YES Input TRS: 1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2)) 2: U12(tt(),V2) -> U13(isNat(activate(V2))) 3: U13(tt()) -> tt() 4: U21(tt(),V1) -> U22(isNat(activate(V1))) 5: U22(tt()) -> tt() 6: U31(tt(),V1,V2) -> U32(isNat(activate(V1)),activate(V2)) 7: U32(tt(),V2) -> U33(isNat(activate(V2))) 8: U33(tt()) -> tt() 9: U41(tt(),N) -> activate(N) 10: U51(tt(),M,N) -> s(plus(activate(N),activate(M))) 11: U61(tt()) -> 0() 12: U71(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N)) 13: and(tt(),X) -> activate(X) 14: isNat(n__0()) -> tt() 15: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) 16: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1)) 17: isNat(n__x(V1,V2)) -> U31(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) 18: isNatKind(n__0()) -> tt() 19: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2))) 20: isNatKind(n__s(V1)) -> isNatKind(activate(V1)) 21: isNatKind(n__x(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2))) 22: plus(N,0()) -> U41(and(isNat(N),n__isNatKind(N)),N) 23: plus(N,s(M)) -> U51(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) 24: x(N,0()) -> U61(and(isNat(N),n__isNatKind(N))) 25: x(N,s(M)) -> U71(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) 26: 0() -> n__0() 27: plus(X1,X2) -> n__plus(X1,X2) 28: isNatKind(X) -> n__isNatKind(X) 29: s(X) -> n__s(X) 30: x(X1,X2) -> n__x(X1,X2) 31: and(X1,X2) -> n__and(X1,X2) 32: activate(n__0()) -> 0() 33: activate(n__plus(X1,X2)) -> plus(X1,X2) 34: activate(n__isNatKind(X)) -> isNatKind(X) 35: activate(n__s(X)) -> s(X) 36: activate(n__x(X1,X2)) -> x(X1,X2) 37: activate(n__and(X1,X2)) -> and(X1,X2) 38: activate(X) -> X Number of strict rules: 38 Direct POLO(bPol) ... failed. Uncurrying ... failed. Dependency Pairs: #1: #U12(tt(),V2) -> #U13(isNat(activate(V2))) #2: #U12(tt(),V2) -> #isNat(activate(V2)) #3: #U12(tt(),V2) -> #activate(V2) #4: #activate(n__s(X)) -> #s(X) #5: #activate(n__and(X1,X2)) -> #and(X1,X2) #6: #U31(tt(),V1,V2) -> #U32(isNat(activate(V1)),activate(V2)) #7: #U31(tt(),V1,V2) -> #isNat(activate(V1)) #8: #U31(tt(),V1,V2) -> #activate(V1) #9: #U31(tt(),V1,V2) -> #activate(V2) #10: #and(tt(),X) -> #activate(X) #11: #U41(tt(),N) -> #activate(N) #12: #U61(tt()) -> #0() #13: #x(N,0()) -> #U61(and(isNat(N),n__isNatKind(N))) #14: #x(N,0()) -> #and(isNat(N),n__isNatKind(N)) #15: #x(N,0()) -> #isNat(N) #16: #plus(N,s(M)) -> #U51(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) #17: #plus(N,s(M)) -> #and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))) #18: #plus(N,s(M)) -> #and(isNat(M),n__isNatKind(M)) #19: #plus(N,s(M)) -> #isNat(M) #20: #plus(N,s(M)) -> #isNat(N) #21: #U71(tt(),M,N) -> #plus(x(activate(N),activate(M)),activate(N)) #22: #U71(tt(),M,N) -> #x(activate(N),activate(M)) #23: #U71(tt(),M,N) -> #activate(N) #24: #U71(tt(),M,N) -> #activate(M) #25: #U71(tt(),M,N) -> #activate(N) #26: #x(N,s(M)) -> #U71(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) #27: #x(N,s(M)) -> #and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))) #28: #x(N,s(M)) -> #and(isNat(M),n__isNatKind(M)) #29: #x(N,s(M)) -> #isNat(M) #30: #x(N,s(M)) -> #isNat(N) #31: #isNatKind(n__s(V1)) -> #isNatKind(activate(V1)) #32: #isNatKind(n__s(V1)) -> #activate(V1) #33: #U32(tt(),V2) -> #U33(isNat(activate(V2))) #34: #U32(tt(),V2) -> #isNat(activate(V2)) #35: #U32(tt(),V2) -> #activate(V2) #36: #U51(tt(),M,N) -> #s(plus(activate(N),activate(M))) #37: #U51(tt(),M,N) -> #plus(activate(N),activate(M)) #38: #U51(tt(),M,N) -> #activate(N) #39: #U51(tt(),M,N) -> #activate(M) #40: #activate(n__plus(X1,X2)) -> #plus(X1,X2) #41: #plus(N,0()) -> #U41(and(isNat(N),n__isNatKind(N)),N) #42: #plus(N,0()) -> #and(isNat(N),n__isNatKind(N)) #43: #plus(N,0()) -> #isNat(N) #44: #activate(n__isNatKind(X)) -> #isNatKind(X) #45: #isNat(n__x(V1,V2)) -> #U31(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) #46: #isNat(n__x(V1,V2)) -> #and(isNatKind(activate(V1)),n__isNatKind(activate(V2))) #47: #isNat(n__x(V1,V2)) -> #isNatKind(activate(V1)) #48: #isNat(n__x(V1,V2)) -> #activate(V1) #49: #isNat(n__x(V1,V2)) -> #activate(V2) #50: #isNat(n__x(V1,V2)) -> #activate(V1) #51: #isNat(n__x(V1,V2)) -> #activate(V2) #52: #activate(n__0()) -> #0() #53: #isNatKind(n__plus(V1,V2)) -> #and(isNatKind(activate(V1)),n__isNatKind(activate(V2))) #54: #isNatKind(n__plus(V1,V2)) -> #isNatKind(activate(V1)) #55: #isNatKind(n__plus(V1,V2)) -> #activate(V1) #56: #isNatKind(n__plus(V1,V2)) -> #activate(V2) #57: #activate(n__x(X1,X2)) -> #x(X1,X2) #58: #isNatKind(n__x(V1,V2)) -> #and(isNatKind(activate(V1)),n__isNatKind(activate(V2))) #59: #isNatKind(n__x(V1,V2)) -> #isNatKind(activate(V1)) #60: #isNatKind(n__x(V1,V2)) -> #activate(V1) #61: #isNatKind(n__x(V1,V2)) -> #activate(V2) #62: #isNat(n__s(V1)) -> #U21(isNatKind(activate(V1)),activate(V1)) #63: #isNat(n__s(V1)) -> #isNatKind(activate(V1)) #64: #isNat(n__s(V1)) -> #activate(V1) #65: #isNat(n__s(V1)) -> #activate(V1) #66: #U11(tt(),V1,V2) -> #U12(isNat(activate(V1)),activate(V2)) #67: #U11(tt(),V1,V2) -> #isNat(activate(V1)) #68: #U11(tt(),V1,V2) -> #activate(V1) #69: #U11(tt(),V1,V2) -> #activate(V2) #70: #isNat(n__plus(V1,V2)) -> #U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) #71: #isNat(n__plus(V1,V2)) -> #and(isNatKind(activate(V1)),n__isNatKind(activate(V2))) #72: #isNat(n__plus(V1,V2)) -> #isNatKind(activate(V1)) #73: #isNat(n__plus(V1,V2)) -> #activate(V1) #74: #isNat(n__plus(V1,V2)) -> #activate(V2) #75: #isNat(n__plus(V1,V2)) -> #activate(V1) #76: #isNat(n__plus(V1,V2)) -> #activate(V2) #77: #U21(tt(),V1) -> #U22(isNat(activate(V1))) #78: #U21(tt(),V1) -> #isNat(activate(V1)) #79: #U21(tt(),V1) -> #activate(V1) Number of SCCs: 1, DPs: 71 SCC { #2 #3 #5..11 #14..32 #34 #35 #37..51 #53..76 #78 #79 } POLO(Sum)... POLO(max)... succeeded. #0 w: 0 #U32 w: max(x2) isNatKind w: x1 U21 w: max(x1) U11 w: max(x1, x2) s w: x1 #isNat w: x1 activate w: x1 U71 w: max(x2, x3 + 1) n__isNatKind w: x1 and w: max(x1, x2) #plus w: max(x1, x2) #activate w: x1 #U13 w: 0 U12 w: max(x1) #U33 w: 0 x w: max(x1 + 1, x2) n__s w: x1 #U12 w: max(x2) 0 w: 12331 #x w: max(x1 + 1, x2) #s w: 0 n__plus w: max(x1, x2) U32 w: max(x2) U33 w: x1 n__0 w: 12331 isNat w: x1 n__x w: max(x1 + 1, x2) plus w: max(x1, x2) U61 w: 12331 #U51 w: max(x2, x3) #U11 w: max(x2, x3) U31 w: max(x3) #U41 w: max(x1, x2) #U21 w: max(x1, x2) #U22 w: 0 tt w: 1 n__and w: max(x1, x2) #U71 w: max(x2, x3 + 1) U13 w: 1 U22 w: 1 U51 w: max(x2, x3) #isNatKind w: x1 U41 w: max(x2) #U31 w: max(x2, x3) #and w: max(x2) #U61 w: 0 USABLE RULES: { 1..38 } Removed DPs: #14 #15 #23 #25 #27 #30 #47 #48 #50 #59 #60 Number of SCCs: 1, DPs: 60 SCC { #2 #3 #5..11 #16..22 #24 #26 #28 #29 #31 #32 #34 #35 #37..46 #49 #51 #53..58 #61..76 #78 #79 } POLO(Sum)... POLO(max)... succeeded. #0 w: 0 #U32 w: max(x2 + 2) isNatKind w: x1 U21 w: max(x1) U11 w: max(x1, x2) s w: x1 #isNat w: x1 + 2 activate w: x1 U71 w: max(x2, x3 + 1) n__isNatKind w: x1 and w: max(x1, x2) #plus w: max(x1 + 2, x2 + 2) #activate w: x1 + 2 #U13 w: 0 U12 w: max(x1) #U33 w: 0 x w: max(x1 + 1, x2) n__s w: x1 #U12 w: max(x2 + 2) 0 w: 15944 #x w: max(x1 + 3, x2 + 2) #s w: 0 n__plus w: max(x1, x2) U32 w: max(x2) U33 w: x1 n__0 w: 15944 isNat w: x1 n__x w: max(x1 + 1, x2) plus w: max(x1, x2) U61 w: 15944 #U51 w: max(x1 + 2, x2 + 2, x3 + 2) #U11 w: max(x1 + 2, x2 + 2, x3 + 2) U31 w: max(x3) #U41 w: max(x1 + 2, x2 + 2) #U21 w: max(x1 + 1, x2 + 2) #U22 w: 0 tt w: 2 n__and w: max(x1, x2) #U71 w: max(x2 + 2, x3 + 3) U13 w: 2 U22 w: 2 U51 w: max(x2, x3) #isNatKind w: x1 + 2 U41 w: max(x2) #U31 w: max(x1 + 1, x2 + 3, x3 + 2) #and w: max(x2 + 2) #U61 w: 0 USABLE RULES: { 1..38 } Removed DPs: #7 #8 Number of SCCs: 1, DPs: 58 SCC { #2 #3 #5 #6 #9..11 #16..22 #24 #26 #28 #29 #31 #32 #34 #35 #37..46 #49 #51 #53..58 #61..76 #78 #79 } POLO(Sum)... POLO(max)... succeeded. #0 w: 0 #U32 w: max(x2 + 3) isNatKind w: x1 U21 w: max(x1 + 1, x2 + 2) U11 w: max(x1 + 5, x3 + 8) s w: x1 #isNat w: x1 + 2 activate w: x1 U71 w: max(x2 + 5, x3 + 4) n__isNatKind w: x1 and w: max(x2) #plus w: max(x1 + 2, x2 + 5) #activate w: x1 + 2 #U13 w: 0 U12 w: max(x1 + 5, x2 + 8) #U33 w: 0 x w: max(x1 + 4, x2 + 5) n__s w: x1 #U12 w: max(x2 + 4) 0 w: 8856 #x w: max(x1 + 6, x2 + 7) #s w: 0 n__plus w: max(x1, x2 + 4) U32 w: 0 U33 w: 0 n__0 w: 8856 isNat w: x1 + 4 n__x w: max(x1 + 4, x2 + 5) plus w: max(x1, x2 + 4) U61 w: 8861 #U51 w: max(x2 + 5, x3 + 2) #U11 w: max(x1 + 3, x2 + 2, x3 + 5) U31 w: max(x1 + 10) #U41 w: max(x1 + 1, x2 + 2) #U21 w: max(x1 + 1, x2 + 2) #U22 w: 0 tt w: 2 n__and w: max(x2) #U71 w: max(x2 + 7, x3 + 6) U13 w: 8 U22 w: 4 U51 w: max(x2 + 4, x3) #isNatKind w: x1 + 2 U41 w: max(x2) #U31 w: max(x3 + 5) #and w: max(x2 + 2) #U61 w: 0 USABLE RULES: { 2 3 5 9..14 18..38 } Removed DPs: #2 #3 #6 #9 #18 #19 #24 #28 #29 #34 #35 #39 #45 #46 #49 #51 #53 #56 #58 #61 #66 #69 #71 #74 #76 Number of SCCs: 1, DPs: 33 SCC { #5 #10 #11 #16 #17 #20..22 #26 #31 #32 #37 #38 #40..44 #54 #55 #57 #62..65 #67 #68 #70 #72 #73 #75 #78 #79 } POLO(Sum)... POLO(max)... QLPOS... succeeded. #0 s: [] p: 0 #U32 s: 2 isNatKind s: [1] p: 1 U21 s: [1,2] p: 4 U11 s: [3] p: 3 s s: [1] p: 3 #isNat s: 1 activate s: 1 U71 s: [2,3,1] p: 7 n__isNatKind s: [1] p: 1 and s: 2 #plus s: [1] p: 3 #activate s: 1 #U13 s: [] p: 0 U12 s: [2] p: 3 #U33 s: 1 x s: [2,1] p: 7 n__s s: [1] p: 3 #U12 s: [2,1] p: 0 0 s: [] p: 0 #x s: [2,1] p: 7 #s s: [] p: 0 n__plus s: [2,1] p: 4 U32 s: 2 U33 s: [1] p: 5 n__0 s: [] p: 0 isNat s: [] p: 3 n__x s: [2,1] p: 7 plus s: [2,1] p: 4 U61 s: [1] p: 0 #U51 s: [3] p: 3 #U11 s: [2] p: 2 U31 s: [1,2,3] p: 3 #U41 s: [1,2] p: 2 #U21 s: [2,1] p: 2 #U22 s: [] p: 0 tt s: [] p: 1 n__and s: 2 #U71 s: [2,3,1] p: 7 U13 s: [] p: 3 U22 s: [] p: 1 U51 s: [2,3,1] p: 4 #isNatKind s: [1] p: 0 U41 s: 2 #U31 s: [3,2,1] p: 0 #and s: 2 #U61 s: [] p: 0 USABLE RULES: { 9..13 18..38 } Removed DPs: #11 #17 #20..22 #26 #31 #32 #38 #40..44 #54 #55 #62..65 #67 #68 #70 #72 #73 #75 #78 #79 Number of SCCs: 2, DPs: 4 SCC { #5 #10 } POLO(Sum)... succeeded. #0 w: 0 #U32 w: 2 isNatKind w: x1 + 7 U21 w: x1 U11 w: x1 + x2 s w: 2457 #isNat w: 3 activate w: x1 + 5 U71 w: x2 + 4 n__isNatKind w: 1 and w: 10 #plus w: 3 #activate w: x1 + 3 #U13 w: 0 U12 w: x1 + 2 #U33 w: 0 x w: 1 n__s w: 2451 #U12 w: 2 0 w: 3 #x w: 3 #s w: 0 n__plus w: 1 U32 w: x1 + x2 + 3 U33 w: x1 + 6 n__0 w: 1 isNat w: 1 n__x w: x1 + 2 plus w: x2 U61 w: 2 #U51 w: 0 #U11 w: 3 U31 w: x1 + x2 + x3 + 6 #U41 w: 3 #U21 w: 3 #U22 w: 0 tt w: 2 n__and w: x2 + 2 #U71 w: 3 U13 w: x1 + 4 U22 w: 3 U51 w: x1 + 2454 #isNatKind w: 3 U41 w: x2 + 4 #U31 w: 1 #and w: x2 + 4 #U61 w: 0 USABLE RULES: { } Removed DPs: #5 #10 Number of SCCs: 1, DPs: 2 SCC { #16 #37 } POLO(Sum)... POLO(max)... QLPOS... succeeded. #0 s: [] p: 0 #U32 s: 2 isNatKind s: [1] p: 1 U21 s: [1,2] p: 4 U11 s: [3] p: 3 s s: [1] p: 3 #isNat s: 1 activate s: 1 U71 s: [2,3,1] p: 7 n__isNatKind s: [1] p: 1 and s: 2 #plus s: [2] p: 1 #activate s: 1 #U13 s: [] p: 0 U12 s: [2] p: 3 #U33 s: 1 x s: [2,1] p: 7 n__s s: [1] p: 3 #U12 s: [2,1] p: 0 0 s: [] p: 0 #x s: [2,1] p: 7 #s s: [] p: 0 n__plus s: [2,1] p: 4 U32 s: 2 U33 s: [1] p: 5 n__0 s: [] p: 0 isNat s: [] p: 3 n__x s: [2,1] p: 7 plus s: [2,1] p: 4 U61 s: [1] p: 0 #U51 s: [2] p: 2 #U11 s: [2] p: 2 U31 s: [1,2,3] p: 3 #U41 s: [1,2] p: 2 #U21 s: [2,1] p: 2 #U22 s: [] p: 0 tt s: [] p: 1 n__and s: 2 #U71 s: [2,3,1] p: 7 U13 s: [] p: 3 U22 s: [] p: 1 U51 s: [2,3,1] p: 4 #isNatKind s: [1] p: 0 U41 s: 2 #U31 s: [3,2,1] p: 0 #and s: 2 #U61 s: [] p: 0 USABLE RULES: { 9..13 18..38 } Removed DPs: #16 #37 Number of SCCs: 0, DPs: 0