* Step 1: Sum WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: a() -> b() a() -> c() div(x,y,0()) -> divisible(x,y) div(0(),y,s(z)) -> false() div(s(x),y,s(z)) -> div(x,y,z) divisible(0(),s(y)) -> true() divisible(s(x),s(y)) -> div(s(x),s(y),s(y)) ge(x,0()) -> true() ge(0(),s(y)) -> false() ge(s(x),s(y)) -> ge(x,y) if(false(),x,y,z,u) -> if2(divisible(z,y),x,y,z,u) if(true(),x,y,z,u) -> z if2(false(),x,y,z,u) -> lcmIter(x,y,plus(x,z),u) if2(true(),x,y,z,u) -> z ifTimes(false(),x,y) -> plus(y,times(y,p(x))) ifTimes(true(),x,y) -> 0() lcm(x,y) -> lcmIter(x,y,0(),times(x,y)) lcmIter(x,y,z,u) -> if(or(ge(0(),x),ge(z,u)),x,y,z,u) or(false(),y) -> y or(true(),y) -> true() p(0()) -> s(s(0())) p(s(x)) -> x plus(0(),y) -> y plus(s(x),y) -> s(plus(x,y)) times(x,y) -> ifTimes(ge(0(),x),x,y) - Signature: {a/0,div/3,divisible/2,ge/2,if/5,if2/5,ifTimes/3,lcm/2,lcmIter/4,or/2,p/1,plus/2,times/2} / {0/0,b/0,c/0 ,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a,div,divisible,ge,if,if2,ifTimes,lcm,lcmIter,or,p,plus ,times} and constructors {0,b,c,false,s,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: a() -> b() a() -> c() div(x,y,0()) -> divisible(x,y) div(0(),y,s(z)) -> false() div(s(x),y,s(z)) -> div(x,y,z) divisible(0(),s(y)) -> true() divisible(s(x),s(y)) -> div(s(x),s(y),s(y)) ge(x,0()) -> true() ge(0(),s(y)) -> false() ge(s(x),s(y)) -> ge(x,y) if(false(),x,y,z,u) -> if2(divisible(z,y),x,y,z,u) if(true(),x,y,z,u) -> z if2(false(),x,y,z,u) -> lcmIter(x,y,plus(x,z),u) if2(true(),x,y,z,u) -> z ifTimes(false(),x,y) -> plus(y,times(y,p(x))) ifTimes(true(),x,y) -> 0() lcm(x,y) -> lcmIter(x,y,0(),times(x,y)) lcmIter(x,y,z,u) -> if(or(ge(0(),x),ge(z,u)),x,y,z,u) or(false(),y) -> y or(true(),y) -> true() p(0()) -> s(s(0())) p(s(x)) -> x plus(0(),y) -> y plus(s(x),y) -> s(plus(x,y)) times(x,y) -> ifTimes(ge(0(),x),x,y) - Signature: {a/0,div/3,divisible/2,ge/2,if/5,if2/5,ifTimes/3,lcm/2,lcmIter/4,or/2,p/1,plus/2,times/2} / {0/0,b/0,c/0 ,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a,div,divisible,ge,if,if2,ifTimes,lcm,lcmIter,or,p,plus ,times} and constructors {0,b,c,false,s,true} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: div(x,y,z){x -> s(x),z -> s(z)} = div(s(x),y,s(z)) ->^+ div(x,y,z) = C[div(x,y,z) = div(x,y,z){}] WORST_CASE(Omega(n^1),?)