/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 252 ms] (4) CpxRelTRS (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (6) TRS for Loop Detection (7) DecreasingLoopProof [LOWER BOUND(ID), 5 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: min(x, 0) -> 0 min(0, y) -> 0 min(s(x), s(y)) -> s(min(x, y)) max(x, 0) -> x max(0, y) -> y max(s(x), s(y)) -> s(max(x, y)) minus(x, 0) -> x minus(s(x), s(y)) -> s(minus(x, y)) gcd(s(x), s(y)) -> gcd(minus(max(x, y), min(x, transform(y))), s(min(x, y))) transform(x) -> s(s(x)) transform(cons(x, y)) -> cons(cons(x, x), x) transform(cons(x, y)) -> y transform(s(x)) -> s(s(transform(x))) cons(x, y) -> y cons(x, cons(y, s(z))) -> cons(y, x) cons(cons(x, z), s(y)) -> transform(x) 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(s(x_1)) -> s(encArg(x_1)) encArg(cons_min(x_1, x_2)) -> min(encArg(x_1), encArg(x_2)) encArg(cons_max(x_1, x_2)) -> max(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_gcd(x_1, x_2)) -> gcd(encArg(x_1), encArg(x_2)) encArg(cons_transform(x_1)) -> transform(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encode_min(x_1, x_2) -> min(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_max(x_1, x_2) -> max(encArg(x_1), encArg(x_2)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_gcd(x_1, x_2) -> gcd(encArg(x_1), encArg(x_2)) encode_transform(x_1) -> transform(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) ---------------------------------------- (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: min(x, 0) -> 0 min(0, y) -> 0 min(s(x), s(y)) -> s(min(x, y)) max(x, 0) -> x max(0, y) -> y max(s(x), s(y)) -> s(max(x, y)) minus(x, 0) -> x minus(s(x), s(y)) -> s(minus(x, y)) gcd(s(x), s(y)) -> gcd(minus(max(x, y), min(x, transform(y))), s(min(x, y))) transform(x) -> s(s(x)) transform(cons(x, y)) -> cons(cons(x, x), x) transform(cons(x, y)) -> y transform(s(x)) -> s(s(transform(x))) cons(x, y) -> y cons(x, cons(y, s(z))) -> cons(y, x) cons(cons(x, z), s(y)) -> transform(x) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_min(x_1, x_2)) -> min(encArg(x_1), encArg(x_2)) encArg(cons_max(x_1, x_2)) -> max(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_gcd(x_1, x_2)) -> gcd(encArg(x_1), encArg(x_2)) encArg(cons_transform(x_1)) -> transform(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encode_min(x_1, x_2) -> min(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_max(x_1, x_2) -> max(encArg(x_1), encArg(x_2)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_gcd(x_1, x_2) -> gcd(encArg(x_1), encArg(x_2)) encode_transform(x_1) -> transform(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) 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: min(x, 0) -> 0 min(0, y) -> 0 min(s(x), s(y)) -> s(min(x, y)) max(x, 0) -> x max(0, y) -> y max(s(x), s(y)) -> s(max(x, y)) minus(x, 0) -> x minus(s(x), s(y)) -> s(minus(x, y)) gcd(s(x), s(y)) -> gcd(minus(max(x, y), min(x, transform(y))), s(min(x, y))) transform(x) -> s(s(x)) transform(cons(x, y)) -> cons(cons(x, x), x) transform(cons(x, y)) -> y transform(s(x)) -> s(s(transform(x))) cons(x, y) -> y cons(x, cons(y, s(z))) -> cons(y, x) cons(cons(x, z), s(y)) -> transform(x) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_min(x_1, x_2)) -> min(encArg(x_1), encArg(x_2)) encArg(cons_max(x_1, x_2)) -> max(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_gcd(x_1, x_2)) -> gcd(encArg(x_1), encArg(x_2)) encArg(cons_transform(x_1)) -> transform(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encode_min(x_1, x_2) -> min(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_max(x_1, x_2) -> max(encArg(x_1), encArg(x_2)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_gcd(x_1, x_2) -> gcd(encArg(x_1), encArg(x_2)) encode_transform(x_1) -> transform(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) 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: min(x, 0) -> 0 min(0, y) -> 0 min(s(x), s(y)) -> s(min(x, y)) max(x, 0) -> x max(0, y) -> y max(s(x), s(y)) -> s(max(x, y)) minus(x, 0) -> x minus(s(x), s(y)) -> s(minus(x, y)) gcd(s(x), s(y)) -> gcd(minus(max(x, y), min(x, transform(y))), s(min(x, y))) transform(x) -> s(s(x)) transform(cons(x, y)) -> cons(cons(x, x), x) transform(cons(x, y)) -> y transform(s(x)) -> s(s(transform(x))) cons(x, y) -> y cons(x, cons(y, s(z))) -> cons(y, x) cons(cons(x, z), s(y)) -> transform(x) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_min(x_1, x_2)) -> min(encArg(x_1), encArg(x_2)) encArg(cons_max(x_1, x_2)) -> max(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_gcd(x_1, x_2)) -> gcd(encArg(x_1), encArg(x_2)) encArg(cons_transform(x_1)) -> transform(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encode_min(x_1, x_2) -> min(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_max(x_1, x_2) -> max(encArg(x_1), encArg(x_2)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_gcd(x_1, x_2) -> gcd(encArg(x_1), encArg(x_2)) encode_transform(x_1) -> transform(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) 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 minus(s(x), s(y)) ->^+ s(minus(x, y)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 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: min(x, 0) -> 0 min(0, y) -> 0 min(s(x), s(y)) -> s(min(x, y)) max(x, 0) -> x max(0, y) -> y max(s(x), s(y)) -> s(max(x, y)) minus(x, 0) -> x minus(s(x), s(y)) -> s(minus(x, y)) gcd(s(x), s(y)) -> gcd(minus(max(x, y), min(x, transform(y))), s(min(x, y))) transform(x) -> s(s(x)) transform(cons(x, y)) -> cons(cons(x, x), x) transform(cons(x, y)) -> y transform(s(x)) -> s(s(transform(x))) cons(x, y) -> y cons(x, cons(y, s(z))) -> cons(y, x) cons(cons(x, z), s(y)) -> transform(x) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_min(x_1, x_2)) -> min(encArg(x_1), encArg(x_2)) encArg(cons_max(x_1, x_2)) -> max(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_gcd(x_1, x_2)) -> gcd(encArg(x_1), encArg(x_2)) encArg(cons_transform(x_1)) -> transform(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encode_min(x_1, x_2) -> min(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_max(x_1, x_2) -> max(encArg(x_1), encArg(x_2)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_gcd(x_1, x_2) -> gcd(encArg(x_1), encArg(x_2)) encode_transform(x_1) -> transform(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) 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: min(x, 0) -> 0 min(0, y) -> 0 min(s(x), s(y)) -> s(min(x, y)) max(x, 0) -> x max(0, y) -> y max(s(x), s(y)) -> s(max(x, y)) minus(x, 0) -> x minus(s(x), s(y)) -> s(minus(x, y)) gcd(s(x), s(y)) -> gcd(minus(max(x, y), min(x, transform(y))), s(min(x, y))) transform(x) -> s(s(x)) transform(cons(x, y)) -> cons(cons(x, x), x) transform(cons(x, y)) -> y transform(s(x)) -> s(s(transform(x))) cons(x, y) -> y cons(x, cons(y, s(z))) -> cons(y, x) cons(cons(x, z), s(y)) -> transform(x) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_min(x_1, x_2)) -> min(encArg(x_1), encArg(x_2)) encArg(cons_max(x_1, x_2)) -> max(encArg(x_1), encArg(x_2)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_gcd(x_1, x_2)) -> gcd(encArg(x_1), encArg(x_2)) encArg(cons_transform(x_1)) -> transform(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encode_min(x_1, x_2) -> min(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_max(x_1, x_2) -> max(encArg(x_1), encArg(x_2)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_gcd(x_1, x_2) -> gcd(encArg(x_1), encArg(x_2)) encode_transform(x_1) -> transform(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) Rewrite Strategy: INNERMOST