WORST_CASE(Omega(n^1), ?) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 518 ms] (4) CpxRelTRS (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (6) TRS for Loop Detection (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (8) BEST (9) proven lower bound (10) LowerBoundPropagationProof [FINISHED, 0 ms] (11) BOUNDS(n^1, INF) (12) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: din(der(plus(X, Y))) -> u21(din(der(X)), X, Y) u21(dout(DX), X, Y) -> u22(din(der(Y)), X, Y, DX) u22(dout(DY), X, Y, DX) -> dout(plus(DX, DY)) din(der(times(X, Y))) -> u31(din(der(X)), X, Y) u31(dout(DX), X, Y) -> u32(din(der(Y)), X, Y, DX) u32(dout(DY), X, Y, DX) -> dout(plus(times(X, DY), times(Y, DX))) din(der(der(X))) -> u41(din(der(X)), X) u41(dout(DX), X) -> u42(din(der(DX)), X, DX) u42(dout(DDX), X, DX) -> dout(DDX) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(der(x_1)) -> der(encArg(x_1)) encArg(plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(dout(x_1)) -> dout(encArg(x_1)) encArg(times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_din(x_1)) -> din(encArg(x_1)) encArg(cons_u21(x_1, x_2, x_3)) -> u21(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_u22(x_1, x_2, x_3, x_4)) -> u22(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_u31(x_1, x_2, x_3)) -> u31(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_u32(x_1, x_2, x_3, x_4)) -> u32(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_u41(x_1, x_2)) -> u41(encArg(x_1), encArg(x_2)) encArg(cons_u42(x_1, x_2, x_3)) -> u42(encArg(x_1), encArg(x_2), encArg(x_3)) encode_din(x_1) -> din(encArg(x_1)) encode_der(x_1) -> der(encArg(x_1)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_u21(x_1, x_2, x_3) -> u21(encArg(x_1), encArg(x_2), encArg(x_3)) encode_dout(x_1) -> dout(encArg(x_1)) encode_u22(x_1, x_2, x_3, x_4) -> u22(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_u31(x_1, x_2, x_3) -> u31(encArg(x_1), encArg(x_2), encArg(x_3)) encode_u32(x_1, x_2, x_3, x_4) -> u32(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_u41(x_1, x_2) -> u41(encArg(x_1), encArg(x_2)) encode_u42(x_1, x_2, x_3) -> u42(encArg(x_1), encArg(x_2), encArg(x_3)) ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: din(der(plus(X, Y))) -> u21(din(der(X)), X, Y) u21(dout(DX), X, Y) -> u22(din(der(Y)), X, Y, DX) u22(dout(DY), X, Y, DX) -> dout(plus(DX, DY)) din(der(times(X, Y))) -> u31(din(der(X)), X, Y) u31(dout(DX), X, Y) -> u32(din(der(Y)), X, Y, DX) u32(dout(DY), X, Y, DX) -> dout(plus(times(X, DY), times(Y, DX))) din(der(der(X))) -> u41(din(der(X)), X) u41(dout(DX), X) -> u42(din(der(DX)), X, DX) u42(dout(DDX), X, DX) -> dout(DDX) The (relative) TRS S consists of the following rules: encArg(der(x_1)) -> der(encArg(x_1)) encArg(plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(dout(x_1)) -> dout(encArg(x_1)) encArg(times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_din(x_1)) -> din(encArg(x_1)) encArg(cons_u21(x_1, x_2, x_3)) -> u21(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_u22(x_1, x_2, x_3, x_4)) -> u22(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_u31(x_1, x_2, x_3)) -> u31(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_u32(x_1, x_2, x_3, x_4)) -> u32(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_u41(x_1, x_2)) -> u41(encArg(x_1), encArg(x_2)) encArg(cons_u42(x_1, x_2, x_3)) -> u42(encArg(x_1), encArg(x_2), encArg(x_3)) encode_din(x_1) -> din(encArg(x_1)) encode_der(x_1) -> der(encArg(x_1)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_u21(x_1, x_2, x_3) -> u21(encArg(x_1), encArg(x_2), encArg(x_3)) encode_dout(x_1) -> dout(encArg(x_1)) encode_u22(x_1, x_2, x_3, x_4) -> u22(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_u31(x_1, x_2, x_3) -> u31(encArg(x_1), encArg(x_2), encArg(x_3)) encode_u32(x_1, x_2, x_3, x_4) -> u32(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_u41(x_1, x_2) -> u41(encArg(x_1), encArg(x_2)) encode_u42(x_1, x_2, x_3) -> u42(encArg(x_1), encArg(x_2), encArg(x_3)) Rewrite Strategy: INNERMOST ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: din(der(plus(X, Y))) -> u21(din(der(X)), X, Y) u21(dout(DX), X, Y) -> u22(din(der(Y)), X, Y, DX) u22(dout(DY), X, Y, DX) -> dout(plus(DX, DY)) din(der(times(X, Y))) -> u31(din(der(X)), X, Y) u31(dout(DX), X, Y) -> u32(din(der(Y)), X, Y, DX) u32(dout(DY), X, Y, DX) -> dout(plus(times(X, DY), times(Y, DX))) din(der(der(X))) -> u41(din(der(X)), X) u41(dout(DX), X) -> u42(din(der(DX)), X, DX) u42(dout(DDX), X, DX) -> dout(DDX) The (relative) TRS S consists of the following rules: encArg(der(x_1)) -> der(encArg(x_1)) encArg(plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(dout(x_1)) -> dout(encArg(x_1)) encArg(times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_din(x_1)) -> din(encArg(x_1)) encArg(cons_u21(x_1, x_2, x_3)) -> u21(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_u22(x_1, x_2, x_3, x_4)) -> u22(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_u31(x_1, x_2, x_3)) -> u31(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_u32(x_1, x_2, x_3, x_4)) -> u32(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_u41(x_1, x_2)) -> u41(encArg(x_1), encArg(x_2)) encArg(cons_u42(x_1, x_2, x_3)) -> u42(encArg(x_1), encArg(x_2), encArg(x_3)) encode_din(x_1) -> din(encArg(x_1)) encode_der(x_1) -> der(encArg(x_1)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_u21(x_1, x_2, x_3) -> u21(encArg(x_1), encArg(x_2), encArg(x_3)) encode_dout(x_1) -> dout(encArg(x_1)) encode_u22(x_1, x_2, x_3, x_4) -> u22(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_u31(x_1, x_2, x_3) -> u31(encArg(x_1), encArg(x_2), encArg(x_3)) encode_u32(x_1, x_2, x_3, x_4) -> u32(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_u41(x_1, x_2) -> u41(encArg(x_1), encArg(x_2)) encode_u42(x_1, x_2, x_3) -> u42(encArg(x_1), encArg(x_2), encArg(x_3)) Rewrite Strategy: INNERMOST ---------------------------------------- (5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (6) Obligation: Analyzing the following TRS for decreasing loops: 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: din(der(plus(X, Y))) -> u21(din(der(X)), X, Y) u21(dout(DX), X, Y) -> u22(din(der(Y)), X, Y, DX) u22(dout(DY), X, Y, DX) -> dout(plus(DX, DY)) din(der(times(X, Y))) -> u31(din(der(X)), X, Y) u31(dout(DX), X, Y) -> u32(din(der(Y)), X, Y, DX) u32(dout(DY), X, Y, DX) -> dout(plus(times(X, DY), times(Y, DX))) din(der(der(X))) -> u41(din(der(X)), X) u41(dout(DX), X) -> u42(din(der(DX)), X, DX) u42(dout(DDX), X, DX) -> dout(DDX) The (relative) TRS S consists of the following rules: encArg(der(x_1)) -> der(encArg(x_1)) encArg(plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(dout(x_1)) -> dout(encArg(x_1)) encArg(times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_din(x_1)) -> din(encArg(x_1)) encArg(cons_u21(x_1, x_2, x_3)) -> u21(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_u22(x_1, x_2, x_3, x_4)) -> u22(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_u31(x_1, x_2, x_3)) -> u31(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_u32(x_1, x_2, x_3, x_4)) -> u32(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_u41(x_1, x_2)) -> u41(encArg(x_1), encArg(x_2)) encArg(cons_u42(x_1, x_2, x_3)) -> u42(encArg(x_1), encArg(x_2), encArg(x_3)) encode_din(x_1) -> din(encArg(x_1)) encode_der(x_1) -> der(encArg(x_1)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_u21(x_1, x_2, x_3) -> u21(encArg(x_1), encArg(x_2), encArg(x_3)) encode_dout(x_1) -> dout(encArg(x_1)) encode_u22(x_1, x_2, x_3, x_4) -> u22(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_u31(x_1, x_2, x_3) -> u31(encArg(x_1), encArg(x_2), encArg(x_3)) encode_u32(x_1, x_2, x_3, x_4) -> u32(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_u41(x_1, x_2) -> u41(encArg(x_1), encArg(x_2)) encode_u42(x_1, x_2, x_3) -> u42(encArg(x_1), encArg(x_2), encArg(x_3)) Rewrite Strategy: INNERMOST ---------------------------------------- (7) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence din(der(plus(X, Y))) ->^+ u21(din(der(X)), X, Y) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [X / plus(X, Y)]. The result substitution is [ ]. ---------------------------------------- (8) Complex Obligation (BEST) ---------------------------------------- (9) Obligation: Proved the lower bound n^1 for the following 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: din(der(plus(X, Y))) -> u21(din(der(X)), X, Y) u21(dout(DX), X, Y) -> u22(din(der(Y)), X, Y, DX) u22(dout(DY), X, Y, DX) -> dout(plus(DX, DY)) din(der(times(X, Y))) -> u31(din(der(X)), X, Y) u31(dout(DX), X, Y) -> u32(din(der(Y)), X, Y, DX) u32(dout(DY), X, Y, DX) -> dout(plus(times(X, DY), times(Y, DX))) din(der(der(X))) -> u41(din(der(X)), X) u41(dout(DX), X) -> u42(din(der(DX)), X, DX) u42(dout(DDX), X, DX) -> dout(DDX) The (relative) TRS S consists of the following rules: encArg(der(x_1)) -> der(encArg(x_1)) encArg(plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(dout(x_1)) -> dout(encArg(x_1)) encArg(times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_din(x_1)) -> din(encArg(x_1)) encArg(cons_u21(x_1, x_2, x_3)) -> u21(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_u22(x_1, x_2, x_3, x_4)) -> u22(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_u31(x_1, x_2, x_3)) -> u31(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_u32(x_1, x_2, x_3, x_4)) -> u32(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_u41(x_1, x_2)) -> u41(encArg(x_1), encArg(x_2)) encArg(cons_u42(x_1, x_2, x_3)) -> u42(encArg(x_1), encArg(x_2), encArg(x_3)) encode_din(x_1) -> din(encArg(x_1)) encode_der(x_1) -> der(encArg(x_1)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_u21(x_1, x_2, x_3) -> u21(encArg(x_1), encArg(x_2), encArg(x_3)) encode_dout(x_1) -> dout(encArg(x_1)) encode_u22(x_1, x_2, x_3, x_4) -> u22(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_u31(x_1, x_2, x_3) -> u31(encArg(x_1), encArg(x_2), encArg(x_3)) encode_u32(x_1, x_2, x_3, x_4) -> u32(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_u41(x_1, x_2) -> u41(encArg(x_1), encArg(x_2)) encode_u42(x_1, x_2, x_3) -> u42(encArg(x_1), encArg(x_2), encArg(x_3)) Rewrite Strategy: INNERMOST ---------------------------------------- (10) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (11) BOUNDS(n^1, INF) ---------------------------------------- (12) Obligation: Analyzing the following TRS for decreasing loops: 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: din(der(plus(X, Y))) -> u21(din(der(X)), X, Y) u21(dout(DX), X, Y) -> u22(din(der(Y)), X, Y, DX) u22(dout(DY), X, Y, DX) -> dout(plus(DX, DY)) din(der(times(X, Y))) -> u31(din(der(X)), X, Y) u31(dout(DX), X, Y) -> u32(din(der(Y)), X, Y, DX) u32(dout(DY), X, Y, DX) -> dout(plus(times(X, DY), times(Y, DX))) din(der(der(X))) -> u41(din(der(X)), X) u41(dout(DX), X) -> u42(din(der(DX)), X, DX) u42(dout(DDX), X, DX) -> dout(DDX) The (relative) TRS S consists of the following rules: encArg(der(x_1)) -> der(encArg(x_1)) encArg(plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(dout(x_1)) -> dout(encArg(x_1)) encArg(times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_din(x_1)) -> din(encArg(x_1)) encArg(cons_u21(x_1, x_2, x_3)) -> u21(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_u22(x_1, x_2, x_3, x_4)) -> u22(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_u31(x_1, x_2, x_3)) -> u31(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_u32(x_1, x_2, x_3, x_4)) -> u32(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_u41(x_1, x_2)) -> u41(encArg(x_1), encArg(x_2)) encArg(cons_u42(x_1, x_2, x_3)) -> u42(encArg(x_1), encArg(x_2), encArg(x_3)) encode_din(x_1) -> din(encArg(x_1)) encode_der(x_1) -> der(encArg(x_1)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_u21(x_1, x_2, x_3) -> u21(encArg(x_1), encArg(x_2), encArg(x_3)) encode_dout(x_1) -> dout(encArg(x_1)) encode_u22(x_1, x_2, x_3, x_4) -> u22(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_u31(x_1, x_2, x_3) -> u31(encArg(x_1), encArg(x_2), encArg(x_3)) encode_u32(x_1, x_2, x_3, x_4) -> u32(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_u41(x_1, x_2) -> u41(encArg(x_1), encArg(x_2)) encode_u42(x_1, x_2, x_3) -> u42(encArg(x_1), encArg(x_2), encArg(x_3)) Rewrite Strategy: INNERMOST