* Step 1: Sum WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
and(false(),y) -> false()
and(true(),y) -> y
eq(0(),0()) -> true()
eq(0(),s(x)) -> false()
eq(s(x),0()) -> false()
eq(s(x),s(y)) -> eq(x,y)
if1(false(),x,y,c,i,j) -> if2(le(c,size(j)),x,y,c,i,j)
if1(true(),x,y,c,i,j) -> true()
if2(false(),x,y,c,i,j) -> false()
if2(true(),x,y,c,edge(u,v,i),j) -> or(if2(true(),x,y,c,i,j),and(eq(x,u),reach(v,y,s(c),j,j)))
if2(true(),x,y,c,empty(),j) -> false()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
or(false(),y) -> y
or(true(),y) -> true()
reach(x,y,c,i,j) -> if1(eq(x,y),x,y,c,i,j)
reachable(x,y,i) -> reach(x,y,0(),i,i)
size(edge(x,y,i)) -> s(size(i))
size(empty()) -> 0()
- Signature:
{and/2,eq/2,if1/6,if2/6,le/2,or/2,reach/5,reachable/3,size/1} / {0/0,edge/3,empty/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {and,eq,if1,if2,le,or,reach,reachable
,size} and constructors {0,edge,empty,false,s,true}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
* Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
and(false(),y) -> false()
and(true(),y) -> y
eq(0(),0()) -> true()
eq(0(),s(x)) -> false()
eq(s(x),0()) -> false()
eq(s(x),s(y)) -> eq(x,y)
if1(false(),x,y,c,i,j) -> if2(le(c,size(j)),x,y,c,i,j)
if1(true(),x,y,c,i,j) -> true()
if2(false(),x,y,c,i,j) -> false()
if2(true(),x,y,c,edge(u,v,i),j) -> or(if2(true(),x,y,c,i,j),and(eq(x,u),reach(v,y,s(c),j,j)))
if2(true(),x,y,c,empty(),j) -> false()
le(0(),y) -> true()
le(s(x),0()) -> false()
le(s(x),s(y)) -> le(x,y)
or(false(),y) -> y
or(true(),y) -> true()
reach(x,y,c,i,j) -> if1(eq(x,y),x,y,c,i,j)
reachable(x,y,i) -> reach(x,y,0(),i,i)
size(edge(x,y,i)) -> s(size(i))
size(empty()) -> 0()
- Signature:
{and/2,eq/2,if1/6,if2/6,le/2,or/2,reach/5,reachable/3,size/1} / {0/0,edge/3,empty/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {and,eq,if1,if2,le,or,reach,reachable
,size} and constructors {0,edge,empty,false,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),?)