YES
Problem:
a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) -> a__U16(a__isNat(V2))
a__U16(tt()) -> tt()
a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) -> a__U23(a__isNat(V1))
a__U23(tt()) -> tt()
a__U31(tt(),V2) -> a__U32(a__isNatKind(V2))
a__U32(tt()) -> tt()
a__U41(tt()) -> tt()
a__U51(tt(),N) -> a__U52(a__isNatKind(N),N)
a__U52(tt(),N) -> mark(N)
a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
a__isNat(0()) -> tt()
a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) -> tt()
a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1))
a__plus(N,0()) -> a__U51(a__isNat(N),N)
a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) -> a__isNatKind(X)
mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) -> a__U15(mark(X1),X2)
mark(isNat(X)) -> a__isNat(X)
mark(U16(X)) -> a__U16(mark(X))
mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
mark(U23(X)) -> a__U23(mark(X))
mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
mark(U32(X)) -> a__U32(mark(X))
mark(U41(X)) -> a__U41(mark(X))
mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
mark(tt()) -> tt()
mark(s(X)) -> s(mark(X))
mark(0()) -> 0()
a__U11(X1,X2,X3) -> U11(X1,X2,X3)
a__U12(X1,X2,X3) -> U12(X1,X2,X3)
a__isNatKind(X) -> isNatKind(X)
a__U13(X1,X2,X3) -> U13(X1,X2,X3)
a__U14(X1,X2,X3) -> U14(X1,X2,X3)
a__U15(X1,X2) -> U15(X1,X2)
a__isNat(X) -> isNat(X)
a__U16(X) -> U16(X)
a__U21(X1,X2) -> U21(X1,X2)
a__U22(X1,X2) -> U22(X1,X2)
a__U23(X) -> U23(X)
a__U31(X1,X2) -> U31(X1,X2)
a__U32(X) -> U32(X)
a__U41(X) -> U41(X)
a__U51(X1,X2) -> U51(X1,X2)
a__U52(X1,X2) -> U52(X1,X2)
a__U61(X1,X2,X3) -> U61(X1,X2,X3)
a__U62(X1,X2,X3) -> U62(X1,X2,X3)
a__U63(X1,X2,X3) -> U63(X1,X2,X3)
a__U64(X1,X2,X3) -> U64(X1,X2,X3)
a__plus(X1,X2) -> plus(X1,X2)
Proof:
DP Processor:
DPs:
a__U11#(tt(),V1,V2) -> a__isNatKind#(V1)
a__U11#(tt(),V1,V2) -> a__U12#(a__isNatKind(V1),V1,V2)
a__U12#(tt(),V1,V2) -> a__isNatKind#(V2)
a__U12#(tt(),V1,V2) -> a__U13#(a__isNatKind(V2),V1,V2)
a__U13#(tt(),V1,V2) -> a__isNatKind#(V2)
a__U13#(tt(),V1,V2) -> a__U14#(a__isNatKind(V2),V1,V2)
a__U14#(tt(),V1,V2) -> a__isNat#(V1)
a__U14#(tt(),V1,V2) -> a__U15#(a__isNat(V1),V2)
a__U15#(tt(),V2) -> a__isNat#(V2)
a__U15#(tt(),V2) -> a__U16#(a__isNat(V2))
a__U21#(tt(),V1) -> a__isNatKind#(V1)
a__U21#(tt(),V1) -> a__U22#(a__isNatKind(V1),V1)
a__U22#(tt(),V1) -> a__isNat#(V1)
a__U22#(tt(),V1) -> a__U23#(a__isNat(V1))
a__U31#(tt(),V2) -> a__isNatKind#(V2)
a__U31#(tt(),V2) -> a__U32#(a__isNatKind(V2))
a__U51#(tt(),N) -> a__isNatKind#(N)
a__U51#(tt(),N) -> a__U52#(a__isNatKind(N),N)
a__U52#(tt(),N) -> mark#(N)
a__U61#(tt(),M,N) -> a__isNatKind#(M)
a__U61#(tt(),M,N) -> a__U62#(a__isNatKind(M),M,N)
a__U62#(tt(),M,N) -> a__isNat#(N)
a__U62#(tt(),M,N) -> a__U63#(a__isNat(N),M,N)
a__U63#(tt(),M,N) -> a__isNatKind#(N)
a__U63#(tt(),M,N) -> a__U64#(a__isNatKind(N),M,N)
a__U64#(tt(),M,N) -> mark#(M)
a__U64#(tt(),M,N) -> mark#(N)
a__U64#(tt(),M,N) -> a__plus#(mark(N),mark(M))
a__isNat#(plus(V1,V2)) -> a__isNatKind#(V1)
a__isNat#(plus(V1,V2)) -> a__U11#(a__isNatKind(V1),V1,V2)
a__isNat#(s(V1)) -> a__isNatKind#(V1)
a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1)
a__isNatKind#(plus(V1,V2)) -> a__U31#(a__isNatKind(V1),V2)
a__isNatKind#(s(V1)) -> a__isNatKind#(V1)
a__isNatKind#(s(V1)) -> a__U41#(a__isNatKind(V1))
a__plus#(N,0()) -> a__isNat#(N)
a__plus#(N,0()) -> a__U51#(a__isNat(N),N)
a__plus#(N,s(M)) -> a__isNat#(M)
a__plus#(N,s(M)) -> a__U61#(a__isNat(M),M,N)
mark#(U11(X1,X2,X3)) -> mark#(X1)
mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3)
mark#(U12(X1,X2,X3)) -> mark#(X1)
mark#(U12(X1,X2,X3)) -> a__U12#(mark(X1),X2,X3)
mark#(isNatKind(X)) -> a__isNatKind#(X)
mark#(U13(X1,X2,X3)) -> mark#(X1)
mark#(U13(X1,X2,X3)) -> a__U13#(mark(X1),X2,X3)
mark#(U14(X1,X2,X3)) -> mark#(X1)
mark#(U14(X1,X2,X3)) -> a__U14#(mark(X1),X2,X3)
mark#(U15(X1,X2)) -> mark#(X1)
mark#(U15(X1,X2)) -> a__U15#(mark(X1),X2)
mark#(isNat(X)) -> a__isNat#(X)
mark#(U16(X)) -> mark#(X)
mark#(U16(X)) -> a__U16#(mark(X))
mark#(U21(X1,X2)) -> mark#(X1)
mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2)
mark#(U22(X1,X2)) -> mark#(X1)
mark#(U22(X1,X2)) -> a__U22#(mark(X1),X2)
mark#(U23(X)) -> mark#(X)
mark#(U23(X)) -> a__U23#(mark(X))
mark#(U31(X1,X2)) -> mark#(X1)
mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2)
mark#(U32(X)) -> mark#(X)
mark#(U32(X)) -> a__U32#(mark(X))
mark#(U41(X)) -> mark#(X)
mark#(U41(X)) -> a__U41#(mark(X))
mark#(U51(X1,X2)) -> mark#(X1)
mark#(U51(X1,X2)) -> a__U51#(mark(X1),X2)
mark#(U52(X1,X2)) -> mark#(X1)
mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2)
mark#(U61(X1,X2,X3)) -> mark#(X1)
mark#(U61(X1,X2,X3)) -> a__U61#(mark(X1),X2,X3)
mark#(U62(X1,X2,X3)) -> mark#(X1)
mark#(U62(X1,X2,X3)) -> a__U62#(mark(X1),X2,X3)
mark#(U63(X1,X2,X3)) -> mark#(X1)
mark#(U63(X1,X2,X3)) -> a__U63#(mark(X1),X2,X3)
mark#(U64(X1,X2,X3)) -> mark#(X1)
mark#(U64(X1,X2,X3)) -> a__U64#(mark(X1),X2,X3)
mark#(plus(X1,X2)) -> mark#(X2)
mark#(plus(X1,X2)) -> mark#(X1)
mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
mark#(s(X)) -> mark#(X)
TRS:
a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) -> a__U16(a__isNat(V2))
a__U16(tt()) -> tt()
a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) -> a__U23(a__isNat(V1))
a__U23(tt()) -> tt()
a__U31(tt(),V2) -> a__U32(a__isNatKind(V2))
a__U32(tt()) -> tt()
a__U41(tt()) -> tt()
a__U51(tt(),N) -> a__U52(a__isNatKind(N),N)
a__U52(tt(),N) -> mark(N)
a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
a__isNat(0()) -> tt()
a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) -> tt()
a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1))
a__plus(N,0()) -> a__U51(a__isNat(N),N)
a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) -> a__isNatKind(X)
mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) -> a__U15(mark(X1),X2)
mark(isNat(X)) -> a__isNat(X)
mark(U16(X)) -> a__U16(mark(X))
mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
mark(U23(X)) -> a__U23(mark(X))
mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
mark(U32(X)) -> a__U32(mark(X))
mark(U41(X)) -> a__U41(mark(X))
mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
mark(tt()) -> tt()
mark(s(X)) -> s(mark(X))
mark(0()) -> 0()
a__U11(X1,X2,X3) -> U11(X1,X2,X3)
a__U12(X1,X2,X3) -> U12(X1,X2,X3)
a__isNatKind(X) -> isNatKind(X)
a__U13(X1,X2,X3) -> U13(X1,X2,X3)
a__U14(X1,X2,X3) -> U14(X1,X2,X3)
a__U15(X1,X2) -> U15(X1,X2)
a__isNat(X) -> isNat(X)
a__U16(X) -> U16(X)
a__U21(X1,X2) -> U21(X1,X2)
a__U22(X1,X2) -> U22(X1,X2)
a__U23(X) -> U23(X)
a__U31(X1,X2) -> U31(X1,X2)
a__U32(X) -> U32(X)
a__U41(X) -> U41(X)
a__U51(X1,X2) -> U51(X1,X2)
a__U52(X1,X2) -> U52(X1,X2)
a__U61(X1,X2,X3) -> U61(X1,X2,X3)
a__U62(X1,X2,X3) -> U62(X1,X2,X3)
a__U63(X1,X2,X3) -> U63(X1,X2,X3)
a__U64(X1,X2,X3) -> U64(X1,X2,X3)
a__plus(X1,X2) -> plus(X1,X2)
TDG Processor:
DPs:
a__U11#(tt(),V1,V2) -> a__isNatKind#(V1)
a__U11#(tt(),V1,V2) -> a__U12#(a__isNatKind(V1),V1,V2)
a__U12#(tt(),V1,V2) -> a__isNatKind#(V2)
a__U12#(tt(),V1,V2) -> a__U13#(a__isNatKind(V2),V1,V2)
a__U13#(tt(),V1,V2) -> a__isNatKind#(V2)
a__U13#(tt(),V1,V2) -> a__U14#(a__isNatKind(V2),V1,V2)
a__U14#(tt(),V1,V2) -> a__isNat#(V1)
a__U14#(tt(),V1,V2) -> a__U15#(a__isNat(V1),V2)
a__U15#(tt(),V2) -> a__isNat#(V2)
a__U15#(tt(),V2) -> a__U16#(a__isNat(V2))
a__U21#(tt(),V1) -> a__isNatKind#(V1)
a__U21#(tt(),V1) -> a__U22#(a__isNatKind(V1),V1)
a__U22#(tt(),V1) -> a__isNat#(V1)
a__U22#(tt(),V1) -> a__U23#(a__isNat(V1))
a__U31#(tt(),V2) -> a__isNatKind#(V2)
a__U31#(tt(),V2) -> a__U32#(a__isNatKind(V2))
a__U51#(tt(),N) -> a__isNatKind#(N)
a__U51#(tt(),N) -> a__U52#(a__isNatKind(N),N)
a__U52#(tt(),N) -> mark#(N)
a__U61#(tt(),M,N) -> a__isNatKind#(M)
a__U61#(tt(),M,N) -> a__U62#(a__isNatKind(M),M,N)
a__U62#(tt(),M,N) -> a__isNat#(N)
a__U62#(tt(),M,N) -> a__U63#(a__isNat(N),M,N)
a__U63#(tt(),M,N) -> a__isNatKind#(N)
a__U63#(tt(),M,N) -> a__U64#(a__isNatKind(N),M,N)
a__U64#(tt(),M,N) -> mark#(M)
a__U64#(tt(),M,N) -> mark#(N)
a__U64#(tt(),M,N) -> a__plus#(mark(N),mark(M))
a__isNat#(plus(V1,V2)) -> a__isNatKind#(V1)
a__isNat#(plus(V1,V2)) -> a__U11#(a__isNatKind(V1),V1,V2)
a__isNat#(s(V1)) -> a__isNatKind#(V1)
a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1)
a__isNatKind#(plus(V1,V2)) -> a__U31#(a__isNatKind(V1),V2)
a__isNatKind#(s(V1)) -> a__isNatKind#(V1)
a__isNatKind#(s(V1)) -> a__U41#(a__isNatKind(V1))
a__plus#(N,0()) -> a__isNat#(N)
a__plus#(N,0()) -> a__U51#(a__isNat(N),N)
a__plus#(N,s(M)) -> a__isNat#(M)
a__plus#(N,s(M)) -> a__U61#(a__isNat(M),M,N)
mark#(U11(X1,X2,X3)) -> mark#(X1)
mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3)
mark#(U12(X1,X2,X3)) -> mark#(X1)
mark#(U12(X1,X2,X3)) -> a__U12#(mark(X1),X2,X3)
mark#(isNatKind(X)) -> a__isNatKind#(X)
mark#(U13(X1,X2,X3)) -> mark#(X1)
mark#(U13(X1,X2,X3)) -> a__U13#(mark(X1),X2,X3)
mark#(U14(X1,X2,X3)) -> mark#(X1)
mark#(U14(X1,X2,X3)) -> a__U14#(mark(X1),X2,X3)
mark#(U15(X1,X2)) -> mark#(X1)
mark#(U15(X1,X2)) -> a__U15#(mark(X1),X2)
mark#(isNat(X)) -> a__isNat#(X)
mark#(U16(X)) -> mark#(X)
mark#(U16(X)) -> a__U16#(mark(X))
mark#(U21(X1,X2)) -> mark#(X1)
mark#(U21(X1,X2)) -> a__U21#(mark(X1),X2)
mark#(U22(X1,X2)) -> mark#(X1)
mark#(U22(X1,X2)) -> a__U22#(mark(X1),X2)
mark#(U23(X)) -> mark#(X)
mark#(U23(X)) -> a__U23#(mark(X))
mark#(U31(X1,X2)) -> mark#(X1)
mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2)
mark#(U32(X)) -> mark#(X)
mark#(U32(X)) -> a__U32#(mark(X))
mark#(U41(X)) -> mark#(X)
mark#(U41(X)) -> a__U41#(mark(X))
mark#(U51(X1,X2)) -> mark#(X1)
mark#(U51(X1,X2)) -> a__U51#(mark(X1),X2)
mark#(U52(X1,X2)) -> mark#(X1)
mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2)
mark#(U61(X1,X2,X3)) -> mark#(X1)
mark#(U61(X1,X2,X3)) -> a__U61#(mark(X1),X2,X3)
mark#(U62(X1,X2,X3)) -> mark#(X1)
mark#(U62(X1,X2,X3)) -> a__U62#(mark(X1),X2,X3)
mark#(U63(X1,X2,X3)) -> mark#(X1)
mark#(U63(X1,X2,X3)) -> a__U63#(mark(X1),X2,X3)
mark#(U64(X1,X2,X3)) -> mark#(X1)
mark#(U64(X1,X2,X3)) -> a__U64#(mark(X1),X2,X3)
mark#(plus(X1,X2)) -> mark#(X2)
mark#(plus(X1,X2)) -> mark#(X1)
mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
mark#(s(X)) -> mark#(X)
TRS:
a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) -> a__U16(a__isNat(V2))
a__U16(tt()) -> tt()
a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) -> a__U23(a__isNat(V1))
a__U23(tt()) -> tt()
a__U31(tt(),V2) -> a__U32(a__isNatKind(V2))
a__U32(tt()) -> tt()
a__U41(tt()) -> tt()
a__U51(tt(),N) -> a__U52(a__isNatKind(N),N)
a__U52(tt(),N) -> mark(N)
a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
a__isNat(0()) -> tt()
a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) -> tt()
a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1))
a__plus(N,0()) -> a__U51(a__isNat(N),N)
a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) -> a__isNatKind(X)
mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) -> a__U15(mark(X1),X2)
mark(isNat(X)) -> a__isNat(X)
mark(U16(X)) -> a__U16(mark(X))
mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
mark(U23(X)) -> a__U23(mark(X))
mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
mark(U32(X)) -> a__U32(mark(X))
mark(U41(X)) -> a__U41(mark(X))
mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
mark(tt()) -> tt()
mark(s(X)) -> s(mark(X))
mark(0()) -> 0()
a__U11(X1,X2,X3) -> U11(X1,X2,X3)
a__U12(X1,X2,X3) -> U12(X1,X2,X3)
a__isNatKind(X) -> isNatKind(X)
a__U13(X1,X2,X3) -> U13(X1,X2,X3)
a__U14(X1,X2,X3) -> U14(X1,X2,X3)
a__U15(X1,X2) -> U15(X1,X2)
a__isNat(X) -> isNat(X)
a__U16(X) -> U16(X)
a__U21(X1,X2) -> U21(X1,X2)
a__U22(X1,X2) -> U22(X1,X2)
a__U23(X) -> U23(X)
a__U31(X1,X2) -> U31(X1,X2)
a__U32(X) -> U32(X)
a__U41(X) -> U41(X)
a__U51(X1,X2) -> U51(X1,X2)
a__U52(X1,X2) -> U52(X1,X2)
a__U61(X1,X2,X3) -> U61(X1,X2,X3)
a__U62(X1,X2,X3) -> U62(X1,X2,X3)
a__U63(X1,X2,X3) -> U63(X1,X2,X3)
a__U64(X1,X2,X3) -> U64(X1,X2,X3)
a__plus(X1,X2) -> plus(X1,X2)
graph:
...
SCC Processor:
#sccs: 3
#rules: 52
#arcs: 1152/6724
DPs:
a__plus#(N,0()) -> a__U51#(a__isNat(N),N)
a__U51#(tt(),N) -> a__U52#(a__isNatKind(N),N)
a__U52#(tt(),N) -> mark#(N)
mark#(U11(X1,X2,X3)) -> mark#(X1)
mark#(U12(X1,X2,X3)) -> mark#(X1)
mark#(U13(X1,X2,X3)) -> mark#(X1)
mark#(U14(X1,X2,X3)) -> mark#(X1)
mark#(U15(X1,X2)) -> mark#(X1)
mark#(U16(X)) -> mark#(X)
mark#(U21(X1,X2)) -> mark#(X1)
mark#(U22(X1,X2)) -> mark#(X1)
mark#(U23(X)) -> mark#(X)
mark#(U31(X1,X2)) -> mark#(X1)
mark#(U32(X)) -> mark#(X)
mark#(U41(X)) -> mark#(X)
mark#(U51(X1,X2)) -> mark#(X1)
mark#(U51(X1,X2)) -> a__U51#(mark(X1),X2)
mark#(U52(X1,X2)) -> mark#(X1)
mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2)
mark#(U61(X1,X2,X3)) -> mark#(X1)
mark#(U61(X1,X2,X3)) -> a__U61#(mark(X1),X2,X3)
a__U61#(tt(),M,N) -> a__U62#(a__isNatKind(M),M,N)
a__U62#(tt(),M,N) -> a__U63#(a__isNat(N),M,N)
a__U63#(tt(),M,N) -> a__U64#(a__isNatKind(N),M,N)
a__U64#(tt(),M,N) -> mark#(M)
mark#(U62(X1,X2,X3)) -> mark#(X1)
mark#(U62(X1,X2,X3)) -> a__U62#(mark(X1),X2,X3)
mark#(U63(X1,X2,X3)) -> mark#(X1)
mark#(U63(X1,X2,X3)) -> a__U63#(mark(X1),X2,X3)
mark#(U64(X1,X2,X3)) -> mark#(X1)
mark#(U64(X1,X2,X3)) -> a__U64#(mark(X1),X2,X3)
a__U64#(tt(),M,N) -> mark#(N)
mark#(plus(X1,X2)) -> mark#(X2)
mark#(plus(X1,X2)) -> mark#(X1)
mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
a__plus#(N,s(M)) -> a__U61#(a__isNat(M),M,N)
mark#(s(X)) -> mark#(X)
a__U64#(tt(),M,N) -> a__plus#(mark(N),mark(M))
TRS:
a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) -> a__U16(a__isNat(V2))
a__U16(tt()) -> tt()
a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) -> a__U23(a__isNat(V1))
a__U23(tt()) -> tt()
a__U31(tt(),V2) -> a__U32(a__isNatKind(V2))
a__U32(tt()) -> tt()
a__U41(tt()) -> tt()
a__U51(tt(),N) -> a__U52(a__isNatKind(N),N)
a__U52(tt(),N) -> mark(N)
a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
a__isNat(0()) -> tt()
a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) -> tt()
a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1))
a__plus(N,0()) -> a__U51(a__isNat(N),N)
a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) -> a__isNatKind(X)
mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) -> a__U15(mark(X1),X2)
mark(isNat(X)) -> a__isNat(X)
mark(U16(X)) -> a__U16(mark(X))
mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
mark(U23(X)) -> a__U23(mark(X))
mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
mark(U32(X)) -> a__U32(mark(X))
mark(U41(X)) -> a__U41(mark(X))
mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
mark(tt()) -> tt()
mark(s(X)) -> s(mark(X))
mark(0()) -> 0()
a__U11(X1,X2,X3) -> U11(X1,X2,X3)
a__U12(X1,X2,X3) -> U12(X1,X2,X3)
a__isNatKind(X) -> isNatKind(X)
a__U13(X1,X2,X3) -> U13(X1,X2,X3)
a__U14(X1,X2,X3) -> U14(X1,X2,X3)
a__U15(X1,X2) -> U15(X1,X2)
a__isNat(X) -> isNat(X)
a__U16(X) -> U16(X)
a__U21(X1,X2) -> U21(X1,X2)
a__U22(X1,X2) -> U22(X1,X2)
a__U23(X) -> U23(X)
a__U31(X1,X2) -> U31(X1,X2)
a__U32(X) -> U32(X)
a__U41(X) -> U41(X)
a__U51(X1,X2) -> U51(X1,X2)
a__U52(X1,X2) -> U52(X1,X2)
a__U61(X1,X2,X3) -> U61(X1,X2,X3)
a__U62(X1,X2,X3) -> U62(X1,X2,X3)
a__U63(X1,X2,X3) -> U63(X1,X2,X3)
a__U64(X1,X2,X3) -> U64(X1,X2,X3)
a__plus(X1,X2) -> plus(X1,X2)
Matrix Interpretation Processor:
dimension: 1
interpretation:
[a__plus#](x0, x1) = x0 + x1 + 1,
[a__U64#](x0, x1, x2) = x0 + x1 + x2,
[a__U63#](x0, x1, x2) = x0 + x1 + x2,
[a__U62#](x0, x1, x2) = x0 + x1 + x2,
[a__U61#](x0, x1, x2) = x0 + x1 + x2 + 1,
[mark#](x0) = x0,
[a__U52#](x0, x1) = x0 + x1 + 1,
[a__U51#](x0, x1) = x0 + x1 + 1,
[U64](x0, x1, x2) = x0 + x1 + x2 + 1,
[U63](x0, x1, x2) = x0 + x1 + x2 + 1,
[U62](x0, x1, x2) = x0 + x1 + x2 + 1,
[U61](x0, x1, x2) = x0 + x1 + x2 + 1,
[U52](x0, x1) = x0 + x1 + 1,
[U51](x0, x1) = x0 + x1 + 1,
[U41](x0) = x0,
[U32](x0) = x0,
[U31](x0, x1) = x0,
[U23](x0) = x0,
[U22](x0, x1) = x0,
[U21](x0, x1) = x0,
[U16](x0) = x0,
[isNat](x0) = 1,
[U15](x0, x1) = x0,
[U14](x0, x1, x2) = x0,
[U13](x0, x1, x2) = x0,
[isNatKind](x0) = 1,
[U12](x0, x1, x2) = x0,
[U11](x0, x1, x2) = x0,
[plus](x0, x1) = x0 + x1 + 1,
[0] = 1,
[s](x0) = x0 + 1,
[a__plus](x0, x1) = x0 + x1 + 1,
[a__U64](x0, x1, x2) = x0 + x1 + x2 + 1,
[a__U63](x0, x1, x2) = x0 + x1 + x2 + 1,
[a__U62](x0, x1, x2) = x0 + x1 + x2 + 1,
[a__U61](x0, x1, x2) = x0 + x1 + x2 + 1,
[mark](x0) = x0,
[a__U52](x0, x1) = x0 + x1 + 1,
[a__U51](x0, x1) = x0 + x1 + 1,
[a__U41](x0) = x0,
[a__U32](x0) = x0,
[a__U31](x0, x1) = x0,
[a__U23](x0) = x0,
[a__U22](x0, x1) = x0,
[a__U21](x0, x1) = x0,
[a__U16](x0) = x0,
[a__U15](x0, x1) = x0,
[a__isNat](x0) = 1,
[a__U14](x0, x1, x2) = x0,
[a__U13](x0, x1, x2) = x0,
[a__U12](x0, x1, x2) = x0,
[a__isNatKind](x0) = 1,
[a__U11](x0, x1, x2) = x0,
[tt] = 1
orientation:
a__plus#(N,0()) = N + 2 >= N + 2 = a__U51#(a__isNat(N),N)
a__U51#(tt(),N) = N + 2 >= N + 2 = a__U52#(a__isNatKind(N),N)
a__U52#(tt(),N) = N + 2 >= N = mark#(N)
mark#(U11(X1,X2,X3)) = X1 >= X1 = mark#(X1)
mark#(U12(X1,X2,X3)) = X1 >= X1 = mark#(X1)
mark#(U13(X1,X2,X3)) = X1 >= X1 = mark#(X1)
mark#(U14(X1,X2,X3)) = X1 >= X1 = mark#(X1)
mark#(U15(X1,X2)) = X1 >= X1 = mark#(X1)
mark#(U16(X)) = X >= X = mark#(X)
mark#(U21(X1,X2)) = X1 >= X1 = mark#(X1)
mark#(U22(X1,X2)) = X1 >= X1 = mark#(X1)
mark#(U23(X)) = X >= X = mark#(X)
mark#(U31(X1,X2)) = X1 >= X1 = mark#(X1)
mark#(U32(X)) = X >= X = mark#(X)
mark#(U41(X)) = X >= X = mark#(X)
mark#(U51(X1,X2)) = X1 + X2 + 1 >= X1 = mark#(X1)
mark#(U51(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__U51#(mark(X1),X2)
mark#(U52(X1,X2)) = X1 + X2 + 1 >= X1 = mark#(X1)
mark#(U52(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__U52#(mark(X1),X2)
mark#(U61(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 = mark#(X1)
mark#(U61(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = a__U61#(mark(X1),X2,X3)
a__U61#(tt(),M,N) = M + N + 2 >= M + N + 1 = a__U62#(a__isNatKind(M),M,N)
a__U62#(tt(),M,N) = M + N + 1 >= M + N + 1 = a__U63#(a__isNat(N),M,N)
a__U63#(tt(),M,N) = M + N + 1 >= M + N + 1 = a__U64#(a__isNatKind(N),M,N)
a__U64#(tt(),M,N) = M + N + 1 >= M = mark#(M)
mark#(U62(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 = mark#(X1)
mark#(U62(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 = a__U62#(mark(X1),X2,X3)
mark#(U63(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 = mark#(X1)
mark#(U63(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 = a__U63#(mark(X1),X2,X3)
mark#(U64(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 = mark#(X1)
mark#(U64(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 = a__U64#(mark(X1),X2,X3)
a__U64#(tt(),M,N) = M + N + 1 >= N = mark#(N)
mark#(plus(X1,X2)) = X1 + X2 + 1 >= X2 = mark#(X2)
mark#(plus(X1,X2)) = X1 + X2 + 1 >= X1 = mark#(X1)
mark#(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__plus#(mark(X1),mark(X2))
a__plus#(N,s(M)) = M + N + 2 >= M + N + 2 = a__U61#(a__isNat(M),M,N)
mark#(s(X)) = X + 1 >= X = mark#(X)
a__U64#(tt(),M,N) = M + N + 1 >= M + N + 1 = a__plus#(mark(N),mark(M))
a__U11(tt(),V1,V2) = 1 >= 1 = a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) = 1 >= 1 = a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) = 1 >= 1 = a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) = 1 >= 1 = a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) = 1 >= 1 = a__U16(a__isNat(V2))
a__U16(tt()) = 1 >= 1 = tt()
a__U21(tt(),V1) = 1 >= 1 = a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) = 1 >= 1 = a__U23(a__isNat(V1))
a__U23(tt()) = 1 >= 1 = tt()
a__U31(tt(),V2) = 1 >= 1 = a__U32(a__isNatKind(V2))
a__U32(tt()) = 1 >= 1 = tt()
a__U41(tt()) = 1 >= 1 = tt()
a__U51(tt(),N) = N + 2 >= N + 2 = a__U52(a__isNatKind(N),N)
a__U52(tt(),N) = N + 2 >= N = mark(N)
a__U61(tt(),M,N) = M + N + 2 >= M + N + 2 = a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) = M + N + 2 >= M + N + 2 = a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) = M + N + 2 >= M + N + 2 = a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) = M + N + 2 >= M + N + 2 = s(a__plus(mark(N),mark(M)))
a__isNat(0()) = 1 >= 1 = tt()
a__isNat(plus(V1,V2)) = 1 >= 1 = a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) = 1 >= 1 = a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) = 1 >= 1 = tt()
a__isNatKind(plus(V1,V2)) = 1 >= 1 = a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) = 1 >= 1 = a__U41(a__isNatKind(V1))
a__plus(N,0()) = N + 2 >= N + 2 = a__U51(a__isNat(N),N)
a__plus(N,s(M)) = M + N + 2 >= M + N + 2 = a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) = X1 >= X1 = a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) = X1 >= X1 = a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) = 1 >= 1 = a__isNatKind(X)
mark(U13(X1,X2,X3)) = X1 >= X1 = a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) = X1 >= X1 = a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) = X1 >= X1 = a__U15(mark(X1),X2)
mark(isNat(X)) = 1 >= 1 = a__isNat(X)
mark(U16(X)) = X >= X = a__U16(mark(X))
mark(U21(X1,X2)) = X1 >= X1 = a__U21(mark(X1),X2)
mark(U22(X1,X2)) = X1 >= X1 = a__U22(mark(X1),X2)
mark(U23(X)) = X >= X = a__U23(mark(X))
mark(U31(X1,X2)) = X1 >= X1 = a__U31(mark(X1),X2)
mark(U32(X)) = X >= X = a__U32(mark(X))
mark(U41(X)) = X >= X = a__U41(mark(X))
mark(U51(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__U51(mark(X1),X2)
mark(U52(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__plus(mark(X1),mark(X2))
mark(tt()) = 1 >= 1 = tt()
mark(s(X)) = X + 1 >= X + 1 = s(mark(X))
mark(0()) = 1 >= 1 = 0()
a__U11(X1,X2,X3) = X1 >= X1 = U11(X1,X2,X3)
a__U12(X1,X2,X3) = X1 >= X1 = U12(X1,X2,X3)
a__isNatKind(X) = 1 >= 1 = isNatKind(X)
a__U13(X1,X2,X3) = X1 >= X1 = U13(X1,X2,X3)
a__U14(X1,X2,X3) = X1 >= X1 = U14(X1,X2,X3)
a__U15(X1,X2) = X1 >= X1 = U15(X1,X2)
a__isNat(X) = 1 >= 1 = isNat(X)
a__U16(X) = X >= X = U16(X)
a__U21(X1,X2) = X1 >= X1 = U21(X1,X2)
a__U22(X1,X2) = X1 >= X1 = U22(X1,X2)
a__U23(X) = X >= X = U23(X)
a__U31(X1,X2) = X1 >= X1 = U31(X1,X2)
a__U32(X) = X >= X = U32(X)
a__U41(X) = X >= X = U41(X)
a__U51(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = U51(X1,X2)
a__U52(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = U52(X1,X2)
a__U61(X1,X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U61(X1,X2,X3)
a__U62(X1,X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U62(X1,X2,X3)
a__U63(X1,X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U63(X1,X2,X3)
a__U64(X1,X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U64(X1,X2,X3)
a__plus(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2)
problem:
DPs:
a__plus#(N,0()) -> a__U51#(a__isNat(N),N)
a__U51#(tt(),N) -> a__U52#(a__isNatKind(N),N)
mark#(U11(X1,X2,X3)) -> mark#(X1)
mark#(U12(X1,X2,X3)) -> mark#(X1)
mark#(U13(X1,X2,X3)) -> mark#(X1)
mark#(U14(X1,X2,X3)) -> mark#(X1)
mark#(U15(X1,X2)) -> mark#(X1)
mark#(U16(X)) -> mark#(X)
mark#(U21(X1,X2)) -> mark#(X1)
mark#(U22(X1,X2)) -> mark#(X1)
mark#(U23(X)) -> mark#(X)
mark#(U31(X1,X2)) -> mark#(X1)
mark#(U32(X)) -> mark#(X)
mark#(U41(X)) -> mark#(X)
mark#(U51(X1,X2)) -> a__U51#(mark(X1),X2)
mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2)
mark#(U61(X1,X2,X3)) -> a__U61#(mark(X1),X2,X3)
a__U62#(tt(),M,N) -> a__U63#(a__isNat(N),M,N)
a__U63#(tt(),M,N) -> a__U64#(a__isNatKind(N),M,N)
mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2))
a__plus#(N,s(M)) -> a__U61#(a__isNat(M),M,N)
a__U64#(tt(),M,N) -> a__plus#(mark(N),mark(M))
TRS:
a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) -> a__U16(a__isNat(V2))
a__U16(tt()) -> tt()
a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) -> a__U23(a__isNat(V1))
a__U23(tt()) -> tt()
a__U31(tt(),V2) -> a__U32(a__isNatKind(V2))
a__U32(tt()) -> tt()
a__U41(tt()) -> tt()
a__U51(tt(),N) -> a__U52(a__isNatKind(N),N)
a__U52(tt(),N) -> mark(N)
a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
a__isNat(0()) -> tt()
a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) -> tt()
a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1))
a__plus(N,0()) -> a__U51(a__isNat(N),N)
a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) -> a__isNatKind(X)
mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) -> a__U15(mark(X1),X2)
mark(isNat(X)) -> a__isNat(X)
mark(U16(X)) -> a__U16(mark(X))
mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
mark(U23(X)) -> a__U23(mark(X))
mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
mark(U32(X)) -> a__U32(mark(X))
mark(U41(X)) -> a__U41(mark(X))
mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
mark(tt()) -> tt()
mark(s(X)) -> s(mark(X))
mark(0()) -> 0()
a__U11(X1,X2,X3) -> U11(X1,X2,X3)
a__U12(X1,X2,X3) -> U12(X1,X2,X3)
a__isNatKind(X) -> isNatKind(X)
a__U13(X1,X2,X3) -> U13(X1,X2,X3)
a__U14(X1,X2,X3) -> U14(X1,X2,X3)
a__U15(X1,X2) -> U15(X1,X2)
a__isNat(X) -> isNat(X)
a__U16(X) -> U16(X)
a__U21(X1,X2) -> U21(X1,X2)
a__U22(X1,X2) -> U22(X1,X2)
a__U23(X) -> U23(X)
a__U31(X1,X2) -> U31(X1,X2)
a__U32(X) -> U32(X)
a__U41(X) -> U41(X)
a__U51(X1,X2) -> U51(X1,X2)
a__U52(X1,X2) -> U52(X1,X2)
a__U61(X1,X2,X3) -> U61(X1,X2,X3)
a__U62(X1,X2,X3) -> U62(X1,X2,X3)
a__U63(X1,X2,X3) -> U63(X1,X2,X3)
a__U64(X1,X2,X3) -> U64(X1,X2,X3)
a__plus(X1,X2) -> plus(X1,X2)
Matrix Interpretation Processor:
dimension: 1
interpretation:
[a__plus#](x0, x1) = 1,
[a__U64#](x0, x1, x2) = x1 + 1,
[a__U63#](x0, x1, x2) = x1 + 1,
[a__U62#](x0, x1, x2) = x1 + x2 + 1,
[a__U61#](x0, x1, x2) = 0,
[mark#](x0) = x0 + 1,
[a__U52#](x0, x1) = 1,
[a__U51#](x0, x1) = 1,
[U64](x0, x1, x2) = x1 + x2 + 1,
[U63](x0, x1, x2) = x1 + x2 + 1,
[U62](x0, x1, x2) = x1 + x2 + 1,
[U61](x0, x1, x2) = x1 + x2 + 1,
[U52](x0, x1) = x1,
[U51](x0, x1) = x1,
[U41](x0) = x0,
[U32](x0) = x0,
[U31](x0, x1) = x0,
[U23](x0) = x0,
[U22](x0, x1) = x0 + x1,
[U21](x0, x1) = x0 + x1,
[U16](x0) = x0,
[isNat](x0) = x0,
[U15](x0, x1) = x0 + x1,
[U14](x0, x1, x2) = x0 + x1 + x2 + 1,
[U13](x0, x1, x2) = x0 + x1 + x2 + 1,
[isNatKind](x0) = 0,
[U12](x0, x1, x2) = x0 + x1 + x2 + 1,
[U11](x0, x1, x2) = x0 + x1 + x2 + 1,
[plus](x0, x1) = x0 + x1 + 1,
[0] = 0,
[s](x0) = x0,
[a__plus](x0, x1) = x0 + x1 + 1,
[a__U64](x0, x1, x2) = x1 + x2 + 1,
[a__U63](x0, x1, x2) = x1 + x2 + 1,
[a__U62](x0, x1, x2) = x1 + x2 + 1,
[a__U61](x0, x1, x2) = x1 + x2 + 1,
[mark](x0) = x0,
[a__U52](x0, x1) = x1,
[a__U51](x0, x1) = x1,
[a__U41](x0) = x0,
[a__U32](x0) = x0,
[a__U31](x0, x1) = x0,
[a__U23](x0) = x0,
[a__U22](x0, x1) = x0 + x1,
[a__U21](x0, x1) = x0 + x1,
[a__U16](x0) = x0,
[a__U15](x0, x1) = x0 + x1,
[a__isNat](x0) = x0,
[a__U14](x0, x1, x2) = x0 + x1 + x2 + 1,
[a__U13](x0, x1, x2) = x0 + x1 + x2 + 1,
[a__U12](x0, x1, x2) = x0 + x1 + x2 + 1,
[a__isNatKind](x0) = 0,
[a__U11](x0, x1, x2) = x0 + x1 + x2 + 1,
[tt] = 0
orientation:
a__plus#(N,0()) = 1 >= 1 = a__U51#(a__isNat(N),N)
a__U51#(tt(),N) = 1 >= 1 = a__U52#(a__isNatKind(N),N)
mark#(U11(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X1 + 1 = mark#(X1)
mark#(U12(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X1 + 1 = mark#(X1)
mark#(U13(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X1 + 1 = mark#(X1)
mark#(U14(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X1 + 1 = mark#(X1)
mark#(U15(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = mark#(X1)
mark#(U16(X)) = X + 1 >= X + 1 = mark#(X)
mark#(U21(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = mark#(X1)
mark#(U22(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = mark#(X1)
mark#(U23(X)) = X + 1 >= X + 1 = mark#(X)
mark#(U31(X1,X2)) = X1 + 1 >= X1 + 1 = mark#(X1)
mark#(U32(X)) = X + 1 >= X + 1 = mark#(X)
mark#(U41(X)) = X + 1 >= X + 1 = mark#(X)
mark#(U51(X1,X2)) = X2 + 1 >= 1 = a__U51#(mark(X1),X2)
mark#(U52(X1,X2)) = X2 + 1 >= 1 = a__U52#(mark(X1),X2)
mark#(U61(X1,X2,X3)) = X2 + X3 + 2 >= 0 = a__U61#(mark(X1),X2,X3)
a__U62#(tt(),M,N) = M + N + 1 >= M + 1 = a__U63#(a__isNat(N),M,N)
a__U63#(tt(),M,N) = M + 1 >= M + 1 = a__U64#(a__isNatKind(N),M,N)
mark#(plus(X1,X2)) = X1 + X2 + 2 >= 1 = a__plus#(mark(X1),mark(X2))
a__plus#(N,s(M)) = 1 >= 0 = a__U61#(a__isNat(M),M,N)
a__U64#(tt(),M,N) = M + 1 >= 1 = a__plus#(mark(N),mark(M))
a__U11(tt(),V1,V2) = V1 + V2 + 1 >= V1 + V2 + 1 = a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) = V1 + V2 + 1 >= V1 + V2 + 1 = a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) = V1 + V2 + 1 >= V1 + V2 + 1 = a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) = V1 + V2 + 1 >= V1 + V2 = a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) = V2 >= V2 = a__U16(a__isNat(V2))
a__U16(tt()) = 0 >= 0 = tt()
a__U21(tt(),V1) = V1 >= V1 = a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) = V1 >= V1 = a__U23(a__isNat(V1))
a__U23(tt()) = 0 >= 0 = tt()
a__U31(tt(),V2) = 0 >= 0 = a__U32(a__isNatKind(V2))
a__U32(tt()) = 0 >= 0 = tt()
a__U41(tt()) = 0 >= 0 = tt()
a__U51(tt(),N) = N >= N = a__U52(a__isNatKind(N),N)
a__U52(tt(),N) = N >= N = mark(N)
a__U61(tt(),M,N) = M + N + 1 >= M + N + 1 = a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) = M + N + 1 >= M + N + 1 = a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) = M + N + 1 >= M + N + 1 = a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) = M + N + 1 >= M + N + 1 = s(a__plus(mark(N),mark(M)))
a__isNat(0()) = 0 >= 0 = tt()
a__isNat(plus(V1,V2)) = V1 + V2 + 1 >= V1 + V2 + 1 = a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) = V1 >= V1 = a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) = 0 >= 0 = tt()
a__isNatKind(plus(V1,V2)) = 0 >= 0 = a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) = 0 >= 0 = a__U41(a__isNatKind(V1))
a__plus(N,0()) = N + 1 >= N = a__U51(a__isNat(N),N)
a__plus(N,s(M)) = M + N + 1 >= M + N + 1 = a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) = 0 >= 0 = a__isNatKind(X)
mark(U13(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) = X1 + X2 >= X1 + X2 = a__U15(mark(X1),X2)
mark(isNat(X)) = X >= X = a__isNat(X)
mark(U16(X)) = X >= X = a__U16(mark(X))
mark(U21(X1,X2)) = X1 + X2 >= X1 + X2 = a__U21(mark(X1),X2)
mark(U22(X1,X2)) = X1 + X2 >= X1 + X2 = a__U22(mark(X1),X2)
mark(U23(X)) = X >= X = a__U23(mark(X))
mark(U31(X1,X2)) = X1 >= X1 = a__U31(mark(X1),X2)
mark(U32(X)) = X >= X = a__U32(mark(X))
mark(U41(X)) = X >= X = a__U41(mark(X))
mark(U51(X1,X2)) = X2 >= X2 = a__U51(mark(X1),X2)
mark(U52(X1,X2)) = X2 >= X2 = a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__plus(mark(X1),mark(X2))
mark(tt()) = 0 >= 0 = tt()
mark(s(X)) = X >= X = s(mark(X))
mark(0()) = 0 >= 0 = 0()
a__U11(X1,X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U11(X1,X2,X3)
a__U12(X1,X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U12(X1,X2,X3)
a__isNatKind(X) = 0 >= 0 = isNatKind(X)
a__U13(X1,X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U13(X1,X2,X3)
a__U14(X1,X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U14(X1,X2,X3)
a__U15(X1,X2) = X1 + X2 >= X1 + X2 = U15(X1,X2)
a__isNat(X) = X >= X = isNat(X)
a__U16(X) = X >= X = U16(X)
a__U21(X1,X2) = X1 + X2 >= X1 + X2 = U21(X1,X2)
a__U22(X1,X2) = X1 + X2 >= X1 + X2 = U22(X1,X2)
a__U23(X) = X >= X = U23(X)
a__U31(X1,X2) = X1 >= X1 = U31(X1,X2)
a__U32(X) = X >= X = U32(X)
a__U41(X) = X >= X = U41(X)
a__U51(X1,X2) = X2 >= X2 = U51(X1,X2)
a__U52(X1,X2) = X2 >= X2 = U52(X1,X2)
a__U61(X1,X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3)
a__U62(X1,X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3)
a__U63(X1,X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3)
a__U64(X1,X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3)
a__plus(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2)
problem:
DPs:
a__plus#(N,0()) -> a__U51#(a__isNat(N),N)
a__U51#(tt(),N) -> a__U52#(a__isNatKind(N),N)
mark#(U15(X1,X2)) -> mark#(X1)
mark#(U16(X)) -> mark#(X)
mark#(U21(X1,X2)) -> mark#(X1)
mark#(U22(X1,X2)) -> mark#(X1)
mark#(U23(X)) -> mark#(X)
mark#(U31(X1,X2)) -> mark#(X1)
mark#(U32(X)) -> mark#(X)
mark#(U41(X)) -> mark#(X)
mark#(U51(X1,X2)) -> a__U51#(mark(X1),X2)
mark#(U52(X1,X2)) -> a__U52#(mark(X1),X2)
a__U62#(tt(),M,N) -> a__U63#(a__isNat(N),M,N)
a__U63#(tt(),M,N) -> a__U64#(a__isNatKind(N),M,N)
a__U64#(tt(),M,N) -> a__plus#(mark(N),mark(M))
TRS:
a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) -> a__U16(a__isNat(V2))
a__U16(tt()) -> tt()
a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) -> a__U23(a__isNat(V1))
a__U23(tt()) -> tt()
a__U31(tt(),V2) -> a__U32(a__isNatKind(V2))
a__U32(tt()) -> tt()
a__U41(tt()) -> tt()
a__U51(tt(),N) -> a__U52(a__isNatKind(N),N)
a__U52(tt(),N) -> mark(N)
a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
a__isNat(0()) -> tt()
a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) -> tt()
a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1))
a__plus(N,0()) -> a__U51(a__isNat(N),N)
a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) -> a__isNatKind(X)
mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) -> a__U15(mark(X1),X2)
mark(isNat(X)) -> a__isNat(X)
mark(U16(X)) -> a__U16(mark(X))
mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
mark(U23(X)) -> a__U23(mark(X))
mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
mark(U32(X)) -> a__U32(mark(X))
mark(U41(X)) -> a__U41(mark(X))
mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
mark(tt()) -> tt()
mark(s(X)) -> s(mark(X))
mark(0()) -> 0()
a__U11(X1,X2,X3) -> U11(X1,X2,X3)
a__U12(X1,X2,X3) -> U12(X1,X2,X3)
a__isNatKind(X) -> isNatKind(X)
a__U13(X1,X2,X3) -> U13(X1,X2,X3)
a__U14(X1,X2,X3) -> U14(X1,X2,X3)
a__U15(X1,X2) -> U15(X1,X2)
a__isNat(X) -> isNat(X)
a__U16(X) -> U16(X)
a__U21(X1,X2) -> U21(X1,X2)
a__U22(X1,X2) -> U22(X1,X2)
a__U23(X) -> U23(X)
a__U31(X1,X2) -> U31(X1,X2)
a__U32(X) -> U32(X)
a__U41(X) -> U41(X)
a__U51(X1,X2) -> U51(X1,X2)
a__U52(X1,X2) -> U52(X1,X2)
a__U61(X1,X2,X3) -> U61(X1,X2,X3)
a__U62(X1,X2,X3) -> U62(X1,X2,X3)
a__U63(X1,X2,X3) -> U63(X1,X2,X3)
a__U64(X1,X2,X3) -> U64(X1,X2,X3)
a__plus(X1,X2) -> plus(X1,X2)
Matrix Interpretation Processor:
dimension: 1
interpretation:
[a__plus#](x0, x1) = 0,
[a__U64#](x0, x1, x2) = x0 + 1,
[a__U63#](x0, x1, x2) = x1 + x2 + 1,
[a__U62#](x0, x1, x2) = x1 + x2 + 1,
[mark#](x0) = 1,
[a__U52#](x0, x1) = 0,
[a__U51#](x0, x1) = 0,
[U64](x0, x1, x2) = x2,
[U63](x0, x1, x2) = x2,
[U62](x0, x1, x2) = x2,
[U61](x0, x1, x2) = x2,
[U52](x0, x1) = x1,
[U51](x0, x1) = x1,
[U41](x0) = 0,
[U32](x0) = 0,
[U31](x0, x1) = x0,
[U23](x0) = 0,
[U22](x0, x1) = 0,
[U21](x0, x1) = 1,
[U16](x0) = 0,
[isNat](x0) = x0 + 1,
[U15](x0, x1) = 1,
[U14](x0, x1, x2) = 1,
[U13](x0, x1, x2) = 1,
[isNatKind](x0) = x0,
[U12](x0, x1, x2) = 1,
[U11](x0, x1, x2) = x0 + 1,
[plus](x0, x1) = x0,
[0] = 0,
[s](x0) = x0,
[a__plus](x0, x1) = x0,
[a__U64](x0, x1, x2) = x2,
[a__U63](x0, x1, x2) = x2,
[a__U62](x0, x1, x2) = x2,
[a__U61](x0, x1, x2) = x2,
[mark](x0) = x0,
[a__U52](x0, x1) = x1,
[a__U51](x0, x1) = x1,
[a__U41](x0) = 0,
[a__U32](x0) = 0,
[a__U31](x0, x1) = x0,
[a__U23](x0) = 0,
[a__U22](x0, x1) = 0,
[a__U21](x0, x1) = 1,
[a__U16](x0) = 0,
[a__U15](x0, x1) = 1,
[a__isNat](x0) = x0 + 1,
[a__U14](x0, x1, x2) = 1,
[a__U13](x0, x1, x2) = 1,
[a__U12](x0, x1, x2) = 1,
[a__isNatKind](x0) = x0,
[a__U11](x0, x1, x2) = x0 + 1,
[tt] = 0
orientation:
a__plus#(N,0()) = 0 >= 0 = a__U51#(a__isNat(N),N)
a__U51#(tt(),N) = 0 >= 0 = a__U52#(a__isNatKind(N),N)
mark#(U15(X1,X2)) = 1 >= 1 = mark#(X1)
mark#(U16(X)) = 1 >= 1 = mark#(X)
mark#(U21(X1,X2)) = 1 >= 1 = mark#(X1)
mark#(U22(X1,X2)) = 1 >= 1 = mark#(X1)
mark#(U23(X)) = 1 >= 1 = mark#(X)
mark#(U31(X1,X2)) = 1 >= 1 = mark#(X1)
mark#(U32(X)) = 1 >= 1 = mark#(X)
mark#(U41(X)) = 1 >= 1 = mark#(X)
mark#(U51(X1,X2)) = 1 >= 0 = a__U51#(mark(X1),X2)
mark#(U52(X1,X2)) = 1 >= 0 = a__U52#(mark(X1),X2)
a__U62#(tt(),M,N) = M + N + 1 >= M + N + 1 = a__U63#(a__isNat(N),M,N)
a__U63#(tt(),M,N) = M + N + 1 >= N + 1 = a__U64#(a__isNatKind(N),M,N)
a__U64#(tt(),M,N) = 1 >= 0 = a__plus#(mark(N),mark(M))
a__U11(tt(),V1,V2) = 1 >= 1 = a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) = 1 >= 1 = a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) = 1 >= 1 = a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) = 1 >= 1 = a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) = 1 >= 0 = a__U16(a__isNat(V2))
a__U16(tt()) = 0 >= 0 = tt()
a__U21(tt(),V1) = 1 >= 0 = a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) = 0 >= 0 = a__U23(a__isNat(V1))
a__U23(tt()) = 0 >= 0 = tt()
a__U31(tt(),V2) = 0 >= 0 = a__U32(a__isNatKind(V2))
a__U32(tt()) = 0 >= 0 = tt()
a__U41(tt()) = 0 >= 0 = tt()
a__U51(tt(),N) = N >= N = a__U52(a__isNatKind(N),N)
a__U52(tt(),N) = N >= N = mark(N)
a__U61(tt(),M,N) = N >= N = a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) = N >= N = a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) = N >= N = a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) = N >= N = s(a__plus(mark(N),mark(M)))
a__isNat(0()) = 1 >= 0 = tt()
a__isNat(plus(V1,V2)) = V1 + 1 >= V1 + 1 = a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) = V1 + 1 >= 1 = a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) = 0 >= 0 = tt()
a__isNatKind(plus(V1,V2)) = V1 >= V1 = a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) = V1 >= 0 = a__U41(a__isNatKind(V1))
a__plus(N,0()) = N >= N = a__U51(a__isNat(N),N)
a__plus(N,s(M)) = N >= N = a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) = X1 + 1 >= X1 + 1 = a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) = 1 >= 1 = a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) = X >= X = a__isNatKind(X)
mark(U13(X1,X2,X3)) = 1 >= 1 = a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) = 1 >= 1 = a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) = 1 >= 1 = a__U15(mark(X1),X2)
mark(isNat(X)) = X + 1 >= X + 1 = a__isNat(X)
mark(U16(X)) = 0 >= 0 = a__U16(mark(X))
mark(U21(X1,X2)) = 1 >= 1 = a__U21(mark(X1),X2)
mark(U22(X1,X2)) = 0 >= 0 = a__U22(mark(X1),X2)
mark(U23(X)) = 0 >= 0 = a__U23(mark(X))
mark(U31(X1,X2)) = X1 >= X1 = a__U31(mark(X1),X2)
mark(U32(X)) = 0 >= 0 = a__U32(mark(X))
mark(U41(X)) = 0 >= 0 = a__U41(mark(X))
mark(U51(X1,X2)) = X2 >= X2 = a__U51(mark(X1),X2)
mark(U52(X1,X2)) = X2 >= X2 = a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) = X3 >= X3 = a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) = X3 >= X3 = a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) = X3 >= X3 = a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) = X3 >= X3 = a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) = X1 >= X1 = a__plus(mark(X1),mark(X2))
mark(tt()) = 0 >= 0 = tt()
mark(s(X)) = X >= X = s(mark(X))
mark(0()) = 0 >= 0 = 0()
a__U11(X1,X2,X3) = X1 + 1 >= X1 + 1 = U11(X1,X2,X3)
a__U12(X1,X2,X3) = 1 >= 1 = U12(X1,X2,X3)
a__isNatKind(X) = X >= X = isNatKind(X)
a__U13(X1,X2,X3) = 1 >= 1 = U13(X1,X2,X3)
a__U14(X1,X2,X3) = 1 >= 1 = U14(X1,X2,X3)
a__U15(X1,X2) = 1 >= 1 = U15(X1,X2)
a__isNat(X) = X + 1 >= X + 1 = isNat(X)
a__U16(X) = 0 >= 0 = U16(X)
a__U21(X1,X2) = 1 >= 1 = U21(X1,X2)
a__U22(X1,X2) = 0 >= 0 = U22(X1,X2)
a__U23(X) = 0 >= 0 = U23(X)
a__U31(X1,X2) = X1 >= X1 = U31(X1,X2)
a__U32(X) = 0 >= 0 = U32(X)
a__U41(X) = 0 >= 0 = U41(X)
a__U51(X1,X2) = X2 >= X2 = U51(X1,X2)
a__U52(X1,X2) = X2 >= X2 = U52(X1,X2)
a__U61(X1,X2,X3) = X3 >= X3 = U61(X1,X2,X3)
a__U62(X1,X2,X3) = X3 >= X3 = U62(X1,X2,X3)
a__U63(X1,X2,X3) = X3 >= X3 = U63(X1,X2,X3)
a__U64(X1,X2,X3) = X3 >= X3 = U64(X1,X2,X3)
a__plus(X1,X2) = X1 >= X1 = plus(X1,X2)
problem:
DPs:
a__plus#(N,0()) -> a__U51#(a__isNat(N),N)
a__U51#(tt(),N) -> a__U52#(a__isNatKind(N),N)
mark#(U15(X1,X2)) -> mark#(X1)
mark#(U16(X)) -> mark#(X)
mark#(U21(X1,X2)) -> mark#(X1)
mark#(U22(X1,X2)) -> mark#(X1)
mark#(U23(X)) -> mark#(X)
mark#(U31(X1,X2)) -> mark#(X1)
mark#(U32(X)) -> mark#(X)
mark#(U41(X)) -> mark#(X)
a__U62#(tt(),M,N) -> a__U63#(a__isNat(N),M,N)
a__U63#(tt(),M,N) -> a__U64#(a__isNatKind(N),M,N)
TRS:
a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) -> a__U16(a__isNat(V2))
a__U16(tt()) -> tt()
a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) -> a__U23(a__isNat(V1))
a__U23(tt()) -> tt()
a__U31(tt(),V2) -> a__U32(a__isNatKind(V2))
a__U32(tt()) -> tt()
a__U41(tt()) -> tt()
a__U51(tt(),N) -> a__U52(a__isNatKind(N),N)
a__U52(tt(),N) -> mark(N)
a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
a__isNat(0()) -> tt()
a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) -> tt()
a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1))
a__plus(N,0()) -> a__U51(a__isNat(N),N)
a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) -> a__isNatKind(X)
mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) -> a__U15(mark(X1),X2)
mark(isNat(X)) -> a__isNat(X)
mark(U16(X)) -> a__U16(mark(X))
mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
mark(U23(X)) -> a__U23(mark(X))
mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
mark(U32(X)) -> a__U32(mark(X))
mark(U41(X)) -> a__U41(mark(X))
mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
mark(tt()) -> tt()
mark(s(X)) -> s(mark(X))
mark(0()) -> 0()
a__U11(X1,X2,X3) -> U11(X1,X2,X3)
a__U12(X1,X2,X3) -> U12(X1,X2,X3)
a__isNatKind(X) -> isNatKind(X)
a__U13(X1,X2,X3) -> U13(X1,X2,X3)
a__U14(X1,X2,X3) -> U14(X1,X2,X3)
a__U15(X1,X2) -> U15(X1,X2)
a__isNat(X) -> isNat(X)
a__U16(X) -> U16(X)
a__U21(X1,X2) -> U21(X1,X2)
a__U22(X1,X2) -> U22(X1,X2)
a__U23(X) -> U23(X)
a__U31(X1,X2) -> U31(X1,X2)
a__U32(X) -> U32(X)
a__U41(X) -> U41(X)
a__U51(X1,X2) -> U51(X1,X2)
a__U52(X1,X2) -> U52(X1,X2)
a__U61(X1,X2,X3) -> U61(X1,X2,X3)
a__U62(X1,X2,X3) -> U62(X1,X2,X3)
a__U63(X1,X2,X3) -> U63(X1,X2,X3)
a__U64(X1,X2,X3) -> U64(X1,X2,X3)
a__plus(X1,X2) -> plus(X1,X2)
Matrix Interpretation Processor:
dimension: 1
interpretation:
[a__plus#](x0, x1) = 0,
[a__U64#](x0, x1, x2) = 0,
[a__U63#](x0, x1, x2) = 0,
[a__U62#](x0, x1, x2) = 1,
[mark#](x0) = 0,
[a__U52#](x0, x1) = 0,
[a__U51#](x0, x1) = 0,
[U64](x0, x1, x2) = 0,
[U63](x0, x1, x2) = 0,
[U62](x0, x1, x2) = 0,
[U61](x0, x1, x2) = 0,
[U52](x0, x1) = 0,
[U51](x0, x1) = 0,
[U41](x0) = 0,
[U32](x0) = 0,
[U31](x0, x1) = 0,
[U23](x0) = 0,
[U22](x0, x1) = 0,
[U21](x0, x1) = 0,
[U16](x0) = 0,
[isNat](x0) = 0,
[U15](x0, x1) = 0,
[U14](x0, x1, x2) = 0,
[U13](x0, x1, x2) = 0,
[isNatKind](x0) = 0,
[U12](x0, x1, x2) = 0,
[U11](x0, x1, x2) = 0,
[plus](x0, x1) = 0,
[0] = 0,
[s](x0) = 0,
[a__plus](x0, x1) = 0,
[a__U64](x0, x1, x2) = 0,
[a__U63](x0, x1, x2) = 0,
[a__U62](x0, x1, x2) = 0,
[a__U61](x0, x1, x2) = 0,
[mark](x0) = 0,
[a__U52](x0, x1) = 0,
[a__U51](x0, x1) = 0,
[a__U41](x0) = 0,
[a__U32](x0) = 0,
[a__U31](x0, x1) = 0,
[a__U23](x0) = 0,
[a__U22](x0, x1) = 0,
[a__U21](x0, x1) = 0,
[a__U16](x0) = 0,
[a__U15](x0, x1) = 0,
[a__isNat](x0) = 0,
[a__U14](x0, x1, x2) = 0,
[a__U13](x0, x1, x2) = 0,
[a__U12](x0, x1, x2) = 0,
[a__isNatKind](x0) = 0,
[a__U11](x0, x1, x2) = 0,
[tt] = 0
orientation:
a__plus#(N,0()) = 0 >= 0 = a__U51#(a__isNat(N),N)
a__U51#(tt(),N) = 0 >= 0 = a__U52#(a__isNatKind(N),N)
mark#(U15(X1,X2)) = 0 >= 0 = mark#(X1)
mark#(U16(X)) = 0 >= 0 = mark#(X)
mark#(U21(X1,X2)) = 0 >= 0 = mark#(X1)
mark#(U22(X1,X2)) = 0 >= 0 = mark#(X1)
mark#(U23(X)) = 0 >= 0 = mark#(X)
mark#(U31(X1,X2)) = 0 >= 0 = mark#(X1)
mark#(U32(X)) = 0 >= 0 = mark#(X)
mark#(U41(X)) = 0 >= 0 = mark#(X)
a__U62#(tt(),M,N) = 1 >= 0 = a__U63#(a__isNat(N),M,N)
a__U63#(tt(),M,N) = 0 >= 0 = a__U64#(a__isNatKind(N),M,N)
a__U11(tt(),V1,V2) = 0 >= 0 = a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) = 0 >= 0 = a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) = 0 >= 0 = a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) = 0 >= 0 = a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) = 0 >= 0 = a__U16(a__isNat(V2))
a__U16(tt()) = 0 >= 0 = tt()
a__U21(tt(),V1) = 0 >= 0 = a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) = 0 >= 0 = a__U23(a__isNat(V1))
a__U23(tt()) = 0 >= 0 = tt()
a__U31(tt(),V2) = 0 >= 0 = a__U32(a__isNatKind(V2))
a__U32(tt()) = 0 >= 0 = tt()
a__U41(tt()) = 0 >= 0 = tt()
a__U51(tt(),N) = 0 >= 0 = a__U52(a__isNatKind(N),N)
a__U52(tt(),N) = 0 >= 0 = mark(N)
a__U61(tt(),M,N) = 0 >= 0 = a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) = 0 >= 0 = a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) = 0 >= 0 = a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) = 0 >= 0 = s(a__plus(mark(N),mark(M)))
a__isNat(0()) = 0 >= 0 = tt()
a__isNat(plus(V1,V2)) = 0 >= 0 = a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) = 0 >= 0 = a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) = 0 >= 0 = tt()
a__isNatKind(plus(V1,V2)) = 0 >= 0 = a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) = 0 >= 0 = a__U41(a__isNatKind(V1))
a__plus(N,0()) = 0 >= 0 = a__U51(a__isNat(N),N)
a__plus(N,s(M)) = 0 >= 0 = a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) = 0 >= 0 = a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) = 0 >= 0 = a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) = 0 >= 0 = a__isNatKind(X)
mark(U13(X1,X2,X3)) = 0 >= 0 = a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) = 0 >= 0 = a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) = 0 >= 0 = a__U15(mark(X1),X2)
mark(isNat(X)) = 0 >= 0 = a__isNat(X)
mark(U16(X)) = 0 >= 0 = a__U16(mark(X))
mark(U21(X1,X2)) = 0 >= 0 = a__U21(mark(X1),X2)
mark(U22(X1,X2)) = 0 >= 0 = a__U22(mark(X1),X2)
mark(U23(X)) = 0 >= 0 = a__U23(mark(X))
mark(U31(X1,X2)) = 0 >= 0 = a__U31(mark(X1),X2)
mark(U32(X)) = 0 >= 0 = a__U32(mark(X))
mark(U41(X)) = 0 >= 0 = a__U41(mark(X))
mark(U51(X1,X2)) = 0 >= 0 = a__U51(mark(X1),X2)
mark(U52(X1,X2)) = 0 >= 0 = a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) = 0 >= 0 = a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) = 0 >= 0 = a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) = 0 >= 0 = a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) = 0 >= 0 = a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) = 0 >= 0 = a__plus(mark(X1),mark(X2))
mark(tt()) = 0 >= 0 = tt()
mark(s(X)) = 0 >= 0 = s(mark(X))
mark(0()) = 0 >= 0 = 0()
a__U11(X1,X2,X3) = 0 >= 0 = U11(X1,X2,X3)
a__U12(X1,X2,X3) = 0 >= 0 = U12(X1,X2,X3)
a__isNatKind(X) = 0 >= 0 = isNatKind(X)
a__U13(X1,X2,X3) = 0 >= 0 = U13(X1,X2,X3)
a__U14(X1,X2,X3) = 0 >= 0 = U14(X1,X2,X3)
a__U15(X1,X2) = 0 >= 0 = U15(X1,X2)
a__isNat(X) = 0 >= 0 = isNat(X)
a__U16(X) = 0 >= 0 = U16(X)
a__U21(X1,X2) = 0 >= 0 = U21(X1,X2)
a__U22(X1,X2) = 0 >= 0 = U22(X1,X2)
a__U23(X) = 0 >= 0 = U23(X)
a__U31(X1,X2) = 0 >= 0 = U31(X1,X2)
a__U32(X) = 0 >= 0 = U32(X)
a__U41(X) = 0 >= 0 = U41(X)
a__U51(X1,X2) = 0 >= 0 = U51(X1,X2)
a__U52(X1,X2) = 0 >= 0 = U52(X1,X2)
a__U61(X1,X2,X3) = 0 >= 0 = U61(X1,X2,X3)
a__U62(X1,X2,X3) = 0 >= 0 = U62(X1,X2,X3)
a__U63(X1,X2,X3) = 0 >= 0 = U63(X1,X2,X3)
a__U64(X1,X2,X3) = 0 >= 0 = U64(X1,X2,X3)
a__plus(X1,X2) = 0 >= 0 = plus(X1,X2)
problem:
DPs:
a__plus#(N,0()) -> a__U51#(a__isNat(N),N)
a__U51#(tt(),N) -> a__U52#(a__isNatKind(N),N)
mark#(U15(X1,X2)) -> mark#(X1)
mark#(U16(X)) -> mark#(X)
mark#(U21(X1,X2)) -> mark#(X1)
mark#(U22(X1,X2)) -> mark#(X1)
mark#(U23(X)) -> mark#(X)
mark#(U31(X1,X2)) -> mark#(X1)
mark#(U32(X)) -> mark#(X)
mark#(U41(X)) -> mark#(X)
a__U63#(tt(),M,N) -> a__U64#(a__isNatKind(N),M,N)
TRS:
a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) -> a__U16(a__isNat(V2))
a__U16(tt()) -> tt()
a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) -> a__U23(a__isNat(V1))
a__U23(tt()) -> tt()
a__U31(tt(),V2) -> a__U32(a__isNatKind(V2))
a__U32(tt()) -> tt()
a__U41(tt()) -> tt()
a__U51(tt(),N) -> a__U52(a__isNatKind(N),N)
a__U52(tt(),N) -> mark(N)
a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
a__isNat(0()) -> tt()
a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) -> tt()
a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1))
a__plus(N,0()) -> a__U51(a__isNat(N),N)
a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) -> a__isNatKind(X)
mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) -> a__U15(mark(X1),X2)
mark(isNat(X)) -> a__isNat(X)
mark(U16(X)) -> a__U16(mark(X))
mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
mark(U23(X)) -> a__U23(mark(X))
mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
mark(U32(X)) -> a__U32(mark(X))
mark(U41(X)) -> a__U41(mark(X))
mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
mark(tt()) -> tt()
mark(s(X)) -> s(mark(X))
mark(0()) -> 0()
a__U11(X1,X2,X3) -> U11(X1,X2,X3)
a__U12(X1,X2,X3) -> U12(X1,X2,X3)
a__isNatKind(X) -> isNatKind(X)
a__U13(X1,X2,X3) -> U13(X1,X2,X3)
a__U14(X1,X2,X3) -> U14(X1,X2,X3)
a__U15(X1,X2) -> U15(X1,X2)
a__isNat(X) -> isNat(X)
a__U16(X) -> U16(X)
a__U21(X1,X2) -> U21(X1,X2)
a__U22(X1,X2) -> U22(X1,X2)
a__U23(X) -> U23(X)
a__U31(X1,X2) -> U31(X1,X2)
a__U32(X) -> U32(X)
a__U41(X) -> U41(X)
a__U51(X1,X2) -> U51(X1,X2)
a__U52(X1,X2) -> U52(X1,X2)
a__U61(X1,X2,X3) -> U61(X1,X2,X3)
a__U62(X1,X2,X3) -> U62(X1,X2,X3)
a__U63(X1,X2,X3) -> U63(X1,X2,X3)
a__U64(X1,X2,X3) -> U64(X1,X2,X3)
a__plus(X1,X2) -> plus(X1,X2)
Matrix Interpretation Processor:
dimension: 1
interpretation:
[a__plus#](x0, x1) = x0,
[a__U64#](x0, x1, x2) = 0,
[a__U63#](x0, x1, x2) = 1,
[mark#](x0) = 0,
[a__U52#](x0, x1) = 0,
[a__U51#](x0, x1) = 0,
[U64](x0, x1, x2) = x0,
[U63](x0, x1, x2) = x0,
[U62](x0, x1, x2) = x0,
[U61](x0, x1, x2) = x0,
[U52](x0, x1) = 1,
[U51](x0, x1) = 1,
[U41](x0) = 0,
[U32](x0) = x0,
[U31](x0, x1) = 0,
[U23](x0) = 0,
[U22](x0, x1) = 0,
[U21](x0, x1) = 0,
[U16](x0) = 0,
[isNat](x0) = 0,
[U15](x0, x1) = x0,
[U14](x0, x1, x2) = x0,
[U13](x0, x1, x2) = x0,
[isNatKind](x0) = 0,
[U12](x0, x1, x2) = x0,
[U11](x0, x1, x2) = x0,
[plus](x0, x1) = x1,
[0] = 1,
[s](x0) = 0,
[a__plus](x0, x1) = x1,
[a__U64](x0, x1, x2) = x0,
[a__U63](x0, x1, x2) = x0,
[a__U62](x0, x1, x2) = x0,
[a__U61](x0, x1, x2) = x0,
[mark](x0) = 1,
[a__U52](x0, x1) = 1,
[a__U51](x0, x1) = 1,
[a__U41](x0) = 0,
[a__U32](x0) = x0,
[a__U31](x0, x1) = 0,
[a__U23](x0) = 0,
[a__U22](x0, x1) = 0,
[a__U21](x0, x1) = 0,
[a__U16](x0) = 0,
[a__U15](x0, x1) = x0,
[a__isNat](x0) = 0,
[a__U14](x0, x1, x2) = x0,
[a__U13](x0, x1, x2) = x0,
[a__U12](x0, x1, x2) = x0,
[a__isNatKind](x0) = 0,
[a__U11](x0, x1, x2) = x0,
[tt] = 0
orientation:
a__plus#(N,0()) = N >= 0 = a__U51#(a__isNat(N),N)
a__U51#(tt(),N) = 0 >= 0 = a__U52#(a__isNatKind(N),N)
mark#(U15(X1,X2)) = 0 >= 0 = mark#(X1)
mark#(U16(X)) = 0 >= 0 = mark#(X)
mark#(U21(X1,X2)) = 0 >= 0 = mark#(X1)
mark#(U22(X1,X2)) = 0 >= 0 = mark#(X1)
mark#(U23(X)) = 0 >= 0 = mark#(X)
mark#(U31(X1,X2)) = 0 >= 0 = mark#(X1)
mark#(U32(X)) = 0 >= 0 = mark#(X)
mark#(U41(X)) = 0 >= 0 = mark#(X)
a__U63#(tt(),M,N) = 1 >= 0 = a__U64#(a__isNatKind(N),M,N)
a__U11(tt(),V1,V2) = 0 >= 0 = a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) = 0 >= 0 = a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) = 0 >= 0 = a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) = 0 >= 0 = a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) = 0 >= 0 = a__U16(a__isNat(V2))
a__U16(tt()) = 0 >= 0 = tt()
a__U21(tt(),V1) = 0 >= 0 = a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) = 0 >= 0 = a__U23(a__isNat(V1))
a__U23(tt()) = 0 >= 0 = tt()
a__U31(tt(),V2) = 0 >= 0 = a__U32(a__isNatKind(V2))
a__U32(tt()) = 0 >= 0 = tt()
a__U41(tt()) = 0 >= 0 = tt()
a__U51(tt(),N) = 1 >= 1 = a__U52(a__isNatKind(N),N)
a__U52(tt(),N) = 1 >= 1 = mark(N)
a__U61(tt(),M,N) = 0 >= 0 = a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) = 0 >= 0 = a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) = 0 >= 0 = a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) = 0 >= 0 = s(a__plus(mark(N),mark(M)))
a__isNat(0()) = 0 >= 0 = tt()
a__isNat(plus(V1,V2)) = 0 >= 0 = a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) = 0 >= 0 = a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) = 0 >= 0 = tt()
a__isNatKind(plus(V1,V2)) = 0 >= 0 = a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) = 0 >= 0 = a__U41(a__isNatKind(V1))
a__plus(N,0()) = 1 >= 1 = a__U51(a__isNat(N),N)
a__plus(N,s(M)) = 0 >= 0 = a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) = 1 >= 1 = a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) = 1 >= 1 = a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) = 1 >= 0 = a__isNatKind(X)
mark(U13(X1,X2,X3)) = 1 >= 1 = a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) = 1 >= 1 = a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) = 1 >= 1 = a__U15(mark(X1),X2)
mark(isNat(X)) = 1 >= 0 = a__isNat(X)
mark(U16(X)) = 1 >= 0 = a__U16(mark(X))
mark(U21(X1,X2)) = 1 >= 0 = a__U21(mark(X1),X2)
mark(U22(X1,X2)) = 1 >= 0 = a__U22(mark(X1),X2)
mark(U23(X)) = 1 >= 0 = a__U23(mark(X))
mark(U31(X1,X2)) = 1 >= 0 = a__U31(mark(X1),X2)
mark(U32(X)) = 1 >= 1 = a__U32(mark(X))
mark(U41(X)) = 1 >= 0 = a__U41(mark(X))
mark(U51(X1,X2)) = 1 >= 1 = a__U51(mark(X1),X2)
mark(U52(X1,X2)) = 1 >= 1 = a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) = 1 >= 1 = a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) = 1 >= 1 = a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) = 1 >= 1 = a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) = 1 >= 1 = a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) = 1 >= 1 = a__plus(mark(X1),mark(X2))
mark(tt()) = 1 >= 0 = tt()
mark(s(X)) = 1 >= 0 = s(mark(X))
mark(0()) = 1 >= 1 = 0()
a__U11(X1,X2,X3) = X1 >= X1 = U11(X1,X2,X3)
a__U12(X1,X2,X3) = X1 >= X1 = U12(X1,X2,X3)
a__isNatKind(X) = 0 >= 0 = isNatKind(X)
a__U13(X1,X2,X3) = X1 >= X1 = U13(X1,X2,X3)
a__U14(X1,X2,X3) = X1 >= X1 = U14(X1,X2,X3)
a__U15(X1,X2) = X1 >= X1 = U15(X1,X2)
a__isNat(X) = 0 >= 0 = isNat(X)
a__U16(X) = 0 >= 0 = U16(X)
a__U21(X1,X2) = 0 >= 0 = U21(X1,X2)
a__U22(X1,X2) = 0 >= 0 = U22(X1,X2)
a__U23(X) = 0 >= 0 = U23(X)
a__U31(X1,X2) = 0 >= 0 = U31(X1,X2)
a__U32(X) = X >= X = U32(X)
a__U41(X) = 0 >= 0 = U41(X)
a__U51(X1,X2) = 1 >= 1 = U51(X1,X2)
a__U52(X1,X2) = 1 >= 1 = U52(X1,X2)
a__U61(X1,X2,X3) = X1 >= X1 = U61(X1,X2,X3)
a__U62(X1,X2,X3) = X1 >= X1 = U62(X1,X2,X3)
a__U63(X1,X2,X3) = X1 >= X1 = U63(X1,X2,X3)
a__U64(X1,X2,X3) = X1 >= X1 = U64(X1,X2,X3)
a__plus(X1,X2) = X2 >= X2 = plus(X1,X2)
problem:
DPs:
a__plus#(N,0()) -> a__U51#(a__isNat(N),N)
a__U51#(tt(),N) -> a__U52#(a__isNatKind(N),N)
mark#(U15(X1,X2)) -> mark#(X1)
mark#(U16(X)) -> mark#(X)
mark#(U21(X1,X2)) -> mark#(X1)
mark#(U22(X1,X2)) -> mark#(X1)
mark#(U23(X)) -> mark#(X)
mark#(U31(X1,X2)) -> mark#(X1)
mark#(U32(X)) -> mark#(X)
mark#(U41(X)) -> mark#(X)
TRS:
a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) -> a__U16(a__isNat(V2))
a__U16(tt()) -> tt()
a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) -> a__U23(a__isNat(V1))
a__U23(tt()) -> tt()
a__U31(tt(),V2) -> a__U32(a__isNatKind(V2))
a__U32(tt()) -> tt()
a__U41(tt()) -> tt()
a__U51(tt(),N) -> a__U52(a__isNatKind(N),N)
a__U52(tt(),N) -> mark(N)
a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
a__isNat(0()) -> tt()
a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) -> tt()
a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1))
a__plus(N,0()) -> a__U51(a__isNat(N),N)
a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) -> a__isNatKind(X)
mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) -> a__U15(mark(X1),X2)
mark(isNat(X)) -> a__isNat(X)
mark(U16(X)) -> a__U16(mark(X))
mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
mark(U23(X)) -> a__U23(mark(X))
mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
mark(U32(X)) -> a__U32(mark(X))
mark(U41(X)) -> a__U41(mark(X))
mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
mark(tt()) -> tt()
mark(s(X)) -> s(mark(X))
mark(0()) -> 0()
a__U11(X1,X2,X3) -> U11(X1,X2,X3)
a__U12(X1,X2,X3) -> U12(X1,X2,X3)
a__isNatKind(X) -> isNatKind(X)
a__U13(X1,X2,X3) -> U13(X1,X2,X3)
a__U14(X1,X2,X3) -> U14(X1,X2,X3)
a__U15(X1,X2) -> U15(X1,X2)
a__isNat(X) -> isNat(X)
a__U16(X) -> U16(X)
a__U21(X1,X2) -> U21(X1,X2)
a__U22(X1,X2) -> U22(X1,X2)
a__U23(X) -> U23(X)
a__U31(X1,X2) -> U31(X1,X2)
a__U32(X) -> U32(X)
a__U41(X) -> U41(X)
a__U51(X1,X2) -> U51(X1,X2)
a__U52(X1,X2) -> U52(X1,X2)
a__U61(X1,X2,X3) -> U61(X1,X2,X3)
a__U62(X1,X2,X3) -> U62(X1,X2,X3)
a__U63(X1,X2,X3) -> U63(X1,X2,X3)
a__U64(X1,X2,X3) -> U64(X1,X2,X3)
a__plus(X1,X2) -> plus(X1,X2)
Matrix Interpretation Processor:
dimension: 1
interpretation:
[a__plus#](x0, x1) = 1,
[mark#](x0) = 1,
[a__U52#](x0, x1) = 0,
[a__U51#](x0, x1) = 1,
[U64](x0, x1, x2) = x0,
[U63](x0, x1, x2) = 0,
[U62](x0, x1, x2) = x0,
[U61](x0, x1, x2) = 0,
[U52](x0, x1) = 1,
[U51](x0, x1) = 1,
[U41](x0) = 0,
[U32](x0) = x0,
[U31](x0, x1) = x0,
[U23](x0) = 0,
[U22](x0, x1) = x0,
[U21](x0, x1) = 0,
[U16](x0) = 0,
[isNat](x0) = 1,
[U15](x0, x1) = 0,
[U14](x0, x1, x2) = 0,
[U13](x0, x1, x2) = 0,
[isNatKind](x0) = 0,
[U12](x0, x1, x2) = 0,
[U11](x0, x1, x2) = 1,
[plus](x0, x1) = x1,
[0] = 1,
[s](x0) = 0,
[a__plus](x0, x1) = x1,
[a__U64](x0, x1, x2) = x0,
[a__U63](x0, x1, x2) = 0,
[a__U62](x0, x1, x2) = x0,
[a__U61](x0, x1, x2) = 0,
[mark](x0) = 1,
[a__U52](x0, x1) = 1,
[a__U51](x0, x1) = 1,
[a__U41](x0) = 0,
[a__U32](x0) = x0,
[a__U31](x0, x1) = x0,
[a__U23](x0) = 0,
[a__U22](x0, x1) = x0,
[a__U21](x0, x1) = 0,
[a__U16](x0) = 0,
[a__U15](x0, x1) = 0,
[a__isNat](x0) = 1,
[a__U14](x0, x1, x2) = 0,
[a__U13](x0, x1, x2) = 0,
[a__U12](x0, x1, x2) = 0,
[a__isNatKind](x0) = 0,
[a__U11](x0, x1, x2) = 1,
[tt] = 0
orientation:
a__plus#(N,0()) = 1 >= 1 = a__U51#(a__isNat(N),N)
a__U51#(tt(),N) = 1 >= 0 = a__U52#(a__isNatKind(N),N)
mark#(U15(X1,X2)) = 1 >= 1 = mark#(X1)
mark#(U16(X)) = 1 >= 1 = mark#(X)
mark#(U21(X1,X2)) = 1 >= 1 = mark#(X1)
mark#(U22(X1,X2)) = 1 >= 1 = mark#(X1)
mark#(U23(X)) = 1 >= 1 = mark#(X)
mark#(U31(X1,X2)) = 1 >= 1 = mark#(X1)
mark#(U32(X)) = 1 >= 1 = mark#(X)
mark#(U41(X)) = 1 >= 1 = mark#(X)
a__U11(tt(),V1,V2) = 1 >= 0 = a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) = 0 >= 0 = a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) = 0 >= 0 = a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) = 0 >= 0 = a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) = 0 >= 0 = a__U16(a__isNat(V2))
a__U16(tt()) = 0 >= 0 = tt()
a__U21(tt(),V1) = 0 >= 0 = a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) = 0 >= 0 = a__U23(a__isNat(V1))
a__U23(tt()) = 0 >= 0 = tt()
a__U31(tt(),V2) = 0 >= 0 = a__U32(a__isNatKind(V2))
a__U32(tt()) = 0 >= 0 = tt()
a__U41(tt()) = 0 >= 0 = tt()
a__U51(tt(),N) = 1 >= 1 = a__U52(a__isNatKind(N),N)
a__U52(tt(),N) = 1 >= 1 = mark(N)
a__U61(tt(),M,N) = 0 >= 0 = a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) = 0 >= 0 = a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) = 0 >= 0 = a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) = 0 >= 0 = s(a__plus(mark(N),mark(M)))
a__isNat(0()) = 1 >= 0 = tt()
a__isNat(plus(V1,V2)) = 1 >= 1 = a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) = 1 >= 0 = a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) = 0 >= 0 = tt()
a__isNatKind(plus(V1,V2)) = 0 >= 0 = a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) = 0 >= 0 = a__U41(a__isNatKind(V1))
a__plus(N,0()) = 1 >= 1 = a__U51(a__isNat(N),N)
a__plus(N,s(M)) = 0 >= 0 = a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) = 1 >= 1 = a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) = 1 >= 0 = a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) = 1 >= 0 = a__isNatKind(X)
mark(U13(X1,X2,X3)) = 1 >= 0 = a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) = 1 >= 0 = a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) = 1 >= 0 = a__U15(mark(X1),X2)
mark(isNat(X)) = 1 >= 1 = a__isNat(X)
mark(U16(X)) = 1 >= 0 = a__U16(mark(X))
mark(U21(X1,X2)) = 1 >= 0 = a__U21(mark(X1),X2)
mark(U22(X1,X2)) = 1 >= 1 = a__U22(mark(X1),X2)
mark(U23(X)) = 1 >= 0 = a__U23(mark(X))
mark(U31(X1,X2)) = 1 >= 1 = a__U31(mark(X1),X2)
mark(U32(X)) = 1 >= 1 = a__U32(mark(X))
mark(U41(X)) = 1 >= 0 = a__U41(mark(X))
mark(U51(X1,X2)) = 1 >= 1 = a__U51(mark(X1),X2)
mark(U52(X1,X2)) = 1 >= 1 = a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) = 1 >= 0 = a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) = 1 >= 1 = a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) = 1 >= 0 = a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) = 1 >= 1 = a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) = 1 >= 1 = a__plus(mark(X1),mark(X2))
mark(tt()) = 1 >= 0 = tt()
mark(s(X)) = 1 >= 0 = s(mark(X))
mark(0()) = 1 >= 1 = 0()
a__U11(X1,X2,X3) = 1 >= 1 = U11(X1,X2,X3)
a__U12(X1,X2,X3) = 0 >= 0 = U12(X1,X2,X3)
a__isNatKind(X) = 0 >= 0 = isNatKind(X)
a__U13(X1,X2,X3) = 0 >= 0 = U13(X1,X2,X3)
a__U14(X1,X2,X3) = 0 >= 0 = U14(X1,X2,X3)
a__U15(X1,X2) = 0 >= 0 = U15(X1,X2)
a__isNat(X) = 1 >= 1 = isNat(X)
a__U16(X) = 0 >= 0 = U16(X)
a__U21(X1,X2) = 0 >= 0 = U21(X1,X2)
a__U22(X1,X2) = X1 >= X1 = U22(X1,X2)
a__U23(X) = 0 >= 0 = U23(X)
a__U31(X1,X2) = X1 >= X1 = U31(X1,X2)
a__U32(X) = X >= X = U32(X)
a__U41(X) = 0 >= 0 = U41(X)
a__U51(X1,X2) = 1 >= 1 = U51(X1,X2)
a__U52(X1,X2) = 1 >= 1 = U52(X1,X2)
a__U61(X1,X2,X3) = 0 >= 0 = U61(X1,X2,X3)
a__U62(X1,X2,X3) = X1 >= X1 = U62(X1,X2,X3)
a__U63(X1,X2,X3) = 0 >= 0 = U63(X1,X2,X3)
a__U64(X1,X2,X3) = X1 >= X1 = U64(X1,X2,X3)
a__plus(X1,X2) = X2 >= X2 = plus(X1,X2)
problem:
DPs:
a__plus#(N,0()) -> a__U51#(a__isNat(N),N)
mark#(U15(X1,X2)) -> mark#(X1)
mark#(U16(X)) -> mark#(X)
mark#(U21(X1,X2)) -> mark#(X1)
mark#(U22(X1,X2)) -> mark#(X1)
mark#(U23(X)) -> mark#(X)
mark#(U31(X1,X2)) -> mark#(X1)
mark#(U32(X)) -> mark#(X)
mark#(U41(X)) -> mark#(X)
TRS:
a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) -> a__U16(a__isNat(V2))
a__U16(tt()) -> tt()
a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) -> a__U23(a__isNat(V1))
a__U23(tt()) -> tt()
a__U31(tt(),V2) -> a__U32(a__isNatKind(V2))
a__U32(tt()) -> tt()
a__U41(tt()) -> tt()
a__U51(tt(),N) -> a__U52(a__isNatKind(N),N)
a__U52(tt(),N) -> mark(N)
a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
a__isNat(0()) -> tt()
a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) -> tt()
a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1))
a__plus(N,0()) -> a__U51(a__isNat(N),N)
a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) -> a__isNatKind(X)
mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) -> a__U15(mark(X1),X2)
mark(isNat(X)) -> a__isNat(X)
mark(U16(X)) -> a__U16(mark(X))
mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
mark(U23(X)) -> a__U23(mark(X))
mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
mark(U32(X)) -> a__U32(mark(X))
mark(U41(X)) -> a__U41(mark(X))
mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
mark(tt()) -> tt()
mark(s(X)) -> s(mark(X))
mark(0()) -> 0()
a__U11(X1,X2,X3) -> U11(X1,X2,X3)
a__U12(X1,X2,X3) -> U12(X1,X2,X3)
a__isNatKind(X) -> isNatKind(X)
a__U13(X1,X2,X3) -> U13(X1,X2,X3)
a__U14(X1,X2,X3) -> U14(X1,X2,X3)
a__U15(X1,X2) -> U15(X1,X2)
a__isNat(X) -> isNat(X)
a__U16(X) -> U16(X)
a__U21(X1,X2) -> U21(X1,X2)
a__U22(X1,X2) -> U22(X1,X2)
a__U23(X) -> U23(X)
a__U31(X1,X2) -> U31(X1,X2)
a__U32(X) -> U32(X)
a__U41(X) -> U41(X)
a__U51(X1,X2) -> U51(X1,X2)
a__U52(X1,X2) -> U52(X1,X2)
a__U61(X1,X2,X3) -> U61(X1,X2,X3)
a__U62(X1,X2,X3) -> U62(X1,X2,X3)
a__U63(X1,X2,X3) -> U63(X1,X2,X3)
a__U64(X1,X2,X3) -> U64(X1,X2,X3)
a__plus(X1,X2) -> plus(X1,X2)
Matrix Interpretation Processor:
dimension: 1
interpretation:
[a__plus#](x0, x1) = x1,
[mark#](x0) = 0,
[a__U51#](x0, x1) = 0,
[U64](x0, x1, x2) = 0,
[U63](x0, x1, x2) = x2,
[U62](x0, x1, x2) = x2,
[U61](x0, x1, x2) = x2 + 1,
[U52](x0, x1) = x1 + 1,
[U51](x0, x1) = x1 + 1,
[U41](x0) = x0,
[U32](x0) = x0,
[U31](x0, x1) = 1,
[U23](x0) = 0,
[U22](x0, x1) = 1,
[U21](x0, x1) = 1,
[U16](x0) = 0,
[isNat](x0) = x0 + 1,
[U15](x0, x1) = 0,
[U14](x0, x1, x2) = 0,
[U13](x0, x1, x2) = x0,
[isNatKind](x0) = 1,
[U12](x0, x1, x2) = 1,
[U11](x0, x1, x2) = x1 + 1,
[plus](x0, x1) = x0 + 1,
[0] = 1,
[s](x0) = 0,
[a__plus](x0, x1) = x0 + 1,
[a__U64](x0, x1, x2) = 0,
[a__U63](x0, x1, x2) = x2,
[a__U62](x0, x1, x2) = x2,
[a__U61](x0, x1, x2) = x2 + 1,
[mark](x0) = x0,
[a__U52](x0, x1) = x1 + 1,
[a__U51](x0, x1) = x1 + 1,
[a__U41](x0) = x0,
[a__U32](x0) = x0,
[a__U31](x0, x1) = 1,
[a__U23](x0) = 0,
[a__U22](x0, x1) = 1,
[a__U21](x0, x1) = 1,
[a__U16](x0) = 0,
[a__U15](x0, x1) = 0,
[a__isNat](x0) = x0 + 1,
[a__U14](x0, x1, x2) = 0,
[a__U13](x0, x1, x2) = x0,
[a__U12](x0, x1, x2) = 1,
[a__isNatKind](x0) = 1,
[a__U11](x0, x1, x2) = x1 + 1,
[tt] = 0
orientation:
a__plus#(N,0()) = 1 >= 0 = a__U51#(a__isNat(N),N)
mark#(U15(X1,X2)) = 0 >= 0 = mark#(X1)
mark#(U16(X)) = 0 >= 0 = mark#(X)
mark#(U21(X1,X2)) = 0 >= 0 = mark#(X1)
mark#(U22(X1,X2)) = 0 >= 0 = mark#(X1)
mark#(U23(X)) = 0 >= 0 = mark#(X)
mark#(U31(X1,X2)) = 0 >= 0 = mark#(X1)
mark#(U32(X)) = 0 >= 0 = mark#(X)
mark#(U41(X)) = 0 >= 0 = mark#(X)
a__U11(tt(),V1,V2) = V1 + 1 >= 1 = a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) = 1 >= 1 = a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) = 0 >= 0 = a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) = 0 >= 0 = a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) = 0 >= 0 = a__U16(a__isNat(V2))
a__U16(tt()) = 0 >= 0 = tt()
a__U21(tt(),V1) = 1 >= 1 = a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) = 1 >= 0 = a__U23(a__isNat(V1))
a__U23(tt()) = 0 >= 0 = tt()
a__U31(tt(),V2) = 1 >= 1 = a__U32(a__isNatKind(V2))
a__U32(tt()) = 0 >= 0 = tt()
a__U41(tt()) = 0 >= 0 = tt()
a__U51(tt(),N) = N + 1 >= N + 1 = a__U52(a__isNatKind(N),N)
a__U52(tt(),N) = N + 1 >= N = mark(N)
a__U61(tt(),M,N) = N + 1 >= N = a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) = N >= N = a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) = N >= 0 = a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) = 0 >= 0 = s(a__plus(mark(N),mark(M)))
a__isNat(0()) = 2 >= 0 = tt()
a__isNat(plus(V1,V2)) = V1 + 2 >= V1 + 1 = a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) = 1 >= 1 = a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) = 1 >= 0 = tt()
a__isNatKind(plus(V1,V2)) = 1 >= 1 = a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) = 1 >= 1 = a__U41(a__isNatKind(V1))
a__plus(N,0()) = N + 1 >= N + 1 = a__U51(a__isNat(N),N)
a__plus(N,s(M)) = N + 1 >= N + 1 = a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) = X2 + 1 >= X2 + 1 = a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) = 1 >= 1 = a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) = 1 >= 1 = a__isNatKind(X)
mark(U13(X1,X2,X3)) = X1 >= X1 = a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) = 0 >= 0 = a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) = 0 >= 0 = a__U15(mark(X1),X2)
mark(isNat(X)) = X + 1 >= X + 1 = a__isNat(X)
mark(U16(X)) = 0 >= 0 = a__U16(mark(X))
mark(U21(X1,X2)) = 1 >= 1 = a__U21(mark(X1),X2)
mark(U22(X1,X2)) = 1 >= 1 = a__U22(mark(X1),X2)
mark(U23(X)) = 0 >= 0 = a__U23(mark(X))
mark(U31(X1,X2)) = 1 >= 1 = a__U31(mark(X1),X2)
mark(U32(X)) = X >= X = a__U32(mark(X))
mark(U41(X)) = X >= X = a__U41(mark(X))
mark(U51(X1,X2)) = X2 + 1 >= X2 + 1 = a__U51(mark(X1),X2)
mark(U52(X1,X2)) = X2 + 1 >= X2 + 1 = a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) = X3 + 1 >= X3 + 1 = a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) = X3 >= X3 = a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) = X3 >= X3 = a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) = 0 >= 0 = a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) = X1 + 1 >= X1 + 1 = a__plus(mark(X1),mark(X2))
mark(tt()) = 0 >= 0 = tt()
mark(s(X)) = 0 >= 0 = s(mark(X))
mark(0()) = 1 >= 1 = 0()
a__U11(X1,X2,X3) = X2 + 1 >= X2 + 1 = U11(X1,X2,X3)
a__U12(X1,X2,X3) = 1 >= 1 = U12(X1,X2,X3)
a__isNatKind(X) = 1 >= 1 = isNatKind(X)
a__U13(X1,X2,X3) = X1 >= X1 = U13(X1,X2,X3)
a__U14(X1,X2,X3) = 0 >= 0 = U14(X1,X2,X3)
a__U15(X1,X2) = 0 >= 0 = U15(X1,X2)
a__isNat(X) = X + 1 >= X + 1 = isNat(X)
a__U16(X) = 0 >= 0 = U16(X)
a__U21(X1,X2) = 1 >= 1 = U21(X1,X2)
a__U22(X1,X2) = 1 >= 1 = U22(X1,X2)
a__U23(X) = 0 >= 0 = U23(X)
a__U31(X1,X2) = 1 >= 1 = U31(X1,X2)
a__U32(X) = X >= X = U32(X)
a__U41(X) = X >= X = U41(X)
a__U51(X1,X2) = X2 + 1 >= X2 + 1 = U51(X1,X2)
a__U52(X1,X2) = X2 + 1 >= X2 + 1 = U52(X1,X2)
a__U61(X1,X2,X3) = X3 + 1 >= X3 + 1 = U61(X1,X2,X3)
a__U62(X1,X2,X3) = X3 >= X3 = U62(X1,X2,X3)
a__U63(X1,X2,X3) = X3 >= X3 = U63(X1,X2,X3)
a__U64(X1,X2,X3) = 0 >= 0 = U64(X1,X2,X3)
a__plus(X1,X2) = X1 + 1 >= X1 + 1 = plus(X1,X2)
problem:
DPs:
mark#(U15(X1,X2)) -> mark#(X1)
mark#(U16(X)) -> mark#(X)
mark#(U21(X1,X2)) -> mark#(X1)
mark#(U22(X1,X2)) -> mark#(X1)
mark#(U23(X)) -> mark#(X)
mark#(U31(X1,X2)) -> mark#(X1)
mark#(U32(X)) -> mark#(X)
mark#(U41(X)) -> mark#(X)
TRS:
a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) -> a__U16(a__isNat(V2))
a__U16(tt()) -> tt()
a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) -> a__U23(a__isNat(V1))
a__U23(tt()) -> tt()
a__U31(tt(),V2) -> a__U32(a__isNatKind(V2))
a__U32(tt()) -> tt()
a__U41(tt()) -> tt()
a__U51(tt(),N) -> a__U52(a__isNatKind(N),N)
a__U52(tt(),N) -> mark(N)
a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
a__isNat(0()) -> tt()
a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) -> tt()
a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1))
a__plus(N,0()) -> a__U51(a__isNat(N),N)
a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) -> a__isNatKind(X)
mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) -> a__U15(mark(X1),X2)
mark(isNat(X)) -> a__isNat(X)
mark(U16(X)) -> a__U16(mark(X))
mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
mark(U23(X)) -> a__U23(mark(X))
mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
mark(U32(X)) -> a__U32(mark(X))
mark(U41(X)) -> a__U41(mark(X))
mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
mark(tt()) -> tt()
mark(s(X)) -> s(mark(X))
mark(0()) -> 0()
a__U11(X1,X2,X3) -> U11(X1,X2,X3)
a__U12(X1,X2,X3) -> U12(X1,X2,X3)
a__isNatKind(X) -> isNatKind(X)
a__U13(X1,X2,X3) -> U13(X1,X2,X3)
a__U14(X1,X2,X3) -> U14(X1,X2,X3)
a__U15(X1,X2) -> U15(X1,X2)
a__isNat(X) -> isNat(X)
a__U16(X) -> U16(X)
a__U21(X1,X2) -> U21(X1,X2)
a__U22(X1,X2) -> U22(X1,X2)
a__U23(X) -> U23(X)
a__U31(X1,X2) -> U31(X1,X2)
a__U32(X) -> U32(X)
a__U41(X) -> U41(X)
a__U51(X1,X2) -> U51(X1,X2)
a__U52(X1,X2) -> U52(X1,X2)
a__U61(X1,X2,X3) -> U61(X1,X2,X3)
a__U62(X1,X2,X3) -> U62(X1,X2,X3)
a__U63(X1,X2,X3) -> U63(X1,X2,X3)
a__U64(X1,X2,X3) -> U64(X1,X2,X3)
a__plus(X1,X2) -> plus(X1,X2)
Matrix Interpretation Processor:
dimension: 1
interpretation:
[mark#](x0) = x0,
[U64](x0, x1, x2) = x1 + x2 + 1,
[U63](x0, x1, x2) = x1 + x2 + 1,
[U62](x0, x1, x2) = x1 + x2 + 1,
[U61](x0, x1, x2) = x1 + x2 + 1,
[U52](x0, x1) = x1,
[U51](x0, x1) = x1,
[U41](x0) = x0,
[U32](x0) = x0,
[U31](x0, x1) = x0,
[U23](x0) = x0,
[U22](x0, x1) = x0 + x1,
[U21](x0, x1) = x0 + x1 + 1,
[U16](x0) = x0,
[isNat](x0) = x0,
[U15](x0, x1) = x0 + x1,
[U14](x0, x1, x2) = x1 + x2,
[U13](x0, x1, x2) = x1 + x2,
[isNatKind](x0) = 0,
[U12](x0, x1, x2) = x1 + x2,
[U11](x0, x1, x2) = x1 + x2,
[plus](x0, x1) = x0 + x1,
[0] = 0,
[s](x0) = x0 + 1,
[a__plus](x0, x1) = x0 + x1,
[a__U64](x0, x1, x2) = x1 + x2 + 1,
[a__U63](x0, x1, x2) = x1 + x2 + 1,
[a__U62](x0, x1, x2) = x1 + x2 + 1,
[a__U61](x0, x1, x2) = x1 + x2 + 1,
[mark](x0) = x0,
[a__U52](x0, x1) = x1,
[a__U51](x0, x1) = x1,
[a__U41](x0) = x0,
[a__U32](x0) = x0,
[a__U31](x0, x1) = x0,
[a__U23](x0) = x0,
[a__U22](x0, x1) = x0 + x1,
[a__U21](x0, x1) = x0 + x1 + 1,
[a__U16](x0) = x0,
[a__U15](x0, x1) = x0 + x1,
[a__isNat](x0) = x0,
[a__U14](x0, x1, x2) = x1 + x2,
[a__U13](x0, x1, x2) = x1 + x2,
[a__U12](x0, x1, x2) = x1 + x2,
[a__isNatKind](x0) = 0,
[a__U11](x0, x1, x2) = x1 + x2,
[tt] = 0
orientation:
mark#(U15(X1,X2)) = X1 + X2 >= X1 = mark#(X1)
mark#(U16(X)) = X >= X = mark#(X)
mark#(U21(X1,X2)) = X1 + X2 + 1 >= X1 = mark#(X1)
mark#(U22(X1,X2)) = X1 + X2 >= X1 = mark#(X1)
mark#(U23(X)) = X >= X = mark#(X)
mark#(U31(X1,X2)) = X1 >= X1 = mark#(X1)
mark#(U32(X)) = X >= X = mark#(X)
mark#(U41(X)) = X >= X = mark#(X)
a__U11(tt(),V1,V2) = V1 + V2 >= V1 + V2 = a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) = V1 + V2 >= V1 + V2 = a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) = V1 + V2 >= V1 + V2 = a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) = V1 + V2 >= V1 + V2 = a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) = V2 >= V2 = a__U16(a__isNat(V2))
a__U16(tt()) = 0 >= 0 = tt()
a__U21(tt(),V1) = V1 + 1 >= V1 = a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) = V1 >= V1 = a__U23(a__isNat(V1))
a__U23(tt()) = 0 >= 0 = tt()
a__U31(tt(),V2) = 0 >= 0 = a__U32(a__isNatKind(V2))
a__U32(tt()) = 0 >= 0 = tt()
a__U41(tt()) = 0 >= 0 = tt()
a__U51(tt(),N) = N >= N = a__U52(a__isNatKind(N),N)
a__U52(tt(),N) = N >= N = mark(N)
a__U61(tt(),M,N) = M + N + 1 >= M + N + 1 = a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) = M + N + 1 >= M + N + 1 = a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) = M + N + 1 >= M + N + 1 = a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) = M + N + 1 >= M + N + 1 = s(a__plus(mark(N),mark(M)))
a__isNat(0()) = 0 >= 0 = tt()
a__isNat(plus(V1,V2)) = V1 + V2 >= V1 + V2 = a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) = V1 + 1 >= V1 + 1 = a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) = 0 >= 0 = tt()
a__isNatKind(plus(V1,V2)) = 0 >= 0 = a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) = 0 >= 0 = a__U41(a__isNatKind(V1))
a__plus(N,0()) = N >= N = a__U51(a__isNat(N),N)
a__plus(N,s(M)) = M + N + 1 >= M + N + 1 = a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) = X2 + X3 >= X2 + X3 = a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) = X2 + X3 >= X2 + X3 = a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) = 0 >= 0 = a__isNatKind(X)
mark(U13(X1,X2,X3)) = X2 + X3 >= X2 + X3 = a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) = X2 + X3 >= X2 + X3 = a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) = X1 + X2 >= X1 + X2 = a__U15(mark(X1),X2)
mark(isNat(X)) = X >= X = a__isNat(X)
mark(U16(X)) = X >= X = a__U16(mark(X))
mark(U21(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__U21(mark(X1),X2)
mark(U22(X1,X2)) = X1 + X2 >= X1 + X2 = a__U22(mark(X1),X2)
mark(U23(X)) = X >= X = a__U23(mark(X))
mark(U31(X1,X2)) = X1 >= X1 = a__U31(mark(X1),X2)
mark(U32(X)) = X >= X = a__U32(mark(X))
mark(U41(X)) = X >= X = a__U41(mark(X))
mark(U51(X1,X2)) = X2 >= X2 = a__U51(mark(X1),X2)
mark(U52(X1,X2)) = X2 >= X2 = a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) = X1 + X2 >= X1 + X2 = a__plus(mark(X1),mark(X2))
mark(tt()) = 0 >= 0 = tt()
mark(s(X)) = X + 1 >= X + 1 = s(mark(X))
mark(0()) = 0 >= 0 = 0()
a__U11(X1,X2,X3) = X2 + X3 >= X2 + X3 = U11(X1,X2,X3)
a__U12(X1,X2,X3) = X2 + X3 >= X2 + X3 = U12(X1,X2,X3)
a__isNatKind(X) = 0 >= 0 = isNatKind(X)
a__U13(X1,X2,X3) = X2 + X3 >= X2 + X3 = U13(X1,X2,X3)
a__U14(X1,X2,X3) = X2 + X3 >= X2 + X3 = U14(X1,X2,X3)
a__U15(X1,X2) = X1 + X2 >= X1 + X2 = U15(X1,X2)
a__isNat(X) = X >= X = isNat(X)
a__U16(X) = X >= X = U16(X)
a__U21(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = U21(X1,X2)
a__U22(X1,X2) = X1 + X2 >= X1 + X2 = U22(X1,X2)
a__U23(X) = X >= X = U23(X)
a__U31(X1,X2) = X1 >= X1 = U31(X1,X2)
a__U32(X) = X >= X = U32(X)
a__U41(X) = X >= X = U41(X)
a__U51(X1,X2) = X2 >= X2 = U51(X1,X2)
a__U52(X1,X2) = X2 >= X2 = U52(X1,X2)
a__U61(X1,X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3)
a__U62(X1,X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3)
a__U63(X1,X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3)
a__U64(X1,X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3)
a__plus(X1,X2) = X1 + X2 >= X1 + X2 = plus(X1,X2)
problem:
DPs:
mark#(U15(X1,X2)) -> mark#(X1)
mark#(U16(X)) -> mark#(X)
mark#(U22(X1,X2)) -> mark#(X1)
mark#(U23(X)) -> mark#(X)
mark#(U31(X1,X2)) -> mark#(X1)
mark#(U32(X)) -> mark#(X)
mark#(U41(X)) -> mark#(X)
TRS:
a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) -> a__U16(a__isNat(V2))
a__U16(tt()) -> tt()
a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) -> a__U23(a__isNat(V1))
a__U23(tt()) -> tt()
a__U31(tt(),V2) -> a__U32(a__isNatKind(V2))
a__U32(tt()) -> tt()
a__U41(tt()) -> tt()
a__U51(tt(),N) -> a__U52(a__isNatKind(N),N)
a__U52(tt(),N) -> mark(N)
a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
a__isNat(0()) -> tt()
a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) -> tt()
a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1))
a__plus(N,0()) -> a__U51(a__isNat(N),N)
a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) -> a__isNatKind(X)
mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) -> a__U15(mark(X1),X2)
mark(isNat(X)) -> a__isNat(X)
mark(U16(X)) -> a__U16(mark(X))
mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
mark(U23(X)) -> a__U23(mark(X))
mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
mark(U32(X)) -> a__U32(mark(X))
mark(U41(X)) -> a__U41(mark(X))
mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
mark(tt()) -> tt()
mark(s(X)) -> s(mark(X))
mark(0()) -> 0()
a__U11(X1,X2,X3) -> U11(X1,X2,X3)
a__U12(X1,X2,X3) -> U12(X1,X2,X3)
a__isNatKind(X) -> isNatKind(X)
a__U13(X1,X2,X3) -> U13(X1,X2,X3)
a__U14(X1,X2,X3) -> U14(X1,X2,X3)
a__U15(X1,X2) -> U15(X1,X2)
a__isNat(X) -> isNat(X)
a__U16(X) -> U16(X)
a__U21(X1,X2) -> U21(X1,X2)
a__U22(X1,X2) -> U22(X1,X2)
a__U23(X) -> U23(X)
a__U31(X1,X2) -> U31(X1,X2)
a__U32(X) -> U32(X)
a__U41(X) -> U41(X)
a__U51(X1,X2) -> U51(X1,X2)
a__U52(X1,X2) -> U52(X1,X2)
a__U61(X1,X2,X3) -> U61(X1,X2,X3)
a__U62(X1,X2,X3) -> U62(X1,X2,X3)
a__U63(X1,X2,X3) -> U63(X1,X2,X3)
a__U64(X1,X2,X3) -> U64(X1,X2,X3)
a__plus(X1,X2) -> plus(X1,X2)
Matrix Interpretation Processor:
dimension: 1
interpretation:
[mark#](x0) = x0,
[U64](x0, x1, x2) = x1 + x2 + 1,
[U63](x0, x1, x2) = x1 + x2 + 1,
[U62](x0, x1, x2) = x1 + x2 + 1,
[U61](x0, x1, x2) = x1 + x2 + 1,
[U52](x0, x1) = x1,
[U51](x0, x1) = x1,
[U41](x0) = x0,
[U32](x0) = x0,
[U31](x0, x1) = x0,
[U23](x0) = x0,
[U22](x0, x1) = x0 + x1,
[U21](x0, x1) = x1,
[U16](x0) = x0 + 1,
[isNat](x0) = x0,
[U15](x0, x1) = x0 + x1 + 1,
[U14](x0, x1, x2) = x1 + x2 + 1,
[U13](x0, x1, x2) = x1 + x2 + 1,
[isNatKind](x0) = 0,
[U12](x0, x1, x2) = x1 + x2 + 1,
[U11](x0, x1, x2) = x1 + x2 + 1,
[plus](x0, x1) = x0 + x1 + 1,
[0] = 0,
[s](x0) = x0,
[a__plus](x0, x1) = x0 + x1 + 1,
[a__U64](x0, x1, x2) = x1 + x2 + 1,
[a__U63](x0, x1, x2) = x1 + x2 + 1,
[a__U62](x0, x1, x2) = x1 + x2 + 1,
[a__U61](x0, x1, x2) = x1 + x2 + 1,
[mark](x0) = x0,
[a__U52](x0, x1) = x1,
[a__U51](x0, x1) = x1,
[a__U41](x0) = x0,
[a__U32](x0) = x0,
[a__U31](x0, x1) = x0,
[a__U23](x0) = x0,
[a__U22](x0, x1) = x0 + x1,
[a__U21](x0, x1) = x1,
[a__U16](x0) = x0 + 1,
[a__U15](x0, x1) = x0 + x1 + 1,
[a__isNat](x0) = x0,
[a__U14](x0, x1, x2) = x1 + x2 + 1,
[a__U13](x0, x1, x2) = x1 + x2 + 1,
[a__U12](x0, x1, x2) = x1 + x2 + 1,
[a__isNatKind](x0) = 0,
[a__U11](x0, x1, x2) = x1 + x2 + 1,
[tt] = 0
orientation:
mark#(U15(X1,X2)) = X1 + X2 + 1 >= X1 = mark#(X1)
mark#(U16(X)) = X + 1 >= X = mark#(X)
mark#(U22(X1,X2)) = X1 + X2 >= X1 = mark#(X1)
mark#(U23(X)) = X >= X = mark#(X)
mark#(U31(X1,X2)) = X1 >= X1 = mark#(X1)
mark#(U32(X)) = X >= X = mark#(X)
mark#(U41(X)) = X >= X = mark#(X)
a__U11(tt(),V1,V2) = V1 + V2 + 1 >= V1 + V2 + 1 = a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) = V1 + V2 + 1 >= V1 + V2 + 1 = a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) = V1 + V2 + 1 >= V1 + V2 + 1 = a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) = V1 + V2 + 1 >= V1 + V2 + 1 = a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) = V2 + 1 >= V2 + 1 = a__U16(a__isNat(V2))
a__U16(tt()) = 1 >= 0 = tt()
a__U21(tt(),V1) = V1 >= V1 = a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) = V1 >= V1 = a__U23(a__isNat(V1))
a__U23(tt()) = 0 >= 0 = tt()
a__U31(tt(),V2) = 0 >= 0 = a__U32(a__isNatKind(V2))
a__U32(tt()) = 0 >= 0 = tt()
a__U41(tt()) = 0 >= 0 = tt()
a__U51(tt(),N) = N >= N = a__U52(a__isNatKind(N),N)
a__U52(tt(),N) = N >= N = mark(N)
a__U61(tt(),M,N) = M + N + 1 >= M + N + 1 = a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) = M + N + 1 >= M + N + 1 = a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) = M + N + 1 >= M + N + 1 = a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) = M + N + 1 >= M + N + 1 = s(a__plus(mark(N),mark(M)))
a__isNat(0()) = 0 >= 0 = tt()
a__isNat(plus(V1,V2)) = V1 + V2 + 1 >= V1 + V2 + 1 = a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) = V1 >= V1 = a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) = 0 >= 0 = tt()
a__isNatKind(plus(V1,V2)) = 0 >= 0 = a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) = 0 >= 0 = a__U41(a__isNatKind(V1))
a__plus(N,0()) = N + 1 >= N = a__U51(a__isNat(N),N)
a__plus(N,s(M)) = M + N + 1 >= M + N + 1 = a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) = 0 >= 0 = a__isNatKind(X)
mark(U13(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__U15(mark(X1),X2)
mark(isNat(X)) = X >= X = a__isNat(X)
mark(U16(X)) = X + 1 >= X + 1 = a__U16(mark(X))
mark(U21(X1,X2)) = X2 >= X2 = a__U21(mark(X1),X2)
mark(U22(X1,X2)) = X1 + X2 >= X1 + X2 = a__U22(mark(X1),X2)
mark(U23(X)) = X >= X = a__U23(mark(X))
mark(U31(X1,X2)) = X1 >= X1 = a__U31(mark(X1),X2)
mark(U32(X)) = X >= X = a__U32(mark(X))
mark(U41(X)) = X >= X = a__U41(mark(X))
mark(U51(X1,X2)) = X2 >= X2 = a__U51(mark(X1),X2)
mark(U52(X1,X2)) = X2 >= X2 = a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__plus(mark(X1),mark(X2))
mark(tt()) = 0 >= 0 = tt()
mark(s(X)) = X >= X = s(mark(X))
mark(0()) = 0 >= 0 = 0()
a__U11(X1,X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U11(X1,X2,X3)
a__U12(X1,X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U12(X1,X2,X3)
a__isNatKind(X) = 0 >= 0 = isNatKind(X)
a__U13(X1,X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U13(X1,X2,X3)
a__U14(X1,X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U14(X1,X2,X3)
a__U15(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = U15(X1,X2)
a__isNat(X) = X >= X = isNat(X)
a__U16(X) = X + 1 >= X + 1 = U16(X)
a__U21(X1,X2) = X2 >= X2 = U21(X1,X2)
a__U22(X1,X2) = X1 + X2 >= X1 + X2 = U22(X1,X2)
a__U23(X) = X >= X = U23(X)
a__U31(X1,X2) = X1 >= X1 = U31(X1,X2)
a__U32(X) = X >= X = U32(X)
a__U41(X) = X >= X = U41(X)
a__U51(X1,X2) = X2 >= X2 = U51(X1,X2)
a__U52(X1,X2) = X2 >= X2 = U52(X1,X2)
a__U61(X1,X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3)
a__U62(X1,X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3)
a__U63(X1,X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3)
a__U64(X1,X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3)
a__plus(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2)
problem:
DPs:
mark#(U22(X1,X2)) -> mark#(X1)
mark#(U23(X)) -> mark#(X)
mark#(U31(X1,X2)) -> mark#(X1)
mark#(U32(X)) -> mark#(X)
mark#(U41(X)) -> mark#(X)
TRS:
a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) -> a__U16(a__isNat(V2))
a__U16(tt()) -> tt()
a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) -> a__U23(a__isNat(V1))
a__U23(tt()) -> tt()
a__U31(tt(),V2) -> a__U32(a__isNatKind(V2))
a__U32(tt()) -> tt()
a__U41(tt()) -> tt()
a__U51(tt(),N) -> a__U52(a__isNatKind(N),N)
a__U52(tt(),N) -> mark(N)
a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
a__isNat(0()) -> tt()
a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) -> tt()
a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1))
a__plus(N,0()) -> a__U51(a__isNat(N),N)
a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) -> a__isNatKind(X)
mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) -> a__U15(mark(X1),X2)
mark(isNat(X)) -> a__isNat(X)
mark(U16(X)) -> a__U16(mark(X))
mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
mark(U23(X)) -> a__U23(mark(X))
mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
mark(U32(X)) -> a__U32(mark(X))
mark(U41(X)) -> a__U41(mark(X))
mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
mark(tt()) -> tt()
mark(s(X)) -> s(mark(X))
mark(0()) -> 0()
a__U11(X1,X2,X3) -> U11(X1,X2,X3)
a__U12(X1,X2,X3) -> U12(X1,X2,X3)
a__isNatKind(X) -> isNatKind(X)
a__U13(X1,X2,X3) -> U13(X1,X2,X3)
a__U14(X1,X2,X3) -> U14(X1,X2,X3)
a__U15(X1,X2) -> U15(X1,X2)
a__isNat(X) -> isNat(X)
a__U16(X) -> U16(X)
a__U21(X1,X2) -> U21(X1,X2)
a__U22(X1,X2) -> U22(X1,X2)
a__U23(X) -> U23(X)
a__U31(X1,X2) -> U31(X1,X2)
a__U32(X) -> U32(X)
a__U41(X) -> U41(X)
a__U51(X1,X2) -> U51(X1,X2)
a__U52(X1,X2) -> U52(X1,X2)
a__U61(X1,X2,X3) -> U61(X1,X2,X3)
a__U62(X1,X2,X3) -> U62(X1,X2,X3)
a__U63(X1,X2,X3) -> U63(X1,X2,X3)
a__U64(X1,X2,X3) -> U64(X1,X2,X3)
a__plus(X1,X2) -> plus(X1,X2)
Matrix Interpretation Processor:
dimension: 1
interpretation:
[mark#](x0) = x0 + 1,
[U64](x0, x1, x2) = x1 + x2 + 1,
[U63](x0, x1, x2) = x1 + x2 + 1,
[U62](x0, x1, x2) = x1 + x2 + 1,
[U61](x0, x1, x2) = x1 + x2 + 1,
[U52](x0, x1) = x1 + 1,
[U51](x0, x1) = x1 + 1,
[U41](x0) = x0,
[U32](x0) = x0,
[U31](x0, x1) = x0,
[U23](x0) = x0 + 1,
[U22](x0, x1) = x0 + x1 + 1,
[U21](x0, x1) = x1 + 1,
[U16](x0) = 0,
[isNat](x0) = x0,
[U15](x0, x1) = 0,
[U14](x0, x1, x2) = 0,
[U13](x0, x1, x2) = 0,
[isNatKind](x0) = 0,
[U12](x0, x1, x2) = 0,
[U11](x0, x1, x2) = 0,
[plus](x0, x1) = x0 + x1,
[0] = 1,
[s](x0) = x0 + 1,
[a__plus](x0, x1) = x0 + x1,
[a__U64](x0, x1, x2) = x1 + x2 + 1,
[a__U63](x0, x1, x2) = x1 + x2 + 1,
[a__U62](x0, x1, x2) = x1 + x2 + 1,
[a__U61](x0, x1, x2) = x1 + x2 + 1,
[mark](x0) = x0,
[a__U52](x0, x1) = x1 + 1,
[a__U51](x0, x1) = x1 + 1,
[a__U41](x0) = x0,
[a__U32](x0) = x0,
[a__U31](x0, x1) = x0,
[a__U23](x0) = x0 + 1,
[a__U22](x0, x1) = x0 + x1 + 1,
[a__U21](x0, x1) = x1 + 1,
[a__U16](x0) = 0,
[a__U15](x0, x1) = 0,
[a__isNat](x0) = x0,
[a__U14](x0, x1, x2) = 0,
[a__U13](x0, x1, x2) = 0,
[a__U12](x0, x1, x2) = 0,
[a__isNatKind](x0) = 0,
[a__U11](x0, x1, x2) = 0,
[tt] = 0
orientation:
mark#(U22(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = mark#(X1)
mark#(U23(X)) = X + 2 >= X + 1 = mark#(X)
mark#(U31(X1,X2)) = X1 + 1 >= X1 + 1 = mark#(X1)
mark#(U32(X)) = X + 1 >= X + 1 = mark#(X)
mark#(U41(X)) = X + 1 >= X + 1 = mark#(X)
a__U11(tt(),V1,V2) = 0 >= 0 = a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) = 0 >= 0 = a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) = 0 >= 0 = a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) = 0 >= 0 = a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) = 0 >= 0 = a__U16(a__isNat(V2))
a__U16(tt()) = 0 >= 0 = tt()
a__U21(tt(),V1) = V1 + 1 >= V1 + 1 = a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) = V1 + 1 >= V1 + 1 = a__U23(a__isNat(V1))
a__U23(tt()) = 1 >= 0 = tt()
a__U31(tt(),V2) = 0 >= 0 = a__U32(a__isNatKind(V2))
a__U32(tt()) = 0 >= 0 = tt()
a__U41(tt()) = 0 >= 0 = tt()
a__U51(tt(),N) = N + 1 >= N + 1 = a__U52(a__isNatKind(N),N)
a__U52(tt(),N) = N + 1 >= N = mark(N)
a__U61(tt(),M,N) = M + N + 1 >= M + N + 1 = a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) = M + N + 1 >= M + N + 1 = a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) = M + N + 1 >= M + N + 1 = a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) = M + N + 1 >= M + N + 1 = s(a__plus(mark(N),mark(M)))
a__isNat(0()) = 1 >= 0 = tt()
a__isNat(plus(V1,V2)) = V1 + V2 >= 0 = a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) = V1 + 1 >= V1 + 1 = a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) = 0 >= 0 = tt()
a__isNatKind(plus(V1,V2)) = 0 >= 0 = a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) = 0 >= 0 = a__U41(a__isNatKind(V1))
a__plus(N,0()) = N + 1 >= N + 1 = a__U51(a__isNat(N),N)
a__plus(N,s(M)) = M + N + 1 >= M + N + 1 = a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) = 0 >= 0 = a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) = 0 >= 0 = a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) = 0 >= 0 = a__isNatKind(X)
mark(U13(X1,X2,X3)) = 0 >= 0 = a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) = 0 >= 0 = a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) = 0 >= 0 = a__U15(mark(X1),X2)
mark(isNat(X)) = X >= X = a__isNat(X)
mark(U16(X)) = 0 >= 0 = a__U16(mark(X))
mark(U21(X1,X2)) = X2 + 1 >= X2 + 1 = a__U21(mark(X1),X2)
mark(U22(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__U22(mark(X1),X2)
mark(U23(X)) = X + 1 >= X + 1 = a__U23(mark(X))
mark(U31(X1,X2)) = X1 >= X1 = a__U31(mark(X1),X2)
mark(U32(X)) = X >= X = a__U32(mark(X))
mark(U41(X)) = X >= X = a__U41(mark(X))
mark(U51(X1,X2)) = X2 + 1 >= X2 + 1 = a__U51(mark(X1),X2)
mark(U52(X1,X2)) = X2 + 1 >= X2 + 1 = a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) = X1 + X2 >= X1 + X2 = a__plus(mark(X1),mark(X2))
mark(tt()) = 0 >= 0 = tt()
mark(s(X)) = X + 1 >= X + 1 = s(mark(X))
mark(0()) = 1 >= 1 = 0()
a__U11(X1,X2,X3) = 0 >= 0 = U11(X1,X2,X3)
a__U12(X1,X2,X3) = 0 >= 0 = U12(X1,X2,X3)
a__isNatKind(X) = 0 >= 0 = isNatKind(X)
a__U13(X1,X2,X3) = 0 >= 0 = U13(X1,X2,X3)
a__U14(X1,X2,X3) = 0 >= 0 = U14(X1,X2,X3)
a__U15(X1,X2) = 0 >= 0 = U15(X1,X2)
a__isNat(X) = X >= X = isNat(X)
a__U16(X) = 0 >= 0 = U16(X)
a__U21(X1,X2) = X2 + 1 >= X2 + 1 = U21(X1,X2)
a__U22(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = U22(X1,X2)
a__U23(X) = X + 1 >= X + 1 = U23(X)
a__U31(X1,X2) = X1 >= X1 = U31(X1,X2)
a__U32(X) = X >= X = U32(X)
a__U41(X) = X >= X = U41(X)
a__U51(X1,X2) = X2 + 1 >= X2 + 1 = U51(X1,X2)
a__U52(X1,X2) = X2 + 1 >= X2 + 1 = U52(X1,X2)
a__U61(X1,X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3)
a__U62(X1,X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3)
a__U63(X1,X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3)
a__U64(X1,X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3)
a__plus(X1,X2) = X1 + X2 >= X1 + X2 = plus(X1,X2)
problem:
DPs:
mark#(U31(X1,X2)) -> mark#(X1)
mark#(U32(X)) -> mark#(X)
mark#(U41(X)) -> mark#(X)
TRS:
a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) -> a__U16(a__isNat(V2))
a__U16(tt()) -> tt()
a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) -> a__U23(a__isNat(V1))
a__U23(tt()) -> tt()
a__U31(tt(),V2) -> a__U32(a__isNatKind(V2))
a__U32(tt()) -> tt()
a__U41(tt()) -> tt()
a__U51(tt(),N) -> a__U52(a__isNatKind(N),N)
a__U52(tt(),N) -> mark(N)
a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
a__isNat(0()) -> tt()
a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) -> tt()
a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1))
a__plus(N,0()) -> a__U51(a__isNat(N),N)
a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) -> a__isNatKind(X)
mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) -> a__U15(mark(X1),X2)
mark(isNat(X)) -> a__isNat(X)
mark(U16(X)) -> a__U16(mark(X))
mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
mark(U23(X)) -> a__U23(mark(X))
mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
mark(U32(X)) -> a__U32(mark(X))
mark(U41(X)) -> a__U41(mark(X))
mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
mark(tt()) -> tt()
mark(s(X)) -> s(mark(X))
mark(0()) -> 0()
a__U11(X1,X2,X3) -> U11(X1,X2,X3)
a__U12(X1,X2,X3) -> U12(X1,X2,X3)
a__isNatKind(X) -> isNatKind(X)
a__U13(X1,X2,X3) -> U13(X1,X2,X3)
a__U14(X1,X2,X3) -> U14(X1,X2,X3)
a__U15(X1,X2) -> U15(X1,X2)
a__isNat(X) -> isNat(X)
a__U16(X) -> U16(X)
a__U21(X1,X2) -> U21(X1,X2)
a__U22(X1,X2) -> U22(X1,X2)
a__U23(X) -> U23(X)
a__U31(X1,X2) -> U31(X1,X2)
a__U32(X) -> U32(X)
a__U41(X) -> U41(X)
a__U51(X1,X2) -> U51(X1,X2)
a__U52(X1,X2) -> U52(X1,X2)
a__U61(X1,X2,X3) -> U61(X1,X2,X3)
a__U62(X1,X2,X3) -> U62(X1,X2,X3)
a__U63(X1,X2,X3) -> U63(X1,X2,X3)
a__U64(X1,X2,X3) -> U64(X1,X2,X3)
a__plus(X1,X2) -> plus(X1,X2)
Matrix Interpretation Processor:
dimension: 1
interpretation:
[mark#](x0) = x0,
[U64](x0, x1, x2) = x1 + x2 + 1,
[U63](x0, x1, x2) = x1 + x2 + 1,
[U62](x0, x1, x2) = x1 + x2 + 1,
[U61](x0, x1, x2) = x1 + x2 + 1,
[U52](x0, x1) = x1,
[U51](x0, x1) = x1,
[U41](x0) = x0,
[U32](x0) = x0,
[U31](x0, x1) = x0 + x1 + 1,
[U23](x0) = 0,
[U22](x0, x1) = 0,
[U21](x0, x1) = 0,
[U16](x0) = 0,
[isNat](x0) = x0,
[U15](x0, x1) = 1,
[U14](x0, x1, x2) = 1,
[U13](x0, x1, x2) = 1,
[isNatKind](x0) = x0,
[U12](x0, x1, x2) = x2 + 1,
[U11](x0, x1, x2) = x2 + 1,
[plus](x0, x1) = x0 + x1 + 1,
[0] = 0,
[s](x0) = x0,
[a__plus](x0, x1) = x0 + x1 + 1,
[a__U64](x0, x1, x2) = x1 + x2 + 1,
[a__U63](x0, x1, x2) = x1 + x2 + 1,
[a__U62](x0, x1, x2) = x1 + x2 + 1,
[a__U61](x0, x1, x2) = x1 + x2 + 1,
[mark](x0) = x0,
[a__U52](x0, x1) = x1,
[a__U51](x0, x1) = x1,
[a__U41](x0) = x0,
[a__U32](x0) = x0,
[a__U31](x0, x1) = x0 + x1 + 1,
[a__U23](x0) = 0,
[a__U22](x0, x1) = 0,
[a__U21](x0, x1) = 0,
[a__U16](x0) = 0,
[a__U15](x0, x1) = 1,
[a__isNat](x0) = x0,
[a__U14](x0, x1, x2) = 1,
[a__U13](x0, x1, x2) = 1,
[a__U12](x0, x1, x2) = x2 + 1,
[a__isNatKind](x0) = x0,
[a__U11](x0, x1, x2) = x2 + 1,
[tt] = 0
orientation:
mark#(U31(X1,X2)) = X1 + X2 + 1 >= X1 = mark#(X1)
mark#(U32(X)) = X >= X = mark#(X)
mark#(U41(X)) = X >= X = mark#(X)
a__U11(tt(),V1,V2) = V2 + 1 >= V2 + 1 = a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) = V2 + 1 >= 1 = a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) = 1 >= 1 = a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) = 1 >= 1 = a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) = 1 >= 0 = a__U16(a__isNat(V2))
a__U16(tt()) = 0 >= 0 = tt()
a__U21(tt(),V1) = 0 >= 0 = a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) = 0 >= 0 = a__U23(a__isNat(V1))
a__U23(tt()) = 0 >= 0 = tt()
a__U31(tt(),V2) = V2 + 1 >= V2 = a__U32(a__isNatKind(V2))
a__U32(tt()) = 0 >= 0 = tt()
a__U41(tt()) = 0 >= 0 = tt()
a__U51(tt(),N) = N >= N = a__U52(a__isNatKind(N),N)
a__U52(tt(),N) = N >= N = mark(N)
a__U61(tt(),M,N) = M + N + 1 >= M + N + 1 = a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) = M + N + 1 >= M + N + 1 = a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) = M + N + 1 >= M + N + 1 = a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) = M + N + 1 >= M + N + 1 = s(a__plus(mark(N),mark(M)))
a__isNat(0()) = 0 >= 0 = tt()
a__isNat(plus(V1,V2)) = V1 + V2 + 1 >= V2 + 1 = a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) = V1 >= 0 = a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) = 0 >= 0 = tt()
a__isNatKind(plus(V1,V2)) = V1 + V2 + 1 >= V1 + V2 + 1 = a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) = V1 >= V1 = a__U41(a__isNatKind(V1))
a__plus(N,0()) = N + 1 >= N = a__U51(a__isNat(N),N)
a__plus(N,s(M)) = M + N + 1 >= M + N + 1 = a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) = X3 + 1 >= X3 + 1 = a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) = X3 + 1 >= X3 + 1 = a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) = X >= X = a__isNatKind(X)
mark(U13(X1,X2,X3)) = 1 >= 1 = a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) = 1 >= 1 = a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) = 1 >= 1 = a__U15(mark(X1),X2)
mark(isNat(X)) = X >= X = a__isNat(X)
mark(U16(X)) = 0 >= 0 = a__U16(mark(X))
mark(U21(X1,X2)) = 0 >= 0 = a__U21(mark(X1),X2)
mark(U22(X1,X2)) = 0 >= 0 = a__U22(mark(X1),X2)
mark(U23(X)) = 0 >= 0 = a__U23(mark(X))
mark(U31(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__U31(mark(X1),X2)
mark(U32(X)) = X >= X = a__U32(mark(X))
mark(U41(X)) = X >= X = a__U41(mark(X))
mark(U51(X1,X2)) = X2 >= X2 = a__U51(mark(X1),X2)
mark(U52(X1,X2)) = X2 >= X2 = a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__plus(mark(X1),mark(X2))
mark(tt()) = 0 >= 0 = tt()
mark(s(X)) = X >= X = s(mark(X))
mark(0()) = 0 >= 0 = 0()
a__U11(X1,X2,X3) = X3 + 1 >= X3 + 1 = U11(X1,X2,X3)
a__U12(X1,X2,X3) = X3 + 1 >= X3 + 1 = U12(X1,X2,X3)
a__isNatKind(X) = X >= X = isNatKind(X)
a__U13(X1,X2,X3) = 1 >= 1 = U13(X1,X2,X3)
a__U14(X1,X2,X3) = 1 >= 1 = U14(X1,X2,X3)
a__U15(X1,X2) = 1 >= 1 = U15(X1,X2)
a__isNat(X) = X >= X = isNat(X)
a__U16(X) = 0 >= 0 = U16(X)
a__U21(X1,X2) = 0 >= 0 = U21(X1,X2)
a__U22(X1,X2) = 0 >= 0 = U22(X1,X2)
a__U23(X) = 0 >= 0 = U23(X)
a__U31(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = U31(X1,X2)
a__U32(X) = X >= X = U32(X)
a__U41(X) = X >= X = U41(X)
a__U51(X1,X2) = X2 >= X2 = U51(X1,X2)
a__U52(X1,X2) = X2 >= X2 = U52(X1,X2)
a__U61(X1,X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3)
a__U62(X1,X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3)
a__U63(X1,X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3)
a__U64(X1,X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3)
a__plus(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2)
problem:
DPs:
mark#(U32(X)) -> mark#(X)
mark#(U41(X)) -> mark#(X)
TRS:
a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) -> a__U16(a__isNat(V2))
a__U16(tt()) -> tt()
a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) -> a__U23(a__isNat(V1))
a__U23(tt()) -> tt()
a__U31(tt(),V2) -> a__U32(a__isNatKind(V2))
a__U32(tt()) -> tt()
a__U41(tt()) -> tt()
a__U51(tt(),N) -> a__U52(a__isNatKind(N),N)
a__U52(tt(),N) -> mark(N)
a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
a__isNat(0()) -> tt()
a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) -> tt()
a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1))
a__plus(N,0()) -> a__U51(a__isNat(N),N)
a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) -> a__isNatKind(X)
mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) -> a__U15(mark(X1),X2)
mark(isNat(X)) -> a__isNat(X)
mark(U16(X)) -> a__U16(mark(X))
mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
mark(U23(X)) -> a__U23(mark(X))
mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
mark(U32(X)) -> a__U32(mark(X))
mark(U41(X)) -> a__U41(mark(X))
mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
mark(tt()) -> tt()
mark(s(X)) -> s(mark(X))
mark(0()) -> 0()
a__U11(X1,X2,X3) -> U11(X1,X2,X3)
a__U12(X1,X2,X3) -> U12(X1,X2,X3)
a__isNatKind(X) -> isNatKind(X)
a__U13(X1,X2,X3) -> U13(X1,X2,X3)
a__U14(X1,X2,X3) -> U14(X1,X2,X3)
a__U15(X1,X2) -> U15(X1,X2)
a__isNat(X) -> isNat(X)
a__U16(X) -> U16(X)
a__U21(X1,X2) -> U21(X1,X2)
a__U22(X1,X2) -> U22(X1,X2)
a__U23(X) -> U23(X)
a__U31(X1,X2) -> U31(X1,X2)
a__U32(X) -> U32(X)
a__U41(X) -> U41(X)
a__U51(X1,X2) -> U51(X1,X2)
a__U52(X1,X2) -> U52(X1,X2)
a__U61(X1,X2,X3) -> U61(X1,X2,X3)
a__U62(X1,X2,X3) -> U62(X1,X2,X3)
a__U63(X1,X2,X3) -> U63(X1,X2,X3)
a__U64(X1,X2,X3) -> U64(X1,X2,X3)
a__plus(X1,X2) -> plus(X1,X2)
Matrix Interpretation Processor:
dimension: 1
interpretation:
[mark#](x0) = x0,
[U64](x0, x1, x2) = x1 + x2 + 1,
[U63](x0, x1, x2) = x1 + x2 + 1,
[U62](x0, x1, x2) = x1 + x2 + 1,
[U61](x0, x1, x2) = x1 + x2 + 1,
[U52](x0, x1) = x1,
[U51](x0, x1) = x1,
[U41](x0) = x0 + 1,
[U32](x0) = x0,
[U31](x0, x1) = x1,
[U23](x0) = 0,
[U22](x0, x1) = 0,
[U21](x0, x1) = 0,
[U16](x0) = 0,
[isNat](x0) = 0,
[U15](x0, x1) = 0,
[U14](x0, x1, x2) = 0,
[U13](x0, x1, x2) = 0,
[isNatKind](x0) = x0,
[U12](x0, x1, x2) = 0,
[U11](x0, x1, x2) = 0,
[plus](x0, x1) = x0 + x1,
[0] = 0,
[s](x0) = x0 + 1,
[a__plus](x0, x1) = x0 + x1,
[a__U64](x0, x1, x2) = x1 + x2 + 1,
[a__U63](x0, x1, x2) = x1 + x2 + 1,
[a__U62](x0, x1, x2) = x1 + x2 + 1,
[a__U61](x0, x1, x2) = x1 + x2 + 1,
[mark](x0) = x0,
[a__U52](x0, x1) = x1,
[a__U51](x0, x1) = x1,
[a__U41](x0) = x0 + 1,
[a__U32](x0) = x0,
[a__U31](x0, x1) = x1,
[a__U23](x0) = 0,
[a__U22](x0, x1) = 0,
[a__U21](x0, x1) = 0,
[a__U16](x0) = 0,
[a__U15](x0, x1) = 0,
[a__isNat](x0) = 0,
[a__U14](x0, x1, x2) = 0,
[a__U13](x0, x1, x2) = 0,
[a__U12](x0, x1, x2) = 0,
[a__isNatKind](x0) = x0,
[a__U11](x0, x1, x2) = 0,
[tt] = 0
orientation:
mark#(U32(X)) = X >= X = mark#(X)
mark#(U41(X)) = X + 1 >= X = mark#(X)
a__U11(tt(),V1,V2) = 0 >= 0 = a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) = 0 >= 0 = a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) = 0 >= 0 = a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) = 0 >= 0 = a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) = 0 >= 0 = a__U16(a__isNat(V2))
a__U16(tt()) = 0 >= 0 = tt()
a__U21(tt(),V1) = 0 >= 0 = a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) = 0 >= 0 = a__U23(a__isNat(V1))
a__U23(tt()) = 0 >= 0 = tt()
a__U31(tt(),V2) = V2 >= V2 = a__U32(a__isNatKind(V2))
a__U32(tt()) = 0 >= 0 = tt()
a__U41(tt()) = 1 >= 0 = tt()
a__U51(tt(),N) = N >= N = a__U52(a__isNatKind(N),N)
a__U52(tt(),N) = N >= N = mark(N)
a__U61(tt(),M,N) = M + N + 1 >= M + N + 1 = a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) = M + N + 1 >= M + N + 1 = a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) = M + N + 1 >= M + N + 1 = a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) = M + N + 1 >= M + N + 1 = s(a__plus(mark(N),mark(M)))
a__isNat(0()) = 0 >= 0 = tt()
a__isNat(plus(V1,V2)) = 0 >= 0 = a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) = 0 >= 0 = a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) = 0 >= 0 = tt()
a__isNatKind(plus(V1,V2)) = V1 + V2 >= V2 = a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) = V1 + 1 >= V1 + 1 = a__U41(a__isNatKind(V1))
a__plus(N,0()) = N >= N = a__U51(a__isNat(N),N)
a__plus(N,s(M)) = M + N + 1 >= M + N + 1 = a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) = 0 >= 0 = a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) = 0 >= 0 = a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) = X >= X = a__isNatKind(X)
mark(U13(X1,X2,X3)) = 0 >= 0 = a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) = 0 >= 0 = a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) = 0 >= 0 = a__U15(mark(X1),X2)
mark(isNat(X)) = 0 >= 0 = a__isNat(X)
mark(U16(X)) = 0 >= 0 = a__U16(mark(X))
mark(U21(X1,X2)) = 0 >= 0 = a__U21(mark(X1),X2)
mark(U22(X1,X2)) = 0 >= 0 = a__U22(mark(X1),X2)
mark(U23(X)) = 0 >= 0 = a__U23(mark(X))
mark(U31(X1,X2)) = X2 >= X2 = a__U31(mark(X1),X2)
mark(U32(X)) = X >= X = a__U32(mark(X))
mark(U41(X)) = X + 1 >= X + 1 = a__U41(mark(X))
mark(U51(X1,X2)) = X2 >= X2 = a__U51(mark(X1),X2)
mark(U52(X1,X2)) = X2 >= X2 = a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) = X1 + X2 >= X1 + X2 = a__plus(mark(X1),mark(X2))
mark(tt()) = 0 >= 0 = tt()
mark(s(X)) = X + 1 >= X + 1 = s(mark(X))
mark(0()) = 0 >= 0 = 0()
a__U11(X1,X2,X3) = 0 >= 0 = U11(X1,X2,X3)
a__U12(X1,X2,X3) = 0 >= 0 = U12(X1,X2,X3)
a__isNatKind(X) = X >= X = isNatKind(X)
a__U13(X1,X2,X3) = 0 >= 0 = U13(X1,X2,X3)
a__U14(X1,X2,X3) = 0 >= 0 = U14(X1,X2,X3)
a__U15(X1,X2) = 0 >= 0 = U15(X1,X2)
a__isNat(X) = 0 >= 0 = isNat(X)
a__U16(X) = 0 >= 0 = U16(X)
a__U21(X1,X2) = 0 >= 0 = U21(X1,X2)
a__U22(X1,X2) = 0 >= 0 = U22(X1,X2)
a__U23(X) = 0 >= 0 = U23(X)
a__U31(X1,X2) = X2 >= X2 = U31(X1,X2)
a__U32(X) = X >= X = U32(X)
a__U41(X) = X + 1 >= X + 1 = U41(X)
a__U51(X1,X2) = X2 >= X2 = U51(X1,X2)
a__U52(X1,X2) = X2 >= X2 = U52(X1,X2)
a__U61(X1,X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3)
a__U62(X1,X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3)
a__U63(X1,X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3)
a__U64(X1,X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3)
a__plus(X1,X2) = X1 + X2 >= X1 + X2 = plus(X1,X2)
problem:
DPs:
mark#(U32(X)) -> mark#(X)
TRS:
a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) -> a__U16(a__isNat(V2))
a__U16(tt()) -> tt()
a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) -> a__U23(a__isNat(V1))
a__U23(tt()) -> tt()
a__U31(tt(),V2) -> a__U32(a__isNatKind(V2))
a__U32(tt()) -> tt()
a__U41(tt()) -> tt()
a__U51(tt(),N) -> a__U52(a__isNatKind(N),N)
a__U52(tt(),N) -> mark(N)
a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
a__isNat(0()) -> tt()
a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) -> tt()
a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1))
a__plus(N,0()) -> a__U51(a__isNat(N),N)
a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) -> a__isNatKind(X)
mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) -> a__U15(mark(X1),X2)
mark(isNat(X)) -> a__isNat(X)
mark(U16(X)) -> a__U16(mark(X))
mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
mark(U23(X)) -> a__U23(mark(X))
mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
mark(U32(X)) -> a__U32(mark(X))
mark(U41(X)) -> a__U41(mark(X))
mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
mark(tt()) -> tt()
mark(s(X)) -> s(mark(X))
mark(0()) -> 0()
a__U11(X1,X2,X3) -> U11(X1,X2,X3)
a__U12(X1,X2,X3) -> U12(X1,X2,X3)
a__isNatKind(X) -> isNatKind(X)
a__U13(X1,X2,X3) -> U13(X1,X2,X3)
a__U14(X1,X2,X3) -> U14(X1,X2,X3)
a__U15(X1,X2) -> U15(X1,X2)
a__isNat(X) -> isNat(X)
a__U16(X) -> U16(X)
a__U21(X1,X2) -> U21(X1,X2)
a__U22(X1,X2) -> U22(X1,X2)
a__U23(X) -> U23(X)
a__U31(X1,X2) -> U31(X1,X2)
a__U32(X) -> U32(X)
a__U41(X) -> U41(X)
a__U51(X1,X2) -> U51(X1,X2)
a__U52(X1,X2) -> U52(X1,X2)
a__U61(X1,X2,X3) -> U61(X1,X2,X3)
a__U62(X1,X2,X3) -> U62(X1,X2,X3)
a__U63(X1,X2,X3) -> U63(X1,X2,X3)
a__U64(X1,X2,X3) -> U64(X1,X2,X3)
a__plus(X1,X2) -> plus(X1,X2)
Matrix Interpretation Processor:
dimension: 1
interpretation:
[mark#](x0) = x0 + 1,
[U64](x0, x1, x2) = 1,
[U63](x0, x1, x2) = 1,
[U62](x0, x1, x2) = 1,
[U61](x0, x1, x2) = 1,
[U52](x0, x1) = x1,
[U51](x0, x1) = x1,
[U41](x0) = 0,
[U32](x0) = x0 + 1,
[U31](x0, x1) = x1 + 1,
[U23](x0) = 0,
[U22](x0, x1) = 0,
[U21](x0, x1) = 0,
[U16](x0) = 0,
[isNat](x0) = 0,
[U15](x0, x1) = 0,
[U14](x0, x1, x2) = 0,
[U13](x0, x1, x2) = 0,
[isNatKind](x0) = x0,
[U12](x0, x1, x2) = 0,
[U11](x0, x1, x2) = 0,
[plus](x0, x1) = x0 + x1 + 1,
[0] = 0,
[s](x0) = 1,
[a__plus](x0, x1) = x0 + x1 + 1,
[a__U64](x0, x1, x2) = 1,
[a__U63](x0, x1, x2) = 1,
[a__U62](x0, x1, x2) = 1,
[a__U61](x0, x1, x2) = 1,
[mark](x0) = x0,
[a__U52](x0, x1) = x1,
[a__U51](x0, x1) = x1,
[a__U41](x0) = 0,
[a__U32](x0) = x0 + 1,
[a__U31](x0, x1) = x1 + 1,
[a__U23](x0) = 0,
[a__U22](x0, x1) = 0,
[a__U21](x0, x1) = 0,
[a__U16](x0) = 0,
[a__U15](x0, x1) = 0,
[a__isNat](x0) = 0,
[a__U14](x0, x1, x2) = 0,
[a__U13](x0, x1, x2) = 0,
[a__U12](x0, x1, x2) = 0,
[a__isNatKind](x0) = x0,
[a__U11](x0, x1, x2) = 0,
[tt] = 0
orientation:
mark#(U32(X)) = X + 2 >= X + 1 = mark#(X)
a__U11(tt(),V1,V2) = 0 >= 0 = a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) = 0 >= 0 = a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) = 0 >= 0 = a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) = 0 >= 0 = a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) = 0 >= 0 = a__U16(a__isNat(V2))
a__U16(tt()) = 0 >= 0 = tt()
a__U21(tt(),V1) = 0 >= 0 = a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) = 0 >= 0 = a__U23(a__isNat(V1))
a__U23(tt()) = 0 >= 0 = tt()
a__U31(tt(),V2) = V2 + 1 >= V2 + 1 = a__U32(a__isNatKind(V2))
a__U32(tt()) = 1 >= 0 = tt()
a__U41(tt()) = 0 >= 0 = tt()
a__U51(tt(),N) = N >= N = a__U52(a__isNatKind(N),N)
a__U52(tt(),N) = N >= N = mark(N)
a__U61(tt(),M,N) = 1 >= 1 = a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) = 1 >= 1 = a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) = 1 >= 1 = a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) = 1 >= 1 = s(a__plus(mark(N),mark(M)))
a__isNat(0()) = 0 >= 0 = tt()
a__isNat(plus(V1,V2)) = 0 >= 0 = a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) = 0 >= 0 = a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) = 0 >= 0 = tt()
a__isNatKind(plus(V1,V2)) = V1 + V2 + 1 >= V2 + 1 = a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) = 1 >= 0 = a__U41(a__isNatKind(V1))
a__plus(N,0()) = N + 1 >= N = a__U51(a__isNat(N),N)
a__plus(N,s(M)) = N + 2 >= 1 = a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) = 0 >= 0 = a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) = 0 >= 0 = a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) = X >= X = a__isNatKind(X)
mark(U13(X1,X2,X3)) = 0 >= 0 = a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) = 0 >= 0 = a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) = 0 >= 0 = a__U15(mark(X1),X2)
mark(isNat(X)) = 0 >= 0 = a__isNat(X)
mark(U16(X)) = 0 >= 0 = a__U16(mark(X))
mark(U21(X1,X2)) = 0 >= 0 = a__U21(mark(X1),X2)
mark(U22(X1,X2)) = 0 >= 0 = a__U22(mark(X1),X2)
mark(U23(X)) = 0 >= 0 = a__U23(mark(X))
mark(U31(X1,X2)) = X2 + 1 >= X2 + 1 = a__U31(mark(X1),X2)
mark(U32(X)) = X + 1 >= X + 1 = a__U32(mark(X))
mark(U41(X)) = 0 >= 0 = a__U41(mark(X))
mark(U51(X1,X2)) = X2 >= X2 = a__U51(mark(X1),X2)
mark(U52(X1,X2)) = X2 >= X2 = a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) = 1 >= 1 = a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) = 1 >= 1 = a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) = 1 >= 1 = a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) = 1 >= 1 = a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__plus(mark(X1),mark(X2))
mark(tt()) = 0 >= 0 = tt()
mark(s(X)) = 1 >= 1 = s(mark(X))
mark(0()) = 0 >= 0 = 0()
a__U11(X1,X2,X3) = 0 >= 0 = U11(X1,X2,X3)
a__U12(X1,X2,X3) = 0 >= 0 = U12(X1,X2,X3)
a__isNatKind(X) = X >= X = isNatKind(X)
a__U13(X1,X2,X3) = 0 >= 0 = U13(X1,X2,X3)
a__U14(X1,X2,X3) = 0 >= 0 = U14(X1,X2,X3)
a__U15(X1,X2) = 0 >= 0 = U15(X1,X2)
a__isNat(X) = 0 >= 0 = isNat(X)
a__U16(X) = 0 >= 0 = U16(X)
a__U21(X1,X2) = 0 >= 0 = U21(X1,X2)
a__U22(X1,X2) = 0 >= 0 = U22(X1,X2)
a__U23(X) = 0 >= 0 = U23(X)
a__U31(X1,X2) = X2 + 1 >= X2 + 1 = U31(X1,X2)
a__U32(X) = X + 1 >= X + 1 = U32(X)
a__U41(X) = 0 >= 0 = U41(X)
a__U51(X1,X2) = X2 >= X2 = U51(X1,X2)
a__U52(X1,X2) = X2 >= X2 = U52(X1,X2)
a__U61(X1,X2,X3) = 1 >= 1 = U61(X1,X2,X3)
a__U62(X1,X2,X3) = 1 >= 1 = U62(X1,X2,X3)
a__U63(X1,X2,X3) = 1 >= 1 = U63(X1,X2,X3)
a__U64(X1,X2,X3) = 1 >= 1 = U64(X1,X2,X3)
a__plus(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2)
problem:
DPs:
TRS:
a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) -> a__U16(a__isNat(V2))
a__U16(tt()) -> tt()
a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) -> a__U23(a__isNat(V1))
a__U23(tt()) -> tt()
a__U31(tt(),V2) -> a__U32(a__isNatKind(V2))
a__U32(tt()) -> tt()
a__U41(tt()) -> tt()
a__U51(tt(),N) -> a__U52(a__isNatKind(N),N)
a__U52(tt(),N) -> mark(N)
a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
a__isNat(0()) -> tt()
a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) -> tt()
a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1))
a__plus(N,0()) -> a__U51(a__isNat(N),N)
a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) -> a__isNatKind(X)
mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) -> a__U15(mark(X1),X2)
mark(isNat(X)) -> a__isNat(X)
mark(U16(X)) -> a__U16(mark(X))
mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
mark(U23(X)) -> a__U23(mark(X))
mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
mark(U32(X)) -> a__U32(mark(X))
mark(U41(X)) -> a__U41(mark(X))
mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
mark(tt()) -> tt()
mark(s(X)) -> s(mark(X))
mark(0()) -> 0()
a__U11(X1,X2,X3) -> U11(X1,X2,X3)
a__U12(X1,X2,X3) -> U12(X1,X2,X3)
a__isNatKind(X) -> isNatKind(X)
a__U13(X1,X2,X3) -> U13(X1,X2,X3)
a__U14(X1,X2,X3) -> U14(X1,X2,X3)
a__U15(X1,X2) -> U15(X1,X2)
a__isNat(X) -> isNat(X)
a__U16(X) -> U16(X)
a__U21(X1,X2) -> U21(X1,X2)
a__U22(X1,X2) -> U22(X1,X2)
a__U23(X) -> U23(X)
a__U31(X1,X2) -> U31(X1,X2)
a__U32(X) -> U32(X)
a__U41(X) -> U41(X)
a__U51(X1,X2) -> U51(X1,X2)
a__U52(X1,X2) -> U52(X1,X2)
a__U61(X1,X2,X3) -> U61(X1,X2,X3)
a__U62(X1,X2,X3) -> U62(X1,X2,X3)
a__U63(X1,X2,X3) -> U63(X1,X2,X3)
a__U64(X1,X2,X3) -> U64(X1,X2,X3)
a__plus(X1,X2) -> plus(X1,X2)
Qed
DPs:
a__U11#(tt(),V1,V2) -> a__U12#(a__isNatKind(V1),V1,V2)
a__U12#(tt(),V1,V2) -> a__U13#(a__isNatKind(V2),V1,V2)
a__U13#(tt(),V1,V2) -> a__U14#(a__isNatKind(V2),V1,V2)
a__U14#(tt(),V1,V2) -> a__isNat#(V1)
a__isNat#(plus(V1,V2)) -> a__U11#(a__isNatKind(V1),V1,V2)
a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
a__U21#(tt(),V1) -> a__U22#(a__isNatKind(V1),V1)
a__U22#(tt(),V1) -> a__isNat#(V1)
a__U14#(tt(),V1,V2) -> a__U15#(a__isNat(V1),V2)
a__U15#(tt(),V2) -> a__isNat#(V2)
TRS:
a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) -> a__U16(a__isNat(V2))
a__U16(tt()) -> tt()
a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) -> a__U23(a__isNat(V1))
a__U23(tt()) -> tt()
a__U31(tt(),V2) -> a__U32(a__isNatKind(V2))
a__U32(tt()) -> tt()
a__U41(tt()) -> tt()
a__U51(tt(),N) -> a__U52(a__isNatKind(N),N)
a__U52(tt(),N) -> mark(N)
a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
a__isNat(0()) -> tt()
a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) -> tt()
a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1))
a__plus(N,0()) -> a__U51(a__isNat(N),N)
a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) -> a__isNatKind(X)
mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) -> a__U15(mark(X1),X2)
mark(isNat(X)) -> a__isNat(X)
mark(U16(X)) -> a__U16(mark(X))
mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
mark(U23(X)) -> a__U23(mark(X))
mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
mark(U32(X)) -> a__U32(mark(X))
mark(U41(X)) -> a__U41(mark(X))
mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
mark(tt()) -> tt()
mark(s(X)) -> s(mark(X))
mark(0()) -> 0()
a__U11(X1,X2,X3) -> U11(X1,X2,X3)
a__U12(X1,X2,X3) -> U12(X1,X2,X3)
a__isNatKind(X) -> isNatKind(X)
a__U13(X1,X2,X3) -> U13(X1,X2,X3)
a__U14(X1,X2,X3) -> U14(X1,X2,X3)
a__U15(X1,X2) -> U15(X1,X2)
a__isNat(X) -> isNat(X)
a__U16(X) -> U16(X)
a__U21(X1,X2) -> U21(X1,X2)
a__U22(X1,X2) -> U22(X1,X2)
a__U23(X) -> U23(X)
a__U31(X1,X2) -> U31(X1,X2)
a__U32(X) -> U32(X)
a__U41(X) -> U41(X)
a__U51(X1,X2) -> U51(X1,X2)
a__U52(X1,X2) -> U52(X1,X2)
a__U61(X1,X2,X3) -> U61(X1,X2,X3)
a__U62(X1,X2,X3) -> U62(X1,X2,X3)
a__U63(X1,X2,X3) -> U63(X1,X2,X3)
a__U64(X1,X2,X3) -> U64(X1,X2,X3)
a__plus(X1,X2) -> plus(X1,X2)
Matrix Interpretation Processor:
dimension: 1
interpretation:
[a__U22#](x0, x1) = x1 + 1,
[a__U21#](x0, x1) = x1 + 1,
[a__U15#](x0, x1) = x1,
[a__isNat#](x0) = x0,
[a__U14#](x0, x1, x2) = x1 + x2,
[a__U13#](x0, x1, x2) = x1 + x2,
[a__U12#](x0, x1, x2) = x1 + x2,
[a__U11#](x0, x1, x2) = x1 + x2,
[U64](x0, x1, x2) = x1 + x2 + 1,
[U63](x0, x1, x2) = x1 + x2 + 1,
[U62](x0, x1, x2) = x1 + x2 + 1,
[U61](x0, x1, x2) = x1 + x2 + 1,
[U52](x0, x1) = x1,
[U51](x0, x1) = x1,
[U41](x0) = 0,
[U32](x0) = 0,
[U31](x0, x1) = 0,
[U23](x0) = 0,
[U22](x0, x1) = 0,
[U21](x0, x1) = 0,
[U16](x0) = 0,
[isNat](x0) = 0,
[U15](x0, x1) = 0,
[U14](x0, x1, x2) = 0,
[U13](x0, x1, x2) = 0,
[isNatKind](x0) = 0,
[U12](x0, x1, x2) = 0,
[U11](x0, x1, x2) = 0,
[plus](x0, x1) = x0 + x1,
[0] = 0,
[s](x0) = x0 + 1,
[a__plus](x0, x1) = x0 + x1,
[a__U64](x0, x1, x2) = x1 + x2 + 1,
[a__U63](x0, x1, x2) = x1 + x2 + 1,
[a__U62](x0, x1, x2) = x1 + x2 + 1,
[a__U61](x0, x1, x2) = x1 + x2 + 1,
[mark](x0) = x0,
[a__U52](x0, x1) = x1,
[a__U51](x0, x1) = x1,
[a__U41](x0) = 0,
[a__U32](x0) = 0,
[a__U31](x0, x1) = 0,
[a__U23](x0) = 0,
[a__U22](x0, x1) = 0,
[a__U21](x0, x1) = 0,
[a__U16](x0) = 0,
[a__U15](x0, x1) = 0,
[a__isNat](x0) = 0,
[a__U14](x0, x1, x2) = 0,
[a__U13](x0, x1, x2) = 0,
[a__U12](x0, x1, x2) = 0,
[a__isNatKind](x0) = 0,
[a__U11](x0, x1, x2) = 0,
[tt] = 0
orientation:
a__U11#(tt(),V1,V2) = V1 + V2 >= V1 + V2 = a__U12#(a__isNatKind(V1),V1,V2)
a__U12#(tt(),V1,V2) = V1 + V2 >= V1 + V2 = a__U13#(a__isNatKind(V2),V1,V2)
a__U13#(tt(),V1,V2) = V1 + V2 >= V1 + V2 = a__U14#(a__isNatKind(V2),V1,V2)
a__U14#(tt(),V1,V2) = V1 + V2 >= V1 = a__isNat#(V1)
a__isNat#(plus(V1,V2)) = V1 + V2 >= V1 + V2 = a__U11#(a__isNatKind(V1),V1,V2)
a__isNat#(s(V1)) = V1 + 1 >= V1 + 1 = a__U21#(a__isNatKind(V1),V1)
a__U21#(tt(),V1) = V1 + 1 >= V1 + 1 = a__U22#(a__isNatKind(V1),V1)
a__U22#(tt(),V1) = V1 + 1 >= V1 = a__isNat#(V1)
a__U14#(tt(),V1,V2) = V1 + V2 >= V2 = a__U15#(a__isNat(V1),V2)
a__U15#(tt(),V2) = V2 >= V2 = a__isNat#(V2)
a__U11(tt(),V1,V2) = 0 >= 0 = a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) = 0 >= 0 = a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) = 0 >= 0 = a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) = 0 >= 0 = a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) = 0 >= 0 = a__U16(a__isNat(V2))
a__U16(tt()) = 0 >= 0 = tt()
a__U21(tt(),V1) = 0 >= 0 = a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) = 0 >= 0 = a__U23(a__isNat(V1))
a__U23(tt()) = 0 >= 0 = tt()
a__U31(tt(),V2) = 0 >= 0 = a__U32(a__isNatKind(V2))
a__U32(tt()) = 0 >= 0 = tt()
a__U41(tt()) = 0 >= 0 = tt()
a__U51(tt(),N) = N >= N = a__U52(a__isNatKind(N),N)
a__U52(tt(),N) = N >= N = mark(N)
a__U61(tt(),M,N) = M + N + 1 >= M + N + 1 = a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) = M + N + 1 >= M + N + 1 = a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) = M + N + 1 >= M + N + 1 = a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) = M + N + 1 >= M + N + 1 = s(a__plus(mark(N),mark(M)))
a__isNat(0()) = 0 >= 0 = tt()
a__isNat(plus(V1,V2)) = 0 >= 0 = a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) = 0 >= 0 = a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) = 0 >= 0 = tt()
a__isNatKind(plus(V1,V2)) = 0 >= 0 = a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) = 0 >= 0 = a__U41(a__isNatKind(V1))
a__plus(N,0()) = N >= N = a__U51(a__isNat(N),N)
a__plus(N,s(M)) = M + N + 1 >= M + N + 1 = a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) = 0 >= 0 = a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) = 0 >= 0 = a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) = 0 >= 0 = a__isNatKind(X)
mark(U13(X1,X2,X3)) = 0 >= 0 = a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) = 0 >= 0 = a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) = 0 >= 0 = a__U15(mark(X1),X2)
mark(isNat(X)) = 0 >= 0 = a__isNat(X)
mark(U16(X)) = 0 >= 0 = a__U16(mark(X))
mark(U21(X1,X2)) = 0 >= 0 = a__U21(mark(X1),X2)
mark(U22(X1,X2)) = 0 >= 0 = a__U22(mark(X1),X2)
mark(U23(X)) = 0 >= 0 = a__U23(mark(X))
mark(U31(X1,X2)) = 0 >= 0 = a__U31(mark(X1),X2)
mark(U32(X)) = 0 >= 0 = a__U32(mark(X))
mark(U41(X)) = 0 >= 0 = a__U41(mark(X))
mark(U51(X1,X2)) = X2 >= X2 = a__U51(mark(X1),X2)
mark(U52(X1,X2)) = X2 >= X2 = a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) = X1 + X2 >= X1 + X2 = a__plus(mark(X1),mark(X2))
mark(tt()) = 0 >= 0 = tt()
mark(s(X)) = X + 1 >= X + 1 = s(mark(X))
mark(0()) = 0 >= 0 = 0()
a__U11(X1,X2,X3) = 0 >= 0 = U11(X1,X2,X3)
a__U12(X1,X2,X3) = 0 >= 0 = U12(X1,X2,X3)
a__isNatKind(X) = 0 >= 0 = isNatKind(X)
a__U13(X1,X2,X3) = 0 >= 0 = U13(X1,X2,X3)
a__U14(X1,X2,X3) = 0 >= 0 = U14(X1,X2,X3)
a__U15(X1,X2) = 0 >= 0 = U15(X1,X2)
a__isNat(X) = 0 >= 0 = isNat(X)
a__U16(X) = 0 >= 0 = U16(X)
a__U21(X1,X2) = 0 >= 0 = U21(X1,X2)
a__U22(X1,X2) = 0 >= 0 = U22(X1,X2)
a__U23(X) = 0 >= 0 = U23(X)
a__U31(X1,X2) = 0 >= 0 = U31(X1,X2)
a__U32(X) = 0 >= 0 = U32(X)
a__U41(X) = 0 >= 0 = U41(X)
a__U51(X1,X2) = X2 >= X2 = U51(X1,X2)
a__U52(X1,X2) = X2 >= X2 = U52(X1,X2)
a__U61(X1,X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3)
a__U62(X1,X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3)
a__U63(X1,X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3)
a__U64(X1,X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3)
a__plus(X1,X2) = X1 + X2 >= X1 + X2 = plus(X1,X2)
problem:
DPs:
a__U11#(tt(),V1,V2) -> a__U12#(a__isNatKind(V1),V1,V2)
a__U12#(tt(),V1,V2) -> a__U13#(a__isNatKind(V2),V1,V2)
a__U13#(tt(),V1,V2) -> a__U14#(a__isNatKind(V2),V1,V2)
a__U14#(tt(),V1,V2) -> a__isNat#(V1)
a__isNat#(plus(V1,V2)) -> a__U11#(a__isNatKind(V1),V1,V2)
a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
a__U21#(tt(),V1) -> a__U22#(a__isNatKind(V1),V1)
a__U14#(tt(),V1,V2) -> a__U15#(a__isNat(V1),V2)
a__U15#(tt(),V2) -> a__isNat#(V2)
TRS:
a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) -> a__U16(a__isNat(V2))
a__U16(tt()) -> tt()
a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) -> a__U23(a__isNat(V1))
a__U23(tt()) -> tt()
a__U31(tt(),V2) -> a__U32(a__isNatKind(V2))
a__U32(tt()) -> tt()
a__U41(tt()) -> tt()
a__U51(tt(),N) -> a__U52(a__isNatKind(N),N)
a__U52(tt(),N) -> mark(N)
a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
a__isNat(0()) -> tt()
a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) -> tt()
a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1))
a__plus(N,0()) -> a__U51(a__isNat(N),N)
a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) -> a__isNatKind(X)
mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) -> a__U15(mark(X1),X2)
mark(isNat(X)) -> a__isNat(X)
mark(U16(X)) -> a__U16(mark(X))
mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
mark(U23(X)) -> a__U23(mark(X))
mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
mark(U32(X)) -> a__U32(mark(X))
mark(U41(X)) -> a__U41(mark(X))
mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
mark(tt()) -> tt()
mark(s(X)) -> s(mark(X))
mark(0()) -> 0()
a__U11(X1,X2,X3) -> U11(X1,X2,X3)
a__U12(X1,X2,X3) -> U12(X1,X2,X3)
a__isNatKind(X) -> isNatKind(X)
a__U13(X1,X2,X3) -> U13(X1,X2,X3)
a__U14(X1,X2,X3) -> U14(X1,X2,X3)
a__U15(X1,X2) -> U15(X1,X2)
a__isNat(X) -> isNat(X)
a__U16(X) -> U16(X)
a__U21(X1,X2) -> U21(X1,X2)
a__U22(X1,X2) -> U22(X1,X2)
a__U23(X) -> U23(X)
a__U31(X1,X2) -> U31(X1,X2)
a__U32(X) -> U32(X)
a__U41(X) -> U41(X)
a__U51(X1,X2) -> U51(X1,X2)
a__U52(X1,X2) -> U52(X1,X2)
a__U61(X1,X2,X3) -> U61(X1,X2,X3)
a__U62(X1,X2,X3) -> U62(X1,X2,X3)
a__U63(X1,X2,X3) -> U63(X1,X2,X3)
a__U64(X1,X2,X3) -> U64(X1,X2,X3)
a__plus(X1,X2) -> plus(X1,X2)
Matrix Interpretation Processor:
dimension: 1
interpretation:
[a__U22#](x0, x1) = 0,
[a__U21#](x0, x1) = 1,
[a__U15#](x0, x1) = x1 + 1,
[a__isNat#](x0) = x0 + 1,
[a__U14#](x0, x1, x2) = x1 + x2 + 1,
[a__U13#](x0, x1, x2) = x1 + x2 + 1,
[a__U12#](x0, x1, x2) = x1 + x2 + 1,
[a__U11#](x0, x1, x2) = x1 + x2 + 1,
[U64](x0, x1, x2) = 0,
[U63](x0, x1, x2) = 0,
[U62](x0, x1, x2) = 0,
[U61](x0, x1, x2) = 0,
[U52](x0, x1) = 0,
[U51](x0, x1) = 0,
[U41](x0) = 0,
[U32](x0) = 0,
[U31](x0, x1) = 0,
[U23](x0) = 0,
[U22](x0, x1) = 0,
[U21](x0, x1) = 0,
[U16](x0) = 0,
[isNat](x0) = 0,
[U15](x0, x1) = 0,
[U14](x0, x1, x2) = 0,
[U13](x0, x1, x2) = 0,
[isNatKind](x0) = 0,
[U12](x0, x1, x2) = 0,
[U11](x0, x1, x2) = 0,
[plus](x0, x1) = x0 + x1,
[0] = 0,
[s](x0) = 0,
[a__plus](x0, x1) = x0 + x1,
[a__U64](x0, x1, x2) = 0,
[a__U63](x0, x1, x2) = 0,
[a__U62](x0, x1, x2) = 0,
[a__U61](x0, x1, x2) = 0,
[mark](x0) = 0,
[a__U52](x0, x1) = 0,
[a__U51](x0, x1) = 0,
[a__U41](x0) = 0,
[a__U32](x0) = 0,
[a__U31](x0, x1) = 0,
[a__U23](x0) = 0,
[a__U22](x0, x1) = 0,
[a__U21](x0, x1) = 0,
[a__U16](x0) = 0,
[a__U15](x0, x1) = 0,
[a__isNat](x0) = 0,
[a__U14](x0, x1, x2) = 0,
[a__U13](x0, x1, x2) = 0,
[a__U12](x0, x1, x2) = 0,
[a__isNatKind](x0) = 0,
[a__U11](x0, x1, x2) = 0,
[tt] = 0
orientation:
a__U11#(tt(),V1,V2) = V1 + V2 + 1 >= V1 + V2 + 1 = a__U12#(a__isNatKind(V1),V1,V2)
a__U12#(tt(),V1,V2) = V1 + V2 + 1 >= V1 + V2 + 1 = a__U13#(a__isNatKind(V2),V1,V2)
a__U13#(tt(),V1,V2) = V1 + V2 + 1 >= V1 + V2 + 1 = a__U14#(a__isNatKind(V2),V1,V2)
a__U14#(tt(),V1,V2) = V1 + V2 + 1 >= V1 + 1 = a__isNat#(V1)
a__isNat#(plus(V1,V2)) = V1 + V2 + 1 >= V1 + V2 + 1 = a__U11#(a__isNatKind(V1),V1,V2)
a__isNat#(s(V1)) = 1 >= 1 = a__U21#(a__isNatKind(V1),V1)
a__U21#(tt(),V1) = 1 >= 0 = a__U22#(a__isNatKind(V1),V1)
a__U14#(tt(),V1,V2) = V1 + V2 + 1 >= V2 + 1 = a__U15#(a__isNat(V1),V2)
a__U15#(tt(),V2) = V2 + 1 >= V2 + 1 = a__isNat#(V2)
a__U11(tt(),V1,V2) = 0 >= 0 = a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) = 0 >= 0 = a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) = 0 >= 0 = a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) = 0 >= 0 = a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) = 0 >= 0 = a__U16(a__isNat(V2))
a__U16(tt()) = 0 >= 0 = tt()
a__U21(tt(),V1) = 0 >= 0 = a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) = 0 >= 0 = a__U23(a__isNat(V1))
a__U23(tt()) = 0 >= 0 = tt()
a__U31(tt(),V2) = 0 >= 0 = a__U32(a__isNatKind(V2))
a__U32(tt()) = 0 >= 0 = tt()
a__U41(tt()) = 0 >= 0 = tt()
a__U51(tt(),N) = 0 >= 0 = a__U52(a__isNatKind(N),N)
a__U52(tt(),N) = 0 >= 0 = mark(N)
a__U61(tt(),M,N) = 0 >= 0 = a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) = 0 >= 0 = a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) = 0 >= 0 = a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) = 0 >= 0 = s(a__plus(mark(N),mark(M)))
a__isNat(0()) = 0 >= 0 = tt()
a__isNat(plus(V1,V2)) = 0 >= 0 = a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) = 0 >= 0 = a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) = 0 >= 0 = tt()
a__isNatKind(plus(V1,V2)) = 0 >= 0 = a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) = 0 >= 0 = a__U41(a__isNatKind(V1))
a__plus(N,0()) = N >= 0 = a__U51(a__isNat(N),N)
a__plus(N,s(M)) = N >= 0 = a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) = 0 >= 0 = a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) = 0 >= 0 = a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) = 0 >= 0 = a__isNatKind(X)
mark(U13(X1,X2,X3)) = 0 >= 0 = a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) = 0 >= 0 = a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) = 0 >= 0 = a__U15(mark(X1),X2)
mark(isNat(X)) = 0 >= 0 = a__isNat(X)
mark(U16(X)) = 0 >= 0 = a__U16(mark(X))
mark(U21(X1,X2)) = 0 >= 0 = a__U21(mark(X1),X2)
mark(U22(X1,X2)) = 0 >= 0 = a__U22(mark(X1),X2)
mark(U23(X)) = 0 >= 0 = a__U23(mark(X))
mark(U31(X1,X2)) = 0 >= 0 = a__U31(mark(X1),X2)
mark(U32(X)) = 0 >= 0 = a__U32(mark(X))
mark(U41(X)) = 0 >= 0 = a__U41(mark(X))
mark(U51(X1,X2)) = 0 >= 0 = a__U51(mark(X1),X2)
mark(U52(X1,X2)) = 0 >= 0 = a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) = 0 >= 0 = a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) = 0 >= 0 = a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) = 0 >= 0 = a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) = 0 >= 0 = a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) = 0 >= 0 = a__plus(mark(X1),mark(X2))
mark(tt()) = 0 >= 0 = tt()
mark(s(X)) = 0 >= 0 = s(mark(X))
mark(0()) = 0 >= 0 = 0()
a__U11(X1,X2,X3) = 0 >= 0 = U11(X1,X2,X3)
a__U12(X1,X2,X3) = 0 >= 0 = U12(X1,X2,X3)
a__isNatKind(X) = 0 >= 0 = isNatKind(X)
a__U13(X1,X2,X3) = 0 >= 0 = U13(X1,X2,X3)
a__U14(X1,X2,X3) = 0 >= 0 = U14(X1,X2,X3)
a__U15(X1,X2) = 0 >= 0 = U15(X1,X2)
a__isNat(X) = 0 >= 0 = isNat(X)
a__U16(X) = 0 >= 0 = U16(X)
a__U21(X1,X2) = 0 >= 0 = U21(X1,X2)
a__U22(X1,X2) = 0 >= 0 = U22(X1,X2)
a__U23(X) = 0 >= 0 = U23(X)
a__U31(X1,X2) = 0 >= 0 = U31(X1,X2)
a__U32(X) = 0 >= 0 = U32(X)
a__U41(X) = 0 >= 0 = U41(X)
a__U51(X1,X2) = 0 >= 0 = U51(X1,X2)
a__U52(X1,X2) = 0 >= 0 = U52(X1,X2)
a__U61(X1,X2,X3) = 0 >= 0 = U61(X1,X2,X3)
a__U62(X1,X2,X3) = 0 >= 0 = U62(X1,X2,X3)
a__U63(X1,X2,X3) = 0 >= 0 = U63(X1,X2,X3)
a__U64(X1,X2,X3) = 0 >= 0 = U64(X1,X2,X3)
a__plus(X1,X2) = X1 + X2 >= X1 + X2 = plus(X1,X2)
problem:
DPs:
a__U11#(tt(),V1,V2) -> a__U12#(a__isNatKind(V1),V1,V2)
a__U12#(tt(),V1,V2) -> a__U13#(a__isNatKind(V2),V1,V2)
a__U13#(tt(),V1,V2) -> a__U14#(a__isNatKind(V2),V1,V2)
a__U14#(tt(),V1,V2) -> a__isNat#(V1)
a__isNat#(plus(V1,V2)) -> a__U11#(a__isNatKind(V1),V1,V2)
a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
a__U14#(tt(),V1,V2) -> a__U15#(a__isNat(V1),V2)
a__U15#(tt(),V2) -> a__isNat#(V2)
TRS:
a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) -> a__U16(a__isNat(V2))
a__U16(tt()) -> tt()
a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) -> a__U23(a__isNat(V1))
a__U23(tt()) -> tt()
a__U31(tt(),V2) -> a__U32(a__isNatKind(V2))
a__U32(tt()) -> tt()
a__U41(tt()) -> tt()
a__U51(tt(),N) -> a__U52(a__isNatKind(N),N)
a__U52(tt(),N) -> mark(N)
a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
a__isNat(0()) -> tt()
a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) -> tt()
a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1))
a__plus(N,0()) -> a__U51(a__isNat(N),N)
a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) -> a__isNatKind(X)
mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) -> a__U15(mark(X1),X2)
mark(isNat(X)) -> a__isNat(X)
mark(U16(X)) -> a__U16(mark(X))
mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
mark(U23(X)) -> a__U23(mark(X))
mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
mark(U32(X)) -> a__U32(mark(X))
mark(U41(X)) -> a__U41(mark(X))
mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
mark(tt()) -> tt()
mark(s(X)) -> s(mark(X))
mark(0()) -> 0()
a__U11(X1,X2,X3) -> U11(X1,X2,X3)
a__U12(X1,X2,X3) -> U12(X1,X2,X3)
a__isNatKind(X) -> isNatKind(X)
a__U13(X1,X2,X3) -> U13(X1,X2,X3)
a__U14(X1,X2,X3) -> U14(X1,X2,X3)
a__U15(X1,X2) -> U15(X1,X2)
a__isNat(X) -> isNat(X)
a__U16(X) -> U16(X)
a__U21(X1,X2) -> U21(X1,X2)
a__U22(X1,X2) -> U22(X1,X2)
a__U23(X) -> U23(X)
a__U31(X1,X2) -> U31(X1,X2)
a__U32(X) -> U32(X)
a__U41(X) -> U41(X)
a__U51(X1,X2) -> U51(X1,X2)
a__U52(X1,X2) -> U52(X1,X2)
a__U61(X1,X2,X3) -> U61(X1,X2,X3)
a__U62(X1,X2,X3) -> U62(X1,X2,X3)
a__U63(X1,X2,X3) -> U63(X1,X2,X3)
a__U64(X1,X2,X3) -> U64(X1,X2,X3)
a__plus(X1,X2) -> plus(X1,X2)
Matrix Interpretation Processor:
dimension: 1
interpretation:
[a__U21#](x0, x1) = x0 + 1,
[a__U15#](x0, x1) = x0 + x1 + 1,
[a__isNat#](x0) = x0 + 1,
[a__U14#](x0, x1, x2) = x0 + x1 + x2,
[a__U13#](x0, x1, x2) = x1 + x2 + 1,
[a__U12#](x0, x1, x2) = x0 + x1 + x2,
[a__U11#](x0, x1, x2) = x0 + x1 + x2 + 1,
[U64](x0, x1, x2) = 1,
[U63](x0, x1, x2) = x0,
[U62](x0, x1, x2) = x2,
[U61](x0, x1, x2) = x2,
[U52](x0, x1) = x1,
[U51](x0, x1) = x1,
[U41](x0) = 1,
[U32](x0) = 1,
[U31](x0, x1) = 1,
[U23](x0) = 1,
[U22](x0, x1) = 1,
[U21](x0, x1) = 1,
[U16](x0) = 1,
[isNat](x0) = x0,
[U15](x0, x1) = 1,
[U14](x0, x1, x2) = x0,
[U13](x0, x1, x2) = x0,
[isNatKind](x0) = 1,
[U12](x0, x1, x2) = x0,
[U11](x0, x1, x2) = x0,
[plus](x0, x1) = x0 + x1 + 1,
[0] = 1,
[s](x0) = 1,
[a__plus](x0, x1) = x0 + x1 + 1,
[a__U64](x0, x1, x2) = 1,
[a__U63](x0, x1, x2) = x0,
[a__U62](x0, x1, x2) = x2,
[a__U61](x0, x1, x2) = x2,
[mark](x0) = x0,
[a__U52](x0, x1) = x1,
[a__U51](x0, x1) = x1,
[a__U41](x0) = 1,
[a__U32](x0) = 1,
[a__U31](x0, x1) = 1,
[a__U23](x0) = 1,
[a__U22](x0, x1) = 1,
[a__U21](x0, x1) = 1,
[a__U16](x0) = 1,
[a__U15](x0, x1) = 1,
[a__isNat](x0) = x0,
[a__U14](x0, x1, x2) = x0,
[a__U13](x0, x1, x2) = x0,
[a__U12](x0, x1, x2) = x0,
[a__isNatKind](x0) = 1,
[a__U11](x0, x1, x2) = x0,
[tt] = 1
orientation:
a__U11#(tt(),V1,V2) = V1 + V2 + 2 >= V1 + V2 + 1 = a__U12#(a__isNatKind(V1),V1,V2)
a__U12#(tt(),V1,V2) = V1 + V2 + 1 >= V1 + V2 + 1 = a__U13#(a__isNatKind(V2),V1,V2)
a__U13#(tt(),V1,V2) = V1 + V2 + 1 >= V1 + V2 + 1 = a__U14#(a__isNatKind(V2),V1,V2)
a__U14#(tt(),V1,V2) = V1 + V2 + 1 >= V1 + 1 = a__isNat#(V1)
a__isNat#(plus(V1,V2)) = V1 + V2 + 2 >= V1 + V2 + 2 = a__U11#(a__isNatKind(V1),V1,V2)
a__isNat#(s(V1)) = 2 >= 2 = a__U21#(a__isNatKind(V1),V1)
a__U14#(tt(),V1,V2) = V1 + V2 + 1 >= V1 + V2 + 1 = a__U15#(a__isNat(V1),V2)
a__U15#(tt(),V2) = V2 + 2 >= V2 + 1 = a__isNat#(V2)
a__U11(tt(),V1,V2) = 1 >= 1 = a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) = 1 >= 1 = a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) = 1 >= 1 = a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) = 1 >= 1 = a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) = 1 >= 1 = a__U16(a__isNat(V2))
a__U16(tt()) = 1 >= 1 = tt()
a__U21(tt(),V1) = 1 >= 1 = a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) = 1 >= 1 = a__U23(a__isNat(V1))
a__U23(tt()) = 1 >= 1 = tt()
a__U31(tt(),V2) = 1 >= 1 = a__U32(a__isNatKind(V2))
a__U32(tt()) = 1 >= 1 = tt()
a__U41(tt()) = 1 >= 1 = tt()
a__U51(tt(),N) = N >= N = a__U52(a__isNatKind(N),N)
a__U52(tt(),N) = N >= N = mark(N)
a__U61(tt(),M,N) = N >= N = a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) = N >= N = a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) = 1 >= 1 = a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) = 1 >= 1 = s(a__plus(mark(N),mark(M)))
a__isNat(0()) = 1 >= 1 = tt()
a__isNat(plus(V1,V2)) = V1 + V2 + 1 >= 1 = a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) = 1 >= 1 = a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) = 1 >= 1 = tt()
a__isNatKind(plus(V1,V2)) = 1 >= 1 = a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) = 1 >= 1 = a__U41(a__isNatKind(V1))
a__plus(N,0()) = N + 2 >= N = a__U51(a__isNat(N),N)
a__plus(N,s(M)) = N + 2 >= N = a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) = X1 >= X1 = a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) = X1 >= X1 = a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) = 1 >= 1 = a__isNatKind(X)
mark(U13(X1,X2,X3)) = X1 >= X1 = a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) = X1 >= X1 = a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) = 1 >= 1 = a__U15(mark(X1),X2)
mark(isNat(X)) = X >= X = a__isNat(X)
mark(U16(X)) = 1 >= 1 = a__U16(mark(X))
mark(U21(X1,X2)) = 1 >= 1 = a__U21(mark(X1),X2)
mark(U22(X1,X2)) = 1 >= 1 = a__U22(mark(X1),X2)
mark(U23(X)) = 1 >= 1 = a__U23(mark(X))
mark(U31(X1,X2)) = 1 >= 1 = a__U31(mark(X1),X2)
mark(U32(X)) = 1 >= 1 = a__U32(mark(X))
mark(U41(X)) = 1 >= 1 = a__U41(mark(X))
mark(U51(X1,X2)) = X2 >= X2 = a__U51(mark(X1),X2)
mark(U52(X1,X2)) = X2 >= X2 = a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) = X3 >= X3 = a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) = X3 >= X3 = a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) = X1 >= X1 = a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) = 1 >= 1 = a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__plus(mark(X1),mark(X2))
mark(tt()) = 1 >= 1 = tt()
mark(s(X)) = 1 >= 1 = s(mark(X))
mark(0()) = 1 >= 1 = 0()
a__U11(X1,X2,X3) = X1 >= X1 = U11(X1,X2,X3)
a__U12(X1,X2,X3) = X1 >= X1 = U12(X1,X2,X3)
a__isNatKind(X) = 1 >= 1 = isNatKind(X)
a__U13(X1,X2,X3) = X1 >= X1 = U13(X1,X2,X3)
a__U14(X1,X2,X3) = X1 >= X1 = U14(X1,X2,X3)
a__U15(X1,X2) = 1 >= 1 = U15(X1,X2)
a__isNat(X) = X >= X = isNat(X)
a__U16(X) = 1 >= 1 = U16(X)
a__U21(X1,X2) = 1 >= 1 = U21(X1,X2)
a__U22(X1,X2) = 1 >= 1 = U22(X1,X2)
a__U23(X) = 1 >= 1 = U23(X)
a__U31(X1,X2) = 1 >= 1 = U31(X1,X2)
a__U32(X) = 1 >= 1 = U32(X)
a__U41(X) = 1 >= 1 = U41(X)
a__U51(X1,X2) = X2 >= X2 = U51(X1,X2)
a__U52(X1,X2) = X2 >= X2 = U52(X1,X2)
a__U61(X1,X2,X3) = X3 >= X3 = U61(X1,X2,X3)
a__U62(X1,X2,X3) = X3 >= X3 = U62(X1,X2,X3)
a__U63(X1,X2,X3) = X1 >= X1 = U63(X1,X2,X3)
a__U64(X1,X2,X3) = 1 >= 1 = U64(X1,X2,X3)
a__plus(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2)
problem:
DPs:
a__U12#(tt(),V1,V2) -> a__U13#(a__isNatKind(V2),V1,V2)
a__U13#(tt(),V1,V2) -> a__U14#(a__isNatKind(V2),V1,V2)
a__U14#(tt(),V1,V2) -> a__isNat#(V1)
a__isNat#(plus(V1,V2)) -> a__U11#(a__isNatKind(V1),V1,V2)
a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
a__U14#(tt(),V1,V2) -> a__U15#(a__isNat(V1),V2)
TRS:
a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) -> a__U16(a__isNat(V2))
a__U16(tt()) -> tt()
a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) -> a__U23(a__isNat(V1))
a__U23(tt()) -> tt()
a__U31(tt(),V2) -> a__U32(a__isNatKind(V2))
a__U32(tt()) -> tt()
a__U41(tt()) -> tt()
a__U51(tt(),N) -> a__U52(a__isNatKind(N),N)
a__U52(tt(),N) -> mark(N)
a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
a__isNat(0()) -> tt()
a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) -> tt()
a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1))
a__plus(N,0()) -> a__U51(a__isNat(N),N)
a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) -> a__isNatKind(X)
mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) -> a__U15(mark(X1),X2)
mark(isNat(X)) -> a__isNat(X)
mark(U16(X)) -> a__U16(mark(X))
mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
mark(U23(X)) -> a__U23(mark(X))
mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
mark(U32(X)) -> a__U32(mark(X))
mark(U41(X)) -> a__U41(mark(X))
mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
mark(tt()) -> tt()
mark(s(X)) -> s(mark(X))
mark(0()) -> 0()
a__U11(X1,X2,X3) -> U11(X1,X2,X3)
a__U12(X1,X2,X3) -> U12(X1,X2,X3)
a__isNatKind(X) -> isNatKind(X)
a__U13(X1,X2,X3) -> U13(X1,X2,X3)
a__U14(X1,X2,X3) -> U14(X1,X2,X3)
a__U15(X1,X2) -> U15(X1,X2)
a__isNat(X) -> isNat(X)
a__U16(X) -> U16(X)
a__U21(X1,X2) -> U21(X1,X2)
a__U22(X1,X2) -> U22(X1,X2)
a__U23(X) -> U23(X)
a__U31(X1,X2) -> U31(X1,X2)
a__U32(X) -> U32(X)
a__U41(X) -> U41(X)
a__U51(X1,X2) -> U51(X1,X2)
a__U52(X1,X2) -> U52(X1,X2)
a__U61(X1,X2,X3) -> U61(X1,X2,X3)
a__U62(X1,X2,X3) -> U62(X1,X2,X3)
a__U63(X1,X2,X3) -> U63(X1,X2,X3)
a__U64(X1,X2,X3) -> U64(X1,X2,X3)
a__plus(X1,X2) -> plus(X1,X2)
Matrix Interpretation Processor:
dimension: 1
interpretation:
[a__U21#](x0, x1) = 0,
[a__U15#](x0, x1) = x0 + 1,
[a__isNat#](x0) = 0,
[a__U14#](x0, x1, x2) = x0 + 1,
[a__U13#](x0, x1, x2) = x0 + 1,
[a__U12#](x0, x1, x2) = x0 + 1,
[a__U11#](x0, x1, x2) = 0,
[U64](x0, x1, x2) = 1,
[U63](x0, x1, x2) = 1,
[U62](x0, x1, x2) = 1,
[U61](x0, x1, x2) = 1,
[U52](x0, x1) = 1,
[U51](x0, x1) = 1,
[U41](x0) = x0,
[U32](x0) = 1,
[U31](x0, x1) = 1,
[U23](x0) = 1,
[U22](x0, x1) = 1,
[U21](x0, x1) = 1,
[U16](x0) = 1,
[isNat](x0) = 1,
[U15](x0, x1) = 1,
[U14](x0, x1, x2) = 1,
[U13](x0, x1, x2) = 0,
[isNatKind](x0) = 1,
[U12](x0, x1, x2) = 1,
[U11](x0, x1, x2) = 1,
[plus](x0, x1) = 1,
[0] = 1,
[s](x0) = 1,
[a__plus](x0, x1) = 1,
[a__U64](x0, x1, x2) = 1,
[a__U63](x0, x1, x2) = 1,
[a__U62](x0, x1, x2) = 1,
[a__U61](x0, x1, x2) = 1,
[mark](x0) = 1,
[a__U52](x0, x1) = 1,
[a__U51](x0, x1) = 1,
[a__U41](x0) = x0,
[a__U32](x0) = 1,
[a__U31](x0, x1) = 1,
[a__U23](x0) = 1,
[a__U22](x0, x1) = 1,
[a__U21](x0, x1) = 1,
[a__U16](x0) = 1,
[a__U15](x0, x1) = 1,
[a__isNat](x0) = 1,
[a__U14](x0, x1, x2) = 1,
[a__U13](x0, x1, x2) = x0,
[a__U12](x0, x1, x2) = 1,
[a__isNatKind](x0) = 1,
[a__U11](x0, x1, x2) = 1,
[tt] = 1
orientation:
a__U12#(tt(),V1,V2) = 2 >= 2 = a__U13#(a__isNatKind(V2),V1,V2)
a__U13#(tt(),V1,V2) = 2 >= 2 = a__U14#(a__isNatKind(V2),V1,V2)
a__U14#(tt(),V1,V2) = 2 >= 0 = a__isNat#(V1)
a__isNat#(plus(V1,V2)) = 0 >= 0 = a__U11#(a__isNatKind(V1),V1,V2)
a__isNat#(s(V1)) = 0 >= 0 = a__U21#(a__isNatKind(V1),V1)
a__U14#(tt(),V1,V2) = 2 >= 2 = a__U15#(a__isNat(V1),V2)
a__U11(tt(),V1,V2) = 1 >= 1 = a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) = 1 >= 1 = a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) = 1 >= 1 = a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) = 1 >= 1 = a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) = 1 >= 1 = a__U16(a__isNat(V2))
a__U16(tt()) = 1 >= 1 = tt()
a__U21(tt(),V1) = 1 >= 1 = a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) = 1 >= 1 = a__U23(a__isNat(V1))
a__U23(tt()) = 1 >= 1 = tt()
a__U31(tt(),V2) = 1 >= 1 = a__U32(a__isNatKind(V2))
a__U32(tt()) = 1 >= 1 = tt()
a__U41(tt()) = 1 >= 1 = tt()
a__U51(tt(),N) = 1 >= 1 = a__U52(a__isNatKind(N),N)
a__U52(tt(),N) = 1 >= 1 = mark(N)
a__U61(tt(),M,N) = 1 >= 1 = a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) = 1 >= 1 = a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) = 1 >= 1 = a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) = 1 >= 1 = s(a__plus(mark(N),mark(M)))
a__isNat(0()) = 1 >= 1 = tt()
a__isNat(plus(V1,V2)) = 1 >= 1 = a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) = 1 >= 1 = a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) = 1 >= 1 = tt()
a__isNatKind(plus(V1,V2)) = 1 >= 1 = a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) = 1 >= 1 = a__U41(a__isNatKind(V1))
a__plus(N,0()) = 1 >= 1 = a__U51(a__isNat(N),N)
a__plus(N,s(M)) = 1 >= 1 = a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) = 1 >= 1 = a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) = 1 >= 1 = a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) = 1 >= 1 = a__isNatKind(X)
mark(U13(X1,X2,X3)) = 1 >= 1 = a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) = 1 >= 1 = a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) = 1 >= 1 = a__U15(mark(X1),X2)
mark(isNat(X)) = 1 >= 1 = a__isNat(X)
mark(U16(X)) = 1 >= 1 = a__U16(mark(X))
mark(U21(X1,X2)) = 1 >= 1 = a__U21(mark(X1),X2)
mark(U22(X1,X2)) = 1 >= 1 = a__U22(mark(X1),X2)
mark(U23(X)) = 1 >= 1 = a__U23(mark(X))
mark(U31(X1,X2)) = 1 >= 1 = a__U31(mark(X1),X2)
mark(U32(X)) = 1 >= 1 = a__U32(mark(X))
mark(U41(X)) = 1 >= 1 = a__U41(mark(X))
mark(U51(X1,X2)) = 1 >= 1 = a__U51(mark(X1),X2)
mark(U52(X1,X2)) = 1 >= 1 = a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) = 1 >= 1 = a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) = 1 >= 1 = a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) = 1 >= 1 = a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) = 1 >= 1 = a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) = 1 >= 1 = a__plus(mark(X1),mark(X2))
mark(tt()) = 1 >= 1 = tt()
mark(s(X)) = 1 >= 1 = s(mark(X))
mark(0()) = 1 >= 1 = 0()
a__U11(X1,X2,X3) = 1 >= 1 = U11(X1,X2,X3)
a__U12(X1,X2,X3) = 1 >= 1 = U12(X1,X2,X3)
a__isNatKind(X) = 1 >= 1 = isNatKind(X)
a__U13(X1,X2,X3) = X1 >= 0 = U13(X1,X2,X3)
a__U14(X1,X2,X3) = 1 >= 1 = U14(X1,X2,X3)
a__U15(X1,X2) = 1 >= 1 = U15(X1,X2)
a__isNat(X) = 1 >= 1 = isNat(X)
a__U16(X) = 1 >= 1 = U16(X)
a__U21(X1,X2) = 1 >= 1 = U21(X1,X2)
a__U22(X1,X2) = 1 >= 1 = U22(X1,X2)
a__U23(X) = 1 >= 1 = U23(X)
a__U31(X1,X2) = 1 >= 1 = U31(X1,X2)
a__U32(X) = 1 >= 1 = U32(X)
a__U41(X) = X >= X = U41(X)
a__U51(X1,X2) = 1 >= 1 = U51(X1,X2)
a__U52(X1,X2) = 1 >= 1 = U52(X1,X2)
a__U61(X1,X2,X3) = 1 >= 1 = U61(X1,X2,X3)
a__U62(X1,X2,X3) = 1 >= 1 = U62(X1,X2,X3)
a__U63(X1,X2,X3) = 1 >= 1 = U63(X1,X2,X3)
a__U64(X1,X2,X3) = 1 >= 1 = U64(X1,X2,X3)
a__plus(X1,X2) = 1 >= 1 = plus(X1,X2)
problem:
DPs:
a__U12#(tt(),V1,V2) -> a__U13#(a__isNatKind(V2),V1,V2)
a__U13#(tt(),V1,V2) -> a__U14#(a__isNatKind(V2),V1,V2)
a__isNat#(plus(V1,V2)) -> a__U11#(a__isNatKind(V1),V1,V2)
a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
a__U14#(tt(),V1,V2) -> a__U15#(a__isNat(V1),V2)
TRS:
a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) -> a__U16(a__isNat(V2))
a__U16(tt()) -> tt()
a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) -> a__U23(a__isNat(V1))
a__U23(tt()) -> tt()
a__U31(tt(),V2) -> a__U32(a__isNatKind(V2))
a__U32(tt()) -> tt()
a__U41(tt()) -> tt()
a__U51(tt(),N) -> a__U52(a__isNatKind(N),N)
a__U52(tt(),N) -> mark(N)
a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
a__isNat(0()) -> tt()
a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) -> tt()
a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1))
a__plus(N,0()) -> a__U51(a__isNat(N),N)
a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) -> a__isNatKind(X)
mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) -> a__U15(mark(X1),X2)
mark(isNat(X)) -> a__isNat(X)
mark(U16(X)) -> a__U16(mark(X))
mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
mark(U23(X)) -> a__U23(mark(X))
mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
mark(U32(X)) -> a__U32(mark(X))
mark(U41(X)) -> a__U41(mark(X))
mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
mark(tt()) -> tt()
mark(s(X)) -> s(mark(X))
mark(0()) -> 0()
a__U11(X1,X2,X3) -> U11(X1,X2,X3)
a__U12(X1,X2,X3) -> U12(X1,X2,X3)
a__isNatKind(X) -> isNatKind(X)
a__U13(X1,X2,X3) -> U13(X1,X2,X3)
a__U14(X1,X2,X3) -> U14(X1,X2,X3)
a__U15(X1,X2) -> U15(X1,X2)
a__isNat(X) -> isNat(X)
a__U16(X) -> U16(X)
a__U21(X1,X2) -> U21(X1,X2)
a__U22(X1,X2) -> U22(X1,X2)
a__U23(X) -> U23(X)
a__U31(X1,X2) -> U31(X1,X2)
a__U32(X) -> U32(X)
a__U41(X) -> U41(X)
a__U51(X1,X2) -> U51(X1,X2)
a__U52(X1,X2) -> U52(X1,X2)
a__U61(X1,X2,X3) -> U61(X1,X2,X3)
a__U62(X1,X2,X3) -> U62(X1,X2,X3)
a__U63(X1,X2,X3) -> U63(X1,X2,X3)
a__U64(X1,X2,X3) -> U64(X1,X2,X3)
a__plus(X1,X2) -> plus(X1,X2)
Matrix Interpretation Processor:
dimension: 1
interpretation:
[a__U21#](x0, x1) = 0,
[a__U15#](x0, x1) = 0,
[a__isNat#](x0) = 0,
[a__U14#](x0, x1, x2) = x0,
[a__U13#](x0, x1, x2) = 1,
[a__U12#](x0, x1, x2) = 1,
[a__U11#](x0, x1, x2) = 0,
[U64](x0, x1, x2) = 1,
[U63](x0, x1, x2) = 1,
[U62](x0, x1, x2) = 1,
[U61](x0, x1, x2) = 1,
[U52](x0, x1) = x0,
[U51](x0, x1) = 1,
[U41](x0) = x0,
[U32](x0) = 1,
[U31](x0, x1) = 1,
[U23](x0) = 1,
[U22](x0, x1) = 1,
[U21](x0, x1) = 1,
[U16](x0) = 1,
[isNat](x0) = 1,
[U15](x0, x1) = 1,
[U14](x0, x1, x2) = 1,
[U13](x0, x1, x2) = 1,
[isNatKind](x0) = 1,
[U12](x0, x1, x2) = 1,
[U11](x0, x1, x2) = 0,
[plus](x0, x1) = 1,
[0] = 1,
[s](x0) = 1,
[a__plus](x0, x1) = 1,
[a__U64](x0, x1, x2) = 1,
[a__U63](x0, x1, x2) = 1,
[a__U62](x0, x1, x2) = 1,
[a__U61](x0, x1, x2) = 1,
[mark](x0) = 1,
[a__U52](x0, x1) = x0,
[a__U51](x0, x1) = 1,
[a__U41](x0) = x0,
[a__U32](x0) = 1,
[a__U31](x0, x1) = 1,
[a__U23](x0) = 1,
[a__U22](x0, x1) = 1,
[a__U21](x0, x1) = 1,
[a__U16](x0) = 1,
[a__U15](x0, x1) = 1,
[a__isNat](x0) = 1,
[a__U14](x0, x1, x2) = 1,
[a__U13](x0, x1, x2) = 1,
[a__U12](x0, x1, x2) = 1,
[a__isNatKind](x0) = 1,
[a__U11](x0, x1, x2) = x0,
[tt] = 1
orientation:
a__U12#(tt(),V1,V2) = 1 >= 1 = a__U13#(a__isNatKind(V2),V1,V2)
a__U13#(tt(),V1,V2) = 1 >= 1 = a__U14#(a__isNatKind(V2),V1,V2)
a__isNat#(plus(V1,V2)) = 0 >= 0 = a__U11#(a__isNatKind(V1),V1,V2)
a__isNat#(s(V1)) = 0 >= 0 = a__U21#(a__isNatKind(V1),V1)
a__U14#(tt(),V1,V2) = 1 >= 0 = a__U15#(a__isNat(V1),V2)
a__U11(tt(),V1,V2) = 1 >= 1 = a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) = 1 >= 1 = a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) = 1 >= 1 = a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) = 1 >= 1 = a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) = 1 >= 1 = a__U16(a__isNat(V2))
a__U16(tt()) = 1 >= 1 = tt()
a__U21(tt(),V1) = 1 >= 1 = a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) = 1 >= 1 = a__U23(a__isNat(V1))
a__U23(tt()) = 1 >= 1 = tt()
a__U31(tt(),V2) = 1 >= 1 = a__U32(a__isNatKind(V2))
a__U32(tt()) = 1 >= 1 = tt()
a__U41(tt()) = 1 >= 1 = tt()
a__U51(tt(),N) = 1 >= 1 = a__U52(a__isNatKind(N),N)
a__U52(tt(),N) = 1 >= 1 = mark(N)
a__U61(tt(),M,N) = 1 >= 1 = a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) = 1 >= 1 = a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) = 1 >= 1 = a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) = 1 >= 1 = s(a__plus(mark(N),mark(M)))
a__isNat(0()) = 1 >= 1 = tt()
a__isNat(plus(V1,V2)) = 1 >= 1 = a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) = 1 >= 1 = a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) = 1 >= 1 = tt()
a__isNatKind(plus(V1,V2)) = 1 >= 1 = a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) = 1 >= 1 = a__U41(a__isNatKind(V1))
a__plus(N,0()) = 1 >= 1 = a__U51(a__isNat(N),N)
a__plus(N,s(M)) = 1 >= 1 = a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) = 1 >= 1 = a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) = 1 >= 1 = a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) = 1 >= 1 = a__isNatKind(X)
mark(U13(X1,X2,X3)) = 1 >= 1 = a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) = 1 >= 1 = a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) = 1 >= 1 = a__U15(mark(X1),X2)
mark(isNat(X)) = 1 >= 1 = a__isNat(X)
mark(U16(X)) = 1 >= 1 = a__U16(mark(X))
mark(U21(X1,X2)) = 1 >= 1 = a__U21(mark(X1),X2)
mark(U22(X1,X2)) = 1 >= 1 = a__U22(mark(X1),X2)
mark(U23(X)) = 1 >= 1 = a__U23(mark(X))
mark(U31(X1,X2)) = 1 >= 1 = a__U31(mark(X1),X2)
mark(U32(X)) = 1 >= 1 = a__U32(mark(X))
mark(U41(X)) = 1 >= 1 = a__U41(mark(X))
mark(U51(X1,X2)) = 1 >= 1 = a__U51(mark(X1),X2)
mark(U52(X1,X2)) = 1 >= 1 = a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) = 1 >= 1 = a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) = 1 >= 1 = a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) = 1 >= 1 = a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) = 1 >= 1 = a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) = 1 >= 1 = a__plus(mark(X1),mark(X2))
mark(tt()) = 1 >= 1 = tt()
mark(s(X)) = 1 >= 1 = s(mark(X))
mark(0()) = 1 >= 1 = 0()
a__U11(X1,X2,X3) = X1 >= 0 = U11(X1,X2,X3)
a__U12(X1,X2,X3) = 1 >= 1 = U12(X1,X2,X3)
a__isNatKind(X) = 1 >= 1 = isNatKind(X)
a__U13(X1,X2,X3) = 1 >= 1 = U13(X1,X2,X3)
a__U14(X1,X2,X3) = 1 >= 1 = U14(X1,X2,X3)
a__U15(X1,X2) = 1 >= 1 = U15(X1,X2)
a__isNat(X) = 1 >= 1 = isNat(X)
a__U16(X) = 1 >= 1 = U16(X)
a__U21(X1,X2) = 1 >= 1 = U21(X1,X2)
a__U22(X1,X2) = 1 >= 1 = U22(X1,X2)
a__U23(X) = 1 >= 1 = U23(X)
a__U31(X1,X2) = 1 >= 1 = U31(X1,X2)
a__U32(X) = 1 >= 1 = U32(X)
a__U41(X) = X >= X = U41(X)
a__U51(X1,X2) = 1 >= 1 = U51(X1,X2)
a__U52(X1,X2) = X1 >= X1 = U52(X1,X2)
a__U61(X1,X2,X3) = 1 >= 1 = U61(X1,X2,X3)
a__U62(X1,X2,X3) = 1 >= 1 = U62(X1,X2,X3)
a__U63(X1,X2,X3) = 1 >= 1 = U63(X1,X2,X3)
a__U64(X1,X2,X3) = 1 >= 1 = U64(X1,X2,X3)
a__plus(X1,X2) = 1 >= 1 = plus(X1,X2)
problem:
DPs:
a__U12#(tt(),V1,V2) -> a__U13#(a__isNatKind(V2),V1,V2)
a__U13#(tt(),V1,V2) -> a__U14#(a__isNatKind(V2),V1,V2)
a__isNat#(plus(V1,V2)) -> a__U11#(a__isNatKind(V1),V1,V2)
a__isNat#(s(V1)) -> a__U21#(a__isNatKind(V1),V1)
TRS:
a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) -> a__U16(a__isNat(V2))
a__U16(tt()) -> tt()
a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) -> a__U23(a__isNat(V1))
a__U23(tt()) -> tt()
a__U31(tt(),V2) -> a__U32(a__isNatKind(V2))
a__U32(tt()) -> tt()
a__U41(tt()) -> tt()
a__U51(tt(),N) -> a__U52(a__isNatKind(N),N)
a__U52(tt(),N) -> mark(N)
a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
a__isNat(0()) -> tt()
a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) -> tt()
a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1))
a__plus(N,0()) -> a__U51(a__isNat(N),N)
a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) -> a__isNatKind(X)
mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) -> a__U15(mark(X1),X2)
mark(isNat(X)) -> a__isNat(X)
mark(U16(X)) -> a__U16(mark(X))
mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
mark(U23(X)) -> a__U23(mark(X))
mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
mark(U32(X)) -> a__U32(mark(X))
mark(U41(X)) -> a__U41(mark(X))
mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
mark(tt()) -> tt()
mark(s(X)) -> s(mark(X))
mark(0()) -> 0()
a__U11(X1,X2,X3) -> U11(X1,X2,X3)
a__U12(X1,X2,X3) -> U12(X1,X2,X3)
a__isNatKind(X) -> isNatKind(X)
a__U13(X1,X2,X3) -> U13(X1,X2,X3)
a__U14(X1,X2,X3) -> U14(X1,X2,X3)
a__U15(X1,X2) -> U15(X1,X2)
a__isNat(X) -> isNat(X)
a__U16(X) -> U16(X)
a__U21(X1,X2) -> U21(X1,X2)
a__U22(X1,X2) -> U22(X1,X2)
a__U23(X) -> U23(X)
a__U31(X1,X2) -> U31(X1,X2)
a__U32(X) -> U32(X)
a__U41(X) -> U41(X)
a__U51(X1,X2) -> U51(X1,X2)
a__U52(X1,X2) -> U52(X1,X2)
a__U61(X1,X2,X3) -> U61(X1,X2,X3)
a__U62(X1,X2,X3) -> U62(X1,X2,X3)
a__U63(X1,X2,X3) -> U63(X1,X2,X3)
a__U64(X1,X2,X3) -> U64(X1,X2,X3)
a__plus(X1,X2) -> plus(X1,X2)
Matrix Interpretation Processor:
dimension: 1
interpretation:
[a__U21#](x0, x1) = 0,
[a__isNat#](x0) = 1,
[a__U14#](x0, x1, x2) = 0,
[a__U13#](x0, x1, x2) = 0,
[a__U12#](x0, x1, x2) = 0,
[a__U11#](x0, x1, x2) = 1,
[U64](x0, x1, x2) = 0,
[U63](x0, x1, x2) = 0,
[U62](x0, x1, x2) = 0,
[U61](x0, x1, x2) = 0,
[U52](x0, x1) = 0,
[U51](x0, x1) = 0,
[U41](x0) = 0,
[U32](x0) = 0,
[U31](x0, x1) = 0,
[U23](x0) = 0,
[U22](x0, x1) = 0,
[U21](x0, x1) = 0,
[U16](x0) = 0,
[isNat](x0) = 0,
[U15](x0, x1) = 0,
[U14](x0, x1, x2) = 0,
[U13](x0, x1, x2) = 0,
[isNatKind](x0) = 0,
[U12](x0, x1, x2) = 0,
[U11](x0, x1, x2) = 0,
[plus](x0, x1) = 0,
[0] = 0,
[s](x0) = 0,
[a__plus](x0, x1) = 0,
[a__U64](x0, x1, x2) = 0,
[a__U63](x0, x1, x2) = 0,
[a__U62](x0, x1, x2) = 0,
[a__U61](x0, x1, x2) = 0,
[mark](x0) = 0,
[a__U52](x0, x1) = 0,
[a__U51](x0, x1) = 0,
[a__U41](x0) = 0,
[a__U32](x0) = 0,
[a__U31](x0, x1) = 0,
[a__U23](x0) = 0,
[a__U22](x0, x1) = 0,
[a__U21](x0, x1) = 0,
[a__U16](x0) = 0,
[a__U15](x0, x1) = 0,
[a__isNat](x0) = 0,
[a__U14](x0, x1, x2) = 0,
[a__U13](x0, x1, x2) = 0,
[a__U12](x0, x1, x2) = 0,
[a__isNatKind](x0) = 0,
[a__U11](x0, x1, x2) = 0,
[tt] = 0
orientation:
a__U12#(tt(),V1,V2) = 0 >= 0 = a__U13#(a__isNatKind(V2),V1,V2)
a__U13#(tt(),V1,V2) = 0 >= 0 = a__U14#(a__isNatKind(V2),V1,V2)
a__isNat#(plus(V1,V2)) = 1 >= 1 = a__U11#(a__isNatKind(V1),V1,V2)
a__isNat#(s(V1)) = 1 >= 0 = a__U21#(a__isNatKind(V1),V1)
a__U11(tt(),V1,V2) = 0 >= 0 = a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) = 0 >= 0 = a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) = 0 >= 0 = a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) = 0 >= 0 = a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) = 0 >= 0 = a__U16(a__isNat(V2))
a__U16(tt()) = 0 >= 0 = tt()
a__U21(tt(),V1) = 0 >= 0 = a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) = 0 >= 0 = a__U23(a__isNat(V1))
a__U23(tt()) = 0 >= 0 = tt()
a__U31(tt(),V2) = 0 >= 0 = a__U32(a__isNatKind(V2))
a__U32(tt()) = 0 >= 0 = tt()
a__U41(tt()) = 0 >= 0 = tt()
a__U51(tt(),N) = 0 >= 0 = a__U52(a__isNatKind(N),N)
a__U52(tt(),N) = 0 >= 0 = mark(N)
a__U61(tt(),M,N) = 0 >= 0 = a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) = 0 >= 0 = a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) = 0 >= 0 = a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) = 0 >= 0 = s(a__plus(mark(N),mark(M)))
a__isNat(0()) = 0 >= 0 = tt()
a__isNat(plus(V1,V2)) = 0 >= 0 = a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) = 0 >= 0 = a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) = 0 >= 0 = tt()
a__isNatKind(plus(V1,V2)) = 0 >= 0 = a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) = 0 >= 0 = a__U41(a__isNatKind(V1))
a__plus(N,0()) = 0 >= 0 = a__U51(a__isNat(N),N)
a__plus(N,s(M)) = 0 >= 0 = a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) = 0 >= 0 = a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) = 0 >= 0 = a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) = 0 >= 0 = a__isNatKind(X)
mark(U13(X1,X2,X3)) = 0 >= 0 = a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) = 0 >= 0 = a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) = 0 >= 0 = a__U15(mark(X1),X2)
mark(isNat(X)) = 0 >= 0 = a__isNat(X)
mark(U16(X)) = 0 >= 0 = a__U16(mark(X))
mark(U21(X1,X2)) = 0 >= 0 = a__U21(mark(X1),X2)
mark(U22(X1,X2)) = 0 >= 0 = a__U22(mark(X1),X2)
mark(U23(X)) = 0 >= 0 = a__U23(mark(X))
mark(U31(X1,X2)) = 0 >= 0 = a__U31(mark(X1),X2)
mark(U32(X)) = 0 >= 0 = a__U32(mark(X))
mark(U41(X)) = 0 >= 0 = a__U41(mark(X))
mark(U51(X1,X2)) = 0 >= 0 = a__U51(mark(X1),X2)
mark(U52(X1,X2)) = 0 >= 0 = a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) = 0 >= 0 = a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) = 0 >= 0 = a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) = 0 >= 0 = a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) = 0 >= 0 = a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) = 0 >= 0 = a__plus(mark(X1),mark(X2))
mark(tt()) = 0 >= 0 = tt()
mark(s(X)) = 0 >= 0 = s(mark(X))
mark(0()) = 0 >= 0 = 0()
a__U11(X1,X2,X3) = 0 >= 0 = U11(X1,X2,X3)
a__U12(X1,X2,X3) = 0 >= 0 = U12(X1,X2,X3)
a__isNatKind(X) = 0 >= 0 = isNatKind(X)
a__U13(X1,X2,X3) = 0 >= 0 = U13(X1,X2,X3)
a__U14(X1,X2,X3) = 0 >= 0 = U14(X1,X2,X3)
a__U15(X1,X2) = 0 >= 0 = U15(X1,X2)
a__isNat(X) = 0 >= 0 = isNat(X)
a__U16(X) = 0 >= 0 = U16(X)
a__U21(X1,X2) = 0 >= 0 = U21(X1,X2)
a__U22(X1,X2) = 0 >= 0 = U22(X1,X2)
a__U23(X) = 0 >= 0 = U23(X)
a__U31(X1,X2) = 0 >= 0 = U31(X1,X2)
a__U32(X) = 0 >= 0 = U32(X)
a__U41(X) = 0 >= 0 = U41(X)
a__U51(X1,X2) = 0 >= 0 = U51(X1,X2)
a__U52(X1,X2) = 0 >= 0 = U52(X1,X2)
a__U61(X1,X2,X3) = 0 >= 0 = U61(X1,X2,X3)
a__U62(X1,X2,X3) = 0 >= 0 = U62(X1,X2,X3)
a__U63(X1,X2,X3) = 0 >= 0 = U63(X1,X2,X3)
a__U64(X1,X2,X3) = 0 >= 0 = U64(X1,X2,X3)
a__plus(X1,X2) = 0 >= 0 = plus(X1,X2)
problem:
DPs:
a__U12#(tt(),V1,V2) -> a__U13#(a__isNatKind(V2),V1,V2)
a__U13#(tt(),V1,V2) -> a__U14#(a__isNatKind(V2),V1,V2)
a__isNat#(plus(V1,V2)) -> a__U11#(a__isNatKind(V1),V1,V2)
TRS:
a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) -> a__U16(a__isNat(V2))
a__U16(tt()) -> tt()
a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) -> a__U23(a__isNat(V1))
a__U23(tt()) -> tt()
a__U31(tt(),V2) -> a__U32(a__isNatKind(V2))
a__U32(tt()) -> tt()
a__U41(tt()) -> tt()
a__U51(tt(),N) -> a__U52(a__isNatKind(N),N)
a__U52(tt(),N) -> mark(N)
a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
a__isNat(0()) -> tt()
a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) -> tt()
a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1))
a__plus(N,0()) -> a__U51(a__isNat(N),N)
a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) -> a__isNatKind(X)
mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) -> a__U15(mark(X1),X2)
mark(isNat(X)) -> a__isNat(X)
mark(U16(X)) -> a__U16(mark(X))
mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
mark(U23(X)) -> a__U23(mark(X))
mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
mark(U32(X)) -> a__U32(mark(X))
mark(U41(X)) -> a__U41(mark(X))
mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
mark(tt()) -> tt()
mark(s(X)) -> s(mark(X))
mark(0()) -> 0()
a__U11(X1,X2,X3) -> U11(X1,X2,X3)
a__U12(X1,X2,X3) -> U12(X1,X2,X3)
a__isNatKind(X) -> isNatKind(X)
a__U13(X1,X2,X3) -> U13(X1,X2,X3)
a__U14(X1,X2,X3) -> U14(X1,X2,X3)
a__U15(X1,X2) -> U15(X1,X2)
a__isNat(X) -> isNat(X)
a__U16(X) -> U16(X)
a__U21(X1,X2) -> U21(X1,X2)
a__U22(X1,X2) -> U22(X1,X2)
a__U23(X) -> U23(X)
a__U31(X1,X2) -> U31(X1,X2)
a__U32(X) -> U32(X)
a__U41(X) -> U41(X)
a__U51(X1,X2) -> U51(X1,X2)
a__U52(X1,X2) -> U52(X1,X2)
a__U61(X1,X2,X3) -> U61(X1,X2,X3)
a__U62(X1,X2,X3) -> U62(X1,X2,X3)
a__U63(X1,X2,X3) -> U63(X1,X2,X3)
a__U64(X1,X2,X3) -> U64(X1,X2,X3)
a__plus(X1,X2) -> plus(X1,X2)
Matrix Interpretation Processor:
dimension: 1
interpretation:
[a__isNat#](x0) = 1,
[a__U14#](x0, x1, x2) = 0,
[a__U13#](x0, x1, x2) = 0,
[a__U12#](x0, x1, x2) = 0,
[a__U11#](x0, x1, x2) = 0,
[U64](x0, x1, x2) = 0,
[U63](x0, x1, x2) = 0,
[U62](x0, x1, x2) = 0,
[U61](x0, x1, x2) = 0,
[U52](x0, x1) = 0,
[U51](x0, x1) = 0,
[U41](x0) = 0,
[U32](x0) = 0,
[U31](x0, x1) = 0,
[U23](x0) = 0,
[U22](x0, x1) = 0,
[U21](x0, x1) = 0,
[U16](x0) = 0,
[isNat](x0) = 0,
[U15](x0, x1) = 0,
[U14](x0, x1, x2) = 0,
[U13](x0, x1, x2) = 0,
[isNatKind](x0) = 0,
[U12](x0, x1, x2) = 0,
[U11](x0, x1, x2) = 0,
[plus](x0, x1) = 0,
[0] = 0,
[s](x0) = 0,
[a__plus](x0, x1) = 0,
[a__U64](x0, x1, x2) = 0,
[a__U63](x0, x1, x2) = 0,
[a__U62](x0, x1, x2) = 0,
[a__U61](x0, x1, x2) = 0,
[mark](x0) = 0,
[a__U52](x0, x1) = 0,
[a__U51](x0, x1) = 0,
[a__U41](x0) = 0,
[a__U32](x0) = 0,
[a__U31](x0, x1) = 0,
[a__U23](x0) = 0,
[a__U22](x0, x1) = 0,
[a__U21](x0, x1) = 0,
[a__U16](x0) = 0,
[a__U15](x0, x1) = 0,
[a__isNat](x0) = 0,
[a__U14](x0, x1, x2) = 0,
[a__U13](x0, x1, x2) = 0,
[a__U12](x0, x1, x2) = 0,
[a__isNatKind](x0) = 0,
[a__U11](x0, x1, x2) = 0,
[tt] = 0
orientation:
a__U12#(tt(),V1,V2) = 0 >= 0 = a__U13#(a__isNatKind(V2),V1,V2)
a__U13#(tt(),V1,V2) = 0 >= 0 = a__U14#(a__isNatKind(V2),V1,V2)
a__isNat#(plus(V1,V2)) = 1 >= 0 = a__U11#(a__isNatKind(V1),V1,V2)
a__U11(tt(),V1,V2) = 0 >= 0 = a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) = 0 >= 0 = a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) = 0 >= 0 = a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) = 0 >= 0 = a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) = 0 >= 0 = a__U16(a__isNat(V2))
a__U16(tt()) = 0 >= 0 = tt()
a__U21(tt(),V1) = 0 >= 0 = a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) = 0 >= 0 = a__U23(a__isNat(V1))
a__U23(tt()) = 0 >= 0 = tt()
a__U31(tt(),V2) = 0 >= 0 = a__U32(a__isNatKind(V2))
a__U32(tt()) = 0 >= 0 = tt()
a__U41(tt()) = 0 >= 0 = tt()
a__U51(tt(),N) = 0 >= 0 = a__U52(a__isNatKind(N),N)
a__U52(tt(),N) = 0 >= 0 = mark(N)
a__U61(tt(),M,N) = 0 >= 0 = a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) = 0 >= 0 = a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) = 0 >= 0 = a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) = 0 >= 0 = s(a__plus(mark(N),mark(M)))
a__isNat(0()) = 0 >= 0 = tt()
a__isNat(plus(V1,V2)) = 0 >= 0 = a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) = 0 >= 0 = a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) = 0 >= 0 = tt()
a__isNatKind(plus(V1,V2)) = 0 >= 0 = a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) = 0 >= 0 = a__U41(a__isNatKind(V1))
a__plus(N,0()) = 0 >= 0 = a__U51(a__isNat(N),N)
a__plus(N,s(M)) = 0 >= 0 = a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) = 0 >= 0 = a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) = 0 >= 0 = a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) = 0 >= 0 = a__isNatKind(X)
mark(U13(X1,X2,X3)) = 0 >= 0 = a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) = 0 >= 0 = a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) = 0 >= 0 = a__U15(mark(X1),X2)
mark(isNat(X)) = 0 >= 0 = a__isNat(X)
mark(U16(X)) = 0 >= 0 = a__U16(mark(X))
mark(U21(X1,X2)) = 0 >= 0 = a__U21(mark(X1),X2)
mark(U22(X1,X2)) = 0 >= 0 = a__U22(mark(X1),X2)
mark(U23(X)) = 0 >= 0 = a__U23(mark(X))
mark(U31(X1,X2)) = 0 >= 0 = a__U31(mark(X1),X2)
mark(U32(X)) = 0 >= 0 = a__U32(mark(X))
mark(U41(X)) = 0 >= 0 = a__U41(mark(X))
mark(U51(X1,X2)) = 0 >= 0 = a__U51(mark(X1),X2)
mark(U52(X1,X2)) = 0 >= 0 = a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) = 0 >= 0 = a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) = 0 >= 0 = a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) = 0 >= 0 = a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) = 0 >= 0 = a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) = 0 >= 0 = a__plus(mark(X1),mark(X2))
mark(tt()) = 0 >= 0 = tt()
mark(s(X)) = 0 >= 0 = s(mark(X))
mark(0()) = 0 >= 0 = 0()
a__U11(X1,X2,X3) = 0 >= 0 = U11(X1,X2,X3)
a__U12(X1,X2,X3) = 0 >= 0 = U12(X1,X2,X3)
a__isNatKind(X) = 0 >= 0 = isNatKind(X)
a__U13(X1,X2,X3) = 0 >= 0 = U13(X1,X2,X3)
a__U14(X1,X2,X3) = 0 >= 0 = U14(X1,X2,X3)
a__U15(X1,X2) = 0 >= 0 = U15(X1,X2)
a__isNat(X) = 0 >= 0 = isNat(X)
a__U16(X) = 0 >= 0 = U16(X)
a__U21(X1,X2) = 0 >= 0 = U21(X1,X2)
a__U22(X1,X2) = 0 >= 0 = U22(X1,X2)
a__U23(X) = 0 >= 0 = U23(X)
a__U31(X1,X2) = 0 >= 0 = U31(X1,X2)
a__U32(X) = 0 >= 0 = U32(X)
a__U41(X) = 0 >= 0 = U41(X)
a__U51(X1,X2) = 0 >= 0 = U51(X1,X2)
a__U52(X1,X2) = 0 >= 0 = U52(X1,X2)
a__U61(X1,X2,X3) = 0 >= 0 = U61(X1,X2,X3)
a__U62(X1,X2,X3) = 0 >= 0 = U62(X1,X2,X3)
a__U63(X1,X2,X3) = 0 >= 0 = U63(X1,X2,X3)
a__U64(X1,X2,X3) = 0 >= 0 = U64(X1,X2,X3)
a__plus(X1,X2) = 0 >= 0 = plus(X1,X2)
problem:
DPs:
a__U12#(tt(),V1,V2) -> a__U13#(a__isNatKind(V2),V1,V2)
a__U13#(tt(),V1,V2) -> a__U14#(a__isNatKind(V2),V1,V2)
TRS:
a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) -> a__U16(a__isNat(V2))
a__U16(tt()) -> tt()
a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) -> a__U23(a__isNat(V1))
a__U23(tt()) -> tt()
a__U31(tt(),V2) -> a__U32(a__isNatKind(V2))
a__U32(tt()) -> tt()
a__U41(tt()) -> tt()
a__U51(tt(),N) -> a__U52(a__isNatKind(N),N)
a__U52(tt(),N) -> mark(N)
a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
a__isNat(0()) -> tt()
a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) -> tt()
a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1))
a__plus(N,0()) -> a__U51(a__isNat(N),N)
a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) -> a__isNatKind(X)
mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) -> a__U15(mark(X1),X2)
mark(isNat(X)) -> a__isNat(X)
mark(U16(X)) -> a__U16(mark(X))
mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
mark(U23(X)) -> a__U23(mark(X))
mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
mark(U32(X)) -> a__U32(mark(X))
mark(U41(X)) -> a__U41(mark(X))
mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
mark(tt()) -> tt()
mark(s(X)) -> s(mark(X))
mark(0()) -> 0()
a__U11(X1,X2,X3) -> U11(X1,X2,X3)
a__U12(X1,X2,X3) -> U12(X1,X2,X3)
a__isNatKind(X) -> isNatKind(X)
a__U13(X1,X2,X3) -> U13(X1,X2,X3)
a__U14(X1,X2,X3) -> U14(X1,X2,X3)
a__U15(X1,X2) -> U15(X1,X2)
a__isNat(X) -> isNat(X)
a__U16(X) -> U16(X)
a__U21(X1,X2) -> U21(X1,X2)
a__U22(X1,X2) -> U22(X1,X2)
a__U23(X) -> U23(X)
a__U31(X1,X2) -> U31(X1,X2)
a__U32(X) -> U32(X)
a__U41(X) -> U41(X)
a__U51(X1,X2) -> U51(X1,X2)
a__U52(X1,X2) -> U52(X1,X2)
a__U61(X1,X2,X3) -> U61(X1,X2,X3)
a__U62(X1,X2,X3) -> U62(X1,X2,X3)
a__U63(X1,X2,X3) -> U63(X1,X2,X3)
a__U64(X1,X2,X3) -> U64(X1,X2,X3)
a__plus(X1,X2) -> plus(X1,X2)
Matrix Interpretation Processor:
dimension: 1
interpretation:
[a__U14#](x0, x1, x2) = 0,
[a__U13#](x0, x1, x2) = 0,
[a__U12#](x0, x1, x2) = x0 + 1,
[U64](x0, x1, x2) = 0,
[U63](x0, x1, x2) = 0,
[U62](x0, x1, x2) = 0,
[U61](x0, x1, x2) = 0,
[U52](x0, x1) = 1,
[U51](x0, x1) = 0,
[U41](x0) = 1,
[U32](x0) = 1,
[U31](x0, x1) = 1,
[U23](x0) = 1,
[U22](x0, x1) = 1,
[U21](x0, x1) = 1,
[U16](x0) = 1,
[isNat](x0) = 1,
[U15](x0, x1) = 1,
[U14](x0, x1, x2) = 0,
[U13](x0, x1, x2) = 0,
[isNatKind](x0) = 1,
[U12](x0, x1, x2) = 1,
[U11](x0, x1, x2) = x0,
[plus](x0, x1) = x1,
[0] = 1,
[s](x0) = 0,
[a__plus](x0, x1) = x1,
[a__U64](x0, x1, x2) = 0,
[a__U63](x0, x1, x2) = 0,
[a__U62](x0, x1, x2) = 0,
[a__U61](x0, x1, x2) = 0,
[mark](x0) = 1,
[a__U52](x0, x1) = 1,
[a__U51](x0, x1) = x0,
[a__U41](x0) = 1,
[a__U32](x0) = 1,
[a__U31](x0, x1) = 1,
[a__U23](x0) = 1,
[a__U22](x0, x1) = 1,
[a__U21](x0, x1) = 1,
[a__U16](x0) = 1,
[a__U15](x0, x1) = 1,
[a__isNat](x0) = 1,
[a__U14](x0, x1, x2) = 1,
[a__U13](x0, x1, x2) = 1,
[a__U12](x0, x1, x2) = 1,
[a__isNatKind](x0) = 1,
[a__U11](x0, x1, x2) = x0,
[tt] = 1
orientation:
a__U12#(tt(),V1,V2) = 2 >= 0 = a__U13#(a__isNatKind(V2),V1,V2)
a__U13#(tt(),V1,V2) = 0 >= 0 = a__U14#(a__isNatKind(V2),V1,V2)
a__U11(tt(),V1,V2) = 1 >= 1 = a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) = 1 >= 1 = a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) = 1 >= 1 = a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) = 1 >= 1 = a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) = 1 >= 1 = a__U16(a__isNat(V2))
a__U16(tt()) = 1 >= 1 = tt()
a__U21(tt(),V1) = 1 >= 1 = a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) = 1 >= 1 = a__U23(a__isNat(V1))
a__U23(tt()) = 1 >= 1 = tt()
a__U31(tt(),V2) = 1 >= 1 = a__U32(a__isNatKind(V2))
a__U32(tt()) = 1 >= 1 = tt()
a__U41(tt()) = 1 >= 1 = tt()
a__U51(tt(),N) = 1 >= 1 = a__U52(a__isNatKind(N),N)
a__U52(tt(),N) = 1 >= 1 = mark(N)
a__U61(tt(),M,N) = 0 >= 0 = a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) = 0 >= 0 = a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) = 0 >= 0 = a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) = 0 >= 0 = s(a__plus(mark(N),mark(M)))
a__isNat(0()) = 1 >= 1 = tt()
a__isNat(plus(V1,V2)) = 1 >= 1 = a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) = 1 >= 1 = a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) = 1 >= 1 = tt()
a__isNatKind(plus(V1,V2)) = 1 >= 1 = a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) = 1 >= 1 = a__U41(a__isNatKind(V1))
a__plus(N,0()) = 1 >= 1 = a__U51(a__isNat(N),N)
a__plus(N,s(M)) = 0 >= 0 = a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) = 1 >= 1 = a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) = 1 >= 1 = a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) = 1 >= 1 = a__isNatKind(X)
mark(U13(X1,X2,X3)) = 1 >= 1 = a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) = 1 >= 1 = a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) = 1 >= 1 = a__U15(mark(X1),X2)
mark(isNat(X)) = 1 >= 1 = a__isNat(X)
mark(U16(X)) = 1 >= 1 = a__U16(mark(X))
mark(U21(X1,X2)) = 1 >= 1 = a__U21(mark(X1),X2)
mark(U22(X1,X2)) = 1 >= 1 = a__U22(mark(X1),X2)
mark(U23(X)) = 1 >= 1 = a__U23(mark(X))
mark(U31(X1,X2)) = 1 >= 1 = a__U31(mark(X1),X2)
mark(U32(X)) = 1 >= 1 = a__U32(mark(X))
mark(U41(X)) = 1 >= 1 = a__U41(mark(X))
mark(U51(X1,X2)) = 1 >= 1 = a__U51(mark(X1),X2)
mark(U52(X1,X2)) = 1 >= 1 = a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) = 1 >= 0 = a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) = 1 >= 0 = a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) = 1 >= 0 = a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) = 1 >= 0 = a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) = 1 >= 1 = a__plus(mark(X1),mark(X2))
mark(tt()) = 1 >= 1 = tt()
mark(s(X)) = 1 >= 0 = s(mark(X))
mark(0()) = 1 >= 1 = 0()
a__U11(X1,X2,X3) = X1 >= X1 = U11(X1,X2,X3)
a__U12(X1,X2,X3) = 1 >= 1 = U12(X1,X2,X3)
a__isNatKind(X) = 1 >= 1 = isNatKind(X)
a__U13(X1,X2,X3) = 1 >= 0 = U13(X1,X2,X3)
a__U14(X1,X2,X3) = 1 >= 0 = U14(X1,X2,X3)
a__U15(X1,X2) = 1 >= 1 = U15(X1,X2)
a__isNat(X) = 1 >= 1 = isNat(X)
a__U16(X) = 1 >= 1 = U16(X)
a__U21(X1,X2) = 1 >= 1 = U21(X1,X2)
a__U22(X1,X2) = 1 >= 1 = U22(X1,X2)
a__U23(X) = 1 >= 1 = U23(X)
a__U31(X1,X2) = 1 >= 1 = U31(X1,X2)
a__U32(X) = 1 >= 1 = U32(X)
a__U41(X) = 1 >= 1 = U41(X)
a__U51(X1,X2) = X1 >= 0 = U51(X1,X2)
a__U52(X1,X2) = 1 >= 1 = U52(X1,X2)
a__U61(X1,X2,X3) = 0 >= 0 = U61(X1,X2,X3)
a__U62(X1,X2,X3) = 0 >= 0 = U62(X1,X2,X3)
a__U63(X1,X2,X3) = 0 >= 0 = U63(X1,X2,X3)
a__U64(X1,X2,X3) = 0 >= 0 = U64(X1,X2,X3)
a__plus(X1,X2) = X2 >= X2 = plus(X1,X2)
problem:
DPs:
a__U13#(tt(),V1,V2) -> a__U14#(a__isNatKind(V2),V1,V2)
TRS:
a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) -> a__U16(a__isNat(V2))
a__U16(tt()) -> tt()
a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) -> a__U23(a__isNat(V1))
a__U23(tt()) -> tt()
a__U31(tt(),V2) -> a__U32(a__isNatKind(V2))
a__U32(tt()) -> tt()
a__U41(tt()) -> tt()
a__U51(tt(),N) -> a__U52(a__isNatKind(N),N)
a__U52(tt(),N) -> mark(N)
a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
a__isNat(0()) -> tt()
a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) -> tt()
a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1))
a__plus(N,0()) -> a__U51(a__isNat(N),N)
a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) -> a__isNatKind(X)
mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) -> a__U15(mark(X1),X2)
mark(isNat(X)) -> a__isNat(X)
mark(U16(X)) -> a__U16(mark(X))
mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
mark(U23(X)) -> a__U23(mark(X))
mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
mark(U32(X)) -> a__U32(mark(X))
mark(U41(X)) -> a__U41(mark(X))
mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
mark(tt()) -> tt()
mark(s(X)) -> s(mark(X))
mark(0()) -> 0()
a__U11(X1,X2,X3) -> U11(X1,X2,X3)
a__U12(X1,X2,X3) -> U12(X1,X2,X3)
a__isNatKind(X) -> isNatKind(X)
a__U13(X1,X2,X3) -> U13(X1,X2,X3)
a__U14(X1,X2,X3) -> U14(X1,X2,X3)
a__U15(X1,X2) -> U15(X1,X2)
a__isNat(X) -> isNat(X)
a__U16(X) -> U16(X)
a__U21(X1,X2) -> U21(X1,X2)
a__U22(X1,X2) -> U22(X1,X2)
a__U23(X) -> U23(X)
a__U31(X1,X2) -> U31(X1,X2)
a__U32(X) -> U32(X)
a__U41(X) -> U41(X)
a__U51(X1,X2) -> U51(X1,X2)
a__U52(X1,X2) -> U52(X1,X2)
a__U61(X1,X2,X3) -> U61(X1,X2,X3)
a__U62(X1,X2,X3) -> U62(X1,X2,X3)
a__U63(X1,X2,X3) -> U63(X1,X2,X3)
a__U64(X1,X2,X3) -> U64(X1,X2,X3)
a__plus(X1,X2) -> plus(X1,X2)
Matrix Interpretation Processor:
dimension: 1
interpretation:
[a__U14#](x0, x1, x2) = 0,
[a__U13#](x0, x1, x2) = x2 + 1,
[U64](x0, x1, x2) = 0,
[U63](x0, x1, x2) = 0,
[U62](x0, x1, x2) = 0,
[U61](x0, x1, x2) = 0,
[U52](x0, x1) = 0,
[U51](x0, x1) = 0,
[U41](x0) = 0,
[U32](x0) = 0,
[U31](x0, x1) = 0,
[U23](x0) = 0,
[U22](x0, x1) = 0,
[U21](x0, x1) = 0,
[U16](x0) = 0,
[isNat](x0) = 0,
[U15](x0, x1) = 0,
[U14](x0, x1, x2) = 0,
[U13](x0, x1, x2) = 0,
[isNatKind](x0) = 0,
[U12](x0, x1, x2) = 0,
[U11](x0, x1, x2) = 0,
[plus](x0, x1) = 0,
[0] = 0,
[s](x0) = 0,
[a__plus](x0, x1) = 0,
[a__U64](x0, x1, x2) = 0,
[a__U63](x0, x1, x2) = 0,
[a__U62](x0, x1, x2) = 0,
[a__U61](x0, x1, x2) = x0,
[mark](x0) = 0,
[a__U52](x0, x1) = 0,
[a__U51](x0, x1) = 0,
[a__U41](x0) = 0,
[a__U32](x0) = 0,
[a__U31](x0, x1) = 0,
[a__U23](x0) = 0,
[a__U22](x0, x1) = 0,
[a__U21](x0, x1) = 0,
[a__U16](x0) = 0,
[a__U15](x0, x1) = 0,
[a__isNat](x0) = 0,
[a__U14](x0, x1, x2) = 0,
[a__U13](x0, x1, x2) = 0,
[a__U12](x0, x1, x2) = 0,
[a__isNatKind](x0) = 0,
[a__U11](x0, x1, x2) = 0,
[tt] = 0
orientation:
a__U13#(tt(),V1,V2) = V2 + 1 >= 0 = a__U14#(a__isNatKind(V2),V1,V2)
a__U11(tt(),V1,V2) = 0 >= 0 = a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) = 0 >= 0 = a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) = 0 >= 0 = a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) = 0 >= 0 = a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) = 0 >= 0 = a__U16(a__isNat(V2))
a__U16(tt()) = 0 >= 0 = tt()
a__U21(tt(),V1) = 0 >= 0 = a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) = 0 >= 0 = a__U23(a__isNat(V1))
a__U23(tt()) = 0 >= 0 = tt()
a__U31(tt(),V2) = 0 >= 0 = a__U32(a__isNatKind(V2))
a__U32(tt()) = 0 >= 0 = tt()
a__U41(tt()) = 0 >= 0 = tt()
a__U51(tt(),N) = 0 >= 0 = a__U52(a__isNatKind(N),N)
a__U52(tt(),N) = 0 >= 0 = mark(N)
a__U61(tt(),M,N) = 0 >= 0 = a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) = 0 >= 0 = a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) = 0 >= 0 = a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) = 0 >= 0 = s(a__plus(mark(N),mark(M)))
a__isNat(0()) = 0 >= 0 = tt()
a__isNat(plus(V1,V2)) = 0 >= 0 = a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) = 0 >= 0 = a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) = 0 >= 0 = tt()
a__isNatKind(plus(V1,V2)) = 0 >= 0 = a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) = 0 >= 0 = a__U41(a__isNatKind(V1))
a__plus(N,0()) = 0 >= 0 = a__U51(a__isNat(N),N)
a__plus(N,s(M)) = 0 >= 0 = a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) = 0 >= 0 = a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) = 0 >= 0 = a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) = 0 >= 0 = a__isNatKind(X)
mark(U13(X1,X2,X3)) = 0 >= 0 = a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) = 0 >= 0 = a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) = 0 >= 0 = a__U15(mark(X1),X2)
mark(isNat(X)) = 0 >= 0 = a__isNat(X)
mark(U16(X)) = 0 >= 0 = a__U16(mark(X))
mark(U21(X1,X2)) = 0 >= 0 = a__U21(mark(X1),X2)
mark(U22(X1,X2)) = 0 >= 0 = a__U22(mark(X1),X2)
mark(U23(X)) = 0 >= 0 = a__U23(mark(X))
mark(U31(X1,X2)) = 0 >= 0 = a__U31(mark(X1),X2)
mark(U32(X)) = 0 >= 0 = a__U32(mark(X))
mark(U41(X)) = 0 >= 0 = a__U41(mark(X))
mark(U51(X1,X2)) = 0 >= 0 = a__U51(mark(X1),X2)
mark(U52(X1,X2)) = 0 >= 0 = a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) = 0 >= 0 = a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) = 0 >= 0 = a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) = 0 >= 0 = a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) = 0 >= 0 = a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) = 0 >= 0 = a__plus(mark(X1),mark(X2))
mark(tt()) = 0 >= 0 = tt()
mark(s(X)) = 0 >= 0 = s(mark(X))
mark(0()) = 0 >= 0 = 0()
a__U11(X1,X2,X3) = 0 >= 0 = U11(X1,X2,X3)
a__U12(X1,X2,X3) = 0 >= 0 = U12(X1,X2,X3)
a__isNatKind(X) = 0 >= 0 = isNatKind(X)
a__U13(X1,X2,X3) = 0 >= 0 = U13(X1,X2,X3)
a__U14(X1,X2,X3) = 0 >= 0 = U14(X1,X2,X3)
a__U15(X1,X2) = 0 >= 0 = U15(X1,X2)
a__isNat(X) = 0 >= 0 = isNat(X)
a__U16(X) = 0 >= 0 = U16(X)
a__U21(X1,X2) = 0 >= 0 = U21(X1,X2)
a__U22(X1,X2) = 0 >= 0 = U22(X1,X2)
a__U23(X) = 0 >= 0 = U23(X)
a__U31(X1,X2) = 0 >= 0 = U31(X1,X2)
a__U32(X) = 0 >= 0 = U32(X)
a__U41(X) = 0 >= 0 = U41(X)
a__U51(X1,X2) = 0 >= 0 = U51(X1,X2)
a__U52(X1,X2) = 0 >= 0 = U52(X1,X2)
a__U61(X1,X2,X3) = X1 >= 0 = U61(X1,X2,X3)
a__U62(X1,X2,X3) = 0 >= 0 = U62(X1,X2,X3)
a__U63(X1,X2,X3) = 0 >= 0 = U63(X1,X2,X3)
a__U64(X1,X2,X3) = 0 >= 0 = U64(X1,X2,X3)
a__plus(X1,X2) = 0 >= 0 = plus(X1,X2)
problem:
DPs:
TRS:
a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) -> a__U16(a__isNat(V2))
a__U16(tt()) -> tt()
a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) -> a__U23(a__isNat(V1))
a__U23(tt()) -> tt()
a__U31(tt(),V2) -> a__U32(a__isNatKind(V2))
a__U32(tt()) -> tt()
a__U41(tt()) -> tt()
a__U51(tt(),N) -> a__U52(a__isNatKind(N),N)
a__U52(tt(),N) -> mark(N)
a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
a__isNat(0()) -> tt()
a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) -> tt()
a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1))
a__plus(N,0()) -> a__U51(a__isNat(N),N)
a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) -> a__isNatKind(X)
mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) -> a__U15(mark(X1),X2)
mark(isNat(X)) -> a__isNat(X)
mark(U16(X)) -> a__U16(mark(X))
mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
mark(U23(X)) -> a__U23(mark(X))
mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
mark(U32(X)) -> a__U32(mark(X))
mark(U41(X)) -> a__U41(mark(X))
mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
mark(tt()) -> tt()
mark(s(X)) -> s(mark(X))
mark(0()) -> 0()
a__U11(X1,X2,X3) -> U11(X1,X2,X3)
a__U12(X1,X2,X3) -> U12(X1,X2,X3)
a__isNatKind(X) -> isNatKind(X)
a__U13(X1,X2,X3) -> U13(X1,X2,X3)
a__U14(X1,X2,X3) -> U14(X1,X2,X3)
a__U15(X1,X2) -> U15(X1,X2)
a__isNat(X) -> isNat(X)
a__U16(X) -> U16(X)
a__U21(X1,X2) -> U21(X1,X2)
a__U22(X1,X2) -> U22(X1,X2)
a__U23(X) -> U23(X)
a__U31(X1,X2) -> U31(X1,X2)
a__U32(X) -> U32(X)
a__U41(X) -> U41(X)
a__U51(X1,X2) -> U51(X1,X2)
a__U52(X1,X2) -> U52(X1,X2)
a__U61(X1,X2,X3) -> U61(X1,X2,X3)
a__U62(X1,X2,X3) -> U62(X1,X2,X3)
a__U63(X1,X2,X3) -> U63(X1,X2,X3)
a__U64(X1,X2,X3) -> U64(X1,X2,X3)
a__plus(X1,X2) -> plus(X1,X2)
Qed
DPs:
a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1)
a__isNatKind#(plus(V1,V2)) -> a__U31#(a__isNatKind(V1),V2)
a__U31#(tt(),V2) -> a__isNatKind#(V2)
a__isNatKind#(s(V1)) -> a__isNatKind#(V1)
TRS:
a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) -> a__U16(a__isNat(V2))
a__U16(tt()) -> tt()
a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) -> a__U23(a__isNat(V1))
a__U23(tt()) -> tt()
a__U31(tt(),V2) -> a__U32(a__isNatKind(V2))
a__U32(tt()) -> tt()
a__U41(tt()) -> tt()
a__U51(tt(),N) -> a__U52(a__isNatKind(N),N)
a__U52(tt(),N) -> mark(N)
a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
a__isNat(0()) -> tt()
a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) -> tt()
a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1))
a__plus(N,0()) -> a__U51(a__isNat(N),N)
a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) -> a__isNatKind(X)
mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) -> a__U15(mark(X1),X2)
mark(isNat(X)) -> a__isNat(X)
mark(U16(X)) -> a__U16(mark(X))
mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
mark(U23(X)) -> a__U23(mark(X))
mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
mark(U32(X)) -> a__U32(mark(X))
mark(U41(X)) -> a__U41(mark(X))
mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
mark(tt()) -> tt()
mark(s(X)) -> s(mark(X))
mark(0()) -> 0()
a__U11(X1,X2,X3) -> U11(X1,X2,X3)
a__U12(X1,X2,X3) -> U12(X1,X2,X3)
a__isNatKind(X) -> isNatKind(X)
a__U13(X1,X2,X3) -> U13(X1,X2,X3)
a__U14(X1,X2,X3) -> U14(X1,X2,X3)
a__U15(X1,X2) -> U15(X1,X2)
a__isNat(X) -> isNat(X)
a__U16(X) -> U16(X)
a__U21(X1,X2) -> U21(X1,X2)
a__U22(X1,X2) -> U22(X1,X2)
a__U23(X) -> U23(X)
a__U31(X1,X2) -> U31(X1,X2)
a__U32(X) -> U32(X)
a__U41(X) -> U41(X)
a__U51(X1,X2) -> U51(X1,X2)
a__U52(X1,X2) -> U52(X1,X2)
a__U61(X1,X2,X3) -> U61(X1,X2,X3)
a__U62(X1,X2,X3) -> U62(X1,X2,X3)
a__U63(X1,X2,X3) -> U63(X1,X2,X3)
a__U64(X1,X2,X3) -> U64(X1,X2,X3)
a__plus(X1,X2) -> plus(X1,X2)
Matrix Interpretation Processor:
dimension: 1
interpretation:
[a__U31#](x0, x1) = x1,
[a__isNatKind#](x0) = x0,
[U64](x0, x1, x2) = x1 + x2 + 1,
[U63](x0, x1, x2) = x1 + x2 + 1,
[U62](x0, x1, x2) = x1 + x2 + 1,
[U61](x0, x1, x2) = x1 + x2 + 1,
[U52](x0, x1) = x1,
[U51](x0, x1) = x1,
[U41](x0) = 0,
[U32](x0) = 0,
[U31](x0, x1) = 0,
[U23](x0) = 0,
[U22](x0, x1) = 0,
[U21](x0, x1) = x1 + 1,
[U16](x0) = 0,
[isNat](x0) = x0,
[U15](x0, x1) = 0,
[U14](x0, x1, x2) = 0,
[U13](x0, x1, x2) = 0,
[isNatKind](x0) = x0 + 1,
[U12](x0, x1, x2) = x2,
[U11](x0, x1, x2) = x2,
[plus](x0, x1) = x0 + x1,
[0] = 0,
[s](x0) = x0 + 1,
[a__plus](x0, x1) = x0 + x1,
[a__U64](x0, x1, x2) = x1 + x2 + 1,
[a__U63](x0, x1, x2) = x1 + x2 + 1,
[a__U62](x0, x1, x2) = x1 + x2 + 1,
[a__U61](x0, x1, x2) = x1 + x2 + 1,
[mark](x0) = x0,
[a__U52](x0, x1) = x1,
[a__U51](x0, x1) = x1,
[a__U41](x0) = 0,
[a__U32](x0) = 0,
[a__U31](x0, x1) = 0,
[a__U23](x0) = 0,
[a__U22](x0, x1) = 0,
[a__U21](x0, x1) = x1 + 1,
[a__U16](x0) = 0,
[a__U15](x0, x1) = 0,
[a__isNat](x0) = x0,
[a__U14](x0, x1, x2) = 0,
[a__U13](x0, x1, x2) = 0,
[a__U12](x0, x1, x2) = x2,
[a__isNatKind](x0) = x0 + 1,
[a__U11](x0, x1, x2) = x2,
[tt] = 0
orientation:
a__isNatKind#(plus(V1,V2)) = V1 + V2 >= V1 = a__isNatKind#(V1)
a__isNatKind#(plus(V1,V2)) = V1 + V2 >= V2 = a__U31#(a__isNatKind(V1),V2)
a__U31#(tt(),V2) = V2 >= V2 = a__isNatKind#(V2)
a__isNatKind#(s(V1)) = V1 + 1 >= V1 = a__isNatKind#(V1)
a__U11(tt(),V1,V2) = V2 >= V2 = a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) = V2 >= 0 = a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) = 0 >= 0 = a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) = 0 >= 0 = a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) = 0 >= 0 = a__U16(a__isNat(V2))
a__U16(tt()) = 0 >= 0 = tt()
a__U21(tt(),V1) = V1 + 1 >= 0 = a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) = 0 >= 0 = a__U23(a__isNat(V1))
a__U23(tt()) = 0 >= 0 = tt()
a__U31(tt(),V2) = 0 >= 0 = a__U32(a__isNatKind(V2))
a__U32(tt()) = 0 >= 0 = tt()
a__U41(tt()) = 0 >= 0 = tt()
a__U51(tt(),N) = N >= N = a__U52(a__isNatKind(N),N)
a__U52(tt(),N) = N >= N = mark(N)
a__U61(tt(),M,N) = M + N + 1 >= M + N + 1 = a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) = M + N + 1 >= M + N + 1 = a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) = M + N + 1 >= M + N + 1 = a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) = M + N + 1 >= M + N + 1 = s(a__plus(mark(N),mark(M)))
a__isNat(0()) = 0 >= 0 = tt()
a__isNat(plus(V1,V2)) = V1 + V2 >= V2 = a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) = V1 + 1 >= V1 + 1 = a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) = 1 >= 0 = tt()
a__isNatKind(plus(V1,V2)) = V1 + V2 + 1 >= 0 = a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) = V1 + 2 >= 0 = a__U41(a__isNatKind(V1))
a__plus(N,0()) = N >= N = a__U51(a__isNat(N),N)
a__plus(N,s(M)) = M + N + 1 >= M + N + 1 = a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) = X3 >= X3 = a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) = X3 >= X3 = a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) = X + 1 >= X + 1 = a__isNatKind(X)
mark(U13(X1,X2,X3)) = 0 >= 0 = a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) = 0 >= 0 = a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) = 0 >= 0 = a__U15(mark(X1),X2)
mark(isNat(X)) = X >= X = a__isNat(X)
mark(U16(X)) = 0 >= 0 = a__U16(mark(X))
mark(U21(X1,X2)) = X2 + 1 >= X2 + 1 = a__U21(mark(X1),X2)
mark(U22(X1,X2)) = 0 >= 0 = a__U22(mark(X1),X2)
mark(U23(X)) = 0 >= 0 = a__U23(mark(X))
mark(U31(X1,X2)) = 0 >= 0 = a__U31(mark(X1),X2)
mark(U32(X)) = 0 >= 0 = a__U32(mark(X))
mark(U41(X)) = 0 >= 0 = a__U41(mark(X))
mark(U51(X1,X2)) = X2 >= X2 = a__U51(mark(X1),X2)
mark(U52(X1,X2)) = X2 >= X2 = a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) = X2 + X3 + 1 >= X2 + X3 + 1 = a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) = X1 + X2 >= X1 + X2 = a__plus(mark(X1),mark(X2))
mark(tt()) = 0 >= 0 = tt()
mark(s(X)) = X + 1 >= X + 1 = s(mark(X))
mark(0()) = 0 >= 0 = 0()
a__U11(X1,X2,X3) = X3 >= X3 = U11(X1,X2,X3)
a__U12(X1,X2,X3) = X3 >= X3 = U12(X1,X2,X3)
a__isNatKind(X) = X + 1 >= X + 1 = isNatKind(X)
a__U13(X1,X2,X3) = 0 >= 0 = U13(X1,X2,X3)
a__U14(X1,X2,X3) = 0 >= 0 = U14(X1,X2,X3)
a__U15(X1,X2) = 0 >= 0 = U15(X1,X2)
a__isNat(X) = X >= X = isNat(X)
a__U16(X) = 0 >= 0 = U16(X)
a__U21(X1,X2) = X2 + 1 >= X2 + 1 = U21(X1,X2)
a__U22(X1,X2) = 0 >= 0 = U22(X1,X2)
a__U23(X) = 0 >= 0 = U23(X)
a__U31(X1,X2) = 0 >= 0 = U31(X1,X2)
a__U32(X) = 0 >= 0 = U32(X)
a__U41(X) = 0 >= 0 = U41(X)
a__U51(X1,X2) = X2 >= X2 = U51(X1,X2)
a__U52(X1,X2) = X2 >= X2 = U52(X1,X2)
a__U61(X1,X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U61(X1,X2,X3)
a__U62(X1,X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U62(X1,X2,X3)
a__U63(X1,X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U63(X1,X2,X3)
a__U64(X1,X2,X3) = X2 + X3 + 1 >= X2 + X3 + 1 = U64(X1,X2,X3)
a__plus(X1,X2) = X1 + X2 >= X1 + X2 = plus(X1,X2)
problem:
DPs:
a__isNatKind#(plus(V1,V2)) -> a__isNatKind#(V1)
a__isNatKind#(plus(V1,V2)) -> a__U31#(a__isNatKind(V1),V2)
a__U31#(tt(),V2) -> a__isNatKind#(V2)
TRS:
a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) -> a__U16(a__isNat(V2))
a__U16(tt()) -> tt()
a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) -> a__U23(a__isNat(V1))
a__U23(tt()) -> tt()
a__U31(tt(),V2) -> a__U32(a__isNatKind(V2))
a__U32(tt()) -> tt()
a__U41(tt()) -> tt()
a__U51(tt(),N) -> a__U52(a__isNatKind(N),N)
a__U52(tt(),N) -> mark(N)
a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
a__isNat(0()) -> tt()
a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) -> tt()
a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1))
a__plus(N,0()) -> a__U51(a__isNat(N),N)
a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) -> a__isNatKind(X)
mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) -> a__U15(mark(X1),X2)
mark(isNat(X)) -> a__isNat(X)
mark(U16(X)) -> a__U16(mark(X))
mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
mark(U23(X)) -> a__U23(mark(X))
mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
mark(U32(X)) -> a__U32(mark(X))
mark(U41(X)) -> a__U41(mark(X))
mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
mark(tt()) -> tt()
mark(s(X)) -> s(mark(X))
mark(0()) -> 0()
a__U11(X1,X2,X3) -> U11(X1,X2,X3)
a__U12(X1,X2,X3) -> U12(X1,X2,X3)
a__isNatKind(X) -> isNatKind(X)
a__U13(X1,X2,X3) -> U13(X1,X2,X3)
a__U14(X1,X2,X3) -> U14(X1,X2,X3)
a__U15(X1,X2) -> U15(X1,X2)
a__isNat(X) -> isNat(X)
a__U16(X) -> U16(X)
a__U21(X1,X2) -> U21(X1,X2)
a__U22(X1,X2) -> U22(X1,X2)
a__U23(X) -> U23(X)
a__U31(X1,X2) -> U31(X1,X2)
a__U32(X) -> U32(X)
a__U41(X) -> U41(X)
a__U51(X1,X2) -> U51(X1,X2)
a__U52(X1,X2) -> U52(X1,X2)
a__U61(X1,X2,X3) -> U61(X1,X2,X3)
a__U62(X1,X2,X3) -> U62(X1,X2,X3)
a__U63(X1,X2,X3) -> U63(X1,X2,X3)
a__U64(X1,X2,X3) -> U64(X1,X2,X3)
a__plus(X1,X2) -> plus(X1,X2)
Matrix Interpretation Processor:
dimension: 1
interpretation:
[a__U31#](x0, x1) = x0 + x1 + 1,
[a__isNatKind#](x0) = x0 + 1,
[U64](x0, x1, x2) = 0,
[U63](x0, x1, x2) = 0,
[U62](x0, x1, x2) = 0,
[U61](x0, x1, x2) = 0,
[U52](x0, x1) = x1,
[U51](x0, x1) = x1,
[U41](x0) = 0,
[U32](x0) = 0,
[U31](x0, x1) = 0,
[U23](x0) = 0,
[U22](x0, x1) = 0,
[U21](x0, x1) = 0,
[U16](x0) = 0,
[isNat](x0) = x0,
[U15](x0, x1) = 0,
[U14](x0, x1, x2) = 0,
[U13](x0, x1, x2) = 0,
[isNatKind](x0) = x0 + 1,
[U12](x0, x1, x2) = x2,
[U11](x0, x1, x2) = x2,
[plus](x0, x1) = x0 + x1 + 1,
[0] = 0,
[s](x0) = 0,
[a__plus](x0, x1) = x0 + x1 + 1,
[a__U64](x0, x1, x2) = 0,
[a__U63](x0, x1, x2) = 0,
[a__U62](x0, x1, x2) = 0,
[a__U61](x0, x1, x2) = 0,
[mark](x0) = x0,
[a__U52](x0, x1) = x1,
[a__U51](x0, x1) = x1,
[a__U41](x0) = 0,
[a__U32](x0) = 0,
[a__U31](x0, x1) = 0,
[a__U23](x0) = 0,
[a__U22](x0, x1) = 0,
[a__U21](x0, x1) = 0,
[a__U16](x0) = 0,
[a__U15](x0, x1) = 0,
[a__isNat](x0) = x0,
[a__U14](x0, x1, x2) = 0,
[a__U13](x0, x1, x2) = 0,
[a__U12](x0, x1, x2) = x2,
[a__isNatKind](x0) = x0 + 1,
[a__U11](x0, x1, x2) = x2,
[tt] = 0
orientation:
a__isNatKind#(plus(V1,V2)) = V1 + V2 + 2 >= V1 + 1 = a__isNatKind#(V1)
a__isNatKind#(plus(V1,V2)) = V1 + V2 + 2 >= V1 + V2 + 2 = a__U31#(a__isNatKind(V1),V2)
a__U31#(tt(),V2) = V2 + 1 >= V2 + 1 = a__isNatKind#(V2)
a__U11(tt(),V1,V2) = V2 >= V2 = a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) = V2 >= 0 = a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) = 0 >= 0 = a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) = 0 >= 0 = a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) = 0 >= 0 = a__U16(a__isNat(V2))
a__U16(tt()) = 0 >= 0 = tt()
a__U21(tt(),V1) = 0 >= 0 = a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) = 0 >= 0 = a__U23(a__isNat(V1))
a__U23(tt()) = 0 >= 0 = tt()
a__U31(tt(),V2) = 0 >= 0 = a__U32(a__isNatKind(V2))
a__U32(tt()) = 0 >= 0 = tt()
a__U41(tt()) = 0 >= 0 = tt()
a__U51(tt(),N) = N >= N = a__U52(a__isNatKind(N),N)
a__U52(tt(),N) = N >= N = mark(N)
a__U61(tt(),M,N) = 0 >= 0 = a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) = 0 >= 0 = a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) = 0 >= 0 = a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) = 0 >= 0 = s(a__plus(mark(N),mark(M)))
a__isNat(0()) = 0 >= 0 = tt()
a__isNat(plus(V1,V2)) = V1 + V2 + 1 >= V2 = a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) = 0 >= 0 = a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) = 1 >= 0 = tt()
a__isNatKind(plus(V1,V2)) = V1 + V2 + 2 >= 0 = a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) = 1 >= 0 = a__U41(a__isNatKind(V1))
a__plus(N,0()) = N + 1 >= N = a__U51(a__isNat(N),N)
a__plus(N,s(M)) = N + 1 >= 0 = a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) = X3 >= X3 = a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) = X3 >= X3 = a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) = X + 1 >= X + 1 = a__isNatKind(X)
mark(U13(X1,X2,X3)) = 0 >= 0 = a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) = 0 >= 0 = a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) = 0 >= 0 = a__U15(mark(X1),X2)
mark(isNat(X)) = X >= X = a__isNat(X)
mark(U16(X)) = 0 >= 0 = a__U16(mark(X))
mark(U21(X1,X2)) = 0 >= 0 = a__U21(mark(X1),X2)
mark(U22(X1,X2)) = 0 >= 0 = a__U22(mark(X1),X2)
mark(U23(X)) = 0 >= 0 = a__U23(mark(X))
mark(U31(X1,X2)) = 0 >= 0 = a__U31(mark(X1),X2)
mark(U32(X)) = 0 >= 0 = a__U32(mark(X))
mark(U41(X)) = 0 >= 0 = a__U41(mark(X))
mark(U51(X1,X2)) = X2 >= X2 = a__U51(mark(X1),X2)
mark(U52(X1,X2)) = X2 >= X2 = a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) = 0 >= 0 = a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) = 0 >= 0 = a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) = 0 >= 0 = a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) = 0 >= 0 = a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__plus(mark(X1),mark(X2))
mark(tt()) = 0 >= 0 = tt()
mark(s(X)) = 0 >= 0 = s(mark(X))
mark(0()) = 0 >= 0 = 0()
a__U11(X1,X2,X3) = X3 >= X3 = U11(X1,X2,X3)
a__U12(X1,X2,X3) = X3 >= X3 = U12(X1,X2,X3)
a__isNatKind(X) = X + 1 >= X + 1 = isNatKind(X)
a__U13(X1,X2,X3) = 0 >= 0 = U13(X1,X2,X3)
a__U14(X1,X2,X3) = 0 >= 0 = U14(X1,X2,X3)
a__U15(X1,X2) = 0 >= 0 = U15(X1,X2)
a__isNat(X) = X >= X = isNat(X)
a__U16(X) = 0 >= 0 = U16(X)
a__U21(X1,X2) = 0 >= 0 = U21(X1,X2)
a__U22(X1,X2) = 0 >= 0 = U22(X1,X2)
a__U23(X) = 0 >= 0 = U23(X)
a__U31(X1,X2) = 0 >= 0 = U31(X1,X2)
a__U32(X) = 0 >= 0 = U32(X)
a__U41(X) = 0 >= 0 = U41(X)
a__U51(X1,X2) = X2 >= X2 = U51(X1,X2)
a__U52(X1,X2) = X2 >= X2 = U52(X1,X2)
a__U61(X1,X2,X3) = 0 >= 0 = U61(X1,X2,X3)
a__U62(X1,X2,X3) = 0 >= 0 = U62(X1,X2,X3)
a__U63(X1,X2,X3) = 0 >= 0 = U63(X1,X2,X3)
a__U64(X1,X2,X3) = 0 >= 0 = U64(X1,X2,X3)
a__plus(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2)
problem:
DPs:
a__isNatKind#(plus(V1,V2)) -> a__U31#(a__isNatKind(V1),V2)
a__U31#(tt(),V2) -> a__isNatKind#(V2)
TRS:
a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) -> a__U16(a__isNat(V2))
a__U16(tt()) -> tt()
a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) -> a__U23(a__isNat(V1))
a__U23(tt()) -> tt()
a__U31(tt(),V2) -> a__U32(a__isNatKind(V2))
a__U32(tt()) -> tt()
a__U41(tt()) -> tt()
a__U51(tt(),N) -> a__U52(a__isNatKind(N),N)
a__U52(tt(),N) -> mark(N)
a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
a__isNat(0()) -> tt()
a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) -> tt()
a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1))
a__plus(N,0()) -> a__U51(a__isNat(N),N)
a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) -> a__isNatKind(X)
mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) -> a__U15(mark(X1),X2)
mark(isNat(X)) -> a__isNat(X)
mark(U16(X)) -> a__U16(mark(X))
mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
mark(U23(X)) -> a__U23(mark(X))
mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
mark(U32(X)) -> a__U32(mark(X))
mark(U41(X)) -> a__U41(mark(X))
mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
mark(tt()) -> tt()
mark(s(X)) -> s(mark(X))
mark(0()) -> 0()
a__U11(X1,X2,X3) -> U11(X1,X2,X3)
a__U12(X1,X2,X3) -> U12(X1,X2,X3)
a__isNatKind(X) -> isNatKind(X)
a__U13(X1,X2,X3) -> U13(X1,X2,X3)
a__U14(X1,X2,X3) -> U14(X1,X2,X3)
a__U15(X1,X2) -> U15(X1,X2)
a__isNat(X) -> isNat(X)
a__U16(X) -> U16(X)
a__U21(X1,X2) -> U21(X1,X2)
a__U22(X1,X2) -> U22(X1,X2)
a__U23(X) -> U23(X)
a__U31(X1,X2) -> U31(X1,X2)
a__U32(X) -> U32(X)
a__U41(X) -> U41(X)
a__U51(X1,X2) -> U51(X1,X2)
a__U52(X1,X2) -> U52(X1,X2)
a__U61(X1,X2,X3) -> U61(X1,X2,X3)
a__U62(X1,X2,X3) -> U62(X1,X2,X3)
a__U63(X1,X2,X3) -> U63(X1,X2,X3)
a__U64(X1,X2,X3) -> U64(X1,X2,X3)
a__plus(X1,X2) -> plus(X1,X2)
Matrix Interpretation Processor:
dimension: 1
interpretation:
[a__U31#](x0, x1) = x0 + x1,
[a__isNatKind#](x0) = x0,
[U64](x0, x1, x2) = x2 + 1,
[U63](x0, x1, x2) = x2 + 1,
[U62](x0, x1, x2) = x2 + 1,
[U61](x0, x1, x2) = x2 + 1,
[U52](x0, x1) = x1,
[U51](x0, x1) = x1,
[U41](x0) = 1,
[U32](x0) = 1,
[U31](x0, x1) = x0,
[U23](x0) = 1,
[U22](x0, x1) = 1,
[U21](x0, x1) = 1,
[U16](x0) = 1,
[isNat](x0) = x0 + 1,
[U15](x0, x1) = 1,
[U14](x0, x1, x2) = x0,
[U13](x0, x1, x2) = x2,
[isNatKind](x0) = x0,
[U12](x0, x1, x2) = x2,
[U11](x0, x1, x2) = x2 + 1,
[plus](x0, x1) = x0 + x1,
[0] = 1,
[s](x0) = 1,
[a__plus](x0, x1) = x0 + x1,
[a__U64](x0, x1, x2) = x2 + 1,
[a__U63](x0, x1, x2) = x2 + 1,
[a__U62](x0, x1, x2) = x2 + 1,
[a__U61](x0, x1, x2) = x2 + 1,
[mark](x0) = x0,
[a__U52](x0, x1) = x1,
[a__U51](x0, x1) = x1,
[a__U41](x0) = 1,
[a__U32](x0) = 1,
[a__U31](x0, x1) = x0,
[a__U23](x0) = 1,
[a__U22](x0, x1) = 1,
[a__U21](x0, x1) = 1,
[a__U16](x0) = 1,
[a__U15](x0, x1) = 1,
[a__isNat](x0) = x0 + 1,
[a__U14](x0, x1, x2) = x0,
[a__U13](x0, x1, x2) = x2,
[a__U12](x0, x1, x2) = x2,
[a__isNatKind](x0) = x0,
[a__U11](x0, x1, x2) = x2 + 1,
[tt] = 1
orientation:
a__isNatKind#(plus(V1,V2)) = V1 + V2 >= V1 + V2 = a__U31#(a__isNatKind(V1),V2)
a__U31#(tt(),V2) = V2 + 1 >= V2 = a__isNatKind#(V2)
a__U11(tt(),V1,V2) = V2 + 1 >= V2 = a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) = V2 >= V2 = a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) = V2 >= V2 = a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) = 1 >= 1 = a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) = 1 >= 1 = a__U16(a__isNat(V2))
a__U16(tt()) = 1 >= 1 = tt()
a__U21(tt(),V1) = 1 >= 1 = a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) = 1 >= 1 = a__U23(a__isNat(V1))
a__U23(tt()) = 1 >= 1 = tt()
a__U31(tt(),V2) = 1 >= 1 = a__U32(a__isNatKind(V2))
a__U32(tt()) = 1 >= 1 = tt()
a__U41(tt()) = 1 >= 1 = tt()
a__U51(tt(),N) = N >= N = a__U52(a__isNatKind(N),N)
a__U52(tt(),N) = N >= N = mark(N)
a__U61(tt(),M,N) = N + 1 >= N + 1 = a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) = N + 1 >= N + 1 = a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) = N + 1 >= N + 1 = a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) = N + 1 >= 1 = s(a__plus(mark(N),mark(M)))
a__isNat(0()) = 2 >= 1 = tt()
a__isNat(plus(V1,V2)) = V1 + V2 + 1 >= V2 + 1 = a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) = 2 >= 1 = a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) = 1 >= 1 = tt()
a__isNatKind(plus(V1,V2)) = V1 + V2 >= V1 = a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) = 1 >= 1 = a__U41(a__isNatKind(V1))
a__plus(N,0()) = N + 1 >= N = a__U51(a__isNat(N),N)
a__plus(N,s(M)) = N + 1 >= N + 1 = a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) = X3 + 1 >= X3 + 1 = a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) = X3 >= X3 = a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) = X >= X = a__isNatKind(X)
mark(U13(X1,X2,X3)) = X3 >= X3 = a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) = X1 >= X1 = a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) = 1 >= 1 = a__U15(mark(X1),X2)
mark(isNat(X)) = X + 1 >= X + 1 = a__isNat(X)
mark(U16(X)) = 1 >= 1 = a__U16(mark(X))
mark(U21(X1,X2)) = 1 >= 1 = a__U21(mark(X1),X2)
mark(U22(X1,X2)) = 1 >= 1 = a__U22(mark(X1),X2)
mark(U23(X)) = 1 >= 1 = a__U23(mark(X))
mark(U31(X1,X2)) = X1 >= X1 = a__U31(mark(X1),X2)
mark(U32(X)) = 1 >= 1 = a__U32(mark(X))
mark(U41(X)) = 1 >= 1 = a__U41(mark(X))
mark(U51(X1,X2)) = X2 >= X2 = a__U51(mark(X1),X2)
mark(U52(X1,X2)) = X2 >= X2 = a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) = X3 + 1 >= X3 + 1 = a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) = X3 + 1 >= X3 + 1 = a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) = X3 + 1 >= X3 + 1 = a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) = X3 + 1 >= X3 + 1 = a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) = X1 + X2 >= X1 + X2 = a__plus(mark(X1),mark(X2))
mark(tt()) = 1 >= 1 = tt()
mark(s(X)) = 1 >= 1 = s(mark(X))
mark(0()) = 1 >= 1 = 0()
a__U11(X1,X2,X3) = X3 + 1 >= X3 + 1 = U11(X1,X2,X3)
a__U12(X1,X2,X3) = X3 >= X3 = U12(X1,X2,X3)
a__isNatKind(X) = X >= X = isNatKind(X)
a__U13(X1,X2,X3) = X3 >= X3 = U13(X1,X2,X3)
a__U14(X1,X2,X3) = X1 >= X1 = U14(X1,X2,X3)
a__U15(X1,X2) = 1 >= 1 = U15(X1,X2)
a__isNat(X) = X + 1 >= X + 1 = isNat(X)
a__U16(X) = 1 >= 1 = U16(X)
a__U21(X1,X2) = 1 >= 1 = U21(X1,X2)
a__U22(X1,X2) = 1 >= 1 = U22(X1,X2)
a__U23(X) = 1 >= 1 = U23(X)
a__U31(X1,X2) = X1 >= X1 = U31(X1,X2)
a__U32(X) = 1 >= 1 = U32(X)
a__U41(X) = 1 >= 1 = U41(X)
a__U51(X1,X2) = X2 >= X2 = U51(X1,X2)
a__U52(X1,X2) = X2 >= X2 = U52(X1,X2)
a__U61(X1,X2,X3) = X3 + 1 >= X3 + 1 = U61(X1,X2,X3)
a__U62(X1,X2,X3) = X3 + 1 >= X3 + 1 = U62(X1,X2,X3)
a__U63(X1,X2,X3) = X3 + 1 >= X3 + 1 = U63(X1,X2,X3)
a__U64(X1,X2,X3) = X3 + 1 >= X3 + 1 = U64(X1,X2,X3)
a__plus(X1,X2) = X1 + X2 >= X1 + X2 = plus(X1,X2)
problem:
DPs:
a__isNatKind#(plus(V1,V2)) -> a__U31#(a__isNatKind(V1),V2)
TRS:
a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) -> a__U16(a__isNat(V2))
a__U16(tt()) -> tt()
a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) -> a__U23(a__isNat(V1))
a__U23(tt()) -> tt()
a__U31(tt(),V2) -> a__U32(a__isNatKind(V2))
a__U32(tt()) -> tt()
a__U41(tt()) -> tt()
a__U51(tt(),N) -> a__U52(a__isNatKind(N),N)
a__U52(tt(),N) -> mark(N)
a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
a__isNat(0()) -> tt()
a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) -> tt()
a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1))
a__plus(N,0()) -> a__U51(a__isNat(N),N)
a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) -> a__isNatKind(X)
mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) -> a__U15(mark(X1),X2)
mark(isNat(X)) -> a__isNat(X)
mark(U16(X)) -> a__U16(mark(X))
mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
mark(U23(X)) -> a__U23(mark(X))
mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
mark(U32(X)) -> a__U32(mark(X))
mark(U41(X)) -> a__U41(mark(X))
mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
mark(tt()) -> tt()
mark(s(X)) -> s(mark(X))
mark(0()) -> 0()
a__U11(X1,X2,X3) -> U11(X1,X2,X3)
a__U12(X1,X2,X3) -> U12(X1,X2,X3)
a__isNatKind(X) -> isNatKind(X)
a__U13(X1,X2,X3) -> U13(X1,X2,X3)
a__U14(X1,X2,X3) -> U14(X1,X2,X3)
a__U15(X1,X2) -> U15(X1,X2)
a__isNat(X) -> isNat(X)
a__U16(X) -> U16(X)
a__U21(X1,X2) -> U21(X1,X2)
a__U22(X1,X2) -> U22(X1,X2)
a__U23(X) -> U23(X)
a__U31(X1,X2) -> U31(X1,X2)
a__U32(X) -> U32(X)
a__U41(X) -> U41(X)
a__U51(X1,X2) -> U51(X1,X2)
a__U52(X1,X2) -> U52(X1,X2)
a__U61(X1,X2,X3) -> U61(X1,X2,X3)
a__U62(X1,X2,X3) -> U62(X1,X2,X3)
a__U63(X1,X2,X3) -> U63(X1,X2,X3)
a__U64(X1,X2,X3) -> U64(X1,X2,X3)
a__plus(X1,X2) -> plus(X1,X2)
Matrix Interpretation Processor:
dimension: 1
interpretation:
[a__U31#](x0, x1) = 0,
[a__isNatKind#](x0) = 1,
[U64](x0, x1, x2) = 0,
[U63](x0, x1, x2) = 0,
[U62](x0, x1, x2) = 0,
[U61](x0, x1, x2) = 0,
[U52](x0, x1) = 0,
[U51](x0, x1) = 0,
[U41](x0) = 0,
[U32](x0) = 0,
[U31](x0, x1) = 0,
[U23](x0) = 0,
[U22](x0, x1) = 0,
[U21](x0, x1) = 0,
[U16](x0) = 0,
[isNat](x0) = 0,
[U15](x0, x1) = 0,
[U14](x0, x1, x2) = 0,
[U13](x0, x1, x2) = 0,
[isNatKind](x0) = 0,
[U12](x0, x1, x2) = 0,
[U11](x0, x1, x2) = 0,
[plus](x0, x1) = 0,
[0] = 0,
[s](x0) = 0,
[a__plus](x0, x1) = 0,
[a__U64](x0, x1, x2) = 0,
[a__U63](x0, x1, x2) = 0,
[a__U62](x0, x1, x2) = 0,
[a__U61](x0, x1, x2) = 0,
[mark](x0) = 0,
[a__U52](x0, x1) = 0,
[a__U51](x0, x1) = 0,
[a__U41](x0) = 0,
[a__U32](x0) = 0,
[a__U31](x0, x1) = 0,
[a__U23](x0) = 0,
[a__U22](x0, x1) = 0,
[a__U21](x0, x1) = 0,
[a__U16](x0) = 0,
[a__U15](x0, x1) = 0,
[a__isNat](x0) = 0,
[a__U14](x0, x1, x2) = 0,
[a__U13](x0, x1, x2) = 0,
[a__U12](x0, x1, x2) = 0,
[a__isNatKind](x0) = 0,
[a__U11](x0, x1, x2) = x0,
[tt] = 0
orientation:
a__isNatKind#(plus(V1,V2)) = 1 >= 0 = a__U31#(a__isNatKind(V1),V2)
a__U11(tt(),V1,V2) = 0 >= 0 = a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) = 0 >= 0 = a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) = 0 >= 0 = a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) = 0 >= 0 = a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) = 0 >= 0 = a__U16(a__isNat(V2))
a__U16(tt()) = 0 >= 0 = tt()
a__U21(tt(),V1) = 0 >= 0 = a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) = 0 >= 0 = a__U23(a__isNat(V1))
a__U23(tt()) = 0 >= 0 = tt()
a__U31(tt(),V2) = 0 >= 0 = a__U32(a__isNatKind(V2))
a__U32(tt()) = 0 >= 0 = tt()
a__U41(tt()) = 0 >= 0 = tt()
a__U51(tt(),N) = 0 >= 0 = a__U52(a__isNatKind(N),N)
a__U52(tt(),N) = 0 >= 0 = mark(N)
a__U61(tt(),M,N) = 0 >= 0 = a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) = 0 >= 0 = a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) = 0 >= 0 = a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) = 0 >= 0 = s(a__plus(mark(N),mark(M)))
a__isNat(0()) = 0 >= 0 = tt()
a__isNat(plus(V1,V2)) = 0 >= 0 = a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) = 0 >= 0 = a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) = 0 >= 0 = tt()
a__isNatKind(plus(V1,V2)) = 0 >= 0 = a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) = 0 >= 0 = a__U41(a__isNatKind(V1))
a__plus(N,0()) = 0 >= 0 = a__U51(a__isNat(N),N)
a__plus(N,s(M)) = 0 >= 0 = a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) = 0 >= 0 = a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) = 0 >= 0 = a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) = 0 >= 0 = a__isNatKind(X)
mark(U13(X1,X2,X3)) = 0 >= 0 = a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) = 0 >= 0 = a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) = 0 >= 0 = a__U15(mark(X1),X2)
mark(isNat(X)) = 0 >= 0 = a__isNat(X)
mark(U16(X)) = 0 >= 0 = a__U16(mark(X))
mark(U21(X1,X2)) = 0 >= 0 = a__U21(mark(X1),X2)
mark(U22(X1,X2)) = 0 >= 0 = a__U22(mark(X1),X2)
mark(U23(X)) = 0 >= 0 = a__U23(mark(X))
mark(U31(X1,X2)) = 0 >= 0 = a__U31(mark(X1),X2)
mark(U32(X)) = 0 >= 0 = a__U32(mark(X))
mark(U41(X)) = 0 >= 0 = a__U41(mark(X))
mark(U51(X1,X2)) = 0 >= 0 = a__U51(mark(X1),X2)
mark(U52(X1,X2)) = 0 >= 0 = a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) = 0 >= 0 = a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) = 0 >= 0 = a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) = 0 >= 0 = a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) = 0 >= 0 = a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) = 0 >= 0 = a__plus(mark(X1),mark(X2))
mark(tt()) = 0 >= 0 = tt()
mark(s(X)) = 0 >= 0 = s(mark(X))
mark(0()) = 0 >= 0 = 0()
a__U11(X1,X2,X3) = X1 >= 0 = U11(X1,X2,X3)
a__U12(X1,X2,X3) = 0 >= 0 = U12(X1,X2,X3)
a__isNatKind(X) = 0 >= 0 = isNatKind(X)
a__U13(X1,X2,X3) = 0 >= 0 = U13(X1,X2,X3)
a__U14(X1,X2,X3) = 0 >= 0 = U14(X1,X2,X3)
a__U15(X1,X2) = 0 >= 0 = U15(X1,X2)
a__isNat(X) = 0 >= 0 = isNat(X)
a__U16(X) = 0 >= 0 = U16(X)
a__U21(X1,X2) = 0 >= 0 = U21(X1,X2)
a__U22(X1,X2) = 0 >= 0 = U22(X1,X2)
a__U23(X) = 0 >= 0 = U23(X)
a__U31(X1,X2) = 0 >= 0 = U31(X1,X2)
a__U32(X) = 0 >= 0 = U32(X)
a__U41(X) = 0 >= 0 = U41(X)
a__U51(X1,X2) = 0 >= 0 = U51(X1,X2)
a__U52(X1,X2) = 0 >= 0 = U52(X1,X2)
a__U61(X1,X2,X3) = 0 >= 0 = U61(X1,X2,X3)
a__U62(X1,X2,X3) = 0 >= 0 = U62(X1,X2,X3)
a__U63(X1,X2,X3) = 0 >= 0 = U63(X1,X2,X3)
a__U64(X1,X2,X3) = 0 >= 0 = U64(X1,X2,X3)
a__plus(X1,X2) = 0 >= 0 = plus(X1,X2)
problem:
DPs:
TRS:
a__U11(tt(),V1,V2) -> a__U12(a__isNatKind(V1),V1,V2)
a__U12(tt(),V1,V2) -> a__U13(a__isNatKind(V2),V1,V2)
a__U13(tt(),V1,V2) -> a__U14(a__isNatKind(V2),V1,V2)
a__U14(tt(),V1,V2) -> a__U15(a__isNat(V1),V2)
a__U15(tt(),V2) -> a__U16(a__isNat(V2))
a__U16(tt()) -> tt()
a__U21(tt(),V1) -> a__U22(a__isNatKind(V1),V1)
a__U22(tt(),V1) -> a__U23(a__isNat(V1))
a__U23(tt()) -> tt()
a__U31(tt(),V2) -> a__U32(a__isNatKind(V2))
a__U32(tt()) -> tt()
a__U41(tt()) -> tt()
a__U51(tt(),N) -> a__U52(a__isNatKind(N),N)
a__U52(tt(),N) -> mark(N)
a__U61(tt(),M,N) -> a__U62(a__isNatKind(M),M,N)
a__U62(tt(),M,N) -> a__U63(a__isNat(N),M,N)
a__U63(tt(),M,N) -> a__U64(a__isNatKind(N),M,N)
a__U64(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
a__isNat(0()) -> tt()
a__isNat(plus(V1,V2)) -> a__U11(a__isNatKind(V1),V1,V2)
a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
a__isNatKind(0()) -> tt()
a__isNatKind(plus(V1,V2)) -> a__U31(a__isNatKind(V1),V2)
a__isNatKind(s(V1)) -> a__U41(a__isNatKind(V1))
a__plus(N,0()) -> a__U51(a__isNat(N),N)
a__plus(N,s(M)) -> a__U61(a__isNat(M),M,N)
mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
mark(isNatKind(X)) -> a__isNatKind(X)
mark(U13(X1,X2,X3)) -> a__U13(mark(X1),X2,X3)
mark(U14(X1,X2,X3)) -> a__U14(mark(X1),X2,X3)
mark(U15(X1,X2)) -> a__U15(mark(X1),X2)
mark(isNat(X)) -> a__isNat(X)
mark(U16(X)) -> a__U16(mark(X))
mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
mark(U22(X1,X2)) -> a__U22(mark(X1),X2)
mark(U23(X)) -> a__U23(mark(X))
mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
mark(U32(X)) -> a__U32(mark(X))
mark(U41(X)) -> a__U41(mark(X))
mark(U51(X1,X2)) -> a__U51(mark(X1),X2)
mark(U52(X1,X2)) -> a__U52(mark(X1),X2)
mark(U61(X1,X2,X3)) -> a__U61(mark(X1),X2,X3)
mark(U62(X1,X2,X3)) -> a__U62(mark(X1),X2,X3)
mark(U63(X1,X2,X3)) -> a__U63(mark(X1),X2,X3)
mark(U64(X1,X2,X3)) -> a__U64(mark(X1),X2,X3)
mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
mark(tt()) -> tt()
mark(s(X)) -> s(mark(X))
mark(0()) -> 0()
a__U11(X1,X2,X3) -> U11(X1,X2,X3)
a__U12(X1,X2,X3) -> U12(X1,X2,X3)
a__isNatKind(X) -> isNatKind(X)
a__U13(X1,X2,X3) -> U13(X1,X2,X3)
a__U14(X1,X2,X3) -> U14(X1,X2,X3)
a__U15(X1,X2) -> U15(X1,X2)
a__isNat(X) -> isNat(X)
a__U16(X) -> U16(X)
a__U21(X1,X2) -> U21(X1,X2)
a__U22(X1,X2) -> U22(X1,X2)
a__U23(X) -> U23(X)
a__U31(X1,X2) -> U31(X1,X2)
a__U32(X) -> U32(X)
a__U41(X) -> U41(X)
a__U51(X1,X2) -> U51(X1,X2)
a__U52(X1,X2) -> U52(X1,X2)
a__U61(X1,X2,X3) -> U61(X1,X2,X3)
a__U62(X1,X2,X3) -> U62(X1,X2,X3)
a__U63(X1,X2,X3) -> U63(X1,X2,X3)
a__U64(X1,X2,X3) -> U64(X1,X2,X3)
a__plus(X1,X2) -> plus(X1,X2)
Qed