/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(NON_POLY, ?) 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(INF, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 335 ms] (4) CpxRelTRS (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (6) TRS for Loop Detection (7) InfiniteLowerBoundProof [FINISHED, 65 ms] (8) BOUNDS(INF, INF) ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(INF, INF). The TRS R consists of the following rules: app(app(app(if, true), xs), ys) -> xs app(app(app(if, false), xs), ys) -> ys app(app(lt, app(s, x)), app(s, y)) -> app(app(lt, x), y) app(app(lt, 0), app(s, y)) -> true app(app(lt, y), 0) -> false app(app(eq, x), x) -> true app(app(eq, app(s, x)), 0) -> false app(app(eq, 0), app(s, x)) -> false app(app(merge, xs), nil) -> xs app(app(merge, nil), ys) -> ys app(app(merge, app(app(cons, x), xs)), app(app(cons, y), ys)) -> app(app(app(if, app(app(lt, x), y)), app(app(cons, x), app(app(merge, xs), app(app(cons, y), ys)))), app(app(app(if, app(app(eq, x), y)), app(app(cons, x), app(app(merge, xs), ys))), app(app(cons, y), app(app(merge, app(app(cons, x), xs)), ys)))) app(app(map, f), nil) -> nil app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs)) app(app(mult, 0), x) -> 0 app(app(mult, app(s, x)), y) -> app(app(plus, y), app(app(mult, x), y)) app(app(plus, 0), x) -> 0 app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y)) list1 -> app(app(map, app(mult, app(s, app(s, 0)))), hamming) list2 -> app(app(map, app(mult, app(s, app(s, app(s, 0))))), hamming) list3 -> app(app(map, app(mult, app(s, app(s, app(s, app(s, app(s, 0))))))), hamming) hamming -> app(app(cons, app(s, 0)), app(app(merge, list1), app(app(merge, list2), list3))) 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(if) -> if encArg(true) -> true encArg(false) -> false encArg(lt) -> lt encArg(s) -> s encArg(0) -> 0 encArg(eq) -> eq encArg(merge) -> merge encArg(nil) -> nil encArg(cons) -> cons encArg(map) -> map encArg(mult) -> mult encArg(plus) -> plus encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encArg(cons_list1) -> list1 encArg(cons_list2) -> list2 encArg(cons_list3) -> list3 encArg(cons_hamming) -> hamming encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_if -> if encode_true -> true encode_false -> false encode_lt -> lt encode_s -> s encode_0 -> 0 encode_eq -> eq encode_merge -> merge encode_nil -> nil encode_cons -> cons encode_map -> map encode_mult -> mult encode_plus -> plus encode_list1 -> list1 encode_hamming -> hamming encode_list2 -> list2 encode_list3 -> list3 ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(INF, INF). The TRS R consists of the following rules: app(app(app(if, true), xs), ys) -> xs app(app(app(if, false), xs), ys) -> ys app(app(lt, app(s, x)), app(s, y)) -> app(app(lt, x), y) app(app(lt, 0), app(s, y)) -> true app(app(lt, y), 0) -> false app(app(eq, x), x) -> true app(app(eq, app(s, x)), 0) -> false app(app(eq, 0), app(s, x)) -> false app(app(merge, xs), nil) -> xs app(app(merge, nil), ys) -> ys app(app(merge, app(app(cons, x), xs)), app(app(cons, y), ys)) -> app(app(app(if, app(app(lt, x), y)), app(app(cons, x), app(app(merge, xs), app(app(cons, y), ys)))), app(app(app(if, app(app(eq, x), y)), app(app(cons, x), app(app(merge, xs), ys))), app(app(cons, y), app(app(merge, app(app(cons, x), xs)), ys)))) app(app(map, f), nil) -> nil app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs)) app(app(mult, 0), x) -> 0 app(app(mult, app(s, x)), y) -> app(app(plus, y), app(app(mult, x), y)) app(app(plus, 0), x) -> 0 app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y)) list1 -> app(app(map, app(mult, app(s, app(s, 0)))), hamming) list2 -> app(app(map, app(mult, app(s, app(s, app(s, 0))))), hamming) list3 -> app(app(map, app(mult, app(s, app(s, app(s, app(s, app(s, 0))))))), hamming) hamming -> app(app(cons, app(s, 0)), app(app(merge, list1), app(app(merge, list2), list3))) The (relative) TRS S consists of the following rules: encArg(if) -> if encArg(true) -> true encArg(false) -> false encArg(lt) -> lt encArg(s) -> s encArg(0) -> 0 encArg(eq) -> eq encArg(merge) -> merge encArg(nil) -> nil encArg(cons) -> cons encArg(map) -> map encArg(mult) -> mult encArg(plus) -> plus encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encArg(cons_list1) -> list1 encArg(cons_list2) -> list2 encArg(cons_list3) -> list3 encArg(cons_hamming) -> hamming encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_if -> if encode_true -> true encode_false -> false encode_lt -> lt encode_s -> s encode_0 -> 0 encode_eq -> eq encode_merge -> merge encode_nil -> nil encode_cons -> cons encode_map -> map encode_mult -> mult encode_plus -> plus encode_list1 -> list1 encode_hamming -> hamming encode_list2 -> list2 encode_list3 -> list3 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(INF, INF). The TRS R consists of the following rules: app(app(app(if, true), xs), ys) -> xs app(app(app(if, false), xs), ys) -> ys app(app(lt, app(s, x)), app(s, y)) -> app(app(lt, x), y) app(app(lt, 0), app(s, y)) -> true app(app(lt, y), 0) -> false app(app(eq, x), x) -> true app(app(eq, app(s, x)), 0) -> false app(app(eq, 0), app(s, x)) -> false app(app(merge, xs), nil) -> xs app(app(merge, nil), ys) -> ys app(app(merge, app(app(cons, x), xs)), app(app(cons, y), ys)) -> app(app(app(if, app(app(lt, x), y)), app(app(cons, x), app(app(merge, xs), app(app(cons, y), ys)))), app(app(app(if, app(app(eq, x), y)), app(app(cons, x), app(app(merge, xs), ys))), app(app(cons, y), app(app(merge, app(app(cons, x), xs)), ys)))) app(app(map, f), nil) -> nil app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs)) app(app(mult, 0), x) -> 0 app(app(mult, app(s, x)), y) -> app(app(plus, y), app(app(mult, x), y)) app(app(plus, 0), x) -> 0 app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y)) list1 -> app(app(map, app(mult, app(s, app(s, 0)))), hamming) list2 -> app(app(map, app(mult, app(s, app(s, app(s, 0))))), hamming) list3 -> app(app(map, app(mult, app(s, app(s, app(s, app(s, app(s, 0))))))), hamming) hamming -> app(app(cons, app(s, 0)), app(app(merge, list1), app(app(merge, list2), list3))) The (relative) TRS S consists of the following rules: encArg(if) -> if encArg(true) -> true encArg(false) -> false encArg(lt) -> lt encArg(s) -> s encArg(0) -> 0 encArg(eq) -> eq encArg(merge) -> merge encArg(nil) -> nil encArg(cons) -> cons encArg(map) -> map encArg(mult) -> mult encArg(plus) -> plus encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encArg(cons_list1) -> list1 encArg(cons_list2) -> list2 encArg(cons_list3) -> list3 encArg(cons_hamming) -> hamming encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_if -> if encode_true -> true encode_false -> false encode_lt -> lt encode_s -> s encode_0 -> 0 encode_eq -> eq encode_merge -> merge encode_nil -> nil encode_cons -> cons encode_map -> map encode_mult -> mult encode_plus -> plus encode_list1 -> list1 encode_hamming -> hamming encode_list2 -> list2 encode_list3 -> list3 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(INF, INF). The TRS R consists of the following rules: app(app(app(if, true), xs), ys) -> xs app(app(app(if, false), xs), ys) -> ys app(app(lt, app(s, x)), app(s, y)) -> app(app(lt, x), y) app(app(lt, 0), app(s, y)) -> true app(app(lt, y), 0) -> false app(app(eq, x), x) -> true app(app(eq, app(s, x)), 0) -> false app(app(eq, 0), app(s, x)) -> false app(app(merge, xs), nil) -> xs app(app(merge, nil), ys) -> ys app(app(merge, app(app(cons, x), xs)), app(app(cons, y), ys)) -> app(app(app(if, app(app(lt, x), y)), app(app(cons, x), app(app(merge, xs), app(app(cons, y), ys)))), app(app(app(if, app(app(eq, x), y)), app(app(cons, x), app(app(merge, xs), ys))), app(app(cons, y), app(app(merge, app(app(cons, x), xs)), ys)))) app(app(map, f), nil) -> nil app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs)) app(app(mult, 0), x) -> 0 app(app(mult, app(s, x)), y) -> app(app(plus, y), app(app(mult, x), y)) app(app(plus, 0), x) -> 0 app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y)) list1 -> app(app(map, app(mult, app(s, app(s, 0)))), hamming) list2 -> app(app(map, app(mult, app(s, app(s, app(s, 0))))), hamming) list3 -> app(app(map, app(mult, app(s, app(s, app(s, app(s, app(s, 0))))))), hamming) hamming -> app(app(cons, app(s, 0)), app(app(merge, list1), app(app(merge, list2), list3))) The (relative) TRS S consists of the following rules: encArg(if) -> if encArg(true) -> true encArg(false) -> false encArg(lt) -> lt encArg(s) -> s encArg(0) -> 0 encArg(eq) -> eq encArg(merge) -> merge encArg(nil) -> nil encArg(cons) -> cons encArg(map) -> map encArg(mult) -> mult encArg(plus) -> plus encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encArg(cons_list1) -> list1 encArg(cons_list2) -> list2 encArg(cons_list3) -> list3 encArg(cons_hamming) -> hamming encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_if -> if encode_true -> true encode_false -> false encode_lt -> lt encode_s -> s encode_0 -> 0 encode_eq -> eq encode_merge -> merge encode_nil -> nil encode_cons -> cons encode_map -> map encode_mult -> mult encode_plus -> plus encode_list1 -> list1 encode_hamming -> hamming encode_list2 -> list2 encode_list3 -> list3 Rewrite Strategy: INNERMOST ---------------------------------------- (7) InfiniteLowerBoundProof (FINISHED) The following loop proves infinite runtime complexity: The rewrite sequence list3 ->^+ app(app(map, app(mult, app(s, app(s, app(s, app(s, app(s, 0))))))), app(app(cons, app(s, 0)), app(app(merge, list1), app(app(merge, list2), list3)))) gives rise to a decreasing loop by considering the right hand sides subterm at position [1,1,1,1]. The pumping substitution is [ ]. The result substitution is [ ]. ---------------------------------------- (8) BOUNDS(INF, INF)