* Step 1: Sum WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: eq(0(),0()) -> true() eq(0(),s(x)) -> false() eq(s(x),0()) -> false() eq(s(x),s(y)) -> eq(x,y) from(edge(x,y,i)) -> x if1(false(),b1,b2,b3,x,y,i,h) -> if2(b1,b2,b3,x,y,i,h) if1(true(),b1,b2,b3,x,y,i,h) -> true() if2(false(),b2,b3,x,y,i,h) -> if3(b2,b3,x,y,i,h) if2(true(),b2,b3,x,y,i,h) -> false() if3(false(),b3,x,y,i,h) -> reach(x,y,rest(i),edge(from(i),to(i),h)) if3(true(),b3,x,y,i,h) -> if4(b3,x,y,i,h) if4(false(),x,y,i,h) -> or(reach(x,y,rest(i),h),reach(to(i),y,union(rest(i),h),empty())) if4(true(),x,y,i,h) -> true() isEmpty(edge(x,y,i)) -> false() isEmpty(empty()) -> true() or(false(),y) -> y or(true(),y) -> true() reach(x,y,i,h) -> if1(eq(x,y),isEmpty(i),eq(x,from(i)),eq(y,to(i)),x,y,i,h) rest(edge(x,y,i)) -> i rest(empty()) -> empty() to(edge(x,y,i)) -> y union(edge(x,y,i),h) -> edge(x,y,union(i,h)) union(empty(),h) -> h - Signature: {eq/2,from/1,if1/8,if2/7,if3/6,if4/5,isEmpty/1,or/2,reach/4,rest/1,to/1,union/2} / {0/0,edge/3,empty/0 ,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq,from,if1,if2,if3,if4,isEmpty,or,reach,rest,to ,union} and constructors {0,edge,empty,false,s,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: eq(0(),0()) -> true() eq(0(),s(x)) -> false() eq(s(x),0()) -> false() eq(s(x),s(y)) -> eq(x,y) from(edge(x,y,i)) -> x if1(false(),b1,b2,b3,x,y,i,h) -> if2(b1,b2,b3,x,y,i,h) if1(true(),b1,b2,b3,x,y,i,h) -> true() if2(false(),b2,b3,x,y,i,h) -> if3(b2,b3,x,y,i,h) if2(true(),b2,b3,x,y,i,h) -> false() if3(false(),b3,x,y,i,h) -> reach(x,y,rest(i),edge(from(i),to(i),h)) if3(true(),b3,x,y,i,h) -> if4(b3,x,y,i,h) if4(false(),x,y,i,h) -> or(reach(x,y,rest(i),h),reach(to(i),y,union(rest(i),h),empty())) if4(true(),x,y,i,h) -> true() isEmpty(edge(x,y,i)) -> false() isEmpty(empty()) -> true() or(false(),y) -> y or(true(),y) -> true() reach(x,y,i,h) -> if1(eq(x,y),isEmpty(i),eq(x,from(i)),eq(y,to(i)),x,y,i,h) rest(edge(x,y,i)) -> i rest(empty()) -> empty() to(edge(x,y,i)) -> y union(edge(x,y,i),h) -> edge(x,y,union(i,h)) union(empty(),h) -> h - Signature: {eq/2,from/1,if1/8,if2/7,if3/6,if4/5,isEmpty/1,or/2,reach/4,rest/1,to/1,union/2} / {0/0,edge/3,empty/0 ,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq,from,if1,if2,if3,if4,isEmpty,or,reach,rest,to ,union} and constructors {0,edge,empty,false,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),?)