s77 + s22 :|: s77 >= 0, s77 <= 3 * (1 + (z' - 1)) + 28 * (1 + (z - 1)) + 4 * 0 + 4 * ((1 + (z' - 1)) * (1 + (z - 1))) + 6 * ((1 + (z - 1)) * (1 + (z - 1))) + 6 * (0 * (1 + (z - 1))) + 35, s42 >= 0, s42 <= 2, s22 >= 0, s22 <= 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0, z'' >= 0 DIV(z, z', z'') -{ 168 + 1199*z + 52*z*z' + 90*z^2 + 8*z' }-> 1 + s78 + s23 :|: s78 >= 0, s78 <= 3 * z' + 28 * z + 4 * (1 + 0) + 4 * (z' * z) + 6 * (z * z) + 6 * ((1 + 0) * z) + 35, s23 >= 0, s23 <= z', z'' = 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 168 + 12*s4 + 90*s4*z + 1199*z + 52*z*z' + 90*z^2 + 8*z' }-> 1 + s79 + s24 :|: s79 >= 0, s79 <= 3 * z' + 28 * z + 4 * (1 + s4) + 4 * (z' * z) + 6 * (z * z) + 6 * ((1 + s4) * z) + 35, s24 >= 0, s24 <= z', s4 >= 0, s4 <= z'' - 1 + 1, z'' - 1 >= 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 156 + 1109*z + 52*z*z' + 90*z^2 + 8*z' }-> 1 + s80 + s25 :|: s80 >= 0, s80 <= 3 * z' + 28 * z + 4 * 0 + 4 * (z' * z) + 6 * (z * z) + 6 * (0 * z) + 35, s25 >= 0, s25 <= z', z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 167 + 1199*z + 90*z^2 }-> 1 + s81 + s27 :|: s81 >= 0, s81 <= 3 * 0 + 28 * z + 4 * (1 + 0) + 4 * (0 * z) + 6 * (z * z) + 6 * ((1 + 0) * z) + 35, s27 >= 0, s27 <= 0, z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 167 + 12*s5 + 90*s5*z + 1199*z + 90*z^2 + z'' }-> 1 + s82 + s28 :|: s82 >= 0, s82 <= 3 * 0 + 28 * z + 4 * (1 + s5) + 4 * (0 * z) + 6 * (z * z) + 6 * ((1 + s5) * z) + 35, s28 >= 0, s28 <= 1 + (z'' - 1), s5 >= 0, s5 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 155 + 1109*z + 90*z^2 + z'' }-> 1 + s83 + s29 :|: s83 >= 0, s83 <= 3 * 0 + 28 * z + 4 * 0 + 4 * (0 * z) + 6 * (z * z) + 6 * (0 * z) + 35, s29 >= 0, s29 <= z'', z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 167 + 7*z' }-> 1 + s84 + s30 :|: s84 >= 0, s84 <= 3 * (1 + (z' - 1)) + 28 * 0 + 4 * (1 + 0) + 4 * ((1 + (z' - 1)) * 0) + 6 * (0 * 0) + 6 * ((1 + 0) * 0) + 35, s30 >= 0, s30 <= 0, z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 167 + 12*s6 + 7*z' + z'' }-> 1 + s85 + s31 :|: s85 >= 0, s85 <= 3 * (1 + (z' - 1)) + 28 * 0 + 4 * (1 + s6) + 4 * ((1 + (z' - 1)) * 0) + 6 * (0 * 0) + 6 * ((1 + s6) * 0) + 35, s31 >= 0, s31 <= 1 + (z'' - 1), s6 >= 0, s6 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 155 + 7*z' + z'' }-> 1 + s86 + s32 :|: s86 >= 0, s86 <= 3 * (1 + (z' - 1)) + 28 * 0 + 4 * 0 + 4 * ((1 + (z' - 1)) * 0) + 6 * (0 * 0) + 6 * (0 * 0) + 35, s32 >= 0, s32 <= z'', z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 167 + 1199*z + 52*z*z' + 90*z^2 + 7*z' }-> 1 + s87 + s33 :|: s87 >= 0, s87 <= 3 * (1 + (z' - 1)) + 28 * (1 + (z - 1)) + 4 * (1 + 0) + 4 * ((1 + (z' - 1)) * (1 + (z - 1))) + 6 * ((1 + (z - 1)) * (1 + (z - 1))) + 6 * ((1 + 0) * (1 + (z - 1))) + 35, s43 >= 0, s43 <= 2, s33 >= 0, s33 <= 0, z'' = 0, z - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 167 + 12*s7 + 90*s7*z + 1199*z + 52*z*z' + 90*z^2 + 7*z' + z'' }-> 1 + s88 + s34 :|: s88 >= 0, s88 <= 3 * (1 + (z' - 1)) + 28 * (1 + (z - 1)) + 4 * (1 + s7) + 4 * ((1 + (z' - 1)) * (1 + (z - 1))) + 6 * ((1 + (z - 1)) * (1 + (z - 1))) + 6 * ((1 + s7) * (1 + (z - 1))) + 35, s44 >= 0, s44 <= 2, s34 >= 0, s34 <= 1 + (z'' - 1), s7 >= 0, s7 <= z'' - 1 + 1, z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 155 + 1109*z + 52*z*z' + 90*z^2 + 7*z' + z'' }-> 1 + s89 + s35 :|: s89 >= 0, s89 <= 3 * (1 + (z' - 1)) + 28 * (1 + (z - 1)) + 4 * 0 + 4 * ((1 + (z' - 1)) * (1 + (z - 1))) + 6 * ((1 + (z - 1)) * (1 + (z - 1))) + 6 * (0 * (1 + (z - 1))) + 35, s45 >= 0, s45 <= 2, s35 >= 0, s35 <= z'', z - 1 >= 0, z'' >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 167 + 1199*z + 52*z*z' + 90*z^2 + 7*z' }-> 1 + s90 + s36 :|: s90 >= 0, s90 <= 3 * z' + 28 * z + 4 * (1 + 0) + 4 * (z' * z) + 6 * (z * z) + 6 * ((1 + 0) * z) + 35, s36 >= 0, s36 <= 0, z'' = 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 167 + 12*s8 + 90*s8*z + 1199*z + 52*z*z' + 90*z^2 + 7*z' + z'' }-> 1 + s91 + s37 :|: s91 >= 0, s91 <= 3 * z' + 28 * z + 4 * (1 + s8) + 4 * (z' * z) + 6 * (z * z) + 6 * ((1 + s8) * z) + 35, s37 >= 0, s37 <= 1 + (z'' - 1), s8 >= 0, s8 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 155 + 1109*z + 52*z*z' + 90*z^2 + 7*z' + z'' }-> 1 + s92 + s38 :|: s92 >= 0, s92 <= 3 * z' + 28 * z + 4 * 0 + 4 * (z' * z) + 6 * (z * z) + 6 * (0 * z) + 35, s38 >= 0, s38 <= z'', z' >= 0, z >= 0, z'' >= 0 DIVISION(z, z') -{ 3925 + 21042*z + 624*z*z' + 1620*z^2 + 135*z' }-> 1 + s68 :|: s68 >= 0, s68 <= 576 * z + 108 * (z * z) + 40 * 0 + 36 * (0 * z) + 63 * z' + 48 * (z' * z) + 928, z' >= 0, z >= 0 IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 IF(z, z', z'', z3) -{ 3926 + 21042*s + 624*s*z'' + 540*s*z3 + 1620*s^2 + 136*z'' + 104*z3 }-> 1 + s93 + s50 :|: s93 >= 0, s93 <= 576 * s + 108 * (s * s) + 40 * z3 + 36 * (z3 * s) + 63 * (1 + (z'' - 1)) + 48 * ((1 + (z'' - 1)) * s) + 928, s50 >= 0, s50 <= 1 + (z'' - 1), s >= 0, s <= z' - 1, z' - 1 >= 0, z'' - 1 >= 0, z = 1, z3 >= 0 IF(z, z', z'', z3) -{ 3926 + 136*z'' + 104*z3 }-> 1 + s94 + s51 :|: s94 >= 0, s94 <= 576 * 0 + 108 * (0 * 0) + 40 * z3 + 36 * (z3 * 0) + 63 * (1 + (z'' - 1)) + 48 * ((1 + (z'' - 1)) * 0) + 928, s51 >= 0, s51 <= 1 + (z'' - 1), z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 + z }-> 1 + s39 :|: s39 >= 0, s39 <= z - 1, z - 1 >= 0 LT(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 LT(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 LT(z, z') -{ 2 + z' }-> 1 + s26 :|: s26 >= 0, s26 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 MINUS(z, z') -{ 1 + z' }-> 1 + s52 :|: s52 >= 0, s52 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> s54 :|: s54 >= 0, s54 <= 1 + 0 + 1 + z, z'' = 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> s55 :|: s55 >= 0, s55 <= 1 + s9 + 1 + z, s9 >= 0, s9 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> s56 :|: s56 >= 0, s56 <= 0 + 1 + z, z >= 0, z'' >= 0, z' = 0 div(z, z', z'') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 1 + 0 + 1 + 0, z'' = 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 1 + s10 + 1 + 0, s10 >= 0, s10 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> s59 :|: s59 >= 0, s59 <= 0 + 1 + 0, z' - 1 >= 0, z = 0, z'' >= 0 div(z, z', z'') -{ 0 }-> s60 :|: s60 >= 0, s60 <= 1 + 0 + 1 + (1 + (z - 1)), s46 >= 0, s46 <= 2, z'' = 0, z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> s61 :|: s61 >= 0, s61 <= 1 + s11 + 1 + (1 + (z - 1)), s47 >= 0, s47 <= 2, s11 >= 0, s11 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> s62 :|: s62 >= 0, s62 <= 0 + 1 + (1 + (z - 1)), s48 >= 0, s48 <= 2, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> s63 :|: s63 >= 0, s63 <= 1 + 0 + 1 + z, z'' = 0, z' >= 0, z >= 0 div(z, z', z'') -{ 0 }-> s64 :|: s64 >= 0, s64 <= 1 + s12 + 1 + z, s12 >= 0, s12 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> s65 :|: s65 >= 0, s65 <= 0 + 1 + z, z' >= 0, z >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 division(z, z') -{ 0 }-> s53 :|: s53 >= 0, s53 <= z + 0 + 1, z' >= 0, z >= 0 division(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'', z3) -{ 0 }-> s66 :|: s66 >= 0, s66 <= s' + z3 + 1, s' >= 0, s' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> s67 :|: s67 >= 0, s67 <= 0 + z3 + 1, z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s13 :|: s13 >= 0, s13 <= z - 1 + 1, z - 1 >= 0 inc(z) -{ 0 }-> 1 + 0 :|: z = 0 lt(z, z') -{ 0 }-> s49 :|: s49 >= 0, s49 <= 2, z' - 1 >= 0, z - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {DIVISION} Previous analysis results are: minus: runtime: O(1) [0], size: O(n^1) [z] inc: runtime: O(1) [0], size: O(n^1) [1 + z] LT: runtime: O(n^1) [2 + z'], size: O(n^1) [z'] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] lt: runtime: O(1) [0], size: O(1) [2] MINUS: runtime: O(n^1) [1 + z'], size: O(n^1) [z'] div: runtime: O(1) [0], size: O(n^1) [1 + z + z''] if: runtime: O(1) [0], size: O(n^1) [1 + z' + z3] IF: runtime: O(n^2) [153 + 1109*z' + 52*z'*z'' + 90*z'*z3 + 90*z'^2 + 7*z'' + 12*z3], size: O(n^2) [35 + 28*z' + 4*z'*z'' + 6*z'*z3 + 6*z'^2 + 3*z'' + 4*z3] DIV: runtime: O(n^2) [3924 + 21042*z + 624*z*z' + 540*z*z'' + 1620*z^2 + 135*z' + 104*z''], size: O(n^2) [928 + 576*z + 48*z*z' + 36*z*z'' + 108*z^2 + 63*z' + 40*z''] division: runtime: O(1) [0], size: O(n^1) [1 + z] ---------------------------------------- (71) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (72) Obligation: Complexity RNTS consisting of the following rules: DIV(z, z', z'') -{ 168 + 1199*z + 90*z^2 }-> 1 + s69 + s14 :|: s69 >= 0, s69 <= 3 * 0 + 28 * z + 4 * (1 + 0) + 4 * (0 * z) + 6 * (z * z) + 6 * ((1 + 0) * z) + 35, s14 >= 0, s14 <= 0, z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 168 + 12*s1 + 90*s1*z + 1199*z + 90*z^2 }-> 1 + s70 + s15 :|: s70 >= 0, s70 <= 3 * 0 + 28 * z + 4 * (1 + s1) + 4 * (0 * z) + 6 * (z * z) + 6 * ((1 + s1) * z) + 35, s15 >= 0, s15 <= 0, s1 >= 0, s1 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 156 + 1109*z + 90*z^2 }-> 1 + s71 + s16 :|: s71 >= 0, s71 <= 3 * 0 + 28 * z + 4 * 0 + 4 * (0 * z) + 6 * (z * z) + 6 * (0 * z) + 35, s16 >= 0, s16 <= 0, z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 168 + 8*z' }-> 1 + s72 + s17 :|: s72 >= 0, s72 <= 3 * (1 + (z' - 1)) + 28 * 0 + 4 * (1 + 0) + 4 * ((1 + (z' - 1)) * 0) + 6 * (0 * 0) + 6 * ((1 + 0) * 0) + 35, s17 >= 0, s17 <= 1 + (z' - 1), z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 168 + 12*s2 + 8*z' }-> 1 + s73 + s18 :|: s73 >= 0, s73 <= 3 * (1 + (z' - 1)) + 28 * 0 + 4 * (1 + s2) + 4 * ((1 + (z' - 1)) * 0) + 6 * (0 * 0) + 6 * ((1 + s2) * 0) + 35, s18 >= 0, s18 <= 1 + (z' - 1), s2 >= 0, s2 <= z'' - 1 + 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 156 + 8*z' }-> 1 + s74 + s19 :|: s74 >= 0, s74 <= 3 * (1 + (z' - 1)) + 28 * 0 + 4 * 0 + 4 * ((1 + (z' - 1)) * 0) + 6 * (0 * 0) + 6 * (0 * 0) + 35, s19 >= 0, s19 <= 1 + (z' - 1), z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 168 + 1199*z + 52*z*z' + 90*z^2 + 8*z' }-> 1 + s75 + s20 :|: s75 >= 0, s75 <= 3 * (1 + (z' - 1)) + 28 * (1 + (z - 1)) + 4 * (1 + 0) + 4 * ((1 + (z' - 1)) * (1 + (z - 1))) + 6 * ((1 + (z - 1)) * (1 + (z - 1))) + 6 * ((1 + 0) * (1 + (z - 1))) + 35, s40 >= 0, s40 <= 2, s20 >= 0, s20 <= 1 + (z' - 1), z'' = 0, z' - 1 >= 0, z - 1 >= 0 DIV(z, z', z'') -{ 168 + 12*s3 + 90*s3*z + 1199*z + 52*z*z' + 90*z^2 + 8*z' }-> 1 + s76 + s21 :|: s76 >= 0, s76 <= 3 * (1 + (z' - 1)) + 28 * (1 + (z - 1)) + 4 * (1 + s3) + 4 * ((1 + (z' - 1)) * (1 + (z - 1))) + 6 * ((1 + (z - 1)) * (1 + (z - 1))) + 6 * ((1 + s3) * (1 + (z - 1))) + 35, s41 >= 0, s41 <= 2, s21 >= 0, s21 <= 1 + (z' - 1), s3 >= 0, s3 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 156 + 1109*z + 52*z*z' + 90*z^2 + 8*z' }-> 1 + s77 + s22 :|: s77 >= 0, s77 <= 3 * (1 + (z' - 1)) + 28 * (1 + (z - 1)) + 4 * 0 + 4 * ((1 + (z' - 1)) * (1 + (z - 1))) + 6 * ((1 + (z - 1)) * (1 + (z - 1))) + 6 * (0 * (1 + (z - 1))) + 35, s42 >= 0, s42 <= 2, s22 >= 0, s22 <= 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0, z'' >= 0 DIV(z, z', z'') -{ 168 + 1199*z + 52*z*z' + 90*z^2 + 8*z' }-> 1 + s78 + s23 :|: s78 >= 0, s78 <= 3 * z' + 28 * z + 4 * (1 + 0) + 4 * (z' * z) + 6 * (z * z) + 6 * ((1 + 0) * z) + 35, s23 >= 0, s23 <= z', z'' = 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 168 + 12*s4 + 90*s4*z + 1199*z + 52*z*z' + 90*z^2 + 8*z' }-> 1 + s79 + s24 :|: s79 >= 0, s79 <= 3 * z' + 28 * z + 4 * (1 + s4) + 4 * (z' * z) + 6 * (z * z) + 6 * ((1 + s4) * z) + 35, s24 >= 0, s24 <= z', s4 >= 0, s4 <= z'' - 1 + 1, z'' - 1 >= 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 156 + 1109*z + 52*z*z' + 90*z^2 + 8*z' }-> 1 + s80 + s25 :|: s80 >= 0, s80 <= 3 * z' + 28 * z + 4 * 0 + 4 * (z' * z) + 6 * (z * z) + 6 * (0 * z) + 35, s25 >= 0, s25 <= z', z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 167 + 1199*z + 90*z^2 }-> 1 + s81 + s27 :|: s81 >= 0, s81 <= 3 * 0 + 28 * z + 4 * (1 + 0) + 4 * (0 * z) + 6 * (z * z) + 6 * ((1 + 0) * z) + 35, s27 >= 0, s27 <= 0, z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 167 + 12*s5 + 90*s5*z + 1199*z + 90*z^2 + z'' }-> 1 + s82 + s28 :|: s82 >= 0, s82 <= 3 * 0 + 28 * z + 4 * (1 + s5) + 4 * (0 * z) + 6 * (z * z) + 6 * ((1 + s5) * z) + 35, s28 >= 0, s28 <= 1 + (z'' - 1), s5 >= 0, s5 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 155 + 1109*z + 90*z^2 + z'' }-> 1 + s83 + s29 :|: s83 >= 0, s83 <= 3 * 0 + 28 * z + 4 * 0 + 4 * (0 * z) + 6 * (z * z) + 6 * (0 * z) + 35, s29 >= 0, s29 <= z'', z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 167 + 7*z' }-> 1 + s84 + s30 :|: s84 >= 0, s84 <= 3 * (1 + (z' - 1)) + 28 * 0 + 4 * (1 + 0) + 4 * ((1 + (z' - 1)) * 0) + 6 * (0 * 0) + 6 * ((1 + 0) * 0) + 35, s30 >= 0, s30 <= 0, z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 167 + 12*s6 + 7*z' + z'' }-> 1 + s85 + s31 :|: s85 >= 0, s85 <= 3 * (1 + (z' - 1)) + 28 * 0 + 4 * (1 + s6) + 4 * ((1 + (z' - 1)) * 0) + 6 * (0 * 0) + 6 * ((1 + s6) * 0) + 35, s31 >= 0, s31 <= 1 + (z'' - 1), s6 >= 0, s6 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 155 + 7*z' + z'' }-> 1 + s86 + s32 :|: s86 >= 0, s86 <= 3 * (1 + (z' - 1)) + 28 * 0 + 4 * 0 + 4 * ((1 + (z' - 1)) * 0) + 6 * (0 * 0) + 6 * (0 * 0) + 35, s32 >= 0, s32 <= z'', z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 167 + 1199*z + 52*z*z' + 90*z^2 + 7*z' }-> 1 + s87 + s33 :|: s87 >= 0, s87 <= 3 * (1 + (z' - 1)) + 28 * (1 + (z - 1)) + 4 * (1 + 0) + 4 * ((1 + (z' - 1)) * (1 + (z - 1))) + 6 * ((1 + (z - 1)) * (1 + (z - 1))) + 6 * ((1 + 0) * (1 + (z - 1))) + 35, s43 >= 0, s43 <= 2, s33 >= 0, s33 <= 0, z'' = 0, z - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 167 + 12*s7 + 90*s7*z + 1199*z + 52*z*z' + 90*z^2 + 7*z' + z'' }-> 1 + s88 + s34 :|: s88 >= 0, s88 <= 3 * (1 + (z' - 1)) + 28 * (1 + (z - 1)) + 4 * (1 + s7) + 4 * ((1 + (z' - 1)) * (1 + (z - 1))) + 6 * ((1 + (z - 1)) * (1 + (z - 1))) + 6 * ((1 + s7) * (1 + (z - 1))) + 35, s44 >= 0, s44 <= 2, s34 >= 0, s34 <= 1 + (z'' - 1), s7 >= 0, s7 <= z'' - 1 + 1, z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 155 + 1109*z + 52*z*z' + 90*z^2 + 7*z' + z'' }-> 1 + s89 + s35 :|: s89 >= 0, s89 <= 3 * (1 + (z' - 1)) + 28 * (1 + (z - 1)) + 4 * 0 + 4 * ((1 + (z' - 1)) * (1 + (z - 1))) + 6 * ((1 + (z - 1)) * (1 + (z - 1))) + 6 * (0 * (1 + (z - 1))) + 35, s45 >= 0, s45 <= 2, s35 >= 0, s35 <= z'', z - 1 >= 0, z'' >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 167 + 1199*z + 52*z*z' + 90*z^2 + 7*z' }-> 1 + s90 + s36 :|: s90 >= 0, s90 <= 3 * z' + 28 * z + 4 * (1 + 0) + 4 * (z' * z) + 6 * (z * z) + 6 * ((1 + 0) * z) + 35, s36 >= 0, s36 <= 0, z'' = 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 167 + 12*s8 + 90*s8*z + 1199*z + 52*z*z' + 90*z^2 + 7*z' + z'' }-> 1 + s91 + s37 :|: s91 >= 0, s91 <= 3 * z' + 28 * z + 4 * (1 + s8) + 4 * (z' * z) + 6 * (z * z) + 6 * ((1 + s8) * z) + 35, s37 >= 0, s37 <= 1 + (z'' - 1), s8 >= 0, s8 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 155 + 1109*z + 52*z*z' + 90*z^2 + 7*z' + z'' }-> 1 + s92 + s38 :|: s92 >= 0, s92 <= 3 * z' + 28 * z + 4 * 0 + 4 * (z' * z) + 6 * (z * z) + 6 * (0 * z) + 35, s38 >= 0, s38 <= z'', z' >= 0, z >= 0, z'' >= 0 DIVISION(z, z') -{ 3925 + 21042*z + 624*z*z' + 1620*z^2 + 135*z' }-> 1 + s68 :|: s68 >= 0, s68 <= 576 * z + 108 * (z * z) + 40 * 0 + 36 * (0 * z) + 63 * z' + 48 * (z' * z) + 928, z' >= 0, z >= 0 IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 IF(z, z', z'', z3) -{ 3926 + 21042*s + 624*s*z'' + 540*s*z3 + 1620*s^2 + 136*z'' + 104*z3 }-> 1 + s93 + s50 :|: s93 >= 0, s93 <= 576 * s + 108 * (s * s) + 40 * z3 + 36 * (z3 * s) + 63 * (1 + (z'' - 1)) + 48 * ((1 + (z'' - 1)) * s) + 928, s50 >= 0, s50 <= 1 + (z'' - 1), s >= 0, s <= z' - 1, z' - 1 >= 0, z'' - 1 >= 0, z = 1, z3 >= 0 IF(z, z', z'', z3) -{ 3926 + 136*z'' + 104*z3 }-> 1 + s94 + s51 :|: s94 >= 0, s94 <= 576 * 0 + 108 * (0 * 0) + 40 * z3 + 36 * (z3 * 0) + 63 * (1 + (z'' - 1)) + 48 * ((1 + (z'' - 1)) * 0) + 928, s51 >= 0, s51 <= 1 + (z'' - 1), z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 + z }-> 1 + s39 :|: s39 >= 0, s39 <= z - 1, z - 1 >= 0 LT(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 LT(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 LT(z, z') -{ 2 + z' }-> 1 + s26 :|: s26 >= 0, s26 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 MINUS(z, z') -{ 1 + z' }-> 1 + s52 :|: s52 >= 0, s52 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> s54 :|: s54 >= 0, s54 <= 1 + 0 + 1 + z, z'' = 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> s55 :|: s55 >= 0, s55 <= 1 + s9 + 1 + z, s9 >= 0, s9 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> s56 :|: s56 >= 0, s56 <= 0 + 1 + z, z >= 0, z'' >= 0, z' = 0 div(z, z', z'') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 1 + 0 + 1 + 0, z'' = 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 1 + s10 + 1 + 0, s10 >= 0, s10 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> s59 :|: s59 >= 0, s59 <= 0 + 1 + 0, z' - 1 >= 0, z = 0, z'' >= 0 div(z, z', z'') -{ 0 }-> s60 :|: s60 >= 0, s60 <= 1 + 0 + 1 + (1 + (z - 1)), s46 >= 0, s46 <= 2, z'' = 0, z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> s61 :|: s61 >= 0, s61 <= 1 + s11 + 1 + (1 + (z - 1)), s47 >= 0, s47 <= 2, s11 >= 0, s11 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> s62 :|: s62 >= 0, s62 <= 0 + 1 + (1 + (z - 1)), s48 >= 0, s48 <= 2, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> s63 :|: s63 >= 0, s63 <= 1 + 0 + 1 + z, z'' = 0, z' >= 0, z >= 0 div(z, z', z'') -{ 0 }-> s64 :|: s64 >= 0, s64 <= 1 + s12 + 1 + z, s12 >= 0, s12 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> s65 :|: s65 >= 0, s65 <= 0 + 1 + z, z' >= 0, z >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 division(z, z') -{ 0 }-> s53 :|: s53 >= 0, s53 <= z + 0 + 1, z' >= 0, z >= 0 division(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'', z3) -{ 0 }-> s66 :|: s66 >= 0, s66 <= s' + z3 + 1, s' >= 0, s' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> s67 :|: s67 >= 0, s67 <= 0 + z3 + 1, z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s13 :|: s13 >= 0, s13 <= z - 1 + 1, z - 1 >= 0 inc(z) -{ 0 }-> 1 + 0 :|: z = 0 lt(z, z') -{ 0 }-> s49 :|: s49 >= 0, s49 <= 2, z' - 1 >= 0, z - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {DIVISION} Previous analysis results are: minus: runtime: O(1) [0], size: O(n^1) [z] inc: runtime: O(1) [0], size: O(n^1) [1 + z] LT: runtime: O(n^1) [2 + z'], size: O(n^1) [z'] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] lt: runtime: O(1) [0], size: O(1) [2] MINUS: runtime: O(n^1) [1 + z'], size: O(n^1) [z'] div: runtime: O(1) [0], size: O(n^1) [1 + z + z''] if: runtime: O(1) [0], size: O(n^1) [1 + z' + z3] IF: runtime: O(n^2) [153 + 1109*z' + 52*z'*z'' + 90*z'*z3 + 90*z'^2 + 7*z'' + 12*z3], size: O(n^2) [35 + 28*z' + 4*z'*z'' + 6*z'*z3 + 6*z'^2 + 3*z'' + 4*z3] DIV: runtime: O(n^2) [3924 + 21042*z + 624*z*z' + 540*z*z'' + 1620*z^2 + 135*z' + 104*z''], size: O(n^2) [928 + 576*z + 48*z*z' + 36*z*z'' + 108*z^2 + 63*z' + 40*z''] division: runtime: O(1) [0], size: O(n^1) [1 + z] ---------------------------------------- (73) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: DIVISION after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 929 + 576*z + 48*z*z' + 108*z^2 + 63*z' ---------------------------------------- (74) Obligation: Complexity RNTS consisting of the following rules: DIV(z, z', z'') -{ 168 + 1199*z + 90*z^2 }-> 1 + s69 + s14 :|: s69 >= 0, s69 <= 3 * 0 + 28 * z + 4 * (1 + 0) + 4 * (0 * z) + 6 * (z * z) + 6 * ((1 + 0) * z) + 35, s14 >= 0, s14 <= 0, z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 168 + 12*s1 + 90*s1*z + 1199*z + 90*z^2 }-> 1 + s70 + s15 :|: s70 >= 0, s70 <= 3 * 0 + 28 * z + 4 * (1 + s1) + 4 * (0 * z) + 6 * (z * z) + 6 * ((1 + s1) * z) + 35, s15 >= 0, s15 <= 0, s1 >= 0, s1 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 156 + 1109*z + 90*z^2 }-> 1 + s71 + s16 :|: s71 >= 0, s71 <= 3 * 0 + 28 * z + 4 * 0 + 4 * (0 * z) + 6 * (z * z) + 6 * (0 * z) + 35, s16 >= 0, s16 <= 0, z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 168 + 8*z' }-> 1 + s72 + s17 :|: s72 >= 0, s72 <= 3 * (1 + (z' - 1)) + 28 * 0 + 4 * (1 + 0) + 4 * ((1 + (z' - 1)) * 0) + 6 * (0 * 0) + 6 * ((1 + 0) * 0) + 35, s17 >= 0, s17 <= 1 + (z' - 1), z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 168 + 12*s2 + 8*z' }-> 1 + s73 + s18 :|: s73 >= 0, s73 <= 3 * (1 + (z' - 1)) + 28 * 0 + 4 * (1 + s2) + 4 * ((1 + (z' - 1)) * 0) + 6 * (0 * 0) + 6 * ((1 + s2) * 0) + 35, s18 >= 0, s18 <= 1 + (z' - 1), s2 >= 0, s2 <= z'' - 1 + 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 156 + 8*z' }-> 1 + s74 + s19 :|: s74 >= 0, s74 <= 3 * (1 + (z' - 1)) + 28 * 0 + 4 * 0 + 4 * ((1 + (z' - 1)) * 0) + 6 * (0 * 0) + 6 * (0 * 0) + 35, s19 >= 0, s19 <= 1 + (z' - 1), z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 168 + 1199*z + 52*z*z' + 90*z^2 + 8*z' }-> 1 + s75 + s20 :|: s75 >= 0, s75 <= 3 * (1 + (z' - 1)) + 28 * (1 + (z - 1)) + 4 * (1 + 0) + 4 * ((1 + (z' - 1)) * (1 + (z - 1))) + 6 * ((1 + (z - 1)) * (1 + (z - 1))) + 6 * ((1 + 0) * (1 + (z - 1))) + 35, s40 >= 0, s40 <= 2, s20 >= 0, s20 <= 1 + (z' - 1), z'' = 0, z' - 1 >= 0, z - 1 >= 0 DIV(z, z', z'') -{ 168 + 12*s3 + 90*s3*z + 1199*z + 52*z*z' + 90*z^2 + 8*z' }-> 1 + s76 + s21 :|: s76 >= 0, s76 <= 3 * (1 + (z' - 1)) + 28 * (1 + (z - 1)) + 4 * (1 + s3) + 4 * ((1 + (z' - 1)) * (1 + (z - 1))) + 6 * ((1 + (z - 1)) * (1 + (z - 1))) + 6 * ((1 + s3) * (1 + (z - 1))) + 35, s41 >= 0, s41 <= 2, s21 >= 0, s21 <= 1 + (z' - 1), s3 >= 0, s3 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 156 + 1109*z + 52*z*z' + 90*z^2 + 8*z' }-> 1 + s77 + s22 :|: s77 >= 0, s77 <= 3 * (1 + (z' - 1)) + 28 * (1 + (z - 1)) + 4 * 0 + 4 * ((1 + (z' - 1)) * (1 + (z - 1))) + 6 * ((1 + (z - 1)) * (1 + (z - 1))) + 6 * (0 * (1 + (z - 1))) + 35, s42 >= 0, s42 <= 2, s22 >= 0, s22 <= 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0, z'' >= 0 DIV(z, z', z'') -{ 168 + 1199*z + 52*z*z' + 90*z^2 + 8*z' }-> 1 + s78 + s23 :|: s78 >= 0, s78 <= 3 * z' + 28 * z + 4 * (1 + 0) + 4 * (z' * z) + 6 * (z * z) + 6 * ((1 + 0) * z) + 35, s23 >= 0, s23 <= z', z'' = 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 168 + 12*s4 + 90*s4*z + 1199*z + 52*z*z' + 90*z^2 + 8*z' }-> 1 + s79 + s24 :|: s79 >= 0, s79 <= 3 * z' + 28 * z + 4 * (1 + s4) + 4 * (z' * z) + 6 * (z * z) + 6 * ((1 + s4) * z) + 35, s24 >= 0, s24 <= z', s4 >= 0, s4 <= z'' - 1 + 1, z'' - 1 >= 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 156 + 1109*z + 52*z*z' + 90*z^2 + 8*z' }-> 1 + s80 + s25 :|: s80 >= 0, s80 <= 3 * z' + 28 * z + 4 * 0 + 4 * (z' * z) + 6 * (z * z) + 6 * (0 * z) + 35, s25 >= 0, s25 <= z', z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 167 + 1199*z + 90*z^2 }-> 1 + s81 + s27 :|: s81 >= 0, s81 <= 3 * 0 + 28 * z + 4 * (1 + 0) + 4 * (0 * z) + 6 * (z * z) + 6 * ((1 + 0) * z) + 35, s27 >= 0, s27 <= 0, z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 167 + 12*s5 + 90*s5*z + 1199*z + 90*z^2 + z'' }-> 1 + s82 + s28 :|: s82 >= 0, s82 <= 3 * 0 + 28 * z + 4 * (1 + s5) + 4 * (0 * z) + 6 * (z * z) + 6 * ((1 + s5) * z) + 35, s28 >= 0, s28 <= 1 + (z'' - 1), s5 >= 0, s5 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 155 + 1109*z + 90*z^2 + z'' }-> 1 + s83 + s29 :|: s83 >= 0, s83 <= 3 * 0 + 28 * z + 4 * 0 + 4 * (0 * z) + 6 * (z * z) + 6 * (0 * z) + 35, s29 >= 0, s29 <= z'', z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 167 + 7*z' }-> 1 + s84 + s30 :|: s84 >= 0, s84 <= 3 * (1 + (z' - 1)) + 28 * 0 + 4 * (1 + 0) + 4 * ((1 + (z' - 1)) * 0) + 6 * (0 * 0) + 6 * ((1 + 0) * 0) + 35, s30 >= 0, s30 <= 0, z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 167 + 12*s6 + 7*z' + z'' }-> 1 + s85 + s31 :|: s85 >= 0, s85 <= 3 * (1 + (z' - 1)) + 28 * 0 + 4 * (1 + s6) + 4 * ((1 + (z' - 1)) * 0) + 6 * (0 * 0) + 6 * ((1 + s6) * 0) + 35, s31 >= 0, s31 <= 1 + (z'' - 1), s6 >= 0, s6 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 155 + 7*z' + z'' }-> 1 + s86 + s32 :|: s86 >= 0, s86 <= 3 * (1 + (z' - 1)) + 28 * 0 + 4 * 0 + 4 * ((1 + (z' - 1)) * 0) + 6 * (0 * 0) + 6 * (0 * 0) + 35, s32 >= 0, s32 <= z'', z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 167 + 1199*z + 52*z*z' + 90*z^2 + 7*z' }-> 1 + s87 + s33 :|: s87 >= 0, s87 <= 3 * (1 + (z' - 1)) + 28 * (1 + (z - 1)) + 4 * (1 + 0) + 4 * ((1 + (z' - 1)) * (1 + (z - 1))) + 6 * ((1 + (z - 1)) * (1 + (z - 1))) + 6 * ((1 + 0) * (1 + (z - 1))) + 35, s43 >= 0, s43 <= 2, s33 >= 0, s33 <= 0, z'' = 0, z - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 167 + 12*s7 + 90*s7*z + 1199*z + 52*z*z' + 90*z^2 + 7*z' + z'' }-> 1 + s88 + s34 :|: s88 >= 0, s88 <= 3 * (1 + (z' - 1)) + 28 * (1 + (z - 1)) + 4 * (1 + s7) + 4 * ((1 + (z' - 1)) * (1 + (z - 1))) + 6 * ((1 + (z - 1)) * (1 + (z - 1))) + 6 * ((1 + s7) * (1 + (z - 1))) + 35, s44 >= 0, s44 <= 2, s34 >= 0, s34 <= 1 + (z'' - 1), s7 >= 0, s7 <= z'' - 1 + 1, z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 155 + 1109*z + 52*z*z' + 90*z^2 + 7*z' + z'' }-> 1 + s89 + s35 :|: s89 >= 0, s89 <= 3 * (1 + (z' - 1)) + 28 * (1 + (z - 1)) + 4 * 0 + 4 * ((1 + (z' - 1)) * (1 + (z - 1))) + 6 * ((1 + (z - 1)) * (1 + (z - 1))) + 6 * (0 * (1 + (z - 1))) + 35, s45 >= 0, s45 <= 2, s35 >= 0, s35 <= z'', z - 1 >= 0, z'' >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 167 + 1199*z + 52*z*z' + 90*z^2 + 7*z' }-> 1 + s90 + s36 :|: s90 >= 0, s90 <= 3 * z' + 28 * z + 4 * (1 + 0) + 4 * (z' * z) + 6 * (z * z) + 6 * ((1 + 0) * z) + 35, s36 >= 0, s36 <= 0, z'' = 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 167 + 12*s8 + 90*s8*z + 1199*z + 52*z*z' + 90*z^2 + 7*z' + z'' }-> 1 + s91 + s37 :|: s91 >= 0, s91 <= 3 * z' + 28 * z + 4 * (1 + s8) + 4 * (z' * z) + 6 * (z * z) + 6 * ((1 + s8) * z) + 35, s37 >= 0, s37 <= 1 + (z'' - 1), s8 >= 0, s8 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 155 + 1109*z + 52*z*z' + 90*z^2 + 7*z' + z'' }-> 1 + s92 + s38 :|: s92 >= 0, s92 <= 3 * z' + 28 * z + 4 * 0 + 4 * (z' * z) + 6 * (z * z) + 6 * (0 * z) + 35, s38 >= 0, s38 <= z'', z' >= 0, z >= 0, z'' >= 0 DIVISION(z, z') -{ 3925 + 21042*z + 624*z*z' + 1620*z^2 + 135*z' }-> 1 + s68 :|: s68 >= 0, s68 <= 576 * z + 108 * (z * z) + 40 * 0 + 36 * (0 * z) + 63 * z' + 48 * (z' * z) + 928, z' >= 0, z >= 0 IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 IF(z, z', z'', z3) -{ 3926 + 21042*s + 624*s*z'' + 540*s*z3 + 1620*s^2 + 136*z'' + 104*z3 }-> 1 + s93 + s50 :|: s93 >= 0, s93 <= 576 * s + 108 * (s * s) + 40 * z3 + 36 * (z3 * s) + 63 * (1 + (z'' - 1)) + 48 * ((1 + (z'' - 1)) * s) + 928, s50 >= 0, s50 <= 1 + (z'' - 1), s >= 0, s <= z' - 1, z' - 1 >= 0, z'' - 1 >= 0, z = 1, z3 >= 0 IF(z, z', z'', z3) -{ 3926 + 136*z'' + 104*z3 }-> 1 + s94 + s51 :|: s94 >= 0, s94 <= 576 * 0 + 108 * (0 * 0) + 40 * z3 + 36 * (z3 * 0) + 63 * (1 + (z'' - 1)) + 48 * ((1 + (z'' - 1)) * 0) + 928, s51 >= 0, s51 <= 1 + (z'' - 1), z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 + z }-> 1 + s39 :|: s39 >= 0, s39 <= z - 1, z - 1 >= 0 LT(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 LT(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 LT(z, z') -{ 2 + z' }-> 1 + s26 :|: s26 >= 0, s26 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 MINUS(z, z') -{ 1 + z' }-> 1 + s52 :|: s52 >= 0, s52 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> s54 :|: s54 >= 0, s54 <= 1 + 0 + 1 + z, z'' = 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> s55 :|: s55 >= 0, s55 <= 1 + s9 + 1 + z, s9 >= 0, s9 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> s56 :|: s56 >= 0, s56 <= 0 + 1 + z, z >= 0, z'' >= 0, z' = 0 div(z, z', z'') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 1 + 0 + 1 + 0, z'' = 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 1 + s10 + 1 + 0, s10 >= 0, s10 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> s59 :|: s59 >= 0, s59 <= 0 + 1 + 0, z' - 1 >= 0, z = 0, z'' >= 0 div(z, z', z'') -{ 0 }-> s60 :|: s60 >= 0, s60 <= 1 + 0 + 1 + (1 + (z - 1)), s46 >= 0, s46 <= 2, z'' = 0, z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> s61 :|: s61 >= 0, s61 <= 1 + s11 + 1 + (1 + (z - 1)), s47 >= 0, s47 <= 2, s11 >= 0, s11 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> s62 :|: s62 >= 0, s62 <= 0 + 1 + (1 + (z - 1)), s48 >= 0, s48 <= 2, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> s63 :|: s63 >= 0, s63 <= 1 + 0 + 1 + z, z'' = 0, z' >= 0, z >= 0 div(z, z', z'') -{ 0 }-> s64 :|: s64 >= 0, s64 <= 1 + s12 + 1 + z, s12 >= 0, s12 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> s65 :|: s65 >= 0, s65 <= 0 + 1 + z, z' >= 0, z >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 division(z, z') -{ 0 }-> s53 :|: s53 >= 0, s53 <= z + 0 + 1, z' >= 0, z >= 0 division(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'', z3) -{ 0 }-> s66 :|: s66 >= 0, s66 <= s' + z3 + 1, s' >= 0, s' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> s67 :|: s67 >= 0, s67 <= 0 + z3 + 1, z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s13 :|: s13 >= 0, s13 <= z - 1 + 1, z - 1 >= 0 inc(z) -{ 0 }-> 1 + 0 :|: z = 0 lt(z, z') -{ 0 }-> s49 :|: s49 >= 0, s49 <= 2, z' - 1 >= 0, z - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {DIVISION} Previous analysis results are: minus: runtime: O(1) [0], size: O(n^1) [z] inc: runtime: O(1) [0], size: O(n^1) [1 + z] LT: runtime: O(n^1) [2 + z'], size: O(n^1) [z'] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] lt: runtime: O(1) [0], size: O(1) [2] MINUS: runtime: O(n^1) [1 + z'], size: O(n^1) [z'] div: runtime: O(1) [0], size: O(n^1) [1 + z + z''] if: runtime: O(1) [0], size: O(n^1) [1 + z' + z3] IF: runtime: O(n^2) [153 + 1109*z' + 52*z'*z'' + 90*z'*z3 + 90*z'^2 + 7*z'' + 12*z3], size: O(n^2) [35 + 28*z' + 4*z'*z'' + 6*z'*z3 + 6*z'^2 + 3*z'' + 4*z3] DIV: runtime: O(n^2) [3924 + 21042*z + 624*z*z' + 540*z*z'' + 1620*z^2 + 135*z' + 104*z''], size: O(n^2) [928 + 576*z + 48*z*z' + 36*z*z'' + 108*z^2 + 63*z' + 40*z''] division: runtime: O(1) [0], size: O(n^1) [1 + z] DIVISION: runtime: ?, size: O(n^2) [929 + 576*z + 48*z*z' + 108*z^2 + 63*z'] ---------------------------------------- (75) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: DIVISION after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 3925 + 21042*z + 624*z*z' + 1620*z^2 + 135*z' ---------------------------------------- (76) Obligation: Complexity RNTS consisting of the following rules: DIV(z, z', z'') -{ 168 + 1199*z + 90*z^2 }-> 1 + s69 + s14 :|: s69 >= 0, s69 <= 3 * 0 + 28 * z + 4 * (1 + 0) + 4 * (0 * z) + 6 * (z * z) + 6 * ((1 + 0) * z) + 35, s14 >= 0, s14 <= 0, z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 168 + 12*s1 + 90*s1*z + 1199*z + 90*z^2 }-> 1 + s70 + s15 :|: s70 >= 0, s70 <= 3 * 0 + 28 * z + 4 * (1 + s1) + 4 * (0 * z) + 6 * (z * z) + 6 * ((1 + s1) * z) + 35, s15 >= 0, s15 <= 0, s1 >= 0, s1 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 156 + 1109*z + 90*z^2 }-> 1 + s71 + s16 :|: s71 >= 0, s71 <= 3 * 0 + 28 * z + 4 * 0 + 4 * (0 * z) + 6 * (z * z) + 6 * (0 * z) + 35, s16 >= 0, s16 <= 0, z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 168 + 8*z' }-> 1 + s72 + s17 :|: s72 >= 0, s72 <= 3 * (1 + (z' - 1)) + 28 * 0 + 4 * (1 + 0) + 4 * ((1 + (z' - 1)) * 0) + 6 * (0 * 0) + 6 * ((1 + 0) * 0) + 35, s17 >= 0, s17 <= 1 + (z' - 1), z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 168 + 12*s2 + 8*z' }-> 1 + s73 + s18 :|: s73 >= 0, s73 <= 3 * (1 + (z' - 1)) + 28 * 0 + 4 * (1 + s2) + 4 * ((1 + (z' - 1)) * 0) + 6 * (0 * 0) + 6 * ((1 + s2) * 0) + 35, s18 >= 0, s18 <= 1 + (z' - 1), s2 >= 0, s2 <= z'' - 1 + 1, z' - 1 >= 0, z'' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 156 + 8*z' }-> 1 + s74 + s19 :|: s74 >= 0, s74 <= 3 * (1 + (z' - 1)) + 28 * 0 + 4 * 0 + 4 * ((1 + (z' - 1)) * 0) + 6 * (0 * 0) + 6 * (0 * 0) + 35, s19 >= 0, s19 <= 1 + (z' - 1), z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 168 + 1199*z + 52*z*z' + 90*z^2 + 8*z' }-> 1 + s75 + s20 :|: s75 >= 0, s75 <= 3 * (1 + (z' - 1)) + 28 * (1 + (z - 1)) + 4 * (1 + 0) + 4 * ((1 + (z' - 1)) * (1 + (z - 1))) + 6 * ((1 + (z - 1)) * (1 + (z - 1))) + 6 * ((1 + 0) * (1 + (z - 1))) + 35, s40 >= 0, s40 <= 2, s20 >= 0, s20 <= 1 + (z' - 1), z'' = 0, z' - 1 >= 0, z - 1 >= 0 DIV(z, z', z'') -{ 168 + 12*s3 + 90*s3*z + 1199*z + 52*z*z' + 90*z^2 + 8*z' }-> 1 + s76 + s21 :|: s76 >= 0, s76 <= 3 * (1 + (z' - 1)) + 28 * (1 + (z - 1)) + 4 * (1 + s3) + 4 * ((1 + (z' - 1)) * (1 + (z - 1))) + 6 * ((1 + (z - 1)) * (1 + (z - 1))) + 6 * ((1 + s3) * (1 + (z - 1))) + 35, s41 >= 0, s41 <= 2, s21 >= 0, s21 <= 1 + (z' - 1), s3 >= 0, s3 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 156 + 1109*z + 52*z*z' + 90*z^2 + 8*z' }-> 1 + s77 + s22 :|: s77 >= 0, s77 <= 3 * (1 + (z' - 1)) + 28 * (1 + (z - 1)) + 4 * 0 + 4 * ((1 + (z' - 1)) * (1 + (z - 1))) + 6 * ((1 + (z - 1)) * (1 + (z - 1))) + 6 * (0 * (1 + (z - 1))) + 35, s42 >= 0, s42 <= 2, s22 >= 0, s22 <= 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0, z'' >= 0 DIV(z, z', z'') -{ 168 + 1199*z + 52*z*z' + 90*z^2 + 8*z' }-> 1 + s78 + s23 :|: s78 >= 0, s78 <= 3 * z' + 28 * z + 4 * (1 + 0) + 4 * (z' * z) + 6 * (z * z) + 6 * ((1 + 0) * z) + 35, s23 >= 0, s23 <= z', z'' = 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 168 + 12*s4 + 90*s4*z + 1199*z + 52*z*z' + 90*z^2 + 8*z' }-> 1 + s79 + s24 :|: s79 >= 0, s79 <= 3 * z' + 28 * z + 4 * (1 + s4) + 4 * (z' * z) + 6 * (z * z) + 6 * ((1 + s4) * z) + 35, s24 >= 0, s24 <= z', s4 >= 0, s4 <= z'' - 1 + 1, z'' - 1 >= 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 156 + 1109*z + 52*z*z' + 90*z^2 + 8*z' }-> 1 + s80 + s25 :|: s80 >= 0, s80 <= 3 * z' + 28 * z + 4 * 0 + 4 * (z' * z) + 6 * (z * z) + 6 * (0 * z) + 35, s25 >= 0, s25 <= z', z' >= 0, z >= 0, z'' >= 0 DIV(z, z', z'') -{ 167 + 1199*z + 90*z^2 }-> 1 + s81 + s27 :|: s81 >= 0, s81 <= 3 * 0 + 28 * z + 4 * (1 + 0) + 4 * (0 * z) + 6 * (z * z) + 6 * ((1 + 0) * z) + 35, s27 >= 0, s27 <= 0, z'' = 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 167 + 12*s5 + 90*s5*z + 1199*z + 90*z^2 + z'' }-> 1 + s82 + s28 :|: s82 >= 0, s82 <= 3 * 0 + 28 * z + 4 * (1 + s5) + 4 * (0 * z) + 6 * (z * z) + 6 * ((1 + s5) * z) + 35, s28 >= 0, s28 <= 1 + (z'' - 1), s5 >= 0, s5 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 DIV(z, z', z'') -{ 155 + 1109*z + 90*z^2 + z'' }-> 1 + s83 + s29 :|: s83 >= 0, s83 <= 3 * 0 + 28 * z + 4 * 0 + 4 * (0 * z) + 6 * (z * z) + 6 * (0 * z) + 35, s29 >= 0, s29 <= z'', z >= 0, z'' >= 0, z' = 0 DIV(z, z', z'') -{ 167 + 7*z' }-> 1 + s84 + s30 :|: s84 >= 0, s84 <= 3 * (1 + (z' - 1)) + 28 * 0 + 4 * (1 + 0) + 4 * ((1 + (z' - 1)) * 0) + 6 * (0 * 0) + 6 * ((1 + 0) * 0) + 35, s30 >= 0, s30 <= 0, z'' = 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 167 + 12*s6 + 7*z' + z'' }-> 1 + s85 + s31 :|: s85 >= 0, s85 <= 3 * (1 + (z' - 1)) + 28 * 0 + 4 * (1 + s6) + 4 * ((1 + (z' - 1)) * 0) + 6 * (0 * 0) + 6 * ((1 + s6) * 0) + 35, s31 >= 0, s31 <= 1 + (z'' - 1), s6 >= 0, s6 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0 DIV(z, z', z'') -{ 155 + 7*z' + z'' }-> 1 + s86 + s32 :|: s86 >= 0, s86 <= 3 * (1 + (z' - 1)) + 28 * 0 + 4 * 0 + 4 * ((1 + (z' - 1)) * 0) + 6 * (0 * 0) + 6 * (0 * 0) + 35, s32 >= 0, s32 <= z'', z' - 1 >= 0, z = 0, z'' >= 0 DIV(z, z', z'') -{ 167 + 1199*z + 52*z*z' + 90*z^2 + 7*z' }-> 1 + s87 + s33 :|: s87 >= 0, s87 <= 3 * (1 + (z' - 1)) + 28 * (1 + (z - 1)) + 4 * (1 + 0) + 4 * ((1 + (z' - 1)) * (1 + (z - 1))) + 6 * ((1 + (z - 1)) * (1 + (z - 1))) + 6 * ((1 + 0) * (1 + (z - 1))) + 35, s43 >= 0, s43 <= 2, s33 >= 0, s33 <= 0, z'' = 0, z - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 167 + 12*s7 + 90*s7*z + 1199*z + 52*z*z' + 90*z^2 + 7*z' + z'' }-> 1 + s88 + s34 :|: s88 >= 0, s88 <= 3 * (1 + (z' - 1)) + 28 * (1 + (z - 1)) + 4 * (1 + s7) + 4 * ((1 + (z' - 1)) * (1 + (z - 1))) + 6 * ((1 + (z - 1)) * (1 + (z - 1))) + 6 * ((1 + s7) * (1 + (z - 1))) + 35, s44 >= 0, s44 <= 2, s34 >= 0, s34 <= 1 + (z'' - 1), s7 >= 0, s7 <= z'' - 1 + 1, z - 1 >= 0, z'' - 1 >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 155 + 1109*z + 52*z*z' + 90*z^2 + 7*z' + z'' }-> 1 + s89 + s35 :|: s89 >= 0, s89 <= 3 * (1 + (z' - 1)) + 28 * (1 + (z - 1)) + 4 * 0 + 4 * ((1 + (z' - 1)) * (1 + (z - 1))) + 6 * ((1 + (z - 1)) * (1 + (z - 1))) + 6 * (0 * (1 + (z - 1))) + 35, s45 >= 0, s45 <= 2, s35 >= 0, s35 <= z'', z - 1 >= 0, z'' >= 0, z' - 1 >= 0 DIV(z, z', z'') -{ 167 + 1199*z + 52*z*z' + 90*z^2 + 7*z' }-> 1 + s90 + s36 :|: s90 >= 0, s90 <= 3 * z' + 28 * z + 4 * (1 + 0) + 4 * (z' * z) + 6 * (z * z) + 6 * ((1 + 0) * z) + 35, s36 >= 0, s36 <= 0, z'' = 0, z' >= 0, z >= 0 DIV(z, z', z'') -{ 167 + 12*s8 + 90*s8*z + 1199*z + 52*z*z' + 90*z^2 + 7*z' + z'' }-> 1 + s91 + s37 :|: s91 >= 0, s91 <= 3 * z' + 28 * z + 4 * (1 + s8) + 4 * (z' * z) + 6 * (z * z) + 6 * ((1 + s8) * z) + 35, s37 >= 0, s37 <= 1 + (z'' - 1), s8 >= 0, s8 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0 DIV(z, z', z'') -{ 155 + 1109*z + 52*z*z' + 90*z^2 + 7*z' + z'' }-> 1 + s92 + s38 :|: s92 >= 0, s92 <= 3 * z' + 28 * z + 4 * 0 + 4 * (z' * z) + 6 * (z * z) + 6 * (0 * z) + 35, s38 >= 0, s38 <= z'', z' >= 0, z >= 0, z'' >= 0 DIVISION(z, z') -{ 3925 + 21042*z + 624*z*z' + 1620*z^2 + 135*z' }-> 1 + s68 :|: s68 >= 0, s68 <= 576 * z + 108 * (z * z) + 40 * 0 + 36 * (0 * z) + 63 * z' + 48 * (z' * z) + 928, z' >= 0, z >= 0 IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 IF(z, z', z'', z3) -{ 3926 + 21042*s + 624*s*z'' + 540*s*z3 + 1620*s^2 + 136*z'' + 104*z3 }-> 1 + s93 + s50 :|: s93 >= 0, s93 <= 576 * s + 108 * (s * s) + 40 * z3 + 36 * (z3 * s) + 63 * (1 + (z'' - 1)) + 48 * ((1 + (z'' - 1)) * s) + 928, s50 >= 0, s50 <= 1 + (z'' - 1), s >= 0, s <= z' - 1, z' - 1 >= 0, z'' - 1 >= 0, z = 1, z3 >= 0 IF(z, z', z'', z3) -{ 3926 + 136*z'' + 104*z3 }-> 1 + s94 + s51 :|: s94 >= 0, s94 <= 576 * 0 + 108 * (0 * 0) + 40 * z3 + 36 * (z3 * 0) + 63 * (1 + (z'' - 1)) + 48 * ((1 + (z'' - 1)) * 0) + 928, s51 >= 0, s51 <= 1 + (z'' - 1), z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 INC(z) -{ 1 }-> 0 :|: z = 0 INC(z) -{ 1 + z }-> 1 + s39 :|: s39 >= 0, s39 <= z - 1, z - 1 >= 0 LT(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 LT(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 LT(z, z') -{ 2 + z' }-> 1 + s26 :|: s26 >= 0, s26 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 MINUS(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 MINUS(z, z') -{ 1 + z' }-> 1 + s52 :|: s52 >= 0, s52 <= z' - 1, z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> s54 :|: s54 >= 0, s54 <= 1 + 0 + 1 + z, z'' = 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> s55 :|: s55 >= 0, s55 <= 1 + s9 + 1 + z, s9 >= 0, s9 <= z'' - 1 + 1, z'' - 1 >= 0, z >= 0, z' = 0 div(z, z', z'') -{ 0 }-> s56 :|: s56 >= 0, s56 <= 0 + 1 + z, z >= 0, z'' >= 0, z' = 0 div(z, z', z'') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 1 + 0 + 1 + 0, z'' = 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 1 + s10 + 1 + 0, s10 >= 0, s10 <= z'' - 1 + 1, z'' - 1 >= 0, z' - 1 >= 0, z = 0 div(z, z', z'') -{ 0 }-> s59 :|: s59 >= 0, s59 <= 0 + 1 + 0, z' - 1 >= 0, z = 0, z'' >= 0 div(z, z', z'') -{ 0 }-> s60 :|: s60 >= 0, s60 <= 1 + 0 + 1 + (1 + (z - 1)), s46 >= 0, s46 <= 2, z'' = 0, z' - 1 >= 0, z - 1 >= 0 div(z, z', z'') -{ 0 }-> s61 :|: s61 >= 0, s61 <= 1 + s11 + 1 + (1 + (z - 1)), s47 >= 0, s47 <= 2, s11 >= 0, s11 <= z'' - 1 + 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> s62 :|: s62 >= 0, s62 <= 0 + 1 + (1 + (z - 1)), s48 >= 0, s48 <= 2, z' - 1 >= 0, z - 1 >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> s63 :|: s63 >= 0, s63 <= 1 + 0 + 1 + z, z'' = 0, z' >= 0, z >= 0 div(z, z', z'') -{ 0 }-> s64 :|: s64 >= 0, s64 <= 1 + s12 + 1 + z, s12 >= 0, s12 <= z'' - 1 + 1, z' >= 0, z >= 0, z'' - 1 >= 0 div(z, z', z'') -{ 0 }-> s65 :|: s65 >= 0, s65 <= 0 + 1 + z, z' >= 0, z >= 0, z'' >= 0 div(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 division(z, z') -{ 0 }-> s53 :|: s53 >= 0, s53 <= z + 0 + 1, z' >= 0, z >= 0 division(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 if(z, z', z'', z3) -{ 0 }-> s66 :|: s66 >= 0, s66 <= s' + z3 + 1, s' >= 0, s' <= z' - 1, z'' - 1 >= 0, z = 1, z' - 1 >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> s67 :|: s67 >= 0, s67 <= 0 + z3 + 1, z'' - 1 >= 0, z = 1, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> z3 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0 if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0 inc(z) -{ 0 }-> 0 :|: z >= 0 inc(z) -{ 0 }-> 1 + s13 :|: s13 >= 0, s13 <= z - 1 + 1, z - 1 >= 0 inc(z) -{ 0 }-> 1 + 0 :|: z = 0 lt(z, z') -{ 0 }-> s49 :|: s49 >= 0, s49 <= 2, z' - 1 >= 0, z - 1 >= 0 lt(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 lt(z, z') -{ 0 }-> 1 :|: z >= 0, z' = 0 lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 minus(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= z - 1, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: Previous analysis results are: minus: runtime: O(1) [0], size: O(n^1) [z] inc: runtime: O(1) [0], size: O(n^1) [1 + z] LT: runtime: O(n^1) [2 + z'], size: O(n^1) [z'] INC: runtime: O(n^1) [1 + z], size: O(n^1) [z] lt: runtime: O(1) [0], size: O(1) [2] MINUS: runtime: O(n^1) [1 + z'], size: O(n^1) [z'] div: runtime: O(1) [0], size: O(n^1) [1 + z + z''] if: runtime: O(1) [0], size: O(n^1) [1 + z' + z3] IF: runtime: O(n^2) [153 + 1109*z' + 52*z'*z'' + 90*z'*z3 + 90*z'^2 + 7*z'' + 12*z3], size: O(n^2) [35 + 28*z' + 4*z'*z'' + 6*z'*z3 + 6*z'^2 + 3*z'' + 4*z3] DIV: runtime: O(n^2) [3924 + 21042*z + 624*z*z' + 540*z*z'' + 1620*z^2 + 135*z' + 104*z''], size: O(n^2) [928 + 576*z + 48*z*z' + 36*z*z'' + 108*z^2 + 63*z' + 40*z''] division: runtime: O(1) [0], size: O(n^1) [1 + z] DIVISION: runtime: O(n^2) [3925 + 21042*z + 624*z*z' + 1620*z^2 + 135*z'], size: O(n^2) [929 + 576*z + 48*z*z' + 108*z^2 + 63*z'] ---------------------------------------- (77) FinalProof (FINISHED) Computed overall runtime complexity ---------------------------------------- (78) BOUNDS(1, n^2) ---------------------------------------- (79) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (80) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: DIVISION(z0, z1) -> c(DIV(z0, z1, 0')) DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1)) DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2)) IF(true, z0, z1, z2) -> c3 IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1))) MINUS(z0, 0') -> c5 MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1)) LT(z0, 0') -> c7 LT(0', s(z0)) -> c8 LT(s(z0), s(z1)) -> c9(LT(z0, z1)) INC(0') -> c10 INC(s(z0)) -> c11(INC(z0)) The (relative) TRS S consists of the following rules: division(z0, z1) -> div(z0, z1, 0') div(z0, z1, z2) -> if(lt(z0, z1), z0, z1, inc(z2)) if(true, z0, z1, z2) -> z2 if(false, z0, s(z1), z2) -> div(minus(z0, s(z1)), s(z1), z2) minus(z0, 0') -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) lt(z0, 0') -> false lt(0', s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) inc(0') -> s(0') inc(s(z0)) -> s(inc(z0)) Rewrite Strategy: INNERMOST ---------------------------------------- (81) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (82) Obligation: Innermost TRS: Rules: DIVISION(z0, z1) -> c(DIV(z0, z1, 0')) DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1)) DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2)) IF(true, z0, z1, z2) -> c3 IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1))) MINUS(z0, 0') -> c5 MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1)) LT(z0, 0') -> c7 LT(0', s(z0)) -> c8 LT(s(z0), s(z1)) -> c9(LT(z0, z1)) INC(0') -> c10 INC(s(z0)) -> c11(INC(z0)) division(z0, z1) -> div(z0, z1, 0') div(z0, z1, z2) -> if(lt(z0, z1), z0, z1, inc(z2)) if(true, z0, z1, z2) -> z2 if(false, z0, s(z1), z2) -> div(minus(z0, s(z1)), s(z1), z2) minus(z0, 0') -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) lt(z0, 0') -> false lt(0', s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) inc(0') -> s(0') inc(s(z0)) -> s(inc(z0)) Types: DIVISION :: 0':s -> 0':s -> c c :: c1:c2 -> c DIV :: 0':s -> 0':s -> 0':s -> c1:c2 0' :: 0':s c1 :: c3:c4 -> c7:c8:c9 -> c1:c2 IF :: true:false -> 0':s -> 0':s -> 0':s -> c3:c4 lt :: 0':s -> 0':s -> true:false inc :: 0':s -> 0':s LT :: 0':s -> 0':s -> c7:c8:c9 c2 :: c3:c4 -> c10:c11 -> c1:c2 INC :: 0':s -> c10:c11 true :: true:false c3 :: c3:c4 false :: true:false s :: 0':s -> 0':s c4 :: c1:c2 -> c5:c6 -> c3:c4 minus :: 0':s -> 0':s -> 0':s MINUS :: 0':s -> 0':s -> c5:c6 c5 :: c5:c6 c6 :: c5:c6 -> c5:c6 c7 :: c7:c8:c9 c8 :: c7:c8:c9 c9 :: c7:c8:c9 -> c7:c8:c9 c10 :: c10:c11 c11 :: c10:c11 -> c10:c11 division :: 0':s -> 0':s -> 0':s div :: 0':s -> 0':s -> 0':s -> 0':s if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s hole_c1_12 :: c hole_0':s2_12 :: 0':s hole_c1:c23_12 :: c1:c2 hole_c3:c44_12 :: c3:c4 hole_c7:c8:c95_12 :: c7:c8:c9 hole_true:false6_12 :: true:false hole_c10:c117_12 :: c10:c11 hole_c5:c68_12 :: c5:c6 gen_0':s9_12 :: Nat -> 0':s gen_c7:c8:c910_12 :: Nat -> c7:c8:c9 gen_c10:c1111_12 :: Nat -> c10:c11 gen_c5:c612_12 :: Nat -> c5:c6 ---------------------------------------- (83) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: DIV, lt, inc, LT, INC, minus, MINUS, div They will be analysed ascendingly in the following order: lt < DIV inc < DIV LT < DIV INC < DIV minus < DIV MINUS < DIV lt < div inc < div minus < div ---------------------------------------- (84) Obligation: Innermost TRS: Rules: DIVISION(z0, z1) -> c(DIV(z0, z1, 0')) DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1)) DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2)) IF(true, z0, z1, z2) -> c3 IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1))) MINUS(z0, 0') -> c5 MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1)) LT(z0, 0') -> c7 LT(0', s(z0)) -> c8 LT(s(z0), s(z1)) -> c9(LT(z0, z1)) INC(0') -> c10 INC(s(z0)) -> c11(INC(z0)) division(z0, z1) -> div(z0, z1, 0') div(z0, z1, z2) -> if(lt(z0, z1), z0, z1, inc(z2)) if(true, z0, z1, z2) -> z2 if(false, z0, s(z1), z2) -> div(minus(z0, s(z1)), s(z1), z2) minus(z0, 0') -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) lt(z0, 0') -> false lt(0', s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) inc(0') -> s(0') inc(s(z0)) -> s(inc(z0)) Types: DIVISION :: 0':s -> 0':s -> c c :: c1:c2 -> c DIV :: 0':s -> 0':s -> 0':s -> c1:c2 0' :: 0':s c1 :: c3:c4 -> c7:c8:c9 -> c1:c2 IF :: true:false -> 0':s -> 0':s -> 0':s -> c3:c4 lt :: 0':s -> 0':s -> true:false inc :: 0':s -> 0':s LT :: 0':s -> 0':s -> c7:c8:c9 c2 :: c3:c4 -> c10:c11 -> c1:c2 INC :: 0':s -> c10:c11 true :: true:false c3 :: c3:c4 false :: true:false s :: 0':s -> 0':s c4 :: c1:c2 -> c5:c6 -> c3:c4 minus :: 0':s -> 0':s -> 0':s MINUS :: 0':s -> 0':s -> c5:c6 c5 :: c5:c6 c6 :: c5:c6 -> c5:c6 c7 :: c7:c8:c9 c8 :: c7:c8:c9 c9 :: c7:c8:c9 -> c7:c8:c9 c10 :: c10:c11 c11 :: c10:c11 -> c10:c11 division :: 0':s -> 0':s -> 0':s div :: 0':s -> 0':s -> 0':s -> 0':s if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s hole_c1_12 :: c hole_0':s2_12 :: 0':s hole_c1:c23_12 :: c1:c2 hole_c3:c44_12 :: c3:c4 hole_c7:c8:c95_12 :: c7:c8:c9 hole_true:false6_12 :: true:false hole_c10:c117_12 :: c10:c11 hole_c5:c68_12 :: c5:c6 gen_0':s9_12 :: Nat -> 0':s gen_c7:c8:c910_12 :: Nat -> c7:c8:c9 gen_c10:c1111_12 :: Nat -> c10:c11 gen_c5:c612_12 :: Nat -> c5:c6 Generator Equations: gen_0':s9_12(0) <=> 0' gen_0':s9_12(+(x, 1)) <=> s(gen_0':s9_12(x)) gen_c7:c8:c910_12(0) <=> c7 gen_c7:c8:c910_12(+(x, 1)) <=> c9(gen_c7:c8:c910_12(x)) gen_c10:c1111_12(0) <=> c10 gen_c10:c1111_12(+(x, 1)) <=> c11(gen_c10:c1111_12(x)) gen_c5:c612_12(0) <=> c5 gen_c5:c612_12(+(x, 1)) <=> c6(gen_c5:c612_12(x)) The following defined symbols remain to be analysed: lt, DIV, inc, LT, INC, minus, MINUS, div They will be analysed ascendingly in the following order: lt < DIV inc < DIV LT < DIV INC < DIV minus < DIV MINUS < DIV lt < div inc < div minus < div ---------------------------------------- (85) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: lt(gen_0':s9_12(n14_12), gen_0':s9_12(n14_12)) -> false, rt in Omega(0) Induction Base: lt(gen_0':s9_12(0), gen_0':s9_12(0)) ->_R^Omega(0) false Induction Step: lt(gen_0':s9_12(+(n14_12, 1)), gen_0':s9_12(+(n14_12, 1))) ->_R^Omega(0) lt(gen_0':s9_12(n14_12), gen_0':s9_12(n14_12)) ->_IH false We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (86) Obligation: Innermost TRS: Rules: DIVISION(z0, z1) -> c(DIV(z0, z1, 0')) DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1)) DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2)) IF(true, z0, z1, z2) -> c3 IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1))) MINUS(z0, 0') -> c5 MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1)) LT(z0, 0') -> c7 LT(0', s(z0)) -> c8 LT(s(z0), s(z1)) -> c9(LT(z0, z1)) INC(0') -> c10 INC(s(z0)) -> c11(INC(z0)) division(z0, z1) -> div(z0, z1, 0') div(z0, z1, z2) -> if(lt(z0, z1), z0, z1, inc(z2)) if(true, z0, z1, z2) -> z2 if(false, z0, s(z1), z2) -> div(minus(z0, s(z1)), s(z1), z2) minus(z0, 0') -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) lt(z0, 0') -> false lt(0', s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) inc(0') -> s(0') inc(s(z0)) -> s(inc(z0)) Types: DIVISION :: 0':s -> 0':s -> c c :: c1:c2 -> c DIV :: 0':s -> 0':s -> 0':s -> c1:c2 0' :: 0':s c1 :: c3:c4 -> c7:c8:c9 -> c1:c2 IF :: true:false -> 0':s -> 0':s -> 0':s -> c3:c4 lt :: 0':s -> 0':s -> true:false inc :: 0':s -> 0':s LT :: 0':s -> 0':s -> c7:c8:c9 c2 :: c3:c4 -> c10:c11 -> c1:c2 INC :: 0':s -> c10:c11 true :: true:false c3 :: c3:c4 false :: true:false s :: 0':s -> 0':s c4 :: c1:c2 -> c5:c6 -> c3:c4 minus :: 0':s -> 0':s -> 0':s MINUS :: 0':s -> 0':s -> c5:c6 c5 :: c5:c6 c6 :: c5:c6 -> c5:c6 c7 :: c7:c8:c9 c8 :: c7:c8:c9 c9 :: c7:c8:c9 -> c7:c8:c9 c10 :: c10:c11 c11 :: c10:c11 -> c10:c11 division :: 0':s -> 0':s -> 0':s div :: 0':s -> 0':s -> 0':s -> 0':s if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s hole_c1_12 :: c hole_0':s2_12 :: 0':s hole_c1:c23_12 :: c1:c2 hole_c3:c44_12 :: c3:c4 hole_c7:c8:c95_12 :: c7:c8:c9 hole_true:false6_12 :: true:false hole_c10:c117_12 :: c10:c11 hole_c5:c68_12 :: c5:c6 gen_0':s9_12 :: Nat -> 0':s gen_c7:c8:c910_12 :: Nat -> c7:c8:c9 gen_c10:c1111_12 :: Nat -> c10:c11 gen_c5:c612_12 :: Nat -> c5:c6 Lemmas: lt(gen_0':s9_12(n14_12), gen_0':s9_12(n14_12)) -> false, rt in Omega(0) Generator Equations: gen_0':s9_12(0) <=> 0' gen_0':s9_12(+(x, 1)) <=> s(gen_0':s9_12(x)) gen_c7:c8:c910_12(0) <=> c7 gen_c7:c8:c910_12(+(x, 1)) <=> c9(gen_c7:c8:c910_12(x)) gen_c10:c1111_12(0) <=> c10 gen_c10:c1111_12(+(x, 1)) <=> c11(gen_c10:c1111_12(x)) gen_c5:c612_12(0) <=> c5 gen_c5:c612_12(+(x, 1)) <=> c6(gen_c5:c612_12(x)) The following defined symbols remain to be analysed: inc, DIV, LT, INC, minus, MINUS, div They will be analysed ascendingly in the following order: inc < DIV LT < DIV INC < DIV minus < DIV MINUS < DIV inc < div minus < div ---------------------------------------- (87) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: inc(gen_0':s9_12(n398_12)) -> gen_0':s9_12(+(1, n398_12)), rt in Omega(0) Induction Base: inc(gen_0':s9_12(0)) ->_R^Omega(0) s(0') Induction Step: inc(gen_0':s9_12(+(n398_12, 1))) ->_R^Omega(0) s(inc(gen_0':s9_12(n398_12))) ->_IH s(gen_0':s9_12(+(1, c399_12))) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (88) Obligation: Innermost TRS: Rules: DIVISION(z0, z1) -> c(DIV(z0, z1, 0')) DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1)) DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2)) IF(true, z0, z1, z2) -> c3 IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1))) MINUS(z0, 0') -> c5 MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1)) LT(z0, 0') -> c7 LT(0', s(z0)) -> c8 LT(s(z0), s(z1)) -> c9(LT(z0, z1)) INC(0') -> c10 INC(s(z0)) -> c11(INC(z0)) division(z0, z1) -> div(z0, z1, 0') div(z0, z1, z2) -> if(lt(z0, z1), z0, z1, inc(z2)) if(true, z0, z1, z2) -> z2 if(false, z0, s(z1), z2) -> div(minus(z0, s(z1)), s(z1), z2) minus(z0, 0') -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) lt(z0, 0') -> false lt(0', s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) inc(0') -> s(0') inc(s(z0)) -> s(inc(z0)) Types: DIVISION :: 0':s -> 0':s -> c c :: c1:c2 -> c DIV :: 0':s -> 0':s -> 0':s -> c1:c2 0' :: 0':s c1 :: c3:c4 -> c7:c8:c9 -> c1:c2 IF :: true:false -> 0':s -> 0':s -> 0':s -> c3:c4 lt :: 0':s -> 0':s -> true:false inc :: 0':s -> 0':s LT :: 0':s -> 0':s -> c7:c8:c9 c2 :: c3:c4 -> c10:c11 -> c1:c2 INC :: 0':s -> c10:c11 true :: true:false c3 :: c3:c4 false :: true:false s :: 0':s -> 0':s c4 :: c1:c2 -> c5:c6 -> c3:c4 minus :: 0':s -> 0':s -> 0':s MINUS :: 0':s -> 0':s -> c5:c6 c5 :: c5:c6 c6 :: c5:c6 -> c5:c6 c7 :: c7:c8:c9 c8 :: c7:c8:c9 c9 :: c7:c8:c9 -> c7:c8:c9 c10 :: c10:c11 c11 :: c10:c11 -> c10:c11 division :: 0':s -> 0':s -> 0':s div :: 0':s -> 0':s -> 0':s -> 0':s if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s hole_c1_12 :: c hole_0':s2_12 :: 0':s hole_c1:c23_12 :: c1:c2 hole_c3:c44_12 :: c3:c4 hole_c7:c8:c95_12 :: c7:c8:c9 hole_true:false6_12 :: true:false hole_c10:c117_12 :: c10:c11 hole_c5:c68_12 :: c5:c6 gen_0':s9_12 :: Nat -> 0':s gen_c7:c8:c910_12 :: Nat -> c7:c8:c9 gen_c10:c1111_12 :: Nat -> c10:c11 gen_c5:c612_12 :: Nat -> c5:c6 Lemmas: lt(gen_0':s9_12(n14_12), gen_0':s9_12(n14_12)) -> false, rt in Omega(0) inc(gen_0':s9_12(n398_12)) -> gen_0':s9_12(+(1, n398_12)), rt in Omega(0) Generator Equations: gen_0':s9_12(0) <=> 0' gen_0':s9_12(+(x, 1)) <=> s(gen_0':s9_12(x)) gen_c7:c8:c910_12(0) <=> c7 gen_c7:c8:c910_12(+(x, 1)) <=> c9(gen_c7:c8:c910_12(x)) gen_c10:c1111_12(0) <=> c10 gen_c10:c1111_12(+(x, 1)) <=> c11(gen_c10:c1111_12(x)) gen_c5:c612_12(0) <=> c5 gen_c5:c612_12(+(x, 1)) <=> c6(gen_c5:c612_12(x)) The following defined symbols remain to be analysed: LT, DIV, INC, minus, MINUS, div They will be analysed ascendingly in the following order: LT < DIV INC < DIV minus < DIV MINUS < DIV minus < div ---------------------------------------- (89) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: LT(gen_0':s9_12(n694_12), gen_0':s9_12(n694_12)) -> gen_c7:c8:c910_12(n694_12), rt in Omega(1 + n694_12) Induction Base: LT(gen_0':s9_12(0), gen_0':s9_12(0)) ->_R^Omega(1) c7 Induction Step: LT(gen_0':s9_12(+(n694_12, 1)), gen_0':s9_12(+(n694_12, 1))) ->_R^Omega(1) c9(LT(gen_0':s9_12(n694_12), gen_0':s9_12(n694_12))) ->_IH c9(gen_c7:c8:c910_12(c695_12)) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (90) Complex Obligation (BEST) ---------------------------------------- (91) Obligation: Proved the lower bound n^1 for the following obligation: Innermost TRS: Rules: DIVISION(z0, z1) -> c(DIV(z0, z1, 0')) DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1)) DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2)) IF(true, z0, z1, z2) -> c3 IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1))) MINUS(z0, 0') -> c5 MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1)) LT(z0, 0') -> c7 LT(0', s(z0)) -> c8 LT(s(z0), s(z1)) -> c9(LT(z0, z1)) INC(0') -> c10 INC(s(z0)) -> c11(INC(z0)) division(z0, z1) -> div(z0, z1, 0') div(z0, z1, z2) -> if(lt(z0, z1), z0, z1, inc(z2)) if(true, z0, z1, z2) -> z2 if(false, z0, s(z1), z2) -> div(minus(z0, s(z1)), s(z1), z2) minus(z0, 0') -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) lt(z0, 0') -> false lt(0', s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) inc(0') -> s(0') inc(s(z0)) -> s(inc(z0)) Types: DIVISION :: 0':s -> 0':s -> c c :: c1:c2 -> c DIV :: 0':s -> 0':s -> 0':s -> c1:c2 0' :: 0':s c1 :: c3:c4 -> c7:c8:c9 -> c1:c2 IF :: true:false -> 0':s -> 0':s -> 0':s -> c3:c4 lt :: 0':s -> 0':s -> true:false inc :: 0':s -> 0':s LT :: 0':s -> 0':s -> c7:c8:c9 c2 :: c3:c4 -> c10:c11 -> c1:c2 INC :: 0':s -> c10:c11 true :: true:false c3 :: c3:c4 false :: true:false s :: 0':s -> 0':s c4 :: c1:c2 -> c5:c6 -> c3:c4 minus :: 0':s -> 0':s -> 0':s MINUS :: 0':s -> 0':s -> c5:c6 c5 :: c5:c6 c6 :: c5:c6 -> c5:c6 c7 :: c7:c8:c9 c8 :: c7:c8:c9 c9 :: c7:c8:c9 -> c7:c8:c9 c10 :: c10:c11 c11 :: c10:c11 -> c10:c11 division :: 0':s -> 0':s -> 0':s div :: 0':s -> 0':s -> 0':s -> 0':s if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s hole_c1_12 :: c hole_0':s2_12 :: 0':s hole_c1:c23_12 :: c1:c2 hole_c3:c44_12 :: c3:c4 hole_c7:c8:c95_12 :: c7:c8:c9 hole_true:false6_12 :: true:false hole_c10:c117_12 :: c10:c11 hole_c5:c68_12 :: c5:c6 gen_0':s9_12 :: Nat -> 0':s gen_c7:c8:c910_12 :: Nat -> c7:c8:c9 gen_c10:c1111_12 :: Nat -> c10:c11 gen_c5:c612_12 :: Nat -> c5:c6 Lemmas: lt(gen_0':s9_12(n14_12), gen_0':s9_12(n14_12)) -> false, rt in Omega(0) inc(gen_0':s9_12(n398_12)) -> gen_0':s9_12(+(1, n398_12)), rt in Omega(0) Generator Equations: gen_0':s9_12(0) <=> 0' gen_0':s9_12(+(x, 1)) <=> s(gen_0':s9_12(x)) gen_c7:c8:c910_12(0) <=> c7 gen_c7:c8:c910_12(+(x, 1)) <=> c9(gen_c7:c8:c910_12(x)) gen_c10:c1111_12(0) <=> c10 gen_c10:c1111_12(+(x, 1)) <=> c11(gen_c10:c1111_12(x)) gen_c5:c612_12(0) <=> c5 gen_c5:c612_12(+(x, 1)) <=> c6(gen_c5:c612_12(x)) The following defined symbols remain to be analysed: LT, DIV, INC, minus, MINUS, div They will be analysed ascendingly in the following order: LT < DIV INC < DIV minus < DIV MINUS < DIV minus < div ---------------------------------------- (92) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (93) BOUNDS(n^1, INF) ---------------------------------------- (94) Obligation: Innermost TRS: Rules: DIVISION(z0, z1) -> c(DIV(z0, z1, 0')) DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1)) DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2)) IF(true, z0, z1, z2) -> c3 IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1))) MINUS(z0, 0') -> c5 MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1)) LT(z0, 0') -> c7 LT(0', s(z0)) -> c8 LT(s(z0), s(z1)) -> c9(LT(z0, z1)) INC(0') -> c10 INC(s(z0)) -> c11(INC(z0)) division(z0, z1) -> div(z0, z1, 0') div(z0, z1, z2) -> if(lt(z0, z1), z0, z1, inc(z2)) if(true, z0, z1, z2) -> z2 if(false, z0, s(z1), z2) -> div(minus(z0, s(z1)), s(z1), z2) minus(z0, 0') -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) lt(z0, 0') -> false lt(0', s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) inc(0') -> s(0') inc(s(z0)) -> s(inc(z0)) Types: DIVISION :: 0':s -> 0':s -> c c :: c1:c2 -> c DIV :: 0':s -> 0':s -> 0':s -> c1:c2 0' :: 0':s c1 :: c3:c4 -> c7:c8:c9 -> c1:c2 IF :: true:false -> 0':s -> 0':s -> 0':s -> c3:c4 lt :: 0':s -> 0':s -> true:false inc :: 0':s -> 0':s LT :: 0':s -> 0':s -> c7:c8:c9 c2 :: c3:c4 -> c10:c11 -> c1:c2 INC :: 0':s -> c10:c11 true :: true:false c3 :: c3:c4 false :: true:false s :: 0':s -> 0':s c4 :: c1:c2 -> c5:c6 -> c3:c4 minus :: 0':s -> 0':s -> 0':s MINUS :: 0':s -> 0':s -> c5:c6 c5 :: c5:c6 c6 :: c5:c6 -> c5:c6 c7 :: c7:c8:c9 c8 :: c7:c8:c9 c9 :: c7:c8:c9 -> c7:c8:c9 c10 :: c10:c11 c11 :: c10:c11 -> c10:c11 division :: 0':s -> 0':s -> 0':s div :: 0':s -> 0':s -> 0':s -> 0':s if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s hole_c1_12 :: c hole_0':s2_12 :: 0':s hole_c1:c23_12 :: c1:c2 hole_c3:c44_12 :: c3:c4 hole_c7:c8:c95_12 :: c7:c8:c9 hole_true:false6_12 :: true:false hole_c10:c117_12 :: c10:c11 hole_c5:c68_12 :: c5:c6 gen_0':s9_12 :: Nat -> 0':s gen_c7:c8:c910_12 :: Nat -> c7:c8:c9 gen_c10:c1111_12 :: Nat -> c10:c11 gen_c5:c612_12 :: Nat -> c5:c6 Lemmas: lt(gen_0':s9_12(n14_12), gen_0':s9_12(n14_12)) -> false, rt in Omega(0) inc(gen_0':s9_12(n398_12)) -> gen_0':s9_12(+(1, n398_12)), rt in Omega(0) LT(gen_0':s9_12(n694_12), gen_0':s9_12(n694_12)) -> gen_c7:c8:c910_12(n694_12), rt in Omega(1 + n694_12) Generator Equations: gen_0':s9_12(0) <=> 0' gen_0':s9_12(+(x, 1)) <=> s(gen_0':s9_12(x)) gen_c7:c8:c910_12(0) <=> c7 gen_c7:c8:c910_12(+(x, 1)) <=> c9(gen_c7:c8:c910_12(x)) gen_c10:c1111_12(0) <=> c10 gen_c10:c1111_12(+(x, 1)) <=> c11(gen_c10:c1111_12(x)) gen_c5:c612_12(0) <=> c5 gen_c5:c612_12(+(x, 1)) <=> c6(gen_c5:c612_12(x)) The following defined symbols remain to be analysed: INC, DIV, minus, MINUS, div They will be analysed ascendingly in the following order: INC < DIV minus < DIV MINUS < DIV minus < div ---------------------------------------- (95) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: INC(gen_0':s9_12(n1356_12)) -> gen_c10:c1111_12(n1356_12), rt in Omega(1 + n1356_12) Induction Base: INC(gen_0':s9_12(0)) ->_R^Omega(1) c10 Induction Step: INC(gen_0':s9_12(+(n1356_12, 1))) ->_R^Omega(1) c11(INC(gen_0':s9_12(n1356_12))) ->_IH c11(gen_c10:c1111_12(c1357_12)) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (96) Obligation: Innermost TRS: Rules: DIVISION(z0, z1) -> c(DIV(z0, z1, 0')) DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1)) DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2)) IF(true, z0, z1, z2) -> c3 IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1))) MINUS(z0, 0') -> c5 MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1)) LT(z0, 0') -> c7 LT(0', s(z0)) -> c8 LT(s(z0), s(z1)) -> c9(LT(z0, z1)) INC(0') -> c10 INC(s(z0)) -> c11(INC(z0)) division(z0, z1) -> div(z0, z1, 0') div(z0, z1, z2) -> if(lt(z0, z1), z0, z1, inc(z2)) if(true, z0, z1, z2) -> z2 if(false, z0, s(z1), z2) -> div(minus(z0, s(z1)), s(z1), z2) minus(z0, 0') -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) lt(z0, 0') -> false lt(0', s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) inc(0') -> s(0') inc(s(z0)) -> s(inc(z0)) Types: DIVISION :: 0':s -> 0':s -> c c :: c1:c2 -> c DIV :: 0':s -> 0':s -> 0':s -> c1:c2 0' :: 0':s c1 :: c3:c4 -> c7:c8:c9 -> c1:c2 IF :: true:false -> 0':s -> 0':s -> 0':s -> c3:c4 lt :: 0':s -> 0':s -> true:false inc :: 0':s -> 0':s LT :: 0':s -> 0':s -> c7:c8:c9 c2 :: c3:c4 -> c10:c11 -> c1:c2 INC :: 0':s -> c10:c11 true :: true:false c3 :: c3:c4 false :: true:false s :: 0':s -> 0':s c4 :: c1:c2 -> c5:c6 -> c3:c4 minus :: 0':s -> 0':s -> 0':s MINUS :: 0':s -> 0':s -> c5:c6 c5 :: c5:c6 c6 :: c5:c6 -> c5:c6 c7 :: c7:c8:c9 c8 :: c7:c8:c9 c9 :: c7:c8:c9 -> c7:c8:c9 c10 :: c10:c11 c11 :: c10:c11 -> c10:c11 division :: 0':s -> 0':s -> 0':s div :: 0':s -> 0':s -> 0':s -> 0':s if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s hole_c1_12 :: c hole_0':s2_12 :: 0':s hole_c1:c23_12 :: c1:c2 hole_c3:c44_12 :: c3:c4 hole_c7:c8:c95_12 :: c7:c8:c9 hole_true:false6_12 :: true:false hole_c10:c117_12 :: c10:c11 hole_c5:c68_12 :: c5:c6 gen_0':s9_12 :: Nat -> 0':s gen_c7:c8:c910_12 :: Nat -> c7:c8:c9 gen_c10:c1111_12 :: Nat -> c10:c11 gen_c5:c612_12 :: Nat -> c5:c6 Lemmas: lt(gen_0':s9_12(n14_12), gen_0':s9_12(n14_12)) -> false, rt in Omega(0) inc(gen_0':s9_12(n398_12)) -> gen_0':s9_12(+(1, n398_12)), rt in Omega(0) LT(gen_0':s9_12(n694_12), gen_0':s9_12(n694_12)) -> gen_c7:c8:c910_12(n694_12), rt in Omega(1 + n694_12) INC(gen_0':s9_12(n1356_12)) -> gen_c10:c1111_12(n1356_12), rt in Omega(1 + n1356_12) Generator Equations: gen_0':s9_12(0) <=> 0' gen_0':s9_12(+(x, 1)) <=> s(gen_0':s9_12(x)) gen_c7:c8:c910_12(0) <=> c7 gen_c7:c8:c910_12(+(x, 1)) <=> c9(gen_c7:c8:c910_12(x)) gen_c10:c1111_12(0) <=> c10 gen_c10:c1111_12(+(x, 1)) <=> c11(gen_c10:c1111_12(x)) gen_c5:c612_12(0) <=> c5 gen_c5:c612_12(+(x, 1)) <=> c6(gen_c5:c612_12(x)) The following defined symbols remain to be analysed: minus, DIV, MINUS, div They will be analysed ascendingly in the following order: minus < DIV MINUS < DIV minus < div ---------------------------------------- (97) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: minus(gen_0':s9_12(n1724_12), gen_0':s9_12(n1724_12)) -> gen_0':s9_12(0), rt in Omega(0) Induction Base: minus(gen_0':s9_12(0), gen_0':s9_12(0)) ->_R^Omega(0) gen_0':s9_12(0) Induction Step: minus(gen_0':s9_12(+(n1724_12, 1)), gen_0':s9_12(+(n1724_12, 1))) ->_R^Omega(0) minus(gen_0':s9_12(n1724_12), gen_0':s9_12(n1724_12)) ->_IH gen_0':s9_12(0) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (98) Obligation: Innermost TRS: Rules: DIVISION(z0, z1) -> c(DIV(z0, z1, 0')) DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1)) DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2)) IF(true, z0, z1, z2) -> c3 IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1))) MINUS(z0, 0') -> c5 MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1)) LT(z0, 0') -> c7 LT(0', s(z0)) -> c8 LT(s(z0), s(z1)) -> c9(LT(z0, z1)) INC(0') -> c10 INC(s(z0)) -> c11(INC(z0)) division(z0, z1) -> div(z0, z1, 0') div(z0, z1, z2) -> if(lt(z0, z1), z0, z1, inc(z2)) if(true, z0, z1, z2) -> z2 if(false, z0, s(z1), z2) -> div(minus(z0, s(z1)), s(z1), z2) minus(z0, 0') -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) lt(z0, 0') -> false lt(0', s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) inc(0') -> s(0') inc(s(z0)) -> s(inc(z0)) Types: DIVISION :: 0':s -> 0':s -> c c :: c1:c2 -> c DIV :: 0':s -> 0':s -> 0':s -> c1:c2 0' :: 0':s c1 :: c3:c4 -> c7:c8:c9 -> c1:c2 IF :: true:false -> 0':s -> 0':s -> 0':s -> c3:c4 lt :: 0':s -> 0':s -> true:false inc :: 0':s -> 0':s LT :: 0':s -> 0':s -> c7:c8:c9 c2 :: c3:c4 -> c10:c11 -> c1:c2 INC :: 0':s -> c10:c11 true :: true:false c3 :: c3:c4 false :: true:false s :: 0':s -> 0':s c4 :: c1:c2 -> c5:c6 -> c3:c4 minus :: 0':s -> 0':s -> 0':s MINUS :: 0':s -> 0':s -> c5:c6 c5 :: c5:c6 c6 :: c5:c6 -> c5:c6 c7 :: c7:c8:c9 c8 :: c7:c8:c9 c9 :: c7:c8:c9 -> c7:c8:c9 c10 :: c10:c11 c11 :: c10:c11 -> c10:c11 division :: 0':s -> 0':s -> 0':s div :: 0':s -> 0':s -> 0':s -> 0':s if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s hole_c1_12 :: c hole_0':s2_12 :: 0':s hole_c1:c23_12 :: c1:c2 hole_c3:c44_12 :: c3:c4 hole_c7:c8:c95_12 :: c7:c8:c9 hole_true:false6_12 :: true:false hole_c10:c117_12 :: c10:c11 hole_c5:c68_12 :: c5:c6 gen_0':s9_12 :: Nat -> 0':s gen_c7:c8:c910_12 :: Nat -> c7:c8:c9 gen_c10:c1111_12 :: Nat -> c10:c11 gen_c5:c612_12 :: Nat -> c5:c6 Lemmas: lt(gen_0':s9_12(n14_12), gen_0':s9_12(n14_12)) -> false, rt in Omega(0) inc(gen_0':s9_12(n398_12)) -> gen_0':s9_12(+(1, n398_12)), rt in Omega(0) LT(gen_0':s9_12(n694_12), gen_0':s9_12(n694_12)) -> gen_c7:c8:c910_12(n694_12), rt in Omega(1 + n694_12) INC(gen_0':s9_12(n1356_12)) -> gen_c10:c1111_12(n1356_12), rt in Omega(1 + n1356_12) minus(gen_0':s9_12(n1724_12), gen_0':s9_12(n1724_12)) -> gen_0':s9_12(0), rt in Omega(0) Generator Equations: gen_0':s9_12(0) <=> 0' gen_0':s9_12(+(x, 1)) <=> s(gen_0':s9_12(x)) gen_c7:c8:c910_12(0) <=> c7 gen_c7:c8:c910_12(+(x, 1)) <=> c9(gen_c7:c8:c910_12(x)) gen_c10:c1111_12(0) <=> c10 gen_c10:c1111_12(+(x, 1)) <=> c11(gen_c10:c1111_12(x)) gen_c5:c612_12(0) <=> c5 gen_c5:c612_12(+(x, 1)) <=> c6(gen_c5:c612_12(x)) The following defined symbols remain to be analysed: MINUS, DIV, div They will be analysed ascendingly in the following order: MINUS < DIV ---------------------------------------- (99) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: MINUS(gen_0':s9_12(n2396_12), gen_0':s9_12(n2396_12)) -> gen_c5:c612_12(n2396_12), rt in Omega(1 + n2396_12) Induction Base: MINUS(gen_0':s9_12(0), gen_0':s9_12(0)) ->_R^Omega(1) c5 Induction Step: MINUS(gen_0':s9_12(+(n2396_12, 1)), gen_0':s9_12(+(n2396_12, 1))) ->_R^Omega(1) c6(MINUS(gen_0':s9_12(n2396_12), gen_0':s9_12(n2396_12))) ->_IH c6(gen_c5:c612_12(c2397_12)) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (100) Obligation: Innermost TRS: Rules: DIVISION(z0, z1) -> c(DIV(z0, z1, 0')) DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1)) DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2)) IF(true, z0, z1, z2) -> c3 IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1))) MINUS(z0, 0') -> c5 MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1)) LT(z0, 0') -> c7 LT(0', s(z0)) -> c8 LT(s(z0), s(z1)) -> c9(LT(z0, z1)) INC(0') -> c10 INC(s(z0)) -> c11(INC(z0)) division(z0, z1) -> div(z0, z1, 0') div(z0, z1, z2) -> if(lt(z0, z1), z0, z1, inc(z2)) if(true, z0, z1, z2) -> z2 if(false, z0, s(z1), z2) -> div(minus(z0, s(z1)), s(z1), z2) minus(z0, 0') -> z0 minus(s(z0), s(z1)) -> minus(z0, z1) lt(z0, 0') -> false lt(0', s(z0)) -> true lt(s(z0), s(z1)) -> lt(z0, z1) inc(0') -> s(0') inc(s(z0)) -> s(inc(z0)) Types: DIVISION :: 0':s -> 0':s -> c c :: c1:c2 -> c DIV :: 0':s -> 0':s -> 0':s -> c1:c2 0' :: 0':s c1 :: c3:c4 -> c7:c8:c9 -> c1:c2 IF :: true:false -> 0':s -> 0':s -> 0':s -> c3:c4 lt :: 0':s -> 0':s -> true:false inc :: 0':s -> 0':s LT :: 0':s -> 0':s -> c7:c8:c9 c2 :: c3:c4 -> c10:c11 -> c1:c2 INC :: 0':s -> c10:c11 true :: true:false c3 :: c3:c4 false :: true:false s :: 0':s -> 0':s c4 :: c1:c2 -> c5:c6 -> c3:c4 minus :: 0':s -> 0':s -> 0':s MINUS :: 0':s -> 0':s -> c5:c6 c5 :: c5:c6 c6 :: c5:c6 -> c5:c6 c7 :: c7:c8:c9 c8 :: c7:c8:c9 c9 :: c7:c8:c9 -> c7:c8:c9 c10 :: c10:c11 c11 :: c10:c11 -> c10:c11 division :: 0':s -> 0':s -> 0':s div :: 0':s -> 0':s -> 0':s -> 0':s if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s hole_c1_12 :: c hole_0':s2_12 :: 0':s hole_c1:c23_12 :: c1:c2 hole_c3:c44_12 :: c3:c4 hole_c7:c8:c95_12 :: c7:c8:c9 hole_true:false6_12 :: true:false hole_c10:c117_12 :: c10:c11 hole_c5:c68_12 :: c5:c6 gen_0':s9_12 :: Nat -> 0':s gen_c7:c8:c910_12 :: Nat -> c7:c8:c9 gen_c10:c1111_12 :: Nat -> c10:c11 gen_c5:c612_12 :: Nat -> c5:c6 Lemmas: lt(gen_0':s9_12(n14_12), gen_0':s9_12(n14_12)) -> false, rt in Omega(0) inc(gen_0':s9_12(n398_12)) -> gen_0':s9_12(+(1, n398_12)), rt in Omega(0) LT(gen_0':s9_12(n694_12), gen_0':s9_12(n694_12)) -> gen_c7:c8:c910_12(n694_12), rt in Omega(1 + n694_12) INC(gen_0':s9_12(n1356_12)) -> gen_c10:c1111_12(n1356_12), rt in Omega(1 + n1356_12) minus(gen_0':s9_12(n1724_12), gen_0':s9_12(n1724_12)) -> gen_0':s9_12(0), rt in Omega(0) MINUS(gen_0':s9_12(n2396_12), gen_0':s9_12(n2396_12)) -> gen_c5:c612_12(n2396_12), rt in Omega(1 + n2396_12) Generator Equations: gen_0':s9_12(0) <=> 0' gen_0':s9_12(+(x, 1)) <=> s(gen_0':s9_12(x)) gen_c7:c8:c910_12(0) <=> c7 gen_c7:c8:c910_12(+(x, 1)) <=> c9(gen_c7:c8:c910_12(x)) gen_c10:c1111_12(0) <=> c10 gen_c10:c1111_12(+(x, 1)) <=> c11(gen_c10:c1111_12(x)) gen_c5:c612_12(0) <=> c5 gen_c5:c612_12(+(x, 1)) <=> c6(gen_c5:c612_12(x)) The following defined symbols remain to be analysed: DIV, div