/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 299 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: D(t) -> 1 D(constant) -> 0 D(+(x, y)) -> +(D(x), D(y)) D(*(x, y)) -> +(*(y, D(x)), *(x, D(y))) D(-(x, y)) -> -(D(x), D(y)) D(minus(x)) -> minus(D(x)) D(div(x, y)) -> -(div(D(x), y), div(*(x, D(y)), pow(y, 2))) D(ln(x)) -> div(D(x), x) D(pow(x, y)) -> +(*(*(y, pow(x, -(y, 1))), D(x)), *(*(pow(x, y), ln(x)), D(y))) 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(t) -> t encArg(1) -> 1 encArg(constant) -> constant encArg(0) -> 0 encArg(+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) encArg(*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2)) encArg(-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2)) encArg(minus(x_1)) -> minus(encArg(x_1)) encArg(div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2)) encArg(pow(x_1, x_2)) -> pow(encArg(x_1), encArg(x_2)) encArg(2) -> 2 encArg(ln(x_1)) -> ln(encArg(x_1)) encArg(cons_D(x_1)) -> D(encArg(x_1)) encode_D(x_1) -> D(encArg(x_1)) encode_t -> t encode_1 -> 1 encode_constant -> constant encode_0 -> 0 encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2)) encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2)) encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2)) encode_minus(x_1) -> minus(encArg(x_1)) encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2)) encode_pow(x_1, x_2) -> pow(encArg(x_1), encArg(x_2)) encode_2 -> 2 encode_ln(x_1) -> ln(encArg(x_1)) ---------------------------------------- (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: D(t) -> 1 D(constant) -> 0 D(+(x, y)) -> +(D(x), D(y)) D(*(x, y)) -> +(*(y, D(x)), *(x, D(y))) D(-(x, y)) -> -(D(x), D(y)) D(minus(x)) -> minus(D(x)) D(div(x, y)) -> -(div(D(x), y), div(*(x, D(y)), pow(y, 2))) D(ln(x)) -> div(D(x), x) D(pow(x, y)) -> +(*(*(y, pow(x, -(y, 1))), D(x)), *(*(pow(x, y), ln(x)), D(y))) The (relative) TRS S consists of the following rules: encArg(t) -> t encArg(1) -> 1 encArg(constant) -> constant encArg(0) -> 0 encArg(+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) encArg(*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2)) encArg(-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2)) encArg(minus(x_1)) -> minus(encArg(x_1)) encArg(div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2)) encArg(pow(x_1, x_2)) -> pow(encArg(x_1), encArg(x_2)) encArg(2) -> 2 encArg(ln(x_1)) -> ln(encArg(x_1)) encArg(cons_D(x_1)) -> D(encArg(x_1)) encode_D(x_1) -> D(encArg(x_1)) encode_t -> t encode_1 -> 1 encode_constant -> constant encode_0 -> 0 encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2)) encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2)) encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2)) encode_minus(x_1) -> minus(encArg(x_1)) encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2)) encode_pow(x_1, x_2) -> pow(encArg(x_1), encArg(x_2)) encode_2 -> 2 encode_ln(x_1) -> ln(encArg(x_1)) 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: D(t) -> 1 D(constant) -> 0 D(+(x, y)) -> +(D(x), D(y)) D(*(x, y)) -> +(*(y, D(x)), *(x, D(y))) D(-(x, y)) -> -(D(x), D(y)) D(minus(x)) -> minus(D(x)) D(div(x, y)) -> -(div(D(x), y), div(*(x, D(y)), pow(y, 2))) D(ln(x)) -> div(D(x), x) D(pow(x, y)) -> +(*(*(y, pow(x, -(y, 1))), D(x)), *(*(pow(x, y), ln(x)), D(y))) The (relative) TRS S consists of the following rules: encArg(t) -> t encArg(1) -> 1 encArg(constant) -> constant encArg(0) -> 0 encArg(+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) encArg(*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2)) encArg(-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2)) encArg(minus(x_1)) -> minus(encArg(x_1)) encArg(div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2)) encArg(pow(x_1, x_2)) -> pow(encArg(x_1), encArg(x_2)) encArg(2) -> 2 encArg(ln(x_1)) -> ln(encArg(x_1)) encArg(cons_D(x_1)) -> D(encArg(x_1)) encode_D(x_1) -> D(encArg(x_1)) encode_t -> t encode_1 -> 1 encode_constant -> constant encode_0 -> 0 encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2)) encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2)) encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2)) encode_minus(x_1) -> minus(encArg(x_1)) encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2)) encode_pow(x_1, x_2) -> pow(encArg(x_1), encArg(x_2)) encode_2 -> 2 encode_ln(x_1) -> ln(encArg(x_1)) 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: D(t) -> 1 D(constant) -> 0 D(+(x, y)) -> +(D(x), D(y)) D(*(x, y)) -> +(*(y, D(x)), *(x, D(y))) D(-(x, y)) -> -(D(x), D(y)) D(minus(x)) -> minus(D(x)) D(div(x, y)) -> -(div(D(x), y), div(*(x, D(y)), pow(y, 2))) D(ln(x)) -> div(D(x), x) D(pow(x, y)) -> +(*(*(y, pow(x, -(y, 1))), D(x)), *(*(pow(x, y), ln(x)), D(y))) The (relative) TRS S consists of the following rules: encArg(t) -> t encArg(1) -> 1 encArg(constant) -> constant encArg(0) -> 0 encArg(+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) encArg(*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2)) encArg(-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2)) encArg(minus(x_1)) -> minus(encArg(x_1)) encArg(div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2)) encArg(pow(x_1, x_2)) -> pow(encArg(x_1), encArg(x_2)) encArg(2) -> 2 encArg(ln(x_1)) -> ln(encArg(x_1)) encArg(cons_D(x_1)) -> D(encArg(x_1)) encode_D(x_1) -> D(encArg(x_1)) encode_t -> t encode_1 -> 1 encode_constant -> constant encode_0 -> 0 encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2)) encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2)) encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2)) encode_minus(x_1) -> minus(encArg(x_1)) encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2)) encode_pow(x_1, x_2) -> pow(encArg(x_1), encArg(x_2)) encode_2 -> 2 encode_ln(x_1) -> ln(encArg(x_1)) 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 D(+(x, y)) ->^+ +(D(x), D(y)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [x / +(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: D(t) -> 1 D(constant) -> 0 D(+(x, y)) -> +(D(x), D(y)) D(*(x, y)) -> +(*(y, D(x)), *(x, D(y))) D(-(x, y)) -> -(D(x), D(y)) D(minus(x)) -> minus(D(x)) D(div(x, y)) -> -(div(D(x), y), div(*(x, D(y)), pow(y, 2))) D(ln(x)) -> div(D(x), x) D(pow(x, y)) -> +(*(*(y, pow(x, -(y, 1))), D(x)), *(*(pow(x, y), ln(x)), D(y))) The (relative) TRS S consists of the following rules: encArg(t) -> t encArg(1) -> 1 encArg(constant) -> constant encArg(0) -> 0 encArg(+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) encArg(*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2)) encArg(-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2)) encArg(minus(x_1)) -> minus(encArg(x_1)) encArg(div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2)) encArg(pow(x_1, x_2)) -> pow(encArg(x_1), encArg(x_2)) encArg(2) -> 2 encArg(ln(x_1)) -> ln(encArg(x_1)) encArg(cons_D(x_1)) -> D(encArg(x_1)) encode_D(x_1) -> D(encArg(x_1)) encode_t -> t encode_1 -> 1 encode_constant -> constant encode_0 -> 0 encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2)) encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2)) encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2)) encode_minus(x_1) -> minus(encArg(x_1)) encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2)) encode_pow(x_1, x_2) -> pow(encArg(x_1), encArg(x_2)) encode_2 -> 2 encode_ln(x_1) -> ln(encArg(x_1)) 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: D(t) -> 1 D(constant) -> 0 D(+(x, y)) -> +(D(x), D(y)) D(*(x, y)) -> +(*(y, D(x)), *(x, D(y))) D(-(x, y)) -> -(D(x), D(y)) D(minus(x)) -> minus(D(x)) D(div(x, y)) -> -(div(D(x), y), div(*(x, D(y)), pow(y, 2))) D(ln(x)) -> div(D(x), x) D(pow(x, y)) -> +(*(*(y, pow(x, -(y, 1))), D(x)), *(*(pow(x, y), ln(x)), D(y))) The (relative) TRS S consists of the following rules: encArg(t) -> t encArg(1) -> 1 encArg(constant) -> constant encArg(0) -> 0 encArg(+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) encArg(*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2)) encArg(-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2)) encArg(minus(x_1)) -> minus(encArg(x_1)) encArg(div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2)) encArg(pow(x_1, x_2)) -> pow(encArg(x_1), encArg(x_2)) encArg(2) -> 2 encArg(ln(x_1)) -> ln(encArg(x_1)) encArg(cons_D(x_1)) -> D(encArg(x_1)) encode_D(x_1) -> D(encArg(x_1)) encode_t -> t encode_1 -> 1 encode_constant -> constant encode_0 -> 0 encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2)) encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2)) encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2)) encode_minus(x_1) -> minus(encArg(x_1)) encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2)) encode_pow(x_1, x_2) -> pow(encArg(x_1), encArg(x_2)) encode_2 -> 2 encode_ln(x_1) -> ln(encArg(x_1)) Rewrite Strategy: INNERMOST