Input TRS: 1: and(tt(),T) -> T 2: isNatIList(IL) -> isNatList(activate(IL)) 3: isNat(n__0()) -> tt() 4: isNat(n__s(N)) -> isNat(activate(N)) 5: isNat(n__length(L)) -> isNatList(activate(L)) 6: isNatIList(n__zeros()) -> tt() 7: isNatIList(n__cons(N,IL)) -> and(isNat(activate(N)),isNatIList(activate(IL))) 8: isNatList(n__nil()) -> tt() 9: isNatList(n__cons(N,L)) -> and(isNat(activate(N)),isNatList(activate(L))) 10: isNatList(n__take(N,IL)) -> and(isNat(activate(N)),isNatIList(activate(IL))) 11: zeros() -> cons(|0|(),n__zeros()) 12: take(|0|(),IL) -> uTake1(isNatIList(IL)) 13: uTake1(tt()) -> nil() 14: take(s(M),cons(N,IL)) -> uTake2(and(isNat(M),and(isNat(N),isNatIList(activate(IL)))),M,N,activate(IL)) 15: uTake2(tt(),M,N,IL) -> cons(activate(N),n__take(activate(M),activate(IL))) 16: length(cons(N,L)) -> uLength(and(isNat(N),isNatList(activate(L))),activate(L)) 17: uLength(tt(),L) -> s(length(activate(L))) 18: |0|() -> n__0() 19: s(X) -> n__s(X) 20: length(X) -> n__length(X) 21: zeros() -> n__zeros() 22: cons(X1,X2) -> n__cons(X1,X2) 23: nil() -> n__nil() 24: take(X1,X2) -> n__take(X1,X2) 25: activate(n__0()) -> |0|() 26: activate(n__s(X)) -> s(X) 27: activate(n__length(X)) -> length(X) 28: activate(n__zeros()) -> zeros() 29: activate(n__cons(X1,X2)) -> cons(X1,X2) 30: activate(n__nil()) -> nil() 31: activate(n__take(X1,X2)) -> take(X1,X2) 32: activate(X) -> X Number of strict rules: 32 Direct Order(PosReal,>,Poly) ... failed. Freezing ... failed. Dependency Pairs: #1: #isNatIList(IL) -> #isNatList(activate(IL)) #2: #isNatIList(IL) -> #activate(IL) #3: #activate(n__cons(X1,X2)) -> #cons(X1,X2) #4: #uTake1(tt()) -> #nil() #5: #isNatList(n__cons(N,L)) -> #and(isNat(activate(N)),isNatList(activate(L))) #6: #isNatList(n__cons(N,L)) -> #isNat(activate(N)) #7: #isNatList(n__cons(N,L)) -> #activate(N) #8: #isNatList(n__cons(N,L)) -> #isNatList(activate(L)) #9: #isNatList(n__cons(N,L)) -> #activate(L) #10: #zeros() -> #cons(|0|(),n__zeros()) #11: #zeros() -> #|0|() #12: #take(|0|(),IL) -> #uTake1(isNatIList(IL)) #13: #take(|0|(),IL) -> #isNatIList(IL) #14: #activate(n__take(X1,X2)) -> #take(X1,X2) #15: #take(s(M),cons(N,IL)) -> #uTake2(and(isNat(M),and(isNat(N),isNatIList(activate(IL)))),M,N,activate(IL)) #16: #take(s(M),cons(N,IL)) -> #and(isNat(M),and(isNat(N),isNatIList(activate(IL)))) #17: #take(s(M),cons(N,IL)) -> #isNat(M) #18: #take(s(M),cons(N,IL)) -> #and(isNat(N),isNatIList(activate(IL))) #19: #take(s(M),cons(N,IL)) -> #isNat(N) #20: #take(s(M),cons(N,IL)) -> #isNatIList(activate(IL)) #21: #take(s(M),cons(N,IL)) -> #activate(IL) #22: #take(s(M),cons(N,IL)) -> #activate(IL) #23: #activate(n__nil()) -> #nil() #24: #activate(n__0()) -> #|0|() #25: #isNatIList(n__cons(N,IL)) -> #and(isNat(activate(N)),isNatIList(activate(IL))) #26: #isNatIList(n__cons(N,IL)) -> #isNat(activate(N)) #27: #isNatIList(n__cons(N,IL)) -> #activate(N) #28: #isNatIList(n__cons(N,IL)) -> #isNatIList(activate(IL)) #29: #isNatIList(n__cons(N,IL)) -> #activate(IL) #30: #isNatList(n__take(N,IL)) -> #and(isNat(activate(N)),isNatIList(activate(IL))) #31: #isNatList(n__take(N,IL)) -> #isNat(activate(N)) #32: #isNatList(n__take(N,IL)) -> #activate(N) #33: #isNatList(n__take(N,IL)) -> #isNatIList(activate(IL)) #34: #isNatList(n__take(N,IL)) -> #activate(IL) #35: #isNat(n__length(L)) -> #isNatList(activate(L)) #36: #isNat(n__length(L)) -> #activate(L) #37: #activate(n__zeros()) -> #zeros() #38: #activate(n__length(X)) -> #length(X) #39: #uLength(tt(),L) -> #s(length(activate(L))) #40: #uLength(tt(),L) -> #length(activate(L)) #41: #uLength(tt(),L) -> #activate(L) #42: #activate(n__s(X)) -> #s(X) #43: #length(cons(N,L)) -> #uLength(and(isNat(N),isNatList(activate(L))),activate(L)) #44: #length(cons(N,L)) -> #and(isNat(N),isNatList(activate(L))) #45: #length(cons(N,L)) -> #isNat(N) #46: #length(cons(N,L)) -> #isNatList(activate(L)) #47: #length(cons(N,L)) -> #activate(L) #48: #length(cons(N,L)) -> #activate(L) #49: #uTake2(tt(),M,N,IL) -> #cons(activate(N),n__take(activate(M),activate(IL))) #50: #uTake2(tt(),M,N,IL) -> #activate(N) #51: #uTake2(tt(),M,N,IL) -> #activate(M) #52: #uTake2(tt(),M,N,IL) -> #activate(IL) #53: #isNat(n__s(N)) -> #isNat(activate(N)) #54: #isNat(n__s(N)) -> #activate(N) Number of SCCs: 1, DPs: 37, edges: 140 SCC { #1 #2 #6..9 #13..15 #17 #19..22 #26..29 #31..36 #38 #40 #41 #43 #45..48 #50..54 } Removing DPs: Order(PosReal,>,Sum)... succeeded. |0|() weight: 0 #uTake2(x1,x2,x3,x4) weight: (/ 1 8) + x2 + x3 + x4 isNatList(x1) weight: (/ 1 4) + x1 #|0|() weight: 0 #cons(x1,x2) weight: 0 s(x1) weight: x1 #isNat(x1) weight: x1 #take(x1,x2) weight: (/ 1 4) + x1 + x2 activate(x1) weight: x1 take(x1,x2) weight: (/ 3 8) + x1 + x2 #uTake1(x1) weight: 0 and(x1,x2) weight: (/ 3 8) n__zeros() weight: 0 isNatIList(x1) weight: (/ 1 8) #activate(x1) weight: x1 zeros() weight: 0 n__nil() weight: 0 uTake2(x1,x2,x3,x4) weight: (/ 3 8) + x2 + x3 + x4 n__s(x1) weight: x1 uLength(x1,x2) weight: x2 #zeros() weight: 0 n__take(x1,x2) weight: (/ 3 8) + x1 + x2 #isNatList(x1) weight: x1 #s(x1) weight: 0 n__cons(x1,x2) weight: x1 + x2 nil() weight: 0 #nil() weight: 0 n__0() weight: 0 n__length(x1) weight: x1 isNat(x1) weight: (/ 1 8) cons(x1,x2) weight: x1 + x2 #isNatIList(x1) weight: (/ 1 8) + x1 tt() weight: 0 uTake1(x1) weight: 0 length(x1) weight: x1 #length(x1) weight: x1 #and(x1,x2) weight: 0 #uLength(x1,x2) weight: x2 Usable rules: { 11..32 } Removed DPs: #1 #2 #13..15 #17 #19..22 #26 #27 #29 #31..34 #50..52 Number of SCCs: 2, DPs: 17, edges: 44 SCC { #28 } Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... Order(PosReal,>,MaxSum)... QWPOpS(PosReal,>,MaxSum)... Order(PosReal,>,Sum-Sum; PosReal,≥,Sum-Sum)... Order(PosReal,>,Sum-Sum; NegReal,≥,Sum)... Order(PosReal,>,MaxSum-Sum; NegReal,≥,Sum)... failed. Removing edges: failed. Finding a loop... found. #isNatIList(n__cons(N,n__zeros())) -#28-> #isNatIList(activate(n__zeros())) --->* #isNatIList(n__cons(|0|(),n__zeros())) Looping with: [ N := |0|(); ] NO