* Step 1: Sum WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
cond1(false(),x,y) -> cond2(gt(x,y),x,y)
cond1(true(),x,y) -> 0()
cond2(false(),x,y) -> s(diff(s(x),y))
cond2(true(),x,y) -> s(diff(x,s(y)))
diff(x,y) -> cond1(equal(x,y),x,y)
equal(0(),0()) -> true()
equal(0(),s(y)) -> false()
equal(s(x),0()) -> false()
equal(s(x),s(y)) -> equal(x,y)
gt(0(),v) -> false()
gt(s(u),0()) -> true()
gt(s(u),s(v)) -> gt(u,v)
- Signature:
{cond1/3,cond2/3,diff/2,equal/2,gt/2} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {cond1,cond2,diff,equal,gt} 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:
cond1(false(),x,y) -> cond2(gt(x,y),x,y)
cond1(true(),x,y) -> 0()
cond2(false(),x,y) -> s(diff(s(x),y))
cond2(true(),x,y) -> s(diff(x,s(y)))
diff(x,y) -> cond1(equal(x,y),x,y)
equal(0(),0()) -> true()
equal(0(),s(y)) -> false()
equal(s(x),0()) -> false()
equal(s(x),s(y)) -> equal(x,y)
gt(0(),v) -> false()
gt(s(u),0()) -> true()
gt(s(u),s(v)) -> gt(u,v)
- Signature:
{cond1/3,cond2/3,diff/2,equal/2,gt/2} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {cond1,cond2,diff,equal,gt} and constructors {0,false,s
,true}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
equal(x,y){x -> s(x),y -> s(y)} =
equal(s(x),s(y)) ->^+ equal(x,y)
= C[equal(x,y) = equal(x,y){}]
WORST_CASE(Omega(n^1),?)