* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
+ Considered Problem:
- Strict TRS:
U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3))
U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
active(U11(tt(),M,N)) -> mark(U12(tt(),M,N))
active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3)
active(U12(tt(),M,N)) -> mark(s(plus(N,M)))
active(plus(N,0())) -> mark(N)
active(plus(N,s(M))) -> mark(U11(tt(),M,N))
active(plus(X1,X2)) -> plus(X1,active(X2))
active(plus(X1,X2)) -> plus(active(X1),X2)
active(s(X)) -> s(active(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))
proper(0()) -> ok(0())
proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
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:
{U11/3,U12/3,active/1,plus/2,proper/1,s/1,top/1} / {0/0,mark/1,ok/1,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {U11,U12,active,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:
U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3))
U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
active(U11(tt(),M,N)) -> mark(U12(tt(),M,N))
active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3)
active(U12(tt(),M,N)) -> mark(s(plus(N,M)))
active(plus(N,0())) -> mark(N)
active(plus(N,s(M))) -> mark(U11(tt(),M,N))
active(plus(X1,X2)) -> plus(X1,active(X2))
active(plus(X1,X2)) -> plus(active(X1),X2)
active(s(X)) -> s(active(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))
proper(0()) -> ok(0())
proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
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:
{U11/3,U12/3,active/1,plus/2,proper/1,s/1,top/1} / {0/0,mark/1,ok/1,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {U11,U12,active,plus,proper,s,top} and constructors {0
,mark,ok,tt}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
U11(x,y,z){x -> mark(x)} =
U11(mark(x),y,z) ->^+ mark(U11(x,y,z))
= C[U11(x,y,z) = U11(x,y,z){}]
** Step 1.b:1: Bounds WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3))
U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
active(U11(tt(),M,N)) -> mark(U12(tt(),M,N))
active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3)
active(U12(tt(),M,N)) -> mark(s(plus(N,M)))
active(plus(N,0())) -> mark(N)
active(plus(N,s(M))) -> mark(U11(tt(),M,N))
active(plus(X1,X2)) -> plus(X1,active(X2))
active(plus(X1,X2)) -> plus(active(X1),X2)
active(s(X)) -> s(active(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))
proper(0()) -> ok(0())
proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
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:
{U11/3,U12/3,active/1,plus/2,proper/1,s/1,top/1} / {0/0,mark/1,ok/1,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {U11,U12,active,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() -> 15
U11_0(1,1,1) -> 2
U11_0(1,1,5) -> 2
U11_0(1,1,6) -> 2
U11_0(1,1,11) -> 2
U11_0(1,5,1) -> 2
U11_0(1,5,5) -> 2
U11_0(1,5,6) -> 2
U11_0(1,5,11) -> 2
U11_0(1,6,1) -> 2
U11_0(1,6,5) -> 2
U11_0(1,6,6) -> 2
U11_0(1,6,11) -> 2
U11_0(1,11,1) -> 2
U11_0(1,11,5) -> 2
U11_0(1,11,6) -> 2
U11_0(1,11,11) -> 2
U11_0(5,1,1) -> 2
U11_0(5,1,5) -> 2
U11_0(5,1,6) -> 2
U11_0(5,1,11) -> 2
U11_0(5,5,1) -> 2
U11_0(5,5,5) -> 2
U11_0(5,5,6) -> 2
U11_0(5,5,11) -> 2
U11_0(5,6,1) -> 2
U11_0(5,6,5) -> 2
U11_0(5,6,6) -> 2
U11_0(5,6,11) -> 2
U11_0(5,11,1) -> 2
U11_0(5,11,5) -> 2
U11_0(5,11,6) -> 2
U11_0(5,11,11) -> 2
U11_0(6,1,1) -> 2
U11_0(6,1,5) -> 2
U11_0(6,1,6) -> 2
U11_0(6,1,11) -> 2
U11_0(6,5,1) -> 2
U11_0(6,5,5) -> 2
U11_0(6,5,6) -> 2
U11_0(6,5,11) -> 2
U11_0(6,6,1) -> 2
U11_0(6,6,5) -> 2
U11_0(6,6,6) -> 2
U11_0(6,6,11) -> 2
U11_0(6,11,1) -> 2
U11_0(6,11,5) -> 2
U11_0(6,11,6) -> 2
U11_0(6,11,11) -> 2
U11_0(11,1,1) -> 2
U11_0(11,1,5) -> 2
U11_0(11,1,6) -> 2
U11_0(11,1,11) -> 2
U11_0(11,5,1) -> 2
U11_0(11,5,5) -> 2
U11_0(11,5,6) -> 2
U11_0(11,5,11) -> 2
U11_0(11,6,1) -> 2
U11_0(11,6,5) -> 2
U11_0(11,6,6) -> 2
U11_0(11,6,11) -> 2
U11_0(11,11,1) -> 2
U11_0(11,11,5) -> 2
U11_0(11,11,6) -> 2
U11_0(11,11,11) -> 2
U11_1(1,1,1) -> 12
U11_1(1,1,5) -> 12
U11_1(1,1,6) -> 12
U11_1(1,1,11) -> 12
U11_1(1,5,1) -> 12
U11_1(1,5,5) -> 12
U11_1(1,5,6) -> 12
U11_1(1,5,11) -> 12
U11_1(1,6,1) -> 12
U11_1(1,6,5) -> 12
U11_1(1,6,6) -> 12
U11_1(1,6,11) -> 12
U11_1(1,11,1) -> 12
U11_1(1,11,5) -> 12
U11_1(1,11,6) -> 12
U11_1(1,11,11) -> 12
U11_1(5,1,1) -> 12
U11_1(5,1,5) -> 12
U11_1(5,1,6) -> 12
U11_1(5,1,11) -> 12
U11_1(5,5,1) -> 12
U11_1(5,5,5) -> 12
U11_1(5,5,6) -> 12
U11_1(5,5,11) -> 12
U11_1(5,6,1) -> 12
U11_1(5,6,5) -> 12
U11_1(5,6,6) -> 12
U11_1(5,6,11) -> 12
U11_1(5,11,1) -> 12
U11_1(5,11,5) -> 12
U11_1(5,11,6) -> 12
U11_1(5,11,11) -> 12
U11_1(6,1,1) -> 12
U11_1(6,1,5) -> 12
U11_1(6,1,6) -> 12
U11_1(6,1,11) -> 12
U11_1(6,5,1) -> 12
U11_1(6,5,5) -> 12
U11_1(6,5,6) -> 12
U11_1(6,5,11) -> 12
U11_1(6,6,1) -> 12
U11_1(6,6,5) -> 12
U11_1(6,6,6) -> 12
U11_1(6,6,11) -> 12
U11_1(6,11,1) -> 12
U11_1(6,11,5) -> 12
U11_1(6,11,6) -> 12
U11_1(6,11,11) -> 12
U11_1(11,1,1) -> 12
U11_1(11,1,5) -> 12
U11_1(11,1,6) -> 12
U11_1(11,1,11) -> 12
U11_1(11,5,1) -> 12
U11_1(11,5,5) -> 12
U11_1(11,5,6) -> 12
U11_1(11,5,11) -> 12
U11_1(11,6,1) -> 12
U11_1(11,6,5) -> 12
U11_1(11,6,6) -> 12
U11_1(11,6,11) -> 12
U11_1(11,11,1) -> 12
U11_1(11,11,5) -> 12
U11_1(11,11,6) -> 12
U11_1(11,11,11) -> 12
U12_0(1,1,1) -> 3
U12_0(1,1,5) -> 3
U12_0(1,1,6) -> 3
U12_0(1,1,11) -> 3
U12_0(1,5,1) -> 3
U12_0(1,5,5) -> 3
U12_0(1,5,6) -> 3
U12_0(1,5,11) -> 3
U12_0(1,6,1) -> 3
U12_0(1,6,5) -> 3
U12_0(1,6,6) -> 3
U12_0(1,6,11) -> 3
U12_0(1,11,1) -> 3
U12_0(1,11,5) -> 3
U12_0(1,11,6) -> 3
U12_0(1,11,11) -> 3
U12_0(5,1,1) -> 3
U12_0(5,1,5) -> 3
U12_0(5,1,6) -> 3
U12_0(5,1,11) -> 3
U12_0(5,5,1) -> 3
U12_0(5,5,5) -> 3
U12_0(5,5,6) -> 3
U12_0(5,5,11) -> 3
U12_0(5,6,1) -> 3
U12_0(5,6,5) -> 3
U12_0(5,6,6) -> 3
U12_0(5,6,11) -> 3
U12_0(5,11,1) -> 3
U12_0(5,11,5) -> 3
U12_0(5,11,6) -> 3
U12_0(5,11,11) -> 3
U12_0(6,1,1) -> 3
U12_0(6,1,5) -> 3
U12_0(6,1,6) -> 3
U12_0(6,1,11) -> 3
U12_0(6,5,1) -> 3
U12_0(6,5,5) -> 3
U12_0(6,5,6) -> 3
U12_0(6,5,11) -> 3
U12_0(6,6,1) -> 3
U12_0(6,6,5) -> 3
U12_0(6,6,6) -> 3
U12_0(6,6,11) -> 3
U12_0(6,11,1) -> 3
U12_0(6,11,5) -> 3
U12_0(6,11,6) -> 3
U12_0(6,11,11) -> 3
U12_0(11,1,1) -> 3
U12_0(11,1,5) -> 3
U12_0(11,1,6) -> 3
U12_0(11,1,11) -> 3
U12_0(11,5,1) -> 3
U12_0(11,5,5) -> 3
U12_0(11,5,6) -> 3
U12_0(11,5,11) -> 3
U12_0(11,6,1) -> 3
U12_0(11,6,5) -> 3
U12_0(11,6,6) -> 3
U12_0(11,6,11) -> 3
U12_0(11,11,1) -> 3
U12_0(11,11,5) -> 3
U12_0(11,11,6) -> 3
U12_0(11,11,11) -> 3
U12_1(1,1,1) -> 13
U12_1(1,1,5) -> 13
U12_1(1,1,6) -> 13
U12_1(1,1,11) -> 13
U12_1(1,5,1) -> 13
U12_1(1,5,5) -> 13
U12_1(1,5,6) -> 13
U12_1(1,5,11) -> 13
U12_1(1,6,1) -> 13
U12_1(1,6,5) -> 13
U12_1(1,6,6) -> 13
U12_1(1,6,11) -> 13
U12_1(1,11,1) -> 13
U12_1(1,11,5) -> 13
U12_1(1,11,6) -> 13
U12_1(1,11,11) -> 13
U12_1(5,1,1) -> 13
U12_1(5,1,5) -> 13
U12_1(5,1,6) -> 13
U12_1(5,1,11) -> 13
U12_1(5,5,1) -> 13
U12_1(5,5,5) -> 13
U12_1(5,5,6) -> 13
U12_1(5,5,11) -> 13
U12_1(5,6,1) -> 13
U12_1(5,6,5) -> 13
U12_1(5,6,6) -> 13
U12_1(5,6,11) -> 13
U12_1(5,11,1) -> 13
U12_1(5,11,5) -> 13
U12_1(5,11,6) -> 13
U12_1(5,11,11) -> 13
U12_1(6,1,1) -> 13
U12_1(6,1,5) -> 13
U12_1(6,1,6) -> 13
U12_1(6,1,11) -> 13
U12_1(6,5,1) -> 13
U12_1(6,5,5) -> 13
U12_1(6,5,6) -> 13
U12_1(6,5,11) -> 13
U12_1(6,6,1) -> 13
U12_1(6,6,5) -> 13
U12_1(6,6,6) -> 13
U12_1(6,6,11) -> 13
U12_1(6,11,1) -> 13
U12_1(6,11,5) -> 13
U12_1(6,11,6) -> 13
U12_1(6,11,11) -> 13
U12_1(11,1,1) -> 13
U12_1(11,1,5) -> 13
U12_1(11,1,6) -> 13
U12_1(11,1,11) -> 13
U12_1(11,5,1) -> 13
U12_1(11,5,5) -> 13
U12_1(11,5,6) -> 13
U12_1(11,5,11) -> 13
U12_1(11,6,1) -> 13
U12_1(11,6,5) -> 13
U12_1(11,6,6) -> 13
U12_1(11,6,11) -> 13
U12_1(11,11,1) -> 13
U12_1(11,11,5) -> 13
U12_1(11,11,6) -> 13
U12_1(11,11,11) -> 13
active_0(1) -> 4
active_0(5) -> 4
active_0(6) -> 4
active_0(11) -> 4
active_1(1) -> 17
active_1(5) -> 17
active_1(6) -> 17
active_1(11) -> 17
active_2(15) -> 18
mark_0(1) -> 5
mark_0(5) -> 5
mark_0(6) -> 5
mark_0(11) -> 5
mark_1(12) -> 2
mark_1(12) -> 12
mark_1(13) -> 3
mark_1(13) -> 13
mark_1(14) -> 7
mark_1(14) -> 14
mark_1(16) -> 9
mark_1(16) -> 16
ok_0(1) -> 6
ok_0(5) -> 6
ok_0(6) -> 6
ok_0(11) -> 6
ok_1(12) -> 2
ok_1(12) -> 12
ok_1(13) -> 3
ok_1(13) -> 13
ok_1(14) -> 7
ok_1(14) -> 14
ok_1(15) -> 8
ok_1(15) -> 17
ok_1(16) -> 9
ok_1(16) -> 16
plus_0(1,1) -> 7
plus_0(1,5) -> 7
plus_0(1,6) -> 7
plus_0(1,11) -> 7
plus_0(5,1) -> 7
plus_0(5,5) -> 7
plus_0(5,6) -> 7
plus_0(5,11) -> 7
plus_0(6,1) -> 7
plus_0(6,5) -> 7
plus_0(6,6) -> 7
plus_0(6,11) -> 7
plus_0(11,1) -> 7
plus_0(11,5) -> 7
plus_0(11,6) -> 7
plus_0(11,11) -> 7
plus_1(1,1) -> 14
plus_1(1,5) -> 14
plus_1(1,6) -> 14
plus_1(1,11) -> 14
plus_1(5,1) -> 14
plus_1(5,5) -> 14
plus_1(5,6) -> 14
plus_1(5,11) -> 14
plus_1(6,1) -> 14
plus_1(6,5) -> 14
plus_1(6,6) -> 14
plus_1(6,11) -> 14
plus_1(11,1) -> 14
plus_1(11,5) -> 14
plus_1(11,6) -> 14
plus_1(11,11) -> 14
proper_0(1) -> 8
proper_0(5) -> 8
proper_0(6) -> 8
proper_0(11) -> 8
proper_1(1) -> 17
proper_1(5) -> 17
proper_1(6) -> 17
proper_1(11) -> 17
s_0(1) -> 9
s_0(5) -> 9
s_0(6) -> 9
s_0(11) -> 9
s_1(1) -> 16
s_1(5) -> 16
s_1(6) -> 16
s_1(11) -> 16
top_0(1) -> 10
top_0(5) -> 10
top_0(6) -> 10
top_0(11) -> 10
top_1(17) -> 10
top_2(18) -> 10
tt_0() -> 11
tt_1() -> 15
** Step 1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3))
U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3))
active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
active(U11(tt(),M,N)) -> mark(U12(tt(),M,N))
active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3)
active(U12(tt(),M,N)) -> mark(s(plus(N,M)))
active(plus(N,0())) -> mark(N)
active(plus(N,s(M))) -> mark(U11(tt(),M,N))
active(plus(X1,X2)) -> plus(X1,active(X2))
active(plus(X1,X2)) -> plus(active(X1),X2)
active(s(X)) -> s(active(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))
proper(0()) -> ok(0())
proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3))
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:
{U11/3,U12/3,active/1,plus/2,proper/1,s/1,top/1} / {0/0,mark/1,ok/1,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {U11,U12,active,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))