/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 310 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) -> s(h) D(constant) -> h D(b(x, y)) -> b(D(x), D(y)) D(c(x, y)) -> b(c(y, D(x)), c(x, D(y))) D(m(x, y)) -> m(D(x), D(y)) D(opp(x)) -> opp(D(x)) D(div(x, y)) -> m(div(D(x), y), div(c(x, D(y)), pow(y, 2))) D(ln(x)) -> div(D(x), x) D(pow(x, y)) -> b(c(c(y, pow(x, m(y, 1))), D(x)), c(c(pow(x, y), ln(x)), D(y))) b(h, x) -> x b(x, h) -> x b(s(x), s(y)) -> s(s(b(x, y))) b(b(x, y), z) -> b(x, b(y, z)) 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(s(x_1)) -> s(encArg(x_1)) encArg(h) -> h encArg(constant) -> constant encArg(c(x_1, x_2)) -> c(encArg(x_1), encArg(x_2)) encArg(m(x_1, x_2)) -> m(encArg(x_1), encArg(x_2)) encArg(opp(x_1)) -> opp(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(1) -> 1 encArg(cons_D(x_1)) -> D(encArg(x_1)) encArg(cons_b(x_1, x_2)) -> b(encArg(x_1), encArg(x_2)) encode_D(x_1) -> D(encArg(x_1)) encode_t -> t encode_s(x_1) -> s(encArg(x_1)) encode_h -> h encode_constant -> constant encode_b(x_1, x_2) -> b(encArg(x_1), encArg(x_2)) encode_c(x_1, x_2) -> c(encArg(x_1), encArg(x_2)) encode_m(x_1, x_2) -> m(encArg(x_1), encArg(x_2)) encode_opp(x_1) -> opp(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)) encode_1 -> 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) -> s(h) D(constant) -> h D(b(x, y)) -> b(D(x), D(y)) D(c(x, y)) -> b(c(y, D(x)), c(x, D(y))) D(m(x, y)) -> m(D(x), D(y)) D(opp(x)) -> opp(D(x)) D(div(x, y)) -> m(div(D(x), y), div(c(x, D(y)), pow(y, 2))) D(ln(x)) -> div(D(x), x) D(pow(x, y)) -> b(c(c(y, pow(x, m(y, 1))), D(x)), c(c(pow(x, y), ln(x)), D(y))) b(h, x) -> x b(x, h) -> x b(s(x), s(y)) -> s(s(b(x, y))) b(b(x, y), z) -> b(x, b(y, z)) The (relative) TRS S consists of the following rules: encArg(t) -> t encArg(s(x_1)) -> s(encArg(x_1)) encArg(h) -> h encArg(constant) -> constant encArg(c(x_1, x_2)) -> c(encArg(x_1), encArg(x_2)) encArg(m(x_1, x_2)) -> m(encArg(x_1), encArg(x_2)) encArg(opp(x_1)) -> opp(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(1) -> 1 encArg(cons_D(x_1)) -> D(encArg(x_1)) encArg(cons_b(x_1, x_2)) -> b(encArg(x_1), encArg(x_2)) encode_D(x_1) -> D(encArg(x_1)) encode_t -> t encode_s(x_1) -> s(encArg(x_1)) encode_h -> h encode_constant -> constant encode_b(x_1, x_2) -> b(encArg(x_1), encArg(x_2)) encode_c(x_1, x_2) -> c(encArg(x_1), encArg(x_2)) encode_m(x_1, x_2) -> m(encArg(x_1), encArg(x_2)) encode_opp(x_1) -> opp(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)) encode_1 -> 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) -> s(h) D(constant) -> h D(b(x, y)) -> b(D(x), D(y)) D(c(x, y)) -> b(c(y, D(x)), c(x, D(y))) D(m(x, y)) -> m(D(x), D(y)) D(opp(x)) -> opp(D(x)) D(div(x, y)) -> m(div(D(x), y), div(c(x, D(y)), pow(y, 2))) D(ln(x)) -> div(D(x), x) D(pow(x, y)) -> b(c(c(y, pow(x, m(y, 1))), D(x)), c(c(pow(x, y), ln(x)), D(y))) b(h, x) -> x b(x, h) -> x b(s(x), s(y)) -> s(s(b(x, y))) b(b(x, y), z) -> b(x, b(y, z)) The (relative) TRS S consists of the following rules: encArg(t) -> t encArg(s(x_1)) -> s(encArg(x_1)) encArg(h) -> h encArg(constant) -> constant encArg(c(x_1, x_2)) -> c(encArg(x_1), encArg(x_2)) encArg(m(x_1, x_2)) -> m(encArg(x_1), encArg(x_2)) encArg(opp(x_1)) -> opp(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(1) -> 1 encArg(cons_D(x_1)) -> D(encArg(x_1)) encArg(cons_b(x_1, x_2)) -> b(encArg(x_1), encArg(x_2)) encode_D(x_1) -> D(encArg(x_1)) encode_t -> t encode_s(x_1) -> s(encArg(x_1)) encode_h -> h encode_constant -> constant encode_b(x_1, x_2) -> b(encArg(x_1), encArg(x_2)) encode_c(x_1, x_2) -> c(encArg(x_1), encArg(x_2)) encode_m(x_1, x_2) -> m(encArg(x_1), encArg(x_2)) encode_opp(x_1) -> opp(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)) encode_1 -> 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) -> s(h) D(constant) -> h D(b(x, y)) -> b(D(x), D(y)) D(c(x, y)) -> b(c(y, D(x)), c(x, D(y))) D(m(x, y)) -> m(D(x), D(y)) D(opp(x)) -> opp(D(x)) D(div(x, y)) -> m(div(D(x), y), div(c(x, D(y)), pow(y, 2))) D(ln(x)) -> div(D(x), x) D(pow(x, y)) -> b(c(c(y, pow(x, m(y, 1))), D(x)), c(c(pow(x, y), ln(x)), D(y))) b(h, x) -> x b(x, h) -> x b(s(x), s(y)) -> s(s(b(x, y))) b(b(x, y), z) -> b(x, b(y, z)) The (relative) TRS S consists of the following rules: encArg(t) -> t encArg(s(x_1)) -> s(encArg(x_1)) encArg(h) -> h encArg(constant) -> constant encArg(c(x_1, x_2)) -> c(encArg(x_1), encArg(x_2)) encArg(m(x_1, x_2)) -> m(encArg(x_1), encArg(x_2)) encArg(opp(x_1)) -> opp(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(1) -> 1 encArg(cons_D(x_1)) -> D(encArg(x_1)) encArg(cons_b(x_1, x_2)) -> b(encArg(x_1), encArg(x_2)) encode_D(x_1) -> D(encArg(x_1)) encode_t -> t encode_s(x_1) -> s(encArg(x_1)) encode_h -> h encode_constant -> constant encode_b(x_1, x_2) -> b(encArg(x_1), encArg(x_2)) encode_c(x_1, x_2) -> c(encArg(x_1), encArg(x_2)) encode_m(x_1, x_2) -> m(encArg(x_1), encArg(x_2)) encode_opp(x_1) -> opp(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)) encode_1 -> 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(div(x, y)) ->^+ m(div(D(x), y), div(c(x, D(y)), pow(y, 2))) gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0]. The pumping substitution is [x / div(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) -> s(h) D(constant) -> h D(b(x, y)) -> b(D(x), D(y)) D(c(x, y)) -> b(c(y, D(x)), c(x, D(y))) D(m(x, y)) -> m(D(x), D(y)) D(opp(x)) -> opp(D(x)) D(div(x, y)) -> m(div(D(x), y), div(c(x, D(y)), pow(y, 2))) D(ln(x)) -> div(D(x), x) D(pow(x, y)) -> b(c(c(y, pow(x, m(y, 1))), D(x)), c(c(pow(x, y), ln(x)), D(y))) b(h, x) -> x b(x, h) -> x b(s(x), s(y)) -> s(s(b(x, y))) b(b(x, y), z) -> b(x, b(y, z)) The (relative) TRS S consists of the following rules: encArg(t) -> t encArg(s(x_1)) -> s(encArg(x_1)) encArg(h) -> h encArg(constant) -> constant encArg(c(x_1, x_2)) -> c(encArg(x_1), encArg(x_2)) encArg(m(x_1, x_2)) -> m(encArg(x_1), encArg(x_2)) encArg(opp(x_1)) -> opp(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(1) -> 1 encArg(cons_D(x_1)) -> D(encArg(x_1)) encArg(cons_b(x_1, x_2)) -> b(encArg(x_1), encArg(x_2)) encode_D(x_1) -> D(encArg(x_1)) encode_t -> t encode_s(x_1) -> s(encArg(x_1)) encode_h -> h encode_constant -> constant encode_b(x_1, x_2) -> b(encArg(x_1), encArg(x_2)) encode_c(x_1, x_2) -> c(encArg(x_1), encArg(x_2)) encode_m(x_1, x_2) -> m(encArg(x_1), encArg(x_2)) encode_opp(x_1) -> opp(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)) encode_1 -> 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) -> s(h) D(constant) -> h D(b(x, y)) -> b(D(x), D(y)) D(c(x, y)) -> b(c(y, D(x)), c(x, D(y))) D(m(x, y)) -> m(D(x), D(y)) D(opp(x)) -> opp(D(x)) D(div(x, y)) -> m(div(D(x), y), div(c(x, D(y)), pow(y, 2))) D(ln(x)) -> div(D(x), x) D(pow(x, y)) -> b(c(c(y, pow(x, m(y, 1))), D(x)), c(c(pow(x, y), ln(x)), D(y))) b(h, x) -> x b(x, h) -> x b(s(x), s(y)) -> s(s(b(x, y))) b(b(x, y), z) -> b(x, b(y, z)) The (relative) TRS S consists of the following rules: encArg(t) -> t encArg(s(x_1)) -> s(encArg(x_1)) encArg(h) -> h encArg(constant) -> constant encArg(c(x_1, x_2)) -> c(encArg(x_1), encArg(x_2)) encArg(m(x_1, x_2)) -> m(encArg(x_1), encArg(x_2)) encArg(opp(x_1)) -> opp(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(1) -> 1 encArg(cons_D(x_1)) -> D(encArg(x_1)) encArg(cons_b(x_1, x_2)) -> b(encArg(x_1), encArg(x_2)) encode_D(x_1) -> D(encArg(x_1)) encode_t -> t encode_s(x_1) -> s(encArg(x_1)) encode_h -> h encode_constant -> constant encode_b(x_1, x_2) -> b(encArg(x_1), encArg(x_2)) encode_c(x_1, x_2) -> c(encArg(x_1), encArg(x_2)) encode_m(x_1, x_2) -> m(encArg(x_1), encArg(x_2)) encode_opp(x_1) -> opp(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)) encode_1 -> 1 Rewrite Strategy: INNERMOST