* Step 1: Sum WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
eq(0(),0()) -> true()
eq(0(),s(y)) -> false()
eq(s(x),0()) -> false()
eq(s(x),s(y)) -> eq(x,y)
if1(false(),x,y,xs) -> min(cons(y,xs))
if1(true(),x,y,xs) -> min(cons(x,xs))
if2(false(),x,y,xs) -> cons(y,rm(x,xs))
if2(true(),x,y,xs) -> rm(x,xs)
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
min(cons(x,cons(y,xs))) -> if1(le(x,y),x,y,xs)
min(cons(x,nil())) -> x
min(nil()) -> 0()
minsort(cons(x,xs)) -> cons(min(cons(x,xs)),minsort(rm(min(cons(x,xs)),cons(x,xs))))
minsort(nil()) -> nil()
rm(x,cons(y,xs)) -> if2(eq(x,y),x,y,xs)
rm(x,nil()) -> nil()
- Signature:
{eq/2,if1/4,if2/4,le/2,min/1,minsort/1,rm/2} / {0/0,cons/2,false/0,nil/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {eq,if1,if2,le,min,minsort,rm} 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:
eq(0(),0()) -> true()
eq(0(),s(y)) -> false()
eq(s(x),0()) -> false()
eq(s(x),s(y)) -> eq(x,y)
if1(false(),x,y,xs) -> min(cons(y,xs))
if1(true(),x,y,xs) -> min(cons(x,xs))
if2(false(),x,y,xs) -> cons(y,rm(x,xs))
if2(true(),x,y,xs) -> rm(x,xs)
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
min(cons(x,cons(y,xs))) -> if1(le(x,y),x,y,xs)
min(cons(x,nil())) -> x
min(nil()) -> 0()
minsort(cons(x,xs)) -> cons(min(cons(x,xs)),minsort(rm(min(cons(x,xs)),cons(x,xs))))
minsort(nil()) -> nil()
rm(x,cons(y,xs)) -> if2(eq(x,y),x,y,xs)
rm(x,nil()) -> nil()
- Signature:
{eq/2,if1/4,if2/4,le/2,min/1,minsort/1,rm/2} / {0/0,cons/2,false/0,nil/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {eq,if1,if2,le,min,minsort,rm} and constructors {0,cons
,false,nil,s,true}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
eq(x,y){x -> s(x),y -> s(y)} =
eq(s(x),s(y)) ->^+ eq(x,y)
= C[eq(x,y) = eq(x,y){}]
WORST_CASE(Omega(n^1),?)