* Step 1: Sum WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
2ndsneg(X1,mark(X2)) -> mark(2ndsneg(X1,X2))
2ndsneg(mark(X1),X2) -> mark(2ndsneg(X1,X2))
2ndsneg(ok(X1),ok(X2)) -> ok(2ndsneg(X1,X2))
2ndspos(X1,mark(X2)) -> mark(2ndspos(X1,X2))
2ndspos(mark(X1),X2) -> mark(2ndspos(X1,X2))
2ndspos(ok(X1),ok(X2)) -> ok(2ndspos(X1,X2))
active(2ndsneg(X1,X2)) -> 2ndsneg(X1,active(X2))
active(2ndsneg(X1,X2)) -> 2ndsneg(active(X1),X2)
active(2ndsneg(0(),Z)) -> mark(rnil())
active(2ndsneg(s(N),cons(X,Z))) -> mark(2ndsneg(s(N),cons2(X,Z)))
active(2ndsneg(s(N),cons2(X,cons(Y,Z)))) -> mark(rcons(negrecip(Y),2ndspos(N,Z)))
active(2ndspos(X1,X2)) -> 2ndspos(X1,active(X2))
active(2ndspos(X1,X2)) -> 2ndspos(active(X1),X2)
active(2ndspos(0(),Z)) -> mark(rnil())
active(2ndspos(s(N),cons(X,Z))) -> mark(2ndspos(s(N),cons2(X,Z)))
active(2ndspos(s(N),cons2(X,cons(Y,Z)))) -> mark(rcons(posrecip(Y),2ndsneg(N,Z)))
active(cons(X1,X2)) -> cons(active(X1),X2)
active(cons2(X1,X2)) -> cons2(X1,active(X2))
active(from(X)) -> from(active(X))
active(from(X)) -> mark(cons(X,from(s(X))))
active(negrecip(X)) -> negrecip(active(X))
active(pi(X)) -> mark(2ndspos(X,from(0())))
active(pi(X)) -> pi(active(X))
active(plus(X1,X2)) -> plus(X1,active(X2))
active(plus(X1,X2)) -> plus(active(X1),X2)
active(plus(0(),Y)) -> mark(Y)
active(plus(s(X),Y)) -> mark(s(plus(X,Y)))
active(posrecip(X)) -> posrecip(active(X))
active(rcons(X1,X2)) -> rcons(X1,active(X2))
active(rcons(X1,X2)) -> rcons(active(X1),X2)
active(s(X)) -> s(active(X))
active(square(X)) -> mark(times(X,X))
active(square(X)) -> square(active(X))
active(times(X1,X2)) -> times(X1,active(X2))
active(times(X1,X2)) -> times(active(X1),X2)
active(times(0(),Y)) -> mark(0())
active(times(s(X),Y)) -> mark(plus(Y,times(X,Y)))
cons(mark(X1),X2) -> mark(cons(X1,X2))
cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
cons2(X1,mark(X2)) -> mark(cons2(X1,X2))
cons2(ok(X1),ok(X2)) -> ok(cons2(X1,X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
negrecip(mark(X)) -> mark(negrecip(X))
negrecip(ok(X)) -> ok(negrecip(X))
pi(mark(X)) -> mark(pi(X))
pi(ok(X)) -> ok(pi(X))
plus(X1,mark(X2)) -> mark(plus(X1,X2))
plus(mark(X1),X2) -> mark(plus(X1,X2))
plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
posrecip(mark(X)) -> mark(posrecip(X))
posrecip(ok(X)) -> ok(posrecip(X))
proper(0()) -> ok(0())
proper(2ndsneg(X1,X2)) -> 2ndsneg(proper(X1),proper(X2))
proper(2ndspos(X1,X2)) -> 2ndspos(proper(X1),proper(X2))
proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
proper(cons2(X1,X2)) -> cons2(proper(X1),proper(X2))
proper(from(X)) -> from(proper(X))
proper(negrecip(X)) -> negrecip(proper(X))
proper(nil()) -> ok(nil())
proper(pi(X)) -> pi(proper(X))
proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
proper(posrecip(X)) -> posrecip(proper(X))
proper(rcons(X1,X2)) -> rcons(proper(X1),proper(X2))
proper(rnil()) -> ok(rnil())
proper(s(X)) -> s(proper(X))
proper(square(X)) -> square(proper(X))
proper(times(X1,X2)) -> times(proper(X1),proper(X2))
rcons(X1,mark(X2)) -> mark(rcons(X1,X2))
rcons(mark(X1),X2) -> mark(rcons(X1,X2))
rcons(ok(X1),ok(X2)) -> ok(rcons(X1,X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
square(mark(X)) -> mark(square(X))
square(ok(X)) -> ok(square(X))
times(X1,mark(X2)) -> mark(times(X1,X2))
times(mark(X1),X2) -> mark(times(X1,X2))
times(ok(X1),ok(X2)) -> ok(times(X1,X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
- Signature:
{2ndsneg/2,2ndspos/2,active/1,cons/2,cons2/2,from/1,negrecip/1,pi/1,plus/2,posrecip/1,proper/1,rcons/2,s/1
,square/1,times/2,top/1} / {0/0,mark/1,nil/0,ok/1,rnil/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,active,cons,cons2,from,negrecip,pi,plus
,posrecip,proper,rcons,s,square,times,top} and constructors {0,mark,nil,ok,rnil}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
* Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
2ndsneg(X1,mark(X2)) -> mark(2ndsneg(X1,X2))
2ndsneg(mark(X1),X2) -> mark(2ndsneg(X1,X2))
2ndsneg(ok(X1),ok(X2)) -> ok(2ndsneg(X1,X2))
2ndspos(X1,mark(X2)) -> mark(2ndspos(X1,X2))
2ndspos(mark(X1),X2) -> mark(2ndspos(X1,X2))
2ndspos(ok(X1),ok(X2)) -> ok(2ndspos(X1,X2))
active(2ndsneg(X1,X2)) -> 2ndsneg(X1,active(X2))
active(2ndsneg(X1,X2)) -> 2ndsneg(active(X1),X2)
active(2ndsneg(0(),Z)) -> mark(rnil())
active(2ndsneg(s(N),cons(X,Z))) -> mark(2ndsneg(s(N),cons2(X,Z)))
active(2ndsneg(s(N),cons2(X,cons(Y,Z)))) -> mark(rcons(negrecip(Y),2ndspos(N,Z)))
active(2ndspos(X1,X2)) -> 2ndspos(X1,active(X2))
active(2ndspos(X1,X2)) -> 2ndspos(active(X1),X2)
active(2ndspos(0(),Z)) -> mark(rnil())
active(2ndspos(s(N),cons(X,Z))) -> mark(2ndspos(s(N),cons2(X,Z)))
active(2ndspos(s(N),cons2(X,cons(Y,Z)))) -> mark(rcons(posrecip(Y),2ndsneg(N,Z)))
active(cons(X1,X2)) -> cons(active(X1),X2)
active(cons2(X1,X2)) -> cons2(X1,active(X2))
active(from(X)) -> from(active(X))
active(from(X)) -> mark(cons(X,from(s(X))))
active(negrecip(X)) -> negrecip(active(X))
active(pi(X)) -> mark(2ndspos(X,from(0())))
active(pi(X)) -> pi(active(X))
active(plus(X1,X2)) -> plus(X1,active(X2))
active(plus(X1,X2)) -> plus(active(X1),X2)
active(plus(0(),Y)) -> mark(Y)
active(plus(s(X),Y)) -> mark(s(plus(X,Y)))
active(posrecip(X)) -> posrecip(active(X))
active(rcons(X1,X2)) -> rcons(X1,active(X2))
active(rcons(X1,X2)) -> rcons(active(X1),X2)
active(s(X)) -> s(active(X))
active(square(X)) -> mark(times(X,X))
active(square(X)) -> square(active(X))
active(times(X1,X2)) -> times(X1,active(X2))
active(times(X1,X2)) -> times(active(X1),X2)
active(times(0(),Y)) -> mark(0())
active(times(s(X),Y)) -> mark(plus(Y,times(X,Y)))
cons(mark(X1),X2) -> mark(cons(X1,X2))
cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
cons2(X1,mark(X2)) -> mark(cons2(X1,X2))
cons2(ok(X1),ok(X2)) -> ok(cons2(X1,X2))
from(mark(X)) -> mark(from(X))
from(ok(X)) -> ok(from(X))
negrecip(mark(X)) -> mark(negrecip(X))
negrecip(ok(X)) -> ok(negrecip(X))
pi(mark(X)) -> mark(pi(X))
pi(ok(X)) -> ok(pi(X))
plus(X1,mark(X2)) -> mark(plus(X1,X2))
plus(mark(X1),X2) -> mark(plus(X1,X2))
plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
posrecip(mark(X)) -> mark(posrecip(X))
posrecip(ok(X)) -> ok(posrecip(X))
proper(0()) -> ok(0())
proper(2ndsneg(X1,X2)) -> 2ndsneg(proper(X1),proper(X2))
proper(2ndspos(X1,X2)) -> 2ndspos(proper(X1),proper(X2))
proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
proper(cons2(X1,X2)) -> cons2(proper(X1),proper(X2))
proper(from(X)) -> from(proper(X))
proper(negrecip(X)) -> negrecip(proper(X))
proper(nil()) -> ok(nil())
proper(pi(X)) -> pi(proper(X))
proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
proper(posrecip(X)) -> posrecip(proper(X))
proper(rcons(X1,X2)) -> rcons(proper(X1),proper(X2))
proper(rnil()) -> ok(rnil())
proper(s(X)) -> s(proper(X))
proper(square(X)) -> square(proper(X))
proper(times(X1,X2)) -> times(proper(X1),proper(X2))
rcons(X1,mark(X2)) -> mark(rcons(X1,X2))
rcons(mark(X1),X2) -> mark(rcons(X1,X2))
rcons(ok(X1),ok(X2)) -> ok(rcons(X1,X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
square(mark(X)) -> mark(square(X))
square(ok(X)) -> ok(square(X))
times(X1,mark(X2)) -> mark(times(X1,X2))
times(mark(X1),X2) -> mark(times(X1,X2))
times(ok(X1),ok(X2)) -> ok(times(X1,X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
- Signature:
{2ndsneg/2,2ndspos/2,active/1,cons/2,cons2/2,from/1,negrecip/1,pi/1,plus/2,posrecip/1,proper/1,rcons/2,s/1
,square/1,times/2,top/1} / {0/0,mark/1,nil/0,ok/1,rnil/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {2ndsneg,2ndspos,active,cons,cons2,from,negrecip,pi,plus
,posrecip,proper,rcons,s,square,times,top} and constructors {0,mark,nil,ok,rnil}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
2ndsneg(x,y){y -> mark(y)} =
2ndsneg(x,mark(y)) ->^+ mark(2ndsneg(x,y))
= C[2ndsneg(x,y) = 2ndsneg(x,y){}]
WORST_CASE(Omega(n^1),?)