*** 1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
0() -> n__0()
U101(tt(),M,N) -> U102(isNatKind(activate(M)),activate(M),activate(N))
U102(tt(),M,N) -> U103(isNat(activate(N)),activate(M),activate(N))
U103(tt(),M,N) -> U104(isNatKind(activate(N)),activate(M),activate(N))
U104(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N))
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2))
U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2))
U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2))
U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2))
U35(tt(),V2) -> U36(isNat(activate(V2)))
U36(tt()) -> tt()
U41(tt(),V2) -> U42(isNatKind(activate(V2)))
U42(tt()) -> tt()
U51(tt()) -> tt()
U61(tt(),V2) -> U62(isNatKind(activate(V2)))
U62(tt()) -> tt()
U71(tt(),N) -> U72(isNatKind(activate(N)),activate(N))
U72(tt(),N) -> activate(N)
U81(tt(),M,N) -> U82(isNatKind(activate(M)),activate(M),activate(N))
U82(tt(),M,N) -> U83(isNat(activate(N)),activate(M),activate(N))
U83(tt(),M,N) -> U84(isNatKind(activate(N)),activate(M),activate(N))
U84(tt(),M,N) -> s(plus(activate(N),activate(M)))
U91(tt(),N) -> U92(isNatKind(activate(N)))
U92(tt()) -> 0()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1)))
isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2))
plus(N,0()) -> U71(isNat(N),N)
plus(N,s(M)) -> U81(isNat(M),M,N)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(N,0()) -> U91(isNat(N),N)
x(N,s(M)) -> U101(isNat(M),M,N)
x(X1,X2) -> n__x(X1,X2)
Weak DP Rules:
Weak TRS Rules:
Signature:
{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,activate/1,isNat/1,isNatKind/1,plus/2,s/1,x/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0}
Obligation:
Innermost
basic terms: {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,activate,isNat,isNatKind,plus,s,x}/{n__0,n__plus,n__s,n__x,tt}
Applied Processor:
InnermostRuleRemoval
Proof:
Arguments of following rules are not normal-forms.
plus(N,0()) -> U71(isNat(N),N)
plus(N,s(M)) -> U81(isNat(M),M,N)
x(N,0()) -> U91(isNat(N),N)
x(N,s(M)) -> U101(isNat(M),M,N)
All above mentioned rules can be savely removed.
*** 1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
0() -> n__0()
U101(tt(),M,N) -> U102(isNatKind(activate(M)),activate(M),activate(N))
U102(tt(),M,N) -> U103(isNat(activate(N)),activate(M),activate(N))
U103(tt(),M,N) -> U104(isNatKind(activate(N)),activate(M),activate(N))
U104(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N))
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2))
U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2))
U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2))
U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2))
U35(tt(),V2) -> U36(isNat(activate(V2)))
U36(tt()) -> tt()
U41(tt(),V2) -> U42(isNatKind(activate(V2)))
U42(tt()) -> tt()
U51(tt()) -> tt()
U61(tt(),V2) -> U62(isNatKind(activate(V2)))
U62(tt()) -> tt()
U71(tt(),N) -> U72(isNatKind(activate(N)),activate(N))
U72(tt(),N) -> activate(N)
U81(tt(),M,N) -> U82(isNatKind(activate(M)),activate(M),activate(N))
U82(tt(),M,N) -> U83(isNat(activate(N)),activate(M),activate(N))
U83(tt(),M,N) -> U84(isNatKind(activate(N)),activate(M),activate(N))
U84(tt(),M,N) -> s(plus(activate(N),activate(M)))
U91(tt(),N) -> U92(isNatKind(activate(N)))
U92(tt()) -> 0()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1)))
isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
Weak DP Rules:
Weak TRS Rules:
Signature:
{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,activate/1,isNat/1,isNatKind/1,plus/2,s/1,x/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0}
Obligation:
Innermost
basic terms: {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,activate,isNat,isNatKind,plus,s,x}/{n__0,n__plus,n__s,n__x,tt}
Applied Processor:
DependencyPairs {dpKind_ = DT}
Proof:
We add the following dependency tuples:
Strict DPs
0#() -> c_1()
U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)),activate#(M),activate#(M),activate#(N))
U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U103#(tt(),M,N) -> c_4(U104#(isNatKind(activate(N)),activate(M),activate(N)),isNatKind#(activate(N)),activate#(N),activate#(M),activate#(N))
U104#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N)),x#(activate(N),activate(M)),activate#(N),activate#(M),activate#(N))
U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2))
U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)),activate#(V2),activate#(V1),activate#(V2))
U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)),activate#(V2),activate#(V1),activate#(V2))
U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
U15#(tt(),V2) -> c_10(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2))
U16#(tt()) -> c_11()
U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)),activate#(V1),activate#(V1))
U22#(tt(),V1) -> c_13(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1))
U23#(tt()) -> c_14()
U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2))
U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)),activate#(V2),activate#(V1),activate#(V2))
U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)),activate#(V2),activate#(V1),activate#(V2))
U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
U35#(tt(),V2) -> c_19(U36#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2))
U36#(tt()) -> c_20()
U41#(tt(),V2) -> c_21(U42#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2))
U42#(tt()) -> c_22()
U51#(tt()) -> c_23()
U61#(tt(),V2) -> c_24(U62#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2))
U62#(tt()) -> c_25()
U71#(tt(),N) -> c_26(U72#(isNatKind(activate(N)),activate(N)),isNatKind#(activate(N)),activate#(N),activate#(N))
U72#(tt(),N) -> c_27(activate#(N))
U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)),activate#(M),activate#(M),activate#(N))
U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U83#(tt(),M,N) -> c_30(U84#(isNatKind(activate(N)),activate(M),activate(N)),isNatKind#(activate(N)),activate#(N),activate#(M),activate#(N))
U84#(tt(),M,N) -> c_31(s#(plus(activate(N),activate(M))),plus#(activate(N),activate(M)),activate#(N),activate#(M))
U91#(tt(),N) -> c_32(U92#(isNatKind(activate(N))),isNatKind#(activate(N)),activate#(N))
U92#(tt()) -> c_33(0#())
activate#(X) -> c_34()
activate#(n__0()) -> c_35(0#())
activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2))
activate#(n__s(X)) -> c_37(s#(X))
activate#(n__x(X1,X2)) -> c_38(x#(X1,X2))
isNat#(n__0()) -> c_39()
isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2))
isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)),activate#(V1),activate#(V1))
isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2))
isNatKind#(n__0()) -> c_43()
isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2))
isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1))
isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2))
plus#(X1,X2) -> c_47()
s#(X) -> c_48()
x#(X1,X2) -> c_49()
Weak DPs
and mark the set of starting terms.
*** 1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
0#() -> c_1()
U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)),activate#(M),activate#(M),activate#(N))
U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U103#(tt(),M,N) -> c_4(U104#(isNatKind(activate(N)),activate(M),activate(N)),isNatKind#(activate(N)),activate#(N),activate#(M),activate#(N))
U104#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N)),x#(activate(N),activate(M)),activate#(N),activate#(M),activate#(N))
U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2))
U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)),activate#(V2),activate#(V1),activate#(V2))
U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)),activate#(V2),activate#(V1),activate#(V2))
U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
U15#(tt(),V2) -> c_10(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2))
U16#(tt()) -> c_11()
U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)),activate#(V1),activate#(V1))
U22#(tt(),V1) -> c_13(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1))
U23#(tt()) -> c_14()
U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2))
U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)),activate#(V2),activate#(V1),activate#(V2))
U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)),activate#(V2),activate#(V1),activate#(V2))
U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
U35#(tt(),V2) -> c_19(U36#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2))
U36#(tt()) -> c_20()
U41#(tt(),V2) -> c_21(U42#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2))
U42#(tt()) -> c_22()
U51#(tt()) -> c_23()
U61#(tt(),V2) -> c_24(U62#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2))
U62#(tt()) -> c_25()
U71#(tt(),N) -> c_26(U72#(isNatKind(activate(N)),activate(N)),isNatKind#(activate(N)),activate#(N),activate#(N))
U72#(tt(),N) -> c_27(activate#(N))
U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)),activate#(M),activate#(M),activate#(N))
U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U83#(tt(),M,N) -> c_30(U84#(isNatKind(activate(N)),activate(M),activate(N)),isNatKind#(activate(N)),activate#(N),activate#(M),activate#(N))
U84#(tt(),M,N) -> c_31(s#(plus(activate(N),activate(M))),plus#(activate(N),activate(M)),activate#(N),activate#(M))
U91#(tt(),N) -> c_32(U92#(isNatKind(activate(N))),isNatKind#(activate(N)),activate#(N))
U92#(tt()) -> c_33(0#())
activate#(X) -> c_34()
activate#(n__0()) -> c_35(0#())
activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2))
activate#(n__s(X)) -> c_37(s#(X))
activate#(n__x(X1,X2)) -> c_38(x#(X1,X2))
isNat#(n__0()) -> c_39()
isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2))
isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)),activate#(V1),activate#(V1))
isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2))
isNatKind#(n__0()) -> c_43()
isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2))
isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1))
isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2))
plus#(X1,X2) -> c_47()
s#(X) -> c_48()
x#(X1,X2) -> c_49()
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
0() -> n__0()
U101(tt(),M,N) -> U102(isNatKind(activate(M)),activate(M),activate(N))
U102(tt(),M,N) -> U103(isNat(activate(N)),activate(M),activate(N))
U103(tt(),M,N) -> U104(isNatKind(activate(N)),activate(M),activate(N))
U104(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N))
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2))
U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2))
U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2))
U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2))
U35(tt(),V2) -> U36(isNat(activate(V2)))
U36(tt()) -> tt()
U41(tt(),V2) -> U42(isNatKind(activate(V2)))
U42(tt()) -> tt()
U51(tt()) -> tt()
U61(tt(),V2) -> U62(isNatKind(activate(V2)))
U62(tt()) -> tt()
U71(tt(),N) -> U72(isNatKind(activate(N)),activate(N))
U72(tt(),N) -> activate(N)
U81(tt(),M,N) -> U82(isNatKind(activate(M)),activate(M),activate(N))
U82(tt(),M,N) -> U83(isNat(activate(N)),activate(M),activate(N))
U83(tt(),M,N) -> U84(isNatKind(activate(N)),activate(M),activate(N))
U84(tt(),M,N) -> s(plus(activate(N),activate(M)))
U91(tt(),N) -> U92(isNatKind(activate(N)))
U92(tt()) -> 0()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1)))
isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
Signature:
{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,activate/1,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/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,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/5,c_6/5,c_7/5,c_8/5,c_9/4,c_10/3,c_11/0,c_12/4,c_13/3,c_14/0,c_15/5,c_16/5,c_17/5,c_18/4,c_19/3,c_20/0,c_21/3,c_22/0,c_23/0,c_24/3,c_25/0,c_26/4,c_27/1,c_28/5,c_29/5,c_30/5,c_31/4,c_32/3,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/5,c_41/4,c_42/5,c_43/0,c_44/4,c_45/3,c_46/4,c_47/0,c_48/0,c_49/0}
Obligation:
Innermost
basic terms: {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#,activate#,isNat#,isNatKind#,plus#,s#,x#}/{n__0,n__plus,n__s,n__x,tt}
Applied Processor:
UsableRules
Proof:
We replace rewrite rules by usable rules:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2))
U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2))
U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2))
U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2))
U35(tt(),V2) -> U36(isNat(activate(V2)))
U36(tt()) -> tt()
U41(tt(),V2) -> U42(isNatKind(activate(V2)))
U42(tt()) -> tt()
U51(tt()) -> tt()
U61(tt(),V2) -> U62(isNatKind(activate(V2)))
U62(tt()) -> tt()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1)))
isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
0#() -> c_1()
U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)),activate#(M),activate#(M),activate#(N))
U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U103#(tt(),M,N) -> c_4(U104#(isNatKind(activate(N)),activate(M),activate(N)),isNatKind#(activate(N)),activate#(N),activate#(M),activate#(N))
U104#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N)),x#(activate(N),activate(M)),activate#(N),activate#(M),activate#(N))
U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2))
U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)),activate#(V2),activate#(V1),activate#(V2))
U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)),activate#(V2),activate#(V1),activate#(V2))
U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
U15#(tt(),V2) -> c_10(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2))
U16#(tt()) -> c_11()
U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)),activate#(V1),activate#(V1))
U22#(tt(),V1) -> c_13(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1))
U23#(tt()) -> c_14()
U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2))
U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)),activate#(V2),activate#(V1),activate#(V2))
U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)),activate#(V2),activate#(V1),activate#(V2))
U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
U35#(tt(),V2) -> c_19(U36#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2))
U36#(tt()) -> c_20()
U41#(tt(),V2) -> c_21(U42#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2))
U42#(tt()) -> c_22()
U51#(tt()) -> c_23()
U61#(tt(),V2) -> c_24(U62#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2))
U62#(tt()) -> c_25()
U71#(tt(),N) -> c_26(U72#(isNatKind(activate(N)),activate(N)),isNatKind#(activate(N)),activate#(N),activate#(N))
U72#(tt(),N) -> c_27(activate#(N))
U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)),activate#(M),activate#(M),activate#(N))
U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U83#(tt(),M,N) -> c_30(U84#(isNatKind(activate(N)),activate(M),activate(N)),isNatKind#(activate(N)),activate#(N),activate#(M),activate#(N))
U84#(tt(),M,N) -> c_31(s#(plus(activate(N),activate(M))),plus#(activate(N),activate(M)),activate#(N),activate#(M))
U91#(tt(),N) -> c_32(U92#(isNatKind(activate(N))),isNatKind#(activate(N)),activate#(N))
U92#(tt()) -> c_33(0#())
activate#(X) -> c_34()
activate#(n__0()) -> c_35(0#())
activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2))
activate#(n__s(X)) -> c_37(s#(X))
activate#(n__x(X1,X2)) -> c_38(x#(X1,X2))
isNat#(n__0()) -> c_39()
isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2))
isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)),activate#(V1),activate#(V1))
isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2))
isNatKind#(n__0()) -> c_43()
isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2))
isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1))
isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2))
plus#(X1,X2) -> c_47()
s#(X) -> c_48()
x#(X1,X2) -> c_49()
*** 1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
0#() -> c_1()
U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)),activate#(M),activate#(M),activate#(N))
U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U103#(tt(),M,N) -> c_4(U104#(isNatKind(activate(N)),activate(M),activate(N)),isNatKind#(activate(N)),activate#(N),activate#(M),activate#(N))
U104#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N)),x#(activate(N),activate(M)),activate#(N),activate#(M),activate#(N))
U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2))
U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)),activate#(V2),activate#(V1),activate#(V2))
U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)),activate#(V2),activate#(V1),activate#(V2))
U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
U15#(tt(),V2) -> c_10(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2))
U16#(tt()) -> c_11()
U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)),activate#(V1),activate#(V1))
U22#(tt(),V1) -> c_13(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1))
U23#(tt()) -> c_14()
U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2))
U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)),activate#(V2),activate#(V1),activate#(V2))
U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)),activate#(V2),activate#(V1),activate#(V2))
U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
U35#(tt(),V2) -> c_19(U36#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2))
U36#(tt()) -> c_20()
U41#(tt(),V2) -> c_21(U42#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2))
U42#(tt()) -> c_22()
U51#(tt()) -> c_23()
U61#(tt(),V2) -> c_24(U62#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2))
U62#(tt()) -> c_25()
U71#(tt(),N) -> c_26(U72#(isNatKind(activate(N)),activate(N)),isNatKind#(activate(N)),activate#(N),activate#(N))
U72#(tt(),N) -> c_27(activate#(N))
U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)),activate#(M),activate#(M),activate#(N))
U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U83#(tt(),M,N) -> c_30(U84#(isNatKind(activate(N)),activate(M),activate(N)),isNatKind#(activate(N)),activate#(N),activate#(M),activate#(N))
U84#(tt(),M,N) -> c_31(s#(plus(activate(N),activate(M))),plus#(activate(N),activate(M)),activate#(N),activate#(M))
U91#(tt(),N) -> c_32(U92#(isNatKind(activate(N))),isNatKind#(activate(N)),activate#(N))
U92#(tt()) -> c_33(0#())
activate#(X) -> c_34()
activate#(n__0()) -> c_35(0#())
activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2))
activate#(n__s(X)) -> c_37(s#(X))
activate#(n__x(X1,X2)) -> c_38(x#(X1,X2))
isNat#(n__0()) -> c_39()
isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2))
isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)),activate#(V1),activate#(V1))
isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2))
isNatKind#(n__0()) -> c_43()
isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2))
isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1))
isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2))
plus#(X1,X2) -> c_47()
s#(X) -> c_48()
x#(X1,X2) -> c_49()
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2))
U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2))
U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2))
U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2))
U35(tt(),V2) -> U36(isNat(activate(V2)))
U36(tt()) -> tt()
U41(tt(),V2) -> U42(isNatKind(activate(V2)))
U42(tt()) -> tt()
U51(tt()) -> tt()
U61(tt(),V2) -> U62(isNatKind(activate(V2)))
U62(tt()) -> tt()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1)))
isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
Signature:
{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,activate/1,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/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,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/5,c_6/5,c_7/5,c_8/5,c_9/4,c_10/3,c_11/0,c_12/4,c_13/3,c_14/0,c_15/5,c_16/5,c_17/5,c_18/4,c_19/3,c_20/0,c_21/3,c_22/0,c_23/0,c_24/3,c_25/0,c_26/4,c_27/1,c_28/5,c_29/5,c_30/5,c_31/4,c_32/3,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/5,c_41/4,c_42/5,c_43/0,c_44/4,c_45/3,c_46/4,c_47/0,c_48/0,c_49/0}
Obligation:
Innermost
basic terms: {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#,activate#,isNat#,isNatKind#,plus#,s#,x#}/{n__0,n__plus,n__s,n__x,tt}
Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
Proof:
We estimate the number of application of
{1,11,14,20,22,23,25,34,39,43,47,48,49}
by application of
Pre({1,11,14,20,22,23,25,34,39,43,47,48,49}) = {2,3,4,5,6,7,8,9,10,12,13,15,16,17,18,19,21,24,26,27,28,29,30,31,32,33,35,36,37,38,40,41,42,44,45,46}.
Here rules are labelled as follows:
1: 0#() -> c_1()
2: U101#(tt(),M,N) ->
c_2(U102#(isNatKind(activate(M))
,activate(M)
,activate(N))
,isNatKind#(activate(M))
,activate#(M)
,activate#(M)
,activate#(N))
3: U102#(tt(),M,N) ->
c_3(U103#(isNat(activate(N))
,activate(M)
,activate(N))
,isNat#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
4: U103#(tt(),M,N) ->
c_4(U104#(isNatKind(activate(N))
,activate(M)
,activate(N))
,isNatKind#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
5: U104#(tt(),M,N) ->
c_5(plus#(x(activate(N)
,activate(M))
,activate(N))
,x#(activate(N),activate(M))
,activate#(N)
,activate#(M)
,activate#(N))
6: U11#(tt(),V1,V2) ->
c_6(U12#(isNatKind(activate(V1))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
7: U12#(tt(),V1,V2) ->
c_7(U13#(isNatKind(activate(V2))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
8: U13#(tt(),V1,V2) ->
c_8(U14#(isNatKind(activate(V2))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
9: U14#(tt(),V1,V2) ->
c_9(U15#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
10: U15#(tt(),V2) ->
c_10(U16#(isNat(activate(V2)))
,isNat#(activate(V2))
,activate#(V2))
11: U16#(tt()) -> c_11()
12: U21#(tt(),V1) ->
c_12(U22#(isNatKind(activate(V1))
,activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1))
13: U22#(tt(),V1) ->
c_13(U23#(isNat(activate(V1)))
,isNat#(activate(V1))
,activate#(V1))
14: U23#(tt()) -> c_14()
15: U31#(tt(),V1,V2) ->
c_15(U32#(isNatKind(activate(V1))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
16: U32#(tt(),V1,V2) ->
c_16(U33#(isNatKind(activate(V2))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
17: U33#(tt(),V1,V2) ->
c_17(U34#(isNatKind(activate(V2))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
18: U34#(tt(),V1,V2) ->
c_18(U35#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
19: U35#(tt(),V2) ->
c_19(U36#(isNat(activate(V2)))
,isNat#(activate(V2))
,activate#(V2))
20: U36#(tt()) -> c_20()
21: U41#(tt(),V2) ->
c_21(U42#(isNatKind(activate(V2)))
,isNatKind#(activate(V2))
,activate#(V2))
22: U42#(tt()) -> c_22()
23: U51#(tt()) -> c_23()
24: U61#(tt(),V2) ->
c_24(U62#(isNatKind(activate(V2)))
,isNatKind#(activate(V2))
,activate#(V2))
25: U62#(tt()) -> c_25()
26: U71#(tt(),N) ->
c_26(U72#(isNatKind(activate(N))
,activate(N))
,isNatKind#(activate(N))
,activate#(N)
,activate#(N))
27: U72#(tt(),N) ->
c_27(activate#(N))
28: U81#(tt(),M,N) ->
c_28(U82#(isNatKind(activate(M))
,activate(M)
,activate(N))
,isNatKind#(activate(M))
,activate#(M)
,activate#(M)
,activate#(N))
29: U82#(tt(),M,N) ->
c_29(U83#(isNat(activate(N))
,activate(M)
,activate(N))
,isNat#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
30: U83#(tt(),M,N) ->
c_30(U84#(isNatKind(activate(N))
,activate(M)
,activate(N))
,isNatKind#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
31: U84#(tt(),M,N) ->
c_31(s#(plus(activate(N)
,activate(M)))
,plus#(activate(N),activate(M))
,activate#(N)
,activate#(M))
32: U91#(tt(),N) ->
c_32(U92#(isNatKind(activate(N)))
,isNatKind#(activate(N))
,activate#(N))
33: U92#(tt()) -> c_33(0#())
34: activate#(X) -> c_34()
35: activate#(n__0()) -> c_35(0#())
36: activate#(n__plus(X1,X2)) ->
c_36(plus#(X1,X2))
37: activate#(n__s(X)) ->
c_37(s#(X))
38: activate#(n__x(X1,X2)) ->
c_38(x#(X1,X2))
39: isNat#(n__0()) -> c_39()
40: isNat#(n__plus(V1,V2)) ->
c_40(U11#(isNatKind(activate(V1))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
41: isNat#(n__s(V1)) ->
c_41(U21#(isNatKind(activate(V1))
,activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1))
42: isNat#(n__x(V1,V2)) ->
c_42(U31#(isNatKind(activate(V1))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
43: isNatKind#(n__0()) -> c_43()
44: isNatKind#(n__plus(V1,V2)) ->
c_44(U41#(isNatKind(activate(V1))
,activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2))
45: isNatKind#(n__s(V1)) ->
c_45(U51#(isNatKind(activate(V1)))
,isNatKind#(activate(V1))
,activate#(V1))
46: isNatKind#(n__x(V1,V2)) ->
c_46(U61#(isNatKind(activate(V1))
,activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2))
47: plus#(X1,X2) -> c_47()
48: s#(X) -> c_48()
49: x#(X1,X2) -> c_49()
*** 1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)),activate#(M),activate#(M),activate#(N))
U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U103#(tt(),M,N) -> c_4(U104#(isNatKind(activate(N)),activate(M),activate(N)),isNatKind#(activate(N)),activate#(N),activate#(M),activate#(N))
U104#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N)),x#(activate(N),activate(M)),activate#(N),activate#(M),activate#(N))
U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2))
U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)),activate#(V2),activate#(V1),activate#(V2))
U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)),activate#(V2),activate#(V1),activate#(V2))
U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
U15#(tt(),V2) -> c_10(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2))
U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)),activate#(V1),activate#(V1))
U22#(tt(),V1) -> c_13(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1))
U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2))
U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)),activate#(V2),activate#(V1),activate#(V2))
U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)),activate#(V2),activate#(V1),activate#(V2))
U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
U35#(tt(),V2) -> c_19(U36#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2))
U41#(tt(),V2) -> c_21(U42#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2))
U61#(tt(),V2) -> c_24(U62#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2))
U71#(tt(),N) -> c_26(U72#(isNatKind(activate(N)),activate(N)),isNatKind#(activate(N)),activate#(N),activate#(N))
U72#(tt(),N) -> c_27(activate#(N))
U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)),activate#(M),activate#(M),activate#(N))
U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U83#(tt(),M,N) -> c_30(U84#(isNatKind(activate(N)),activate(M),activate(N)),isNatKind#(activate(N)),activate#(N),activate#(M),activate#(N))
U84#(tt(),M,N) -> c_31(s#(plus(activate(N),activate(M))),plus#(activate(N),activate(M)),activate#(N),activate#(M))
U91#(tt(),N) -> c_32(U92#(isNatKind(activate(N))),isNatKind#(activate(N)),activate#(N))
U92#(tt()) -> c_33(0#())
activate#(n__0()) -> c_35(0#())
activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2))
activate#(n__s(X)) -> c_37(s#(X))
activate#(n__x(X1,X2)) -> c_38(x#(X1,X2))
isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2))
isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)),activate#(V1),activate#(V1))
isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2))
isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2))
isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1))
isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2))
Strict TRS Rules:
Weak DP Rules:
0#() -> c_1()
U16#(tt()) -> c_11()
U23#(tt()) -> c_14()
U36#(tt()) -> c_20()
U42#(tt()) -> c_22()
U51#(tt()) -> c_23()
U62#(tt()) -> c_25()
activate#(X) -> c_34()
isNat#(n__0()) -> c_39()
isNatKind#(n__0()) -> c_43()
plus#(X1,X2) -> c_47()
s#(X) -> c_48()
x#(X1,X2) -> c_49()
Weak TRS Rules:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2))
U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2))
U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2))
U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2))
U35(tt(),V2) -> U36(isNat(activate(V2)))
U36(tt()) -> tt()
U41(tt(),V2) -> U42(isNatKind(activate(V2)))
U42(tt()) -> tt()
U51(tt()) -> tt()
U61(tt(),V2) -> U62(isNatKind(activate(V2)))
U62(tt()) -> tt()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1)))
isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
Signature:
{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,activate/1,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/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,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/5,c_6/5,c_7/5,c_8/5,c_9/4,c_10/3,c_11/0,c_12/4,c_13/3,c_14/0,c_15/5,c_16/5,c_17/5,c_18/4,c_19/3,c_20/0,c_21/3,c_22/0,c_23/0,c_24/3,c_25/0,c_26/4,c_27/1,c_28/5,c_29/5,c_30/5,c_31/4,c_32/3,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/5,c_41/4,c_42/5,c_43/0,c_44/4,c_45/3,c_46/4,c_47/0,c_48/0,c_49/0}
Obligation:
Innermost
basic terms: {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#,activate#,isNat#,isNatKind#,plus#,s#,x#}/{n__0,n__plus,n__s,n__x,tt}
Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
Proof:
We estimate the number of application of
{26,27,28,29,30}
by application of
Pre({26,27,28,29,30}) = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,31,32,33,34,35,36}.
Here rules are labelled as follows:
1: U101#(tt(),M,N) ->
c_2(U102#(isNatKind(activate(M))
,activate(M)
,activate(N))
,isNatKind#(activate(M))
,activate#(M)
,activate#(M)
,activate#(N))
2: U102#(tt(),M,N) ->
c_3(U103#(isNat(activate(N))
,activate(M)
,activate(N))
,isNat#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
3: U103#(tt(),M,N) ->
c_4(U104#(isNatKind(activate(N))
,activate(M)
,activate(N))
,isNatKind#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
4: U104#(tt(),M,N) ->
c_5(plus#(x(activate(N)
,activate(M))
,activate(N))
,x#(activate(N),activate(M))
,activate#(N)
,activate#(M)
,activate#(N))
5: U11#(tt(),V1,V2) ->
c_6(U12#(isNatKind(activate(V1))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
6: U12#(tt(),V1,V2) ->
c_7(U13#(isNatKind(activate(V2))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
7: U13#(tt(),V1,V2) ->
c_8(U14#(isNatKind(activate(V2))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
8: U14#(tt(),V1,V2) ->
c_9(U15#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
9: U15#(tt(),V2) ->
c_10(U16#(isNat(activate(V2)))
,isNat#(activate(V2))
,activate#(V2))
10: U21#(tt(),V1) ->
c_12(U22#(isNatKind(activate(V1))
,activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1))
11: U22#(tt(),V1) ->
c_13(U23#(isNat(activate(V1)))
,isNat#(activate(V1))
,activate#(V1))
12: U31#(tt(),V1,V2) ->
c_15(U32#(isNatKind(activate(V1))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
13: U32#(tt(),V1,V2) ->
c_16(U33#(isNatKind(activate(V2))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
14: U33#(tt(),V1,V2) ->
c_17(U34#(isNatKind(activate(V2))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
15: U34#(tt(),V1,V2) ->
c_18(U35#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
16: U35#(tt(),V2) ->
c_19(U36#(isNat(activate(V2)))
,isNat#(activate(V2))
,activate#(V2))
17: U41#(tt(),V2) ->
c_21(U42#(isNatKind(activate(V2)))
,isNatKind#(activate(V2))
,activate#(V2))
18: U61#(tt(),V2) ->
c_24(U62#(isNatKind(activate(V2)))
,isNatKind#(activate(V2))
,activate#(V2))
19: U71#(tt(),N) ->
c_26(U72#(isNatKind(activate(N))
,activate(N))
,isNatKind#(activate(N))
,activate#(N)
,activate#(N))
20: U72#(tt(),N) ->
c_27(activate#(N))
21: U81#(tt(),M,N) ->
c_28(U82#(isNatKind(activate(M))
,activate(M)
,activate(N))
,isNatKind#(activate(M))
,activate#(M)
,activate#(M)
,activate#(N))
22: U82#(tt(),M,N) ->
c_29(U83#(isNat(activate(N))
,activate(M)
,activate(N))
,isNat#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
23: U83#(tt(),M,N) ->
c_30(U84#(isNatKind(activate(N))
,activate(M)
,activate(N))
,isNatKind#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
24: U84#(tt(),M,N) ->
c_31(s#(plus(activate(N)
,activate(M)))
,plus#(activate(N),activate(M))
,activate#(N)
,activate#(M))
25: U91#(tt(),N) ->
c_32(U92#(isNatKind(activate(N)))
,isNatKind#(activate(N))
,activate#(N))
26: U92#(tt()) -> c_33(0#())
27: activate#(n__0()) -> c_35(0#())
28: activate#(n__plus(X1,X2)) ->
c_36(plus#(X1,X2))
29: activate#(n__s(X)) ->
c_37(s#(X))
30: activate#(n__x(X1,X2)) ->
c_38(x#(X1,X2))
31: isNat#(n__plus(V1,V2)) ->
c_40(U11#(isNatKind(activate(V1))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
32: isNat#(n__s(V1)) ->
c_41(U21#(isNatKind(activate(V1))
,activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1))
33: isNat#(n__x(V1,V2)) ->
c_42(U31#(isNatKind(activate(V1))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
34: isNatKind#(n__plus(V1,V2)) ->
c_44(U41#(isNatKind(activate(V1))
,activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2))
35: isNatKind#(n__s(V1)) ->
c_45(U51#(isNatKind(activate(V1)))
,isNatKind#(activate(V1))
,activate#(V1))
36: isNatKind#(n__x(V1,V2)) ->
c_46(U61#(isNatKind(activate(V1))
,activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2))
37: 0#() -> c_1()
38: U16#(tt()) -> c_11()
39: U23#(tt()) -> c_14()
40: U36#(tt()) -> c_20()
41: U42#(tt()) -> c_22()
42: U51#(tt()) -> c_23()
43: U62#(tt()) -> c_25()
44: activate#(X) -> c_34()
45: isNat#(n__0()) -> c_39()
46: isNatKind#(n__0()) -> c_43()
47: plus#(X1,X2) -> c_47()
48: s#(X) -> c_48()
49: x#(X1,X2) -> c_49()
*** 1.1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)),activate#(M),activate#(M),activate#(N))
U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U103#(tt(),M,N) -> c_4(U104#(isNatKind(activate(N)),activate(M),activate(N)),isNatKind#(activate(N)),activate#(N),activate#(M),activate#(N))
U104#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N)),x#(activate(N),activate(M)),activate#(N),activate#(M),activate#(N))
U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2))
U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)),activate#(V2),activate#(V1),activate#(V2))
U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)),activate#(V2),activate#(V1),activate#(V2))
U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
U15#(tt(),V2) -> c_10(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2))
U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)),activate#(V1),activate#(V1))
U22#(tt(),V1) -> c_13(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1))
U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2))
U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)),activate#(V2),activate#(V1),activate#(V2))
U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)),activate#(V2),activate#(V1),activate#(V2))
U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
U35#(tt(),V2) -> c_19(U36#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2))
U41#(tt(),V2) -> c_21(U42#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2))
U61#(tt(),V2) -> c_24(U62#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2))
U71#(tt(),N) -> c_26(U72#(isNatKind(activate(N)),activate(N)),isNatKind#(activate(N)),activate#(N),activate#(N))
U72#(tt(),N) -> c_27(activate#(N))
U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)),activate#(M),activate#(M),activate#(N))
U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U83#(tt(),M,N) -> c_30(U84#(isNatKind(activate(N)),activate(M),activate(N)),isNatKind#(activate(N)),activate#(N),activate#(M),activate#(N))
U84#(tt(),M,N) -> c_31(s#(plus(activate(N),activate(M))),plus#(activate(N),activate(M)),activate#(N),activate#(M))
U91#(tt(),N) -> c_32(U92#(isNatKind(activate(N))),isNatKind#(activate(N)),activate#(N))
isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2))
isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)),activate#(V1),activate#(V1))
isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2))
isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2))
isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1))
isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2))
Strict TRS Rules:
Weak DP Rules:
0#() -> c_1()
U16#(tt()) -> c_11()
U23#(tt()) -> c_14()
U36#(tt()) -> c_20()
U42#(tt()) -> c_22()
U51#(tt()) -> c_23()
U62#(tt()) -> c_25()
U92#(tt()) -> c_33(0#())
activate#(X) -> c_34()
activate#(n__0()) -> c_35(0#())
activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2))
activate#(n__s(X)) -> c_37(s#(X))
activate#(n__x(X1,X2)) -> c_38(x#(X1,X2))
isNat#(n__0()) -> c_39()
isNatKind#(n__0()) -> c_43()
plus#(X1,X2) -> c_47()
s#(X) -> c_48()
x#(X1,X2) -> c_49()
Weak TRS Rules:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2))
U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2))
U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2))
U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2))
U35(tt(),V2) -> U36(isNat(activate(V2)))
U36(tt()) -> tt()
U41(tt(),V2) -> U42(isNatKind(activate(V2)))
U42(tt()) -> tt()
U51(tt()) -> tt()
U61(tt(),V2) -> U62(isNatKind(activate(V2)))
U62(tt()) -> tt()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1)))
isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
Signature:
{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,activate/1,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/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,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/5,c_6/5,c_7/5,c_8/5,c_9/4,c_10/3,c_11/0,c_12/4,c_13/3,c_14/0,c_15/5,c_16/5,c_17/5,c_18/4,c_19/3,c_20/0,c_21/3,c_22/0,c_23/0,c_24/3,c_25/0,c_26/4,c_27/1,c_28/5,c_29/5,c_30/5,c_31/4,c_32/3,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/5,c_41/4,c_42/5,c_43/0,c_44/4,c_45/3,c_46/4,c_47/0,c_48/0,c_49/0}
Obligation:
Innermost
basic terms: {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#,activate#,isNat#,isNatKind#,plus#,s#,x#}/{n__0,n__plus,n__s,n__x,tt}
Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
Proof:
We estimate the number of application of
{4,20,24}
by application of
Pre({4,20,24}) = {3,19,23}.
Here rules are labelled as follows:
1: U101#(tt(),M,N) ->
c_2(U102#(isNatKind(activate(M))
,activate(M)
,activate(N))
,isNatKind#(activate(M))
,activate#(M)
,activate#(M)
,activate#(N))
2: U102#(tt(),M,N) ->
c_3(U103#(isNat(activate(N))
,activate(M)
,activate(N))
,isNat#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
3: U103#(tt(),M,N) ->
c_4(U104#(isNatKind(activate(N))
,activate(M)
,activate(N))
,isNatKind#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
4: U104#(tt(),M,N) ->
c_5(plus#(x(activate(N)
,activate(M))
,activate(N))
,x#(activate(N),activate(M))
,activate#(N)
,activate#(M)
,activate#(N))
5: U11#(tt(),V1,V2) ->
c_6(U12#(isNatKind(activate(V1))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
6: U12#(tt(),V1,V2) ->
c_7(U13#(isNatKind(activate(V2))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
7: U13#(tt(),V1,V2) ->
c_8(U14#(isNatKind(activate(V2))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
8: U14#(tt(),V1,V2) ->
c_9(U15#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
9: U15#(tt(),V2) ->
c_10(U16#(isNat(activate(V2)))
,isNat#(activate(V2))
,activate#(V2))
10: U21#(tt(),V1) ->
c_12(U22#(isNatKind(activate(V1))
,activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1))
11: U22#(tt(),V1) ->
c_13(U23#(isNat(activate(V1)))
,isNat#(activate(V1))
,activate#(V1))
12: U31#(tt(),V1,V2) ->
c_15(U32#(isNatKind(activate(V1))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
13: U32#(tt(),V1,V2) ->
c_16(U33#(isNatKind(activate(V2))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
14: U33#(tt(),V1,V2) ->
c_17(U34#(isNatKind(activate(V2))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
15: U34#(tt(),V1,V2) ->
c_18(U35#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
16: U35#(tt(),V2) ->
c_19(U36#(isNat(activate(V2)))
,isNat#(activate(V2))
,activate#(V2))
17: U41#(tt(),V2) ->
c_21(U42#(isNatKind(activate(V2)))
,isNatKind#(activate(V2))
,activate#(V2))
18: U61#(tt(),V2) ->
c_24(U62#(isNatKind(activate(V2)))
,isNatKind#(activate(V2))
,activate#(V2))
19: U71#(tt(),N) ->
c_26(U72#(isNatKind(activate(N))
,activate(N))
,isNatKind#(activate(N))
,activate#(N)
,activate#(N))
20: U72#(tt(),N) ->
c_27(activate#(N))
21: U81#(tt(),M,N) ->
c_28(U82#(isNatKind(activate(M))
,activate(M)
,activate(N))
,isNatKind#(activate(M))
,activate#(M)
,activate#(M)
,activate#(N))
22: U82#(tt(),M,N) ->
c_29(U83#(isNat(activate(N))
,activate(M)
,activate(N))
,isNat#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
23: U83#(tt(),M,N) ->
c_30(U84#(isNatKind(activate(N))
,activate(M)
,activate(N))
,isNatKind#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
24: U84#(tt(),M,N) ->
c_31(s#(plus(activate(N)
,activate(M)))
,plus#(activate(N),activate(M))
,activate#(N)
,activate#(M))
25: U91#(tt(),N) ->
c_32(U92#(isNatKind(activate(N)))
,isNatKind#(activate(N))
,activate#(N))
26: isNat#(n__plus(V1,V2)) ->
c_40(U11#(isNatKind(activate(V1))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
27: isNat#(n__s(V1)) ->
c_41(U21#(isNatKind(activate(V1))
,activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1))
28: isNat#(n__x(V1,V2)) ->
c_42(U31#(isNatKind(activate(V1))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
29: isNatKind#(n__plus(V1,V2)) ->
c_44(U41#(isNatKind(activate(V1))
,activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2))
30: isNatKind#(n__s(V1)) ->
c_45(U51#(isNatKind(activate(V1)))
,isNatKind#(activate(V1))
,activate#(V1))
31: isNatKind#(n__x(V1,V2)) ->
c_46(U61#(isNatKind(activate(V1))
,activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2))
32: 0#() -> c_1()
33: U16#(tt()) -> c_11()
34: U23#(tt()) -> c_14()
35: U36#(tt()) -> c_20()
36: U42#(tt()) -> c_22()
37: U51#(tt()) -> c_23()
38: U62#(tt()) -> c_25()
39: U92#(tt()) -> c_33(0#())
40: activate#(X) -> c_34()
41: activate#(n__0()) -> c_35(0#())
42: activate#(n__plus(X1,X2)) ->
c_36(plus#(X1,X2))
43: activate#(n__s(X)) ->
c_37(s#(X))
44: activate#(n__x(X1,X2)) ->
c_38(x#(X1,X2))
45: isNat#(n__0()) -> c_39()
46: isNatKind#(n__0()) -> c_43()
47: plus#(X1,X2) -> c_47()
48: s#(X) -> c_48()
49: x#(X1,X2) -> c_49()
*** 1.1.1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)),activate#(M),activate#(M),activate#(N))
U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U103#(tt(),M,N) -> c_4(U104#(isNatKind(activate(N)),activate(M),activate(N)),isNatKind#(activate(N)),activate#(N),activate#(M),activate#(N))
U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2))
U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)),activate#(V2),activate#(V1),activate#(V2))
U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)),activate#(V2),activate#(V1),activate#(V2))
U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
U15#(tt(),V2) -> c_10(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2))
U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)),activate#(V1),activate#(V1))
U22#(tt(),V1) -> c_13(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1))
U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2))
U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)),activate#(V2),activate#(V1),activate#(V2))
U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)),activate#(V2),activate#(V1),activate#(V2))
U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
U35#(tt(),V2) -> c_19(U36#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2))
U41#(tt(),V2) -> c_21(U42#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2))
U61#(tt(),V2) -> c_24(U62#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2))
U71#(tt(),N) -> c_26(U72#(isNatKind(activate(N)),activate(N)),isNatKind#(activate(N)),activate#(N),activate#(N))
U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)),activate#(M),activate#(M),activate#(N))
U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U83#(tt(),M,N) -> c_30(U84#(isNatKind(activate(N)),activate(M),activate(N)),isNatKind#(activate(N)),activate#(N),activate#(M),activate#(N))
U91#(tt(),N) -> c_32(U92#(isNatKind(activate(N))),isNatKind#(activate(N)),activate#(N))
isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2))
isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)),activate#(V1),activate#(V1))
isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2))
isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2))
isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1))
isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2))
Strict TRS Rules:
Weak DP Rules:
0#() -> c_1()
U104#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N)),x#(activate(N),activate(M)),activate#(N),activate#(M),activate#(N))
U16#(tt()) -> c_11()
U23#(tt()) -> c_14()
U36#(tt()) -> c_20()
U42#(tt()) -> c_22()
U51#(tt()) -> c_23()
U62#(tt()) -> c_25()
U72#(tt(),N) -> c_27(activate#(N))
U84#(tt(),M,N) -> c_31(s#(plus(activate(N),activate(M))),plus#(activate(N),activate(M)),activate#(N),activate#(M))
U92#(tt()) -> c_33(0#())
activate#(X) -> c_34()
activate#(n__0()) -> c_35(0#())
activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2))
activate#(n__s(X)) -> c_37(s#(X))
activate#(n__x(X1,X2)) -> c_38(x#(X1,X2))
isNat#(n__0()) -> c_39()
isNatKind#(n__0()) -> c_43()
plus#(X1,X2) -> c_47()
s#(X) -> c_48()
x#(X1,X2) -> c_49()
Weak TRS Rules:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2))
U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2))
U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2))
U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2))
U35(tt(),V2) -> U36(isNat(activate(V2)))
U36(tt()) -> tt()
U41(tt(),V2) -> U42(isNatKind(activate(V2)))
U42(tt()) -> tt()
U51(tt()) -> tt()
U61(tt(),V2) -> U62(isNatKind(activate(V2)))
U62(tt()) -> tt()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1)))
isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
Signature:
{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,activate/1,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/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,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/5,c_6/5,c_7/5,c_8/5,c_9/4,c_10/3,c_11/0,c_12/4,c_13/3,c_14/0,c_15/5,c_16/5,c_17/5,c_18/4,c_19/3,c_20/0,c_21/3,c_22/0,c_23/0,c_24/3,c_25/0,c_26/4,c_27/1,c_28/5,c_29/5,c_30/5,c_31/4,c_32/3,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/5,c_41/4,c_42/5,c_43/0,c_44/4,c_45/3,c_46/4,c_47/0,c_48/0,c_49/0}
Obligation:
Innermost
basic terms: {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#,activate#,isNat#,isNatKind#,plus#,s#,x#}/{n__0,n__plus,n__s,n__x,tt}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S:U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)),activate#(M),activate#(M),activate#(N))
-->_5 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_4 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_5 activate#(n__s(X)) -> c_37(s#(X)):43
-->_4 activate#(n__s(X)) -> c_37(s#(X)):43
-->_3 activate#(n__s(X)) -> c_37(s#(X)):43
-->_5 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_4 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_5 activate#(n__0()) -> c_35(0#()):41
-->_4 activate#(n__0()) -> c_35(0#()):41
-->_3 activate#(n__0()) -> c_35(0#()):41
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):28
-->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):26
-->_1 U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N)):2
-->_2 isNatKind#(n__0()) -> c_43():46
-->_5 activate#(X) -> c_34():40
-->_4 activate#(X) -> c_34():40
-->_3 activate#(X) -> c_34():40
2:S:U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
-->_5 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_4 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_5 activate#(n__s(X)) -> c_37(s#(X)):43
-->_4 activate#(n__s(X)) -> c_37(s#(X)):43
-->_3 activate#(n__s(X)) -> c_37(s#(X)):43
-->_5 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_4 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_5 activate#(n__0()) -> c_35(0#()):41
-->_4 activate#(n__0()) -> c_35(0#()):41
-->_3 activate#(n__0()) -> c_35(0#()):41
-->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2)):25
-->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)),activate#(V1),activate#(V1)):24
-->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2)):23
-->_1 U103#(tt(),M,N) -> c_4(U104#(isNatKind(activate(N)),activate(M),activate(N)),isNatKind#(activate(N)),activate#(N),activate#(M),activate#(N)):3
-->_2 isNat#(n__0()) -> c_39():45
-->_5 activate#(X) -> c_34():40
-->_4 activate#(X) -> c_34():40
-->_3 activate#(X) -> c_34():40
3:S:U103#(tt(),M,N) -> c_4(U104#(isNatKind(activate(N)),activate(M),activate(N)),isNatKind#(activate(N)),activate#(N),activate#(M),activate#(N))
-->_5 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_4 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_5 activate#(n__s(X)) -> c_37(s#(X)):43
-->_4 activate#(n__s(X)) -> c_37(s#(X)):43
-->_3 activate#(n__s(X)) -> c_37(s#(X)):43
-->_5 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_4 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_5 activate#(n__0()) -> c_35(0#()):41
-->_4 activate#(n__0()) -> c_35(0#()):41
-->_3 activate#(n__0()) -> c_35(0#()):41
-->_1 U104#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N)),x#(activate(N),activate(M)),activate#(N),activate#(M),activate#(N)):30
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):28
-->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):26
-->_2 isNatKind#(n__0()) -> c_43():46
-->_5 activate#(X) -> c_34():40
-->_4 activate#(X) -> c_34():40
-->_3 activate#(X) -> c_34():40
4:S:U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2))
-->_5 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_4 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_5 activate#(n__s(X)) -> c_37(s#(X)):43
-->_4 activate#(n__s(X)) -> c_37(s#(X)):43
-->_3 activate#(n__s(X)) -> c_37(s#(X)):43
-->_5 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_4 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_5 activate#(n__0()) -> c_35(0#()):41
-->_4 activate#(n__0()) -> c_35(0#()):41
-->_3 activate#(n__0()) -> c_35(0#()):41
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):28
-->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):26
-->_1 U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)),activate#(V2),activate#(V1),activate#(V2)):5
-->_2 isNatKind#(n__0()) -> c_43():46
-->_5 activate#(X) -> c_34():40
-->_4 activate#(X) -> c_34():40
-->_3 activate#(X) -> c_34():40
5:S:U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)),activate#(V2),activate#(V1),activate#(V2))
-->_5 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_4 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_5 activate#(n__s(X)) -> c_37(s#(X)):43
-->_4 activate#(n__s(X)) -> c_37(s#(X)):43
-->_3 activate#(n__s(X)) -> c_37(s#(X)):43
-->_5 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_4 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_5 activate#(n__0()) -> c_35(0#()):41
-->_4 activate#(n__0()) -> c_35(0#()):41
-->_3 activate#(n__0()) -> c_35(0#()):41
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):28
-->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):26
-->_1 U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)),activate#(V2),activate#(V1),activate#(V2)):6
-->_2 isNatKind#(n__0()) -> c_43():46
-->_5 activate#(X) -> c_34():40
-->_4 activate#(X) -> c_34():40
-->_3 activate#(X) -> c_34():40
6:S:U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)),activate#(V2),activate#(V1),activate#(V2))
-->_5 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_4 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_5 activate#(n__s(X)) -> c_37(s#(X)):43
-->_4 activate#(n__s(X)) -> c_37(s#(X)):43
-->_3 activate#(n__s(X)) -> c_37(s#(X)):43
-->_5 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_4 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_5 activate#(n__0()) -> c_35(0#()):41
-->_4 activate#(n__0()) -> c_35(0#()):41
-->_3 activate#(n__0()) -> c_35(0#()):41
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):28
-->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):26
-->_1 U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):7
-->_2 isNatKind#(n__0()) -> c_43():46
-->_5 activate#(X) -> c_34():40
-->_4 activate#(X) -> c_34():40
-->_3 activate#(X) -> c_34():40
7:S:U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
-->_4 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_4 activate#(n__s(X)) -> c_37(s#(X)):43
-->_3 activate#(n__s(X)) -> c_37(s#(X)):43
-->_4 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_4 activate#(n__0()) -> c_35(0#()):41
-->_3 activate#(n__0()) -> c_35(0#()):41
-->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2)):25
-->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)),activate#(V1),activate#(V1)):24
-->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2)):23
-->_1 U15#(tt(),V2) -> c_10(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)):8
-->_2 isNat#(n__0()) -> c_39():45
-->_4 activate#(X) -> c_34():40
-->_3 activate#(X) -> c_34():40
8:S:U15#(tt(),V2) -> c_10(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2))
-->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_3 activate#(n__s(X)) -> c_37(s#(X)):43
-->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_3 activate#(n__0()) -> c_35(0#()):41
-->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2)):25
-->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)),activate#(V1),activate#(V1)):24
-->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2)):23
-->_2 isNat#(n__0()) -> c_39():45
-->_3 activate#(X) -> c_34():40
-->_1 U16#(tt()) -> c_11():31
9:S:U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)),activate#(V1),activate#(V1))
-->_4 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_4 activate#(n__s(X)) -> c_37(s#(X)):43
-->_3 activate#(n__s(X)) -> c_37(s#(X)):43
-->_4 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_4 activate#(n__0()) -> c_35(0#()):41
-->_3 activate#(n__0()) -> c_35(0#()):41
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):28
-->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):26
-->_1 U22#(tt(),V1) -> c_13(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):10
-->_2 isNatKind#(n__0()) -> c_43():46
-->_4 activate#(X) -> c_34():40
-->_3 activate#(X) -> c_34():40
10:S:U22#(tt(),V1) -> c_13(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1))
-->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_3 activate#(n__s(X)) -> c_37(s#(X)):43
-->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_3 activate#(n__0()) -> c_35(0#()):41
-->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2)):25
-->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)),activate#(V1),activate#(V1)):24
-->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2)):23
-->_2 isNat#(n__0()) -> c_39():45
-->_3 activate#(X) -> c_34():40
-->_1 U23#(tt()) -> c_14():32
11:S:U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2))
-->_5 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_4 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_5 activate#(n__s(X)) -> c_37(s#(X)):43
-->_4 activate#(n__s(X)) -> c_37(s#(X)):43
-->_3 activate#(n__s(X)) -> c_37(s#(X)):43
-->_5 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_4 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_5 activate#(n__0()) -> c_35(0#()):41
-->_4 activate#(n__0()) -> c_35(0#()):41
-->_3 activate#(n__0()) -> c_35(0#()):41
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):28
-->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):26
-->_1 U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)),activate#(V2),activate#(V1),activate#(V2)):12
-->_2 isNatKind#(n__0()) -> c_43():46
-->_5 activate#(X) -> c_34():40
-->_4 activate#(X) -> c_34():40
-->_3 activate#(X) -> c_34():40
12:S:U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)),activate#(V2),activate#(V1),activate#(V2))
-->_5 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_4 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_5 activate#(n__s(X)) -> c_37(s#(X)):43
-->_4 activate#(n__s(X)) -> c_37(s#(X)):43
-->_3 activate#(n__s(X)) -> c_37(s#(X)):43
-->_5 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_4 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_5 activate#(n__0()) -> c_35(0#()):41
-->_4 activate#(n__0()) -> c_35(0#()):41
-->_3 activate#(n__0()) -> c_35(0#()):41
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):28
-->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):26
-->_1 U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)),activate#(V2),activate#(V1),activate#(V2)):13
-->_2 isNatKind#(n__0()) -> c_43():46
-->_5 activate#(X) -> c_34():40
-->_4 activate#(X) -> c_34():40
-->_3 activate#(X) -> c_34():40
13:S:U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)),activate#(V2),activate#(V1),activate#(V2))
-->_5 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_4 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_5 activate#(n__s(X)) -> c_37(s#(X)):43
-->_4 activate#(n__s(X)) -> c_37(s#(X)):43
-->_3 activate#(n__s(X)) -> c_37(s#(X)):43
-->_5 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_4 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_5 activate#(n__0()) -> c_35(0#()):41
-->_4 activate#(n__0()) -> c_35(0#()):41
-->_3 activate#(n__0()) -> c_35(0#()):41
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):28
-->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):26
-->_1 U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):14
-->_2 isNatKind#(n__0()) -> c_43():46
-->_5 activate#(X) -> c_34():40
-->_4 activate#(X) -> c_34():40
-->_3 activate#(X) -> c_34():40
14:S:U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
-->_4 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_4 activate#(n__s(X)) -> c_37(s#(X)):43
-->_3 activate#(n__s(X)) -> c_37(s#(X)):43
-->_4 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_4 activate#(n__0()) -> c_35(0#()):41
-->_3 activate#(n__0()) -> c_35(0#()):41
-->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2)):25
-->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)),activate#(V1),activate#(V1)):24
-->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2)):23
-->_1 U35#(tt(),V2) -> c_19(U36#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)):15
-->_2 isNat#(n__0()) -> c_39():45
-->_4 activate#(X) -> c_34():40
-->_3 activate#(X) -> c_34():40
15:S:U35#(tt(),V2) -> c_19(U36#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2))
-->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_3 activate#(n__s(X)) -> c_37(s#(X)):43
-->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_3 activate#(n__0()) -> c_35(0#()):41
-->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2)):25
-->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)),activate#(V1),activate#(V1)):24
-->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2)):23
-->_2 isNat#(n__0()) -> c_39():45
-->_3 activate#(X) -> c_34():40
-->_1 U36#(tt()) -> c_20():33
16:S:U41#(tt(),V2) -> c_21(U42#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2))
-->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_3 activate#(n__s(X)) -> c_37(s#(X)):43
-->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_3 activate#(n__0()) -> c_35(0#()):41
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):28
-->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):26
-->_2 isNatKind#(n__0()) -> c_43():46
-->_3 activate#(X) -> c_34():40
-->_1 U42#(tt()) -> c_22():34
17:S:U61#(tt(),V2) -> c_24(U62#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2))
-->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_3 activate#(n__s(X)) -> c_37(s#(X)):43
-->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_3 activate#(n__0()) -> c_35(0#()):41
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):28
-->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):26
-->_2 isNatKind#(n__0()) -> c_43():46
-->_3 activate#(X) -> c_34():40
-->_1 U62#(tt()) -> c_25():36
18:S:U71#(tt(),N) -> c_26(U72#(isNatKind(activate(N)),activate(N)),isNatKind#(activate(N)),activate#(N),activate#(N))
-->_4 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_4 activate#(n__s(X)) -> c_37(s#(X)):43
-->_3 activate#(n__s(X)) -> c_37(s#(X)):43
-->_4 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_4 activate#(n__0()) -> c_35(0#()):41
-->_3 activate#(n__0()) -> c_35(0#()):41
-->_1 U72#(tt(),N) -> c_27(activate#(N)):37
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):28
-->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):26
-->_2 isNatKind#(n__0()) -> c_43():46
-->_4 activate#(X) -> c_34():40
-->_3 activate#(X) -> c_34():40
19:S:U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)),activate#(M),activate#(M),activate#(N))
-->_5 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_4 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_5 activate#(n__s(X)) -> c_37(s#(X)):43
-->_4 activate#(n__s(X)) -> c_37(s#(X)):43
-->_3 activate#(n__s(X)) -> c_37(s#(X)):43
-->_5 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_4 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_5 activate#(n__0()) -> c_35(0#()):41
-->_4 activate#(n__0()) -> c_35(0#()):41
-->_3 activate#(n__0()) -> c_35(0#()):41
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):28
-->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):26
-->_1 U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N)):20
-->_2 isNatKind#(n__0()) -> c_43():46
-->_5 activate#(X) -> c_34():40
-->_4 activate#(X) -> c_34():40
-->_3 activate#(X) -> c_34():40
20:S:U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
-->_5 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_4 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_5 activate#(n__s(X)) -> c_37(s#(X)):43
-->_4 activate#(n__s(X)) -> c_37(s#(X)):43
-->_3 activate#(n__s(X)) -> c_37(s#(X)):43
-->_5 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_4 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_5 activate#(n__0()) -> c_35(0#()):41
-->_4 activate#(n__0()) -> c_35(0#()):41
-->_3 activate#(n__0()) -> c_35(0#()):41
-->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2)):25
-->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)),activate#(V1),activate#(V1)):24
-->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2)):23
-->_1 U83#(tt(),M,N) -> c_30(U84#(isNatKind(activate(N)),activate(M),activate(N)),isNatKind#(activate(N)),activate#(N),activate#(M),activate#(N)):21
-->_2 isNat#(n__0()) -> c_39():45
-->_5 activate#(X) -> c_34():40
-->_4 activate#(X) -> c_34():40
-->_3 activate#(X) -> c_34():40
21:S:U83#(tt(),M,N) -> c_30(U84#(isNatKind(activate(N)),activate(M),activate(N)),isNatKind#(activate(N)),activate#(N),activate#(M),activate#(N))
-->_5 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_4 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_5 activate#(n__s(X)) -> c_37(s#(X)):43
-->_4 activate#(n__s(X)) -> c_37(s#(X)):43
-->_3 activate#(n__s(X)) -> c_37(s#(X)):43
-->_5 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_4 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_5 activate#(n__0()) -> c_35(0#()):41
-->_4 activate#(n__0()) -> c_35(0#()):41
-->_3 activate#(n__0()) -> c_35(0#()):41
-->_1 U84#(tt(),M,N) -> c_31(s#(plus(activate(N),activate(M))),plus#(activate(N),activate(M)),activate#(N),activate#(M)):38
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):28
-->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):26
-->_2 isNatKind#(n__0()) -> c_43():46
-->_5 activate#(X) -> c_34():40
-->_4 activate#(X) -> c_34():40
-->_3 activate#(X) -> c_34():40
22:S:U91#(tt(),N) -> c_32(U92#(isNatKind(activate(N))),isNatKind#(activate(N)),activate#(N))
-->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_3 activate#(n__s(X)) -> c_37(s#(X)):43
-->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_3 activate#(n__0()) -> c_35(0#()):41
-->_1 U92#(tt()) -> c_33(0#()):39
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):28
-->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):26
-->_2 isNatKind#(n__0()) -> c_43():46
-->_3 activate#(X) -> c_34():40
23:S:isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2))
-->_5 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_4 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_5 activate#(n__s(X)) -> c_37(s#(X)):43
-->_4 activate#(n__s(X)) -> c_37(s#(X)):43
-->_3 activate#(n__s(X)) -> c_37(s#(X)):43
-->_5 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_4 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_5 activate#(n__0()) -> c_35(0#()):41
-->_4 activate#(n__0()) -> c_35(0#()):41
-->_3 activate#(n__0()) -> c_35(0#()):41
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):28
-->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):26
-->_2 isNatKind#(n__0()) -> c_43():46
-->_5 activate#(X) -> c_34():40
-->_4 activate#(X) -> c_34():40
-->_3 activate#(X) -> c_34():40
-->_1 U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2)):4
24:S:isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)),activate#(V1),activate#(V1))
-->_4 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_4 activate#(n__s(X)) -> c_37(s#(X)):43
-->_3 activate#(n__s(X)) -> c_37(s#(X)):43
-->_4 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_4 activate#(n__0()) -> c_35(0#()):41
-->_3 activate#(n__0()) -> c_35(0#()):41
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):28
-->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):26
-->_2 isNatKind#(n__0()) -> c_43():46
-->_4 activate#(X) -> c_34():40
-->_3 activate#(X) -> c_34():40
-->_1 U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)),activate#(V1),activate#(V1)):9
25:S:isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2))
-->_5 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_4 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_5 activate#(n__s(X)) -> c_37(s#(X)):43
-->_4 activate#(n__s(X)) -> c_37(s#(X)):43
-->_3 activate#(n__s(X)) -> c_37(s#(X)):43
-->_5 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_4 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_5 activate#(n__0()) -> c_35(0#()):41
-->_4 activate#(n__0()) -> c_35(0#()):41
-->_3 activate#(n__0()) -> c_35(0#()):41
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):28
-->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):26
-->_2 isNatKind#(n__0()) -> c_43():46
-->_5 activate#(X) -> c_34():40
-->_4 activate#(X) -> c_34():40
-->_3 activate#(X) -> c_34():40
-->_1 U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2)):11
26:S:isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2))
-->_4 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_4 activate#(n__s(X)) -> c_37(s#(X)):43
-->_3 activate#(n__s(X)) -> c_37(s#(X)):43
-->_4 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_4 activate#(n__0()) -> c_35(0#()):41
-->_3 activate#(n__0()) -> c_35(0#()):41
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):28
-->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27
-->_2 isNatKind#(n__0()) -> c_43():46
-->_4 activate#(X) -> c_34():40
-->_3 activate#(X) -> c_34():40
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):26
-->_1 U41#(tt(),V2) -> c_21(U42#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)):16
27:S:isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1))
-->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_3 activate#(n__s(X)) -> c_37(s#(X)):43
-->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_3 activate#(n__0()) -> c_35(0#()):41
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):28
-->_2 isNatKind#(n__0()) -> c_43():46
-->_3 activate#(X) -> c_34():40
-->_1 U51#(tt()) -> c_23():35
-->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):26
28:S:isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2))
-->_4 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_4 activate#(n__s(X)) -> c_37(s#(X)):43
-->_3 activate#(n__s(X)) -> c_37(s#(X)):43
-->_4 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_4 activate#(n__0()) -> c_35(0#()):41
-->_3 activate#(n__0()) -> c_35(0#()):41
-->_2 isNatKind#(n__0()) -> c_43():46
-->_4 activate#(X) -> c_34():40
-->_3 activate#(X) -> c_34():40
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):28
-->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):26
-->_1 U61#(tt(),V2) -> c_24(U62#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)):17
29:W:0#() -> c_1()
30:W:U104#(tt(),M,N) -> c_5(plus#(x(activate(N),activate(M)),activate(N)),x#(activate(N),activate(M)),activate#(N),activate#(M),activate#(N))
-->_5 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_4 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_5 activate#(n__s(X)) -> c_37(s#(X)):43
-->_4 activate#(n__s(X)) -> c_37(s#(X)):43
-->_3 activate#(n__s(X)) -> c_37(s#(X)):43
-->_5 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_4 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_5 activate#(n__0()) -> c_35(0#()):41
-->_4 activate#(n__0()) -> c_35(0#()):41
-->_3 activate#(n__0()) -> c_35(0#()):41
-->_2 x#(X1,X2) -> c_49():49
-->_1 plus#(X1,X2) -> c_47():47
-->_5 activate#(X) -> c_34():40
-->_4 activate#(X) -> c_34():40
-->_3 activate#(X) -> c_34():40
31:W:U16#(tt()) -> c_11()
32:W:U23#(tt()) -> c_14()
33:W:U36#(tt()) -> c_20()
34:W:U42#(tt()) -> c_22()
35:W:U51#(tt()) -> c_23()
36:W:U62#(tt()) -> c_25()
37:W:U72#(tt(),N) -> c_27(activate#(N))
-->_1 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_1 activate#(n__s(X)) -> c_37(s#(X)):43
-->_1 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_1 activate#(n__0()) -> c_35(0#()):41
-->_1 activate#(X) -> c_34():40
38:W:U84#(tt(),M,N) -> c_31(s#(plus(activate(N),activate(M))),plus#(activate(N),activate(M)),activate#(N),activate#(M))
-->_4 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_3 activate#(n__x(X1,X2)) -> c_38(x#(X1,X2)):44
-->_4 activate#(n__s(X)) -> c_37(s#(X)):43
-->_3 activate#(n__s(X)) -> c_37(s#(X)):43
-->_4 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_3 activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2)):42
-->_4 activate#(n__0()) -> c_35(0#()):41
-->_3 activate#(n__0()) -> c_35(0#()):41
-->_1 s#(X) -> c_48():48
-->_2 plus#(X1,X2) -> c_47():47
-->_4 activate#(X) -> c_34():40
-->_3 activate#(X) -> c_34():40
39:W:U92#(tt()) -> c_33(0#())
-->_1 0#() -> c_1():29
40:W:activate#(X) -> c_34()
41:W:activate#(n__0()) -> c_35(0#())
-->_1 0#() -> c_1():29
42:W:activate#(n__plus(X1,X2)) -> c_36(plus#(X1,X2))
-->_1 plus#(X1,X2) -> c_47():47
43:W:activate#(n__s(X)) -> c_37(s#(X))
-->_1 s#(X) -> c_48():48
44:W:activate#(n__x(X1,X2)) -> c_38(x#(X1,X2))
-->_1 x#(X1,X2) -> c_49():49
45:W:isNat#(n__0()) -> c_39()
46:W:isNatKind#(n__0()) -> c_43()
47:W:plus#(X1,X2) -> c_47()
48:W:s#(X) -> c_48()
49:W:x#(X1,X2) -> c_49()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
39: U92#(tt()) -> c_33(0#())
38: U84#(tt(),M,N) ->
c_31(s#(plus(activate(N)
,activate(M)))
,plus#(activate(N),activate(M))
,activate#(N)
,activate#(M))
37: U72#(tt(),N) ->
c_27(activate#(N))
30: U104#(tt(),M,N) ->
c_5(plus#(x(activate(N)
,activate(M))
,activate(N))
,x#(activate(N),activate(M))
,activate#(N)
,activate#(M)
,activate#(N))
33: U36#(tt()) -> c_20()
32: U23#(tt()) -> c_14()
31: U16#(tt()) -> c_11()
45: isNat#(n__0()) -> c_39()
36: U62#(tt()) -> c_25()
34: U42#(tt()) -> c_22()
35: U51#(tt()) -> c_23()
40: activate#(X) -> c_34()
46: isNatKind#(n__0()) -> c_43()
41: activate#(n__0()) -> c_35(0#())
29: 0#() -> c_1()
42: activate#(n__plus(X1,X2)) ->
c_36(plus#(X1,X2))
47: plus#(X1,X2) -> c_47()
43: activate#(n__s(X)) ->
c_37(s#(X))
48: s#(X) -> c_48()
44: activate#(n__x(X1,X2)) ->
c_38(x#(X1,X2))
49: x#(X1,X2) -> c_49()
*** 1.1.1.1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)),activate#(M),activate#(M),activate#(N))
U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U103#(tt(),M,N) -> c_4(U104#(isNatKind(activate(N)),activate(M),activate(N)),isNatKind#(activate(N)),activate#(N),activate#(M),activate#(N))
U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2))
U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)),activate#(V2),activate#(V1),activate#(V2))
U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)),activate#(V2),activate#(V1),activate#(V2))
U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
U15#(tt(),V2) -> c_10(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2))
U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)),activate#(V1),activate#(V1))
U22#(tt(),V1) -> c_13(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1))
U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2))
U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)),activate#(V2),activate#(V1),activate#(V2))
U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)),activate#(V2),activate#(V1),activate#(V2))
U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
U35#(tt(),V2) -> c_19(U36#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2))
U41#(tt(),V2) -> c_21(U42#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2))
U61#(tt(),V2) -> c_24(U62#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2))
U71#(tt(),N) -> c_26(U72#(isNatKind(activate(N)),activate(N)),isNatKind#(activate(N)),activate#(N),activate#(N))
U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)),activate#(M),activate#(M),activate#(N))
U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
U83#(tt(),M,N) -> c_30(U84#(isNatKind(activate(N)),activate(M),activate(N)),isNatKind#(activate(N)),activate#(N),activate#(M),activate#(N))
U91#(tt(),N) -> c_32(U92#(isNatKind(activate(N))),isNatKind#(activate(N)),activate#(N))
isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2))
isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)),activate#(V1),activate#(V1))
isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2))
isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2))
isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1))
isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2))
U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2))
U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2))
U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2))
U35(tt(),V2) -> U36(isNat(activate(V2)))
U36(tt()) -> tt()
U41(tt(),V2) -> U42(isNatKind(activate(V2)))
U42(tt()) -> tt()
U51(tt()) -> tt()
U61(tt(),V2) -> U62(isNatKind(activate(V2)))
U62(tt()) -> tt()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1)))
isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
Signature:
{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,activate/1,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/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,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/5,c_6/5,c_7/5,c_8/5,c_9/4,c_10/3,c_11/0,c_12/4,c_13/3,c_14/0,c_15/5,c_16/5,c_17/5,c_18/4,c_19/3,c_20/0,c_21/3,c_22/0,c_23/0,c_24/3,c_25/0,c_26/4,c_27/1,c_28/5,c_29/5,c_30/5,c_31/4,c_32/3,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/5,c_41/4,c_42/5,c_43/0,c_44/4,c_45/3,c_46/4,c_47/0,c_48/0,c_49/0}
Obligation:
Innermost
basic terms: {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#,activate#,isNat#,isNatKind#,plus#,s#,x#}/{n__0,n__plus,n__s,n__x,tt}
Applied Processor:
SimplifyRHS
Proof:
Consider the dependency graph
1:S:U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)),activate#(M),activate#(M),activate#(N))
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):28
-->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):26
-->_1 U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N)):2
2:S:U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
-->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2)):25
-->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)),activate#(V1),activate#(V1)):24
-->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2)):23
-->_1 U103#(tt(),M,N) -> c_4(U104#(isNatKind(activate(N)),activate(M),activate(N)),isNatKind#(activate(N)),activate#(N),activate#(M),activate#(N)):3
3:S:U103#(tt(),M,N) -> c_4(U104#(isNatKind(activate(N)),activate(M),activate(N)),isNatKind#(activate(N)),activate#(N),activate#(M),activate#(N))
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):28
-->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):26
4:S:U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2))
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):28
-->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):26
-->_1 U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)),activate#(V2),activate#(V1),activate#(V2)):5
5:S:U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)),activate#(V2),activate#(V1),activate#(V2))
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):28
-->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):26
-->_1 U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)),activate#(V2),activate#(V1),activate#(V2)):6
6:S:U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)),activate#(V2),activate#(V1),activate#(V2))
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):28
-->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):26
-->_1 U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):7
7:S:U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
-->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2)):25
-->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)),activate#(V1),activate#(V1)):24
-->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2)):23
-->_1 U15#(tt(),V2) -> c_10(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)):8
8:S:U15#(tt(),V2) -> c_10(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2))
-->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2)):25
-->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)),activate#(V1),activate#(V1)):24
-->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2)):23
9:S:U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)),activate#(V1),activate#(V1))
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):28
-->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):26
-->_1 U22#(tt(),V1) -> c_13(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):10
10:S:U22#(tt(),V1) -> c_13(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1))
-->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2)):25
-->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)),activate#(V1),activate#(V1)):24
-->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2)):23
11:S:U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2))
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):28
-->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):26
-->_1 U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)),activate#(V2),activate#(V1),activate#(V2)):12
12:S:U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)),activate#(V2),activate#(V1),activate#(V2))
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):28
-->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):26
-->_1 U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)),activate#(V2),activate#(V1),activate#(V2)):13
13:S:U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)),activate#(V2),activate#(V1),activate#(V2))
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):28
-->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):26
-->_1 U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2)):14
14:S:U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)),activate#(V1),activate#(V2))
-->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2)):25
-->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)),activate#(V1),activate#(V1)):24
-->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2)):23
-->_1 U35#(tt(),V2) -> c_19(U36#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)):15
15:S:U35#(tt(),V2) -> c_19(U36#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2))
-->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2)):25
-->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)),activate#(V1),activate#(V1)):24
-->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2)):23
16:S:U41#(tt(),V2) -> c_21(U42#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2))
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):28
-->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):26
17:S:U61#(tt(),V2) -> c_24(U62#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2))
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):28
-->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):26
18:S:U71#(tt(),N) -> c_26(U72#(isNatKind(activate(N)),activate(N)),isNatKind#(activate(N)),activate#(N),activate#(N))
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):28
-->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):26
19:S:U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)),activate#(M),activate#(M),activate#(N))
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):28
-->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):26
-->_1 U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N)):20
20:S:U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)),activate#(N),activate#(M),activate#(N))
-->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2)):25
-->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)),activate#(V1),activate#(V1)):24
-->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2)):23
-->_1 U83#(tt(),M,N) -> c_30(U84#(isNatKind(activate(N)),activate(M),activate(N)),isNatKind#(activate(N)),activate#(N),activate#(M),activate#(N)):21
21:S:U83#(tt(),M,N) -> c_30(U84#(isNatKind(activate(N)),activate(M),activate(N)),isNatKind#(activate(N)),activate#(N),activate#(M),activate#(N))
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):28
-->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):26
22:S:U91#(tt(),N) -> c_32(U92#(isNatKind(activate(N))),isNatKind#(activate(N)),activate#(N))
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):28
-->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):26
23:S:isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2))
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):28
-->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):26
-->_1 U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2)):4
24:S:isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)),activate#(V1),activate#(V1))
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):28
-->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):26
-->_1 U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)),activate#(V1),activate#(V1)):9
25:S:isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2))
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):28
-->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):26
-->_1 U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V1),activate#(V2)):11
26:S:isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2))
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):28
-->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):26
-->_1 U41#(tt(),V2) -> c_21(U42#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)):16
27:S:isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1))
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):28
-->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):26
28:S:isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2))
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):28
-->_2 isNatKind#(n__s(V1)) -> c_45(U51#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)),activate#(V1),activate#(V2)):26
-->_1 U61#(tt(),V2) -> c_24(U62#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)):17
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)))
U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)))
U103#(tt(),M,N) -> c_4(isNatKind#(activate(N)))
U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U15#(tt(),V2) -> c_10(isNat#(activate(V2)))
U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)))
U22#(tt(),V1) -> c_13(isNat#(activate(V1)))
U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U35#(tt(),V2) -> c_19(isNat#(activate(V2)))
U41#(tt(),V2) -> c_21(isNatKind#(activate(V2)))
U61#(tt(),V2) -> c_24(isNatKind#(activate(V2)))
U71#(tt(),N) -> c_26(isNatKind#(activate(N)))
U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)))
U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)))
U83#(tt(),M,N) -> c_30(isNatKind#(activate(N)))
U91#(tt(),N) -> c_32(isNatKind#(activate(N)))
isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)))
isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)))
isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1)))
isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)))
*** 1.1.1.1.1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)))
U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)))
U103#(tt(),M,N) -> c_4(isNatKind#(activate(N)))
U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U15#(tt(),V2) -> c_10(isNat#(activate(V2)))
U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)))
U22#(tt(),V1) -> c_13(isNat#(activate(V1)))
U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U35#(tt(),V2) -> c_19(isNat#(activate(V2)))
U41#(tt(),V2) -> c_21(isNatKind#(activate(V2)))
U61#(tt(),V2) -> c_24(isNatKind#(activate(V2)))
U71#(tt(),N) -> c_26(isNatKind#(activate(N)))
U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)))
U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)))
U83#(tt(),M,N) -> c_30(isNatKind#(activate(N)))
U91#(tt(),N) -> c_32(isNatKind#(activate(N)))
isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)))
isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)))
isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1)))
isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2))
U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2))
U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2))
U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2))
U35(tt(),V2) -> U36(isNat(activate(V2)))
U36(tt()) -> tt()
U41(tt(),V2) -> U42(isNatKind(activate(V2)))
U42(tt()) -> tt()
U51(tt()) -> tt()
U61(tt(),V2) -> U62(isNatKind(activate(V2)))
U62(tt()) -> tt()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1)))
isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
Signature:
{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,activate/1,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/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,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/5,c_6/2,c_7/2,c_8/2,c_9/2,c_10/1,c_11/0,c_12/2,c_13/1,c_14/0,c_15/2,c_16/2,c_17/2,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0,c_26/1,c_27/1,c_28/2,c_29/2,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/2,c_41/2,c_42/2,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0}
Obligation:
Innermost
basic terms: {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#,activate#,isNat#,isNatKind#,plus#,s#,x#}/{n__0,n__plus,n__s,n__x,tt}
Applied Processor:
Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
Proof:
We analyse the complexity of following sub-problems (R) and (S).
Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
Problem (R)
Strict DP Rules:
U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)))
U41#(tt(),V2) -> c_21(isNatKind#(activate(V2)))
U61#(tt(),V2) -> c_24(isNatKind#(activate(V2)))
isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)))
isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1)))
isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)))
Strict TRS Rules:
Weak DP Rules:
U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)))
U103#(tt(),M,N) -> c_4(isNatKind#(activate(N)))
U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U15#(tt(),V2) -> c_10(isNat#(activate(V2)))
U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)))
U22#(tt(),V1) -> c_13(isNat#(activate(V1)))
U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U35#(tt(),V2) -> c_19(isNat#(activate(V2)))
U71#(tt(),N) -> c_26(isNatKind#(activate(N)))
U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)))
U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)))
U83#(tt(),M,N) -> c_30(isNatKind#(activate(N)))
U91#(tt(),N) -> c_32(isNatKind#(activate(N)))
isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)))
isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
Weak TRS Rules:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2))
U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2))
U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2))
U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2))
U35(tt(),V2) -> U36(isNat(activate(V2)))
U36(tt()) -> tt()
U41(tt(),V2) -> U42(isNatKind(activate(V2)))
U42(tt()) -> tt()
U51(tt()) -> tt()
U61(tt(),V2) -> U62(isNatKind(activate(V2)))
U62(tt()) -> tt()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1)))
isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
Signature:
{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,activate/1,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/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,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/5,c_6/2,c_7/2,c_8/2,c_9/2,c_10/1,c_11/0,c_12/2,c_13/1,c_14/0,c_15/2,c_16/2,c_17/2,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0,c_26/1,c_27/1,c_28/2,c_29/2,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/2,c_41/2,c_42/2,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0}
Obligation:
Innermost
basic terms: {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#,activate#,isNat#,isNatKind#,plus#,s#,x#}/{n__0,n__plus,n__s,n__x,tt}
Problem (S)
Strict DP Rules:
U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)))
U103#(tt(),M,N) -> c_4(isNatKind#(activate(N)))
U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U15#(tt(),V2) -> c_10(isNat#(activate(V2)))
U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)))
U22#(tt(),V1) -> c_13(isNat#(activate(V1)))
U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U35#(tt(),V2) -> c_19(isNat#(activate(V2)))
U71#(tt(),N) -> c_26(isNatKind#(activate(N)))
U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)))
U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)))
U83#(tt(),M,N) -> c_30(isNatKind#(activate(N)))
U91#(tt(),N) -> c_32(isNatKind#(activate(N)))
isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)))
isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
Strict TRS Rules:
Weak DP Rules:
U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)))
U41#(tt(),V2) -> c_21(isNatKind#(activate(V2)))
U61#(tt(),V2) -> c_24(isNatKind#(activate(V2)))
isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)))
isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1)))
isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)))
Weak TRS Rules:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2))
U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2))
U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2))
U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2))
U35(tt(),V2) -> U36(isNat(activate(V2)))
U36(tt()) -> tt()
U41(tt(),V2) -> U42(isNatKind(activate(V2)))
U42(tt()) -> tt()
U51(tt()) -> tt()
U61(tt(),V2) -> U62(isNatKind(activate(V2)))
U62(tt()) -> tt()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1)))
isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
Signature:
{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,activate/1,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/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,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/5,c_6/2,c_7/2,c_8/2,c_9/2,c_10/1,c_11/0,c_12/2,c_13/1,c_14/0,c_15/2,c_16/2,c_17/2,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0,c_26/1,c_27/1,c_28/2,c_29/2,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/2,c_41/2,c_42/2,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0}
Obligation:
Innermost
basic terms: {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#,activate#,isNat#,isNatKind#,plus#,s#,x#}/{n__0,n__plus,n__s,n__x,tt}
*** 1.1.1.1.1.1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)))
U41#(tt(),V2) -> c_21(isNatKind#(activate(V2)))
U61#(tt(),V2) -> c_24(isNatKind#(activate(V2)))
isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)))
isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1)))
isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)))
Strict TRS Rules:
Weak DP Rules:
U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)))
U103#(tt(),M,N) -> c_4(isNatKind#(activate(N)))
U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U15#(tt(),V2) -> c_10(isNat#(activate(V2)))
U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)))
U22#(tt(),V1) -> c_13(isNat#(activate(V1)))
U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U35#(tt(),V2) -> c_19(isNat#(activate(V2)))
U71#(tt(),N) -> c_26(isNatKind#(activate(N)))
U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)))
U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)))
U83#(tt(),M,N) -> c_30(isNatKind#(activate(N)))
U91#(tt(),N) -> c_32(isNatKind#(activate(N)))
isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)))
isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
Weak TRS Rules:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2))
U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2))
U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2))
U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2))
U35(tt(),V2) -> U36(isNat(activate(V2)))
U36(tt()) -> tt()
U41(tt(),V2) -> U42(isNatKind(activate(V2)))
U42(tt()) -> tt()
U51(tt()) -> tt()
U61(tt(),V2) -> U62(isNatKind(activate(V2)))
U62(tt()) -> tt()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1)))
isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
Signature:
{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,activate/1,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/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,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/5,c_6/2,c_7/2,c_8/2,c_9/2,c_10/1,c_11/0,c_12/2,c_13/1,c_14/0,c_15/2,c_16/2,c_17/2,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0,c_26/1,c_27/1,c_28/2,c_29/2,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/2,c_41/2,c_42/2,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0}
Obligation:
Innermost
basic terms: {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#,activate#,isNat#,isNatKind#,plus#,s#,x#}/{n__0,n__plus,n__s,n__x,tt}
Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}}
Proof:
We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
1: U101#(tt(),M,N) ->
c_2(U102#(isNatKind(activate(M))
,activate(M)
,activate(N))
,isNatKind#(activate(M)))
26: isNatKind#(n__plus(V1,V2)) ->
c_44(U41#(isNatKind(activate(V1))
,activate(V2))
,isNatKind#(activate(V1)))
28: isNatKind#(n__x(V1,V2)) ->
c_46(U61#(isNatKind(activate(V1))
,activate(V2))
,isNatKind#(activate(V1)))
Consider the set of all dependency pairs
1: U101#(tt(),M,N) ->
c_2(U102#(isNatKind(activate(M))
,activate(M)
,activate(N))
,isNatKind#(activate(M)))
2: U102#(tt(),M,N) ->
c_3(U103#(isNat(activate(N))
,activate(M)
,activate(N))
,isNat#(activate(N)))
3: U103#(tt(),M,N) ->
c_4(isNatKind#(activate(N)))
4: U11#(tt(),V1,V2) ->
c_6(U12#(isNatKind(activate(V1))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V1)))
5: U12#(tt(),V1,V2) ->
c_7(U13#(isNatKind(activate(V2))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V2)))
6: U13#(tt(),V1,V2) ->
c_8(U14#(isNatKind(activate(V2))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V2)))
7: U14#(tt(),V1,V2) ->
c_9(U15#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1)))
8: U15#(tt(),V2) ->
c_10(isNat#(activate(V2)))
9: U21#(tt(),V1) ->
c_12(U22#(isNatKind(activate(V1))
,activate(V1))
,isNatKind#(activate(V1)))
10: U22#(tt(),V1) ->
c_13(isNat#(activate(V1)))
11: U31#(tt(),V1,V2) ->
c_15(U32#(isNatKind(activate(V1))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V1)))
12: U32#(tt(),V1,V2) ->
c_16(U33#(isNatKind(activate(V2))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V2)))
13: U33#(tt(),V1,V2) ->
c_17(U34#(isNatKind(activate(V2))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V2)))
14: U34#(tt(),V1,V2) ->
c_18(U35#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1)))
15: U35#(tt(),V2) ->
c_19(isNat#(activate(V2)))
16: U41#(tt(),V2) ->
c_21(isNatKind#(activate(V2)))
17: U61#(tt(),V2) ->
c_24(isNatKind#(activate(V2)))
18: U71#(tt(),N) ->
c_26(isNatKind#(activate(N)))
19: U81#(tt(),M,N) ->
c_28(U82#(isNatKind(activate(M))
,activate(M)
,activate(N))
,isNatKind#(activate(M)))
20: U82#(tt(),M,N) ->
c_29(U83#(isNat(activate(N))
,activate(M)
,activate(N))
,isNat#(activate(N)))
21: U83#(tt(),M,N) ->
c_30(isNatKind#(activate(N)))
22: U91#(tt(),N) ->
c_32(isNatKind#(activate(N)))
23: isNat#(n__plus(V1,V2)) ->
c_40(U11#(isNatKind(activate(V1))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V1)))
24: isNat#(n__s(V1)) ->
c_41(U21#(isNatKind(activate(V1))
,activate(V1))
,isNatKind#(activate(V1)))
25: isNat#(n__x(V1,V2)) ->
c_42(U31#(isNatKind(activate(V1))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V1)))
26: isNatKind#(n__plus(V1,V2)) ->
c_44(U41#(isNatKind(activate(V1))
,activate(V2))
,isNatKind#(activate(V1)))
27: isNatKind#(n__s(V1)) ->
c_45(isNatKind#(activate(V1)))
28: isNatKind#(n__x(V1,V2)) ->
c_46(U61#(isNatKind(activate(V1))
,activate(V2))
,isNatKind#(activate(V1)))
Processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}induces the complexity certificateTIME (?,O(n^2))
SPACE(?,?)on application of the dependency pairs
{1,26,28}
These cover all (indirect) predecessors of dependency pairs
{1,2,3,16,17,18,19,20,21,22,26,28}
their number of applications is equally bounded.
The dependency pairs are shifted into the weak component.
*** 1.1.1.1.1.1.1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)))
U41#(tt(),V2) -> c_21(isNatKind#(activate(V2)))
U61#(tt(),V2) -> c_24(isNatKind#(activate(V2)))
isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)))
isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1)))
isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)))
Strict TRS Rules:
Weak DP Rules:
U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)))
U103#(tt(),M,N) -> c_4(isNatKind#(activate(N)))
U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U15#(tt(),V2) -> c_10(isNat#(activate(V2)))
U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)))
U22#(tt(),V1) -> c_13(isNat#(activate(V1)))
U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U35#(tt(),V2) -> c_19(isNat#(activate(V2)))
U71#(tt(),N) -> c_26(isNatKind#(activate(N)))
U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)))
U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)))
U83#(tt(),M,N) -> c_30(isNatKind#(activate(N)))
U91#(tt(),N) -> c_32(isNatKind#(activate(N)))
isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)))
isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
Weak TRS Rules:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2))
U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2))
U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2))
U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2))
U35(tt(),V2) -> U36(isNat(activate(V2)))
U36(tt()) -> tt()
U41(tt(),V2) -> U42(isNatKind(activate(V2)))
U42(tt()) -> tt()
U51(tt()) -> tt()
U61(tt(),V2) -> U62(isNatKind(activate(V2)))
U62(tt()) -> tt()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1)))
isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
Signature:
{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,activate/1,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/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,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/5,c_6/2,c_7/2,c_8/2,c_9/2,c_10/1,c_11/0,c_12/2,c_13/1,c_14/0,c_15/2,c_16/2,c_17/2,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0,c_26/1,c_27/1,c_28/2,c_29/2,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/2,c_41/2,c_42/2,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0}
Obligation:
Innermost
basic terms: {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#,activate#,isNat#,isNatKind#,plus#,s#,x#}/{n__0,n__plus,n__s,n__x,tt}
Applied Processor:
NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
Proof:
We apply a polynomial interpretation of kind constructor-based(mixed(2)):
The following argument positions are considered usable:
uargs(c_2) = {1,2},
uargs(c_3) = {1,2},
uargs(c_4) = {1},
uargs(c_6) = {1,2},
uargs(c_7) = {1,2},
uargs(c_8) = {1,2},
uargs(c_9) = {1,2},
uargs(c_10) = {1},
uargs(c_12) = {1,2},
uargs(c_13) = {1},
uargs(c_15) = {1,2},
uargs(c_16) = {1,2},
uargs(c_17) = {1,2},
uargs(c_18) = {1,2},
uargs(c_19) = {1},
uargs(c_21) = {1},
uargs(c_24) = {1},
uargs(c_26) = {1},
uargs(c_28) = {1,2},
uargs(c_29) = {1,2},
uargs(c_30) = {1},
uargs(c_32) = {1},
uargs(c_40) = {1,2},
uargs(c_41) = {1,2},
uargs(c_42) = {1,2},
uargs(c_44) = {1,2},
uargs(c_45) = {1},
uargs(c_46) = {1,2}
Following symbols are considered usable:
{0,U41,U42,U51,U61,U62,activate,isNatKind,plus,s,x,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#,activate#,isNat#,isNatKind#,plus#,s#,x#}
TcT has computed the following interpretation:
p(0) = 1
p(U101) = 0
p(U102) = 0
p(U103) = 0
p(U104) = 0
p(U11) = 1
p(U12) = x1 + x1^2 + x2^2 + x3^2
p(U13) = x2*x3
p(U14) = x1 + x1*x2 + x1^2 + x2
p(U15) = 0
p(U16) = 0
p(U21) = x1 + x1^2 + x2^2
p(U22) = x1 + x2 + x2^2
p(U23) = 0
p(U31) = x1*x2 + x2*x3
p(U32) = 1 + x1 + x1^2 + x2^2
p(U33) = x1*x3 + x2 + x3^2
p(U34) = x2*x3
p(U35) = x1*x2 + x2
p(U36) = 1 + x1 + x1^2
p(U41) = 1 + x2
p(U42) = x1
p(U51) = 1 + x1
p(U61) = 1 + x1 + x2
p(U62) = 1
p(U71) = 0
p(U72) = 0
p(U81) = 0
p(U82) = 0
p(U83) = 0
p(U84) = 0
p(U91) = 0
p(U92) = 0
p(activate) = x1
p(isNat) = 0
p(isNatKind) = x1
p(n__0) = 1
p(n__plus) = 1 + x1 + x2
p(n__s) = 1 + x1
p(n__x) = 1 + x1 + x2
p(plus) = 1 + x1 + x2
p(s) = 1 + x1
p(tt) = 1
p(x) = 1 + x1 + x2
p(0#) = 0
p(U101#) = x1 + x1*x2 + x1^2 + x2*x3 + x3^2
p(U102#) = x1*x3 + x3^2
p(U103#) = x3
p(U104#) = 0
p(U11#) = 1 + x2 + x2*x3 + x2^2 + x3 + x3^2
p(U12#) = 1 + x1*x3 + x2^2 + x3 + x3^2
p(U13#) = 1 + x2^2 + x3 + x3^2
p(U14#) = 1 + x2^2 + x3^2
p(U15#) = x2^2
p(U16#) = 0
p(U21#) = x2 + x2^2
p(U22#) = x2^2
p(U23#) = 0
p(U31#) = x2 + x2*x3 + x2^2 + x3 + x3^2
p(U32#) = x1*x3 + x2^2 + x3 + x3^2
p(U33#) = x2^2 + x3 + x3^2
p(U34#) = x2^2 + x3^2
p(U35#) = x2^2
p(U36#) = 0
p(U41#) = x2
p(U42#) = 0
p(U51#) = 0
p(U61#) = x2
p(U62#) = 0
p(U71#) = 1 + x1*x2 + x2
p(U72#) = 0
p(U81#) = x1*x2 + x1*x3 + x2 + x2*x3 + x2^2 + x3 + x3^2
p(U82#) = x1 + x1*x2 + x1*x3 + x3^2
p(U83#) = 1 + x2 + x3
p(U84#) = 0
p(U91#) = x1*x2 + x2
p(U92#) = 0
p(activate#) = 0
p(isNat#) = x1^2
p(isNatKind#) = x1
p(plus#) = 0
p(s#) = 0
p(x#) = 0
p(c_1) = 0
p(c_2) = 1 + x1 + x2
p(c_3) = x1 + x2
p(c_4) = x1
p(c_5) = 0
p(c_6) = x1 + x2
p(c_7) = x1 + x2
p(c_8) = x1 + x2
p(c_9) = x1 + x2
p(c_10) = x1
p(c_11) = 0
p(c_12) = x1 + x2
p(c_13) = x1
p(c_14) = 0
p(c_15) = x1 + x2
p(c_16) = x1 + x2
p(c_17) = x1 + x2
p(c_18) = x1 + x2
p(c_19) = x1
p(c_20) = 0
p(c_21) = x1
p(c_22) = 0
p(c_23) = 0
p(c_24) = x1
p(c_25) = 0
p(c_26) = 1 + x1
p(c_27) = 0
p(c_28) = x1 + x2
p(c_29) = x1 + x2
p(c_30) = 1 + x1
p(c_31) = 0
p(c_32) = x1
p(c_33) = 0
p(c_34) = 0
p(c_35) = 0
p(c_36) = 0
p(c_37) = 0
p(c_38) = 0
p(c_39) = 0
p(c_40) = x1 + x2
p(c_41) = x1 + x2
p(c_42) = 1 + x1 + x2
p(c_43) = 0
p(c_44) = x1 + x2
p(c_45) = 1 + x1
p(c_46) = x1 + x2
p(c_47) = 0
p(c_48) = 0
p(c_49) = 0
Following rules are strictly oriented:
U101#(tt(),M,N) = 2 + M + M*N + N^2
> 1 + M + M*N + N^2
= c_2(U102#(isNatKind(activate(M))
,activate(M)
,activate(N))
,isNatKind#(activate(M)))
isNatKind#(n__plus(V1,V2)) = 1 + V1 + V2
> V1 + V2
= c_44(U41#(isNatKind(activate(V1))
,activate(V2))
,isNatKind#(activate(V1)))
isNatKind#(n__x(V1,V2)) = 1 + V1 + V2
> V1 + V2
= c_46(U61#(isNatKind(activate(V1))
,activate(V2))
,isNatKind#(activate(V1)))
Following rules are (at-least) weakly oriented:
U102#(tt(),M,N) = N + N^2
>= N + N^2
= c_3(U103#(isNat(activate(N))
,activate(M)
,activate(N))
,isNat#(activate(N)))
U103#(tt(),M,N) = N
>= N
= c_4(isNatKind#(activate(N)))
U11#(tt(),V1,V2) = 1 + V1 + V1*V2 + V1^2 + V2 + V2^2
>= 1 + V1 + V1*V2 + V1^2 + V2 + V2^2
= c_6(U12#(isNatKind(activate(V1))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V1)))
U12#(tt(),V1,V2) = 1 + V1^2 + 2*V2 + V2^2
>= 1 + V1^2 + 2*V2 + V2^2
= c_7(U13#(isNatKind(activate(V2))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V2)))
U13#(tt(),V1,V2) = 1 + V1^2 + V2 + V2^2
>= 1 + V1^2 + V2 + V2^2
= c_8(U14#(isNatKind(activate(V2))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V2)))
U14#(tt(),V1,V2) = 1 + V1^2 + V2^2
>= V1^2 + V2^2
= c_9(U15#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1)))
U15#(tt(),V2) = V2^2
>= V2^2
= c_10(isNat#(activate(V2)))
U21#(tt(),V1) = V1 + V1^2
>= V1 + V1^2
= c_12(U22#(isNatKind(activate(V1))
,activate(V1))
,isNatKind#(activate(V1)))
U22#(tt(),V1) = V1^2
>= V1^2
= c_13(isNat#(activate(V1)))
U31#(tt(),V1,V2) = V1 + V1*V2 + V1^2 + V2 + V2^2
>= V1 + V1*V2 + V1^2 + V2 + V2^2
= c_15(U32#(isNatKind(activate(V1))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V1)))
U32#(tt(),V1,V2) = V1^2 + 2*V2 + V2^2
>= V1^2 + 2*V2 + V2^2
= c_16(U33#(isNatKind(activate(V2))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V2)))
U33#(tt(),V1,V2) = V1^2 + V2 + V2^2
>= V1^2 + V2 + V2^2
= c_17(U34#(isNatKind(activate(V2))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V2)))
U34#(tt(),V1,V2) = V1^2 + V2^2
>= V1^2 + V2^2
= c_18(U35#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1)))
U35#(tt(),V2) = V2^2
>= V2^2
= c_19(isNat#(activate(V2)))
U41#(tt(),V2) = V2
>= V2
= c_21(isNatKind#(activate(V2)))
U61#(tt(),V2) = V2
>= V2
= c_24(isNatKind#(activate(V2)))
U71#(tt(),N) = 1 + 2*N
>= 1 + N
= c_26(isNatKind#(activate(N)))
U81#(tt(),M,N) = 2*M + M*N + M^2 + 2*N + N^2
>= 2*M + M*N + M^2 + N^2
= c_28(U82#(isNatKind(activate(M))
,activate(M)
,activate(N))
,isNatKind#(activate(M)))
U82#(tt(),M,N) = 1 + M + N + N^2
>= 1 + M + N + N^2
= c_29(U83#(isNat(activate(N))
,activate(M)
,activate(N))
,isNat#(activate(N)))
U83#(tt(),M,N) = 1 + M + N
>= 1 + N
= c_30(isNatKind#(activate(N)))
U91#(tt(),N) = 2*N
>= N
= c_32(isNatKind#(activate(N)))
isNat#(n__plus(V1,V2)) = 1 + 2*V1 + 2*V1*V2 + V1^2 + 2*V2 + V2^2
>= 1 + 2*V1 + V1*V2 + V1^2 + V2 + V2^2
= c_40(U11#(isNatKind(activate(V1))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V1)))
isNat#(n__s(V1)) = 1 + 2*V1 + V1^2
>= 2*V1 + V1^2
= c_41(U21#(isNatKind(activate(V1))
,activate(V1))
,isNatKind#(activate(V1)))
isNat#(n__x(V1,V2)) = 1 + 2*V1 + 2*V1*V2 + V1^2 + 2*V2 + V2^2
>= 1 + 2*V1 + V1*V2 + V1^2 + V2 + V2^2
= c_42(U31#(isNatKind(activate(V1))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V1)))
isNatKind#(n__s(V1)) = 1 + V1
>= 1 + V1
= c_45(isNatKind#(activate(V1)))
0() = 1
>= 1
= n__0()
U41(tt(),V2) = 1 + V2
>= V2
= U42(isNatKind(activate(V2)))
U42(tt()) = 1
>= 1
= tt()
U51(tt()) = 2
>= 1
= tt()
U61(tt(),V2) = 2 + V2
>= 1
= U62(isNatKind(activate(V2)))
U62(tt()) = 1
>= 1
= tt()
activate(X) = X
>= X
= X
activate(n__0()) = 1
>= 1
= 0()
activate(n__plus(X1,X2)) = 1 + X1 + X2
>= 1 + X1 + X2
= plus(X1,X2)
activate(n__s(X)) = 1 + X
>= 1 + X
= s(X)
activate(n__x(X1,X2)) = 1 + X1 + X2
>= 1 + X1 + X2
= x(X1,X2)
isNatKind(n__0()) = 1
>= 1
= tt()
isNatKind(n__plus(V1,V2)) = 1 + V1 + V2
>= 1 + V2
= U41(isNatKind(activate(V1))
,activate(V2))
isNatKind(n__s(V1)) = 1 + V1
>= 1 + V1
= U51(isNatKind(activate(V1)))
isNatKind(n__x(V1,V2)) = 1 + V1 + V2
>= 1 + V1 + V2
= U61(isNatKind(activate(V1))
,activate(V2))
plus(X1,X2) = 1 + X1 + X2
>= 1 + X1 + X2
= n__plus(X1,X2)
s(X) = 1 + X
>= 1 + X
= n__s(X)
x(X1,X2) = 1 + X1 + X2
>= 1 + X1 + X2
= n__x(X1,X2)
*** 1.1.1.1.1.1.1.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
U41#(tt(),V2) -> c_21(isNatKind#(activate(V2)))
U61#(tt(),V2) -> c_24(isNatKind#(activate(V2)))
isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1)))
Strict TRS Rules:
Weak DP Rules:
U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)))
U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)))
U103#(tt(),M,N) -> c_4(isNatKind#(activate(N)))
U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U15#(tt(),V2) -> c_10(isNat#(activate(V2)))
U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)))
U22#(tt(),V1) -> c_13(isNat#(activate(V1)))
U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U35#(tt(),V2) -> c_19(isNat#(activate(V2)))
U71#(tt(),N) -> c_26(isNatKind#(activate(N)))
U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)))
U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)))
U83#(tt(),M,N) -> c_30(isNatKind#(activate(N)))
U91#(tt(),N) -> c_32(isNatKind#(activate(N)))
isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)))
isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)))
isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)))
Weak TRS Rules:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2))
U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2))
U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2))
U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2))
U35(tt(),V2) -> U36(isNat(activate(V2)))
U36(tt()) -> tt()
U41(tt(),V2) -> U42(isNatKind(activate(V2)))
U42(tt()) -> tt()
U51(tt()) -> tt()
U61(tt(),V2) -> U62(isNatKind(activate(V2)))
U62(tt()) -> tt()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1)))
isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
Signature:
{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,activate/1,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/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,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/5,c_6/2,c_7/2,c_8/2,c_9/2,c_10/1,c_11/0,c_12/2,c_13/1,c_14/0,c_15/2,c_16/2,c_17/2,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0,c_26/1,c_27/1,c_28/2,c_29/2,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/2,c_41/2,c_42/2,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0}
Obligation:
Innermost
basic terms: {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#,activate#,isNat#,isNatKind#,plus#,s#,x#}/{n__0,n__plus,n__s,n__x,tt}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.1.1.1.1.2 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1)))
Strict TRS Rules:
Weak DP Rules:
U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)))
U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)))
U103#(tt(),M,N) -> c_4(isNatKind#(activate(N)))
U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U15#(tt(),V2) -> c_10(isNat#(activate(V2)))
U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)))
U22#(tt(),V1) -> c_13(isNat#(activate(V1)))
U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U35#(tt(),V2) -> c_19(isNat#(activate(V2)))
U41#(tt(),V2) -> c_21(isNatKind#(activate(V2)))
U61#(tt(),V2) -> c_24(isNatKind#(activate(V2)))
U71#(tt(),N) -> c_26(isNatKind#(activate(N)))
U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)))
U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)))
U83#(tt(),M,N) -> c_30(isNatKind#(activate(N)))
U91#(tt(),N) -> c_32(isNatKind#(activate(N)))
isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)))
isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)))
isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)))
Weak TRS Rules:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2))
U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2))
U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2))
U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2))
U35(tt(),V2) -> U36(isNat(activate(V2)))
U36(tt()) -> tt()
U41(tt(),V2) -> U42(isNatKind(activate(V2)))
U42(tt()) -> tt()
U51(tt()) -> tt()
U61(tt(),V2) -> U62(isNatKind(activate(V2)))
U62(tt()) -> tt()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1)))
isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
Signature:
{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,activate/1,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/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,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/5,c_6/2,c_7/2,c_8/2,c_9/2,c_10/1,c_11/0,c_12/2,c_13/1,c_14/0,c_15/2,c_16/2,c_17/2,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0,c_26/1,c_27/1,c_28/2,c_29/2,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/2,c_41/2,c_42/2,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0}
Obligation:
Innermost
basic terms: {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#,activate#,isNat#,isNatKind#,plus#,s#,x#}/{n__0,n__plus,n__s,n__x,tt}
Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}}
Proof:
We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
1: isNatKind#(n__s(V1)) ->
c_45(isNatKind#(activate(V1)))
Consider the set of all dependency pairs
1: isNatKind#(n__s(V1)) ->
c_45(isNatKind#(activate(V1)))
2: U101#(tt(),M,N) ->
c_2(U102#(isNatKind(activate(M))
,activate(M)
,activate(N))
,isNatKind#(activate(M)))
3: U102#(tt(),M,N) ->
c_3(U103#(isNat(activate(N))
,activate(M)
,activate(N))
,isNat#(activate(N)))
4: U103#(tt(),M,N) ->
c_4(isNatKind#(activate(N)))
5: U11#(tt(),V1,V2) ->
c_6(U12#(isNatKind(activate(V1))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V1)))
6: U12#(tt(),V1,V2) ->
c_7(U13#(isNatKind(activate(V2))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V2)))
7: U13#(tt(),V1,V2) ->
c_8(U14#(isNatKind(activate(V2))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V2)))
8: U14#(tt(),V1,V2) ->
c_9(U15#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1)))
9: U15#(tt(),V2) ->
c_10(isNat#(activate(V2)))
10: U21#(tt(),V1) ->
c_12(U22#(isNatKind(activate(V1))
,activate(V1))
,isNatKind#(activate(V1)))
11: U22#(tt(),V1) ->
c_13(isNat#(activate(V1)))
12: U31#(tt(),V1,V2) ->
c_15(U32#(isNatKind(activate(V1))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V1)))
13: U32#(tt(),V1,V2) ->
c_16(U33#(isNatKind(activate(V2))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V2)))
14: U33#(tt(),V1,V2) ->
c_17(U34#(isNatKind(activate(V2))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V2)))
15: U34#(tt(),V1,V2) ->
c_18(U35#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1)))
16: U35#(tt(),V2) ->
c_19(isNat#(activate(V2)))
17: U41#(tt(),V2) ->
c_21(isNatKind#(activate(V2)))
18: U61#(tt(),V2) ->
c_24(isNatKind#(activate(V2)))
19: U71#(tt(),N) ->
c_26(isNatKind#(activate(N)))
20: U81#(tt(),M,N) ->
c_28(U82#(isNatKind(activate(M))
,activate(M)
,activate(N))
,isNatKind#(activate(M)))
21: U82#(tt(),M,N) ->
c_29(U83#(isNat(activate(N))
,activate(M)
,activate(N))
,isNat#(activate(N)))
22: U83#(tt(),M,N) ->
c_30(isNatKind#(activate(N)))
23: U91#(tt(),N) ->
c_32(isNatKind#(activate(N)))
24: isNat#(n__plus(V1,V2)) ->
c_40(U11#(isNatKind(activate(V1))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V1)))
25: isNat#(n__s(V1)) ->
c_41(U21#(isNatKind(activate(V1))
,activate(V1))
,isNatKind#(activate(V1)))
26: isNat#(n__x(V1,V2)) ->
c_42(U31#(isNatKind(activate(V1))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V1)))
27: isNatKind#(n__plus(V1,V2)) ->
c_44(U41#(isNatKind(activate(V1))
,activate(V2))
,isNatKind#(activate(V1)))
28: isNatKind#(n__x(V1,V2)) ->
c_46(U61#(isNatKind(activate(V1))
,activate(V2))
,isNatKind#(activate(V1)))
Processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}induces the complexity certificateTIME (?,O(n^2))
SPACE(?,?)on application of the dependency pairs
{1}
These cover all (indirect) predecessors of dependency pairs
{1,2,3,4,19,20,21,22,23}
their number of applications is equally bounded.
The dependency pairs are shifted into the weak component.
*** 1.1.1.1.1.1.1.1.1.1.2.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1)))
Strict TRS Rules:
Weak DP Rules:
U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)))
U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)))
U103#(tt(),M,N) -> c_4(isNatKind#(activate(N)))
U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U15#(tt(),V2) -> c_10(isNat#(activate(V2)))
U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)))
U22#(tt(),V1) -> c_13(isNat#(activate(V1)))
U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U35#(tt(),V2) -> c_19(isNat#(activate(V2)))
U41#(tt(),V2) -> c_21(isNatKind#(activate(V2)))
U61#(tt(),V2) -> c_24(isNatKind#(activate(V2)))
U71#(tt(),N) -> c_26(isNatKind#(activate(N)))
U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)))
U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)))
U83#(tt(),M,N) -> c_30(isNatKind#(activate(N)))
U91#(tt(),N) -> c_32(isNatKind#(activate(N)))
isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)))
isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)))
isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)))
Weak TRS Rules:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2))
U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2))
U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2))
U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2))
U35(tt(),V2) -> U36(isNat(activate(V2)))
U36(tt()) -> tt()
U41(tt(),V2) -> U42(isNatKind(activate(V2)))
U42(tt()) -> tt()
U51(tt()) -> tt()
U61(tt(),V2) -> U62(isNatKind(activate(V2)))
U62(tt()) -> tt()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1)))
isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
Signature:
{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,activate/1,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/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,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/5,c_6/2,c_7/2,c_8/2,c_9/2,c_10/1,c_11/0,c_12/2,c_13/1,c_14/0,c_15/2,c_16/2,c_17/2,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0,c_26/1,c_27/1,c_28/2,c_29/2,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/2,c_41/2,c_42/2,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0}
Obligation:
Innermost
basic terms: {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#,activate#,isNat#,isNatKind#,plus#,s#,x#}/{n__0,n__plus,n__s,n__x,tt}
Applied Processor:
NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
Proof:
We apply a polynomial interpretation of kind constructor-based(mixed(2)):
The following argument positions are considered usable:
uargs(c_2) = {1,2},
uargs(c_3) = {1,2},
uargs(c_4) = {1},
uargs(c_6) = {1,2},
uargs(c_7) = {1,2},
uargs(c_8) = {1,2},
uargs(c_9) = {1,2},
uargs(c_10) = {1},
uargs(c_12) = {1,2},
uargs(c_13) = {1},
uargs(c_15) = {1,2},
uargs(c_16) = {1,2},
uargs(c_17) = {1,2},
uargs(c_18) = {1,2},
uargs(c_19) = {1},
uargs(c_21) = {1},
uargs(c_24) = {1},
uargs(c_26) = {1},
uargs(c_28) = {1,2},
uargs(c_29) = {1,2},
uargs(c_30) = {1},
uargs(c_32) = {1},
uargs(c_40) = {1,2},
uargs(c_41) = {1,2},
uargs(c_42) = {1,2},
uargs(c_44) = {1,2},
uargs(c_45) = {1},
uargs(c_46) = {1,2}
Following symbols are considered usable:
{0,U41,U42,U51,U61,U62,activate,isNatKind,plus,s,x,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#,activate#,isNat#,isNatKind#,plus#,s#,x#}
TcT has computed the following interpretation:
p(0) = 1
p(U101) = 0
p(U102) = 0
p(U103) = 0
p(U104) = 0
p(U11) = x1*x3
p(U12) = x1*x3 + x2^2 + x3^2
p(U13) = x1*x3 + x2*x3 + x2^2
p(U14) = 0
p(U15) = 0
p(U16) = 0
p(U21) = 0
p(U22) = 0
p(U23) = 1
p(U31) = 1 + x1^2
p(U32) = x1 + x1*x3 + x2*x3 + x2^2
p(U33) = x3^2
p(U34) = x1 + x1*x2 + x1*x3 + x2*x3
p(U35) = x1*x2 + x2
p(U36) = x1
p(U41) = x1 + x2
p(U42) = 1
p(U51) = x1
p(U61) = x1
p(U62) = 1
p(U71) = 0
p(U72) = 0
p(U81) = 0
p(U82) = 0
p(U83) = 0
p(U84) = 0
p(U91) = 0
p(U92) = 0
p(activate) = x1
p(isNat) = 0
p(isNatKind) = x1
p(n__0) = 1
p(n__plus) = 1 + x1 + x2
p(n__s) = 1 + x1
p(n__x) = 1 + x1 + x2
p(plus) = 1 + x1 + x2
p(s) = 1 + x1
p(tt) = 1
p(x) = 1 + x1 + x2
p(0#) = 0
p(U101#) = x1^2 + x2 + x2*x3 + x2^2 + x3 + x3^2
p(U102#) = 1 + x1*x3 + x1^2 + x3^2
p(U103#) = 1 + x3
p(U104#) = 0
p(U11#) = 1 + x1*x3 + x2 + x2*x3 + x2^2 + x3^2
p(U12#) = x1*x3 + x2^2 + x3 + x3^2
p(U13#) = x2^2 + x3 + x3^2
p(U14#) = x2^2 + x3^2
p(U15#) = x2^2
p(U16#) = 0
p(U21#) = x2 + x2^2
p(U22#) = x2^2
p(U23#) = 0
p(U31#) = x1*x3 + x2 + x2*x3 + x2^2 + x3^2
p(U32#) = x1*x3 + x2^2 + x3 + x3^2
p(U33#) = x2^2 + x3 + x3^2
p(U34#) = x2^2 + x3^2
p(U35#) = x2^2
p(U36#) = 0
p(U41#) = x2
p(U42#) = 0
p(U51#) = 0
p(U61#) = x2
p(U62#) = 0
p(U71#) = x2
p(U72#) = 0
p(U81#) = x1 + x1*x2 + x1^2 + x2 + x2*x3 + x2^2 + x3 + x3^2
p(U82#) = 1 + x1 + x1*x2 + x1*x3 + x3 + x3^2
p(U83#) = x3
p(U84#) = 0
p(U91#) = x1*x2 + x2
p(U92#) = 0
p(activate#) = 0
p(isNat#) = x1^2
p(isNatKind#) = x1
p(plus#) = 0
p(s#) = 0
p(x#) = 0
p(c_1) = 0
p(c_2) = x1 + x2
p(c_3) = 1 + x1 + x2
p(c_4) = x1
p(c_5) = 0
p(c_6) = 1 + x1 + x2
p(c_7) = x1 + x2
p(c_8) = x1 + x2
p(c_9) = x1 + x2
p(c_10) = x1
p(c_11) = 0
p(c_12) = x1 + x2
p(c_13) = x1
p(c_14) = 0
p(c_15) = x1 + x2
p(c_16) = x1 + x2
p(c_17) = x1 + x2
p(c_18) = x1 + x2
p(c_19) = x1
p(c_20) = 0
p(c_21) = x1
p(c_22) = 0
p(c_23) = 0
p(c_24) = x1
p(c_25) = 0
p(c_26) = x1
p(c_27) = 0
p(c_28) = x1 + x2
p(c_29) = x1 + x2
p(c_30) = x1
p(c_31) = 0
p(c_32) = x1
p(c_33) = 0
p(c_34) = 0
p(c_35) = 0
p(c_36) = 0
p(c_37) = 0
p(c_38) = 0
p(c_39) = 0
p(c_40) = x1 + x2
p(c_41) = x1 + x2
p(c_42) = x1 + x2
p(c_43) = 0
p(c_44) = 1 + x1 + x2
p(c_45) = x1
p(c_46) = x1 + x2
p(c_47) = 0
p(c_48) = 0
p(c_49) = 0
Following rules are strictly oriented:
isNatKind#(n__s(V1)) = 1 + V1
> V1
= c_45(isNatKind#(activate(V1)))
Following rules are (at-least) weakly oriented:
U101#(tt(),M,N) = 1 + M + M*N + M^2 + N + N^2
>= 1 + M + M*N + M^2 + N^2
= c_2(U102#(isNatKind(activate(M))
,activate(M)
,activate(N))
,isNatKind#(activate(M)))
U102#(tt(),M,N) = 2 + N + N^2
>= 2 + N + N^2
= c_3(U103#(isNat(activate(N))
,activate(M)
,activate(N))
,isNat#(activate(N)))
U103#(tt(),M,N) = 1 + N
>= N
= c_4(isNatKind#(activate(N)))
U11#(tt(),V1,V2) = 1 + V1 + V1*V2 + V1^2 + V2 + V2^2
>= 1 + V1 + V1*V2 + V1^2 + V2 + V2^2
= c_6(U12#(isNatKind(activate(V1))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V1)))
U12#(tt(),V1,V2) = V1^2 + 2*V2 + V2^2
>= V1^2 + 2*V2 + V2^2
= c_7(U13#(isNatKind(activate(V2))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V2)))
U13#(tt(),V1,V2) = V1^2 + V2 + V2^2
>= V1^2 + V2 + V2^2
= c_8(U14#(isNatKind(activate(V2))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V2)))
U14#(tt(),V1,V2) = V1^2 + V2^2
>= V1^2 + V2^2
= c_9(U15#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1)))
U15#(tt(),V2) = V2^2
>= V2^2
= c_10(isNat#(activate(V2)))
U21#(tt(),V1) = V1 + V1^2
>= V1 + V1^2
= c_12(U22#(isNatKind(activate(V1))
,activate(V1))
,isNatKind#(activate(V1)))
U22#(tt(),V1) = V1^2
>= V1^2
= c_13(isNat#(activate(V1)))
U31#(tt(),V1,V2) = V1 + V1*V2 + V1^2 + V2 + V2^2
>= V1 + V1*V2 + V1^2 + V2 + V2^2
= c_15(U32#(isNatKind(activate(V1))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V1)))
U32#(tt(),V1,V2) = V1^2 + 2*V2 + V2^2
>= V1^2 + 2*V2 + V2^2
= c_16(U33#(isNatKind(activate(V2))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V2)))
U33#(tt(),V1,V2) = V1^2 + V2 + V2^2
>= V1^2 + V2 + V2^2
= c_17(U34#(isNatKind(activate(V2))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V2)))
U34#(tt(),V1,V2) = V1^2 + V2^2
>= V1^2 + V2^2
= c_18(U35#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1)))
U35#(tt(),V2) = V2^2
>= V2^2
= c_19(isNat#(activate(V2)))
U41#(tt(),V2) = V2
>= V2
= c_21(isNatKind#(activate(V2)))
U61#(tt(),V2) = V2
>= V2
= c_24(isNatKind#(activate(V2)))
U71#(tt(),N) = N
>= N
= c_26(isNatKind#(activate(N)))
U81#(tt(),M,N) = 2 + 2*M + M*N + M^2 + N + N^2
>= 1 + 2*M + M*N + M^2 + N + N^2
= c_28(U82#(isNatKind(activate(M))
,activate(M)
,activate(N))
,isNatKind#(activate(M)))
U82#(tt(),M,N) = 2 + M + 2*N + N^2
>= N + N^2
= c_29(U83#(isNat(activate(N))
,activate(M)
,activate(N))
,isNat#(activate(N)))
U83#(tt(),M,N) = N
>= N
= c_30(isNatKind#(activate(N)))
U91#(tt(),N) = 2*N
>= N
= c_32(isNatKind#(activate(N)))
isNat#(n__plus(V1,V2)) = 1 + 2*V1 + 2*V1*V2 + V1^2 + 2*V2 + V2^2
>= 1 + 2*V1 + 2*V1*V2 + V1^2 + V2^2
= c_40(U11#(isNatKind(activate(V1))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V1)))
isNat#(n__s(V1)) = 1 + 2*V1 + V1^2
>= 2*V1 + V1^2
= c_41(U21#(isNatKind(activate(V1))
,activate(V1))
,isNatKind#(activate(V1)))
isNat#(n__x(V1,V2)) = 1 + 2*V1 + 2*V1*V2 + V1^2 + 2*V2 + V2^2
>= 2*V1 + 2*V1*V2 + V1^2 + V2^2
= c_42(U31#(isNatKind(activate(V1))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V1)))
isNatKind#(n__plus(V1,V2)) = 1 + V1 + V2
>= 1 + V1 + V2
= c_44(U41#(isNatKind(activate(V1))
,activate(V2))
,isNatKind#(activate(V1)))
isNatKind#(n__x(V1,V2)) = 1 + V1 + V2
>= V1 + V2
= c_46(U61#(isNatKind(activate(V1))
,activate(V2))
,isNatKind#(activate(V1)))
0() = 1
>= 1
= n__0()
U41(tt(),V2) = 1 + V2
>= 1
= U42(isNatKind(activate(V2)))
U42(tt()) = 1
>= 1
= tt()
U51(tt()) = 1
>= 1
= tt()
U61(tt(),V2) = 1
>= 1
= U62(isNatKind(activate(V2)))
U62(tt()) = 1
>= 1
= tt()
activate(X) = X
>= X
= X
activate(n__0()) = 1
>= 1
= 0()
activate(n__plus(X1,X2)) = 1 + X1 + X2
>= 1 + X1 + X2
= plus(X1,X2)
activate(n__s(X)) = 1 + X
>= 1 + X
= s(X)
activate(n__x(X1,X2)) = 1 + X1 + X2
>= 1 + X1 + X2
= x(X1,X2)
isNatKind(n__0()) = 1
>= 1
= tt()
isNatKind(n__plus(V1,V2)) = 1 + V1 + V2
>= V1 + V2
= U41(isNatKind(activate(V1))
,activate(V2))
isNatKind(n__s(V1)) = 1 + V1
>= V1
= U51(isNatKind(activate(V1)))
isNatKind(n__x(V1,V2)) = 1 + V1 + V2
>= V1
= U61(isNatKind(activate(V1))
,activate(V2))
plus(X1,X2) = 1 + X1 + X2
>= 1 + X1 + X2
= n__plus(X1,X2)
s(X) = 1 + X
>= 1 + X
= n__s(X)
x(X1,X2) = 1 + X1 + X2
>= 1 + X1 + X2
= n__x(X1,X2)
*** 1.1.1.1.1.1.1.1.1.1.2.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)))
U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)))
U103#(tt(),M,N) -> c_4(isNatKind#(activate(N)))
U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U15#(tt(),V2) -> c_10(isNat#(activate(V2)))
U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)))
U22#(tt(),V1) -> c_13(isNat#(activate(V1)))
U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U35#(tt(),V2) -> c_19(isNat#(activate(V2)))
U41#(tt(),V2) -> c_21(isNatKind#(activate(V2)))
U61#(tt(),V2) -> c_24(isNatKind#(activate(V2)))
U71#(tt(),N) -> c_26(isNatKind#(activate(N)))
U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)))
U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)))
U83#(tt(),M,N) -> c_30(isNatKind#(activate(N)))
U91#(tt(),N) -> c_32(isNatKind#(activate(N)))
isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)))
isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)))
isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1)))
isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)))
Weak TRS Rules:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2))
U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2))
U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2))
U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2))
U35(tt(),V2) -> U36(isNat(activate(V2)))
U36(tt()) -> tt()
U41(tt(),V2) -> U42(isNatKind(activate(V2)))
U42(tt()) -> tt()
U51(tt()) -> tt()
U61(tt(),V2) -> U62(isNatKind(activate(V2)))
U62(tt()) -> tt()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1)))
isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
Signature:
{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,activate/1,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/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,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/5,c_6/2,c_7/2,c_8/2,c_9/2,c_10/1,c_11/0,c_12/2,c_13/1,c_14/0,c_15/2,c_16/2,c_17/2,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0,c_26/1,c_27/1,c_28/2,c_29/2,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/2,c_41/2,c_42/2,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0}
Obligation:
Innermost
basic terms: {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#,activate#,isNat#,isNatKind#,plus#,s#,x#}/{n__0,n__plus,n__s,n__x,tt}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.1.1.1.1.2.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)))
U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)))
U103#(tt(),M,N) -> c_4(isNatKind#(activate(N)))
U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U15#(tt(),V2) -> c_10(isNat#(activate(V2)))
U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)))
U22#(tt(),V1) -> c_13(isNat#(activate(V1)))
U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U35#(tt(),V2) -> c_19(isNat#(activate(V2)))
U41#(tt(),V2) -> c_21(isNatKind#(activate(V2)))
U61#(tt(),V2) -> c_24(isNatKind#(activate(V2)))
U71#(tt(),N) -> c_26(isNatKind#(activate(N)))
U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)))
U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)))
U83#(tt(),M,N) -> c_30(isNatKind#(activate(N)))
U91#(tt(),N) -> c_32(isNatKind#(activate(N)))
isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)))
isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)))
isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1)))
isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)))
Weak TRS Rules:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2))
U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2))
U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2))
U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2))
U35(tt(),V2) -> U36(isNat(activate(V2)))
U36(tt()) -> tt()
U41(tt(),V2) -> U42(isNatKind(activate(V2)))
U42(tt()) -> tt()
U51(tt()) -> tt()
U61(tt(),V2) -> U62(isNatKind(activate(V2)))
U62(tt()) -> tt()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1)))
isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
Signature:
{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,activate/1,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/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,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/5,c_6/2,c_7/2,c_8/2,c_9/2,c_10/1,c_11/0,c_12/2,c_13/1,c_14/0,c_15/2,c_16/2,c_17/2,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0,c_26/1,c_27/1,c_28/2,c_29/2,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/2,c_41/2,c_42/2,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0}
Obligation:
Innermost
basic terms: {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#,activate#,isNat#,isNatKind#,plus#,s#,x#}/{n__0,n__plus,n__s,n__x,tt}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)))
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):28
-->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):26
-->_1 U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))):2
2:W:U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)))
-->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))):25
-->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):24
-->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))):23
-->_1 U103#(tt(),M,N) -> c_4(isNatKind#(activate(N))):3
3:W:U103#(tt(),M,N) -> c_4(isNatKind#(activate(N)))
-->_1 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):28
-->_1 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27
-->_1 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):26
4:W:U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):28
-->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):26
-->_1 U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))):5
5:W:U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):28
-->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):26
-->_1 U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))):6
6:W:U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):28
-->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):26
-->_1 U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):7
7:W:U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
-->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))):25
-->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):24
-->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))):23
-->_1 U15#(tt(),V2) -> c_10(isNat#(activate(V2))):8
8:W:U15#(tt(),V2) -> c_10(isNat#(activate(V2)))
-->_1 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))):25
-->_1 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):24
-->_1 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))):23
9:W:U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)))
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):28
-->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):26
-->_1 U22#(tt(),V1) -> c_13(isNat#(activate(V1))):10
10:W:U22#(tt(),V1) -> c_13(isNat#(activate(V1)))
-->_1 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))):25
-->_1 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):24
-->_1 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))):23
11:W:U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):28
-->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):26
-->_1 U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))):12
12:W:U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):28
-->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):26
-->_1 U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))):13
13:W:U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):28
-->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):26
-->_1 U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):14
14:W:U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
-->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))):25
-->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):24
-->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))):23
-->_1 U35#(tt(),V2) -> c_19(isNat#(activate(V2))):15
15:W:U35#(tt(),V2) -> c_19(isNat#(activate(V2)))
-->_1 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))):25
-->_1 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):24
-->_1 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))):23
16:W:U41#(tt(),V2) -> c_21(isNatKind#(activate(V2)))
-->_1 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):28
-->_1 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27
-->_1 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):26
17:W:U61#(tt(),V2) -> c_24(isNatKind#(activate(V2)))
-->_1 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):28
-->_1 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27
-->_1 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):26
18:W:U71#(tt(),N) -> c_26(isNatKind#(activate(N)))
-->_1 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):28
-->_1 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27
-->_1 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):26
19:W:U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)))
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):28
-->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):26
-->_1 U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))):20
20:W:U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)))
-->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))):25
-->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):24
-->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))):23
-->_1 U83#(tt(),M,N) -> c_30(isNatKind#(activate(N))):21
21:W:U83#(tt(),M,N) -> c_30(isNatKind#(activate(N)))
-->_1 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):28
-->_1 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27
-->_1 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):26
22:W:U91#(tt(),N) -> c_32(isNatKind#(activate(N)))
-->_1 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):28
-->_1 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27
-->_1 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):26
23:W:isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):28
-->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):26
-->_1 U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))):4
24:W:isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)))
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):28
-->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):26
-->_1 U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):9
25:W:isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):28
-->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):26
-->_1 U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))):11
26:W:isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)))
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):28
-->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):26
-->_1 U41#(tt(),V2) -> c_21(isNatKind#(activate(V2))):16
27:W:isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1)))
-->_1 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):28
-->_1 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27
-->_1 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):26
28:W:isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)))
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):28
-->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):26
-->_1 U61#(tt(),V2) -> c_24(isNatKind#(activate(V2))):17
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
22: U91#(tt(),N) ->
c_32(isNatKind#(activate(N)))
19: U81#(tt(),M,N) ->
c_28(U82#(isNatKind(activate(M))
,activate(M)
,activate(N))
,isNatKind#(activate(M)))
20: U82#(tt(),M,N) ->
c_29(U83#(isNat(activate(N))
,activate(M)
,activate(N))
,isNat#(activate(N)))
21: U83#(tt(),M,N) ->
c_30(isNatKind#(activate(N)))
18: U71#(tt(),N) ->
c_26(isNatKind#(activate(N)))
1: U101#(tt(),M,N) ->
c_2(U102#(isNatKind(activate(M))
,activate(M)
,activate(N))
,isNatKind#(activate(M)))
2: U102#(tt(),M,N) ->
c_3(U103#(isNat(activate(N))
,activate(M)
,activate(N))
,isNat#(activate(N)))
3: U103#(tt(),M,N) ->
c_4(isNatKind#(activate(N)))
25: isNat#(n__x(V1,V2)) ->
c_42(U31#(isNatKind(activate(V1))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V1)))
15: U35#(tt(),V2) ->
c_19(isNat#(activate(V2)))
14: U34#(tt(),V1,V2) ->
c_18(U35#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1)))
13: U33#(tt(),V1,V2) ->
c_17(U34#(isNatKind(activate(V2))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V2)))
12: U32#(tt(),V1,V2) ->
c_16(U33#(isNatKind(activate(V2))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V2)))
11: U31#(tt(),V1,V2) ->
c_15(U32#(isNatKind(activate(V1))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V1)))
10: U22#(tt(),V1) ->
c_13(isNat#(activate(V1)))
9: U21#(tt(),V1) ->
c_12(U22#(isNatKind(activate(V1))
,activate(V1))
,isNatKind#(activate(V1)))
24: isNat#(n__s(V1)) ->
c_41(U21#(isNatKind(activate(V1))
,activate(V1))
,isNatKind#(activate(V1)))
8: U15#(tt(),V2) ->
c_10(isNat#(activate(V2)))
7: U14#(tt(),V1,V2) ->
c_9(U15#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1)))
6: U13#(tt(),V1,V2) ->
c_8(U14#(isNatKind(activate(V2))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V2)))
5: U12#(tt(),V1,V2) ->
c_7(U13#(isNatKind(activate(V2))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V2)))
4: U11#(tt(),V1,V2) ->
c_6(U12#(isNatKind(activate(V1))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V1)))
23: isNat#(n__plus(V1,V2)) ->
c_40(U11#(isNatKind(activate(V1))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V1)))
28: isNatKind#(n__x(V1,V2)) ->
c_46(U61#(isNatKind(activate(V1))
,activate(V2))
,isNatKind#(activate(V1)))
27: isNatKind#(n__s(V1)) ->
c_45(isNatKind#(activate(V1)))
26: isNatKind#(n__plus(V1,V2)) ->
c_44(U41#(isNatKind(activate(V1))
,activate(V2))
,isNatKind#(activate(V1)))
17: U61#(tt(),V2) ->
c_24(isNatKind#(activate(V2)))
16: U41#(tt(),V2) ->
c_21(isNatKind#(activate(V2)))
*** 1.1.1.1.1.1.1.1.1.1.2.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2))
U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2))
U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2))
U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2))
U35(tt(),V2) -> U36(isNat(activate(V2)))
U36(tt()) -> tt()
U41(tt(),V2) -> U42(isNatKind(activate(V2)))
U42(tt()) -> tt()
U51(tt()) -> tt()
U61(tt(),V2) -> U62(isNatKind(activate(V2)))
U62(tt()) -> tt()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1)))
isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
Signature:
{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,activate/1,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/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,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/5,c_6/2,c_7/2,c_8/2,c_9/2,c_10/1,c_11/0,c_12/2,c_13/1,c_14/0,c_15/2,c_16/2,c_17/2,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0,c_26/1,c_27/1,c_28/2,c_29/2,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/2,c_41/2,c_42/2,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0}
Obligation:
Innermost
basic terms: {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#,activate#,isNat#,isNatKind#,plus#,s#,x#}/{n__0,n__plus,n__s,n__x,tt}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).
*** 1.1.1.1.1.1.1.1.1.2 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)))
U103#(tt(),M,N) -> c_4(isNatKind#(activate(N)))
U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U15#(tt(),V2) -> c_10(isNat#(activate(V2)))
U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)))
U22#(tt(),V1) -> c_13(isNat#(activate(V1)))
U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U35#(tt(),V2) -> c_19(isNat#(activate(V2)))
U71#(tt(),N) -> c_26(isNatKind#(activate(N)))
U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)))
U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)))
U83#(tt(),M,N) -> c_30(isNatKind#(activate(N)))
U91#(tt(),N) -> c_32(isNatKind#(activate(N)))
isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)))
isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
Strict TRS Rules:
Weak DP Rules:
U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)))
U41#(tt(),V2) -> c_21(isNatKind#(activate(V2)))
U61#(tt(),V2) -> c_24(isNatKind#(activate(V2)))
isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)))
isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1)))
isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)))
Weak TRS Rules:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2))
U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2))
U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2))
U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2))
U35(tt(),V2) -> U36(isNat(activate(V2)))
U36(tt()) -> tt()
U41(tt(),V2) -> U42(isNatKind(activate(V2)))
U42(tt()) -> tt()
U51(tt()) -> tt()
U61(tt(),V2) -> U62(isNatKind(activate(V2)))
U62(tt()) -> tt()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1)))
isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
Signature:
{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,activate/1,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/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,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/5,c_6/2,c_7/2,c_8/2,c_9/2,c_10/1,c_11/0,c_12/2,c_13/1,c_14/0,c_15/2,c_16/2,c_17/2,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0,c_26/1,c_27/1,c_28/2,c_29/2,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/2,c_41/2,c_42/2,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0}
Obligation:
Innermost
basic terms: {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#,activate#,isNat#,isNatKind#,plus#,s#,x#}/{n__0,n__plus,n__s,n__x,tt}
Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
Proof:
We estimate the number of application of
{2,15,18,19}
by application of
Pre({2,15,18,19}) = {1,17}.
Here rules are labelled as follows:
1: U102#(tt(),M,N) ->
c_3(U103#(isNat(activate(N))
,activate(M)
,activate(N))
,isNat#(activate(N)))
2: U103#(tt(),M,N) ->
c_4(isNatKind#(activate(N)))
3: U11#(tt(),V1,V2) ->
c_6(U12#(isNatKind(activate(V1))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V1)))
4: U12#(tt(),V1,V2) ->
c_7(U13#(isNatKind(activate(V2))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V2)))
5: U13#(tt(),V1,V2) ->
c_8(U14#(isNatKind(activate(V2))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V2)))
6: U14#(tt(),V1,V2) ->
c_9(U15#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1)))
7: U15#(tt(),V2) ->
c_10(isNat#(activate(V2)))
8: U21#(tt(),V1) ->
c_12(U22#(isNatKind(activate(V1))
,activate(V1))
,isNatKind#(activate(V1)))
9: U22#(tt(),V1) ->
c_13(isNat#(activate(V1)))
10: U31#(tt(),V1,V2) ->
c_15(U32#(isNatKind(activate(V1))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V1)))
11: U32#(tt(),V1,V2) ->
c_16(U33#(isNatKind(activate(V2))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V2)))
12: U33#(tt(),V1,V2) ->
c_17(U34#(isNatKind(activate(V2))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V2)))
13: U34#(tt(),V1,V2) ->
c_18(U35#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1)))
14: U35#(tt(),V2) ->
c_19(isNat#(activate(V2)))
15: U71#(tt(),N) ->
c_26(isNatKind#(activate(N)))
16: U81#(tt(),M,N) ->
c_28(U82#(isNatKind(activate(M))
,activate(M)
,activate(N))
,isNatKind#(activate(M)))
17: U82#(tt(),M,N) ->
c_29(U83#(isNat(activate(N))
,activate(M)
,activate(N))
,isNat#(activate(N)))
18: U83#(tt(),M,N) ->
c_30(isNatKind#(activate(N)))
19: U91#(tt(),N) ->
c_32(isNatKind#(activate(N)))
20: isNat#(n__plus(V1,V2)) ->
c_40(U11#(isNatKind(activate(V1))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V1)))
21: isNat#(n__s(V1)) ->
c_41(U21#(isNatKind(activate(V1))
,activate(V1))
,isNatKind#(activate(V1)))
22: isNat#(n__x(V1,V2)) ->
c_42(U31#(isNatKind(activate(V1))
,activate(V1)
,activate(V2))
,isNatKind#(activate(V1)))
23: U101#(tt(),M,N) ->
c_2(U102#(isNatKind(activate(M))
,activate(M)
,activate(N))
,isNatKind#(activate(M)))
24: U41#(tt(),V2) ->
c_21(isNatKind#(activate(V2)))
25: U61#(tt(),V2) ->
c_24(isNatKind#(activate(V2)))
26: isNatKind#(n__plus(V1,V2)) ->
c_44(U41#(isNatKind(activate(V1))
,activate(V2))
,isNatKind#(activate(V1)))
27: isNatKind#(n__s(V1)) ->
c_45(isNatKind#(activate(V1)))
28: isNatKind#(n__x(V1,V2)) ->
c_46(U61#(isNatKind(activate(V1))
,activate(V2))
,isNatKind#(activate(V1)))
*** 1.1.1.1.1.1.1.1.1.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)))
U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U15#(tt(),V2) -> c_10(isNat#(activate(V2)))
U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)))
U22#(tt(),V1) -> c_13(isNat#(activate(V1)))
U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U35#(tt(),V2) -> c_19(isNat#(activate(V2)))
U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)))
U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)))
isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)))
isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
Strict TRS Rules:
Weak DP Rules:
U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)))
U103#(tt(),M,N) -> c_4(isNatKind#(activate(N)))
U41#(tt(),V2) -> c_21(isNatKind#(activate(V2)))
U61#(tt(),V2) -> c_24(isNatKind#(activate(V2)))
U71#(tt(),N) -> c_26(isNatKind#(activate(N)))
U83#(tt(),M,N) -> c_30(isNatKind#(activate(N)))
U91#(tt(),N) -> c_32(isNatKind#(activate(N)))
isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)))
isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1)))
isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)))
Weak TRS Rules:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2))
U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2))
U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2))
U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2))
U35(tt(),V2) -> U36(isNat(activate(V2)))
U36(tt()) -> tt()
U41(tt(),V2) -> U42(isNatKind(activate(V2)))
U42(tt()) -> tt()
U51(tt()) -> tt()
U61(tt(),V2) -> U62(isNatKind(activate(V2)))
U62(tt()) -> tt()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1)))
isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
Signature:
{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,activate/1,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/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,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/5,c_6/2,c_7/2,c_8/2,c_9/2,c_10/1,c_11/0,c_12/2,c_13/1,c_14/0,c_15/2,c_16/2,c_17/2,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0,c_26/1,c_27/1,c_28/2,c_29/2,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/2,c_41/2,c_42/2,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0}
Obligation:
Innermost
basic terms: {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#,activate#,isNat#,isNatKind#,plus#,s#,x#}/{n__0,n__plus,n__s,n__x,tt}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S:U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)))
-->_1 U103#(tt(),M,N) -> c_4(isNatKind#(activate(N))):20
-->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))):18
-->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):17
-->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))):16
2:S:U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):28
-->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):26
-->_1 U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))):3
3:S:U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):28
-->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):26
-->_1 U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))):4
4:S:U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):28
-->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):26
-->_1 U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):5
5:S:U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
-->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))):18
-->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):17
-->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))):16
-->_1 U15#(tt(),V2) -> c_10(isNat#(activate(V2))):6
6:S:U15#(tt(),V2) -> c_10(isNat#(activate(V2)))
-->_1 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))):18
-->_1 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):17
-->_1 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))):16
7:S:U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)))
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):28
-->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):26
-->_1 U22#(tt(),V1) -> c_13(isNat#(activate(V1))):8
8:S:U22#(tt(),V1) -> c_13(isNat#(activate(V1)))
-->_1 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))):18
-->_1 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):17
-->_1 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))):16
9:S:U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):28
-->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):26
-->_1 U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))):10
10:S:U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):28
-->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):26
-->_1 U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))):11
11:S:U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):28
-->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):26
-->_1 U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):12
12:S:U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
-->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))):18
-->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):17
-->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))):16
-->_1 U35#(tt(),V2) -> c_19(isNat#(activate(V2))):13
13:S:U35#(tt(),V2) -> c_19(isNat#(activate(V2)))
-->_1 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))):18
-->_1 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):17
-->_1 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))):16
14:S:U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)))
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):28
-->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):26
-->_1 U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))):15
15:S:U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)))
-->_1 U83#(tt(),M,N) -> c_30(isNatKind#(activate(N))):24
-->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))):18
-->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):17
-->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))):16
16:S:isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):28
-->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):26
-->_1 U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))):2
17:S:isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)))
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):28
-->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):26
-->_1 U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):7
18:S:isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):28
-->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):26
-->_1 U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))):9
19:W:U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)))
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):28
-->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):26
-->_1 U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))):1
20:W:U103#(tt(),M,N) -> c_4(isNatKind#(activate(N)))
-->_1 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):28
-->_1 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27
-->_1 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):26
21:W:U41#(tt(),V2) -> c_21(isNatKind#(activate(V2)))
-->_1 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):28
-->_1 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27
-->_1 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):26
22:W:U61#(tt(),V2) -> c_24(isNatKind#(activate(V2)))
-->_1 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):28
-->_1 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27
-->_1 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):26
23:W:U71#(tt(),N) -> c_26(isNatKind#(activate(N)))
-->_1 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):28
-->_1 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27
-->_1 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):26
24:W:U83#(tt(),M,N) -> c_30(isNatKind#(activate(N)))
-->_1 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):28
-->_1 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27
-->_1 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):26
25:W:U91#(tt(),N) -> c_32(isNatKind#(activate(N)))
-->_1 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):28
-->_1 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27
-->_1 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):26
26:W:isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)))
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):28
-->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):26
-->_1 U41#(tt(),V2) -> c_21(isNatKind#(activate(V2))):21
27:W:isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1)))
-->_1 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):28
-->_1 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27
-->_1 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):26
28:W:isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)))
-->_2 isNatKind#(n__x(V1,V2)) -> c_46(U61#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):28
-->_2 isNatKind#(n__s(V1)) -> c_45(isNatKind#(activate(V1))):27
-->_2 isNatKind#(n__plus(V1,V2)) -> c_44(U41#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1))):26
-->_1 U61#(tt(),V2) -> c_24(isNatKind#(activate(V2))):22
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
25: U91#(tt(),N) ->
c_32(isNatKind#(activate(N)))
23: U71#(tt(),N) ->
c_26(isNatKind#(activate(N)))
24: U83#(tt(),M,N) ->
c_30(isNatKind#(activate(N)))
20: U103#(tt(),M,N) ->
c_4(isNatKind#(activate(N)))
28: isNatKind#(n__x(V1,V2)) ->
c_46(U61#(isNatKind(activate(V1))
,activate(V2))
,isNatKind#(activate(V1)))
27: isNatKind#(n__s(V1)) ->
c_45(isNatKind#(activate(V1)))
26: isNatKind#(n__plus(V1,V2)) ->
c_44(U41#(isNatKind(activate(V1))
,activate(V2))
,isNatKind#(activate(V1)))
22: U61#(tt(),V2) ->
c_24(isNatKind#(activate(V2)))
21: U41#(tt(),V2) ->
c_21(isNatKind#(activate(V2)))
*** 1.1.1.1.1.1.1.1.1.2.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)))
U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U15#(tt(),V2) -> c_10(isNat#(activate(V2)))
U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)))
U22#(tt(),V1) -> c_13(isNat#(activate(V1)))
U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U35#(tt(),V2) -> c_19(isNat#(activate(V2)))
U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)))
U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)))
isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)))
isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
Strict TRS Rules:
Weak DP Rules:
U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)))
Weak TRS Rules:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2))
U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2))
U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2))
U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2))
U35(tt(),V2) -> U36(isNat(activate(V2)))
U36(tt()) -> tt()
U41(tt(),V2) -> U42(isNatKind(activate(V2)))
U42(tt()) -> tt()
U51(tt()) -> tt()
U61(tt(),V2) -> U62(isNatKind(activate(V2)))
U62(tt()) -> tt()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1)))
isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
Signature:
{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,activate/1,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/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,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/5,c_6/2,c_7/2,c_8/2,c_9/2,c_10/1,c_11/0,c_12/2,c_13/1,c_14/0,c_15/2,c_16/2,c_17/2,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0,c_26/1,c_27/1,c_28/2,c_29/2,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/2,c_41/2,c_42/2,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0}
Obligation:
Innermost
basic terms: {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#,activate#,isNat#,isNatKind#,plus#,s#,x#}/{n__0,n__plus,n__s,n__x,tt}
Applied Processor:
SimplifyRHS
Proof:
Consider the dependency graph
1:S:U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)))
-->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))):18
-->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):17
-->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))):16
2:S:U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
-->_1 U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))):3
3:S:U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
-->_1 U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))):4
4:S:U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
-->_1 U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):5
5:S:U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
-->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))):18
-->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):17
-->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))):16
-->_1 U15#(tt(),V2) -> c_10(isNat#(activate(V2))):6
6:S:U15#(tt(),V2) -> c_10(isNat#(activate(V2)))
-->_1 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))):18
-->_1 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):17
-->_1 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))):16
7:S:U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)))
-->_1 U22#(tt(),V1) -> c_13(isNat#(activate(V1))):8
8:S:U22#(tt(),V1) -> c_13(isNat#(activate(V1)))
-->_1 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))):18
-->_1 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):17
-->_1 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))):16
9:S:U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
-->_1 U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))):10
10:S:U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
-->_1 U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2))):11
11:S:U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)),isNatKind#(activate(V2)))
-->_1 U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):12
12:S:U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
-->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))):18
-->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):17
-->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))):16
-->_1 U35#(tt(),V2) -> c_19(isNat#(activate(V2))):13
13:S:U35#(tt(),V2) -> c_19(isNat#(activate(V2)))
-->_1 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))):18
-->_1 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):17
-->_1 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))):16
14:S:U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)))
-->_1 U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))):15
15:S:U82#(tt(),M,N) -> c_29(U83#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N)))
-->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))):18
-->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):17
-->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))):16
16:S:isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
-->_1 U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))):2
17:S:isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1)))
-->_1 U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)),isNatKind#(activate(V1))):7
18:S:isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1)))
-->_1 U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)),isNatKind#(activate(V1))):9
19:W:U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)),isNatKind#(activate(M)))
-->_1 U102#(tt(),M,N) -> c_3(U103#(isNat(activate(N)),activate(M),activate(N)),isNat#(activate(N))):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)))
U102#(tt(),M,N) -> c_3(isNat#(activate(N)))
U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)))
U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)))
U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)))
U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)))
U82#(tt(),M,N) -> c_29(isNat#(activate(N)))
isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)))
isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)))
isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)))
*** 1.1.1.1.1.1.1.1.1.2.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
U102#(tt(),M,N) -> c_3(isNat#(activate(N)))
U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)))
U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U15#(tt(),V2) -> c_10(isNat#(activate(V2)))
U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)))
U22#(tt(),V1) -> c_13(isNat#(activate(V1)))
U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)))
U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U35#(tt(),V2) -> c_19(isNat#(activate(V2)))
U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)))
U82#(tt(),M,N) -> c_29(isNat#(activate(N)))
isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)))
isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)))
isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)))
Strict TRS Rules:
Weak DP Rules:
U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)))
Weak TRS Rules:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2))
U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2))
U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2))
U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2))
U35(tt(),V2) -> U36(isNat(activate(V2)))
U36(tt()) -> tt()
U41(tt(),V2) -> U42(isNatKind(activate(V2)))
U42(tt()) -> tt()
U51(tt()) -> tt()
U61(tt(),V2) -> U62(isNatKind(activate(V2)))
U62(tt()) -> tt()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1)))
isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
Signature:
{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,activate/1,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/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,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/5,c_6/1,c_7/1,c_8/1,c_9/2,c_10/1,c_11/0,c_12/1,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0,c_26/1,c_27/1,c_28/1,c_29/1,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/1,c_41/1,c_42/1,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0}
Obligation:
Innermost
basic terms: {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#,activate#,isNat#,isNatKind#,plus#,s#,x#}/{n__0,n__plus,n__s,n__x,tt}
Applied Processor:
Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
Proof:
We analyse the complexity of following sub-problems (R) and (S).
Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
Problem (R)
Strict DP Rules:
U102#(tt(),M,N) -> c_3(isNat#(activate(N)))
U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)))
U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U15#(tt(),V2) -> c_10(isNat#(activate(V2)))
U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)))
U22#(tt(),V1) -> c_13(isNat#(activate(V1)))
U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)))
U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U35#(tt(),V2) -> c_19(isNat#(activate(V2)))
isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)))
isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)))
isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)))
Strict TRS Rules:
Weak DP Rules:
U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)))
U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)))
U82#(tt(),M,N) -> c_29(isNat#(activate(N)))
Weak TRS Rules:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2))
U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2))
U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2))
U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2))
U35(tt(),V2) -> U36(isNat(activate(V2)))
U36(tt()) -> tt()
U41(tt(),V2) -> U42(isNatKind(activate(V2)))
U42(tt()) -> tt()
U51(tt()) -> tt()
U61(tt(),V2) -> U62(isNatKind(activate(V2)))
U62(tt()) -> tt()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1)))
isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
Signature:
{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,activate/1,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/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,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/5,c_6/1,c_7/1,c_8/1,c_9/2,c_10/1,c_11/0,c_12/1,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0,c_26/1,c_27/1,c_28/1,c_29/1,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/1,c_41/1,c_42/1,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0}
Obligation:
Innermost
basic terms: {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#,activate#,isNat#,isNatKind#,plus#,s#,x#}/{n__0,n__plus,n__s,n__x,tt}
Problem (S)
Strict DP Rules:
U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)))
U82#(tt(),M,N) -> c_29(isNat#(activate(N)))
Strict TRS Rules:
Weak DP Rules:
U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)))
U102#(tt(),M,N) -> c_3(isNat#(activate(N)))
U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)))
U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U15#(tt(),V2) -> c_10(isNat#(activate(V2)))
U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)))
U22#(tt(),V1) -> c_13(isNat#(activate(V1)))
U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)))
U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U35#(tt(),V2) -> c_19(isNat#(activate(V2)))
isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)))
isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)))
isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)))
Weak TRS Rules:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2))
U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2))
U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2))
U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2))
U35(tt(),V2) -> U36(isNat(activate(V2)))
U36(tt()) -> tt()
U41(tt(),V2) -> U42(isNatKind(activate(V2)))
U42(tt()) -> tt()
U51(tt()) -> tt()
U61(tt(),V2) -> U62(isNatKind(activate(V2)))
U62(tt()) -> tt()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1)))
isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
Signature:
{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,activate/1,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/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,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/5,c_6/1,c_7/1,c_8/1,c_9/2,c_10/1,c_11/0,c_12/1,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0,c_26/1,c_27/1,c_28/1,c_29/1,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/1,c_41/1,c_42/1,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0}
Obligation:
Innermost
basic terms: {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#,activate#,isNat#,isNatKind#,plus#,s#,x#}/{n__0,n__plus,n__s,n__x,tt}
*** 1.1.1.1.1.1.1.1.1.2.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
U102#(tt(),M,N) -> c_3(isNat#(activate(N)))
U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)))
U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U15#(tt(),V2) -> c_10(isNat#(activate(V2)))
U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)))
U22#(tt(),V1) -> c_13(isNat#(activate(V1)))
U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)))
U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U35#(tt(),V2) -> c_19(isNat#(activate(V2)))
isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)))
isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)))
isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)))
Strict TRS Rules:
Weak DP Rules:
U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)))
U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)))
U82#(tt(),M,N) -> c_29(isNat#(activate(N)))
Weak TRS Rules:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2))
U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2))
U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2))
U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2))
U35(tt(),V2) -> U36(isNat(activate(V2)))
U36(tt()) -> tt()
U41(tt(),V2) -> U42(isNatKind(activate(V2)))
U42(tt()) -> tt()
U51(tt()) -> tt()
U61(tt(),V2) -> U62(isNatKind(activate(V2)))
U62(tt()) -> tt()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1)))
isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
Signature:
{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,activate/1,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/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,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/5,c_6/1,c_7/1,c_8/1,c_9/2,c_10/1,c_11/0,c_12/1,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0,c_26/1,c_27/1,c_28/1,c_29/1,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/1,c_41/1,c_42/1,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0}
Obligation:
Innermost
basic terms: {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#,activate#,isNat#,isNatKind#,plus#,s#,x#}/{n__0,n__plus,n__s,n__x,tt}
Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
16: isNat#(n__plus(V1,V2)) ->
c_40(U11#(isNatKind(activate(V1))
,activate(V1)
,activate(V2)))
17: isNat#(n__s(V1)) ->
c_41(U21#(isNatKind(activate(V1))
,activate(V1)))
Consider the set of all dependency pairs
1: U102#(tt(),M,N) ->
c_3(isNat#(activate(N)))
2: U11#(tt(),V1,V2) ->
c_6(U12#(isNatKind(activate(V1))
,activate(V1)
,activate(V2)))
3: U12#(tt(),V1,V2) ->
c_7(U13#(isNatKind(activate(V2))
,activate(V1)
,activate(V2)))
4: U13#(tt(),V1,V2) ->
c_8(U14#(isNatKind(activate(V2))
,activate(V1)
,activate(V2)))
5: U14#(tt(),V1,V2) ->
c_9(U15#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1)))
6: U15#(tt(),V2) ->
c_10(isNat#(activate(V2)))
7: U21#(tt(),V1) ->
c_12(U22#(isNatKind(activate(V1))
,activate(V1)))
8: U22#(tt(),V1) ->
c_13(isNat#(activate(V1)))
9: U31#(tt(),V1,V2) ->
c_15(U32#(isNatKind(activate(V1))
,activate(V1)
,activate(V2)))
10: U32#(tt(),V1,V2) ->
c_16(U33#(isNatKind(activate(V2))
,activate(V1)
,activate(V2)))
11: U33#(tt(),V1,V2) ->
c_17(U34#(isNatKind(activate(V2))
,activate(V1)
,activate(V2)))
12: U34#(tt(),V1,V2) ->
c_18(U35#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1)))
13: U35#(tt(),V2) ->
c_19(isNat#(activate(V2)))
14: U81#(tt(),M,N) ->
c_28(U82#(isNatKind(activate(M))
,activate(M)
,activate(N)))
15: U82#(tt(),M,N) ->
c_29(isNat#(activate(N)))
16: isNat#(n__plus(V1,V2)) ->
c_40(U11#(isNatKind(activate(V1))
,activate(V1)
,activate(V2)))
17: isNat#(n__s(V1)) ->
c_41(U21#(isNatKind(activate(V1))
,activate(V1)))
18: isNat#(n__x(V1,V2)) ->
c_42(U31#(isNatKind(activate(V1))
,activate(V1)
,activate(V2)))
19: U101#(tt(),M,N) ->
c_2(U102#(isNatKind(activate(M))
,activate(M)
,activate(N)))
Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}induces the complexity certificateTIME (?,O(n^1))
SPACE(?,?)on application of the dependency pairs
{16,17}
These cover all (indirect) predecessors of dependency pairs
{1,2,3,4,5,6,7,8,14,15,16,17,19}
their number of applications is equally bounded.
The dependency pairs are shifted into the weak component.
*** 1.1.1.1.1.1.1.1.1.2.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
U102#(tt(),M,N) -> c_3(isNat#(activate(N)))
U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)))
U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U15#(tt(),V2) -> c_10(isNat#(activate(V2)))
U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)))
U22#(tt(),V1) -> c_13(isNat#(activate(V1)))
U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)))
U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U35#(tt(),V2) -> c_19(isNat#(activate(V2)))
isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)))
isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)))
isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)))
Strict TRS Rules:
Weak DP Rules:
U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)))
U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)))
U82#(tt(),M,N) -> c_29(isNat#(activate(N)))
Weak TRS Rules:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2))
U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2))
U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2))
U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2))
U35(tt(),V2) -> U36(isNat(activate(V2)))
U36(tt()) -> tt()
U41(tt(),V2) -> U42(isNatKind(activate(V2)))
U42(tt()) -> tt()
U51(tt()) -> tt()
U61(tt(),V2) -> U62(isNatKind(activate(V2)))
U62(tt()) -> tt()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1)))
isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
Signature:
{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,activate/1,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/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,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/5,c_6/1,c_7/1,c_8/1,c_9/2,c_10/1,c_11/0,c_12/1,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0,c_26/1,c_27/1,c_28/1,c_29/1,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/1,c_41/1,c_42/1,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0}
Obligation:
Innermost
basic terms: {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#,activate#,isNat#,isNatKind#,plus#,s#,x#}/{n__0,n__plus,n__s,n__x,tt}
Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
Proof:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_2) = {1},
uargs(c_3) = {1},
uargs(c_6) = {1},
uargs(c_7) = {1},
uargs(c_8) = {1},
uargs(c_9) = {1,2},
uargs(c_10) = {1},
uargs(c_12) = {1},
uargs(c_13) = {1},
uargs(c_15) = {1},
uargs(c_16) = {1},
uargs(c_17) = {1},
uargs(c_18) = {1,2},
uargs(c_19) = {1},
uargs(c_28) = {1},
uargs(c_29) = {1},
uargs(c_40) = {1},
uargs(c_41) = {1},
uargs(c_42) = {1}
Following symbols are considered usable:
{0,activate,plus,s,x,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#,activate#,isNat#,isNatKind#,plus#,s#,x#}
TcT has computed the following interpretation:
p(0) = [0]
p(U101) = [0]
p(U102) = [2] x2 + [0]
p(U103) = [1] x1 + [1]
p(U104) = [2] x3 + [1]
p(U11) = [1] x2 + [4]
p(U12) = [0]
p(U13) = [3]
p(U14) = [1] x2 + [0]
p(U15) = [4] x1 + [6]
p(U16) = [2] x1 + [0]
p(U21) = [0]
p(U22) = [0]
p(U23) = [1] x1 + [0]
p(U31) = [5] x2 + [5] x3 + [0]
p(U32) = [6] x2 + [4] x3 + [2]
p(U33) = [4] x3 + [1]
p(U34) = [3] x3 + [0]
p(U35) = [1] x1 + [4] x2 + [5]
p(U36) = [2] x1 + [2]
p(U41) = [1] x1 + [4]
p(U42) = [0]
p(U51) = [4]
p(U61) = [4]
p(U62) = [1] x1 + [1]
p(U71) = [4]
p(U72) = [1] x1 + [0]
p(U81) = [1] x1 + [1] x2 + [1] x3 + [4]
p(U82) = [4] x1 + [1]
p(U83) = [1] x1 + [1] x3 + [0]
p(U84) = [1] x2 + [1]
p(U91) = [2] x2 + [1]
p(U92) = [0]
p(activate) = [1] x1 + [0]
p(isNat) = [0]
p(isNatKind) = [0]
p(n__0) = [0]
p(n__plus) = [1] x1 + [1] x2 + [2]
p(n__s) = [1] x1 + [2]
p(n__x) = [1] x1 + [1] x2 + [0]
p(plus) = [1] x1 + [1] x2 + [2]
p(s) = [1] x1 + [2]
p(tt) = [0]
p(x) = [1] x1 + [1] x2 + [0]
p(0#) = [0]
p(U101#) = [1] x1 + [1] x2 + [4] x3 + [0]
p(U102#) = [4] x3 + [0]
p(U103#) = [4] x2 + [2] x3 + [0]
p(U104#) = [1] x2 + [1]
p(U11#) = [4] x2 + [4] x3 + [4]
p(U12#) = [4] x2 + [4] x3 + [4]
p(U13#) = [4] x2 + [4] x3 + [4]
p(U14#) = [4] x2 + [4] x3 + [4]
p(U15#) = [4] x2 + [4]
p(U16#) = [1] x1 + [0]
p(U21#) = [4] x2 + [0]
p(U22#) = [4] x2 + [0]
p(U23#) = [0]
p(U31#) = [4] x2 + [4] x3 + [0]
p(U32#) = [4] x2 + [4] x3 + [0]
p(U33#) = [4] x2 + [4] x3 + [0]
p(U34#) = [4] x2 + [4] x3 + [0]
p(U35#) = [4] x2 + [0]
p(U36#) = [2]
p(U41#) = [1] x2 + [0]
p(U42#) = [1] x1 + [4]
p(U51#) = [0]
p(U61#) = [4]
p(U62#) = [1] x1 + [1]
p(U71#) = [1]
p(U72#) = [4]
p(U81#) = [4] x3 + [6]
p(U82#) = [4] x3 + [2]
p(U83#) = [1] x2 + [1]
p(U84#) = [1] x2 + [2] x3 + [0]
p(U91#) = [1]
p(U92#) = [1] x1 + [2]
p(activate#) = [1] x1 + [0]
p(isNat#) = [4] x1 + [0]
p(isNatKind#) = [0]
p(plus#) = [1] x2 + [1]
p(s#) = [1]
p(x#) = [1] x1 + [0]
p(c_1) = [4]
p(c_2) = [1] x1 + [0]
p(c_3) = [1] x1 + [0]
p(c_4) = [1]
p(c_5) = [1] x2 + [1] x5 + [0]
p(c_6) = [1] x1 + [0]
p(c_7) = [1] x1 + [0]
p(c_8) = [1] x1 + [0]
p(c_9) = [1] x1 + [1] x2 + [0]
p(c_10) = [1] x1 + [4]
p(c_11) = [1]
p(c_12) = [1] x1 + [0]
p(c_13) = [1] x1 + [0]
p(c_14) = [0]
p(c_15) = [1] x1 + [0]
p(c_16) = [1] x1 + [0]
p(c_17) = [1] x1 + [0]
p(c_18) = [1] x1 + [1] x2 + [0]
p(c_19) = [1] x1 + [0]
p(c_20) = [0]
p(c_21) = [1] x1 + [0]
p(c_22) = [4]
p(c_23) = [0]
p(c_24) = [2] x1 + [1]
p(c_25) = [0]
p(c_26) = [0]
p(c_27) = [1]
p(c_28) = [1] x1 + [4]
p(c_29) = [1] x1 + [2]
p(c_30) = [0]
p(c_31) = [1] x1 + [2]
p(c_32) = [1] x1 + [0]
p(c_33) = [1] x1 + [4]
p(c_34) = [1]
p(c_35) = [2]
p(c_36) = [1] x1 + [0]
p(c_37) = [1]
p(c_38) = [0]
p(c_39) = [4]
p(c_40) = [1] x1 + [2]
p(c_41) = [1] x1 + [7]
p(c_42) = [1] x1 + [0]
p(c_43) = [1]
p(c_44) = [1] x1 + [0]
p(c_45) = [1]
p(c_46) = [1] x2 + [4]
p(c_47) = [4]
p(c_48) = [0]
p(c_49) = [1]
Following rules are strictly oriented:
isNat#(n__plus(V1,V2)) = [4] V1 + [4] V2 + [8]
> [4] V1 + [4] V2 + [6]
= c_40(U11#(isNatKind(activate(V1))
,activate(V1)
,activate(V2)))
isNat#(n__s(V1)) = [4] V1 + [8]
> [4] V1 + [7]
= c_41(U21#(isNatKind(activate(V1))
,activate(V1)))
Following rules are (at-least) weakly oriented:
U101#(tt(),M,N) = [1] M + [4] N + [0]
>= [4] N + [0]
= c_2(U102#(isNatKind(activate(M))
,activate(M)
,activate(N)))
U102#(tt(),M,N) = [4] N + [0]
>= [4] N + [0]
= c_3(isNat#(activate(N)))
U11#(tt(),V1,V2) = [4] V1 + [4] V2 + [4]
>= [4] V1 + [4] V2 + [4]
= c_6(U12#(isNatKind(activate(V1))
,activate(V1)
,activate(V2)))
U12#(tt(),V1,V2) = [4] V1 + [4] V2 + [4]
>= [4] V1 + [4] V2 + [4]
= c_7(U13#(isNatKind(activate(V2))
,activate(V1)
,activate(V2)))
U13#(tt(),V1,V2) = [4] V1 + [4] V2 + [4]
>= [4] V1 + [4] V2 + [4]
= c_8(U14#(isNatKind(activate(V2))
,activate(V1)
,activate(V2)))
U14#(tt(),V1,V2) = [4] V1 + [4] V2 + [4]
>= [4] V1 + [4] V2 + [4]
= c_9(U15#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1)))
U15#(tt(),V2) = [4] V2 + [4]
>= [4] V2 + [4]
= c_10(isNat#(activate(V2)))
U21#(tt(),V1) = [4] V1 + [0]
>= [4] V1 + [0]
= c_12(U22#(isNatKind(activate(V1))
,activate(V1)))
U22#(tt(),V1) = [4] V1 + [0]
>= [4] V1 + [0]
= c_13(isNat#(activate(V1)))
U31#(tt(),V1,V2) = [4] V1 + [4] V2 + [0]
>= [4] V1 + [4] V2 + [0]
= c_15(U32#(isNatKind(activate(V1))
,activate(V1)
,activate(V2)))
U32#(tt(),V1,V2) = [4] V1 + [4] V2 + [0]
>= [4] V1 + [4] V2 + [0]
= c_16(U33#(isNatKind(activate(V2))
,activate(V1)
,activate(V2)))
U33#(tt(),V1,V2) = [4] V1 + [4] V2 + [0]
>= [4] V1 + [4] V2 + [0]
= c_17(U34#(isNatKind(activate(V2))
,activate(V1)
,activate(V2)))
U34#(tt(),V1,V2) = [4] V1 + [4] V2 + [0]
>= [4] V1 + [4] V2 + [0]
= c_18(U35#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1)))
U35#(tt(),V2) = [4] V2 + [0]
>= [4] V2 + [0]
= c_19(isNat#(activate(V2)))
U81#(tt(),M,N) = [4] N + [6]
>= [4] N + [6]
= c_28(U82#(isNatKind(activate(M))
,activate(M)
,activate(N)))
U82#(tt(),M,N) = [4] N + [2]
>= [4] N + [2]
= c_29(isNat#(activate(N)))
isNat#(n__x(V1,V2)) = [4] V1 + [4] V2 + [0]
>= [4] V1 + [4] V2 + [0]
= c_42(U31#(isNatKind(activate(V1))
,activate(V1)
,activate(V2)))
0() = [0]
>= [0]
= n__0()
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__0()) = [0]
>= [0]
= 0()
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [2]
>= [1] X1 + [1] X2 + [2]
= plus(X1,X2)
activate(n__s(X)) = [1] X + [2]
>= [1] X + [2]
= s(X)
activate(n__x(X1,X2)) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= x(X1,X2)
plus(X1,X2) = [1] X1 + [1] X2 + [2]
>= [1] X1 + [1] X2 + [2]
= n__plus(X1,X2)
s(X) = [1] X + [2]
>= [1] X + [2]
= n__s(X)
x(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= n__x(X1,X2)
*** 1.1.1.1.1.1.1.1.1.2.1.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
U102#(tt(),M,N) -> c_3(isNat#(activate(N)))
U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)))
U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U15#(tt(),V2) -> c_10(isNat#(activate(V2)))
U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)))
U22#(tt(),V1) -> c_13(isNat#(activate(V1)))
U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)))
U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U35#(tt(),V2) -> c_19(isNat#(activate(V2)))
isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)))
Strict TRS Rules:
Weak DP Rules:
U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)))
U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)))
U82#(tt(),M,N) -> c_29(isNat#(activate(N)))
isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)))
isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)))
Weak TRS Rules:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2))
U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2))
U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2))
U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2))
U35(tt(),V2) -> U36(isNat(activate(V2)))
U36(tt()) -> tt()
U41(tt(),V2) -> U42(isNatKind(activate(V2)))
U42(tt()) -> tt()
U51(tt()) -> tt()
U61(tt(),V2) -> U62(isNatKind(activate(V2)))
U62(tt()) -> tt()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1)))
isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
Signature:
{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,activate/1,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/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,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/5,c_6/1,c_7/1,c_8/1,c_9/2,c_10/1,c_11/0,c_12/1,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0,c_26/1,c_27/1,c_28/1,c_29/1,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/1,c_41/1,c_42/1,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0}
Obligation:
Innermost
basic terms: {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#,activate#,isNat#,isNatKind#,plus#,s#,x#}/{n__0,n__plus,n__s,n__x,tt}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.1.1.1.2.1.1.1.1.2 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)))
U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U35#(tt(),V2) -> c_19(isNat#(activate(V2)))
isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)))
Strict TRS Rules:
Weak DP Rules:
U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)))
U102#(tt(),M,N) -> c_3(isNat#(activate(N)))
U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)))
U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U15#(tt(),V2) -> c_10(isNat#(activate(V2)))
U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)))
U22#(tt(),V1) -> c_13(isNat#(activate(V1)))
U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)))
U82#(tt(),M,N) -> c_29(isNat#(activate(N)))
isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)))
isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)))
Weak TRS Rules:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2))
U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2))
U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2))
U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2))
U35(tt(),V2) -> U36(isNat(activate(V2)))
U36(tt()) -> tt()
U41(tt(),V2) -> U42(isNatKind(activate(V2)))
U42(tt()) -> tt()
U51(tt()) -> tt()
U61(tt(),V2) -> U62(isNatKind(activate(V2)))
U62(tt()) -> tt()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1)))
isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
Signature:
{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,activate/1,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/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,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/5,c_6/1,c_7/1,c_8/1,c_9/2,c_10/1,c_11/0,c_12/1,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0,c_26/1,c_27/1,c_28/1,c_29/1,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/1,c_41/1,c_42/1,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0}
Obligation:
Innermost
basic terms: {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#,activate#,isNat#,isNatKind#,plus#,s#,x#}/{n__0,n__plus,n__s,n__x,tt}
Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
6: isNat#(n__x(V1,V2)) ->
c_42(U31#(isNatKind(activate(V1))
,activate(V1)
,activate(V2)))
Consider the set of all dependency pairs
1: U31#(tt(),V1,V2) ->
c_15(U32#(isNatKind(activate(V1))
,activate(V1)
,activate(V2)))
2: U32#(tt(),V1,V2) ->
c_16(U33#(isNatKind(activate(V2))
,activate(V1)
,activate(V2)))
3: U33#(tt(),V1,V2) ->
c_17(U34#(isNatKind(activate(V2))
,activate(V1)
,activate(V2)))
4: U34#(tt(),V1,V2) ->
c_18(U35#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1)))
5: U35#(tt(),V2) ->
c_19(isNat#(activate(V2)))
6: isNat#(n__x(V1,V2)) ->
c_42(U31#(isNatKind(activate(V1))
,activate(V1)
,activate(V2)))
7: U101#(tt(),M,N) ->
c_2(U102#(isNatKind(activate(M))
,activate(M)
,activate(N)))
8: U102#(tt(),M,N) ->
c_3(isNat#(activate(N)))
9: U11#(tt(),V1,V2) ->
c_6(U12#(isNatKind(activate(V1))
,activate(V1)
,activate(V2)))
10: U12#(tt(),V1,V2) ->
c_7(U13#(isNatKind(activate(V2))
,activate(V1)
,activate(V2)))
11: U13#(tt(),V1,V2) ->
c_8(U14#(isNatKind(activate(V2))
,activate(V1)
,activate(V2)))
12: U14#(tt(),V1,V2) ->
c_9(U15#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1)))
13: U15#(tt(),V2) ->
c_10(isNat#(activate(V2)))
14: U21#(tt(),V1) ->
c_12(U22#(isNatKind(activate(V1))
,activate(V1)))
15: U22#(tt(),V1) ->
c_13(isNat#(activate(V1)))
16: U81#(tt(),M,N) ->
c_28(U82#(isNatKind(activate(M))
,activate(M)
,activate(N)))
17: U82#(tt(),M,N) ->
c_29(isNat#(activate(N)))
18: isNat#(n__plus(V1,V2)) ->
c_40(U11#(isNatKind(activate(V1))
,activate(V1)
,activate(V2)))
19: isNat#(n__s(V1)) ->
c_41(U21#(isNatKind(activate(V1))
,activate(V1)))
Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}induces the complexity certificateTIME (?,O(n^1))
SPACE(?,?)on application of the dependency pairs
{6}
These cover all (indirect) predecessors of dependency pairs
{1,2,3,4,5,6,7,8,16,17}
their number of applications is equally bounded.
The dependency pairs are shifted into the weak component.
*** 1.1.1.1.1.1.1.1.1.2.1.1.1.1.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)))
U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U35#(tt(),V2) -> c_19(isNat#(activate(V2)))
isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)))
Strict TRS Rules:
Weak DP Rules:
U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)))
U102#(tt(),M,N) -> c_3(isNat#(activate(N)))
U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)))
U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U15#(tt(),V2) -> c_10(isNat#(activate(V2)))
U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)))
U22#(tt(),V1) -> c_13(isNat#(activate(V1)))
U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)))
U82#(tt(),M,N) -> c_29(isNat#(activate(N)))
isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)))
isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)))
Weak TRS Rules:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2))
U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2))
U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2))
U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2))
U35(tt(),V2) -> U36(isNat(activate(V2)))
U36(tt()) -> tt()
U41(tt(),V2) -> U42(isNatKind(activate(V2)))
U42(tt()) -> tt()
U51(tt()) -> tt()
U61(tt(),V2) -> U62(isNatKind(activate(V2)))
U62(tt()) -> tt()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1)))
isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
Signature:
{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,activate/1,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/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,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/5,c_6/1,c_7/1,c_8/1,c_9/2,c_10/1,c_11/0,c_12/1,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0,c_26/1,c_27/1,c_28/1,c_29/1,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/1,c_41/1,c_42/1,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0}
Obligation:
Innermost
basic terms: {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#,activate#,isNat#,isNatKind#,plus#,s#,x#}/{n__0,n__plus,n__s,n__x,tt}
Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
Proof:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_2) = {1},
uargs(c_3) = {1},
uargs(c_6) = {1},
uargs(c_7) = {1},
uargs(c_8) = {1},
uargs(c_9) = {1,2},
uargs(c_10) = {1},
uargs(c_12) = {1},
uargs(c_13) = {1},
uargs(c_15) = {1},
uargs(c_16) = {1},
uargs(c_17) = {1},
uargs(c_18) = {1,2},
uargs(c_19) = {1},
uargs(c_28) = {1},
uargs(c_29) = {1},
uargs(c_40) = {1},
uargs(c_41) = {1},
uargs(c_42) = {1}
Following symbols are considered usable:
{0,activate,plus,s,x,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#,activate#,isNat#,isNatKind#,plus#,s#,x#}
TcT has computed the following interpretation:
p(0) = [0]
p(U101) = [1] x1 + [2] x2 + [0]
p(U102) = [2] x3 + [0]
p(U103) = [2] x3 + [1]
p(U104) = [4] x1 + [1] x3 + [1]
p(U11) = [4] x2 + [6] x3 + [4]
p(U12) = [5] x2 + [4]
p(U13) = [1] x3 + [0]
p(U14) = [2] x2 + [4]
p(U15) = [4] x2 + [5]
p(U16) = [1] x1 + [1]
p(U21) = [2] x2 + [6]
p(U22) = [1] x1 + [4] x2 + [0]
p(U23) = [1] x1 + [1]
p(U31) = [6] x2 + [4] x3 + [1]
p(U32) = [4] x2 + [7]
p(U33) = [1] x1 + [6] x2 + [0]
p(U34) = [4] x1 + [4] x2 + [0]
p(U35) = [1] x1 + [3]
p(U36) = [4]
p(U41) = [0]
p(U42) = [4] x1 + [6]
p(U51) = [1] x1 + [2]
p(U61) = [6]
p(U62) = [5]
p(U71) = [4] x1 + [1] x2 + [1]
p(U72) = [1] x1 + [1] x2 + [1]
p(U81) = [2] x2 + [2]
p(U82) = [2]
p(U83) = [1] x1 + [2] x2 + [0]
p(U84) = [0]
p(U91) = [4] x1 + [0]
p(U92) = [1]
p(activate) = [1] x1 + [0]
p(isNat) = [0]
p(isNatKind) = [0]
p(n__0) = [0]
p(n__plus) = [1] x1 + [1] x2 + [3]
p(n__s) = [1] x1 + [1]
p(n__x) = [1] x1 + [1] x2 + [1]
p(plus) = [1] x1 + [1] x2 + [3]
p(s) = [1] x1 + [1]
p(tt) = [0]
p(x) = [1] x1 + [1] x2 + [1]
p(0#) = [0]
p(U101#) = [1] x1 + [4] x2 + [4] x3 + [7]
p(U102#) = [4] x2 + [1] x3 + [4]
p(U103#) = [1] x1 + [0]
p(U104#) = [1] x1 + [1] x2 + [1]
p(U11#) = [1] x2 + [1] x3 + [3]
p(U12#) = [1] x2 + [1] x3 + [3]
p(U13#) = [1] x2 + [1] x3 + [3]
p(U14#) = [1] x2 + [1] x3 + [3]
p(U15#) = [1] x2 + [1]
p(U16#) = [1]
p(U21#) = [1] x2 + [1]
p(U22#) = [1] x2 + [1]
p(U23#) = [1] x1 + [1]
p(U31#) = [1] x2 + [1] x3 + [0]
p(U32#) = [1] x2 + [1] x3 + [0]
p(U33#) = [1] x2 + [1] x3 + [0]
p(U34#) = [1] x2 + [1] x3 + [0]
p(U35#) = [1] x2 + [0]
p(U36#) = [1]
p(U41#) = [2] x2 + [0]
p(U42#) = [1]
p(U51#) = [1] x1 + [1]
p(U61#) = [1]
p(U62#) = [0]
p(U71#) = [1] x1 + [0]
p(U72#) = [1] x1 + [1]
p(U81#) = [1] x1 + [4] x2 + [1] x3 + [5]
p(U82#) = [1] x2 + [1] x3 + [2]
p(U83#) = [1]
p(U84#) = [1] x1 + [4] x2 + [0]
p(U91#) = [1] x1 + [1]
p(U92#) = [1] x1 + [4]
p(activate#) = [2] x1 + [0]
p(isNat#) = [1] x1 + [0]
p(isNatKind#) = [1] x1 + [1]
p(plus#) = [1] x1 + [0]
p(s#) = [1]
p(x#) = [1]
p(c_1) = [1]
p(c_2) = [1] x1 + [3]
p(c_3) = [1] x1 + [2]
p(c_4) = [4] x1 + [4]
p(c_5) = [2] x3 + [1] x4 + [1] x5 + [0]
p(c_6) = [1] x1 + [0]
p(c_7) = [1] x1 + [0]
p(c_8) = [1] x1 + [0]
p(c_9) = [1] x1 + [1] x2 + [2]
p(c_10) = [1] x1 + [0]
p(c_11) = [1]
p(c_12) = [1] x1 + [0]
p(c_13) = [1] x1 + [1]
p(c_14) = [0]
p(c_15) = [1] x1 + [0]
p(c_16) = [1] x1 + [0]
p(c_17) = [1] x1 + [0]
p(c_18) = [1] x1 + [1] x2 + [0]
p(c_19) = [1] x1 + [0]
p(c_20) = [1]
p(c_21) = [4] x1 + [1]
p(c_22) = [1]
p(c_23) = [2]
p(c_24) = [4] x1 + [0]
p(c_25) = [1]
p(c_26) = [0]
p(c_27) = [1] x1 + [4]
p(c_28) = [1] x1 + [3]
p(c_29) = [1] x1 + [0]
p(c_30) = [0]
p(c_31) = [4] x3 + [0]
p(c_32) = [2] x1 + [0]
p(c_33) = [2] x1 + [1]
p(c_34) = [2]
p(c_35) = [1]
p(c_36) = [2] x1 + [0]
p(c_37) = [1]
p(c_38) = [0]
p(c_39) = [0]
p(c_40) = [1] x1 + [0]
p(c_41) = [1] x1 + [0]
p(c_42) = [1] x1 + [0]
p(c_43) = [1]
p(c_44) = [2] x1 + [2]
p(c_45) = [4] x1 + [0]
p(c_46) = [4] x2 + [1]
p(c_47) = [1]
p(c_48) = [0]
p(c_49) = [1]
Following rules are strictly oriented:
isNat#(n__x(V1,V2)) = [1] V1 + [1] V2 + [1]
> [1] V1 + [1] V2 + [0]
= c_42(U31#(isNatKind(activate(V1))
,activate(V1)
,activate(V2)))
Following rules are (at-least) weakly oriented:
U101#(tt(),M,N) = [4] M + [4] N + [7]
>= [4] M + [1] N + [7]
= c_2(U102#(isNatKind(activate(M))
,activate(M)
,activate(N)))
U102#(tt(),M,N) = [4] M + [1] N + [4]
>= [1] N + [2]
= c_3(isNat#(activate(N)))
U11#(tt(),V1,V2) = [1] V1 + [1] V2 + [3]
>= [1] V1 + [1] V2 + [3]
= c_6(U12#(isNatKind(activate(V1))
,activate(V1)
,activate(V2)))
U12#(tt(),V1,V2) = [1] V1 + [1] V2 + [3]
>= [1] V1 + [1] V2 + [3]
= c_7(U13#(isNatKind(activate(V2))
,activate(V1)
,activate(V2)))
U13#(tt(),V1,V2) = [1] V1 + [1] V2 + [3]
>= [1] V1 + [1] V2 + [3]
= c_8(U14#(isNatKind(activate(V2))
,activate(V1)
,activate(V2)))
U14#(tt(),V1,V2) = [1] V1 + [1] V2 + [3]
>= [1] V1 + [1] V2 + [3]
= c_9(U15#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1)))
U15#(tt(),V2) = [1] V2 + [1]
>= [1] V2 + [0]
= c_10(isNat#(activate(V2)))
U21#(tt(),V1) = [1] V1 + [1]
>= [1] V1 + [1]
= c_12(U22#(isNatKind(activate(V1))
,activate(V1)))
U22#(tt(),V1) = [1] V1 + [1]
>= [1] V1 + [1]
= c_13(isNat#(activate(V1)))
U31#(tt(),V1,V2) = [1] V1 + [1] V2 + [0]
>= [1] V1 + [1] V2 + [0]
= c_15(U32#(isNatKind(activate(V1))
,activate(V1)
,activate(V2)))
U32#(tt(),V1,V2) = [1] V1 + [1] V2 + [0]
>= [1] V1 + [1] V2 + [0]
= c_16(U33#(isNatKind(activate(V2))
,activate(V1)
,activate(V2)))
U33#(tt(),V1,V2) = [1] V1 + [1] V2 + [0]
>= [1] V1 + [1] V2 + [0]
= c_17(U34#(isNatKind(activate(V2))
,activate(V1)
,activate(V2)))
U34#(tt(),V1,V2) = [1] V1 + [1] V2 + [0]
>= [1] V1 + [1] V2 + [0]
= c_18(U35#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1)))
U35#(tt(),V2) = [1] V2 + [0]
>= [1] V2 + [0]
= c_19(isNat#(activate(V2)))
U81#(tt(),M,N) = [4] M + [1] N + [5]
>= [1] M + [1] N + [5]
= c_28(U82#(isNatKind(activate(M))
,activate(M)
,activate(N)))
U82#(tt(),M,N) = [1] M + [1] N + [2]
>= [1] N + [0]
= c_29(isNat#(activate(N)))
isNat#(n__plus(V1,V2)) = [1] V1 + [1] V2 + [3]
>= [1] V1 + [1] V2 + [3]
= c_40(U11#(isNatKind(activate(V1))
,activate(V1)
,activate(V2)))
isNat#(n__s(V1)) = [1] V1 + [1]
>= [1] V1 + [1]
= c_41(U21#(isNatKind(activate(V1))
,activate(V1)))
0() = [0]
>= [0]
= n__0()
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__0()) = [0]
>= [0]
= 0()
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [3]
>= [1] X1 + [1] X2 + [3]
= plus(X1,X2)
activate(n__s(X)) = [1] X + [1]
>= [1] X + [1]
= s(X)
activate(n__x(X1,X2)) = [1] X1 + [1] X2 + [1]
>= [1] X1 + [1] X2 + [1]
= x(X1,X2)
plus(X1,X2) = [1] X1 + [1] X2 + [3]
>= [1] X1 + [1] X2 + [3]
= n__plus(X1,X2)
s(X) = [1] X + [1]
>= [1] X + [1]
= n__s(X)
x(X1,X2) = [1] X1 + [1] X2 + [1]
>= [1] X1 + [1] X2 + [1]
= n__x(X1,X2)
*** 1.1.1.1.1.1.1.1.1.2.1.1.1.1.2.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)))
U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U35#(tt(),V2) -> c_19(isNat#(activate(V2)))
Strict TRS Rules:
Weak DP Rules:
U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)))
U102#(tt(),M,N) -> c_3(isNat#(activate(N)))
U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)))
U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U15#(tt(),V2) -> c_10(isNat#(activate(V2)))
U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)))
U22#(tt(),V1) -> c_13(isNat#(activate(V1)))
U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)))
U82#(tt(),M,N) -> c_29(isNat#(activate(N)))
isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)))
isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)))
isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)))
Weak TRS Rules:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2))
U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2))
U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2))
U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2))
U35(tt(),V2) -> U36(isNat(activate(V2)))
U36(tt()) -> tt()
U41(tt(),V2) -> U42(isNatKind(activate(V2)))
U42(tt()) -> tt()
U51(tt()) -> tt()
U61(tt(),V2) -> U62(isNatKind(activate(V2)))
U62(tt()) -> tt()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1)))
isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
Signature:
{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,activate/1,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/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,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/5,c_6/1,c_7/1,c_8/1,c_9/2,c_10/1,c_11/0,c_12/1,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0,c_26/1,c_27/1,c_28/1,c_29/1,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/1,c_41/1,c_42/1,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0}
Obligation:
Innermost
basic terms: {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#,activate#,isNat#,isNatKind#,plus#,s#,x#}/{n__0,n__plus,n__s,n__x,tt}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.1.1.1.2.1.1.1.1.2.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)))
U102#(tt(),M,N) -> c_3(isNat#(activate(N)))
U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)))
U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U15#(tt(),V2) -> c_10(isNat#(activate(V2)))
U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)))
U22#(tt(),V1) -> c_13(isNat#(activate(V1)))
U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)))
U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U35#(tt(),V2) -> c_19(isNat#(activate(V2)))
U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)))
U82#(tt(),M,N) -> c_29(isNat#(activate(N)))
isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)))
isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)))
isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)))
Weak TRS Rules:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2))
U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2))
U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2))
U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2))
U35(tt(),V2) -> U36(isNat(activate(V2)))
U36(tt()) -> tt()
U41(tt(),V2) -> U42(isNatKind(activate(V2)))
U42(tt()) -> tt()
U51(tt()) -> tt()
U61(tt(),V2) -> U62(isNatKind(activate(V2)))
U62(tt()) -> tt()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1)))
isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
Signature:
{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,activate/1,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/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,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/5,c_6/1,c_7/1,c_8/1,c_9/2,c_10/1,c_11/0,c_12/1,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0,c_26/1,c_27/1,c_28/1,c_29/1,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/1,c_41/1,c_42/1,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0}
Obligation:
Innermost
basic terms: {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#,activate#,isNat#,isNatKind#,plus#,s#,x#}/{n__0,n__plus,n__s,n__x,tt}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)))
-->_1 U102#(tt(),M,N) -> c_3(isNat#(activate(N))):2
2:W:U102#(tt(),M,N) -> c_3(isNat#(activate(N)))
-->_1 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))):19
-->_1 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))):18
-->_1 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))):17
3:W:U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)))
-->_1 U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))):4
4:W:U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)))
-->_1 U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))):5
5:W:U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)))
-->_1 U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):6
6:W:U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
-->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))):19
-->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))):18
-->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))):17
-->_1 U15#(tt(),V2) -> c_10(isNat#(activate(V2))):7
7:W:U15#(tt(),V2) -> c_10(isNat#(activate(V2)))
-->_1 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))):19
-->_1 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))):18
-->_1 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))):17
8:W:U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)))
-->_1 U22#(tt(),V1) -> c_13(isNat#(activate(V1))):9
9:W:U22#(tt(),V1) -> c_13(isNat#(activate(V1)))
-->_1 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))):19
-->_1 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))):18
-->_1 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))):17
10:W:U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)))
-->_1 U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2))):11
11:W:U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)))
-->_1 U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2))):12
12:W:U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)))
-->_1 U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):13
13:W:U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
-->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))):19
-->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))):18
-->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))):17
-->_1 U35#(tt(),V2) -> c_19(isNat#(activate(V2))):14
14:W:U35#(tt(),V2) -> c_19(isNat#(activate(V2)))
-->_1 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))):19
-->_1 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))):18
-->_1 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))):17
15:W:U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)))
-->_1 U82#(tt(),M,N) -> c_29(isNat#(activate(N))):16
16:W:U82#(tt(),M,N) -> c_29(isNat#(activate(N)))
-->_1 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))):19
-->_1 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))):18
-->_1 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))):17
17:W:isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)))
-->_1 U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))):3
18:W:isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)))
-->_1 U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1))):8
19:W:isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)))
-->_1 U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2))):10
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
15: U81#(tt(),M,N) ->
c_28(U82#(isNatKind(activate(M))
,activate(M)
,activate(N)))
16: U82#(tt(),M,N) ->
c_29(isNat#(activate(N)))
1: U101#(tt(),M,N) ->
c_2(U102#(isNatKind(activate(M))
,activate(M)
,activate(N)))
2: U102#(tt(),M,N) ->
c_3(isNat#(activate(N)))
19: isNat#(n__x(V1,V2)) ->
c_42(U31#(isNatKind(activate(V1))
,activate(V1)
,activate(V2)))
14: U35#(tt(),V2) ->
c_19(isNat#(activate(V2)))
13: U34#(tt(),V1,V2) ->
c_18(U35#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1)))
12: U33#(tt(),V1,V2) ->
c_17(U34#(isNatKind(activate(V2))
,activate(V1)
,activate(V2)))
11: U32#(tt(),V1,V2) ->
c_16(U33#(isNatKind(activate(V2))
,activate(V1)
,activate(V2)))
10: U31#(tt(),V1,V2) ->
c_15(U32#(isNatKind(activate(V1))
,activate(V1)
,activate(V2)))
9: U22#(tt(),V1) ->
c_13(isNat#(activate(V1)))
8: U21#(tt(),V1) ->
c_12(U22#(isNatKind(activate(V1))
,activate(V1)))
18: isNat#(n__s(V1)) ->
c_41(U21#(isNatKind(activate(V1))
,activate(V1)))
7: U15#(tt(),V2) ->
c_10(isNat#(activate(V2)))
6: U14#(tt(),V1,V2) ->
c_9(U15#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1)))
5: U13#(tt(),V1,V2) ->
c_8(U14#(isNatKind(activate(V2))
,activate(V1)
,activate(V2)))
4: U12#(tt(),V1,V2) ->
c_7(U13#(isNatKind(activate(V2))
,activate(V1)
,activate(V2)))
3: U11#(tt(),V1,V2) ->
c_6(U12#(isNatKind(activate(V1))
,activate(V1)
,activate(V2)))
17: isNat#(n__plus(V1,V2)) ->
c_40(U11#(isNatKind(activate(V1))
,activate(V1)
,activate(V2)))
*** 1.1.1.1.1.1.1.1.1.2.1.1.1.1.2.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2))
U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2))
U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2))
U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2))
U35(tt(),V2) -> U36(isNat(activate(V2)))
U36(tt()) -> tt()
U41(tt(),V2) -> U42(isNatKind(activate(V2)))
U42(tt()) -> tt()
U51(tt()) -> tt()
U61(tt(),V2) -> U62(isNatKind(activate(V2)))
U62(tt()) -> tt()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1)))
isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
Signature:
{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,activate/1,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/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,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/5,c_6/1,c_7/1,c_8/1,c_9/2,c_10/1,c_11/0,c_12/1,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0,c_26/1,c_27/1,c_28/1,c_29/1,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/1,c_41/1,c_42/1,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0}
Obligation:
Innermost
basic terms: {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#,activate#,isNat#,isNatKind#,plus#,s#,x#}/{n__0,n__plus,n__s,n__x,tt}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).
*** 1.1.1.1.1.1.1.1.1.2.1.1.1.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)))
U82#(tt(),M,N) -> c_29(isNat#(activate(N)))
Strict TRS Rules:
Weak DP Rules:
U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)))
U102#(tt(),M,N) -> c_3(isNat#(activate(N)))
U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)))
U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U15#(tt(),V2) -> c_10(isNat#(activate(V2)))
U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)))
U22#(tt(),V1) -> c_13(isNat#(activate(V1)))
U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)))
U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U35#(tt(),V2) -> c_19(isNat#(activate(V2)))
isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)))
isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)))
isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)))
Weak TRS Rules:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2))
U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2))
U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2))
U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2))
U35(tt(),V2) -> U36(isNat(activate(V2)))
U36(tt()) -> tt()
U41(tt(),V2) -> U42(isNatKind(activate(V2)))
U42(tt()) -> tt()
U51(tt()) -> tt()
U61(tt(),V2) -> U62(isNatKind(activate(V2)))
U62(tt()) -> tt()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1)))
isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
Signature:
{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,activate/1,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/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,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/5,c_6/1,c_7/1,c_8/1,c_9/2,c_10/1,c_11/0,c_12/1,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0,c_26/1,c_27/1,c_28/1,c_29/1,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/1,c_41/1,c_42/1,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0}
Obligation:
Innermost
basic terms: {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#,activate#,isNat#,isNatKind#,plus#,s#,x#}/{n__0,n__plus,n__s,n__x,tt}
Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
Proof:
We estimate the number of application of
{2}
by application of
Pre({2}) = {1}.
Here rules are labelled as follows:
1: U81#(tt(),M,N) ->
c_28(U82#(isNatKind(activate(M))
,activate(M)
,activate(N)))
2: U82#(tt(),M,N) ->
c_29(isNat#(activate(N)))
3: U101#(tt(),M,N) ->
c_2(U102#(isNatKind(activate(M))
,activate(M)
,activate(N)))
4: U102#(tt(),M,N) ->
c_3(isNat#(activate(N)))
5: U11#(tt(),V1,V2) ->
c_6(U12#(isNatKind(activate(V1))
,activate(V1)
,activate(V2)))
6: U12#(tt(),V1,V2) ->
c_7(U13#(isNatKind(activate(V2))
,activate(V1)
,activate(V2)))
7: U13#(tt(),V1,V2) ->
c_8(U14#(isNatKind(activate(V2))
,activate(V1)
,activate(V2)))
8: U14#(tt(),V1,V2) ->
c_9(U15#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1)))
9: U15#(tt(),V2) ->
c_10(isNat#(activate(V2)))
10: U21#(tt(),V1) ->
c_12(U22#(isNatKind(activate(V1))
,activate(V1)))
11: U22#(tt(),V1) ->
c_13(isNat#(activate(V1)))
12: U31#(tt(),V1,V2) ->
c_15(U32#(isNatKind(activate(V1))
,activate(V1)
,activate(V2)))
13: U32#(tt(),V1,V2) ->
c_16(U33#(isNatKind(activate(V2))
,activate(V1)
,activate(V2)))
14: U33#(tt(),V1,V2) ->
c_17(U34#(isNatKind(activate(V2))
,activate(V1)
,activate(V2)))
15: U34#(tt(),V1,V2) ->
c_18(U35#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1)))
16: U35#(tt(),V2) ->
c_19(isNat#(activate(V2)))
17: isNat#(n__plus(V1,V2)) ->
c_40(U11#(isNatKind(activate(V1))
,activate(V1)
,activate(V2)))
18: isNat#(n__s(V1)) ->
c_41(U21#(isNatKind(activate(V1))
,activate(V1)))
19: isNat#(n__x(V1,V2)) ->
c_42(U31#(isNatKind(activate(V1))
,activate(V1)
,activate(V2)))
*** 1.1.1.1.1.1.1.1.1.2.1.1.1.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)))
Strict TRS Rules:
Weak DP Rules:
U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)))
U102#(tt(),M,N) -> c_3(isNat#(activate(N)))
U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)))
U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U15#(tt(),V2) -> c_10(isNat#(activate(V2)))
U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)))
U22#(tt(),V1) -> c_13(isNat#(activate(V1)))
U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)))
U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U35#(tt(),V2) -> c_19(isNat#(activate(V2)))
U82#(tt(),M,N) -> c_29(isNat#(activate(N)))
isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)))
isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)))
isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)))
Weak TRS Rules:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2))
U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2))
U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2))
U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2))
U35(tt(),V2) -> U36(isNat(activate(V2)))
U36(tt()) -> tt()
U41(tt(),V2) -> U42(isNatKind(activate(V2)))
U42(tt()) -> tt()
U51(tt()) -> tt()
U61(tt(),V2) -> U62(isNatKind(activate(V2)))
U62(tt()) -> tt()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1)))
isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
Signature:
{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,activate/1,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/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,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/5,c_6/1,c_7/1,c_8/1,c_9/2,c_10/1,c_11/0,c_12/1,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0,c_26/1,c_27/1,c_28/1,c_29/1,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/1,c_41/1,c_42/1,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0}
Obligation:
Innermost
basic terms: {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#,activate#,isNat#,isNatKind#,plus#,s#,x#}/{n__0,n__plus,n__s,n__x,tt}
Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
Proof:
We estimate the number of application of
{1}
by application of
Pre({1}) = {}.
Here rules are labelled as follows:
1: U81#(tt(),M,N) ->
c_28(U82#(isNatKind(activate(M))
,activate(M)
,activate(N)))
2: U101#(tt(),M,N) ->
c_2(U102#(isNatKind(activate(M))
,activate(M)
,activate(N)))
3: U102#(tt(),M,N) ->
c_3(isNat#(activate(N)))
4: U11#(tt(),V1,V2) ->
c_6(U12#(isNatKind(activate(V1))
,activate(V1)
,activate(V2)))
5: U12#(tt(),V1,V2) ->
c_7(U13#(isNatKind(activate(V2))
,activate(V1)
,activate(V2)))
6: U13#(tt(),V1,V2) ->
c_8(U14#(isNatKind(activate(V2))
,activate(V1)
,activate(V2)))
7: U14#(tt(),V1,V2) ->
c_9(U15#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1)))
8: U15#(tt(),V2) ->
c_10(isNat#(activate(V2)))
9: U21#(tt(),V1) ->
c_12(U22#(isNatKind(activate(V1))
,activate(V1)))
10: U22#(tt(),V1) ->
c_13(isNat#(activate(V1)))
11: U31#(tt(),V1,V2) ->
c_15(U32#(isNatKind(activate(V1))
,activate(V1)
,activate(V2)))
12: U32#(tt(),V1,V2) ->
c_16(U33#(isNatKind(activate(V2))
,activate(V1)
,activate(V2)))
13: U33#(tt(),V1,V2) ->
c_17(U34#(isNatKind(activate(V2))
,activate(V1)
,activate(V2)))
14: U34#(tt(),V1,V2) ->
c_18(U35#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1)))
15: U35#(tt(),V2) ->
c_19(isNat#(activate(V2)))
16: U82#(tt(),M,N) ->
c_29(isNat#(activate(N)))
17: isNat#(n__plus(V1,V2)) ->
c_40(U11#(isNatKind(activate(V1))
,activate(V1)
,activate(V2)))
18: isNat#(n__s(V1)) ->
c_41(U21#(isNatKind(activate(V1))
,activate(V1)))
19: isNat#(n__x(V1,V2)) ->
c_42(U31#(isNatKind(activate(V1))
,activate(V1)
,activate(V2)))
*** 1.1.1.1.1.1.1.1.1.2.1.1.1.2.1.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)))
U102#(tt(),M,N) -> c_3(isNat#(activate(N)))
U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)))
U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U15#(tt(),V2) -> c_10(isNat#(activate(V2)))
U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)))
U22#(tt(),V1) -> c_13(isNat#(activate(V1)))
U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)))
U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U35#(tt(),V2) -> c_19(isNat#(activate(V2)))
U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)))
U82#(tt(),M,N) -> c_29(isNat#(activate(N)))
isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)))
isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)))
isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)))
Weak TRS Rules:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2))
U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2))
U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2))
U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2))
U35(tt(),V2) -> U36(isNat(activate(V2)))
U36(tt()) -> tt()
U41(tt(),V2) -> U42(isNatKind(activate(V2)))
U42(tt()) -> tt()
U51(tt()) -> tt()
U61(tt(),V2) -> U62(isNatKind(activate(V2)))
U62(tt()) -> tt()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1)))
isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
Signature:
{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,activate/1,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/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,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/5,c_6/1,c_7/1,c_8/1,c_9/2,c_10/1,c_11/0,c_12/1,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0,c_26/1,c_27/1,c_28/1,c_29/1,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/1,c_41/1,c_42/1,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0}
Obligation:
Innermost
basic terms: {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#,activate#,isNat#,isNatKind#,plus#,s#,x#}/{n__0,n__plus,n__s,n__x,tt}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:U101#(tt(),M,N) -> c_2(U102#(isNatKind(activate(M)),activate(M),activate(N)))
-->_1 U102#(tt(),M,N) -> c_3(isNat#(activate(N))):2
2:W:U102#(tt(),M,N) -> c_3(isNat#(activate(N)))
-->_1 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))):19
-->_1 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))):18
-->_1 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))):17
3:W:U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)))
-->_1 U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))):4
4:W:U12#(tt(),V1,V2) -> c_7(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)))
-->_1 U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))):5
5:W:U13#(tt(),V1,V2) -> c_8(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)))
-->_1 U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):6
6:W:U14#(tt(),V1,V2) -> c_9(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
-->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))):19
-->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))):18
-->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))):17
-->_1 U15#(tt(),V2) -> c_10(isNat#(activate(V2))):7
7:W:U15#(tt(),V2) -> c_10(isNat#(activate(V2)))
-->_1 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))):19
-->_1 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))):18
-->_1 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))):17
8:W:U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1)))
-->_1 U22#(tt(),V1) -> c_13(isNat#(activate(V1))):9
9:W:U22#(tt(),V1) -> c_13(isNat#(activate(V1)))
-->_1 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))):19
-->_1 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))):18
-->_1 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))):17
10:W:U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2)))
-->_1 U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2))):11
11:W:U32#(tt(),V1,V2) -> c_16(U33#(isNatKind(activate(V2)),activate(V1),activate(V2)))
-->_1 U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2))):12
12:W:U33#(tt(),V1,V2) -> c_17(U34#(isNatKind(activate(V2)),activate(V1),activate(V2)))
-->_1 U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1))):13
13:W:U34#(tt(),V1,V2) -> c_18(U35#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
-->_2 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))):19
-->_2 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))):18
-->_2 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))):17
-->_1 U35#(tt(),V2) -> c_19(isNat#(activate(V2))):14
14:W:U35#(tt(),V2) -> c_19(isNat#(activate(V2)))
-->_1 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))):19
-->_1 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))):18
-->_1 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))):17
15:W:U81#(tt(),M,N) -> c_28(U82#(isNatKind(activate(M)),activate(M),activate(N)))
-->_1 U82#(tt(),M,N) -> c_29(isNat#(activate(N))):16
16:W:U82#(tt(),M,N) -> c_29(isNat#(activate(N)))
-->_1 isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2))):19
-->_1 isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1))):18
-->_1 isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))):17
17:W:isNat#(n__plus(V1,V2)) -> c_40(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)))
-->_1 U11#(tt(),V1,V2) -> c_6(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))):3
18:W:isNat#(n__s(V1)) -> c_41(U21#(isNatKind(activate(V1)),activate(V1)))
-->_1 U21#(tt(),V1) -> c_12(U22#(isNatKind(activate(V1)),activate(V1))):8
19:W:isNat#(n__x(V1,V2)) -> c_42(U31#(isNatKind(activate(V1)),activate(V1),activate(V2)))
-->_1 U31#(tt(),V1,V2) -> c_15(U32#(isNatKind(activate(V1)),activate(V1),activate(V2))):10
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
15: U81#(tt(),M,N) ->
c_28(U82#(isNatKind(activate(M))
,activate(M)
,activate(N)))
16: U82#(tt(),M,N) ->
c_29(isNat#(activate(N)))
1: U101#(tt(),M,N) ->
c_2(U102#(isNatKind(activate(M))
,activate(M)
,activate(N)))
2: U102#(tt(),M,N) ->
c_3(isNat#(activate(N)))
19: isNat#(n__x(V1,V2)) ->
c_42(U31#(isNatKind(activate(V1))
,activate(V1)
,activate(V2)))
14: U35#(tt(),V2) ->
c_19(isNat#(activate(V2)))
13: U34#(tt(),V1,V2) ->
c_18(U35#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1)))
12: U33#(tt(),V1,V2) ->
c_17(U34#(isNatKind(activate(V2))
,activate(V1)
,activate(V2)))
11: U32#(tt(),V1,V2) ->
c_16(U33#(isNatKind(activate(V2))
,activate(V1)
,activate(V2)))
10: U31#(tt(),V1,V2) ->
c_15(U32#(isNatKind(activate(V1))
,activate(V1)
,activate(V2)))
9: U22#(tt(),V1) ->
c_13(isNat#(activate(V1)))
8: U21#(tt(),V1) ->
c_12(U22#(isNatKind(activate(V1))
,activate(V1)))
18: isNat#(n__s(V1)) ->
c_41(U21#(isNatKind(activate(V1))
,activate(V1)))
7: U15#(tt(),V2) ->
c_10(isNat#(activate(V2)))
6: U14#(tt(),V1,V2) ->
c_9(U15#(isNat(activate(V1))
,activate(V2))
,isNat#(activate(V1)))
5: U13#(tt(),V1,V2) ->
c_8(U14#(isNatKind(activate(V2))
,activate(V1)
,activate(V2)))
4: U12#(tt(),V1,V2) ->
c_7(U13#(isNatKind(activate(V2))
,activate(V1)
,activate(V2)))
3: U11#(tt(),V1,V2) ->
c_6(U12#(isNatKind(activate(V1))
,activate(V1)
,activate(V2)))
17: isNat#(n__plus(V1,V2)) ->
c_40(U11#(isNatKind(activate(V1))
,activate(V1)
,activate(V2)))
*** 1.1.1.1.1.1.1.1.1.2.1.1.1.2.1.1.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V1,V2) -> U32(isNatKind(activate(V1)),activate(V1),activate(V2))
U32(tt(),V1,V2) -> U33(isNatKind(activate(V2)),activate(V1),activate(V2))
U33(tt(),V1,V2) -> U34(isNatKind(activate(V2)),activate(V1),activate(V2))
U34(tt(),V1,V2) -> U35(isNat(activate(V1)),activate(V2))
U35(tt(),V2) -> U36(isNat(activate(V2)))
U36(tt()) -> tt()
U41(tt(),V2) -> U42(isNatKind(activate(V2)))
U42(tt()) -> tt()
U51(tt()) -> tt()
U61(tt(),V2) -> U62(isNatKind(activate(V2)))
U62(tt()) -> tt()
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(X1,X2)
activate(n__s(X)) -> s(X)
activate(n__x(X1,X2)) -> x(X1,X2)
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNat(n__x(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V1),activate(V2))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U41(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U51(isNatKind(activate(V1)))
isNatKind(n__x(V1,V2)) -> U61(isNatKind(activate(V1)),activate(V2))
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
x(X1,X2) -> n__x(X1,X2)
Signature:
{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,activate/1,isNat/1,isNatKind/1,plus/2,s/1,x/2,0#/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,activate#/1,isNat#/1,isNatKind#/1,plus#/2,s#/1,x#/2} / {n__0/0,n__plus/2,n__s/1,n__x/2,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/5,c_6/1,c_7/1,c_8/1,c_9/2,c_10/1,c_11/0,c_12/1,c_13/1,c_14/0,c_15/1,c_16/1,c_17/1,c_18/2,c_19/1,c_20/0,c_21/1,c_22/0,c_23/0,c_24/1,c_25/0,c_26/1,c_27/1,c_28/1,c_29/1,c_30/1,c_31/4,c_32/1,c_33/1,c_34/0,c_35/1,c_36/1,c_37/1,c_38/1,c_39/0,c_40/1,c_41/1,c_42/1,c_43/0,c_44/2,c_45/1,c_46/2,c_47/0,c_48/0,c_49/0}
Obligation:
Innermost
basic terms: {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#,activate#,isNat#,isNatKind#,plus#,s#,x#}/{n__0,n__plus,n__s,n__x,tt}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).