* Step 1: Sum WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
eq(0(),0()) -> true()
eq(0(),s(x)) -> false()
eq(s(x),0()) -> false()
eq(s(x),s(y)) -> eq(x,y)
if_reach_1(false(),x,y,edge(u,v,i),h) -> reach(x,y,i,edge(u,v,h))
if_reach_1(true(),x,y,edge(u,v,i),h) -> if_reach_2(eq(y,v),x,y,edge(u,v,i),h)
if_reach_2(false(),x,y,edge(u,v,i),h) -> or(reach(x,y,i,h),reach(v,y,union(i,h),empty()))
if_reach_2(true(),x,y,edge(u,v,i),h) -> true()
or(false(),y) -> y
or(true(),y) -> true()
reach(x,y,edge(u,v,i),h) -> if_reach_1(eq(x,u),x,y,edge(u,v,i),h)
reach(x,y,empty(),h) -> false()
union(edge(x,y,i),h) -> edge(x,y,union(i,h))
union(empty(),h) -> h
- Signature:
{eq/2,if_reach_1/5,if_reach_2/5,or/2,reach/4,union/2} / {0/0,edge/3,empty/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {eq,if_reach_1,if_reach_2,or,reach
,union} 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:
eq(0(),0()) -> true()
eq(0(),s(x)) -> false()
eq(s(x),0()) -> false()
eq(s(x),s(y)) -> eq(x,y)
if_reach_1(false(),x,y,edge(u,v,i),h) -> reach(x,y,i,edge(u,v,h))
if_reach_1(true(),x,y,edge(u,v,i),h) -> if_reach_2(eq(y,v),x,y,edge(u,v,i),h)
if_reach_2(false(),x,y,edge(u,v,i),h) -> or(reach(x,y,i,h),reach(v,y,union(i,h),empty()))
if_reach_2(true(),x,y,edge(u,v,i),h) -> true()
or(false(),y) -> y
or(true(),y) -> true()
reach(x,y,edge(u,v,i),h) -> if_reach_1(eq(x,u),x,y,edge(u,v,i),h)
reach(x,y,empty(),h) -> false()
union(edge(x,y,i),h) -> edge(x,y,union(i,h))
union(empty(),h) -> h
- Signature:
{eq/2,if_reach_1/5,if_reach_2/5,or/2,reach/4,union/2} / {0/0,edge/3,empty/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {eq,if_reach_1,if_reach_2,or,reach
,union} 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),?)