/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 336 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: a__dbl(0) -> 0 a__dbl(s(X)) -> s(s(dbl(X))) a__dbls(nil) -> nil a__dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y)) a__sel(0, cons(X, Y)) -> mark(X) a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z)) a__indx(nil, X) -> nil a__indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z)) a__from(X) -> cons(X, from(s(X))) mark(dbl(X)) -> a__dbl(mark(X)) mark(dbls(X)) -> a__dbls(mark(X)) mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) mark(indx(X1, X2)) -> a__indx(mark(X1), X2) mark(from(X)) -> a__from(X) mark(0) -> 0 mark(s(X)) -> s(X) mark(nil) -> nil mark(cons(X1, X2)) -> cons(X1, X2) a__dbl(X) -> dbl(X) a__dbls(X) -> dbls(X) a__sel(X1, X2) -> sel(X1, X2) a__indx(X1, X2) -> indx(X1, X2) a__from(X) -> from(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(dbl(x_1)) -> dbl(encArg(x_1)) encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(dbls(x_1)) -> dbls(encArg(x_1)) encArg(sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(indx(x_1, x_2)) -> indx(encArg(x_1), encArg(x_2)) encArg(from(x_1)) -> from(encArg(x_1)) encArg(cons_a__dbl(x_1)) -> a__dbl(encArg(x_1)) encArg(cons_a__dbls(x_1)) -> a__dbls(encArg(x_1)) encArg(cons_a__sel(x_1, x_2)) -> a__sel(encArg(x_1), encArg(x_2)) encArg(cons_a__indx(x_1, x_2)) -> a__indx(encArg(x_1), encArg(x_2)) encArg(cons_a__from(x_1)) -> a__from(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__dbl(x_1) -> a__dbl(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_dbl(x_1) -> dbl(encArg(x_1)) encode_a__dbls(x_1) -> a__dbls(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_dbls(x_1) -> dbls(encArg(x_1)) encode_a__sel(x_1, x_2) -> a__sel(encArg(x_1), encArg(x_2)) encode_mark(x_1) -> mark(encArg(x_1)) encode_a__indx(x_1, x_2) -> a__indx(encArg(x_1), encArg(x_2)) encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2)) encode_indx(x_1, x_2) -> indx(encArg(x_1), encArg(x_2)) encode_a__from(x_1) -> a__from(encArg(x_1)) encode_from(x_1) -> from(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: a__dbl(0) -> 0 a__dbl(s(X)) -> s(s(dbl(X))) a__dbls(nil) -> nil a__dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y)) a__sel(0, cons(X, Y)) -> mark(X) a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z)) a__indx(nil, X) -> nil a__indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z)) a__from(X) -> cons(X, from(s(X))) mark(dbl(X)) -> a__dbl(mark(X)) mark(dbls(X)) -> a__dbls(mark(X)) mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) mark(indx(X1, X2)) -> a__indx(mark(X1), X2) mark(from(X)) -> a__from(X) mark(0) -> 0 mark(s(X)) -> s(X) mark(nil) -> nil mark(cons(X1, X2)) -> cons(X1, X2) a__dbl(X) -> dbl(X) a__dbls(X) -> dbls(X) a__sel(X1, X2) -> sel(X1, X2) a__indx(X1, X2) -> indx(X1, X2) a__from(X) -> from(X) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(dbl(x_1)) -> dbl(encArg(x_1)) encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(dbls(x_1)) -> dbls(encArg(x_1)) encArg(sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(indx(x_1, x_2)) -> indx(encArg(x_1), encArg(x_2)) encArg(from(x_1)) -> from(encArg(x_1)) encArg(cons_a__dbl(x_1)) -> a__dbl(encArg(x_1)) encArg(cons_a__dbls(x_1)) -> a__dbls(encArg(x_1)) encArg(cons_a__sel(x_1, x_2)) -> a__sel(encArg(x_1), encArg(x_2)) encArg(cons_a__indx(x_1, x_2)) -> a__indx(encArg(x_1), encArg(x_2)) encArg(cons_a__from(x_1)) -> a__from(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__dbl(x_1) -> a__dbl(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_dbl(x_1) -> dbl(encArg(x_1)) encode_a__dbls(x_1) -> a__dbls(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_dbls(x_1) -> dbls(encArg(x_1)) encode_a__sel(x_1, x_2) -> a__sel(encArg(x_1), encArg(x_2)) encode_mark(x_1) -> mark(encArg(x_1)) encode_a__indx(x_1, x_2) -> a__indx(encArg(x_1), encArg(x_2)) encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2)) encode_indx(x_1, x_2) -> indx(encArg(x_1), encArg(x_2)) encode_a__from(x_1) -> a__from(encArg(x_1)) encode_from(x_1) -> from(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: a__dbl(0) -> 0 a__dbl(s(X)) -> s(s(dbl(X))) a__dbls(nil) -> nil a__dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y)) a__sel(0, cons(X, Y)) -> mark(X) a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z)) a__indx(nil, X) -> nil a__indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z)) a__from(X) -> cons(X, from(s(X))) mark(dbl(X)) -> a__dbl(mark(X)) mark(dbls(X)) -> a__dbls(mark(X)) mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) mark(indx(X1, X2)) -> a__indx(mark(X1), X2) mark(from(X)) -> a__from(X) mark(0) -> 0 mark(s(X)) -> s(X) mark(nil) -> nil mark(cons(X1, X2)) -> cons(X1, X2) a__dbl(X) -> dbl(X) a__dbls(X) -> dbls(X) a__sel(X1, X2) -> sel(X1, X2) a__indx(X1, X2) -> indx(X1, X2) a__from(X) -> from(X) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(dbl(x_1)) -> dbl(encArg(x_1)) encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(dbls(x_1)) -> dbls(encArg(x_1)) encArg(sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(indx(x_1, x_2)) -> indx(encArg(x_1), encArg(x_2)) encArg(from(x_1)) -> from(encArg(x_1)) encArg(cons_a__dbl(x_1)) -> a__dbl(encArg(x_1)) encArg(cons_a__dbls(x_1)) -> a__dbls(encArg(x_1)) encArg(cons_a__sel(x_1, x_2)) -> a__sel(encArg(x_1), encArg(x_2)) encArg(cons_a__indx(x_1, x_2)) -> a__indx(encArg(x_1), encArg(x_2)) encArg(cons_a__from(x_1)) -> a__from(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__dbl(x_1) -> a__dbl(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_dbl(x_1) -> dbl(encArg(x_1)) encode_a__dbls(x_1) -> a__dbls(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_dbls(x_1) -> dbls(encArg(x_1)) encode_a__sel(x_1, x_2) -> a__sel(encArg(x_1), encArg(x_2)) encode_mark(x_1) -> mark(encArg(x_1)) encode_a__indx(x_1, x_2) -> a__indx(encArg(x_1), encArg(x_2)) encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2)) encode_indx(x_1, x_2) -> indx(encArg(x_1), encArg(x_2)) encode_a__from(x_1) -> a__from(encArg(x_1)) encode_from(x_1) -> from(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: a__dbl(0) -> 0 a__dbl(s(X)) -> s(s(dbl(X))) a__dbls(nil) -> nil a__dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y)) a__sel(0, cons(X, Y)) -> mark(X) a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z)) a__indx(nil, X) -> nil a__indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z)) a__from(X) -> cons(X, from(s(X))) mark(dbl(X)) -> a__dbl(mark(X)) mark(dbls(X)) -> a__dbls(mark(X)) mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) mark(indx(X1, X2)) -> a__indx(mark(X1), X2) mark(from(X)) -> a__from(X) mark(0) -> 0 mark(s(X)) -> s(X) mark(nil) -> nil mark(cons(X1, X2)) -> cons(X1, X2) a__dbl(X) -> dbl(X) a__dbls(X) -> dbls(X) a__sel(X1, X2) -> sel(X1, X2) a__indx(X1, X2) -> indx(X1, X2) a__from(X) -> from(X) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(dbl(x_1)) -> dbl(encArg(x_1)) encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(dbls(x_1)) -> dbls(encArg(x_1)) encArg(sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(indx(x_1, x_2)) -> indx(encArg(x_1), encArg(x_2)) encArg(from(x_1)) -> from(encArg(x_1)) encArg(cons_a__dbl(x_1)) -> a__dbl(encArg(x_1)) encArg(cons_a__dbls(x_1)) -> a__dbls(encArg(x_1)) encArg(cons_a__sel(x_1, x_2)) -> a__sel(encArg(x_1), encArg(x_2)) encArg(cons_a__indx(x_1, x_2)) -> a__indx(encArg(x_1), encArg(x_2)) encArg(cons_a__from(x_1)) -> a__from(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__dbl(x_1) -> a__dbl(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_dbl(x_1) -> dbl(encArg(x_1)) encode_a__dbls(x_1) -> a__dbls(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_dbls(x_1) -> dbls(encArg(x_1)) encode_a__sel(x_1, x_2) -> a__sel(encArg(x_1), encArg(x_2)) encode_mark(x_1) -> mark(encArg(x_1)) encode_a__indx(x_1, x_2) -> a__indx(encArg(x_1), encArg(x_2)) encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2)) encode_indx(x_1, x_2) -> indx(encArg(x_1), encArg(x_2)) encode_a__from(x_1) -> a__from(encArg(x_1)) encode_from(x_1) -> from(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 mark(indx(X1, X2)) ->^+ a__indx(mark(X1), X2) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [X1 / indx(X1, X2)]. 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: a__dbl(0) -> 0 a__dbl(s(X)) -> s(s(dbl(X))) a__dbls(nil) -> nil a__dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y)) a__sel(0, cons(X, Y)) -> mark(X) a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z)) a__indx(nil, X) -> nil a__indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z)) a__from(X) -> cons(X, from(s(X))) mark(dbl(X)) -> a__dbl(mark(X)) mark(dbls(X)) -> a__dbls(mark(X)) mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) mark(indx(X1, X2)) -> a__indx(mark(X1), X2) mark(from(X)) -> a__from(X) mark(0) -> 0 mark(s(X)) -> s(X) mark(nil) -> nil mark(cons(X1, X2)) -> cons(X1, X2) a__dbl(X) -> dbl(X) a__dbls(X) -> dbls(X) a__sel(X1, X2) -> sel(X1, X2) a__indx(X1, X2) -> indx(X1, X2) a__from(X) -> from(X) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(dbl(x_1)) -> dbl(encArg(x_1)) encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(dbls(x_1)) -> dbls(encArg(x_1)) encArg(sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(indx(x_1, x_2)) -> indx(encArg(x_1), encArg(x_2)) encArg(from(x_1)) -> from(encArg(x_1)) encArg(cons_a__dbl(x_1)) -> a__dbl(encArg(x_1)) encArg(cons_a__dbls(x_1)) -> a__dbls(encArg(x_1)) encArg(cons_a__sel(x_1, x_2)) -> a__sel(encArg(x_1), encArg(x_2)) encArg(cons_a__indx(x_1, x_2)) -> a__indx(encArg(x_1), encArg(x_2)) encArg(cons_a__from(x_1)) -> a__from(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__dbl(x_1) -> a__dbl(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_dbl(x_1) -> dbl(encArg(x_1)) encode_a__dbls(x_1) -> a__dbls(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_dbls(x_1) -> dbls(encArg(x_1)) encode_a__sel(x_1, x_2) -> a__sel(encArg(x_1), encArg(x_2)) encode_mark(x_1) -> mark(encArg(x_1)) encode_a__indx(x_1, x_2) -> a__indx(encArg(x_1), encArg(x_2)) encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2)) encode_indx(x_1, x_2) -> indx(encArg(x_1), encArg(x_2)) encode_a__from(x_1) -> a__from(encArg(x_1)) encode_from(x_1) -> from(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: a__dbl(0) -> 0 a__dbl(s(X)) -> s(s(dbl(X))) a__dbls(nil) -> nil a__dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y)) a__sel(0, cons(X, Y)) -> mark(X) a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z)) a__indx(nil, X) -> nil a__indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z)) a__from(X) -> cons(X, from(s(X))) mark(dbl(X)) -> a__dbl(mark(X)) mark(dbls(X)) -> a__dbls(mark(X)) mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) mark(indx(X1, X2)) -> a__indx(mark(X1), X2) mark(from(X)) -> a__from(X) mark(0) -> 0 mark(s(X)) -> s(X) mark(nil) -> nil mark(cons(X1, X2)) -> cons(X1, X2) a__dbl(X) -> dbl(X) a__dbls(X) -> dbls(X) a__sel(X1, X2) -> sel(X1, X2) a__indx(X1, X2) -> indx(X1, X2) a__from(X) -> from(X) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(dbl(x_1)) -> dbl(encArg(x_1)) encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(dbls(x_1)) -> dbls(encArg(x_1)) encArg(sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(indx(x_1, x_2)) -> indx(encArg(x_1), encArg(x_2)) encArg(from(x_1)) -> from(encArg(x_1)) encArg(cons_a__dbl(x_1)) -> a__dbl(encArg(x_1)) encArg(cons_a__dbls(x_1)) -> a__dbls(encArg(x_1)) encArg(cons_a__sel(x_1, x_2)) -> a__sel(encArg(x_1), encArg(x_2)) encArg(cons_a__indx(x_1, x_2)) -> a__indx(encArg(x_1), encArg(x_2)) encArg(cons_a__from(x_1)) -> a__from(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__dbl(x_1) -> a__dbl(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_dbl(x_1) -> dbl(encArg(x_1)) encode_a__dbls(x_1) -> a__dbls(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_dbls(x_1) -> dbls(encArg(x_1)) encode_a__sel(x_1, x_2) -> a__sel(encArg(x_1), encArg(x_2)) encode_mark(x_1) -> mark(encArg(x_1)) encode_a__indx(x_1, x_2) -> a__indx(encArg(x_1), encArg(x_2)) encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2)) encode_indx(x_1, x_2) -> indx(encArg(x_1), encArg(x_2)) encode_a__from(x_1) -> a__from(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) Rewrite Strategy: INNERMOST