/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 537 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: empty(nil) -> true empty(cons(x, y)) -> false tail(nil) -> nil tail(cons(x, y)) -> y head(cons(x, y)) -> x zero(0) -> true zero(s(x)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(x))) -> s(p(s(x))) intlist(x) -> if_intlist(empty(x), x) if_intlist(true, x) -> nil if_intlist(false, x) -> cons(s(head(x)), intlist(tail(x))) int(x, y) -> if_int(zero(x), zero(y), x, y) if_int(true, b, x, y) -> if1(b, x, y) if_int(false, b, x, y) -> if2(b, x, y) if1(true, x, y) -> cons(0, nil) if1(false, x, y) -> cons(0, int(s(0), y)) if2(true, x, y) -> nil if2(false, x, y) -> intlist(int(p(x), p(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(nil) -> nil encArg(true) -> true encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(false) -> false encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_empty(x_1)) -> empty(encArg(x_1)) encArg(cons_tail(x_1)) -> tail(encArg(x_1)) encArg(cons_head(x_1)) -> head(encArg(x_1)) encArg(cons_zero(x_1)) -> zero(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_intlist(x_1)) -> intlist(encArg(x_1)) encArg(cons_if_intlist(x_1, x_2)) -> if_intlist(encArg(x_1), encArg(x_2)) encArg(cons_int(x_1, x_2)) -> int(encArg(x_1), encArg(x_2)) encArg(cons_if_int(x_1, x_2, x_3, x_4)) -> if_int(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_if1(x_1, x_2, x_3)) -> if1(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_if2(x_1, x_2, x_3)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_empty(x_1) -> empty(encArg(x_1)) encode_nil -> nil encode_true -> true encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_false -> false encode_tail(x_1) -> tail(encArg(x_1)) encode_head(x_1) -> head(encArg(x_1)) encode_zero(x_1) -> zero(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_intlist(x_1) -> intlist(encArg(x_1)) encode_if_intlist(x_1, x_2) -> if_intlist(encArg(x_1), encArg(x_2)) encode_int(x_1, x_2) -> int(encArg(x_1), encArg(x_2)) encode_if_int(x_1, x_2, x_3, x_4) -> if_int(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_if1(x_1, x_2, x_3) -> if1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_if2(x_1, x_2, x_3) -> if2(encArg(x_1), encArg(x_2), encArg(x_3)) ---------------------------------------- (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: empty(nil) -> true empty(cons(x, y)) -> false tail(nil) -> nil tail(cons(x, y)) -> y head(cons(x, y)) -> x zero(0) -> true zero(s(x)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(x))) -> s(p(s(x))) intlist(x) -> if_intlist(empty(x), x) if_intlist(true, x) -> nil if_intlist(false, x) -> cons(s(head(x)), intlist(tail(x))) int(x, y) -> if_int(zero(x), zero(y), x, y) if_int(true, b, x, y) -> if1(b, x, y) if_int(false, b, x, y) -> if2(b, x, y) if1(true, x, y) -> cons(0, nil) if1(false, x, y) -> cons(0, int(s(0), y)) if2(true, x, y) -> nil if2(false, x, y) -> intlist(int(p(x), p(y))) The (relative) TRS S consists of the following rules: encArg(nil) -> nil encArg(true) -> true encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(false) -> false encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_empty(x_1)) -> empty(encArg(x_1)) encArg(cons_tail(x_1)) -> tail(encArg(x_1)) encArg(cons_head(x_1)) -> head(encArg(x_1)) encArg(cons_zero(x_1)) -> zero(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_intlist(x_1)) -> intlist(encArg(x_1)) encArg(cons_if_intlist(x_1, x_2)) -> if_intlist(encArg(x_1), encArg(x_2)) encArg(cons_int(x_1, x_2)) -> int(encArg(x_1), encArg(x_2)) encArg(cons_if_int(x_1, x_2, x_3, x_4)) -> if_int(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_if1(x_1, x_2, x_3)) -> if1(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_if2(x_1, x_2, x_3)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_empty(x_1) -> empty(encArg(x_1)) encode_nil -> nil encode_true -> true encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_false -> false encode_tail(x_1) -> tail(encArg(x_1)) encode_head(x_1) -> head(encArg(x_1)) encode_zero(x_1) -> zero(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_intlist(x_1) -> intlist(encArg(x_1)) encode_if_intlist(x_1, x_2) -> if_intlist(encArg(x_1), encArg(x_2)) encode_int(x_1, x_2) -> int(encArg(x_1), encArg(x_2)) encode_if_int(x_1, x_2, x_3, x_4) -> if_int(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_if1(x_1, x_2, x_3) -> if1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_if2(x_1, x_2, x_3) -> if2(encArg(x_1), encArg(x_2), encArg(x_3)) 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: empty(nil) -> true empty(cons(x, y)) -> false tail(nil) -> nil tail(cons(x, y)) -> y head(cons(x, y)) -> x zero(0) -> true zero(s(x)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(x))) -> s(p(s(x))) intlist(x) -> if_intlist(empty(x), x) if_intlist(true, x) -> nil if_intlist(false, x) -> cons(s(head(x)), intlist(tail(x))) int(x, y) -> if_int(zero(x), zero(y), x, y) if_int(true, b, x, y) -> if1(b, x, y) if_int(false, b, x, y) -> if2(b, x, y) if1(true, x, y) -> cons(0, nil) if1(false, x, y) -> cons(0, int(s(0), y)) if2(true, x, y) -> nil if2(false, x, y) -> intlist(int(p(x), p(y))) The (relative) TRS S consists of the following rules: encArg(nil) -> nil encArg(true) -> true encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(false) -> false encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_empty(x_1)) -> empty(encArg(x_1)) encArg(cons_tail(x_1)) -> tail(encArg(x_1)) encArg(cons_head(x_1)) -> head(encArg(x_1)) encArg(cons_zero(x_1)) -> zero(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_intlist(x_1)) -> intlist(encArg(x_1)) encArg(cons_if_intlist(x_1, x_2)) -> if_intlist(encArg(x_1), encArg(x_2)) encArg(cons_int(x_1, x_2)) -> int(encArg(x_1), encArg(x_2)) encArg(cons_if_int(x_1, x_2, x_3, x_4)) -> if_int(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_if1(x_1, x_2, x_3)) -> if1(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_if2(x_1, x_2, x_3)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_empty(x_1) -> empty(encArg(x_1)) encode_nil -> nil encode_true -> true encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_false -> false encode_tail(x_1) -> tail(encArg(x_1)) encode_head(x_1) -> head(encArg(x_1)) encode_zero(x_1) -> zero(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_intlist(x_1) -> intlist(encArg(x_1)) encode_if_intlist(x_1, x_2) -> if_intlist(encArg(x_1), encArg(x_2)) encode_int(x_1, x_2) -> int(encArg(x_1), encArg(x_2)) encode_if_int(x_1, x_2, x_3, x_4) -> if_int(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_if1(x_1, x_2, x_3) -> if1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_if2(x_1, x_2, x_3) -> if2(encArg(x_1), encArg(x_2), encArg(x_3)) 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: empty(nil) -> true empty(cons(x, y)) -> false tail(nil) -> nil tail(cons(x, y)) -> y head(cons(x, y)) -> x zero(0) -> true zero(s(x)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(x))) -> s(p(s(x))) intlist(x) -> if_intlist(empty(x), x) if_intlist(true, x) -> nil if_intlist(false, x) -> cons(s(head(x)), intlist(tail(x))) int(x, y) -> if_int(zero(x), zero(y), x, y) if_int(true, b, x, y) -> if1(b, x, y) if_int(false, b, x, y) -> if2(b, x, y) if1(true, x, y) -> cons(0, nil) if1(false, x, y) -> cons(0, int(s(0), y)) if2(true, x, y) -> nil if2(false, x, y) -> intlist(int(p(x), p(y))) The (relative) TRS S consists of the following rules: encArg(nil) -> nil encArg(true) -> true encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(false) -> false encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_empty(x_1)) -> empty(encArg(x_1)) encArg(cons_tail(x_1)) -> tail(encArg(x_1)) encArg(cons_head(x_1)) -> head(encArg(x_1)) encArg(cons_zero(x_1)) -> zero(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_intlist(x_1)) -> intlist(encArg(x_1)) encArg(cons_if_intlist(x_1, x_2)) -> if_intlist(encArg(x_1), encArg(x_2)) encArg(cons_int(x_1, x_2)) -> int(encArg(x_1), encArg(x_2)) encArg(cons_if_int(x_1, x_2, x_3, x_4)) -> if_int(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_if1(x_1, x_2, x_3)) -> if1(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_if2(x_1, x_2, x_3)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_empty(x_1) -> empty(encArg(x_1)) encode_nil -> nil encode_true -> true encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_false -> false encode_tail(x_1) -> tail(encArg(x_1)) encode_head(x_1) -> head(encArg(x_1)) encode_zero(x_1) -> zero(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_intlist(x_1) -> intlist(encArg(x_1)) encode_if_intlist(x_1, x_2) -> if_intlist(encArg(x_1), encArg(x_2)) encode_int(x_1, x_2) -> int(encArg(x_1), encArg(x_2)) encode_if_int(x_1, x_2, x_3, x_4) -> if_int(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_if1(x_1, x_2, x_3) -> if1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_if2(x_1, x_2, x_3) -> if2(encArg(x_1), encArg(x_2), encArg(x_3)) 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 p(s(s(x))) ->^+ s(p(s(x))) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [x / s(x)]. 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: empty(nil) -> true empty(cons(x, y)) -> false tail(nil) -> nil tail(cons(x, y)) -> y head(cons(x, y)) -> x zero(0) -> true zero(s(x)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(x))) -> s(p(s(x))) intlist(x) -> if_intlist(empty(x), x) if_intlist(true, x) -> nil if_intlist(false, x) -> cons(s(head(x)), intlist(tail(x))) int(x, y) -> if_int(zero(x), zero(y), x, y) if_int(true, b, x, y) -> if1(b, x, y) if_int(false, b, x, y) -> if2(b, x, y) if1(true, x, y) -> cons(0, nil) if1(false, x, y) -> cons(0, int(s(0), y)) if2(true, x, y) -> nil if2(false, x, y) -> intlist(int(p(x), p(y))) The (relative) TRS S consists of the following rules: encArg(nil) -> nil encArg(true) -> true encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(false) -> false encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_empty(x_1)) -> empty(encArg(x_1)) encArg(cons_tail(x_1)) -> tail(encArg(x_1)) encArg(cons_head(x_1)) -> head(encArg(x_1)) encArg(cons_zero(x_1)) -> zero(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_intlist(x_1)) -> intlist(encArg(x_1)) encArg(cons_if_intlist(x_1, x_2)) -> if_intlist(encArg(x_1), encArg(x_2)) encArg(cons_int(x_1, x_2)) -> int(encArg(x_1), encArg(x_2)) encArg(cons_if_int(x_1, x_2, x_3, x_4)) -> if_int(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_if1(x_1, x_2, x_3)) -> if1(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_if2(x_1, x_2, x_3)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_empty(x_1) -> empty(encArg(x_1)) encode_nil -> nil encode_true -> true encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_false -> false encode_tail(x_1) -> tail(encArg(x_1)) encode_head(x_1) -> head(encArg(x_1)) encode_zero(x_1) -> zero(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_intlist(x_1) -> intlist(encArg(x_1)) encode_if_intlist(x_1, x_2) -> if_intlist(encArg(x_1), encArg(x_2)) encode_int(x_1, x_2) -> int(encArg(x_1), encArg(x_2)) encode_if_int(x_1, x_2, x_3, x_4) -> if_int(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_if1(x_1, x_2, x_3) -> if1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_if2(x_1, x_2, x_3) -> if2(encArg(x_1), encArg(x_2), encArg(x_3)) 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: empty(nil) -> true empty(cons(x, y)) -> false tail(nil) -> nil tail(cons(x, y)) -> y head(cons(x, y)) -> x zero(0) -> true zero(s(x)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(x))) -> s(p(s(x))) intlist(x) -> if_intlist(empty(x), x) if_intlist(true, x) -> nil if_intlist(false, x) -> cons(s(head(x)), intlist(tail(x))) int(x, y) -> if_int(zero(x), zero(y), x, y) if_int(true, b, x, y) -> if1(b, x, y) if_int(false, b, x, y) -> if2(b, x, y) if1(true, x, y) -> cons(0, nil) if1(false, x, y) -> cons(0, int(s(0), y)) if2(true, x, y) -> nil if2(false, x, y) -> intlist(int(p(x), p(y))) The (relative) TRS S consists of the following rules: encArg(nil) -> nil encArg(true) -> true encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(false) -> false encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_empty(x_1)) -> empty(encArg(x_1)) encArg(cons_tail(x_1)) -> tail(encArg(x_1)) encArg(cons_head(x_1)) -> head(encArg(x_1)) encArg(cons_zero(x_1)) -> zero(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_intlist(x_1)) -> intlist(encArg(x_1)) encArg(cons_if_intlist(x_1, x_2)) -> if_intlist(encArg(x_1), encArg(x_2)) encArg(cons_int(x_1, x_2)) -> int(encArg(x_1), encArg(x_2)) encArg(cons_if_int(x_1, x_2, x_3, x_4)) -> if_int(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_if1(x_1, x_2, x_3)) -> if1(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_if2(x_1, x_2, x_3)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_empty(x_1) -> empty(encArg(x_1)) encode_nil -> nil encode_true -> true encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_false -> false encode_tail(x_1) -> tail(encArg(x_1)) encode_head(x_1) -> head(encArg(x_1)) encode_zero(x_1) -> zero(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_intlist(x_1) -> intlist(encArg(x_1)) encode_if_intlist(x_1, x_2) -> if_intlist(encArg(x_1), encArg(x_2)) encode_int(x_1, x_2) -> int(encArg(x_1), encArg(x_2)) encode_if_int(x_1, x_2, x_3, x_4) -> if_int(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_if1(x_1, x_2, x_3) -> if1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_if2(x_1, x_2, x_3) -> if2(encArg(x_1), encArg(x_2), encArg(x_3)) Rewrite Strategy: INNERMOST