* Step 1: Bounds WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) fst(mark(X)) -> mark(fst(X)) fst(ok(X)) -> ok(fst(X)) head(mark(X)) -> mark(head(X)) head(ok(X)) -> ok(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) snd(mark(X)) -> mark(snd(X)) snd(ok(X)) -> ok(snd(X)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) tail(mark(X)) -> mark(tail(X)) tail(ok(X)) -> ok(tail(X)) take(X1,mark(X2)) -> mark(take(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) u(mark(X1),X2,X3,X4) -> mark(u(X1,X2,X3,X4)) u(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(u(X1,X2,X3,X4)) - Signature: {afterNth/2,cons/2,fst/1,head/1,natsFrom/1,pair/2,proper/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2,top/1 ,u/4} / {0/0,active/1,mark/1,nil/0,ok/1} - Obligation: innermost runtime complexity wrt. defined symbols {afterNth,cons,fst,head,natsFrom,pair,proper,s,sel,snd ,splitAt,tail,take,top,u} and constructors {0,active,mark,nil,ok} + Applied Processor: Bounds {initialAutomaton = minimal, enrichment = match} + Details: The problem is match-bounded by 2. The enriched problem is compatible with follwoing automaton. 0_0() -> 2 0_1() -> 3 active_0(2) -> 2 active_1(2) -> 4 active_2(3) -> 5 afterNth_0(2,2) -> 1 afterNth_1(2,2) -> 3 cons_0(2,2) -> 1 cons_1(2,2) -> 3 fst_0(2) -> 1 fst_1(2) -> 3 head_0(2) -> 1 head_1(2) -> 3 mark_0(2) -> 2 mark_1(3) -> 1 mark_1(3) -> 3 natsFrom_0(2) -> 1 natsFrom_1(2) -> 3 nil_0() -> 2 nil_1() -> 3 ok_0(2) -> 2 ok_1(3) -> 1 ok_1(3) -> 3 ok_1(3) -> 4 pair_0(2,2) -> 1 pair_1(2,2) -> 3 proper_0(2) -> 1 proper_1(2) -> 4 s_0(2) -> 1 s_1(2) -> 3 sel_0(2,2) -> 1 sel_1(2,2) -> 3 snd_0(2) -> 1 snd_1(2) -> 3 splitAt_0(2,2) -> 1 splitAt_1(2,2) -> 3 tail_0(2) -> 1 tail_1(2) -> 3 take_0(2,2) -> 1 take_1(2,2) -> 3 top_0(2) -> 1 top_1(4) -> 1 top_2(5) -> 1 u_0(2,2,2,2) -> 1 u_1(2,2,2,2) -> 3 * Step 2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: afterNth(X1,mark(X2)) -> mark(afterNth(X1,X2)) afterNth(mark(X1),X2) -> mark(afterNth(X1,X2)) afterNth(ok(X1),ok(X2)) -> ok(afterNth(X1,X2)) cons(mark(X1),X2) -> mark(cons(X1,X2)) cons(ok(X1),ok(X2)) -> ok(cons(X1,X2)) fst(mark(X)) -> mark(fst(X)) fst(ok(X)) -> ok(fst(X)) head(mark(X)) -> mark(head(X)) head(ok(X)) -> ok(head(X)) natsFrom(mark(X)) -> mark(natsFrom(X)) natsFrom(ok(X)) -> ok(natsFrom(X)) pair(X1,mark(X2)) -> mark(pair(X1,X2)) pair(mark(X1),X2) -> mark(pair(X1,X2)) pair(ok(X1),ok(X2)) -> ok(pair(X1,X2)) proper(0()) -> ok(0()) proper(nil()) -> ok(nil()) s(mark(X)) -> mark(s(X)) s(ok(X)) -> ok(s(X)) sel(X1,mark(X2)) -> mark(sel(X1,X2)) sel(mark(X1),X2) -> mark(sel(X1,X2)) sel(ok(X1),ok(X2)) -> ok(sel(X1,X2)) snd(mark(X)) -> mark(snd(X)) snd(ok(X)) -> ok(snd(X)) splitAt(X1,mark(X2)) -> mark(splitAt(X1,X2)) splitAt(mark(X1),X2) -> mark(splitAt(X1,X2)) splitAt(ok(X1),ok(X2)) -> ok(splitAt(X1,X2)) tail(mark(X)) -> mark(tail(X)) tail(ok(X)) -> ok(tail(X)) take(X1,mark(X2)) -> mark(take(X1,X2)) take(mark(X1),X2) -> mark(take(X1,X2)) take(ok(X1),ok(X2)) -> ok(take(X1,X2)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) u(mark(X1),X2,X3,X4) -> mark(u(X1,X2,X3,X4)) u(ok(X1),ok(X2),ok(X3),ok(X4)) -> ok(u(X1,X2,X3,X4)) - Signature: {afterNth/2,cons/2,fst/1,head/1,natsFrom/1,pair/2,proper/1,s/1,sel/2,snd/1,splitAt/2,tail/1,take/2,top/1 ,u/4} / {0/0,active/1,mark/1,nil/0,ok/1} - Obligation: innermost runtime complexity wrt. defined symbols {afterNth,cons,fst,head,natsFrom,pair,proper,s,sel,snd ,splitAt,tail,take,top,u} and constructors {0,active,mark,nil,ok} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))