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