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