YES (VAR V2 N M V1 X1 X2 X X3) (RULES a__U11(tt,V2) -> a__U12(a__isNat(V2)) a__U12(tt) -> tt a__U21(tt) -> tt a__U31(tt,V2) -> a__U32(a__isNat(V2)) a__U32(tt) -> tt a__U41(tt,N) -> mark(N) a__U51(tt,M,N) -> a__U52(a__isNat(N),M,N) a__U52(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U61(tt) -> 0 a__U71(tt,M,N) -> a__U72(a__isNat(N),M,N) a__U72(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNat(x(V1,V2)) -> a__U31(a__isNat(V1),V2) a__plus(N,0) -> a__U41(a__isNat(N),N) a__plus(N,s(M)) -> a__U51(a__isNat(M),M,N) a__x(N,0) -> a__U61(a__isNat(N)) a__x(N,s(M)) -> a__U71(a__isNat(M),M,N) mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2,X3)) -> a__U52(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X1,X2,X3)) -> a__U71(mark(X1),X2,X3) mark(U72(X1,X2,X3)) -> a__U72(mark(X1),X2,X3) mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(0) -> 0 a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNat(X) -> isNat(X) a__U21(X) -> U21(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2) -> U41(X1,X2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2,X3) -> U52(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) a__U61(X) -> U61(X) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(X1,X2,X3) -> U72(X1,X2,X3) a__x(X1,X2) -> x(X1,X2) ) Proving termination of rewriting for MYNAT_nokinds_noand_GM: -> Dependency pairs: nF_a__U11(tt,V2) -> nF_a__U12(a__isNat(V2)) nF_a__U11(tt,V2) -> nF_a__isNat(V2) nF_a__U31(tt,V2) -> nF_a__U32(a__isNat(V2)) nF_a__U31(tt,V2) -> nF_a__isNat(V2) nF_a__U41(tt,N) -> nF_mark(N) nF_a__U51(tt,M,N) -> nF_a__U52(a__isNat(N),M,N) nF_a__U51(tt,M,N) -> nF_a__isNat(N) nF_a__U52(tt,M,N) -> nF_a__plus(mark(N),mark(M)) nF_a__U52(tt,M,N) -> nF_mark(N) nF_a__U52(tt,M,N) -> nF_mark(M) nF_a__U71(tt,M,N) -> nF_a__U72(a__isNat(N),M,N) nF_a__U71(tt,M,N) -> nF_a__isNat(N) nF_a__U72(tt,M,N) -> nF_a__plus(a__x(mark(N),mark(M)),mark(N)) nF_a__U72(tt,M,N) -> nF_a__x(mark(N),mark(M)) nF_a__U72(tt,M,N) -> nF_mark(N) nF_a__U72(tt,M,N) -> nF_mark(M) nF_a__isNat(plus(V1,V2)) -> nF_a__U11(a__isNat(V1),V2) nF_a__isNat(plus(V1,V2)) -> nF_a__isNat(V1) nF_a__isNat(s(V1)) -> nF_a__U21(a__isNat(V1)) nF_a__isNat(s(V1)) -> nF_a__isNat(V1) nF_a__isNat(x(V1,V2)) -> nF_a__U31(a__isNat(V1),V2) nF_a__isNat(x(V1,V2)) -> nF_a__isNat(V1) nF_a__plus(N,0) -> nF_a__U41(a__isNat(N),N) nF_a__plus(N,0) -> nF_a__isNat(N) nF_a__plus(N,s(M)) -> nF_a__U51(a__isNat(M),M,N) nF_a__plus(N,s(M)) -> nF_a__isNat(M) nF_a__x(N,0) -> nF_a__U61(a__isNat(N)) nF_a__x(N,0) -> nF_a__isNat(N) nF_a__x(N,s(M)) -> nF_a__U71(a__isNat(M),M,N) nF_a__x(N,s(M)) -> nF_a__isNat(M) nF_mark(U11(X1,X2)) -> nF_a__U11(mark(X1),X2) nF_mark(U11(X1,X2)) -> nF_mark(X1) nF_mark(U12(X)) -> nF_a__U12(mark(X)) nF_mark(U12(X)) -> nF_mark(X) nF_mark(isNat(X)) -> nF_a__isNat(X) nF_mark(U21(X)) -> nF_a__U21(mark(X)) nF_mark(U21(X)) -> nF_mark(X) nF_mark(U31(X1,X2)) -> nF_a__U31(mark(X1),X2) nF_mark(U31(X1,X2)) -> nF_mark(X1) nF_mark(U32(X)) -> nF_a__U32(mark(X)) nF_mark(U32(X)) -> nF_mark(X) nF_mark(U41(X1,X2)) -> nF_a__U41(mark(X1),X2) nF_mark(U41(X1,X2)) -> nF_mark(X1) nF_mark(U51(X1,X2,X3)) -> nF_a__U51(mark(X1),X2,X3) nF_mark(U51(X1,X2,X3)) -> nF_mark(X1) nF_mark(U52(X1,X2,X3)) -> nF_a__U52(mark(X1),X2,X3) nF_mark(U52(X1,X2,X3)) -> nF_mark(X1) nF_mark(plus(X1,X2)) -> nF_a__plus(mark(X1),mark(X2)) nF_mark(plus(X1,X2)) -> nF_mark(X1) nF_mark(plus(X1,X2)) -> nF_mark(X2) nF_mark(U61(X)) -> nF_a__U61(mark(X)) nF_mark(U61(X)) -> nF_mark(X) nF_mark(U71(X1,X2,X3)) -> nF_a__U71(mark(X1),X2,X3) nF_mark(U71(X1,X2,X3)) -> nF_mark(X1) nF_mark(U72(X1,X2,X3)) -> nF_a__U72(mark(X1),X2,X3) nF_mark(U72(X1,X2,X3)) -> nF_mark(X1) nF_mark(x(X1,X2)) -> nF_a__x(mark(X1),mark(X2)) nF_mark(x(X1,X2)) -> nF_mark(X1) nF_mark(x(X1,X2)) -> nF_mark(X2) nF_mark(s(X)) -> nF_mark(X) -> Proof of termination for MYNAT_nokinds_noand_GM_1_1: -> -> Dependency pairs in cycle: nF_a__U41(tt,N) -> nF_mark(N) nF_mark(U41(X1,X2)) -> nF_a__U41(mark(X1),X2) nF_mark(s(X)) -> nF_mark(X) nF_mark(x(X1,X2)) -> nF_mark(X2) nF_mark(x(X1,X2)) -> nF_mark(X1) nF_mark(U72(X1,X2,X3)) -> nF_mark(X1) nF_mark(U71(X1,X2,X3)) -> nF_mark(X1) nF_mark(U61(X)) -> nF_mark(X) nF_mark(plus(X1,X2)) -> nF_mark(X2) nF_mark(plus(X1,X2)) -> nF_mark(X1) nF_mark(U52(X1,X2,X3)) -> nF_mark(X1) nF_mark(U51(X1,X2,X3)) -> nF_mark(X1) nF_mark(U41(X1,X2)) -> nF_mark(X1) nF_mark(U32(X)) -> nF_mark(X) nF_mark(U31(X1,X2)) -> nF_mark(X1) nF_mark(U21(X)) -> nF_mark(X) nF_mark(U12(X)) -> nF_mark(X) nF_mark(U11(X1,X2)) -> nF_mark(X1) nF_a__U72(tt,M,N) -> nF_mark(M) nF_mark(U72(X1,X2,X3)) -> nF_a__U72(mark(X1),X2,X3) nF_a__U72(tt,M,N) -> nF_mark(N) nF_a__U71(tt,M,N) -> nF_a__U72(a__isNat(N),M,N) nF_mark(U71(X1,X2,X3)) -> nF_a__U71(mark(X1),X2,X3) nF_a__U52(tt,M,N) -> nF_mark(M) nF_mark(U52(X1,X2,X3)) -> nF_a__U52(mark(X1),X2,X3) nF_a__U52(tt,M,N) -> nF_mark(N) nF_a__U51(tt,M,N) -> nF_a__U52(a__isNat(N),M,N) nF_mark(U51(X1,X2,X3)) -> nF_a__U51(mark(X1),X2,X3) nF_a__plus(N,s(M)) -> nF_a__U51(a__isNat(M),M,N) nF_mark(plus(X1,X2)) -> nF_a__plus(mark(X1),mark(X2)) nF_a__U72(tt,M,N) -> nF_a__plus(a__x(mark(N),mark(M)),mark(N)) nF_a__U52(tt,M,N) -> nF_a__plus(mark(N),mark(M)) nF_a__x(N,s(M)) -> nF_a__U71(a__isNat(M),M,N) nF_mark(x(X1,X2)) -> nF_a__x(mark(X1),mark(X2)) nF_a__U72(tt,M,N) -> nF_a__x(mark(N),mark(M)) nF_a__plus(N,0) -> nF_a__U41(a__isNat(N),N) UsableRules: a__U11(tt,V2) -> a__U12(a__isNat(V2)) a__U12(tt) -> tt a__U21(tt) -> tt a__U31(tt,V2) -> a__U32(a__isNat(V2)) a__U32(tt) -> tt a__U41(tt,N) -> mark(N) a__U51(tt,M,N) -> a__U52(a__isNat(N),M,N) a__U52(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U61(tt) -> 0 a__U71(tt,M,N) -> a__U72(a__isNat(N),M,N) a__U72(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNat(x(V1,V2)) -> a__U31(a__isNat(V1),V2) a__plus(N,0) -> a__U41(a__isNat(N),N) a__plus(N,s(M)) -> a__U51(a__isNat(M),M,N) a__x(N,0) -> a__U61(a__isNat(N)) a__x(N,s(M)) -> a__U71(a__isNat(M),M,N) mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2,X3)) -> a__U52(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X1,X2,X3)) -> a__U71(mark(X1),X2,X3) mark(U72(X1,X2,X3)) -> a__U72(mark(X1),X2,X3) mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(0) -> 0 a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNat(X) -> isNat(X) a__U21(X) -> U21(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2) -> U41(X1,X2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2,X3) -> U52(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) a__U61(X) -> U61(X) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(X1,X2,X3) -> U72(X1,X2,X3) a__x(X1,X2) -> x(X1,X2) Polynomial Interpretation: [a__U11](X1,X2) = X1 [tt] = 1 [a__U12](X) = X [a__isNat](X) = 1 [a__U21](X) = X [a__U31](X1,X2) = X1 [a__U32](X) = X [a__U41](X1,X2) = X1 + X2 [mark](X) = X [a__U51](X1,X2,X3) = X1.X2 + X1.X3 + X1 + 1 [a__U52](X1,X2,X3) = X1 + X2 + X3 + 1 [s](X) = X + 1 [a__plus](X1,X2) = X1 + X2 + 1 [a__U61](X) = X [0] = 1 [a__U71](X1,X2,X3) = X1.X2.X3 + X1.X2 + X1.X3 + X1 + X3 + 1 [a__U72](X1,X2,X3) = X1.X3 + X2.X3 + X1 + X2 + X3 + 1 [a__x](X1,X2) = X1.X2 + X1 + X2 + 1 [plus](X1,X2) = X1 + X2 + 1 [x](X1,X2) = X1.X2 + X1 + X2 + 1 [U11](X1,X2) = X1 [U12](X) = X [isNat](X) = 1 [U21](X) = X [U31](X1,X2) = X1 [U32](X) = X [U41](X1,X2) = X1 + X2 [U51](X1,X2,X3) = X1.X2 + X1.X3 + X1 + 1 [U52](X1,X2,X3) = X1 + X2 + X3 + 1 [U61](X) = X [U71](X1,X2,X3) = X1.X2.X3 + X1.X2 + X1.X3 + X1 + X3 + 1 [U72](X1,X2,X3) = X1.X3 + X2.X3 + X1 + X2 + X3 + 1 [nF_a__U41](X1,X2) = X2 [nF_mark](X) = X [nF_a__U72](X1,X2,X3) = X1.X3 + X2.X3 + X2 + X3 + 1 [nF_a__U71](X1,X2,X3) = X1.X2.X3 + X1.X2 + X1.X3 + X1 + X3 [nF_a__U52](X1,X2,X3) = X2 + X3 [nF_a__U51](X1,X2,X3) = X1.X2 + X1.X3 + X1 [nF_a__plus](X1,X2) = X1 + X2 [nF_a__x](X1,X2) = X1.X2 + X1 + X2 TIME: 0.532546 -> -> Dependency pairs in cycle: nF_a__U41(tt,N) -> nF_mark(N) nF_mark(U41(X1,X2)) -> nF_a__U41(mark(X1),X2) nF_a__U52(tt,M,N) -> nF_mark(N) nF_a__U51(tt,M,N) -> nF_a__U52(a__isNat(N),M,N) nF_a__plus(N,s(M)) -> nF_a__U51(a__isNat(M),M,N) nF_a__U52(tt,M,N) -> nF_a__plus(mark(N),mark(M)) nF_mark(U52(X1,X2,X3)) -> nF_a__U52(mark(X1),X2,X3) nF_a__U52(tt,M,N) -> nF_mark(M) nF_a__U72(tt,M,N) -> nF_mark(N) nF_a__U71(tt,M,N) -> nF_a__U72(a__isNat(N),M,N) nF_a__x(N,s(M)) -> nF_a__U71(a__isNat(M),M,N) nF_a__U72(tt,M,N) -> nF_a__x(mark(N),mark(M)) nF_mark(U72(X1,X2,X3)) -> nF_a__U72(mark(X1),X2,X3) nF_a__U72(tt,M,N) -> nF_mark(M) nF_mark(U11(X1,X2)) -> nF_mark(X1) nF_mark(U12(X)) -> nF_mark(X) nF_mark(U21(X)) -> nF_mark(X) nF_mark(U31(X1,X2)) -> nF_mark(X1) nF_mark(U32(X)) -> nF_mark(X) nF_mark(U41(X1,X2)) -> nF_mark(X1) nF_mark(U51(X1,X2,X3)) -> nF_mark(X1) nF_mark(U52(X1,X2,X3)) -> nF_mark(X1) nF_mark(plus(X1,X2)) -> nF_mark(X1) nF_mark(plus(X1,X2)) -> nF_mark(X2) nF_mark(U61(X)) -> nF_mark(X) nF_mark(U71(X1,X2,X3)) -> nF_mark(X1) nF_mark(U72(X1,X2,X3)) -> nF_mark(X1) nF_mark(x(X1,X2)) -> nF_mark(X1) nF_mark(x(X1,X2)) -> nF_mark(X2) nF_mark(s(X)) -> nF_mark(X) nF_mark(x(X1,X2)) -> nF_a__x(mark(X1),mark(X2)) nF_mark(U71(X1,X2,X3)) -> nF_a__U71(mark(X1),X2,X3) nF_a__U72(tt,M,N) -> nF_a__plus(a__x(mark(N),mark(M)),mark(N)) nF_mark(plus(X1,X2)) -> nF_a__plus(mark(X1),mark(X2)) nF_mark(U51(X1,X2,X3)) -> nF_a__U51(mark(X1),X2,X3) UsableRules: a__U11(tt,V2) -> a__U12(a__isNat(V2)) a__U12(tt) -> tt a__U21(tt) -> tt a__U31(tt,V2) -> a__U32(a__isNat(V2)) a__U32(tt) -> tt a__U41(tt,N) -> mark(N) a__U51(tt,M,N) -> a__U52(a__isNat(N),M,N) a__U52(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U61(tt) -> 0 a__U71(tt,M,N) -> a__U72(a__isNat(N),M,N) a__U72(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNat(x(V1,V2)) -> a__U31(a__isNat(V1),V2) a__plus(N,0) -> a__U41(a__isNat(N),N) a__plus(N,s(M)) -> a__U51(a__isNat(M),M,N) a__x(N,0) -> a__U61(a__isNat(N)) a__x(N,s(M)) -> a__U71(a__isNat(M),M,N) mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2,X3)) -> a__U52(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X1,X2,X3)) -> a__U71(mark(X1),X2,X3) mark(U72(X1,X2,X3)) -> a__U72(mark(X1),X2,X3) mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(0) -> 0 a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNat(X) -> isNat(X) a__U21(X) -> U21(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2) -> U41(X1,X2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2,X3) -> U52(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) a__U61(X) -> U61(X) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(X1,X2,X3) -> U72(X1,X2,X3) a__x(X1,X2) -> x(X1,X2) Polynomial Interpretation: [a__U11](X1,X2) = X1 [tt] = 1 [a__U12](X) = X [a__isNat](X) = 1 [a__U21](X) = X [a__U31](X1,X2) = X1 [a__U32](X) = X [a__U41](X1,X2) = X1.X2 + X1 [mark](X) = X [a__U51](X1,X2,X3) = X1.X2 + X1.X3 + X1 + 1 [a__U52](X1,X2,X3) = X1 + X2 + X3 + 1 [s](X) = X + 1 [a__plus](X1,X2) = X1 + X2 + 1 [a__U61](X) = X [0] = 1 [a__U71](X1,X2,X3) = X1.X2.X3 + X1.X2 + X1.X3 + X1 + X3 + 1 [a__U72](X1,X2,X3) = X1.X3 + X2.X3 + X1 + X2 + X3 + 1 [a__x](X1,X2) = X1.X2 + X1 + X2 + 1 [plus](X1,X2) = X1 + X2 + 1 [x](X1,X2) = X1.X2 + X1 + X2 + 1 [U11](X1,X2) = X1 [U12](X) = X [isNat](X) = 1 [U21](X) = X [U31](X1,X2) = X1 [U32](X) = X [U41](X1,X2) = X1.X2 + X1 [U51](X1,X2,X3) = X1.X2 + X1.X3 + X1 + 1 [U52](X1,X2,X3) = X1 + X2 + X3 + 1 [U61](X) = X [U71](X1,X2,X3) = X1.X2.X3 + X1.X2 + X1.X3 + X1 + X3 + 1 [U72](X1,X2,X3) = X1.X3 + X2.X3 + X1 + X2 + X3 + 1 [nF_a__U41](X1,X2) = X1.X2 [nF_mark](X) = X [nF_a__U52](X1,X2,X3) = X2 + X3 [nF_a__U51](X1,X2,X3) = X1.X2 + X1.X3 + X1 [nF_a__plus](X1,X2) = X1 + X2 [nF_a__U72](X1,X2,X3) = X1.X3 + X2.X3 + X2 + X3 + 1 [nF_a__U71](X1,X2,X3) = X1.X2.X3 + X1.X2 + X1.X3 + X1 + X3 [nF_a__x](X1,X2) = X1.X2 + X1 + X2 TIME: 0.435163 -> -> Dependency pairs in cycle: nF_a__U41(tt,N) -> nF_mark(N) nF_mark(U41(X1,X2)) -> nF_a__U41(mark(X1),X2) nF_mark(s(X)) -> nF_mark(X) nF_mark(x(X1,X2)) -> nF_mark(X2) nF_mark(x(X1,X2)) -> nF_mark(X1) nF_mark(U72(X1,X2,X3)) -> nF_mark(X1) nF_mark(U71(X1,X2,X3)) -> nF_mark(X1) nF_mark(U61(X)) -> nF_mark(X) nF_mark(plus(X1,X2)) -> nF_mark(X2) nF_mark(plus(X1,X2)) -> nF_mark(X1) nF_mark(U52(X1,X2,X3)) -> nF_mark(X1) nF_mark(U51(X1,X2,X3)) -> nF_mark(X1) nF_mark(U41(X1,X2)) -> nF_mark(X1) nF_mark(U32(X)) -> nF_mark(X) nF_mark(U31(X1,X2)) -> nF_mark(X1) nF_mark(U21(X)) -> nF_mark(X) nF_mark(U12(X)) -> nF_mark(X) nF_mark(U11(X1,X2)) -> nF_mark(X1) nF_a__U72(tt,M,N) -> nF_mark(M) nF_mark(U72(X1,X2,X3)) -> nF_a__U72(mark(X1),X2,X3) nF_a__U72(tt,M,N) -> nF_mark(N) nF_a__U71(tt,M,N) -> nF_a__U72(a__isNat(N),M,N) nF_mark(U71(X1,X2,X3)) -> nF_a__U71(mark(X1),X2,X3) nF_a__U52(tt,M,N) -> nF_mark(M) nF_mark(U52(X1,X2,X3)) -> nF_a__U52(mark(X1),X2,X3) nF_a__U52(tt,M,N) -> nF_mark(N) nF_a__U51(tt,M,N) -> nF_a__U52(a__isNat(N),M,N) nF_a__plus(N,s(M)) -> nF_a__U51(a__isNat(M),M,N) nF_mark(plus(X1,X2)) -> nF_a__plus(mark(X1),mark(X2)) nF_a__U72(tt,M,N) -> nF_a__plus(a__x(mark(N),mark(M)),mark(N)) nF_a__U52(tt,M,N) -> nF_a__plus(mark(N),mark(M)) nF_a__x(N,s(M)) -> nF_a__U71(a__isNat(M),M,N) nF_mark(x(X1,X2)) -> nF_a__x(mark(X1),mark(X2)) nF_a__U72(tt,M,N) -> nF_a__x(mark(N),mark(M)) UsableRules: a__U11(tt,V2) -> a__U12(a__isNat(V2)) a__U12(tt) -> tt a__U21(tt) -> tt a__U31(tt,V2) -> a__U32(a__isNat(V2)) a__U32(tt) -> tt a__U41(tt,N) -> mark(N) a__U51(tt,M,N) -> a__U52(a__isNat(N),M,N) a__U52(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U61(tt) -> 0 a__U71(tt,M,N) -> a__U72(a__isNat(N),M,N) a__U72(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNat(x(V1,V2)) -> a__U31(a__isNat(V1),V2) a__plus(N,0) -> a__U41(a__isNat(N),N) a__plus(N,s(M)) -> a__U51(a__isNat(M),M,N) a__x(N,0) -> a__U61(a__isNat(N)) a__x(N,s(M)) -> a__U71(a__isNat(M),M,N) mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2,X3)) -> a__U52(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X1,X2,X3)) -> a__U71(mark(X1),X2,X3) mark(U72(X1,X2,X3)) -> a__U72(mark(X1),X2,X3) mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(0) -> 0 a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNat(X) -> isNat(X) a__U21(X) -> U21(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2) -> U41(X1,X2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2,X3) -> U52(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) a__U61(X) -> U61(X) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(X1,X2,X3) -> U72(X1,X2,X3) a__x(X1,X2) -> x(X1,X2) Polynomial Interpretation: [a__U11](X1,X2) = X1 [tt] = 1 [a__U12](X) = X [a__isNat](X) = 1 [a__U21](X) = X [a__U31](X1,X2) = X1 [a__U32](X) = X [a__U41](X1,X2) = X1 + X2 [mark](X) = X [a__U51](X1,X2,X3) = X1 + X2 + X3 [a__U52](X1,X2,X3) = X1.X3 + X1 + X2 [s](X) = X + 1 [a__plus](X1,X2) = X1 + X2 [a__U61](X) = X.X + X [0] = 1 [a__U71](X1,X2,X3) = X1.X3 + X2.X3 + X1 + X2 + X3 + 1 [a__U72](X1,X2,X3) = X1.X2.X3 + X1.X3 + X1 + X2 + X3 [a__x](X1,X2) = X1.X2 + X1 + X2 + 1 [plus](X1,X2) = X1 + X2 [x](X1,X2) = X1.X2 + X1 + X2 + 1 [U11](X1,X2) = X1 [U12](X) = X [isNat](X) = 1 [U21](X) = X [U31](X1,X2) = X1 [U32](X) = X [U41](X1,X2) = X1 + X2 [U51](X1,X2,X3) = X1 + X2 + X3 [U52](X1,X2,X3) = X1.X3 + X1 + X2 [U61](X) = X.X + X [U71](X1,X2,X3) = X1.X3 + X2.X3 + X1 + X2 + X3 + 1 [U72](X1,X2,X3) = X1.X2.X3 + X1.X3 + X1 + X2 + X3 [nF_a__U41](X1,X2) = X2 [nF_mark](X) = X [nF_a__U72](X1,X2,X3) = X1.X2.X3 + X1.X3 + X1 + X2 + X3 [nF_a__U71](X1,X2,X3) = X1.X3 + X2.X3 + X2 + X3 + 1 [nF_a__U52](X1,X2,X3) = X1.X3 + X2 [nF_a__U51](X1,X2,X3) = X1.X2 + X1.X3 [nF_a__plus](X1,X2) = X1 + X2 [nF_a__x](X1,X2) = X1.X2 + X1 + X2 TIME: 0.418308 -> -> Dependency pairs in cycle: nF_a__U41(tt,N) -> nF_mark(N) nF_mark(U41(X1,X2)) -> nF_a__U41(mark(X1),X2) nF_a__U52(tt,M,N) -> nF_mark(N) nF_a__U51(tt,M,N) -> nF_a__U52(a__isNat(N),M,N) nF_a__plus(N,s(M)) -> nF_a__U51(a__isNat(M),M,N) nF_a__U52(tt,M,N) -> nF_a__plus(mark(N),mark(M)) nF_mark(U52(X1,X2,X3)) -> nF_a__U52(mark(X1),X2,X3) nF_a__U52(tt,M,N) -> nF_mark(M) nF_a__U72(tt,M,N) -> nF_mark(N) nF_a__U71(tt,M,N) -> nF_a__U72(a__isNat(N),M,N) nF_a__x(N,s(M)) -> nF_a__U71(a__isNat(M),M,N) nF_mark(x(X1,X2)) -> nF_a__x(mark(X1),mark(X2)) nF_a__U72(tt,M,N) -> nF_mark(M) nF_mark(U72(X1,X2,X3)) -> nF_a__U72(mark(X1),X2,X3) nF_mark(U11(X1,X2)) -> nF_mark(X1) nF_mark(U12(X)) -> nF_mark(X) nF_mark(U21(X)) -> nF_mark(X) nF_mark(U31(X1,X2)) -> nF_mark(X1) nF_mark(U32(X)) -> nF_mark(X) nF_mark(U41(X1,X2)) -> nF_mark(X1) nF_mark(U51(X1,X2,X3)) -> nF_mark(X1) nF_mark(U52(X1,X2,X3)) -> nF_mark(X1) nF_mark(plus(X1,X2)) -> nF_mark(X1) nF_mark(plus(X1,X2)) -> nF_mark(X2) nF_mark(U61(X)) -> nF_mark(X) nF_mark(U71(X1,X2,X3)) -> nF_mark(X1) nF_mark(U72(X1,X2,X3)) -> nF_mark(X1) nF_mark(x(X1,X2)) -> nF_mark(X1) nF_mark(x(X1,X2)) -> nF_mark(X2) nF_mark(s(X)) -> nF_mark(X) nF_mark(U71(X1,X2,X3)) -> nF_a__U71(mark(X1),X2,X3) nF_a__U72(tt,M,N) -> nF_a__plus(a__x(mark(N),mark(M)),mark(N)) nF_mark(plus(X1,X2)) -> nF_a__plus(mark(X1),mark(X2)) UsableRules: a__U11(tt,V2) -> a__U12(a__isNat(V2)) a__U12(tt) -> tt a__U21(tt) -> tt a__U31(tt,V2) -> a__U32(a__isNat(V2)) a__U32(tt) -> tt a__U41(tt,N) -> mark(N) a__U51(tt,M,N) -> a__U52(a__isNat(N),M,N) a__U52(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U61(tt) -> 0 a__U71(tt,M,N) -> a__U72(a__isNat(N),M,N) a__U72(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNat(x(V1,V2)) -> a__U31(a__isNat(V1),V2) a__plus(N,0) -> a__U41(a__isNat(N),N) a__plus(N,s(M)) -> a__U51(a__isNat(M),M,N) a__x(N,0) -> a__U61(a__isNat(N)) a__x(N,s(M)) -> a__U71(a__isNat(M),M,N) mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2,X3)) -> a__U52(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X1,X2,X3)) -> a__U71(mark(X1),X2,X3) mark(U72(X1,X2,X3)) -> a__U72(mark(X1),X2,X3) mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(0) -> 0 a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNat(X) -> isNat(X) a__U21(X) -> U21(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2) -> U41(X1,X2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2,X3) -> U52(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) a__U61(X) -> U61(X) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(X1,X2,X3) -> U72(X1,X2,X3) a__x(X1,X2) -> x(X1,X2) Polynomial Interpretation: [a__U11](X1,X2) = X1 [tt] = 1 [a__U12](X) = X [a__isNat](X) = 1 [a__U21](X) = X [a__U31](X1,X2) = X1 [a__U32](X) = X [a__U41](X1,X2) = X1 + X2 [mark](X) = X [a__U51](X1,X2,X3) = X1 + X2 + X3 + 1 [a__U52](X1,X2,X3) = X1.X2 + X1 + X3 + 1 [s](X) = X + 1 [a__plus](X1,X2) = X1 + X2 + 1 [a__U61](X) = X [0] = 1 [a__U71](X1,X2,X3) = X1.X3 + X2.X3 + X1 + X2 + X3 + 1 [a__U72](X1,X2,X3) = X1.X2.X3 + X1.X2 + X1.X3 + X1 + X3 + 1 [a__x](X1,X2) = X1.X2 + X1 + X2 + 1 [plus](X1,X2) = X1 + X2 + 1 [x](X1,X2) = X1.X2 + X1 + X2 + 1 [U11](X1,X2) = X1 [U12](X) = X [isNat](X) = 1 [U21](X) = X [U31](X1,X2) = X1 [U32](X) = X [U41](X1,X2) = X1 + X2 [U51](X1,X2,X3) = X1 + X2 + X3 + 1 [U52](X1,X2,X3) = X1.X2 + X1 + X3 + 1 [U61](X) = X [U71](X1,X2,X3) = X1.X3 + X2.X3 + X1 + X2 + X3 + 1 [U72](X1,X2,X3) = X1.X2.X3 + X1.X2 + X1.X3 + X1 + X3 + 1 [nF_a__U41](X1,X2) = X2 [nF_mark](X) = X [nF_a__U52](X1,X2,X3) = X1.X2 + X3 [nF_a__U51](X1,X2,X3) = X1.X3 + X1 + X2 [nF_a__plus](X1,X2) = X1 + X2 [nF_a__U72](X1,X2,X3) = X1.X2.X3 + X1.X2 + X1.X3 + X3 + 1 [nF_a__U71](X1,X2,X3) = X1.X3 + X2.X3 + X2 + X3 + 1 [nF_a__x](X1,X2) = X1.X2 + X1 + X2 TIME: 0.430984 -> -> Dependency pairs in cycle: nF_a__U41(tt,N) -> nF_mark(N) nF_mark(U41(X1,X2)) -> nF_a__U41(mark(X1),X2) nF_mark(s(X)) -> nF_mark(X) nF_mark(x(X1,X2)) -> nF_mark(X2) nF_mark(x(X1,X2)) -> nF_mark(X1) nF_mark(U72(X1,X2,X3)) -> nF_mark(X1) nF_mark(U71(X1,X2,X3)) -> nF_mark(X1) nF_mark(U61(X)) -> nF_mark(X) nF_mark(plus(X1,X2)) -> nF_mark(X2) nF_mark(plus(X1,X2)) -> nF_mark(X1) nF_mark(U52(X1,X2,X3)) -> nF_mark(X1) nF_mark(U51(X1,X2,X3)) -> nF_mark(X1) nF_mark(U41(X1,X2)) -> nF_mark(X1) nF_mark(U32(X)) -> nF_mark(X) nF_mark(U31(X1,X2)) -> nF_mark(X1) nF_mark(U21(X)) -> nF_mark(X) nF_mark(U12(X)) -> nF_mark(X) nF_mark(U11(X1,X2)) -> nF_mark(X1) nF_a__U72(tt,M,N) -> nF_mark(M) nF_mark(U72(X1,X2,X3)) -> nF_a__U72(mark(X1),X2,X3) nF_a__U72(tt,M,N) -> nF_mark(N) nF_a__U71(tt,M,N) -> nF_a__U72(a__isNat(N),M,N) nF_mark(U71(X1,X2,X3)) -> nF_a__U71(mark(X1),X2,X3) nF_a__U52(tt,M,N) -> nF_mark(M) nF_mark(U52(X1,X2,X3)) -> nF_a__U52(mark(X1),X2,X3) nF_a__U52(tt,M,N) -> nF_mark(N) nF_a__U51(tt,M,N) -> nF_a__U52(a__isNat(N),M,N) nF_a__plus(N,s(M)) -> nF_a__U51(a__isNat(M),M,N) nF_a__U72(tt,M,N) -> nF_a__plus(a__x(mark(N),mark(M)),mark(N)) nF_a__U52(tt,M,N) -> nF_a__plus(mark(N),mark(M)) nF_a__x(N,s(M)) -> nF_a__U71(a__isNat(M),M,N) nF_mark(x(X1,X2)) -> nF_a__x(mark(X1),mark(X2)) UsableRules: a__U11(tt,V2) -> a__U12(a__isNat(V2)) a__U12(tt) -> tt a__U21(tt) -> tt a__U31(tt,V2) -> a__U32(a__isNat(V2)) a__U32(tt) -> tt a__U41(tt,N) -> mark(N) a__U51(tt,M,N) -> a__U52(a__isNat(N),M,N) a__U52(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U61(tt) -> 0 a__U71(tt,M,N) -> a__U72(a__isNat(N),M,N) a__U72(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNat(x(V1,V2)) -> a__U31(a__isNat(V1),V2) a__plus(N,0) -> a__U41(a__isNat(N),N) a__plus(N,s(M)) -> a__U51(a__isNat(M),M,N) a__x(N,0) -> a__U61(a__isNat(N)) a__x(N,s(M)) -> a__U71(a__isNat(M),M,N) mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2,X3)) -> a__U52(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X1,X2,X3)) -> a__U71(mark(X1),X2,X3) mark(U72(X1,X2,X3)) -> a__U72(mark(X1),X2,X3) mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(0) -> 0 a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNat(X) -> isNat(X) a__U21(X) -> U21(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2) -> U41(X1,X2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2,X3) -> U52(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) a__U61(X) -> U61(X) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(X1,X2,X3) -> U72(X1,X2,X3) a__x(X1,X2) -> x(X1,X2) Polynomial Interpretation: [a__U11](X1,X2) = X1 [tt] = 1 [a__U12](X) = X [a__isNat](X) = 1 [a__U21](X) = X [a__U31](X1,X2) = X1 [a__U32](X) = X [a__U41](X1,X2) = X1 + X2 [mark](X) = X [a__U51](X1,X2,X3) = X1 + X2 + X3 [a__U52](X1,X2,X3) = X1.X3 + X1 + X2 [s](X) = X + 1 [a__plus](X1,X2) = X1 + X2 [a__U61](X) = X [0] = 1 [a__U71](X1,X2,X3) = X1.X2.X3 + X1.X2 + X1.X3 + X1 + X3 + 1 [a__U72](X1,X2,X3) = X1.X3 + X2.X3 + X1 + X2 + X3 [a__x](X1,X2) = X1.X2 + X1 + X2 + 1 [plus](X1,X2) = X1 + X2 [x](X1,X2) = X1.X2 + X1 + X2 + 1 [U11](X1,X2) = X1 [U12](X) = X [isNat](X) = 1 [U21](X) = X [U31](X1,X2) = X1 [U32](X) = X [U41](X1,X2) = X1 + X2 [U51](X1,X2,X3) = X1 + X2 + X3 [U52](X1,X2,X3) = X1.X3 + X1 + X2 [U61](X) = X [U71](X1,X2,X3) = X1.X2.X3 + X1.X2 + X1.X3 + X1 + X3 + 1 [U72](X1,X2,X3) = X1.X3 + X2.X3 + X1 + X2 + X3 [nF_a__U41](X1,X2) = X2 [nF_mark](X) = X [nF_a__U72](X1,X2,X3) = X1.X3 + X2.X3 + X1 + X2 + X3 [nF_a__U71](X1,X2,X3) = X1.X2.X3 + X1.X2 + X1.X3 + X3 + 1 [nF_a__U52](X1,X2,X3) = X1.X3 + X2 [nF_a__U51](X1,X2,X3) = X1.X2 + X3 [nF_a__plus](X1,X2) = X1 + X2 [nF_a__x](X1,X2) = X1.X2 + X1 + X2 TIME: 0.450651 -> -> Dependency pairs in cycle: nF_a__U41(tt,N) -> nF_mark(N) nF_mark(U41(X1,X2)) -> nF_a__U41(mark(X1),X2) nF_a__U52(tt,M,N) -> nF_mark(N) nF_a__U51(tt,M,N) -> nF_a__U52(a__isNat(N),M,N) nF_a__plus(N,s(M)) -> nF_a__U51(a__isNat(M),M,N) nF_a__U52(tt,M,N) -> nF_a__plus(mark(N),mark(M)) nF_mark(U52(X1,X2,X3)) -> nF_a__U52(mark(X1),X2,X3) nF_a__U52(tt,M,N) -> nF_mark(M) nF_a__U72(tt,M,N) -> nF_mark(N) nF_a__U71(tt,M,N) -> nF_a__U72(a__isNat(N),M,N) nF_mark(U71(X1,X2,X3)) -> nF_a__U71(mark(X1),X2,X3) nF_a__U72(tt,M,N) -> nF_mark(M) nF_mark(U72(X1,X2,X3)) -> nF_a__U72(mark(X1),X2,X3) nF_mark(U11(X1,X2)) -> nF_mark(X1) nF_mark(U12(X)) -> nF_mark(X) nF_mark(U21(X)) -> nF_mark(X) nF_mark(U31(X1,X2)) -> nF_mark(X1) nF_mark(U32(X)) -> nF_mark(X) nF_mark(U41(X1,X2)) -> nF_mark(X1) nF_mark(U51(X1,X2,X3)) -> nF_mark(X1) nF_mark(U52(X1,X2,X3)) -> nF_mark(X1) nF_mark(plus(X1,X2)) -> nF_mark(X1) nF_mark(plus(X1,X2)) -> nF_mark(X2) nF_mark(U61(X)) -> nF_mark(X) nF_mark(U71(X1,X2,X3)) -> nF_mark(X1) nF_mark(U72(X1,X2,X3)) -> nF_mark(X1) nF_mark(x(X1,X2)) -> nF_mark(X1) nF_mark(x(X1,X2)) -> nF_mark(X2) nF_mark(s(X)) -> nF_mark(X) nF_a__U72(tt,M,N) -> nF_a__plus(a__x(mark(N),mark(M)),mark(N)) UsableRules: a__U11(tt,V2) -> a__U12(a__isNat(V2)) a__U12(tt) -> tt a__U21(tt) -> tt a__U31(tt,V2) -> a__U32(a__isNat(V2)) a__U32(tt) -> tt a__U41(tt,N) -> mark(N) a__U51(tt,M,N) -> a__U52(a__isNat(N),M,N) a__U52(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U61(tt) -> 0 a__U71(tt,M,N) -> a__U72(a__isNat(N),M,N) a__U72(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNat(x(V1,V2)) -> a__U31(a__isNat(V1),V2) a__plus(N,0) -> a__U41(a__isNat(N),N) a__plus(N,s(M)) -> a__U51(a__isNat(M),M,N) a__x(N,0) -> a__U61(a__isNat(N)) a__x(N,s(M)) -> a__U71(a__isNat(M),M,N) mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2,X3)) -> a__U52(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X1,X2,X3)) -> a__U71(mark(X1),X2,X3) mark(U72(X1,X2,X3)) -> a__U72(mark(X1),X2,X3) mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(0) -> 0 a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNat(X) -> isNat(X) a__U21(X) -> U21(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2) -> U41(X1,X2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2,X3) -> U52(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) a__U61(X) -> U61(X) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(X1,X2,X3) -> U72(X1,X2,X3) a__x(X1,X2) -> x(X1,X2) Polynomial Interpretation: [a__U11](X1,X2) = X1 [tt] = 1 [a__U12](X) = X [a__isNat](X) = 1 [a__U21](X) = X [a__U31](X1,X2) = X1 [a__U32](X) = X [a__U41](X1,X2) = X1 + X2 [mark](X) = X [a__U51](X1,X2,X3) = X1 + X2 + X3 [a__U52](X1,X2,X3) = X1.X2 + X1 + X3 [s](X) = X + 1 [a__plus](X1,X2) = X1 + X2 [a__U61](X) = X [0] = 1 [a__U71](X1,X2,X3) = X1.X2.X3 + X1.X2 + X1.X3 + X1 + X3 [a__U72](X1,X2,X3) = X1.X3 + X2.X3 + X1 + X2 + X3 [a__x](X1,X2) = X1.X2 + X1 + X2 [plus](X1,X2) = X1 + X2 [x](X1,X2) = X1.X2 + X1 + X2 [U11](X1,X2) = X1 [U12](X) = X [isNat](X) = 1 [U21](X) = X [U31](X1,X2) = X1 [U32](X) = X [U41](X1,X2) = X1 + X2 [U51](X1,X2,X3) = X1 + X2 + X3 [U52](X1,X2,X3) = X1.X2 + X1 + X3 [U61](X) = X [U71](X1,X2,X3) = X1.X2.X3 + X1.X2 + X1.X3 + X1 + X3 [U72](X1,X2,X3) = X1.X3 + X2.X3 + X1 + X2 + X3 [nF_a__U41](X1,X2) = X2 [nF_mark](X) = X [nF_a__U52](X1,X2,X3) = X1.X2 + X3 [nF_a__U51](X1,X2,X3) = X1.X3 + X1 + X2 [nF_a__plus](X1,X2) = X1 + X2 [nF_a__U72](X1,X2,X3) = X1.X3 + X2.X3 + X1 + X2 + X3 [nF_a__U71](X1,X2,X3) = X1.X2.X3 + X1.X2 + X1.X3 + X1 + X3 TIME: 0.434858 -> -> Dependency pairs in cycle: nF_a__U41(tt,N) -> nF_mark(N) nF_mark(U41(X1,X2)) -> nF_a__U41(mark(X1),X2) nF_mark(s(X)) -> nF_mark(X) nF_mark(x(X1,X2)) -> nF_mark(X2) nF_mark(x(X1,X2)) -> nF_mark(X1) nF_mark(U72(X1,X2,X3)) -> nF_mark(X1) nF_mark(U71(X1,X2,X3)) -> nF_mark(X1) nF_mark(U61(X)) -> nF_mark(X) nF_mark(plus(X1,X2)) -> nF_mark(X2) nF_mark(plus(X1,X2)) -> nF_mark(X1) nF_mark(U52(X1,X2,X3)) -> nF_mark(X1) nF_mark(U51(X1,X2,X3)) -> nF_mark(X1) nF_mark(U41(X1,X2)) -> nF_mark(X1) nF_mark(U32(X)) -> nF_mark(X) nF_mark(U31(X1,X2)) -> nF_mark(X1) nF_mark(U21(X)) -> nF_mark(X) nF_mark(U12(X)) -> nF_mark(X) nF_mark(U11(X1,X2)) -> nF_mark(X1) nF_a__U72(tt,M,N) -> nF_mark(M) nF_mark(U72(X1,X2,X3)) -> nF_a__U72(mark(X1),X2,X3) nF_a__U72(tt,M,N) -> nF_mark(N) nF_a__U71(tt,M,N) -> nF_a__U72(a__isNat(N),M,N) nF_mark(U71(X1,X2,X3)) -> nF_a__U71(mark(X1),X2,X3) nF_a__U52(tt,M,N) -> nF_mark(M) nF_mark(U52(X1,X2,X3)) -> nF_a__U52(mark(X1),X2,X3) nF_a__U52(tt,M,N) -> nF_mark(N) nF_a__U51(tt,M,N) -> nF_a__U52(a__isNat(N),M,N) nF_a__plus(N,s(M)) -> nF_a__U51(a__isNat(M),M,N) nF_a__U52(tt,M,N) -> nF_a__plus(mark(N),mark(M)) UsableRules: a__U11(tt,V2) -> a__U12(a__isNat(V2)) a__U12(tt) -> tt a__U21(tt) -> tt a__U31(tt,V2) -> a__U32(a__isNat(V2)) a__U32(tt) -> tt a__U41(tt,N) -> mark(N) a__U51(tt,M,N) -> a__U52(a__isNat(N),M,N) a__U52(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U61(tt) -> 0 a__U71(tt,M,N) -> a__U72(a__isNat(N),M,N) a__U72(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNat(x(V1,V2)) -> a__U31(a__isNat(V1),V2) a__plus(N,0) -> a__U41(a__isNat(N),N) a__plus(N,s(M)) -> a__U51(a__isNat(M),M,N) a__x(N,0) -> a__U61(a__isNat(N)) a__x(N,s(M)) -> a__U71(a__isNat(M),M,N) mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2,X3)) -> a__U52(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X1,X2,X3)) -> a__U71(mark(X1),X2,X3) mark(U72(X1,X2,X3)) -> a__U72(mark(X1),X2,X3) mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(0) -> 0 a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNat(X) -> isNat(X) a__U21(X) -> U21(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2) -> U41(X1,X2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2,X3) -> U52(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) a__U61(X) -> U61(X) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(X1,X2,X3) -> U72(X1,X2,X3) a__x(X1,X2) -> x(X1,X2) Polynomial Interpretation: [a__U11](X1,X2) = X1 [tt] = 1 [a__U12](X) = X [a__isNat](X) = 1 [a__U21](X) = X [a__U31](X1,X2) = X1 [a__U32](X) = X [a__U41](X1,X2) = X1 + X2 [mark](X) = X [a__U51](X1,X2,X3) = X1 + X2 + X3 [a__U52](X1,X2,X3) = X1.X2 + X1.X3 + X1 [s](X) = X + 1 [a__plus](X1,X2) = X1 + X2 [a__U61](X) = X [0] = 1 [a__U71](X1,X2,X3) = X1.X2.X3 + X1.X2 + X1.X3 + X1 + X3 [a__U72](X1,X2,X3) = X1.X3 + X2.X3 + X1 + X2 + X3 [a__x](X1,X2) = X1.X2 + X1 + X2 + 1 [plus](X1,X2) = X1 + X2 [x](X1,X2) = X1.X2 + X1 + X2 + 1 [U11](X1,X2) = X1 [U12](X) = X [isNat](X) = 1 [U21](X) = X [U31](X1,X2) = X1 [U32](X) = X [U41](X1,X2) = X1 + X2 [U51](X1,X2,X3) = X1 + X2 + X3 [U52](X1,X2,X3) = X1.X2 + X1.X3 + X1 [U61](X) = X [U71](X1,X2,X3) = X1.X2.X3 + X1.X2 + X1.X3 + X1 + X3 [U72](X1,X2,X3) = X1.X3 + X2.X3 + X1 + X2 + X3 [nF_a__U41](X1,X2) = X2 [nF_mark](X) = X [nF_a__U72](X1,X2,X3) = X1.X3 + X2 [nF_a__U71](X1,X2,X3) = X1.X2.X3 + X1.X2 + X1 + X3 [nF_a__U52](X1,X2,X3) = X1.X2 + X1.X3 + X1 [nF_a__U51](X1,X2,X3) = X1.X3 + X2 + 1 [nF_a__plus](X1,X2) = X1 + X2 TIME: 0.37618700000000005 -> -> Dependency pairs in cycle: nF_a__U41(tt,N) -> nF_mark(N) nF_mark(U41(X1,X2)) -> nF_a__U41(mark(X1),X2) nF_a__U52(tt,M,N) -> nF_mark(N) nF_mark(U52(X1,X2,X3)) -> nF_a__U52(mark(X1),X2,X3) nF_a__U52(tt,M,N) -> nF_mark(M) nF_a__U72(tt,M,N) -> nF_mark(N) nF_a__U71(tt,M,N) -> nF_a__U72(a__isNat(N),M,N) nF_mark(U71(X1,X2,X3)) -> nF_a__U71(mark(X1),X2,X3) nF_a__U72(tt,M,N) -> nF_mark(M) nF_mark(U72(X1,X2,X3)) -> nF_a__U72(mark(X1),X2,X3) nF_mark(U11(X1,X2)) -> nF_mark(X1) nF_mark(U12(X)) -> nF_mark(X) nF_mark(U21(X)) -> nF_mark(X) nF_mark(U31(X1,X2)) -> nF_mark(X1) nF_mark(U32(X)) -> nF_mark(X) nF_mark(U41(X1,X2)) -> nF_mark(X1) nF_mark(U51(X1,X2,X3)) -> nF_mark(X1) nF_mark(U52(X1,X2,X3)) -> nF_mark(X1) nF_mark(plus(X1,X2)) -> nF_mark(X1) nF_mark(plus(X1,X2)) -> nF_mark(X2) nF_mark(U61(X)) -> nF_mark(X) nF_mark(U71(X1,X2,X3)) -> nF_mark(X1) nF_mark(U72(X1,X2,X3)) -> nF_mark(X1) nF_mark(x(X1,X2)) -> nF_mark(X1) nF_mark(x(X1,X2)) -> nF_mark(X2) nF_mark(s(X)) -> nF_mark(X) UsableRules: a__U11(tt,V2) -> a__U12(a__isNat(V2)) a__U12(tt) -> tt a__U21(tt) -> tt a__U31(tt,V2) -> a__U32(a__isNat(V2)) a__U32(tt) -> tt a__U41(tt,N) -> mark(N) a__U51(tt,M,N) -> a__U52(a__isNat(N),M,N) a__U52(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U61(tt) -> 0 a__U71(tt,M,N) -> a__U72(a__isNat(N),M,N) a__U72(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNat(x(V1,V2)) -> a__U31(a__isNat(V1),V2) a__plus(N,0) -> a__U41(a__isNat(N),N) a__plus(N,s(M)) -> a__U51(a__isNat(M),M,N) a__x(N,0) -> a__U61(a__isNat(N)) a__x(N,s(M)) -> a__U71(a__isNat(M),M,N) mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2,X3)) -> a__U52(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X1,X2,X3)) -> a__U71(mark(X1),X2,X3) mark(U72(X1,X2,X3)) -> a__U72(mark(X1),X2,X3) mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(0) -> 0 a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNat(X) -> isNat(X) a__U21(X) -> U21(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2) -> U41(X1,X2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2,X3) -> U52(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) a__U61(X) -> U61(X) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(X1,X2,X3) -> U72(X1,X2,X3) a__x(X1,X2) -> x(X1,X2) Polynomial Interpretation: [a__U11](X1,X2) = X1 [tt] = 1 [a__U12](X) = X [a__isNat](X) = 1 [a__U21](X) = X [a__U31](X1,X2) = X1 [a__U32](X) = X [a__U41](X1,X2) = X1.X2 + X1 [mark](X) = X [a__U51](X1,X2,X3) = X1.X2 + X1.X3 + X1 [a__U52](X1,X2,X3) = X1 + X2 + X3 [s](X) = X + 1 [a__plus](X1,X2) = X1 + X2 [a__U61](X) = X.X + X [0] = 1 [a__U71](X1,X2,X3) = X1.X2.X3 + X1.X2 + X1.X3 + X1 + X3 [a__U72](X1,X2,X3) = X1.X3 + X2.X3 + X1 + X2 + X3 [a__x](X1,X2) = X1.X2 + X1 + X2 + 1 [plus](X1,X2) = X1 + X2 [x](X1,X2) = X1.X2 + X1 + X2 + 1 [U11](X1,X2) = X1 [U12](X) = X [isNat](X) = 1 [U21](X) = X [U31](X1,X2) = X1 [U32](X) = X [U41](X1,X2) = X1.X2 + X1 [U51](X1,X2,X3) = X1.X2 + X1.X3 + X1 [U52](X1,X2,X3) = X1 + X2 + X3 [U61](X) = X.X + X [U71](X1,X2,X3) = X1.X2.X3 + X1.X2 + X1.X3 + X1 + X3 [U72](X1,X2,X3) = X1.X3 + X2.X3 + X1 + X2 + X3 [nF_a__U41](X1,X2) = X1.X2 [nF_mark](X) = X [nF_a__U52](X1,X2,X3) = X2 + X3 [nF_a__U72](X1,X2,X3) = X1.X3 + X2 [nF_a__U71](X1,X2,X3) = X1.X2.X3 + X1.X2 + X1 + X3 TIME: 0.370565 -> -> Dependency pairs in cycle: nF_a__U41(tt,N) -> nF_mark(N) nF_mark(U41(X1,X2)) -> nF_a__U41(mark(X1),X2) nF_mark(x(X1,X2)) -> nF_mark(X2) nF_mark(x(X1,X2)) -> nF_mark(X1) nF_mark(U72(X1,X2,X3)) -> nF_mark(X1) nF_mark(U71(X1,X2,X3)) -> nF_mark(X1) nF_mark(U61(X)) -> nF_mark(X) nF_mark(plus(X1,X2)) -> nF_mark(X2) nF_mark(plus(X1,X2)) -> nF_mark(X1) nF_mark(U52(X1,X2,X3)) -> nF_mark(X1) nF_mark(U51(X1,X2,X3)) -> nF_mark(X1) nF_mark(U41(X1,X2)) -> nF_mark(X1) nF_mark(U32(X)) -> nF_mark(X) nF_mark(U31(X1,X2)) -> nF_mark(X1) nF_mark(U21(X)) -> nF_mark(X) nF_mark(U12(X)) -> nF_mark(X) nF_mark(U11(X1,X2)) -> nF_mark(X1) nF_a__U72(tt,M,N) -> nF_mark(M) nF_mark(U72(X1,X2,X3)) -> nF_a__U72(mark(X1),X2,X3) nF_a__U72(tt,M,N) -> nF_mark(N) nF_a__U71(tt,M,N) -> nF_a__U72(a__isNat(N),M,N) nF_mark(U71(X1,X2,X3)) -> nF_a__U71(mark(X1),X2,X3) nF_a__U52(tt,M,N) -> nF_mark(M) nF_mark(U52(X1,X2,X3)) -> nF_a__U52(mark(X1),X2,X3) nF_a__U52(tt,M,N) -> nF_mark(N) UsableRules: a__U11(tt,V2) -> a__U12(a__isNat(V2)) a__U12(tt) -> tt a__U21(tt) -> tt a__U31(tt,V2) -> a__U32(a__isNat(V2)) a__U32(tt) -> tt a__U41(tt,N) -> mark(N) a__U51(tt,M,N) -> a__U52(a__isNat(N),M,N) a__U52(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U61(tt) -> 0 a__U71(tt,M,N) -> a__U72(a__isNat(N),M,N) a__U72(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNat(x(V1,V2)) -> a__U31(a__isNat(V1),V2) a__plus(N,0) -> a__U41(a__isNat(N),N) a__plus(N,s(M)) -> a__U51(a__isNat(M),M,N) a__x(N,0) -> a__U61(a__isNat(N)) a__x(N,s(M)) -> a__U71(a__isNat(M),M,N) mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2,X3)) -> a__U52(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X1,X2,X3)) -> a__U71(mark(X1),X2,X3) mark(U72(X1,X2,X3)) -> a__U72(mark(X1),X2,X3) mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(0) -> 0 a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNat(X) -> isNat(X) a__U21(X) -> U21(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2) -> U41(X1,X2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2,X3) -> U52(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) a__U61(X) -> U61(X) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(X1,X2,X3) -> U72(X1,X2,X3) a__x(X1,X2) -> x(X1,X2) Polynomial Interpretation: [a__U11](X1,X2) = X1 [tt] = 1 [a__U12](X) = X [a__isNat](X) = 1 [a__U21](X) = X [a__U31](X1,X2) = X1 [a__U32](X) = X [a__U41](X1,X2) = X1.X2 + X1 [mark](X) = X [a__U51](X1,X2,X3) = X1.X2 + X1.X3 + X1 [a__U52](X1,X2,X3) = X1 + X2 + X3 [s](X) = X + 1 [a__plus](X1,X2) = X1 + X2 [a__U61](X) = X [0] = 1 [a__U71](X1,X2,X3) = X1.X2.X3 + X1.X2 + X1.X3 + X1 + X3 [a__U72](X1,X2,X3) = X1.X3 + X2.X3 + X1 + X2 + X3 [a__x](X1,X2) = X1.X2 + X1 + X2 + 1 [plus](X1,X2) = X1 + X2 [x](X1,X2) = X1.X2 + X1 + X2 + 1 [U11](X1,X2) = X1 [U12](X) = X [isNat](X) = 1 [U21](X) = X [U31](X1,X2) = X1 [U32](X) = X [U41](X1,X2) = X1.X2 + X1 [U51](X1,X2,X3) = X1.X2 + X1.X3 + X1 [U52](X1,X2,X3) = X1 + X2 + X3 [U61](X) = X [U71](X1,X2,X3) = X1.X2.X3 + X1.X2 + X1.X3 + X1 + X3 [U72](X1,X2,X3) = X1.X3 + X2.X3 + X1 + X2 + X3 [nF_a__U41](X1,X2) = X1.X2 [nF_mark](X) = X [nF_a__U72](X1,X2,X3) = X1.X3 + X2 [nF_a__U71](X1,X2,X3) = X1.X2.X3 + X1.X2 + X1 + X3 [nF_a__U52](X1,X2,X3) = X1 + X2 + X3 TIME: 0.378634 -> -> Dependency pairs in cycle: nF_a__U41(tt,N) -> nF_mark(N) nF_mark(U41(X1,X2)) -> nF_a__U41(mark(X1),X2) nF_a__U52(tt,M,N) -> nF_mark(M) nF_mark(U52(X1,X2,X3)) -> nF_a__U52(mark(X1),X2,X3) nF_a__U72(tt,M,N) -> nF_mark(N) nF_a__U71(tt,M,N) -> nF_a__U72(a__isNat(N),M,N) nF_mark(U71(X1,X2,X3)) -> nF_a__U71(mark(X1),X2,X3) nF_a__U72(tt,M,N) -> nF_mark(M) nF_mark(U72(X1,X2,X3)) -> nF_a__U72(mark(X1),X2,X3) nF_mark(U11(X1,X2)) -> nF_mark(X1) nF_mark(U12(X)) -> nF_mark(X) nF_mark(U21(X)) -> nF_mark(X) nF_mark(U31(X1,X2)) -> nF_mark(X1) nF_mark(U32(X)) -> nF_mark(X) nF_mark(U41(X1,X2)) -> nF_mark(X1) nF_mark(U51(X1,X2,X3)) -> nF_mark(X1) nF_mark(U52(X1,X2,X3)) -> nF_mark(X1) nF_mark(plus(X1,X2)) -> nF_mark(X1) nF_mark(plus(X1,X2)) -> nF_mark(X2) nF_mark(U61(X)) -> nF_mark(X) nF_mark(U71(X1,X2,X3)) -> nF_mark(X1) nF_mark(U72(X1,X2,X3)) -> nF_mark(X1) nF_mark(x(X1,X2)) -> nF_mark(X1) nF_mark(x(X1,X2)) -> nF_mark(X2) UsableRules: a__U11(tt,V2) -> a__U12(a__isNat(V2)) a__U12(tt) -> tt a__U21(tt) -> tt a__U31(tt,V2) -> a__U32(a__isNat(V2)) a__U32(tt) -> tt a__U41(tt,N) -> mark(N) a__U51(tt,M,N) -> a__U52(a__isNat(N),M,N) a__U52(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U61(tt) -> 0 a__U71(tt,M,N) -> a__U72(a__isNat(N),M,N) a__U72(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNat(x(V1,V2)) -> a__U31(a__isNat(V1),V2) a__plus(N,0) -> a__U41(a__isNat(N),N) a__plus(N,s(M)) -> a__U51(a__isNat(M),M,N) a__x(N,0) -> a__U61(a__isNat(N)) a__x(N,s(M)) -> a__U71(a__isNat(M),M,N) mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2,X3)) -> a__U52(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X1,X2,X3)) -> a__U71(mark(X1),X2,X3) mark(U72(X1,X2,X3)) -> a__U72(mark(X1),X2,X3) mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(0) -> 0 a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNat(X) -> isNat(X) a__U21(X) -> U21(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2) -> U41(X1,X2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2,X3) -> U52(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) a__U61(X) -> U61(X) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(X1,X2,X3) -> U72(X1,X2,X3) a__x(X1,X2) -> x(X1,X2) Polynomial Interpretation: [a__U11](X1,X2) = X1 [tt] = 1 [a__U12](X) = X [a__isNat](X) = 1 [a__U21](X) = X [a__U31](X1,X2) = X1 [a__U32](X) = X [a__U41](X1,X2) = X1.X2 + X1 [mark](X) = X [a__U51](X1,X2,X3) = X1.X2 + X1.X3 + X1 [a__U52](X1,X2,X3) = X1 + X2 + X3 [s](X) = X + 1 [a__plus](X1,X2) = X1 + X2 [a__U61](X) = X.X + X [0] = 1 [a__U71](X1,X2,X3) = X1.X2.X3 + X1.X2 + X1.X3 + X1 + X3 [a__U72](X1,X2,X3) = X1.X3 + X2.X3 + X1 + X2 + X3 [a__x](X1,X2) = X1.X2 + X1 + X2 + 1 [plus](X1,X2) = X1 + X2 [x](X1,X2) = X1.X2 + X1 + X2 + 1 [U11](X1,X2) = X1 [U12](X) = X [isNat](X) = 1 [U21](X) = X [U31](X1,X2) = X1 [U32](X) = X [U41](X1,X2) = X1.X2 + X1 [U51](X1,X2,X3) = X1.X2 + X1.X3 + X1 [U52](X1,X2,X3) = X1 + X2 + X3 [U61](X) = X.X + X [U71](X1,X2,X3) = X1.X2.X3 + X1.X2 + X1.X3 + X1 + X3 [U72](X1,X2,X3) = X1.X3 + X2.X3 + X1 + X2 + X3 [nF_a__U41](X1,X2) = X1.X2 [nF_mark](X) = X [nF_a__U52](X1,X2,X3) = X2 [nF_a__U72](X1,X2,X3) = X1.X3 + X2 [nF_a__U71](X1,X2,X3) = X1.X2.X3 + X1.X2 + X1 + X3 TIME: 0.370426 -> -> Dependency pairs in cycle: nF_a__U41(tt,N) -> nF_mark(N) nF_mark(U41(X1,X2)) -> nF_a__U41(mark(X1),X2) nF_mark(x(X1,X2)) -> nF_mark(X1) nF_mark(U72(X1,X2,X3)) -> nF_mark(X1) nF_mark(U71(X1,X2,X3)) -> nF_mark(X1) nF_mark(U61(X)) -> nF_mark(X) nF_mark(plus(X1,X2)) -> nF_mark(X2) nF_mark(plus(X1,X2)) -> nF_mark(X1) nF_mark(U52(X1,X2,X3)) -> nF_mark(X1) nF_mark(U51(X1,X2,X3)) -> nF_mark(X1) nF_mark(U41(X1,X2)) -> nF_mark(X1) nF_mark(U32(X)) -> nF_mark(X) nF_mark(U31(X1,X2)) -> nF_mark(X1) nF_mark(U21(X)) -> nF_mark(X) nF_mark(U12(X)) -> nF_mark(X) nF_mark(U11(X1,X2)) -> nF_mark(X1) nF_a__U72(tt,M,N) -> nF_mark(M) nF_mark(U72(X1,X2,X3)) -> nF_a__U72(mark(X1),X2,X3) nF_a__U72(tt,M,N) -> nF_mark(N) nF_a__U71(tt,M,N) -> nF_a__U72(a__isNat(N),M,N) nF_mark(U71(X1,X2,X3)) -> nF_a__U71(mark(X1),X2,X3) nF_a__U52(tt,M,N) -> nF_mark(M) nF_mark(U52(X1,X2,X3)) -> nF_a__U52(mark(X1),X2,X3) UsableRules: a__U11(tt,V2) -> a__U12(a__isNat(V2)) a__U12(tt) -> tt a__U21(tt) -> tt a__U31(tt,V2) -> a__U32(a__isNat(V2)) a__U32(tt) -> tt a__U41(tt,N) -> mark(N) a__U51(tt,M,N) -> a__U52(a__isNat(N),M,N) a__U52(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U61(tt) -> 0 a__U71(tt,M,N) -> a__U72(a__isNat(N),M,N) a__U72(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNat(x(V1,V2)) -> a__U31(a__isNat(V1),V2) a__plus(N,0) -> a__U41(a__isNat(N),N) a__plus(N,s(M)) -> a__U51(a__isNat(M),M,N) a__x(N,0) -> a__U61(a__isNat(N)) a__x(N,s(M)) -> a__U71(a__isNat(M),M,N) mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2,X3)) -> a__U52(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X1,X2,X3)) -> a__U71(mark(X1),X2,X3) mark(U72(X1,X2,X3)) -> a__U72(mark(X1),X2,X3) mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(0) -> 0 a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNat(X) -> isNat(X) a__U21(X) -> U21(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2) -> U41(X1,X2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2,X3) -> U52(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) a__U61(X) -> U61(X) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(X1,X2,X3) -> U72(X1,X2,X3) a__x(X1,X2) -> x(X1,X2) Polynomial Interpretation: [a__U11](X1,X2) = X1 [tt] = 1 [a__U12](X) = X [a__isNat](X) = 1 [a__U21](X) = X [a__U31](X1,X2) = X1 [a__U32](X) = X [a__U41](X1,X2) = X1.X2 + X1 [mark](X) = X [a__U51](X1,X2,X3) = X1.X2 + X1.X3 + X1 + 1 [a__U52](X1,X2,X3) = X1 + X2 + X3 + 1 [s](X) = X + 1 [a__plus](X1,X2) = X1 + X2 + 1 [a__U61](X) = X [0] = 1 [a__U71](X1,X2,X3) = X1.X2.X3 + X1.X2 + X1.X3 + X1 + X3 + 1 [a__U72](X1,X2,X3) = X1.X3 + X2.X3 + X1 + X2 + X3 + 1 [a__x](X1,X2) = X1.X2 + X1 + X2 + 1 [plus](X1,X2) = X1 + X2 + 1 [x](X1,X2) = X1.X2 + X1 + X2 + 1 [U11](X1,X2) = X1 [U12](X) = X [isNat](X) = 1 [U21](X) = X [U31](X1,X2) = X1 [U32](X) = X [U41](X1,X2) = X1.X2 + X1 [U51](X1,X2,X3) = X1.X2 + X1.X3 + X1 + 1 [U52](X1,X2,X3) = X1 + X2 + X3 + 1 [U61](X) = X [U71](X1,X2,X3) = X1.X2.X3 + X1.X2 + X1.X3 + X1 + X3 + 1 [U72](X1,X2,X3) = X1.X3 + X2.X3 + X1 + X2 + X3 + 1 [nF_a__U41](X1,X2) = X1.X2 [nF_mark](X) = X [nF_a__U72](X1,X2,X3) = X1.X3 + X2 [nF_a__U71](X1,X2,X3) = X1.X2.X3 + X1.X2 + X1 + X3 [nF_a__U52](X1,X2,X3) = X2 TIME: 0.36389 -> -> Dependency pairs in cycle: nF_a__U41(tt,N) -> nF_mark(N) nF_mark(U41(X1,X2)) -> nF_a__U41(mark(X1),X2) nF_a__U72(tt,M,N) -> nF_mark(N) nF_a__U71(tt,M,N) -> nF_a__U72(a__isNat(N),M,N) nF_mark(U71(X1,X2,X3)) -> nF_a__U71(mark(X1),X2,X3) nF_a__U72(tt,M,N) -> nF_mark(M) nF_mark(U72(X1,X2,X3)) -> nF_a__U72(mark(X1),X2,X3) nF_mark(U11(X1,X2)) -> nF_mark(X1) nF_mark(U12(X)) -> nF_mark(X) nF_mark(U21(X)) -> nF_mark(X) nF_mark(U31(X1,X2)) -> nF_mark(X1) nF_mark(U32(X)) -> nF_mark(X) nF_mark(U41(X1,X2)) -> nF_mark(X1) nF_mark(U51(X1,X2,X3)) -> nF_mark(X1) nF_mark(U52(X1,X2,X3)) -> nF_mark(X1) nF_mark(plus(X1,X2)) -> nF_mark(X1) nF_mark(plus(X1,X2)) -> nF_mark(X2) nF_mark(U61(X)) -> nF_mark(X) nF_mark(U71(X1,X2,X3)) -> nF_mark(X1) nF_mark(U72(X1,X2,X3)) -> nF_mark(X1) nF_mark(x(X1,X2)) -> nF_mark(X1) UsableRules: a__U11(tt,V2) -> a__U12(a__isNat(V2)) a__U12(tt) -> tt a__U21(tt) -> tt a__U31(tt,V2) -> a__U32(a__isNat(V2)) a__U32(tt) -> tt a__U41(tt,N) -> mark(N) a__U51(tt,M,N) -> a__U52(a__isNat(N),M,N) a__U52(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U61(tt) -> 0 a__U71(tt,M,N) -> a__U72(a__isNat(N),M,N) a__U72(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNat(x(V1,V2)) -> a__U31(a__isNat(V1),V2) a__plus(N,0) -> a__U41(a__isNat(N),N) a__plus(N,s(M)) -> a__U51(a__isNat(M),M,N) a__x(N,0) -> a__U61(a__isNat(N)) a__x(N,s(M)) -> a__U71(a__isNat(M),M,N) mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2,X3)) -> a__U52(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X1,X2,X3)) -> a__U71(mark(X1),X2,X3) mark(U72(X1,X2,X3)) -> a__U72(mark(X1),X2,X3) mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(0) -> 0 a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNat(X) -> isNat(X) a__U21(X) -> U21(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2) -> U41(X1,X2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2,X3) -> U52(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) a__U61(X) -> U61(X) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(X1,X2,X3) -> U72(X1,X2,X3) a__x(X1,X2) -> x(X1,X2) Polynomial Interpretation: [a__U11](X1,X2) = X1 [tt] = 1 [a__U12](X) = X [a__isNat](X) = 1 [a__U21](X) = X [a__U31](X1,X2) = X1 [a__U32](X) = X [a__U41](X1,X2) = X1 + X2 [mark](X) = X [a__U51](X1,X2,X3) = X1 + X2 + X3 [a__U52](X1,X2,X3) = X1 + X2 + X3 [s](X) = X + 1 [a__plus](X1,X2) = X1 + X2 [a__U61](X) = X [0] = 1 [a__U71](X1,X2,X3) = X1.X2 + X1.X3 + X2.X3 + X1 + X3 [a__U72](X1,X2,X3) = X1.X2.X3 + X1.X3 + X1 + X2 + X3 [a__x](X1,X2) = X1.X2 + X1 + X2 + 1 [plus](X1,X2) = X1 + X2 [x](X1,X2) = X1.X2 + X1 + X2 + 1 [U11](X1,X2) = X1 [U12](X) = X [isNat](X) = 1 [U21](X) = X [U31](X1,X2) = X1 [U32](X) = X [U41](X1,X2) = X1 + X2 [U51](X1,X2,X3) = X1 + X2 + X3 [U52](X1,X2,X3) = X1 + X2 + X3 [U61](X) = X [U71](X1,X2,X3) = X1.X2 + X1.X3 + X2.X3 + X1 + X3 [U72](X1,X2,X3) = X1.X2.X3 + X1.X3 + X1 + X2 + X3 [nF_a__U41](X1,X2) = X2 [nF_mark](X) = X [nF_a__U72](X1,X2,X3) = X1.X3 + X2 [nF_a__U71](X1,X2,X3) = X1.X2 + X1 + X3 TIME: 0.36003799999999997 -> -> Dependency pairs in cycle: nF_a__U41(tt,N) -> nF_mark(N) nF_mark(U41(X1,X2)) -> nF_a__U41(mark(X1),X2) nF_mark(U72(X1,X2,X3)) -> nF_mark(X1) nF_mark(U71(X1,X2,X3)) -> nF_mark(X1) nF_mark(U61(X)) -> nF_mark(X) nF_mark(plus(X1,X2)) -> nF_mark(X2) nF_mark(plus(X1,X2)) -> nF_mark(X1) nF_mark(U52(X1,X2,X3)) -> nF_mark(X1) nF_mark(U51(X1,X2,X3)) -> nF_mark(X1) nF_mark(U41(X1,X2)) -> nF_mark(X1) nF_mark(U32(X)) -> nF_mark(X) nF_mark(U31(X1,X2)) -> nF_mark(X1) nF_mark(U21(X)) -> nF_mark(X) nF_mark(U12(X)) -> nF_mark(X) nF_mark(U11(X1,X2)) -> nF_mark(X1) nF_a__U72(tt,M,N) -> nF_mark(M) nF_mark(U72(X1,X2,X3)) -> nF_a__U72(mark(X1),X2,X3) nF_a__U72(tt,M,N) -> nF_mark(N) nF_a__U71(tt,M,N) -> nF_a__U72(a__isNat(N),M,N) nF_mark(U71(X1,X2,X3)) -> nF_a__U71(mark(X1),X2,X3) UsableRules: a__U11(tt,V2) -> a__U12(a__isNat(V2)) a__U12(tt) -> tt a__U21(tt) -> tt a__U31(tt,V2) -> a__U32(a__isNat(V2)) a__U32(tt) -> tt a__U41(tt,N) -> mark(N) a__U51(tt,M,N) -> a__U52(a__isNat(N),M,N) a__U52(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U61(tt) -> 0 a__U71(tt,M,N) -> a__U72(a__isNat(N),M,N) a__U72(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNat(x(V1,V2)) -> a__U31(a__isNat(V1),V2) a__plus(N,0) -> a__U41(a__isNat(N),N) a__plus(N,s(M)) -> a__U51(a__isNat(M),M,N) a__x(N,0) -> a__U61(a__isNat(N)) a__x(N,s(M)) -> a__U71(a__isNat(M),M,N) mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2,X3)) -> a__U52(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X1,X2,X3)) -> a__U71(mark(X1),X2,X3) mark(U72(X1,X2,X3)) -> a__U72(mark(X1),X2,X3) mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(0) -> 0 a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNat(X) -> isNat(X) a__U21(X) -> U21(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2) -> U41(X1,X2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2,X3) -> U52(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) a__U61(X) -> U61(X) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(X1,X2,X3) -> U72(X1,X2,X3) a__x(X1,X2) -> x(X1,X2) Polynomial Interpretation: [a__U11](X1,X2) = X1 [tt] = 0 [a__U12](X) = X [a__isNat](X) = 0 [a__U21](X) = X.X + X [a__U31](X1,X2) = X1 [a__U32](X) = X.X + X [a__U41](X1,X2) = X1 + X2 [mark](X) = X [a__U51](X1,X2,X3) = X1 + X2 + X3 + 1 [a__U52](X1,X2,X3) = X1 + X2 + X3 + 1 [s](X) = X + 1 [a__plus](X1,X2) = X1 + X2 [a__U61](X) = X.X + X + 1 [0] = 1 [a__U71](X1,X2,X3) = X2.X3 + X1 + X2 + X3 + 1 [a__U72](X1,X2,X3) = X2.X3 + X1 + X2 + X3 + 1 [a__x](X1,X2) = X1.X2 + X2 + 1 [plus](X1,X2) = X1 + X2 [x](X1,X2) = X1.X2 + X2 + 1 [U11](X1,X2) = X1 [U12](X) = X [isNat](X) = 0 [U21](X) = X.X + X [U31](X1,X2) = X1 [U32](X) = X.X + X [U41](X1,X2) = X1 + X2 [U51](X1,X2,X3) = X1 + X2 + X3 + 1 [U52](X1,X2,X3) = X1 + X2 + X3 + 1 [U61](X) = X.X + X + 1 [U71](X1,X2,X3) = X2.X3 + X1 + X2 + X3 + 1 [U72](X1,X2,X3) = X2.X3 + X1 + X2 + X3 + 1 [nF_a__U41](X1,X2) = X2 [nF_mark](X) = X [nF_a__U72](X1,X2,X3) = X2 + X3 [nF_a__U71](X1,X2,X3) = X2 + X3 TIME: 0.359748 -> -> Dependency pairs in cycle: nF_a__U41(tt,N) -> nF_mark(N) nF_mark(U41(X1,X2)) -> nF_a__U41(mark(X1),X2) nF_a__U72(tt,M,N) -> nF_mark(N) nF_mark(U72(X1,X2,X3)) -> nF_a__U72(mark(X1),X2,X3) nF_a__U72(tt,M,N) -> nF_mark(M) nF_mark(U11(X1,X2)) -> nF_mark(X1) nF_mark(U12(X)) -> nF_mark(X) nF_mark(U21(X)) -> nF_mark(X) nF_mark(U31(X1,X2)) -> nF_mark(X1) nF_mark(U32(X)) -> nF_mark(X) nF_mark(U41(X1,X2)) -> nF_mark(X1) nF_mark(U51(X1,X2,X3)) -> nF_mark(X1) nF_mark(U52(X1,X2,X3)) -> nF_mark(X1) nF_mark(plus(X1,X2)) -> nF_mark(X1) nF_mark(plus(X1,X2)) -> nF_mark(X2) nF_mark(U61(X)) -> nF_mark(X) nF_mark(U71(X1,X2,X3)) -> nF_mark(X1) nF_mark(U72(X1,X2,X3)) -> nF_mark(X1) UsableRules: a__U11(tt,V2) -> a__U12(a__isNat(V2)) a__U12(tt) -> tt a__U21(tt) -> tt a__U31(tt,V2) -> a__U32(a__isNat(V2)) a__U32(tt) -> tt a__U41(tt,N) -> mark(N) a__U51(tt,M,N) -> a__U52(a__isNat(N),M,N) a__U52(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U61(tt) -> 0 a__U71(tt,M,N) -> a__U72(a__isNat(N),M,N) a__U72(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNat(x(V1,V2)) -> a__U31(a__isNat(V1),V2) a__plus(N,0) -> a__U41(a__isNat(N),N) a__plus(N,s(M)) -> a__U51(a__isNat(M),M,N) a__x(N,0) -> a__U61(a__isNat(N)) a__x(N,s(M)) -> a__U71(a__isNat(M),M,N) mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2,X3)) -> a__U52(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X1,X2,X3)) -> a__U71(mark(X1),X2,X3) mark(U72(X1,X2,X3)) -> a__U72(mark(X1),X2,X3) mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(0) -> 0 a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNat(X) -> isNat(X) a__U21(X) -> U21(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2) -> U41(X1,X2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2,X3) -> U52(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) a__U61(X) -> U61(X) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(X1,X2,X3) -> U72(X1,X2,X3) a__x(X1,X2) -> x(X1,X2) Polynomial Interpretation: [a__U11](X1,X2) = X1 [tt] = 0 [a__U12](X) = X [a__isNat](X) = 0 [a__U21](X) = X.X + X [a__U31](X1,X2) = X1 [a__U32](X) = X.X + X [a__U41](X1,X2) = X1 + X2 [mark](X) = X [a__U51](X1,X2,X3) = X1 + X2 + X3 + 1 [a__U52](X1,X2,X3) = X1 + X2 + X3 + 1 [s](X) = X + 1 [a__plus](X1,X2) = X1 + X2 [a__U61](X) = X.X + X + 1 [0] = 1 [a__U71](X1,X2,X3) = X2.X3 + X1 + X2 + X3 + 1 [a__U72](X1,X2,X3) = X2.X3 + X1 + X2 + X3 + 1 [a__x](X1,X2) = X1.X2 + X2 + 1 [plus](X1,X2) = X1 + X2 [x](X1,X2) = X1.X2 + X2 + 1 [U11](X1,X2) = X1 [U12](X) = X [isNat](X) = 0 [U21](X) = X.X + X [U31](X1,X2) = X1 [U32](X) = X.X + X [U41](X1,X2) = X1 + X2 [U51](X1,X2,X3) = X1 + X2 + X3 + 1 [U52](X1,X2,X3) = X1 + X2 + X3 + 1 [U61](X) = X.X + X + 1 [U71](X1,X2,X3) = X2.X3 + X1 + X2 + X3 + 1 [U72](X1,X2,X3) = X2.X3 + X1 + X2 + X3 + 1 [nF_a__U41](X1,X2) = X2 [nF_mark](X) = X [nF_a__U72](X1,X2,X3) = X2 + X3 TIME: 0.35484099999999996 -> -> Dependency pairs in cycle: nF_a__U41(tt,N) -> nF_mark(N) nF_mark(U41(X1,X2)) -> nF_a__U41(mark(X1),X2) nF_mark(U71(X1,X2,X3)) -> nF_mark(X1) nF_mark(U61(X)) -> nF_mark(X) nF_mark(plus(X1,X2)) -> nF_mark(X2) nF_mark(plus(X1,X2)) -> nF_mark(X1) nF_mark(U52(X1,X2,X3)) -> nF_mark(X1) nF_mark(U51(X1,X2,X3)) -> nF_mark(X1) nF_mark(U41(X1,X2)) -> nF_mark(X1) nF_mark(U32(X)) -> nF_mark(X) nF_mark(U31(X1,X2)) -> nF_mark(X1) nF_mark(U21(X)) -> nF_mark(X) nF_mark(U12(X)) -> nF_mark(X) nF_mark(U11(X1,X2)) -> nF_mark(X1) nF_a__U72(tt,M,N) -> nF_mark(M) nF_mark(U72(X1,X2,X3)) -> nF_a__U72(mark(X1),X2,X3) nF_a__U72(tt,M,N) -> nF_mark(N) UsableRules: a__U11(tt,V2) -> a__U12(a__isNat(V2)) a__U12(tt) -> tt a__U21(tt) -> tt a__U31(tt,V2) -> a__U32(a__isNat(V2)) a__U32(tt) -> tt a__U41(tt,N) -> mark(N) a__U51(tt,M,N) -> a__U52(a__isNat(N),M,N) a__U52(tt,M,N) -> s(a__plus(mark(N),mark(M))) a__U61(tt) -> 0 a__U71(tt,M,N) -> a__U72(a__isNat(N),M,N) a__U72(tt,M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N)) a__isNat(0) -> tt a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__isNat(x(V1,V2)) -> a__U31(a__isNat(V1),V2) a__plus(N,0) -> a__U41(a__isNat(N),N) a__plus(N,s(M)) -> a__U51(a__isNat(M),M,N) a__x(N,0) -> a__U61(a__isNat(N)) a__x(N,s(M)) -> a__U71(a__isNat(M),M,N) mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U32(X)) -> a__U32(mark(X)) mark(U41(X1,X2)) -> a__U41(mark(X1),X2) mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3) mark(U52(X1,X2,X3)) -> a__U52(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(U61(X)) -> a__U61(mark(X)) mark(U71(X1,X2,X3)) -> a__U71(mark(X1),X2,X3) mark(U72(X1,X2,X3)) -> a__U72(mark(X1),X2,X3) mark(x(X1,X2)) -> a__x(mark(X1),mark(X2)) mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(0) -> 0 a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNat(X) -> isNat(X) a__U21(X) -> U21(X) a__U31(X1,X2) -> U31(X1,X2) a__U32(X) -> U32(X) a__U41(X1,X2) -> U41(X1,X2) a__U51(X1,X2,X3) -> U51(X1,X2,X3) a__U52(X1,X2,X3) -> U52(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) a__U61(X) -> U61(X) a__U71(X1,X2,X3) -> U71(X1,X2,X3) a__U72(X1,X2,X3) -> U72(X1,X2,X3) a__x(X1,X2) -> x(X1,X2) Polynomial Interpretation: [a__U11](X1,X2) = X1 [tt] = 1 [a__U12](X) = X [a__isNat](X) = 1 [a__U21](X) = X [a__U31](X1,X2) = X1 [a__U32](X) = X [a__U41](X1,X2) = X1 + X2 [mark](X) = X [a__U51](X1,X2,X3) = X1.X2 + X1.X3 + X1 [a__U52](X1,X2,X3) = X1 + X2 + X3 [s](X) = X + 1 [a__plus](X1,X2) = X1 + X2 [a__U61](X) = X [0] = 1 [a__U71](X1,X2,X3) = X1.X2.X3 + X1.X2 + X1.X3 + X1 + X3 [a__U72](X1,X2,X3) = X1.X3 + X2.X3 + X1 + X2 + X3 [a__x](X1,X2) = X1.X2 + X1 + X2 + 1 [plus](X1,X2) = X1 + X2 [x](X1,X2) = X1.X2 + X1 + X2 + 1 [U11](X1,X2) = X1 [U12](X) = X [isNat](X) = 1 [U21](X) = X [U31](X1,X2) = X1 [U32](X) = X [U41](X1,X2) = X1 + X2 [U51](X1,X2,X3) = X1.X2 + X1.X3 + X1 [U52](X1,X2,X3) = X1 + X2 + X3 [U61](X) = X [U71](X1,X2,X3) = X1.X2.X3 + X1.X2 + X1.X3 + X1 + X3 [U72](X1,X2,X3) = X1.X3 + X2.X3 + X1 + X2 + X3 [nF_a__U41](X1,X2) = X2 [nF_mark](X) = X [nF_a__U72](X1,X2,X3) = X1.X3 + X1 + X2 TIME: 0.351539 -> -> Dependency pairs in cycle: nF_a__U41(tt,N) -> nF_mark(N) nF_mark(U41(X1,X2)) -> nF_a__U41(mark(X1),X2) nF_a__U72(tt,M,N) -> nF_mark(M) nF_mark(U72(X1,X2,X3)) -> nF_a__U72(mark(X1),X2,X3) nF_mark(U11(X1,X2)) -> nF_mark(X1) nF_mark(U12(X)) -> nF_mark(X) nF_mark(U21(X)) -> nF_mark(X) nF_mark(U31(X1,X2)) -> nF_mark(X1) nF_mark(U32(X)) -> nF_mark(X) nF_mark(U41(X1,X2)) -> nF_mark(X1) nF_mark(U51(X1,X2,X3)) -> nF_mark(X1) nF_mark(U52(X1,X2,X3)) -> nF_mark(X1) nF_mark(plus(X1,X2)) -> nF_mark(X1) nF_mark(plus(X1,X2)) -> nF_mark(X2) nF_mark(U61(X)) -> nF_mark(X) nF_mark(U71(X1,X2,X3)) -> nF_mark(X1) Termination proved: Cycles verify subterm criterion. -> Proof of termination for MYNAT_nokinds_noand_GM_1_2: -> -> Dependency pairs in cycle: nF_a__U11(tt,V2) -> nF_a__isNat(V2) nF_a__isNat(plus(V1,V2)) -> nF_a__U11(a__isNat(V1),V2) nF_a__isNat(x(V1,V2)) -> nF_a__isNat(V1) nF_a__isNat(s(V1)) -> nF_a__isNat(V1) nF_a__isNat(plus(V1,V2)) -> nF_a__isNat(V1) nF_a__U31(tt,V2) -> nF_a__isNat(V2) nF_a__isNat(x(V1,V2)) -> nF_a__U31(a__isNat(V1),V2) Termination proved: Cycles verify subterm criterion. SETTINGS: Base ordering: Polynomial ordering Proof mode: SCCs in DG + base ordering Upper bound for coeffs: 1 Rationals below 1 for all non-replacing args: No Polynomial interpretation: Simple mixed Coeffs in polynomials: No rationals Delta: automatic Termination was proved succesfully.