/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), ?) proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 214 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: 1024 -> 1024_1(0) 1024_1(x) -> if(lt(x, 10), x) if(true, x) -> double(1024_1(s(x))) if(false, x) -> s(0) lt(0, s(y)) -> true lt(x, 0) -> false lt(s(x), s(y)) -> lt(x, y) double(0) -> 0 double(s(x)) -> s(s(double(x))) 10 -> double(s(double(s(s(0))))) 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(0) -> 0 encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(cons_1024) -> 1024 encArg(cons_1024_1(x_1)) -> 1024_1(encArg(x_1)) encArg(cons_if(x_1, x_2)) -> if(encArg(x_1), encArg(x_2)) encArg(cons_lt(x_1, x_2)) -> lt(encArg(x_1), encArg(x_2)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_10) -> 10 encode_1024 -> 1024 encode_1024_1(x_1) -> 1024_1(encArg(x_1)) encode_0 -> 0 encode_if(x_1, x_2) -> if(encArg(x_1), encArg(x_2)) encode_lt(x_1, x_2) -> lt(encArg(x_1), encArg(x_2)) encode_10 -> 10 encode_true -> true encode_double(x_1) -> double(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_false -> false ---------------------------------------- (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: 1024 -> 1024_1(0) 1024_1(x) -> if(lt(x, 10), x) if(true, x) -> double(1024_1(s(x))) if(false, x) -> s(0) lt(0, s(y)) -> true lt(x, 0) -> false lt(s(x), s(y)) -> lt(x, y) double(0) -> 0 double(s(x)) -> s(s(double(x))) 10 -> double(s(double(s(s(0))))) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(cons_1024) -> 1024 encArg(cons_1024_1(x_1)) -> 1024_1(encArg(x_1)) encArg(cons_if(x_1, x_2)) -> if(encArg(x_1), encArg(x_2)) encArg(cons_lt(x_1, x_2)) -> lt(encArg(x_1), encArg(x_2)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_10) -> 10 encode_1024 -> 1024 encode_1024_1(x_1) -> 1024_1(encArg(x_1)) encode_0 -> 0 encode_if(x_1, x_2) -> if(encArg(x_1), encArg(x_2)) encode_lt(x_1, x_2) -> lt(encArg(x_1), encArg(x_2)) encode_10 -> 10 encode_true -> true encode_double(x_1) -> double(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_false -> false 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: 1024 -> 1024_1(0) 1024_1(x) -> if(lt(x, 10), x) if(true, x) -> double(1024_1(s(x))) if(false, x) -> s(0) lt(0, s(y)) -> true lt(x, 0) -> false lt(s(x), s(y)) -> lt(x, y) double(0) -> 0 double(s(x)) -> s(s(double(x))) 10 -> double(s(double(s(s(0))))) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(cons_1024) -> 1024 encArg(cons_1024_1(x_1)) -> 1024_1(encArg(x_1)) encArg(cons_if(x_1, x_2)) -> if(encArg(x_1), encArg(x_2)) encArg(cons_lt(x_1, x_2)) -> lt(encArg(x_1), encArg(x_2)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_10) -> 10 encode_1024 -> 1024 encode_1024_1(x_1) -> 1024_1(encArg(x_1)) encode_0 -> 0 encode_if(x_1, x_2) -> if(encArg(x_1), encArg(x_2)) encode_lt(x_1, x_2) -> lt(encArg(x_1), encArg(x_2)) encode_10 -> 10 encode_true -> true encode_double(x_1) -> double(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_false -> false 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: 1024 -> 1024_1(0) 1024_1(x) -> if(lt(x, 10), x) if(true, x) -> double(1024_1(s(x))) if(false, x) -> s(0) lt(0, s(y)) -> true lt(x, 0) -> false lt(s(x), s(y)) -> lt(x, y) double(0) -> 0 double(s(x)) -> s(s(double(x))) 10 -> double(s(double(s(s(0))))) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(cons_1024) -> 1024 encArg(cons_1024_1(x_1)) -> 1024_1(encArg(x_1)) encArg(cons_if(x_1, x_2)) -> if(encArg(x_1), encArg(x_2)) encArg(cons_lt(x_1, x_2)) -> lt(encArg(x_1), encArg(x_2)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_10) -> 10 encode_1024 -> 1024 encode_1024_1(x_1) -> 1024_1(encArg(x_1)) encode_0 -> 0 encode_if(x_1, x_2) -> if(encArg(x_1), encArg(x_2)) encode_lt(x_1, x_2) -> lt(encArg(x_1), encArg(x_2)) encode_10 -> 10 encode_true -> true encode_double(x_1) -> double(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_false -> false 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 lt(s(x), s(y)) ->^+ lt(x, y) gives rise to a decreasing loop by considering the right hand sides subterm at position []. The pumping substitution is [x / s(x), y / s(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: 1024 -> 1024_1(0) 1024_1(x) -> if(lt(x, 10), x) if(true, x) -> double(1024_1(s(x))) if(false, x) -> s(0) lt(0, s(y)) -> true lt(x, 0) -> false lt(s(x), s(y)) -> lt(x, y) double(0) -> 0 double(s(x)) -> s(s(double(x))) 10 -> double(s(double(s(s(0))))) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(cons_1024) -> 1024 encArg(cons_1024_1(x_1)) -> 1024_1(encArg(x_1)) encArg(cons_if(x_1, x_2)) -> if(encArg(x_1), encArg(x_2)) encArg(cons_lt(x_1, x_2)) -> lt(encArg(x_1), encArg(x_2)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_10) -> 10 encode_1024 -> 1024 encode_1024_1(x_1) -> 1024_1(encArg(x_1)) encode_0 -> 0 encode_if(x_1, x_2) -> if(encArg(x_1), encArg(x_2)) encode_lt(x_1, x_2) -> lt(encArg(x_1), encArg(x_2)) encode_10 -> 10 encode_true -> true encode_double(x_1) -> double(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_false -> false 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: 1024 -> 1024_1(0) 1024_1(x) -> if(lt(x, 10), x) if(true, x) -> double(1024_1(s(x))) if(false, x) -> s(0) lt(0, s(y)) -> true lt(x, 0) -> false lt(s(x), s(y)) -> lt(x, y) double(0) -> 0 double(s(x)) -> s(s(double(x))) 10 -> double(s(double(s(s(0))))) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(cons_1024) -> 1024 encArg(cons_1024_1(x_1)) -> 1024_1(encArg(x_1)) encArg(cons_if(x_1, x_2)) -> if(encArg(x_1), encArg(x_2)) encArg(cons_lt(x_1, x_2)) -> lt(encArg(x_1), encArg(x_2)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_10) -> 10 encode_1024 -> 1024 encode_1024_1(x_1) -> 1024_1(encArg(x_1)) encode_0 -> 0 encode_if(x_1, x_2) -> if(encArg(x_1), encArg(x_2)) encode_lt(x_1, x_2) -> lt(encArg(x_1), encArg(x_2)) encode_10 -> 10 encode_true -> true encode_double(x_1) -> double(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_false -> false Rewrite Strategy: INNERMOST