YES Input TRS: 1: U11(tt(),V2) -> U12(isNat(activate(V2))) 2: U12(tt()) -> tt() 3: U21(tt()) -> tt() 4: U31(tt(),V2) -> U32(isNat(activate(V2))) 5: U32(tt()) -> tt() 6: U41(tt(),N) -> activate(N) 7: U51(tt(),M,N) -> U52(isNat(activate(N)),activate(M),activate(N)) 8: U52(tt(),M,N) -> s(plus(activate(N),activate(M))) 9: U61(tt()) -> 0() 10: U71(tt(),M,N) -> U72(isNat(activate(N)),activate(M),activate(N)) 11: U72(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N)) 12: isNat(n__0()) -> tt() 13: isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) 14: isNat(n__s(V1)) -> U21(isNat(activate(V1))) 15: isNat(n__x(V1,V2)) -> U31(isNat(activate(V1)),activate(V2)) 16: plus(N,0()) -> U41(isNat(N),N) 17: plus(N,s(M)) -> U51(isNat(M),M,N) 18: x(N,0()) -> U61(isNat(N)) 19: x(N,s(M)) -> U71(isNat(M),M,N) 20: 0() -> n__0() 21: plus(X1,X2) -> n__plus(X1,X2) 22: s(X) -> n__s(X) 23: x(X1,X2) -> n__x(X1,X2) 24: activate(n__0()) -> 0() 25: activate(n__plus(X1,X2)) -> plus(X1,X2) 26: activate(n__s(X)) -> s(X) 27: activate(n__x(X1,X2)) -> x(X1,X2) 28: activate(X) -> X Number of strict rules: 28 Direct POLO(bPol) ... failed. Uncurrying ... failed. Dependency Pairs: #1: #U41(tt(),N) -> #activate(N) #2: #isNat(n__plus(V1,V2)) -> #U11(isNat(activate(V1)),activate(V2)) #3: #isNat(n__plus(V1,V2)) -> #isNat(activate(V1)) #4: #isNat(n__plus(V1,V2)) -> #activate(V1) #5: #isNat(n__plus(V1,V2)) -> #activate(V2) #6: #U61(tt()) -> #0() #7: #U72(tt(),M,N) -> #plus(x(activate(N),activate(M)),activate(N)) #8: #U72(tt(),M,N) -> #x(activate(N),activate(M)) #9: #U72(tt(),M,N) -> #activate(N) #10: #U72(tt(),M,N) -> #activate(M) #11: #U72(tt(),M,N) -> #activate(N) #12: #activate(n__0()) -> #0() #13: #isNat(n__s(V1)) -> #U21(isNat(activate(V1))) #14: #isNat(n__s(V1)) -> #isNat(activate(V1)) #15: #isNat(n__s(V1)) -> #activate(V1) #16: #activate(n__plus(X1,X2)) -> #plus(X1,X2) #17: #U51(tt(),M,N) -> #U52(isNat(activate(N)),activate(M),activate(N)) #18: #U51(tt(),M,N) -> #isNat(activate(N)) #19: #U51(tt(),M,N) -> #activate(N) #20: #U51(tt(),M,N) -> #activate(M) #21: #U51(tt(),M,N) -> #activate(N) #22: #U71(tt(),M,N) -> #U72(isNat(activate(N)),activate(M),activate(N)) #23: #U71(tt(),M,N) -> #isNat(activate(N)) #24: #U71(tt(),M,N) -> #activate(N) #25: #U71(tt(),M,N) -> #activate(M) #26: #U71(tt(),M,N) -> #activate(N) #27: #activate(n__x(X1,X2)) -> #x(X1,X2) #28: #plus(N,s(M)) -> #U51(isNat(M),M,N) #29: #plus(N,s(M)) -> #isNat(M) #30: #x(N,s(M)) -> #U71(isNat(M),M,N) #31: #x(N,s(M)) -> #isNat(M) #32: #activate(n__s(X)) -> #s(X) #33: #plus(N,0()) -> #U41(isNat(N),N) #34: #plus(N,0()) -> #isNat(N) #35: #U11(tt(),V2) -> #U12(isNat(activate(V2))) #36: #U11(tt(),V2) -> #isNat(activate(V2)) #37: #U11(tt(),V2) -> #activate(V2) #38: #U52(tt(),M,N) -> #s(plus(activate(N),activate(M))) #39: #U52(tt(),M,N) -> #plus(activate(N),activate(M)) #40: #U52(tt(),M,N) -> #activate(N) #41: #U52(tt(),M,N) -> #activate(M) #42: #isNat(n__x(V1,V2)) -> #U31(isNat(activate(V1)),activate(V2)) #43: #isNat(n__x(V1,V2)) -> #isNat(activate(V1)) #44: #isNat(n__x(V1,V2)) -> #activate(V1) #45: #isNat(n__x(V1,V2)) -> #activate(V2) #46: #U31(tt(),V2) -> #U32(isNat(activate(V2))) #47: #U31(tt(),V2) -> #isNat(activate(V2)) #48: #U31(tt(),V2) -> #activate(V2) #49: #x(N,0()) -> #U61(isNat(N)) #50: #x(N,0()) -> #isNat(N) Number of SCCs: 1, DPs: 42 SCC { #1..5 #7..11 #14..31 #33 #34 #36 #37 #39..45 #47 #48 #50 } POLO(Sum)... POLO(max)... succeeded. #0 w: 0 #U72 w: max(x2 + 18590, x3) #U32 w: 0 U21 w: 0 U11 w: 0 s w: x1 #isNat w: x1 activate w: x1 U71 w: max(x1 + 1, x2 + 18590, x3) #plus w: max(x1, x2) #activate w: x1 U72 w: max(x2 + 18590, x3) #U52 w: max(x2, x3) U12 w: 0 x w: max(x1, x2 + 18590) n__s w: x1 #U12 w: 0 0 w: 17776 #x w: max(x1, x2 + 18590) #s w: 0 n__plus w: max(x1, x2) U32 w: 0 n__0 w: 17776 isNat w: 1 n__x w: max(x1, x2 + 18590) U52 w: max(x2, x3) plus w: max(x1, x2) U61 w: 17776 #U51 w: max(x2, x3) #U11 w: max(x2) U31 w: 0 #U41 w: max(x1 + 17775, x2) #U21 w: 0 tt w: 0 #U71 w: max(x1 + 12457, x2 + 18590, x3) U51 w: max(x2, x3) U41 w: max(x2) #U31 w: max(x1 + 1, x2 + 5023) #U61 w: 0 USABLE RULES: { 1..28 } Removed DPs: #10 #25 #31 #42 #45 #47 #48 Number of SCCs: 1, DPs: 35 SCC { #1..5 #7..9 #11 #14..24 #26..30 #33 #34 #36 #37 #39..41 #43 #44 #50 } POLO(Sum)... POLO(max)... succeeded. #0 w: 0 #U72 w: max(x2 + 9, x3 + 3) #U32 w: 0 U21 w: 0 U11 w: 0 s w: x1 #isNat w: x1 + 2 activate w: x1 U71 w: max(x1 + 4, x2 + 7, x3 + 1) #plus w: max(x1 + 2, x2 + 2) #activate w: x1 + 2 U72 w: max(x1 + 1, x2 + 7, x3 + 1) #U52 w: max(x2 + 2, x3 + 2) U12 w: 0 x w: max(x1 + 1, x2 + 7) n__s w: x1 #U12 w: 0 0 w: 3 #x w: max(x1 + 3, x2 + 9) #s w: 0 n__plus w: max(x1, x2) U32 w: 0 n__0 w: 3 isNat w: 3 n__x w: max(x1 + 1, x2 + 7) U52 w: max(x2, x3) plus w: max(x1, x2) U61 w: 7 #U51 w: max(x2 + 2, x3 + 2) #U11 w: max(x2 + 2) U31 w: 0 #U41 w: max(x1 + 1, x2 + 2) #U21 w: 0 tt w: 0 #U71 w: max(x1 + 1, x2 + 9, x3 + 3) U51 w: max(x2, x3) U41 w: max(x2) #U31 w: max(x1 + 1, x2 + 5023) #U61 w: 0 USABLE RULES: { 1..28 } Removed DPs: #9 #11 #23 #24 #26 #43 #44 #50 Number of SCCs: 1, DPs: 27 SCC { #1..5 #7 #8 #14..22 #27..30 #33 #34 #36 #37 #39..41 } POLO(Sum)... POLO(max)... succeeded. #0 w: 0 #U72 w: max(x1 + 3580, x3 + 3583) #U32 w: 0 U21 w: 0 U11 w: 0 s w: x1 #isNat w: x1 activate w: x1 U71 w: max(x3 + 3583) #plus w: max(x1, x2 + 2) #activate w: x1 U72 w: max(x3 + 3583) #U52 w: max(x1, x2 + 2, x3) U12 w: 0 x w: max(x1 + 3583) n__s w: x1 #U12 w: 0 0 w: 3471 #x w: max(x1 + 3583) #s w: 0 n__plus w: max(x1, x2 + 1725) U32 w: 0 n__0 w: 3471 isNat w: 2 n__x w: max(x1 + 3583) U52 w: max(x1 + 153, x2 + 1725, x3) plus w: max(x1, x2 + 1725) U61 w: 3471 #U51 w: max(x2 + 2, x3) #U11 w: max(x1 + 1722, x2 + 713) U31 w: max(x1) #U41 w: max(x1 + 3471, x2) #U21 w: 0 tt w: 0 #U71 w: max(x1 + 3581, x3 + 3583) U51 w: max(x2 + 1725, x3) U41 w: max(x2) #U31 w: 0 #U61 w: 0 USABLE RULES: { 1..28 } Removed DPs: #2 #5 #20 #29 #36 #37 #41 Number of SCCs: 1, DPs: 20 SCC { #1 #3 #4 #7 #8 #14..19 #21 #22 #27 #28 #30 #33 #34 #39 #40 } POLO(Sum)... POLO(max)... QLPOS... succeeded. #0 s: [] p: 0 #U72 s: [3,2] p: 7 #U32 s: [] p: 0 U21 s: [] p: 3 U11 s: [] p: 3 s s: [1] p: 4 #isNat s: 1 activate s: 1 U71 s: [3,2,1] p: 7 #plus s: 1 #activate s: 1 U72 s: [3,2,1] p: 7 #U52 s: 3 U12 s: 1 x s: [1,2] p: 7 n__s s: [1] p: 4 #U12 s: [] p: 0 0 s: [] p: 2 #x s: [1,2] p: 7 #s s: [] p: 0 n__plus s: [2,1] p: 6 U32 s: 1 n__0 s: [] p: 2 isNat s: [] p: 3 n__x s: [1,2] p: 7 U52 s: [2,3,1] p: 6 plus s: [2,1] p: 6 U61 s: 1 #U51 s: 3 #U11 s: [] p: 0 U31 s: [] p: 3 #U41 s: 2 #U21 s: [] p: 0 tt s: [] p: 3 #U71 s: [3,2,1] p: 7 U51 s: [2,3,1] p: 6 U41 s: [1,2] p: 0 #U31 s: 1 #U61 s: [] p: 0 USABLE RULES: { 1..28 } Removed DPs: #3 #4 #14..16 #22 #30 Number of SCCs: 1, DPs: 3 SCC { #17 #28 #39 } POLO(Sum)... POLO(max)... QLPOS... succeeded. #0 s: [] p: 0 #U72 s: [3,2] p: 7 #U32 s: [] p: 0 U21 s: [] p: 2 U11 s: [] p: 2 s s: [1] p: 3 #isNat s: 1 activate s: 1 U71 s: [3,2,1] p: 7 #plus s: [2,1] p: 4 #activate s: 1 U72 s: [3,2,1] p: 7 #U52 s: [2,3] p: 4 U12 s: 1 x s: [1,2] p: 7 n__s s: [1] p: 3 #U12 s: [] p: 0 0 s: [] p: 1 #x s: [1,2] p: 7 #s s: [] p: 0 n__plus s: [2,1] p: 6 U32 s: 1 n__0 s: [] p: 1 isNat s: [] p: 2 n__x s: [1,2] p: 7 U52 s: [2,3,1] p: 6 plus s: [2,1] p: 6 U61 s: 1 #U51 s: [2,3] p: 4 #U11 s: [] p: 0 U31 s: [] p: 2 #U41 s: 2 #U21 s: [] p: 0 tt s: [] p: 2 #U71 s: [3,2,1] p: 7 U51 s: [2,3,1] p: 6 U41 s: [1,2] p: 0 #U31 s: 1 #U61 s: [] p: 0 USABLE RULES: { 1..28 } Removed DPs: #28 Number of SCCs: 0, DPs: 0