* Step 1: Bounds WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3))
U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3))
U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3))
U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3))
U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3))
U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3))
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))
U13(mark(X1),X2,X3) -> mark(U13(X1,X2,X3))
U13(ok(X1),ok(X2),ok(X3)) -> ok(U13(X1,X2,X3))
U14(mark(X1),X2,X3) -> mark(U14(X1,X2,X3))
U14(ok(X1),ok(X2),ok(X3)) -> ok(U14(X1,X2,X3))
U15(mark(X1),X2) -> mark(U15(X1,X2))
U15(ok(X1),ok(X2)) -> ok(U15(X1,X2))
U16(mark(X)) -> mark(U16(X))
U16(ok(X)) -> ok(U16(X))
U21(mark(X1),X2) -> mark(U21(X1,X2))
U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
U22(mark(X1),X2) -> mark(U22(X1,X2))
U22(ok(X1),ok(X2)) -> ok(U22(X1,X2))
U23(mark(X)) -> mark(U23(X))
U23(ok(X)) -> ok(U23(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,X3) -> mark(U32(X1,X2,X3))
U32(ok(X1),ok(X2),ok(X3)) -> ok(U32(X1,X2,X3))
U33(mark(X1),X2,X3) -> mark(U33(X1,X2,X3))
U33(ok(X1),ok(X2),ok(X3)) -> ok(U33(X1,X2,X3))
U34(mark(X1),X2,X3) -> mark(U34(X1,X2,X3))
U34(ok(X1),ok(X2),ok(X3)) -> ok(U34(X1,X2,X3))
U35(mark(X1),X2) -> mark(U35(X1,X2))
U35(ok(X1),ok(X2)) -> ok(U35(X1,X2))
U36(mark(X)) -> mark(U36(X))
U36(ok(X)) -> ok(U36(X))
U41(mark(X1),X2) -> mark(U41(X1,X2))
U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
U42(mark(X)) -> mark(U42(X))
U42(ok(X)) -> ok(U42(X))
U51(mark(X)) -> mark(U51(X))
U51(ok(X)) -> ok(U51(X))
U61(mark(X1),X2) -> mark(U61(X1,X2))
U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
U62(mark(X)) -> mark(U62(X))
U62(ok(X)) -> ok(U62(X))
U71(mark(X1),X2) -> mark(U71(X1,X2))
U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
U72(mark(X1),X2) -> mark(U72(X1,X2))
U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3))
U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3))
U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3))
U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3))
U83(mark(X1),X2,X3) -> mark(U83(X1,X2,X3))
U83(ok(X1),ok(X2),ok(X3)) -> ok(U83(X1,X2,X3))
U84(mark(X1),X2,X3) -> mark(U84(X1,X2,X3))
U84(ok(X1),ok(X2),ok(X3)) -> ok(U84(X1,X2,X3))
U91(mark(X1),X2) -> mark(U91(X1,X2))
U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
U92(mark(X)) -> mark(U92(X))
U92(ok(X)) -> ok(U92(X))
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:
{U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3,U34/3
,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1,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 {U101,U102,U103,U104,U11,U12,U13,U14,U15,U16,U21,U22,U23
,U31,U32,U33,U34,U35,U36,U41,U42,U51,U61,U62,U71,U72,U81,U82,U83,U84,U91,U92,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
U101_0(2,2,2) -> 1
U101_1(2,2,2) -> 3
U102_0(2,2,2) -> 1
U102_1(2,2,2) -> 3
U103_0(2,2,2) -> 1
U103_1(2,2,2) -> 3
U104_0(2,2,2) -> 1
U104_1(2,2,2) -> 3
U11_0(2,2,2) -> 1
U11_1(2,2,2) -> 3
U12_0(2,2,2) -> 1
U12_1(2,2,2) -> 3
U13_0(2,2,2) -> 1
U13_1(2,2,2) -> 3
U14_0(2,2,2) -> 1
U14_1(2,2,2) -> 3
U15_0(2,2) -> 1
U15_1(2,2) -> 3
U16_0(2) -> 1
U16_1(2) -> 3
U21_0(2,2) -> 1
U21_1(2,2) -> 3
U22_0(2,2) -> 1
U22_1(2,2) -> 3
U23_0(2) -> 1
U23_1(2) -> 3
U31_0(2,2,2) -> 1
U31_1(2,2,2) -> 3
U32_0(2,2,2) -> 1
U32_1(2,2,2) -> 3
U33_0(2,2,2) -> 1
U33_1(2,2,2) -> 3
U34_0(2,2,2) -> 1
U34_1(2,2,2) -> 3
U35_0(2,2) -> 1
U35_1(2,2) -> 3
U36_0(2) -> 1
U36_1(2) -> 3
U41_0(2,2) -> 1
U41_1(2,2) -> 3
U42_0(2) -> 1
U42_1(2) -> 3
U51_0(2) -> 1
U51_1(2) -> 3
U61_0(2,2) -> 1
U61_1(2,2) -> 3
U62_0(2) -> 1
U62_1(2) -> 3
U71_0(2,2) -> 1
U71_1(2,2) -> 3
U72_0(2,2) -> 1
U72_1(2,2) -> 3
U81_0(2,2,2) -> 1
U81_1(2,2,2) -> 3
U82_0(2,2,2) -> 1
U82_1(2,2,2) -> 3
U83_0(2,2,2) -> 1
U83_1(2,2,2) -> 3
U84_0(2,2,2) -> 1
U84_1(2,2,2) -> 3
U91_0(2,2) -> 1
U91_1(2,2) -> 3
U92_0(2) -> 1
U92_1(2) -> 3
active_0(2) -> 2
active_1(2) -> 4
active_2(3) -> 5
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:
U101(mark(X1),X2,X3) -> mark(U101(X1,X2,X3))
U101(ok(X1),ok(X2),ok(X3)) -> ok(U101(X1,X2,X3))
U102(mark(X1),X2,X3) -> mark(U102(X1,X2,X3))
U102(ok(X1),ok(X2),ok(X3)) -> ok(U102(X1,X2,X3))
U103(mark(X1),X2,X3) -> mark(U103(X1,X2,X3))
U103(ok(X1),ok(X2),ok(X3)) -> ok(U103(X1,X2,X3))
U104(mark(X1),X2,X3) -> mark(U104(X1,X2,X3))
U104(ok(X1),ok(X2),ok(X3)) -> ok(U104(X1,X2,X3))
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))
U13(mark(X1),X2,X3) -> mark(U13(X1,X2,X3))
U13(ok(X1),ok(X2),ok(X3)) -> ok(U13(X1,X2,X3))
U14(mark(X1),X2,X3) -> mark(U14(X1,X2,X3))
U14(ok(X1),ok(X2),ok(X3)) -> ok(U14(X1,X2,X3))
U15(mark(X1),X2) -> mark(U15(X1,X2))
U15(ok(X1),ok(X2)) -> ok(U15(X1,X2))
U16(mark(X)) -> mark(U16(X))
U16(ok(X)) -> ok(U16(X))
U21(mark(X1),X2) -> mark(U21(X1,X2))
U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
U22(mark(X1),X2) -> mark(U22(X1,X2))
U22(ok(X1),ok(X2)) -> ok(U22(X1,X2))
U23(mark(X)) -> mark(U23(X))
U23(ok(X)) -> ok(U23(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,X3) -> mark(U32(X1,X2,X3))
U32(ok(X1),ok(X2),ok(X3)) -> ok(U32(X1,X2,X3))
U33(mark(X1),X2,X3) -> mark(U33(X1,X2,X3))
U33(ok(X1),ok(X2),ok(X3)) -> ok(U33(X1,X2,X3))
U34(mark(X1),X2,X3) -> mark(U34(X1,X2,X3))
U34(ok(X1),ok(X2),ok(X3)) -> ok(U34(X1,X2,X3))
U35(mark(X1),X2) -> mark(U35(X1,X2))
U35(ok(X1),ok(X2)) -> ok(U35(X1,X2))
U36(mark(X)) -> mark(U36(X))
U36(ok(X)) -> ok(U36(X))
U41(mark(X1),X2) -> mark(U41(X1,X2))
U41(ok(X1),ok(X2)) -> ok(U41(X1,X2))
U42(mark(X)) -> mark(U42(X))
U42(ok(X)) -> ok(U42(X))
U51(mark(X)) -> mark(U51(X))
U51(ok(X)) -> ok(U51(X))
U61(mark(X1),X2) -> mark(U61(X1,X2))
U61(ok(X1),ok(X2)) -> ok(U61(X1,X2))
U62(mark(X)) -> mark(U62(X))
U62(ok(X)) -> ok(U62(X))
U71(mark(X1),X2) -> mark(U71(X1,X2))
U71(ok(X1),ok(X2)) -> ok(U71(X1,X2))
U72(mark(X1),X2) -> mark(U72(X1,X2))
U72(ok(X1),ok(X2)) -> ok(U72(X1,X2))
U81(mark(X1),X2,X3) -> mark(U81(X1,X2,X3))
U81(ok(X1),ok(X2),ok(X3)) -> ok(U81(X1,X2,X3))
U82(mark(X1),X2,X3) -> mark(U82(X1,X2,X3))
U82(ok(X1),ok(X2),ok(X3)) -> ok(U82(X1,X2,X3))
U83(mark(X1),X2,X3) -> mark(U83(X1,X2,X3))
U83(ok(X1),ok(X2),ok(X3)) -> ok(U83(X1,X2,X3))
U84(mark(X1),X2,X3) -> mark(U84(X1,X2,X3))
U84(ok(X1),ok(X2),ok(X3)) -> ok(U84(X1,X2,X3))
U91(mark(X1),X2) -> mark(U91(X1,X2))
U91(ok(X1),ok(X2)) -> ok(U91(X1,X2))
U92(mark(X)) -> mark(U92(X))
U92(ok(X)) -> ok(U92(X))
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:
{U101/3,U102/3,U103/3,U104/3,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/3,U32/3,U33/3,U34/3
,U35/2,U36/1,U41/2,U42/1,U51/1,U61/2,U62/1,U71/2,U72/2,U81/3,U82/3,U83/3,U84/3,U91/2,U92/1,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 {U101,U102,U103,U104,U11,U12,U13,U14,U15,U16,U21,U22,U23
,U31,U32,U33,U34,U35,U36,U41,U42,U51,U61,U62,U71,U72,U81,U82,U83,U84,U91,U92,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))