* Step 1: Sum WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: empty(cons(n,x)) -> false() empty(nil()) -> true() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) head(cons(n,x)) -> n if(false(),x,y,z) -> sortIter(replace(min(x),head(x),tail(x)),z) if(true(),x,y,z) -> y if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(x,nil())) -> x replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() sort(x) -> sortIter(x,nil()) sortIter(x,y) -> if(empty(x),x,y,append(y,cons(min(x),nil()))) tail(cons(n,x)) -> x tail(nil()) -> nil() - Signature: {empty/1,eq/2,head/1,if/4,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,sortIter/2,tail/1} / {0/0 ,append/2,cons/2,false/0,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {empty,eq,head,if,if_min,if_replace,le,min,replace,sort ,sortIter,tail} and constructors {0,append,cons,false,nil,s,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: empty(cons(n,x)) -> false() empty(nil()) -> true() eq(0(),0()) -> true() eq(0(),s(m)) -> false() eq(s(n),0()) -> false() eq(s(n),s(m)) -> eq(n,m) head(cons(n,x)) -> n if(false(),x,y,z) -> sortIter(replace(min(x),head(x),tail(x)),z) if(true(),x,y,z) -> y if_min(false(),cons(n,cons(m,x))) -> min(cons(m,x)) if_min(true(),cons(n,cons(m,x))) -> min(cons(n,x)) if_replace(false(),n,m,cons(k,x)) -> cons(k,replace(n,m,x)) if_replace(true(),n,m,cons(k,x)) -> cons(m,x) le(0(),m) -> true() le(s(n),0()) -> false() le(s(n),s(m)) -> le(n,m) min(cons(n,cons(m,x))) -> if_min(le(n,m),cons(n,cons(m,x))) min(cons(x,nil())) -> x replace(n,m,cons(k,x)) -> if_replace(eq(n,k),n,m,cons(k,x)) replace(n,m,nil()) -> nil() sort(x) -> sortIter(x,nil()) sortIter(x,y) -> if(empty(x),x,y,append(y,cons(min(x),nil()))) tail(cons(n,x)) -> x tail(nil()) -> nil() - Signature: {empty/1,eq/2,head/1,if/4,if_min/2,if_replace/4,le/2,min/1,replace/3,sort/1,sortIter/2,tail/1} / {0/0 ,append/2,cons/2,false/0,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {empty,eq,head,if,if_min,if_replace,le,min,replace,sort ,sortIter,tail} and constructors {0,append,cons,false,nil,s,true} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: eq(x,y){x -> s(x),y -> s(y)} = eq(s(x),s(y)) ->^+ eq(x,y) = C[eq(x,y) = eq(x,y){}] WORST_CASE(Omega(n^1),?)