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