* Step 1: Sum WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
a__U101(X1,X2,X3) -> U101(X1,X2,X3)
a__U101(tt(),M,N) -> a__U102(a__isNatKind(M),M,N)
a__U102(X1,X2,X3) -> U102(X1,X2,X3)
a__U102(tt(),M,N) -> a__U103(a__isNat(N),M,N)
a__U103(X1,X2,X3) -> U103(X1,X2,X3)
a__U103(tt(),M,N) -> a__U104(a__isNatKind(N),M,N)
a__U104(X1,X2,X3) -> U104(X1,X2,X3)
a__U104(tt(),M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N))
a__U11(X1,X2,X3) -> U11(X1,X2,X3)
a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2)
a__U12(X1,X2,X3) -> U12(X1,X2,X3)
a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2)
a__U13(X1,X2,X3) -> U13(X1,X2,X3)
a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2)
a__U14(X1,X2,X3) -> U14(X1,X2,X3)
a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2)
a__U15(X1,X2) -> U15(X1,X2)
a__U15(tt(),V2) -> a__U16(a__isNat(V2))
a__U16(X) -> U16(X)
a__U16(tt()) -> tt()
a__U21(X1,X2) -> U21(X1,X2)
a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
a__U22(X1,X2) -> U22(X1,X2)
a__U22(tt(),V1) -> a__U23(a__isNat(V1))
a__U23(X) -> U23(X)
a__U23(tt()) -> tt()
a__U31(X1,X2,X3) -> U31(X1,X2,X3)
a__U31(tt(),V1,V2) -> a__U32(a__isNatKind(V1),V1,V2)
a__U32(X1,X2,X3) -> U32(X1,X2,X3)
a__U32(tt(),V1,V2) -> a__U33(a__isNatKind(V2),V1,V2)
a__U33(X1,X2,X3) -> U33(X1,X2,X3)
a__U33(tt(),V1,V2) -> a__U34(a__isNatKind(V2),V1,V2)
a__U34(X1,X2,X3) -> U34(X1,X2,X3)
a__U34(tt(),V1,V2) -> a__U35(a__isNat(V1),V2)
a__U35(X1,X2) -> U35(X1,X2)
a__U35(tt(),V2) -> a__U36(a__isNat(V2))
a__U36(X) -> U36(X)
a__U36(tt()) -> tt()
a__U41(X1,X2) -> U41(X1,X2)
a__U41(tt(),V2) -> a__U42(a__isNatKind(V2))
a__U42(X) -> U42(X)
a__U42(tt()) -> tt()
a__U51(X) -> U51(X)
a__U51(tt()) -> tt()
a__U61(X1,X2) -> U61(X1,X2)
a__U61(tt(),V2) -> a__U62(a__isNatKind(V2))
a__U62(X) -> U62(X)
a__U62(tt()) -> tt()
a__U71(X1,X2) -> U71(X1,X2)
a__U71(tt(),N) -> a__U72(a__isNatKind(N),N)
a__U72(X1,X2) -> U72(X1,X2)
a__U72(tt(),N) -> mark(N)
a__U81(X1,X2,X3) -> U81(X1,X2,X3)
a__U81(tt(),M,N) -> a__U82(a__isNatKind(M),M,N)
a__U82(X1,X2,X3) -> U82(X1,X2,X3)
a__U82(tt(),M,N) -> a__U83(a__isNat(N),M,N)
a__U83(X1,X2,X3) -> U83(X1,X2,X3)
a__U83(tt(),M,N) -> a__U84(a__isNatKind(N),M,N)
a__U84(X1,X2,X3) -> U84(X1,X2,X3)
a__U84(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
a__U91(X1,X2) -> U91(X1,X2)
a__U91(tt(),N) -> a__U92(a__isNatKind(N))
a__U92(X) -> U92(X)
a__U92(tt()) -> 0()
a__isNat(X) -> isNat(X)
a__isNat(0()) -> tt()
a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
a__isNat(x(V1,V2)) -> a__U31(a__isNatKind(V1),V1,V2)
a__isNatKind(X) -> isNatKind(X)
a__isNatKind(0()) -> tt()
a__isNatKind(plus(V1,V2)) -> a__U41(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) -> a__U51(a__isNatKind(V1))
a__isNatKind(x(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
a__plus(N,0()) -> a__U71(a__isNat(N),N)
a__plus(N,s(M)) -> a__U81(a__isNat(M),M,N)
a__plus(X1,X2) -> plus(X1,X2)
a__x(N,0()) -> a__U91(a__isNat(N),N)
a__x(N,s(M)) -> a__U101(a__isNat(M),M,N)
a__x(X1,X2) -> x(X1,X2)
mark(0()) -> 0()
mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) -> a__U15(mark(X1),X2)
mark(U16(X)) -> a__U16(mark(X))
mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
mark(U23(X)) -> a__U23(mark(X))
mark(U31(X1,X2,X3)) -> a__U31(mark(X1),X2,X3)
mark(U32(X1,X2,X3)) -> a__U32(mark(X1),X2,X3)
mark(U33(X1,X2,X3)) -> a__U33(mark(X1),X2,X3)
mark(U34(X1,X2,X3)) -> a__U34(mark(X1),X2,X3)
mark(U35(X1,X2)) -> a__U35(mark(X1),X2)
mark(U36(X)) -> a__U36(mark(X))
mark(U41(X1,X2)) -> a__U41(mark(X1),X2)
mark(U42(X)) -> a__U42(mark(X))
mark(U51(X)) -> a__U51(mark(X))
mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
mark(U62(X)) -> a__U62(mark(X))
mark(U71(X1,X2)) -> a__U71(mark(X1),X2)
mark(U72(X1,X2)) -> a__U72(mark(X1),X2)
mark(U81(X1,X2,X3)) -> a__U81(mark(X1),X2,X3)
mark(U82(X1,X2,X3)) -> a__U82(mark(X1),X2,X3)
mark(U83(X1,X2,X3)) -> a__U83(mark(X1),X2,X3)
mark(U84(X1,X2,X3)) -> a__U84(mark(X1),X2,X3)
mark(U91(X1,X2)) -> a__U91(mark(X1),X2)
mark(U92(X)) -> a__U92(mark(X))
mark(isNat(X)) -> a__isNat(X)
mark(isNatKind(X)) -> a__isNatKind(X)
mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
mark(s(X)) -> s(mark(X))
mark(tt()) -> tt()
mark(x(X1,X2)) -> a__x(mark(X1),mark(X2))
- Signature:
{a__U101/3,a__U102/3,a__U103/3,a__U104/3,a__U11/3,a__U12/3,a__U13/3,a__U14/3,a__U15/2,a__U16/1,a__U21/2
,a__U22/2,a__U23/1,a__U31/3,a__U32/3,a__U33/3,a__U34/3,a__U35/2,a__U36/1,a__U41/2,a__U42/1,a__U51/1,a__U61/2
,a__U62/1,a__U71/2,a__U72/2,a__U81/3,a__U82/3,a__U83/3,a__U84/3,a__U91/2,a__U92/1,a__isNat/1,a__isNatKind/1
,a__plus/2,a__x/2,mark/1} / {0/0,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,s/1,tt/0,x/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__U101,a__U102,a__U103,a__U104,a__U11,a__U12,a__U13
,a__U14,a__U15,a__U16,a__U21,a__U22,a__U23,a__U31,a__U32,a__U33,a__U34,a__U35,a__U36,a__U41,a__U42,a__U51
,a__U61,a__U62,a__U71,a__U72,a__U81,a__U82,a__U83,a__U84,a__U91,a__U92,a__isNat,a__isNatKind,a__plus,a__x
,mark} and constructors {0,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,s,tt,x}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
* Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
a__U101(X1,X2,X3) -> U101(X1,X2,X3)
a__U101(tt(),M,N) -> a__U102(a__isNatKind(M),M,N)
a__U102(X1,X2,X3) -> U102(X1,X2,X3)
a__U102(tt(),M,N) -> a__U103(a__isNat(N),M,N)
a__U103(X1,X2,X3) -> U103(X1,X2,X3)
a__U103(tt(),M,N) -> a__U104(a__isNatKind(N),M,N)
a__U104(X1,X2,X3) -> U104(X1,X2,X3)
a__U104(tt(),M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N))
a__U11(X1,X2,X3) -> U11(X1,X2,X3)
a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2)
a__U12(X1,X2,X3) -> U12(X1,X2,X3)
a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2)
a__U13(X1,X2,X3) -> U13(X1,X2,X3)
a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2)
a__U14(X1,X2,X3) -> U14(X1,X2,X3)
a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2)
a__U15(X1,X2) -> U15(X1,X2)
a__U15(tt(),V2) -> a__U16(a__isNat(V2))
a__U16(X) -> U16(X)
a__U16(tt()) -> tt()
a__U21(X1,X2) -> U21(X1,X2)
a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
a__U22(X1,X2) -> U22(X1,X2)
a__U22(tt(),V1) -> a__U23(a__isNat(V1))
a__U23(X) -> U23(X)
a__U23(tt()) -> tt()
a__U31(X1,X2,X3) -> U31(X1,X2,X3)
a__U31(tt(),V1,V2) -> a__U32(a__isNatKind(V1),V1,V2)
a__U32(X1,X2,X3) -> U32(X1,X2,X3)
a__U32(tt(),V1,V2) -> a__U33(a__isNatKind(V2),V1,V2)
a__U33(X1,X2,X3) -> U33(X1,X2,X3)
a__U33(tt(),V1,V2) -> a__U34(a__isNatKind(V2),V1,V2)
a__U34(X1,X2,X3) -> U34(X1,X2,X3)
a__U34(tt(),V1,V2) -> a__U35(a__isNat(V1),V2)
a__U35(X1,X2) -> U35(X1,X2)
a__U35(tt(),V2) -> a__U36(a__isNat(V2))
a__U36(X) -> U36(X)
a__U36(tt()) -> tt()
a__U41(X1,X2) -> U41(X1,X2)
a__U41(tt(),V2) -> a__U42(a__isNatKind(V2))
a__U42(X) -> U42(X)
a__U42(tt()) -> tt()
a__U51(X) -> U51(X)
a__U51(tt()) -> tt()
a__U61(X1,X2) -> U61(X1,X2)
a__U61(tt(),V2) -> a__U62(a__isNatKind(V2))
a__U62(X) -> U62(X)
a__U62(tt()) -> tt()
a__U71(X1,X2) -> U71(X1,X2)
a__U71(tt(),N) -> a__U72(a__isNatKind(N),N)
a__U72(X1,X2) -> U72(X1,X2)
a__U72(tt(),N) -> mark(N)
a__U81(X1,X2,X3) -> U81(X1,X2,X3)
a__U81(tt(),M,N) -> a__U82(a__isNatKind(M),M,N)
a__U82(X1,X2,X3) -> U82(X1,X2,X3)
a__U82(tt(),M,N) -> a__U83(a__isNat(N),M,N)
a__U83(X1,X2,X3) -> U83(X1,X2,X3)
a__U83(tt(),M,N) -> a__U84(a__isNatKind(N),M,N)
a__U84(X1,X2,X3) -> U84(X1,X2,X3)
a__U84(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
a__U91(X1,X2) -> U91(X1,X2)
a__U91(tt(),N) -> a__U92(a__isNatKind(N))
a__U92(X) -> U92(X)
a__U92(tt()) -> 0()
a__isNat(X) -> isNat(X)
a__isNat(0()) -> tt()
a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
a__isNat(x(V1,V2)) -> a__U31(a__isNatKind(V1),V1,V2)
a__isNatKind(X) -> isNatKind(X)
a__isNatKind(0()) -> tt()
a__isNatKind(plus(V1,V2)) -> a__U41(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) -> a__U51(a__isNatKind(V1))
a__isNatKind(x(V1,V2)) -> a__U61(a__isNatKind(V1),V2)
a__plus(N,0()) -> a__U71(a__isNat(N),N)
a__plus(N,s(M)) -> a__U81(a__isNat(M),M,N)
a__plus(X1,X2) -> plus(X1,X2)
a__x(N,0()) -> a__U91(a__isNat(N),N)
a__x(N,s(M)) -> a__U101(a__isNat(M),M,N)
a__x(X1,X2) -> x(X1,X2)
mark(0()) -> 0()
mark(U101(X1,X2,X3)) -> a__U101(mark(X1),X2,X3)
mark(U102(X1,X2,X3)) -> a__U102(mark(X1),X2,X3)
mark(U103(X1,X2,X3)) -> a__U103(mark(X1),X2,X3)
mark(U104(X1,X2,X3)) -> a__U104(mark(X1),X2,X3)
mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) -> a__U15(mark(X1),X2)
mark(U16(X)) -> a__U16(mark(X))
mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
mark(U23(X)) -> a__U23(mark(X))
mark(U31(X1,X2,X3)) -> a__U31(mark(X1),X2,X3)
mark(U32(X1,X2,X3)) -> a__U32(mark(X1),X2,X3)
mark(U33(X1,X2,X3)) -> a__U33(mark(X1),X2,X3)
mark(U34(X1,X2,X3)) -> a__U34(mark(X1),X2,X3)
mark(U35(X1,X2)) -> a__U35(mark(X1),X2)
mark(U36(X)) -> a__U36(mark(X))
mark(U41(X1,X2)) -> a__U41(mark(X1),X2)
mark(U42(X)) -> a__U42(mark(X))
mark(U51(X)) -> a__U51(mark(X))
mark(U61(X1,X2)) -> a__U61(mark(X1),X2)
mark(U62(X)) -> a__U62(mark(X))
mark(U71(X1,X2)) -> a__U71(mark(X1),X2)
mark(U72(X1,X2)) -> a__U72(mark(X1),X2)
mark(U81(X1,X2,X3)) -> a__U81(mark(X1),X2,X3)
mark(U82(X1,X2,X3)) -> a__U82(mark(X1),X2,X3)
mark(U83(X1,X2,X3)) -> a__U83(mark(X1),X2,X3)
mark(U84(X1,X2,X3)) -> a__U84(mark(X1),X2,X3)
mark(U91(X1,X2)) -> a__U91(mark(X1),X2)
mark(U92(X)) -> a__U92(mark(X))
mark(isNat(X)) -> a__isNat(X)
mark(isNatKind(X)) -> a__isNatKind(X)
mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
mark(s(X)) -> s(mark(X))
mark(tt()) -> tt()
mark(x(X1,X2)) -> a__x(mark(X1),mark(X2))
- Signature:
{a__U101/3,a__U102/3,a__U103/3,a__U104/3,a__U11/3,a__U12/3,a__U13/3,a__U14/3,a__U15/2,a__U16/1,a__U21/2
,a__U22/2,a__U23/1,a__U31/3,a__U32/3,a__U33/3,a__U34/3,a__U35/2,a__U36/1,a__U41/2,a__U42/1,a__U51/1,a__U61/2
,a__U62/1,a__U71/2,a__U72/2,a__U81/3,a__U82/3,a__U83/3,a__U84/3,a__U91/2,a__U92/1,a__isNat/1,a__isNatKind/1
,a__plus/2,a__x/2,mark/1} / {0/0,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,s/1,tt/0,x/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__U101,a__U102,a__U103,a__U104,a__U11,a__U12,a__U13
,a__U14,a__U15,a__U16,a__U21,a__U22,a__U23,a__U31,a__U32,a__U33,a__U34,a__U35,a__U36,a__U41,a__U42,a__U51
,a__U61,a__U62,a__U71,a__U72,a__U81,a__U82,a__U83,a__U84,a__U91,a__U92,a__isNat,a__isNatKind,a__plus,a__x
,mark} and constructors {0,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,s,tt,x}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
a__isNatKind(x){x -> plus(x,y)} =
a__isNatKind(plus(x,y)) ->^+ a__U41(a__isNatKind(x),y)
= C[a__isNatKind(x) = a__isNatKind(x){}]
WORST_CASE(Omega(n^1),?)