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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 506 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: le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) eq(0, 0) -> true eq(0, s(y)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) minsort(nil) -> nil minsort(cons(x, xs)) -> cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs)))) min(nil) -> 0 min(cons(x, nil)) -> x min(cons(x, cons(y, xs))) -> if1(le(x, y), x, y, xs) if1(true, x, y, xs) -> min(cons(x, xs)) if1(false, x, y, xs) -> min(cons(y, xs)) rm(x, nil) -> nil rm(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) if2(true, x, y, xs) -> rm(x, xs) if2(false, x, y, xs) -> cons(y, rm(x, xs)) 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(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2)) encArg(cons_minsort(x_1)) -> minsort(encArg(x_1)) encArg(cons_min(x_1)) -> min(encArg(x_1)) encArg(cons_if1(x_1, x_2, x_3, x_4)) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_rm(x_1, x_2)) -> rm(encArg(x_1), encArg(x_2)) encArg(cons_if2(x_1, x_2, x_3, x_4)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2)) encode_minsort(x_1) -> minsort(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_min(x_1) -> min(encArg(x_1)) encode_rm(x_1, x_2) -> rm(encArg(x_1), encArg(x_2)) encode_if1(x_1, x_2, x_3, x_4) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_if2(x_1, x_2, x_3, x_4) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) ---------------------------------------- (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: le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) eq(0, 0) -> true eq(0, s(y)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) minsort(nil) -> nil minsort(cons(x, xs)) -> cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs)))) min(nil) -> 0 min(cons(x, nil)) -> x min(cons(x, cons(y, xs))) -> if1(le(x, y), x, y, xs) if1(true, x, y, xs) -> min(cons(x, xs)) if1(false, x, y, xs) -> min(cons(y, xs)) rm(x, nil) -> nil rm(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) if2(true, x, y, xs) -> rm(x, xs) if2(false, x, y, xs) -> cons(y, rm(x, xs)) 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(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2)) encArg(cons_minsort(x_1)) -> minsort(encArg(x_1)) encArg(cons_min(x_1)) -> min(encArg(x_1)) encArg(cons_if1(x_1, x_2, x_3, x_4)) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_rm(x_1, x_2)) -> rm(encArg(x_1), encArg(x_2)) encArg(cons_if2(x_1, x_2, x_3, x_4)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2)) encode_minsort(x_1) -> minsort(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_min(x_1) -> min(encArg(x_1)) encode_rm(x_1, x_2) -> rm(encArg(x_1), encArg(x_2)) encode_if1(x_1, x_2, x_3, x_4) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_if2(x_1, x_2, x_3, x_4) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) 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: le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) eq(0, 0) -> true eq(0, s(y)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) minsort(nil) -> nil minsort(cons(x, xs)) -> cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs)))) min(nil) -> 0 min(cons(x, nil)) -> x min(cons(x, cons(y, xs))) -> if1(le(x, y), x, y, xs) if1(true, x, y, xs) -> min(cons(x, xs)) if1(false, x, y, xs) -> min(cons(y, xs)) rm(x, nil) -> nil rm(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) if2(true, x, y, xs) -> rm(x, xs) if2(false, x, y, xs) -> cons(y, rm(x, xs)) 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(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2)) encArg(cons_minsort(x_1)) -> minsort(encArg(x_1)) encArg(cons_min(x_1)) -> min(encArg(x_1)) encArg(cons_if1(x_1, x_2, x_3, x_4)) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_rm(x_1, x_2)) -> rm(encArg(x_1), encArg(x_2)) encArg(cons_if2(x_1, x_2, x_3, x_4)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2)) encode_minsort(x_1) -> minsort(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_min(x_1) -> min(encArg(x_1)) encode_rm(x_1, x_2) -> rm(encArg(x_1), encArg(x_2)) encode_if1(x_1, x_2, x_3, x_4) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_if2(x_1, x_2, x_3, x_4) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) 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: le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) eq(0, 0) -> true eq(0, s(y)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) minsort(nil) -> nil minsort(cons(x, xs)) -> cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs)))) min(nil) -> 0 min(cons(x, nil)) -> x min(cons(x, cons(y, xs))) -> if1(le(x, y), x, y, xs) if1(true, x, y, xs) -> min(cons(x, xs)) if1(false, x, y, xs) -> min(cons(y, xs)) rm(x, nil) -> nil rm(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) if2(true, x, y, xs) -> rm(x, xs) if2(false, x, y, xs) -> cons(y, rm(x, xs)) 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(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2)) encArg(cons_minsort(x_1)) -> minsort(encArg(x_1)) encArg(cons_min(x_1)) -> min(encArg(x_1)) encArg(cons_if1(x_1, x_2, x_3, x_4)) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_rm(x_1, x_2)) -> rm(encArg(x_1), encArg(x_2)) encArg(cons_if2(x_1, x_2, x_3, x_4)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2)) encode_minsort(x_1) -> minsort(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_min(x_1) -> min(encArg(x_1)) encode_rm(x_1, x_2) -> rm(encArg(x_1), encArg(x_2)) encode_if1(x_1, x_2, x_3, x_4) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_if2(x_1, x_2, x_3, x_4) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) 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 le(s(x), s(y)) ->^+ le(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: le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) eq(0, 0) -> true eq(0, s(y)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) minsort(nil) -> nil minsort(cons(x, xs)) -> cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs)))) min(nil) -> 0 min(cons(x, nil)) -> x min(cons(x, cons(y, xs))) -> if1(le(x, y), x, y, xs) if1(true, x, y, xs) -> min(cons(x, xs)) if1(false, x, y, xs) -> min(cons(y, xs)) rm(x, nil) -> nil rm(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) if2(true, x, y, xs) -> rm(x, xs) if2(false, x, y, xs) -> cons(y, rm(x, xs)) 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(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2)) encArg(cons_minsort(x_1)) -> minsort(encArg(x_1)) encArg(cons_min(x_1)) -> min(encArg(x_1)) encArg(cons_if1(x_1, x_2, x_3, x_4)) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_rm(x_1, x_2)) -> rm(encArg(x_1), encArg(x_2)) encArg(cons_if2(x_1, x_2, x_3, x_4)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2)) encode_minsort(x_1) -> minsort(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_min(x_1) -> min(encArg(x_1)) encode_rm(x_1, x_2) -> rm(encArg(x_1), encArg(x_2)) encode_if1(x_1, x_2, x_3, x_4) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_if2(x_1, x_2, x_3, x_4) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) 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: le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) eq(0, 0) -> true eq(0, s(y)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) minsort(nil) -> nil minsort(cons(x, xs)) -> cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs)))) min(nil) -> 0 min(cons(x, nil)) -> x min(cons(x, cons(y, xs))) -> if1(le(x, y), x, y, xs) if1(true, x, y, xs) -> min(cons(x, xs)) if1(false, x, y, xs) -> min(cons(y, xs)) rm(x, nil) -> nil rm(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs) if2(true, x, y, xs) -> rm(x, xs) if2(false, x, y, xs) -> cons(y, rm(x, xs)) 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(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2)) encArg(cons_minsort(x_1)) -> minsort(encArg(x_1)) encArg(cons_min(x_1)) -> min(encArg(x_1)) encArg(cons_if1(x_1, x_2, x_3, x_4)) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_rm(x_1, x_2)) -> rm(encArg(x_1), encArg(x_2)) encArg(cons_if2(x_1, x_2, x_3, x_4)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2)) encode_minsort(x_1) -> minsort(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_min(x_1) -> min(encArg(x_1)) encode_rm(x_1, x_2) -> rm(encArg(x_1), encArg(x_2)) encode_if1(x_1, x_2, x_3, x_4) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_if2(x_1, x_2, x_3, x_4) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) Rewrite Strategy: INNERMOST