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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 487 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: eq(0, 0) -> true eq(0, s(y)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) lt(0, s(y)) -> true lt(x, 0) -> false lt(s(x), s(y)) -> lt(x, y) bin2s(nil) -> 0 bin2s(cons(x, xs)) -> bin2ss(x, xs) bin2ss(x, nil) -> x bin2ss(x, cons(0, xs)) -> bin2ss(double(x), xs) bin2ss(x, cons(1, xs)) -> bin2ss(s(double(x)), xs) half(0) -> 0 half(s(0)) -> 0 half(s(s(x))) -> s(half(x)) log(0) -> 0 log(s(0)) -> 0 log(s(s(x))) -> s(log(half(s(s(x))))) more(nil) -> nil more(cons(xs, ys)) -> cons(cons(0, xs), cons(cons(1, xs), cons(xs, ys))) s2bin(x) -> s2bin1(x, 0, cons(nil, nil)) s2bin1(x, y, lists) -> if1(lt(y, log(x)), x, y, lists) if1(true, x, y, lists) -> s2bin1(x, s(y), more(lists)) if1(false, x, y, lists) -> s2bin2(x, lists) s2bin2(x, nil) -> bug_list_not s2bin2(x, cons(xs, ys)) -> if2(eq(x, bin2s(xs)), x, xs, ys) if2(true, x, xs, ys) -> xs if2(false, x, xs, ys) -> s2bin2(x, ys) 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(double(x_1)) -> double(encArg(x_1)) encArg(1) -> 1 encArg(bug_list_not) -> bug_list_not encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2)) encArg(cons_lt(x_1, x_2)) -> lt(encArg(x_1), encArg(x_2)) encArg(cons_bin2s(x_1)) -> bin2s(encArg(x_1)) encArg(cons_bin2ss(x_1, x_2)) -> bin2ss(encArg(x_1), encArg(x_2)) encArg(cons_half(x_1)) -> half(encArg(x_1)) encArg(cons_log(x_1)) -> log(encArg(x_1)) encArg(cons_more(x_1)) -> more(encArg(x_1)) encArg(cons_s2bin(x_1)) -> s2bin(encArg(x_1)) encArg(cons_s2bin1(x_1, x_2, x_3)) -> s2bin1(encArg(x_1), encArg(x_2), encArg(x_3)) 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_s2bin2(x_1, x_2)) -> s2bin2(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_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_lt(x_1, x_2) -> lt(encArg(x_1), encArg(x_2)) encode_bin2s(x_1) -> bin2s(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_bin2ss(x_1, x_2) -> bin2ss(encArg(x_1), encArg(x_2)) encode_double(x_1) -> double(encArg(x_1)) encode_1 -> 1 encode_half(x_1) -> half(encArg(x_1)) encode_log(x_1) -> log(encArg(x_1)) encode_more(x_1) -> more(encArg(x_1)) encode_s2bin(x_1) -> s2bin(encArg(x_1)) encode_s2bin1(x_1, x_2, x_3) -> s2bin1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_if1(x_1, x_2, x_3, x_4) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_s2bin2(x_1, x_2) -> s2bin2(encArg(x_1), encArg(x_2)) encode_bug_list_not -> bug_list_not 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: eq(0, 0) -> true eq(0, s(y)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) lt(0, s(y)) -> true lt(x, 0) -> false lt(s(x), s(y)) -> lt(x, y) bin2s(nil) -> 0 bin2s(cons(x, xs)) -> bin2ss(x, xs) bin2ss(x, nil) -> x bin2ss(x, cons(0, xs)) -> bin2ss(double(x), xs) bin2ss(x, cons(1, xs)) -> bin2ss(s(double(x)), xs) half(0) -> 0 half(s(0)) -> 0 half(s(s(x))) -> s(half(x)) log(0) -> 0 log(s(0)) -> 0 log(s(s(x))) -> s(log(half(s(s(x))))) more(nil) -> nil more(cons(xs, ys)) -> cons(cons(0, xs), cons(cons(1, xs), cons(xs, ys))) s2bin(x) -> s2bin1(x, 0, cons(nil, nil)) s2bin1(x, y, lists) -> if1(lt(y, log(x)), x, y, lists) if1(true, x, y, lists) -> s2bin1(x, s(y), more(lists)) if1(false, x, y, lists) -> s2bin2(x, lists) s2bin2(x, nil) -> bug_list_not s2bin2(x, cons(xs, ys)) -> if2(eq(x, bin2s(xs)), x, xs, ys) if2(true, x, xs, ys) -> xs if2(false, x, xs, ys) -> s2bin2(x, ys) 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(double(x_1)) -> double(encArg(x_1)) encArg(1) -> 1 encArg(bug_list_not) -> bug_list_not encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2)) encArg(cons_lt(x_1, x_2)) -> lt(encArg(x_1), encArg(x_2)) encArg(cons_bin2s(x_1)) -> bin2s(encArg(x_1)) encArg(cons_bin2ss(x_1, x_2)) -> bin2ss(encArg(x_1), encArg(x_2)) encArg(cons_half(x_1)) -> half(encArg(x_1)) encArg(cons_log(x_1)) -> log(encArg(x_1)) encArg(cons_more(x_1)) -> more(encArg(x_1)) encArg(cons_s2bin(x_1)) -> s2bin(encArg(x_1)) encArg(cons_s2bin1(x_1, x_2, x_3)) -> s2bin1(encArg(x_1), encArg(x_2), encArg(x_3)) 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_s2bin2(x_1, x_2)) -> s2bin2(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_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_lt(x_1, x_2) -> lt(encArg(x_1), encArg(x_2)) encode_bin2s(x_1) -> bin2s(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_bin2ss(x_1, x_2) -> bin2ss(encArg(x_1), encArg(x_2)) encode_double(x_1) -> double(encArg(x_1)) encode_1 -> 1 encode_half(x_1) -> half(encArg(x_1)) encode_log(x_1) -> log(encArg(x_1)) encode_more(x_1) -> more(encArg(x_1)) encode_s2bin(x_1) -> s2bin(encArg(x_1)) encode_s2bin1(x_1, x_2, x_3) -> s2bin1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_if1(x_1, x_2, x_3, x_4) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_s2bin2(x_1, x_2) -> s2bin2(encArg(x_1), encArg(x_2)) encode_bug_list_not -> bug_list_not 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: eq(0, 0) -> true eq(0, s(y)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) lt(0, s(y)) -> true lt(x, 0) -> false lt(s(x), s(y)) -> lt(x, y) bin2s(nil) -> 0 bin2s(cons(x, xs)) -> bin2ss(x, xs) bin2ss(x, nil) -> x bin2ss(x, cons(0, xs)) -> bin2ss(double(x), xs) bin2ss(x, cons(1, xs)) -> bin2ss(s(double(x)), xs) half(0) -> 0 half(s(0)) -> 0 half(s(s(x))) -> s(half(x)) log(0) -> 0 log(s(0)) -> 0 log(s(s(x))) -> s(log(half(s(s(x))))) more(nil) -> nil more(cons(xs, ys)) -> cons(cons(0, xs), cons(cons(1, xs), cons(xs, ys))) s2bin(x) -> s2bin1(x, 0, cons(nil, nil)) s2bin1(x, y, lists) -> if1(lt(y, log(x)), x, y, lists) if1(true, x, y, lists) -> s2bin1(x, s(y), more(lists)) if1(false, x, y, lists) -> s2bin2(x, lists) s2bin2(x, nil) -> bug_list_not s2bin2(x, cons(xs, ys)) -> if2(eq(x, bin2s(xs)), x, xs, ys) if2(true, x, xs, ys) -> xs if2(false, x, xs, ys) -> s2bin2(x, ys) 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(double(x_1)) -> double(encArg(x_1)) encArg(1) -> 1 encArg(bug_list_not) -> bug_list_not encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2)) encArg(cons_lt(x_1, x_2)) -> lt(encArg(x_1), encArg(x_2)) encArg(cons_bin2s(x_1)) -> bin2s(encArg(x_1)) encArg(cons_bin2ss(x_1, x_2)) -> bin2ss(encArg(x_1), encArg(x_2)) encArg(cons_half(x_1)) -> half(encArg(x_1)) encArg(cons_log(x_1)) -> log(encArg(x_1)) encArg(cons_more(x_1)) -> more(encArg(x_1)) encArg(cons_s2bin(x_1)) -> s2bin(encArg(x_1)) encArg(cons_s2bin1(x_1, x_2, x_3)) -> s2bin1(encArg(x_1), encArg(x_2), encArg(x_3)) 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_s2bin2(x_1, x_2)) -> s2bin2(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_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_lt(x_1, x_2) -> lt(encArg(x_1), encArg(x_2)) encode_bin2s(x_1) -> bin2s(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_bin2ss(x_1, x_2) -> bin2ss(encArg(x_1), encArg(x_2)) encode_double(x_1) -> double(encArg(x_1)) encode_1 -> 1 encode_half(x_1) -> half(encArg(x_1)) encode_log(x_1) -> log(encArg(x_1)) encode_more(x_1) -> more(encArg(x_1)) encode_s2bin(x_1) -> s2bin(encArg(x_1)) encode_s2bin1(x_1, x_2, x_3) -> s2bin1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_if1(x_1, x_2, x_3, x_4) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_s2bin2(x_1, x_2) -> s2bin2(encArg(x_1), encArg(x_2)) encode_bug_list_not -> bug_list_not 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: eq(0, 0) -> true eq(0, s(y)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) lt(0, s(y)) -> true lt(x, 0) -> false lt(s(x), s(y)) -> lt(x, y) bin2s(nil) -> 0 bin2s(cons(x, xs)) -> bin2ss(x, xs) bin2ss(x, nil) -> x bin2ss(x, cons(0, xs)) -> bin2ss(double(x), xs) bin2ss(x, cons(1, xs)) -> bin2ss(s(double(x)), xs) half(0) -> 0 half(s(0)) -> 0 half(s(s(x))) -> s(half(x)) log(0) -> 0 log(s(0)) -> 0 log(s(s(x))) -> s(log(half(s(s(x))))) more(nil) -> nil more(cons(xs, ys)) -> cons(cons(0, xs), cons(cons(1, xs), cons(xs, ys))) s2bin(x) -> s2bin1(x, 0, cons(nil, nil)) s2bin1(x, y, lists) -> if1(lt(y, log(x)), x, y, lists) if1(true, x, y, lists) -> s2bin1(x, s(y), more(lists)) if1(false, x, y, lists) -> s2bin2(x, lists) s2bin2(x, nil) -> bug_list_not s2bin2(x, cons(xs, ys)) -> if2(eq(x, bin2s(xs)), x, xs, ys) if2(true, x, xs, ys) -> xs if2(false, x, xs, ys) -> s2bin2(x, ys) 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(double(x_1)) -> double(encArg(x_1)) encArg(1) -> 1 encArg(bug_list_not) -> bug_list_not encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2)) encArg(cons_lt(x_1, x_2)) -> lt(encArg(x_1), encArg(x_2)) encArg(cons_bin2s(x_1)) -> bin2s(encArg(x_1)) encArg(cons_bin2ss(x_1, x_2)) -> bin2ss(encArg(x_1), encArg(x_2)) encArg(cons_half(x_1)) -> half(encArg(x_1)) encArg(cons_log(x_1)) -> log(encArg(x_1)) encArg(cons_more(x_1)) -> more(encArg(x_1)) encArg(cons_s2bin(x_1)) -> s2bin(encArg(x_1)) encArg(cons_s2bin1(x_1, x_2, x_3)) -> s2bin1(encArg(x_1), encArg(x_2), encArg(x_3)) 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_s2bin2(x_1, x_2)) -> s2bin2(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_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_lt(x_1, x_2) -> lt(encArg(x_1), encArg(x_2)) encode_bin2s(x_1) -> bin2s(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_bin2ss(x_1, x_2) -> bin2ss(encArg(x_1), encArg(x_2)) encode_double(x_1) -> double(encArg(x_1)) encode_1 -> 1 encode_half(x_1) -> half(encArg(x_1)) encode_log(x_1) -> log(encArg(x_1)) encode_more(x_1) -> more(encArg(x_1)) encode_s2bin(x_1) -> s2bin(encArg(x_1)) encode_s2bin1(x_1, x_2, x_3) -> s2bin1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_if1(x_1, x_2, x_3, x_4) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_s2bin2(x_1, x_2) -> s2bin2(encArg(x_1), encArg(x_2)) encode_bug_list_not -> bug_list_not 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 bin2ss(x, cons(0, xs)) ->^+ bin2ss(double(x), xs) gives rise to a decreasing loop by considering the right hand sides subterm at position []. The pumping substitution is [xs / cons(0, xs)]. The result substitution is [x / double(x)]. ---------------------------------------- (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: eq(0, 0) -> true eq(0, s(y)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) lt(0, s(y)) -> true lt(x, 0) -> false lt(s(x), s(y)) -> lt(x, y) bin2s(nil) -> 0 bin2s(cons(x, xs)) -> bin2ss(x, xs) bin2ss(x, nil) -> x bin2ss(x, cons(0, xs)) -> bin2ss(double(x), xs) bin2ss(x, cons(1, xs)) -> bin2ss(s(double(x)), xs) half(0) -> 0 half(s(0)) -> 0 half(s(s(x))) -> s(half(x)) log(0) -> 0 log(s(0)) -> 0 log(s(s(x))) -> s(log(half(s(s(x))))) more(nil) -> nil more(cons(xs, ys)) -> cons(cons(0, xs), cons(cons(1, xs), cons(xs, ys))) s2bin(x) -> s2bin1(x, 0, cons(nil, nil)) s2bin1(x, y, lists) -> if1(lt(y, log(x)), x, y, lists) if1(true, x, y, lists) -> s2bin1(x, s(y), more(lists)) if1(false, x, y, lists) -> s2bin2(x, lists) s2bin2(x, nil) -> bug_list_not s2bin2(x, cons(xs, ys)) -> if2(eq(x, bin2s(xs)), x, xs, ys) if2(true, x, xs, ys) -> xs if2(false, x, xs, ys) -> s2bin2(x, ys) 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(double(x_1)) -> double(encArg(x_1)) encArg(1) -> 1 encArg(bug_list_not) -> bug_list_not encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2)) encArg(cons_lt(x_1, x_2)) -> lt(encArg(x_1), encArg(x_2)) encArg(cons_bin2s(x_1)) -> bin2s(encArg(x_1)) encArg(cons_bin2ss(x_1, x_2)) -> bin2ss(encArg(x_1), encArg(x_2)) encArg(cons_half(x_1)) -> half(encArg(x_1)) encArg(cons_log(x_1)) -> log(encArg(x_1)) encArg(cons_more(x_1)) -> more(encArg(x_1)) encArg(cons_s2bin(x_1)) -> s2bin(encArg(x_1)) encArg(cons_s2bin1(x_1, x_2, x_3)) -> s2bin1(encArg(x_1), encArg(x_2), encArg(x_3)) 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_s2bin2(x_1, x_2)) -> s2bin2(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_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_lt(x_1, x_2) -> lt(encArg(x_1), encArg(x_2)) encode_bin2s(x_1) -> bin2s(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_bin2ss(x_1, x_2) -> bin2ss(encArg(x_1), encArg(x_2)) encode_double(x_1) -> double(encArg(x_1)) encode_1 -> 1 encode_half(x_1) -> half(encArg(x_1)) encode_log(x_1) -> log(encArg(x_1)) encode_more(x_1) -> more(encArg(x_1)) encode_s2bin(x_1) -> s2bin(encArg(x_1)) encode_s2bin1(x_1, x_2, x_3) -> s2bin1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_if1(x_1, x_2, x_3, x_4) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_s2bin2(x_1, x_2) -> s2bin2(encArg(x_1), encArg(x_2)) encode_bug_list_not -> bug_list_not 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: eq(0, 0) -> true eq(0, s(y)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) lt(0, s(y)) -> true lt(x, 0) -> false lt(s(x), s(y)) -> lt(x, y) bin2s(nil) -> 0 bin2s(cons(x, xs)) -> bin2ss(x, xs) bin2ss(x, nil) -> x bin2ss(x, cons(0, xs)) -> bin2ss(double(x), xs) bin2ss(x, cons(1, xs)) -> bin2ss(s(double(x)), xs) half(0) -> 0 half(s(0)) -> 0 half(s(s(x))) -> s(half(x)) log(0) -> 0 log(s(0)) -> 0 log(s(s(x))) -> s(log(half(s(s(x))))) more(nil) -> nil more(cons(xs, ys)) -> cons(cons(0, xs), cons(cons(1, xs), cons(xs, ys))) s2bin(x) -> s2bin1(x, 0, cons(nil, nil)) s2bin1(x, y, lists) -> if1(lt(y, log(x)), x, y, lists) if1(true, x, y, lists) -> s2bin1(x, s(y), more(lists)) if1(false, x, y, lists) -> s2bin2(x, lists) s2bin2(x, nil) -> bug_list_not s2bin2(x, cons(xs, ys)) -> if2(eq(x, bin2s(xs)), x, xs, ys) if2(true, x, xs, ys) -> xs if2(false, x, xs, ys) -> s2bin2(x, ys) 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(double(x_1)) -> double(encArg(x_1)) encArg(1) -> 1 encArg(bug_list_not) -> bug_list_not encArg(cons_eq(x_1, x_2)) -> eq(encArg(x_1), encArg(x_2)) encArg(cons_lt(x_1, x_2)) -> lt(encArg(x_1), encArg(x_2)) encArg(cons_bin2s(x_1)) -> bin2s(encArg(x_1)) encArg(cons_bin2ss(x_1, x_2)) -> bin2ss(encArg(x_1), encArg(x_2)) encArg(cons_half(x_1)) -> half(encArg(x_1)) encArg(cons_log(x_1)) -> log(encArg(x_1)) encArg(cons_more(x_1)) -> more(encArg(x_1)) encArg(cons_s2bin(x_1)) -> s2bin(encArg(x_1)) encArg(cons_s2bin1(x_1, x_2, x_3)) -> s2bin1(encArg(x_1), encArg(x_2), encArg(x_3)) 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_s2bin2(x_1, x_2)) -> s2bin2(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_eq(x_1, x_2) -> eq(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_lt(x_1, x_2) -> lt(encArg(x_1), encArg(x_2)) encode_bin2s(x_1) -> bin2s(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_bin2ss(x_1, x_2) -> bin2ss(encArg(x_1), encArg(x_2)) encode_double(x_1) -> double(encArg(x_1)) encode_1 -> 1 encode_half(x_1) -> half(encArg(x_1)) encode_log(x_1) -> log(encArg(x_1)) encode_more(x_1) -> more(encArg(x_1)) encode_s2bin(x_1) -> s2bin(encArg(x_1)) encode_s2bin1(x_1, x_2, x_3) -> s2bin1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_if1(x_1, x_2, x_3, x_4) -> if1(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_s2bin2(x_1, x_2) -> s2bin2(encArg(x_1), encArg(x_2)) encode_bug_list_not -> bug_list_not 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