/export/starexec/sandbox/solver/bin/starexec_run_complexity /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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) SlicingProof [LOWER BOUND(ID), 0 ms] (4) CpxTRS (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (6) typed CpxTrs (7) OrderProof [LOWER BOUND(ID), 0 ms] (8) typed CpxTrs (9) RewriteLemmaProof [LOWER BOUND(ID), 266 ms] (10) BEST (11) proven lower bound (12) LowerBoundPropagationProof [FINISHED, 0 ms] (13) BOUNDS(n^1, INF) (14) typed CpxTrs (15) RewriteLemmaProof [LOWER BOUND(ID), 40 ms] (16) typed CpxTrs (17) RewriteLemmaProof [LOWER BOUND(ID), 21 ms] (18) typed CpxTrs ---------------------------------------- (0) Obligation: The Runtime Complexity (full) of the given CpxTRS 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: FULL ---------------------------------------- (1) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (2) Obligation: The Runtime Complexity (full) of the given CpxTRS 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: FULL ---------------------------------------- (3) SlicingProof (LOWER BOUND(ID)) Sliced the following arguments: double/0 ---------------------------------------- (4) Obligation: The Runtime Complexity (full) of the given CpxTRS 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, xs) bin2ss(x, cons(1', xs)) -> bin2ss(s(double), 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: FULL ---------------------------------------- (5) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (6) Obligation: TRS: 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, xs) bin2ss(x, cons(1', xs)) -> bin2ss(s(double), 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) Types: eq :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> true:false 0' :: 0':s:nil:cons:double:1':bug_list_not true :: true:false s :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not false :: true:false lt :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> true:false bin2s :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not nil :: 0':s:nil:cons:double:1':bug_list_not cons :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not bin2ss :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not double :: 0':s:nil:cons:double:1':bug_list_not 1' :: 0':s:nil:cons:double:1':bug_list_not half :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not log :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not more :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not s2bin :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not s2bin1 :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not if1 :: true:false -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not s2bin2 :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not bug_list_not :: 0':s:nil:cons:double:1':bug_list_not if2 :: true:false -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not hole_true:false1_0 :: true:false hole_0':s:nil:cons:double:1':bug_list_not2_0 :: 0':s:nil:cons:double:1':bug_list_not gen_0':s:nil:cons:double:1':bug_list_not3_0 :: Nat -> 0':s:nil:cons:double:1':bug_list_not ---------------------------------------- (7) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: eq, lt, bin2ss, half, log, s2bin1, s2bin2 They will be analysed ascendingly in the following order: eq < s2bin2 lt < s2bin1 half < log log < s2bin1 s2bin2 < s2bin1 ---------------------------------------- (8) Obligation: TRS: 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, xs) bin2ss(x, cons(1', xs)) -> bin2ss(s(double), 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) Types: eq :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> true:false 0' :: 0':s:nil:cons:double:1':bug_list_not true :: true:false s :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not false :: true:false lt :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> true:false bin2s :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not nil :: 0':s:nil:cons:double:1':bug_list_not cons :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not bin2ss :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not double :: 0':s:nil:cons:double:1':bug_list_not 1' :: 0':s:nil:cons:double:1':bug_list_not half :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not log :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not more :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not s2bin :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not s2bin1 :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not if1 :: true:false -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not s2bin2 :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not bug_list_not :: 0':s:nil:cons:double:1':bug_list_not if2 :: true:false -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not hole_true:false1_0 :: true:false hole_0':s:nil:cons:double:1':bug_list_not2_0 :: 0':s:nil:cons:double:1':bug_list_not gen_0':s:nil:cons:double:1':bug_list_not3_0 :: Nat -> 0':s:nil:cons:double:1':bug_list_not Generator Equations: gen_0':s:nil:cons:double:1':bug_list_not3_0(0) <=> 0' gen_0':s:nil:cons:double:1':bug_list_not3_0(+(x, 1)) <=> s(gen_0':s:nil:cons:double:1':bug_list_not3_0(x)) The following defined symbols remain to be analysed: eq, lt, bin2ss, half, log, s2bin1, s2bin2 They will be analysed ascendingly in the following order: eq < s2bin2 lt < s2bin1 half < log log < s2bin1 s2bin2 < s2bin1 ---------------------------------------- (9) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: eq(gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0), gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0)) -> true, rt in Omega(1 + n5_0) Induction Base: eq(gen_0':s:nil:cons:double:1':bug_list_not3_0(0), gen_0':s:nil:cons:double:1':bug_list_not3_0(0)) ->_R^Omega(1) true Induction Step: eq(gen_0':s:nil:cons:double:1':bug_list_not3_0(+(n5_0, 1)), gen_0':s:nil:cons:double:1':bug_list_not3_0(+(n5_0, 1))) ->_R^Omega(1) eq(gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0), gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0)) ->_IH true We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (10) Complex Obligation (BEST) ---------------------------------------- (11) Obligation: Proved the lower bound n^1 for the following obligation: TRS: 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, xs) bin2ss(x, cons(1', xs)) -> bin2ss(s(double), 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) Types: eq :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> true:false 0' :: 0':s:nil:cons:double:1':bug_list_not true :: true:false s :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not false :: true:false lt :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> true:false bin2s :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not nil :: 0':s:nil:cons:double:1':bug_list_not cons :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not bin2ss :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not double :: 0':s:nil:cons:double:1':bug_list_not 1' :: 0':s:nil:cons:double:1':bug_list_not half :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not log :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not more :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not s2bin :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not s2bin1 :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not if1 :: true:false -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not s2bin2 :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not bug_list_not :: 0':s:nil:cons:double:1':bug_list_not if2 :: true:false -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not hole_true:false1_0 :: true:false hole_0':s:nil:cons:double:1':bug_list_not2_0 :: 0':s:nil:cons:double:1':bug_list_not gen_0':s:nil:cons:double:1':bug_list_not3_0 :: Nat -> 0':s:nil:cons:double:1':bug_list_not Generator Equations: gen_0':s:nil:cons:double:1':bug_list_not3_0(0) <=> 0' gen_0':s:nil:cons:double:1':bug_list_not3_0(+(x, 1)) <=> s(gen_0':s:nil:cons:double:1':bug_list_not3_0(x)) The following defined symbols remain to be analysed: eq, lt, bin2ss, half, log, s2bin1, s2bin2 They will be analysed ascendingly in the following order: eq < s2bin2 lt < s2bin1 half < log log < s2bin1 s2bin2 < s2bin1 ---------------------------------------- (12) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (13) BOUNDS(n^1, INF) ---------------------------------------- (14) Obligation: TRS: 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, xs) bin2ss(x, cons(1', xs)) -> bin2ss(s(double), 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) Types: eq :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> true:false 0' :: 0':s:nil:cons:double:1':bug_list_not true :: true:false s :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not false :: true:false lt :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> true:false bin2s :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not nil :: 0':s:nil:cons:double:1':bug_list_not cons :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not bin2ss :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not double :: 0':s:nil:cons:double:1':bug_list_not 1' :: 0':s:nil:cons:double:1':bug_list_not half :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not log :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not more :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not s2bin :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not s2bin1 :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not if1 :: true:false -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not s2bin2 :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not bug_list_not :: 0':s:nil:cons:double:1':bug_list_not if2 :: true:false -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not hole_true:false1_0 :: true:false hole_0':s:nil:cons:double:1':bug_list_not2_0 :: 0':s:nil:cons:double:1':bug_list_not gen_0':s:nil:cons:double:1':bug_list_not3_0 :: Nat -> 0':s:nil:cons:double:1':bug_list_not Lemmas: eq(gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0), gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0)) -> true, rt in Omega(1 + n5_0) Generator Equations: gen_0':s:nil:cons:double:1':bug_list_not3_0(0) <=> 0' gen_0':s:nil:cons:double:1':bug_list_not3_0(+(x, 1)) <=> s(gen_0':s:nil:cons:double:1':bug_list_not3_0(x)) The following defined symbols remain to be analysed: lt, bin2ss, half, log, s2bin1, s2bin2 They will be analysed ascendingly in the following order: lt < s2bin1 half < log log < s2bin1 s2bin2 < s2bin1 ---------------------------------------- (15) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: lt(gen_0':s:nil:cons:double:1':bug_list_not3_0(n572_0), gen_0':s:nil:cons:double:1':bug_list_not3_0(+(1, n572_0))) -> true, rt in Omega(1 + n572_0) Induction Base: lt(gen_0':s:nil:cons:double:1':bug_list_not3_0(0), gen_0':s:nil:cons:double:1':bug_list_not3_0(+(1, 0))) ->_R^Omega(1) true Induction Step: lt(gen_0':s:nil:cons:double:1':bug_list_not3_0(+(n572_0, 1)), gen_0':s:nil:cons:double:1':bug_list_not3_0(+(1, +(n572_0, 1)))) ->_R^Omega(1) lt(gen_0':s:nil:cons:double:1':bug_list_not3_0(n572_0), gen_0':s:nil:cons:double:1':bug_list_not3_0(+(1, n572_0))) ->_IH true We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (16) Obligation: TRS: 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, xs) bin2ss(x, cons(1', xs)) -> bin2ss(s(double), 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) Types: eq :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> true:false 0' :: 0':s:nil:cons:double:1':bug_list_not true :: true:false s :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not false :: true:false lt :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> true:false bin2s :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not nil :: 0':s:nil:cons:double:1':bug_list_not cons :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not bin2ss :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not double :: 0':s:nil:cons:double:1':bug_list_not 1' :: 0':s:nil:cons:double:1':bug_list_not half :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not log :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not more :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not s2bin :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not s2bin1 :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not if1 :: true:false -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not s2bin2 :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not bug_list_not :: 0':s:nil:cons:double:1':bug_list_not if2 :: true:false -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not hole_true:false1_0 :: true:false hole_0':s:nil:cons:double:1':bug_list_not2_0 :: 0':s:nil:cons:double:1':bug_list_not gen_0':s:nil:cons:double:1':bug_list_not3_0 :: Nat -> 0':s:nil:cons:double:1':bug_list_not Lemmas: eq(gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0), gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0)) -> true, rt in Omega(1 + n5_0) lt(gen_0':s:nil:cons:double:1':bug_list_not3_0(n572_0), gen_0':s:nil:cons:double:1':bug_list_not3_0(+(1, n572_0))) -> true, rt in Omega(1 + n572_0) Generator Equations: gen_0':s:nil:cons:double:1':bug_list_not3_0(0) <=> 0' gen_0':s:nil:cons:double:1':bug_list_not3_0(+(x, 1)) <=> s(gen_0':s:nil:cons:double:1':bug_list_not3_0(x)) The following defined symbols remain to be analysed: bin2ss, half, log, s2bin1, s2bin2 They will be analysed ascendingly in the following order: half < log log < s2bin1 s2bin2 < s2bin1 ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: half(gen_0':s:nil:cons:double:1':bug_list_not3_0(*(2, n952_0))) -> gen_0':s:nil:cons:double:1':bug_list_not3_0(n952_0), rt in Omega(1 + n952_0) Induction Base: half(gen_0':s:nil:cons:double:1':bug_list_not3_0(*(2, 0))) ->_R^Omega(1) 0' Induction Step: half(gen_0':s:nil:cons:double:1':bug_list_not3_0(*(2, +(n952_0, 1)))) ->_R^Omega(1) s(half(gen_0':s:nil:cons:double:1':bug_list_not3_0(*(2, n952_0)))) ->_IH s(gen_0':s:nil:cons:double:1':bug_list_not3_0(c953_0)) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (18) Obligation: TRS: 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, xs) bin2ss(x, cons(1', xs)) -> bin2ss(s(double), 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) Types: eq :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> true:false 0' :: 0':s:nil:cons:double:1':bug_list_not true :: true:false s :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not false :: true:false lt :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> true:false bin2s :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not nil :: 0':s:nil:cons:double:1':bug_list_not cons :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not bin2ss :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not double :: 0':s:nil:cons:double:1':bug_list_not 1' :: 0':s:nil:cons:double:1':bug_list_not half :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not log :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not more :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not s2bin :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not s2bin1 :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not if1 :: true:false -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not s2bin2 :: 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not bug_list_not :: 0':s:nil:cons:double:1':bug_list_not if2 :: true:false -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not -> 0':s:nil:cons:double:1':bug_list_not hole_true:false1_0 :: true:false hole_0':s:nil:cons:double:1':bug_list_not2_0 :: 0':s:nil:cons:double:1':bug_list_not gen_0':s:nil:cons:double:1':bug_list_not3_0 :: Nat -> 0':s:nil:cons:double:1':bug_list_not Lemmas: eq(gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0), gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0)) -> true, rt in Omega(1 + n5_0) lt(gen_0':s:nil:cons:double:1':bug_list_not3_0(n572_0), gen_0':s:nil:cons:double:1':bug_list_not3_0(+(1, n572_0))) -> true, rt in Omega(1 + n572_0) half(gen_0':s:nil:cons:double:1':bug_list_not3_0(*(2, n952_0))) -> gen_0':s:nil:cons:double:1':bug_list_not3_0(n952_0), rt in Omega(1 + n952_0) Generator Equations: gen_0':s:nil:cons:double:1':bug_list_not3_0(0) <=> 0' gen_0':s:nil:cons:double:1':bug_list_not3_0(+(x, 1)) <=> s(gen_0':s:nil:cons:double:1':bug_list_not3_0(x)) The following defined symbols remain to be analysed: log, s2bin1, s2bin2 They will be analysed ascendingly in the following order: log < s2bin1 s2bin2 < s2bin1