* Step 1: Sum WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: f(0(),y,0(),u) -> true() f(0(),y,s(z),u) -> false() f(s(x),0(),z,u) -> f(x,u,minus(z,s(x)),u) f(s(x),s(y),z,u) -> if(le(x,y),f(s(x),minus(y,x),z,u),f(x,u,z,u)) if(false(),x,y) -> y if(true(),x,y) -> x le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),0()) -> s(x) minus(s(x),s(y)) -> minus(x,y) perfectp(0()) -> false() perfectp(s(x)) -> f(x,s(0()),s(x),s(x)) - Signature: {f/4,if/3,le/2,minus/2,perfectp/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {f,if,le,minus,perfectp} and constructors {0,false,s,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: f(0(),y,0(),u) -> true() f(0(),y,s(z),u) -> false() f(s(x),0(),z,u) -> f(x,u,minus(z,s(x)),u) f(s(x),s(y),z,u) -> if(le(x,y),f(s(x),minus(y,x),z,u),f(x,u,z,u)) if(false(),x,y) -> y if(true(),x,y) -> x le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),0()) -> s(x) minus(s(x),s(y)) -> minus(x,y) perfectp(0()) -> false() perfectp(s(x)) -> f(x,s(0()),s(x),s(x)) - Signature: {f/4,if/3,le/2,minus/2,perfectp/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {f,if,le,minus,perfectp} and constructors {0,false,s,true} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: le(x,y){x -> s(x),y -> s(y)} = le(s(x),s(y)) ->^+ le(x,y) = C[le(x,y) = le(x,y){}] WORST_CASE(Omega(n^1),?)