* Step 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) -> mark(U12(X1,X2))
U12(ok(X1),ok(X2)) -> ok(U12(X1,X2))
U13(mark(X)) -> mark(U13(X))
U13(ok(X)) -> ok(U13(X))
U21(mark(X1),X2) -> mark(U21(X1,X2))
U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
U22(mark(X)) -> mark(U22(X))
U22(ok(X)) -> ok(U22(X))
U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3))
U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3))
U32(mark(X1),X2) -> mark(U32(X1,X2))
U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
U33(mark(X)) -> mark(U33(X))
U33(ok(X)) -> ok(U33(X))
U41(mark(X1),X2) -> mark(U41(X1,X2))
U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
U61(mark(X)) -> mark(U61(X))
U61(ok(X)) -> ok(U61(X))
U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
and(mark(X1),X2) -> mark(and(X1,X2))
and(ok(X1),ok(X2)) -> ok(and(X1,X2))
isNat(ok(X)) -> ok(isNat(X))
isNatKind(ok(X)) -> ok(isNatKind(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(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))
x(X1,mark(X2)) -> mark(x(X1,X2))
x(mark(X1),X2) -> mark(x(X1,X2))
x(ok(X1),ok(X2)) -> ok(x(X1,X2))
- Signature:
{U11/3,U12/2,U13/1,U21/2,U22/1,U31/3,U32/2,U33/1,U41/2,U51/3,U61/1,U71/3,and/2,isNat/1,isNatKind/1,plus/2
,proper/1,s/1,top/1,x/2} / {0/0,active/1,mark/1,ok/1,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {U11,U12,U13,U21,U22,U31,U32,U33,U41,U51,U61,U71,and,isNat
,isNatKind,plus,proper,s,top,x} and constructors {0,active,mark,ok,tt}
+ Applied Processor:
Bounds {initialAutomaton = minimal, enrichment = match}
+ Details:
The problem is match-bounded by 2.
The enriched problem is compatible with follwoing automaton.
0_0() -> 2
0_1() -> 3
U11_0(2,2,2) -> 1
U11_1(2,2,2) -> 3
U12_0(2,2) -> 1
U12_1(2,2) -> 3
U13_0(2) -> 1
U13_1(2) -> 3
U21_0(2,2) -> 1
U21_1(2,2) -> 3
U22_0(2) -> 1
U22_1(2) -> 3
U31_0(2,2,2) -> 1
U31_1(2,2,2) -> 3
U32_0(2,2) -> 1
U32_1(2,2) -> 3
U33_0(2) -> 1
U33_1(2) -> 3
U41_0(2,2) -> 1
U41_1(2,2) -> 3
U51_0(2,2,2) -> 1
U51_1(2,2,2) -> 3
U61_0(2) -> 1
U61_1(2) -> 3
U71_0(2,2,2) -> 1
U71_1(2,2,2) -> 3
active_0(2) -> 2
active_1(2) -> 4
active_2(3) -> 5
and_0(2,2) -> 1
and_1(2,2) -> 3
isNat_0(2) -> 1
isNat_1(2) -> 3
isNatKind_0(2) -> 1
isNatKind_1(2) -> 3
mark_0(2) -> 2
mark_1(3) -> 1
mark_1(3) -> 3
ok_0(2) -> 2
ok_1(3) -> 1
ok_1(3) -> 3
ok_1(3) -> 4
plus_0(2,2) -> 1
plus_1(2,2) -> 3
proper_0(2) -> 1
proper_1(2) -> 4
s_0(2) -> 1
s_1(2) -> 3
top_0(2) -> 1
top_1(4) -> 1
top_2(5) -> 1
tt_0() -> 2
tt_1() -> 3
x_0(2,2) -> 1
x_1(2,2) -> 3
* Step 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) -> mark(U12(X1,X2))
U12(ok(X1),ok(X2)) -> ok(U12(X1,X2))
U13(mark(X)) -> mark(U13(X))
U13(ok(X)) -> ok(U13(X))
U21(mark(X1),X2) -> mark(U21(X1,X2))
U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
U22(mark(X)) -> mark(U22(X))
U22(ok(X)) -> ok(U22(X))
U31(mark(X1),X2,X3) -> mark(U31(X1,X2,X3))
U31(ok(X1),ok(X2),ok(X3)) -> ok(U31(X1,X2,X3))
U32(mark(X1),X2) -> mark(U32(X1,X2))
U32(ok(X1),ok(X2)) -> ok(U32(X1,X2))
U33(mark(X)) -> mark(U33(X))
U33(ok(X)) -> ok(U33(X))
U41(mark(X1),X2) -> mark(U41(X1,X2))
U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
U51(mark(X1),X2,X3) -> mark(U51(X1,X2,X3))
U51(ok(X1),ok(X2),ok(X3)) -> ok(U51(X1,X2,X3))
U61(mark(X)) -> mark(U61(X))
U61(ok(X)) -> ok(U61(X))
U71(mark(X1),X2,X3) -> mark(U71(X1,X2,X3))
U71(ok(X1),ok(X2),ok(X3)) -> ok(U71(X1,X2,X3))
and(mark(X1),X2) -> mark(and(X1,X2))
and(ok(X1),ok(X2)) -> ok(and(X1,X2))
isNat(ok(X)) -> ok(isNat(X))
isNatKind(ok(X)) -> ok(isNatKind(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(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))
x(X1,mark(X2)) -> mark(x(X1,X2))
x(mark(X1),X2) -> mark(x(X1,X2))
x(ok(X1),ok(X2)) -> ok(x(X1,X2))
- Signature:
{U11/3,U12/2,U13/1,U21/2,U22/1,U31/3,U32/2,U33/1,U41/2,U51/3,U61/1,U71/3,and/2,isNat/1,isNatKind/1,plus/2
,proper/1,s/1,top/1,x/2} / {0/0,active/1,mark/1,ok/1,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {U11,U12,U13,U21,U22,U31,U32,U33,U41,U51,U61,U71,and,isNat
,isNatKind,plus,proper,s,top,x} and constructors {0,active,mark,ok,tt}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(?,O(n^1))