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(activate(X1),activate(X2)) 26: activate(n__s(X)) -> s(activate(X)) 27: activate(n__x(X1,X2)) -> x(activate(X1),activate(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(activate(X1),activate(X2)) #17: #activate(n__plus(X1,X2)) -> #activate(X1) #18: #activate(n__plus(X1,X2)) -> #activate(X2) #19: #U51(tt(),M,N) -> #U52(isNat(activate(N)),activate(M),activate(N)) #20: #U51(tt(),M,N) -> #isNat(activate(N)) #21: #U51(tt(),M,N) -> #activate(N) #22: #U51(tt(),M,N) -> #activate(M) #23: #U51(tt(),M,N) -> #activate(N) #24: #U71(tt(),M,N) -> #U72(isNat(activate(N)),activate(M),activate(N)) #25: #U71(tt(),M,N) -> #isNat(activate(N)) #26: #U71(tt(),M,N) -> #activate(N) #27: #U71(tt(),M,N) -> #activate(M) #28: #U71(tt(),M,N) -> #activate(N) #29: #activate(n__x(X1,X2)) -> #x(activate(X1),activate(X2)) #30: #activate(n__x(X1,X2)) -> #activate(X1) #31: #activate(n__x(X1,X2)) -> #activate(X2) #32: #plus(N,s(M)) -> #U51(isNat(M),M,N) #33: #plus(N,s(M)) -> #isNat(M) #34: #x(N,s(M)) -> #U71(isNat(M),M,N) #35: #x(N,s(M)) -> #isNat(M) #36: #activate(n__s(X)) -> #s(activate(X)) #37: #activate(n__s(X)) -> #activate(X) #38: #plus(N,0()) -> #U41(isNat(N),N) #39: #plus(N,0()) -> #isNat(N) #40: #U11(tt(),V2) -> #U12(isNat(activate(V2))) #41: #U11(tt(),V2) -> #isNat(activate(V2)) #42: #U11(tt(),V2) -> #activate(V2) #43: #U52(tt(),M,N) -> #s(plus(activate(N),activate(M))) #44: #U52(tt(),M,N) -> #plus(activate(N),activate(M)) #45: #U52(tt(),M,N) -> #activate(N) #46: #U52(tt(),M,N) -> #activate(M) #47: #isNat(n__x(V1,V2)) -> #U31(isNat(activate(V1)),activate(V2)) #48: #isNat(n__x(V1,V2)) -> #isNat(activate(V1)) #49: #isNat(n__x(V1,V2)) -> #activate(V1) #50: #isNat(n__x(V1,V2)) -> #activate(V2) #51: #U31(tt(),V2) -> #U32(isNat(activate(V2))) #52: #U31(tt(),V2) -> #isNat(activate(V2)) #53: #U31(tt(),V2) -> #activate(V2) #54: #x(N,0()) -> #U61(isNat(N)) #55: #x(N,0()) -> #isNat(N) Number of SCCs: 1, DPs: 47 SCC { #1..5 #7..11 #14..35 #37..39 #41 #42 #44..50 #52 #53 #55 } POLO(Sum)... POLO(max)... succeeded. #0 w: 0 #U72 w: max(x2 + 5, x3 + 4) #U32 w: 0 U21 w: x1 + 1 U11 w: max(x1 + 3) s w: x1 #isNat w: x1 + 1 activate w: x1 U71 w: max(x2 + 4, x3 + 3) #plus w: max(x1 + 1, x2 + 3) #activate w: x1 + 1 U72 w: max(x2 + 4, x3 + 3) #U52 w: max(x2 + 3, x3 + 1) U12 w: 1 x w: max(x1 + 3, x2 + 4) n__s w: x1 #U12 w: 0 0 w: 2 #x w: max(x1 + 4, x2 + 5) #s w: 0 n__plus w: max(x1, x2 + 2) U32 w: 6 n__0 w: 2 isNat w: x1 + 2 n__x w: max(x1 + 3, x2 + 4) U52 w: max(x2 + 2, x3) plus w: max(x1, x2 + 2) U61 w: 4 #U51 w: max(x2 + 3, x3 + 1) #U11 w: max(x2 + 2) U31 w: max(x1 + 5, x2 + 6) #U41 w: max(x2 + 1) #U21 w: 0 tt w: 0 #U71 w: max(x2 + 5, x3 + 4) U51 w: max(x2 + 2, x3) U41 w: max(x2) #U31 w: max(x2 + 3) #U61 w: 0 USABLE RULES: { 6..11 16..28 } Removed DPs: #2 #5 #9..11 #18 #22 #25..28 #30 #31 #33 #35 #41 #42 #46..50 #52 #53 #55 Number of SCCs: 1, DPs: 22 SCC { #1 #3 #4 #7 #8 #14..17 #19..21 #23 #24 #29 #32 #34 #37..39 #44 #45 } POLO(Sum)... POLO(max)... QLPOS... succeeded. #0 s: [] p: 0 #U72 s: [3,2,1] p: 7 #U32 s: [] p: 0 U21 s: 1 U11 s: [] p: 1 s s: [1] p: 3 #isNat s: 1 activate s: 1 U71 s: [3,2,1] p: 7 #plus s: [1,2] p: 4 #activate s: 1 U72 s: [3,2,1] p: 7 #U52 s: [3,2] p: 4 U12 s: 1 x s: [1,2] p: 7 n__s s: [1] p: 3 #U12 s: 1 0 s: [] p: 1 #x s: [1,2] p: 7 #s s: [] p: 0 n__plus s: [2,1] p: 5 U32 s: 1 n__0 s: [] p: 1 isNat s: [] p: 1 n__x s: [1,2] p: 7 U52 s: [2,3,1] p: 5 plus s: [2,1] p: 5 U61 s: 1 #U51 s: [3,2] p: 4 #U11 s: 1 U31 s: [] p: 1 #U41 s: 2 #U21 s: 1 tt s: [] p: 1 #U71 s: [3,2,1] p: 7 U51 s: [2,3,1] p: 5 U41 s: 2 #U31 s: 2 #U61 s: 1 USABLE RULES: { 1..28 } Removed DPs: #3 #4 #7 #8 #14..17 #20 #21 #23 #34 #37..39 #45 Number of SCCs: 1, DPs: 3 SCC { #19 #32 #44 } POLO(Sum)... POLO(max)... QLPOS... succeeded. #0 s: [] p: 0 #U72 s: [3,2,1] p: 7 #U32 s: [] p: 0 U21 s: 1 U11 s: [] p: 1 s s: [1] p: 3 #isNat s: 1 activate s: 1 U71 s: [3,2,1] p: 7 #plus s: [1,2] p: 6 #activate s: 1 U72 s: [3,2,1] p: 7 #U52 s: [3,2] p: 6 U12 s: 1 x s: [1,2] p: 7 n__s s: [1] p: 3 #U12 s: 1 0 s: [] p: 1 #x s: [1,2] p: 7 #s s: [] p: 0 n__plus s: [2,1] p: 5 U32 s: 1 n__0 s: [] p: 1 isNat s: [] p: 1 n__x s: [1,2] p: 7 U52 s: [2,3,1] p: 5 plus s: [2,1] p: 5 U61 s: 1 #U51 s: [3,2] p: 6 #U11 s: 1 U31 s: [] p: 1 #U41 s: 2 #U21 s: 1 tt s: [] p: 1 #U71 s: [3,2,1] p: 7 U51 s: [2,3,1] p: 5 U41 s: 2 #U31 s: 2 #U61 s: 1 USABLE RULES: { 1..28 } Removed DPs: #32 Number of SCCs: 0, DPs: 0