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