* Step 1: Sum WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
active(div(X1,X2)) -> div(active(X1),X2)
active(div(0(),s(Y))) -> mark(0())
active(div(s(X),s(Y))) -> mark(if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()))
active(geq(X,0())) -> mark(true())
active(geq(0(),s(Y))) -> mark(false())
active(geq(s(X),s(Y))) -> mark(geq(X,Y))
active(if(X1,X2,X3)) -> if(active(X1),X2,X3)
active(if(false(),X,Y)) -> mark(Y)
active(if(true(),X,Y)) -> mark(X)
active(minus(0(),Y)) -> mark(0())
active(minus(s(X),s(Y))) -> mark(minus(X,Y))
active(s(X)) -> s(active(X))
div(mark(X1),X2) -> mark(div(X1,X2))
div(ok(X1),ok(X2)) -> ok(div(X1,X2))
geq(ok(X1),ok(X2)) -> ok(geq(X1,X2))
if(mark(X1),X2,X3) -> mark(if(X1,X2,X3))
if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3))
minus(ok(X1),ok(X2)) -> ok(minus(X1,X2))
proper(0()) -> ok(0())
proper(div(X1,X2)) -> div(proper(X1),proper(X2))
proper(false()) -> ok(false())
proper(geq(X1,X2)) -> geq(proper(X1),proper(X2))
proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3))
proper(minus(X1,X2)) -> minus(proper(X1),proper(X2))
proper(s(X)) -> s(proper(X))
proper(true()) -> ok(true())
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
- Signature:
{active/1,div/2,geq/2,if/3,minus/2,proper/1,s/1,top/1} / {0/0,false/0,mark/1,ok/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {active,div,geq,if,minus,proper,s,top} and constructors {0
,false,mark,ok,true}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
* Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
active(div(X1,X2)) -> div(active(X1),X2)
active(div(0(),s(Y))) -> mark(0())
active(div(s(X),s(Y))) -> mark(if(geq(X,Y),s(div(minus(X,Y),s(Y))),0()))
active(geq(X,0())) -> mark(true())
active(geq(0(),s(Y))) -> mark(false())
active(geq(s(X),s(Y))) -> mark(geq(X,Y))
active(if(X1,X2,X3)) -> if(active(X1),X2,X3)
active(if(false(),X,Y)) -> mark(Y)
active(if(true(),X,Y)) -> mark(X)
active(minus(0(),Y)) -> mark(0())
active(minus(s(X),s(Y))) -> mark(minus(X,Y))
active(s(X)) -> s(active(X))
div(mark(X1),X2) -> mark(div(X1,X2))
div(ok(X1),ok(X2)) -> ok(div(X1,X2))
geq(ok(X1),ok(X2)) -> ok(geq(X1,X2))
if(mark(X1),X2,X3) -> mark(if(X1,X2,X3))
if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3))
minus(ok(X1),ok(X2)) -> ok(minus(X1,X2))
proper(0()) -> ok(0())
proper(div(X1,X2)) -> div(proper(X1),proper(X2))
proper(false()) -> ok(false())
proper(geq(X1,X2)) -> geq(proper(X1),proper(X2))
proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3))
proper(minus(X1,X2)) -> minus(proper(X1),proper(X2))
proper(s(X)) -> s(proper(X))
proper(true()) -> ok(true())
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
- Signature:
{active/1,div/2,geq/2,if/3,minus/2,proper/1,s/1,top/1} / {0/0,false/0,mark/1,ok/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {active,div,geq,if,minus,proper,s,top} and constructors {0
,false,mark,ok,true}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
div(x,y){x -> mark(x)} =
div(mark(x),y) ->^+ mark(div(x,y))
= C[div(x,y) = div(x,y){}]
WORST_CASE(Omega(n^1),?)