* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
+ Considered Problem:
- Strict TRS:
active(and(X1,X2)) -> and(active(X1),X2)
active(and(tt(),X)) -> mark(X)
active(plus(N,0())) -> mark(N)
active(plus(N,s(M))) -> mark(s(plus(N,M)))
active(plus(X1,X2)) -> plus(X1,active(X2))
active(plus(X1,X2)) -> plus(active(X1),X2)
active(s(X)) -> s(active(X))
and(mark(X1),X2) -> mark(and(X1,X2))
and(ok(X1),ok(X2)) -> ok(and(X1,X2))
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))
proper(0()) -> ok(0())
proper(and(X1,X2)) -> and(proper(X1),proper(X2))
proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
proper(s(X)) -> s(proper(X))
proper(tt()) -> ok(tt())
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,and/2,plus/2,proper/1,s/1,top/1} / {0/0,mark/1,ok/1,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {active,and,plus,proper,s,top} and constructors {0,mark,ok
,tt}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
active(and(X1,X2)) -> and(active(X1),X2)
active(and(tt(),X)) -> mark(X)
active(plus(N,0())) -> mark(N)
active(plus(N,s(M))) -> mark(s(plus(N,M)))
active(plus(X1,X2)) -> plus(X1,active(X2))
active(plus(X1,X2)) -> plus(active(X1),X2)
active(s(X)) -> s(active(X))
and(mark(X1),X2) -> mark(and(X1,X2))
and(ok(X1),ok(X2)) -> ok(and(X1,X2))
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))
proper(0()) -> ok(0())
proper(and(X1,X2)) -> and(proper(X1),proper(X2))
proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
proper(s(X)) -> s(proper(X))
proper(tt()) -> ok(tt())
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,and/2,plus/2,proper/1,s/1,top/1} / {0/0,mark/1,ok/1,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {active,and,plus,proper,s,top} and constructors {0,mark,ok
,tt}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
and(x,y){x -> mark(x)} =
and(mark(x),y) ->^+ mark(and(x,y))
= C[and(x,y) = and(x,y){}]
** Step 1.b:1: Bounds WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
active(and(X1,X2)) -> and(active(X1),X2)
active(and(tt(),X)) -> mark(X)
active(plus(N,0())) -> mark(N)
active(plus(N,s(M))) -> mark(s(plus(N,M)))
active(plus(X1,X2)) -> plus(X1,active(X2))
active(plus(X1,X2)) -> plus(active(X1),X2)
active(s(X)) -> s(active(X))
and(mark(X1),X2) -> mark(and(X1,X2))
and(ok(X1),ok(X2)) -> ok(and(X1,X2))
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))
proper(0()) -> ok(0())
proper(and(X1,X2)) -> and(proper(X1),proper(X2))
proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
proper(s(X)) -> s(proper(X))
proper(tt()) -> ok(tt())
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,and/2,plus/2,proper/1,s/1,top/1} / {0/0,mark/1,ok/1,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {active,and,plus,proper,s,top} and constructors {0,mark,ok
,tt}
+ Applied Processor:
Bounds {initialAutomaton = perSymbol, enrichment = match}
+ Details:
The problem is match-bounded by 2.
The enriched problem is compatible with follwoing automaton.
0_0() -> 1
0_1() -> 13
active_0(1) -> 2
active_0(4) -> 2
active_0(5) -> 2
active_0(10) -> 2
active_1(1) -> 15
active_1(4) -> 15
active_1(5) -> 15
active_1(10) -> 15
active_2(13) -> 16
and_0(1,1) -> 3
and_0(1,4) -> 3
and_0(1,5) -> 3
and_0(1,10) -> 3
and_0(4,1) -> 3
and_0(4,4) -> 3
and_0(4,5) -> 3
and_0(4,10) -> 3
and_0(5,1) -> 3
and_0(5,4) -> 3
and_0(5,5) -> 3
and_0(5,10) -> 3
and_0(10,1) -> 3
and_0(10,4) -> 3
and_0(10,5) -> 3
and_0(10,10) -> 3
and_1(1,1) -> 11
and_1(1,4) -> 11
and_1(1,5) -> 11
and_1(1,10) -> 11
and_1(4,1) -> 11
and_1(4,4) -> 11
and_1(4,5) -> 11
and_1(4,10) -> 11
and_1(5,1) -> 11
and_1(5,4) -> 11
and_1(5,5) -> 11
and_1(5,10) -> 11
and_1(10,1) -> 11
and_1(10,4) -> 11
and_1(10,5) -> 11
and_1(10,10) -> 11
mark_0(1) -> 4
mark_0(4) -> 4
mark_0(5) -> 4
mark_0(10) -> 4
mark_1(11) -> 3
mark_1(11) -> 11
mark_1(12) -> 6
mark_1(12) -> 12
mark_1(14) -> 8
mark_1(14) -> 14
ok_0(1) -> 5
ok_0(4) -> 5
ok_0(5) -> 5
ok_0(10) -> 5
ok_1(11) -> 3
ok_1(11) -> 11
ok_1(12) -> 6
ok_1(12) -> 12
ok_1(13) -> 7
ok_1(13) -> 15
ok_1(14) -> 8
ok_1(14) -> 14
plus_0(1,1) -> 6
plus_0(1,4) -> 6
plus_0(1,5) -> 6
plus_0(1,10) -> 6
plus_0(4,1) -> 6
plus_0(4,4) -> 6
plus_0(4,5) -> 6
plus_0(4,10) -> 6
plus_0(5,1) -> 6
plus_0(5,4) -> 6
plus_0(5,5) -> 6
plus_0(5,10) -> 6
plus_0(10,1) -> 6
plus_0(10,4) -> 6
plus_0(10,5) -> 6
plus_0(10,10) -> 6
plus_1(1,1) -> 12
plus_1(1,4) -> 12
plus_1(1,5) -> 12
plus_1(1,10) -> 12
plus_1(4,1) -> 12
plus_1(4,4) -> 12
plus_1(4,5) -> 12
plus_1(4,10) -> 12
plus_1(5,1) -> 12
plus_1(5,4) -> 12
plus_1(5,5) -> 12
plus_1(5,10) -> 12
plus_1(10,1) -> 12
plus_1(10,4) -> 12
plus_1(10,5) -> 12
plus_1(10,10) -> 12
proper_0(1) -> 7
proper_0(4) -> 7
proper_0(5) -> 7
proper_0(10) -> 7
proper_1(1) -> 15
proper_1(4) -> 15
proper_1(5) -> 15
proper_1(10) -> 15
s_0(1) -> 8
s_0(4) -> 8
s_0(5) -> 8
s_0(10) -> 8
s_1(1) -> 14
s_1(4) -> 14
s_1(5) -> 14
s_1(10) -> 14
top_0(1) -> 9
top_0(4) -> 9
top_0(5) -> 9
top_0(10) -> 9
top_1(15) -> 9
top_2(16) -> 9
tt_0() -> 10
tt_1() -> 13
** Step 1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
active(and(X1,X2)) -> and(active(X1),X2)
active(and(tt(),X)) -> mark(X)
active(plus(N,0())) -> mark(N)
active(plus(N,s(M))) -> mark(s(plus(N,M)))
active(plus(X1,X2)) -> plus(X1,active(X2))
active(plus(X1,X2)) -> plus(active(X1),X2)
active(s(X)) -> s(active(X))
and(mark(X1),X2) -> mark(and(X1,X2))
and(ok(X1),ok(X2)) -> ok(and(X1,X2))
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))
proper(0()) -> ok(0())
proper(and(X1,X2)) -> and(proper(X1),proper(X2))
proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
proper(s(X)) -> s(proper(X))
proper(tt()) -> ok(tt())
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,and/2,plus/2,proper/1,s/1,top/1} / {0/0,mark/1,ok/1,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {active,and,plus,proper,s,top} and constructors {0,mark,ok
,tt}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(Omega(n^1),O(n^1))