* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
+ Considered Problem:
- Strict TRS:
active(f(X)) -> f(active(X))
active(f(X)) -> mark(if(X,c(),f(true())))
active(if(X1,X2,X3)) -> if(X1,active(X2),X3)
active(if(X1,X2,X3)) -> if(active(X1),X2,X3)
active(if(false(),X,Y)) -> mark(Y)
active(if(true(),X,Y)) -> mark(X)
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
if(X1,mark(X2),X3) -> mark(if(X1,X2,X3))
if(mark(X1),X2,X3) -> mark(if(X1,X2,X3))
if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3))
proper(c()) -> ok(c())
proper(f(X)) -> f(proper(X))
proper(false()) -> ok(false())
proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3))
proper(true()) -> ok(true())
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
- Signature:
{active/1,f/1,if/3,proper/1,top/1} / {c/0,false/0,mark/1,ok/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {active,f,if,proper,top} and constructors {c,false,mark,ok
,true}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
active(f(X)) -> f(active(X))
active(f(X)) -> mark(if(X,c(),f(true())))
active(if(X1,X2,X3)) -> if(X1,active(X2),X3)
active(if(X1,X2,X3)) -> if(active(X1),X2,X3)
active(if(false(),X,Y)) -> mark(Y)
active(if(true(),X,Y)) -> mark(X)
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
if(X1,mark(X2),X3) -> mark(if(X1,X2,X3))
if(mark(X1),X2,X3) -> mark(if(X1,X2,X3))
if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3))
proper(c()) -> ok(c())
proper(f(X)) -> f(proper(X))
proper(false()) -> ok(false())
proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3))
proper(true()) -> ok(true())
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
- Signature:
{active/1,f/1,if/3,proper/1,top/1} / {c/0,false/0,mark/1,ok/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {active,f,if,proper,top} and constructors {c,false,mark,ok
,true}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
f(x){x -> mark(x)} =
f(mark(x)) ->^+ mark(f(x))
= C[f(x) = f(x){}]
** Step 1.b:1: Bounds WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
active(f(X)) -> f(active(X))
active(f(X)) -> mark(if(X,c(),f(true())))
active(if(X1,X2,X3)) -> if(X1,active(X2),X3)
active(if(X1,X2,X3)) -> if(active(X1),X2,X3)
active(if(false(),X,Y)) -> mark(Y)
active(if(true(),X,Y)) -> mark(X)
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
if(X1,mark(X2),X3) -> mark(if(X1,X2,X3))
if(mark(X1),X2,X3) -> mark(if(X1,X2,X3))
if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3))
proper(c()) -> ok(c())
proper(f(X)) -> f(proper(X))
proper(false()) -> ok(false())
proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3))
proper(true()) -> ok(true())
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
- Signature:
{active/1,f/1,if/3,proper/1,top/1} / {c/0,false/0,mark/1,ok/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {active,f,if,proper,top} and constructors {c,false,mark,ok
,true}
+ Applied Processor:
Bounds {initialAutomaton = perSymbol, enrichment = match}
+ Details:
The problem is match-bounded by 2.
The enriched problem is compatible with follwoing automaton.
active_0(2) -> 1
active_0(4) -> 1
active_0(6) -> 1
active_0(7) -> 1
active_0(10) -> 1
active_1(2) -> 14
active_1(4) -> 14
active_1(6) -> 14
active_1(7) -> 14
active_1(10) -> 14
active_2(13) -> 15
c_0() -> 2
c_1() -> 13
f_0(2) -> 3
f_0(4) -> 3
f_0(6) -> 3
f_0(7) -> 3
f_0(10) -> 3
f_1(2) -> 11
f_1(4) -> 11
f_1(6) -> 11
f_1(7) -> 11
f_1(10) -> 11
false_0() -> 4
false_1() -> 13
if_0(2,2,2) -> 5
if_0(2,2,4) -> 5
if_0(2,2,6) -> 5
if_0(2,2,7) -> 5
if_0(2,2,10) -> 5
if_0(2,4,2) -> 5
if_0(2,4,4) -> 5
if_0(2,4,6) -> 5
if_0(2,4,7) -> 5
if_0(2,4,10) -> 5
if_0(2,6,2) -> 5
if_0(2,6,4) -> 5
if_0(2,6,6) -> 5
if_0(2,6,7) -> 5
if_0(2,6,10) -> 5
if_0(2,7,2) -> 5
if_0(2,7,4) -> 5
if_0(2,7,6) -> 5
if_0(2,7,7) -> 5
if_0(2,7,10) -> 5
if_0(2,10,2) -> 5
if_0(2,10,4) -> 5
if_0(2,10,6) -> 5
if_0(2,10,7) -> 5
if_0(2,10,10) -> 5
if_0(4,2,2) -> 5
if_0(4,2,4) -> 5
if_0(4,2,6) -> 5
if_0(4,2,7) -> 5
if_0(4,2,10) -> 5
if_0(4,4,2) -> 5
if_0(4,4,4) -> 5
if_0(4,4,6) -> 5
if_0(4,4,7) -> 5
if_0(4,4,10) -> 5
if_0(4,6,2) -> 5
if_0(4,6,4) -> 5
if_0(4,6,6) -> 5
if_0(4,6,7) -> 5
if_0(4,6,10) -> 5
if_0(4,7,2) -> 5
if_0(4,7,4) -> 5
if_0(4,7,6) -> 5
if_0(4,7,7) -> 5
if_0(4,7,10) -> 5
if_0(4,10,2) -> 5
if_0(4,10,4) -> 5
if_0(4,10,6) -> 5
if_0(4,10,7) -> 5
if_0(4,10,10) -> 5
if_0(6,2,2) -> 5
if_0(6,2,4) -> 5
if_0(6,2,6) -> 5
if_0(6,2,7) -> 5
if_0(6,2,10) -> 5
if_0(6,4,2) -> 5
if_0(6,4,4) -> 5
if_0(6,4,6) -> 5
if_0(6,4,7) -> 5
if_0(6,4,10) -> 5
if_0(6,6,2) -> 5
if_0(6,6,4) -> 5
if_0(6,6,6) -> 5
if_0(6,6,7) -> 5
if_0(6,6,10) -> 5
if_0(6,7,2) -> 5
if_0(6,7,4) -> 5
if_0(6,7,6) -> 5
if_0(6,7,7) -> 5
if_0(6,7,10) -> 5
if_0(6,10,2) -> 5
if_0(6,10,4) -> 5
if_0(6,10,6) -> 5
if_0(6,10,7) -> 5
if_0(6,10,10) -> 5
if_0(7,2,2) -> 5
if_0(7,2,4) -> 5
if_0(7,2,6) -> 5
if_0(7,2,7) -> 5
if_0(7,2,10) -> 5
if_0(7,4,2) -> 5
if_0(7,4,4) -> 5
if_0(7,4,6) -> 5
if_0(7,4,7) -> 5
if_0(7,4,10) -> 5
if_0(7,6,2) -> 5
if_0(7,6,4) -> 5
if_0(7,6,6) -> 5
if_0(7,6,7) -> 5
if_0(7,6,10) -> 5
if_0(7,7,2) -> 5
if_0(7,7,4) -> 5
if_0(7,7,6) -> 5
if_0(7,7,7) -> 5
if_0(7,7,10) -> 5
if_0(7,10,2) -> 5
if_0(7,10,4) -> 5
if_0(7,10,6) -> 5
if_0(7,10,7) -> 5
if_0(7,10,10) -> 5
if_0(10,2,2) -> 5
if_0(10,2,4) -> 5
if_0(10,2,6) -> 5
if_0(10,2,7) -> 5
if_0(10,2,10) -> 5
if_0(10,4,2) -> 5
if_0(10,4,4) -> 5
if_0(10,4,6) -> 5
if_0(10,4,7) -> 5
if_0(10,4,10) -> 5
if_0(10,6,2) -> 5
if_0(10,6,4) -> 5
if_0(10,6,6) -> 5
if_0(10,6,7) -> 5
if_0(10,6,10) -> 5
if_0(10,7,2) -> 5
if_0(10,7,4) -> 5
if_0(10,7,6) -> 5
if_0(10,7,7) -> 5
if_0(10,7,10) -> 5
if_0(10,10,2) -> 5
if_0(10,10,4) -> 5
if_0(10,10,6) -> 5
if_0(10,10,7) -> 5
if_0(10,10,10) -> 5
if_1(2,2,2) -> 12
if_1(2,2,4) -> 12
if_1(2,2,6) -> 12
if_1(2,2,7) -> 12
if_1(2,2,10) -> 12
if_1(2,4,2) -> 12
if_1(2,4,4) -> 12
if_1(2,4,6) -> 12
if_1(2,4,7) -> 12
if_1(2,4,10) -> 12
if_1(2,6,2) -> 12
if_1(2,6,4) -> 12
if_1(2,6,6) -> 12
if_1(2,6,7) -> 12
if_1(2,6,10) -> 12
if_1(2,7,2) -> 12
if_1(2,7,4) -> 12
if_1(2,7,6) -> 12
if_1(2,7,7) -> 12
if_1(2,7,10) -> 12
if_1(2,10,2) -> 12
if_1(2,10,4) -> 12
if_1(2,10,6) -> 12
if_1(2,10,7) -> 12
if_1(2,10,10) -> 12
if_1(4,2,2) -> 12
if_1(4,2,4) -> 12
if_1(4,2,6) -> 12
if_1(4,2,7) -> 12
if_1(4,2,10) -> 12
if_1(4,4,2) -> 12
if_1(4,4,4) -> 12
if_1(4,4,6) -> 12
if_1(4,4,7) -> 12
if_1(4,4,10) -> 12
if_1(4,6,2) -> 12
if_1(4,6,4) -> 12
if_1(4,6,6) -> 12
if_1(4,6,7) -> 12
if_1(4,6,10) -> 12
if_1(4,7,2) -> 12
if_1(4,7,4) -> 12
if_1(4,7,6) -> 12
if_1(4,7,7) -> 12
if_1(4,7,10) -> 12
if_1(4,10,2) -> 12
if_1(4,10,4) -> 12
if_1(4,10,6) -> 12
if_1(4,10,7) -> 12
if_1(4,10,10) -> 12
if_1(6,2,2) -> 12
if_1(6,2,4) -> 12
if_1(6,2,6) -> 12
if_1(6,2,7) -> 12
if_1(6,2,10) -> 12
if_1(6,4,2) -> 12
if_1(6,4,4) -> 12
if_1(6,4,6) -> 12
if_1(6,4,7) -> 12
if_1(6,4,10) -> 12
if_1(6,6,2) -> 12
if_1(6,6,4) -> 12
if_1(6,6,6) -> 12
if_1(6,6,7) -> 12
if_1(6,6,10) -> 12
if_1(6,7,2) -> 12
if_1(6,7,4) -> 12
if_1(6,7,6) -> 12
if_1(6,7,7) -> 12
if_1(6,7,10) -> 12
if_1(6,10,2) -> 12
if_1(6,10,4) -> 12
if_1(6,10,6) -> 12
if_1(6,10,7) -> 12
if_1(6,10,10) -> 12
if_1(7,2,2) -> 12
if_1(7,2,4) -> 12
if_1(7,2,6) -> 12
if_1(7,2,7) -> 12
if_1(7,2,10) -> 12
if_1(7,4,2) -> 12
if_1(7,4,4) -> 12
if_1(7,4,6) -> 12
if_1(7,4,7) -> 12
if_1(7,4,10) -> 12
if_1(7,6,2) -> 12
if_1(7,6,4) -> 12
if_1(7,6,6) -> 12
if_1(7,6,7) -> 12
if_1(7,6,10) -> 12
if_1(7,7,2) -> 12
if_1(7,7,4) -> 12
if_1(7,7,6) -> 12
if_1(7,7,7) -> 12
if_1(7,7,10) -> 12
if_1(7,10,2) -> 12
if_1(7,10,4) -> 12
if_1(7,10,6) -> 12
if_1(7,10,7) -> 12
if_1(7,10,10) -> 12
if_1(10,2,2) -> 12
if_1(10,2,4) -> 12
if_1(10,2,6) -> 12
if_1(10,2,7) -> 12
if_1(10,2,10) -> 12
if_1(10,4,2) -> 12
if_1(10,4,4) -> 12
if_1(10,4,6) -> 12
if_1(10,4,7) -> 12
if_1(10,4,10) -> 12
if_1(10,6,2) -> 12
if_1(10,6,4) -> 12
if_1(10,6,6) -> 12
if_1(10,6,7) -> 12
if_1(10,6,10) -> 12
if_1(10,7,2) -> 12
if_1(10,7,4) -> 12
if_1(10,7,6) -> 12
if_1(10,7,7) -> 12
if_1(10,7,10) -> 12
if_1(10,10,2) -> 12
if_1(10,10,4) -> 12
if_1(10,10,6) -> 12
if_1(10,10,7) -> 12
if_1(10,10,10) -> 12
mark_0(2) -> 6
mark_0(4) -> 6
mark_0(6) -> 6
mark_0(7) -> 6
mark_0(10) -> 6
mark_1(11) -> 3
mark_1(11) -> 11
mark_1(12) -> 5
mark_1(12) -> 12
ok_0(2) -> 7
ok_0(4) -> 7
ok_0(6) -> 7
ok_0(7) -> 7
ok_0(10) -> 7
ok_1(11) -> 3
ok_1(11) -> 11
ok_1(12) -> 5
ok_1(12) -> 12
ok_1(13) -> 8
ok_1(13) -> 14
proper_0(2) -> 8
proper_0(4) -> 8
proper_0(6) -> 8
proper_0(7) -> 8
proper_0(10) -> 8
proper_1(2) -> 14
proper_1(4) -> 14
proper_1(6) -> 14
proper_1(7) -> 14
proper_1(10) -> 14
top_0(2) -> 9
top_0(4) -> 9
top_0(6) -> 9
top_0(7) -> 9
top_0(10) -> 9
top_1(14) -> 9
top_2(15) -> 9
true_0() -> 10
true_1() -> 13
** Step 1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
active(f(X)) -> f(active(X))
active(f(X)) -> mark(if(X,c(),f(true())))
active(if(X1,X2,X3)) -> if(X1,active(X2),X3)
active(if(X1,X2,X3)) -> if(active(X1),X2,X3)
active(if(false(),X,Y)) -> mark(Y)
active(if(true(),X,Y)) -> mark(X)
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
if(X1,mark(X2),X3) -> mark(if(X1,X2,X3))
if(mark(X1),X2,X3) -> mark(if(X1,X2,X3))
if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3))
proper(c()) -> ok(c())
proper(f(X)) -> f(proper(X))
proper(false()) -> ok(false())
proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3))
proper(true()) -> ok(true())
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
- Signature:
{active/1,f/1,if/3,proper/1,top/1} / {c/0,false/0,mark/1,ok/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {active,f,if,proper,top} and constructors {c,false,mark,ok
,true}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(Omega(n^1),O(n^1))