* Step 1: Sum WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: a() -> c() a() -> d() ge(x,0()) -> true() ge(0(),s(y)) -> false() ge(s(x),s(y)) -> ge(x,y) gen(x,y,z) -> if(ge(z,x),x,y,z) generate(x,y) -> gen(x,y,0()) head(cons(x,xs)) -> x head(nil()) -> error() if(false(),x,y,z) -> cons(y,gen(x,y,s(z))) if(true(),x,y,z) -> nil() ifsum(false(),b,xs,y) -> ifsum2(b,xs,y) ifsum(true(),b,xs,y) -> y ifsum2(false(),xs,y) -> sum2(cons(p(head(xs)),tail(xs)),s(y)) ifsum2(true(),xs,y) -> sum2(tail(xs),y) isNil(cons(x,xs)) -> false() isNil(nil()) -> true() isZero(0()) -> true() isZero(s(0())) -> false() isZero(s(s(x))) -> isZero(s(x)) p(0()) -> s(s(0())) p(s(0())) -> 0() p(s(s(x))) -> s(p(s(x))) sum(xs) -> sum2(xs,0()) sum2(xs,y) -> ifsum(isNil(xs),isZero(head(xs)),xs,y) tail(cons(x,xs)) -> xs tail(nil()) -> nil() times(x,y) -> sum(generate(x,y)) - Signature: {a/0,ge/2,gen/3,generate/2,head/1,if/4,ifsum/4,ifsum2/3,isNil/1,isZero/1,p/1,sum/1,sum2/2,tail/1 ,times/2} / {0/0,c/0,cons/2,d/0,error/0,false/0,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a,ge,gen,generate,head,if,ifsum,ifsum2,isNil,isZero,p,sum ,sum2,tail,times} and constructors {0,c,cons,d,error,false,nil,s,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: a() -> c() a() -> d() ge(x,0()) -> true() ge(0(),s(y)) -> false() ge(s(x),s(y)) -> ge(x,y) gen(x,y,z) -> if(ge(z,x),x,y,z) generate(x,y) -> gen(x,y,0()) head(cons(x,xs)) -> x head(nil()) -> error() if(false(),x,y,z) -> cons(y,gen(x,y,s(z))) if(true(),x,y,z) -> nil() ifsum(false(),b,xs,y) -> ifsum2(b,xs,y) ifsum(true(),b,xs,y) -> y ifsum2(false(),xs,y) -> sum2(cons(p(head(xs)),tail(xs)),s(y)) ifsum2(true(),xs,y) -> sum2(tail(xs),y) isNil(cons(x,xs)) -> false() isNil(nil()) -> true() isZero(0()) -> true() isZero(s(0())) -> false() isZero(s(s(x))) -> isZero(s(x)) p(0()) -> s(s(0())) p(s(0())) -> 0() p(s(s(x))) -> s(p(s(x))) sum(xs) -> sum2(xs,0()) sum2(xs,y) -> ifsum(isNil(xs),isZero(head(xs)),xs,y) tail(cons(x,xs)) -> xs tail(nil()) -> nil() times(x,y) -> sum(generate(x,y)) - Signature: {a/0,ge/2,gen/3,generate/2,head/1,if/4,ifsum/4,ifsum2/3,isNil/1,isZero/1,p/1,sum/1,sum2/2,tail/1 ,times/2} / {0/0,c/0,cons/2,d/0,error/0,false/0,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a,ge,gen,generate,head,if,ifsum,ifsum2,isNil,isZero,p,sum ,sum2,tail,times} and constructors {0,c,cons,d,error,false,nil,s,true} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: ge(x,y){x -> s(x),y -> s(y)} = ge(s(x),s(y)) ->^+ ge(x,y) = C[ge(x,y) = ge(x,y){}] WORST_CASE(Omega(n^1),?)