* Step 1: Sum WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
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())
if(false(),x,y,z) -> cons(y,gen(x,y,s(z)))
if(true(),x,y,z) -> nil()
sum(cons(0(),xs)) -> sum(xs)
sum(cons(s(x),xs)) -> s(sum(cons(x,xs)))
sum(nil()) -> 0()
times(x,y) -> sum(generate(x,y))
- Signature:
{ge/2,gen/3,generate/2,if/4,sum/1,times/2} / {0/0,cons/2,false/0,nil/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {ge,gen,generate,if,sum,times} and constructors {0,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:
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())
if(false(),x,y,z) -> cons(y,gen(x,y,s(z)))
if(true(),x,y,z) -> nil()
sum(cons(0(),xs)) -> sum(xs)
sum(cons(s(x),xs)) -> s(sum(cons(x,xs)))
sum(nil()) -> 0()
times(x,y) -> sum(generate(x,y))
- Signature:
{ge/2,gen/3,generate/2,if/4,sum/1,times/2} / {0/0,cons/2,false/0,nil/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {ge,gen,generate,if,sum,times} and constructors {0,cons
,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),?)