YES Problem: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) a____(X,nil()) -> mark(X) a____(nil(),X) -> mark(X) a__U11(tt()) -> a__U12(tt()) a__U12(tt()) -> tt() a__isNePal(__(I,__(P,I))) -> a__U11(tt()) mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) mark(U11(X)) -> a__U11(mark(X)) mark(U12(X)) -> a__U12(mark(X)) mark(isNePal(X)) -> a__isNePal(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() a____(X1,X2) -> __(X1,X2) a__U11(X) -> U11(X) a__U12(X) -> U12(X) a__isNePal(X) -> isNePal(X) Proof: DP Processor: DPs: a____#(__(X,Y),Z) -> mark#(Z) a____#(__(X,Y),Z) -> mark#(Y) a____#(__(X,Y),Z) -> a____#(mark(Y),mark(Z)) a____#(__(X,Y),Z) -> mark#(X) a____#(__(X,Y),Z) -> a____#(mark(X),a____(mark(Y),mark(Z))) a____#(X,nil()) -> mark#(X) a____#(nil(),X) -> mark#(X) a__U11#(tt()) -> a__U12#(tt()) a__isNePal#(__(I,__(P,I))) -> a__U11#(tt()) mark#(__(X1,X2)) -> mark#(X2) mark#(__(X1,X2)) -> mark#(X1) mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) mark#(U11(X)) -> mark#(X) mark#(U11(X)) -> a__U11#(mark(X)) mark#(U12(X)) -> mark#(X) mark#(U12(X)) -> a__U12#(mark(X)) mark#(isNePal(X)) -> mark#(X) mark#(isNePal(X)) -> a__isNePal#(mark(X)) TRS: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) a____(X,nil()) -> mark(X) a____(nil(),X) -> mark(X) a__U11(tt()) -> a__U12(tt()) a__U12(tt()) -> tt() a__isNePal(__(I,__(P,I))) -> a__U11(tt()) mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) mark(U11(X)) -> a__U11(mark(X)) mark(U12(X)) -> a__U12(mark(X)) mark(isNePal(X)) -> a__isNePal(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() a____(X1,X2) -> __(X1,X2) a__U11(X) -> U11(X) a__U12(X) -> U12(X) a__isNePal(X) -> isNePal(X) TDG Processor: DPs: a____#(__(X,Y),Z) -> mark#(Z) a____#(__(X,Y),Z) -> mark#(Y) a____#(__(X,Y),Z) -> a____#(mark(Y),mark(Z)) a____#(__(X,Y),Z) -> mark#(X) a____#(__(X,Y),Z) -> a____#(mark(X),a____(mark(Y),mark(Z))) a____#(X,nil()) -> mark#(X) a____#(nil(),X) -> mark#(X) a__U11#(tt()) -> a__U12#(tt()) a__isNePal#(__(I,__(P,I))) -> a__U11#(tt()) mark#(__(X1,X2)) -> mark#(X2) mark#(__(X1,X2)) -> mark#(X1) mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) mark#(U11(X)) -> mark#(X) mark#(U11(X)) -> a__U11#(mark(X)) mark#(U12(X)) -> mark#(X) mark#(U12(X)) -> a__U12#(mark(X)) mark#(isNePal(X)) -> mark#(X) mark#(isNePal(X)) -> a__isNePal#(mark(X)) TRS: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) a____(X,nil()) -> mark(X) a____(nil(),X) -> mark(X) a__U11(tt()) -> a__U12(tt()) a__U12(tt()) -> tt() a__isNePal(__(I,__(P,I))) -> a__U11(tt()) mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) mark(U11(X)) -> a__U11(mark(X)) mark(U12(X)) -> a__U12(mark(X)) mark(isNePal(X)) -> a__isNePal(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() a____(X1,X2) -> __(X1,X2) a__U11(X) -> U11(X) a__U12(X) -> U12(X) a__isNePal(X) -> isNePal(X) graph: a__isNePal#(__(I,__(P,I))) -> a__U11#(tt()) -> a__U11#(tt()) -> a__U12#(tt()) mark#(isNePal(X)) -> a__isNePal#(mark(X)) -> a__isNePal#(__(I,__(P,I))) -> a__U11#(tt()) mark#(isNePal(X)) -> mark#(X) -> mark#(isNePal(X)) -> a__isNePal#(mark(X)) mark#(isNePal(X)) -> mark#(X) -> mark#(isNePal(X)) -> mark#(X) mark#(isNePal(X)) -> mark#(X) -> mark#(U12(X)) -> a__U12#(mark(X)) mark#(isNePal(X)) -> mark#(X) -> mark#(U12(X)) -> mark#(X) mark#(isNePal(X)) -> mark#(X) -> mark#(U11(X)) -> a__U11#(mark(X)) mark#(isNePal(X)) -> mark#(X) -> mark#(U11(X)) -> mark#(X) mark#(isNePal(X)) -> mark#(X) -> mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) mark#(isNePal(X)) -> mark#(X) -> mark#(__(X1,X2)) -> mark#(X1) mark#(isNePal(X)) -> mark#(X) -> mark#(__(X1,X2)) -> mark#(X2) mark#(U12(X)) -> mark#(X) -> mark#(isNePal(X)) -> a__isNePal#(mark(X)) mark#(U12(X)) -> mark#(X) -> mark#(isNePal(X)) -> mark#(X) mark#(U12(X)) -> mark#(X) -> mark#(U12(X)) -> a__U12#(mark(X)) mark#(U12(X)) -> mark#(X) -> mark#(U12(X)) -> mark#(X) mark#(U12(X)) -> mark#(X) -> mark#(U11(X)) -> a__U11#(mark(X)) mark#(U12(X)) -> mark#(X) -> mark#(U11(X)) -> mark#(X) mark#(U12(X)) -> mark#(X) -> mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) mark#(U12(X)) -> mark#(X) -> mark#(__(X1,X2)) -> mark#(X1) mark#(U12(X)) -> mark#(X) -> mark#(__(X1,X2)) -> mark#(X2) mark#(U11(X)) -> a__U11#(mark(X)) -> a__U11#(tt()) -> a__U12#(tt()) mark#(U11(X)) -> mark#(X) -> mark#(isNePal(X)) -> a__isNePal#(mark(X)) mark#(U11(X)) -> mark#(X) -> mark#(isNePal(X)) -> mark#(X) mark#(U11(X)) -> mark#(X) -> mark#(U12(X)) -> a__U12#(mark(X)) mark#(U11(X)) -> mark#(X) -> mark#(U12(X)) -> mark#(X) mark#(U11(X)) -> mark#(X) -> mark#(U11(X)) -> a__U11#(mark(X)) mark#(U11(X)) -> mark#(X) -> mark#(U11(X)) -> mark#(X) mark#(U11(X)) -> mark#(X) -> mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) mark#(U11(X)) -> mark#(X) -> mark#(__(X1,X2)) -> mark#(X1) mark#(U11(X)) -> mark#(X) -> mark#(__(X1,X2)) -> mark#(X2) mark#(__(X1,X2)) -> mark#(X2) -> mark#(isNePal(X)) -> a__isNePal#(mark(X)) mark#(__(X1,X2)) -> mark#(X2) -> mark#(isNePal(X)) -> mark#(X) mark#(__(X1,X2)) -> mark#(X2) -> mark#(U12(X)) -> a__U12#(mark(X)) mark#(__(X1,X2)) -> mark#(X2) -> mark#(U12(X)) -> mark#(X) mark#(__(X1,X2)) -> mark#(X2) -> mark#(U11(X)) -> a__U11#(mark(X)) mark#(__(X1,X2)) -> mark#(X2) -> mark#(U11(X)) -> mark#(X) mark#(__(X1,X2)) -> mark#(X2) -> mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) mark#(__(X1,X2)) -> mark#(X2) -> mark#(__(X1,X2)) -> mark#(X1) mark#(__(X1,X2)) -> mark#(X2) -> mark#(__(X1,X2)) -> mark#(X2) mark#(__(X1,X2)) -> mark#(X1) -> mark#(isNePal(X)) -> a__isNePal#(mark(X)) mark#(__(X1,X2)) -> mark#(X1) -> mark#(isNePal(X)) -> mark#(X) mark#(__(X1,X2)) -> mark#(X1) -> mark#(U12(X)) -> a__U12#(mark(X)) mark#(__(X1,X2)) -> mark#(X1) -> mark#(U12(X)) -> mark#(X) mark#(__(X1,X2)) -> mark#(X1) -> mark#(U11(X)) -> a__U11#(mark(X)) mark#(__(X1,X2)) -> mark#(X1) -> mark#(U11(X)) -> mark#(X) mark#(__(X1,X2)) -> mark#(X1) -> mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) mark#(__(X1,X2)) -> mark#(X1) -> mark#(__(X1,X2)) -> mark#(X1) mark#(__(X1,X2)) -> mark#(X1) -> mark#(__(X1,X2)) -> mark#(X2) mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) -> a____#(nil(),X) -> mark#(X) mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) -> a____#(X,nil()) -> mark#(X) mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) -> a____#(__(X,Y),Z) -> a____#(mark(X),a____(mark(Y),mark(Z))) mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) -> a____#(__(X,Y),Z) -> mark#(X) mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) -> a____#(__(X,Y),Z) -> a____#(mark(Y),mark(Z)) mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) -> a____#(__(X,Y),Z) -> mark#(Y) mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) -> a____#(__(X,Y),Z) -> mark#(Z) a____#(nil(),X) -> mark#(X) -> mark#(isNePal(X)) -> a__isNePal#(mark(X)) a____#(nil(),X) -> mark#(X) -> mark#(isNePal(X)) -> mark#(X) a____#(nil(),X) -> mark#(X) -> mark#(U12(X)) -> a__U12#(mark(X)) a____#(nil(),X) -> mark#(X) -> mark#(U12(X)) -> mark#(X) a____#(nil(),X) -> mark#(X) -> mark#(U11(X)) -> a__U11#(mark(X)) a____#(nil(),X) -> mark#(X) -> mark#(U11(X)) -> mark#(X) a____#(nil(),X) -> mark#(X) -> mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) a____#(nil(),X) -> mark#(X) -> mark#(__(X1,X2)) -> mark#(X1) a____#(nil(),X) -> mark#(X) -> mark#(__(X1,X2)) -> mark#(X2) a____#(__(X,Y),Z) -> mark#(Z) -> mark#(isNePal(X)) -> a__isNePal#(mark(X)) a____#(__(X,Y),Z) -> mark#(Z) -> mark#(isNePal(X)) -> mark#(X) a____#(__(X,Y),Z) -> mark#(Z) -> mark#(U12(X)) -> a__U12#(mark(X)) a____#(__(X,Y),Z) -> mark#(Z) -> mark#(U12(X)) -> mark#(X) a____#(__(X,Y),Z) -> mark#(Z) -> mark#(U11(X)) -> a__U11#(mark(X)) a____#(__(X,Y),Z) -> mark#(Z) -> mark#(U11(X)) -> mark#(X) a____#(__(X,Y),Z) -> mark#(Z) -> mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) a____#(__(X,Y),Z) -> mark#(Z) -> mark#(__(X1,X2)) -> mark#(X1) a____#(__(X,Y),Z) -> mark#(Z) -> mark#(__(X1,X2)) -> mark#(X2) a____#(__(X,Y),Z) -> mark#(Y) -> mark#(isNePal(X)) -> a__isNePal#(mark(X)) a____#(__(X,Y),Z) -> mark#(Y) -> mark#(isNePal(X)) -> mark#(X) a____#(__(X,Y),Z) -> mark#(Y) -> mark#(U12(X)) -> a__U12#(mark(X)) a____#(__(X,Y),Z) -> mark#(Y) -> mark#(U12(X)) -> mark#(X) a____#(__(X,Y),Z) -> mark#(Y) -> mark#(U11(X)) -> a__U11#(mark(X)) a____#(__(X,Y),Z) -> mark#(Y) -> mark#(U11(X)) -> mark#(X) a____#(__(X,Y),Z) -> mark#(Y) -> mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) a____#(__(X,Y),Z) -> mark#(Y) -> mark#(__(X1,X2)) -> mark#(X1) a____#(__(X,Y),Z) -> mark#(Y) -> mark#(__(X1,X2)) -> mark#(X2) a____#(__(X,Y),Z) -> mark#(X) -> mark#(isNePal(X)) -> a__isNePal#(mark(X)) a____#(__(X,Y),Z) -> mark#(X) -> mark#(isNePal(X)) -> mark#(X) a____#(__(X,Y),Z) -> mark#(X) -> mark#(U12(X)) -> a__U12#(mark(X)) a____#(__(X,Y),Z) -> mark#(X) -> mark#(U12(X)) -> mark#(X) a____#(__(X,Y),Z) -> mark#(X) -> mark#(U11(X)) -> a__U11#(mark(X)) a____#(__(X,Y),Z) -> mark#(X) -> mark#(U11(X)) -> mark#(X) a____#(__(X,Y),Z) -> mark#(X) -> mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) a____#(__(X,Y),Z) -> mark#(X) -> mark#(__(X1,X2)) -> mark#(X1) a____#(__(X,Y),Z) -> mark#(X) -> mark#(__(X1,X2)) -> mark#(X2) a____#(__(X,Y),Z) -> a____#(mark(Y),mark(Z)) -> a____#(nil(),X) -> mark#(X) a____#(__(X,Y),Z) -> a____#(mark(Y),mark(Z)) -> a____#(X,nil()) -> mark#(X) a____#(__(X,Y),Z) -> a____#(mark(Y),mark(Z)) -> a____#(__(X,Y),Z) -> a____#(mark(X),a____(mark(Y),mark(Z))) a____#(__(X,Y),Z) -> a____#(mark(Y),mark(Z)) -> a____#(__(X,Y),Z) -> mark#(X) a____#(__(X,Y),Z) -> a____#(mark(Y),mark(Z)) -> a____#(__(X,Y),Z) -> a____#(mark(Y),mark(Z)) a____#(__(X,Y),Z) -> a____#(mark(Y),mark(Z)) -> a____#(__(X,Y),Z) -> mark#(Y) a____#(__(X,Y),Z) -> a____#(mark(Y),mark(Z)) -> a____#(__(X,Y),Z) -> mark#(Z) a____#(__(X,Y),Z) -> a____#(mark(X),a____(mark(Y),mark(Z))) -> a____#(nil(),X) -> mark#(X) a____#(__(X,Y),Z) -> a____#(mark(X),a____(mark(Y),mark(Z))) -> a____#(X,nil()) -> mark#(X) a____#(__(X,Y),Z) -> a____#(mark(X),a____(mark(Y),mark(Z))) -> a____#(__(X,Y),Z) -> a____#(mark(X),a____(mark(Y),mark(Z))) a____#(__(X,Y),Z) -> a____#(mark(X),a____(mark(Y),mark(Z))) -> a____#(__(X,Y),Z) -> mark#(X) a____#(__(X,Y),Z) -> a____#(mark(X),a____(mark(Y),mark(Z))) -> a____#(__(X,Y),Z) -> a____#(mark(Y),mark(Z)) a____#(__(X,Y),Z) -> a____#(mark(X),a____(mark(Y),mark(Z))) -> a____#(__(X,Y),Z) -> mark#(Y) a____#(__(X,Y),Z) -> a____#(mark(X),a____(mark(Y),mark(Z))) -> a____#(__(X,Y),Z) -> mark#(Z) a____#(X,nil()) -> mark#(X) -> mark#(isNePal(X)) -> a__isNePal#(mark(X)) a____#(X,nil()) -> mark#(X) -> mark#(isNePal(X)) -> mark#(X) a____#(X,nil()) -> mark#(X) -> mark#(U12(X)) -> a__U12#(mark(X)) a____#(X,nil()) -> mark#(X) -> mark#(U12(X)) -> mark#(X) a____#(X,nil()) -> mark#(X) -> mark#(U11(X)) -> a__U11#(mark(X)) a____#(X,nil()) -> mark#(X) -> mark#(U11(X)) -> mark#(X) a____#(X,nil()) -> mark#(X) -> mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) a____#(X,nil()) -> mark#(X) -> mark#(__(X1,X2)) -> mark#(X1) a____#(X,nil()) -> mark#(X) -> mark#(__(X1,X2)) -> mark#(X2) SCC Processor: #sccs: 1 #rules: 13 #arcs: 114/324 DPs: mark#(isNePal(X)) -> mark#(X) mark#(__(X1,X2)) -> mark#(X2) mark#(__(X1,X2)) -> mark#(X1) mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) a____#(__(X,Y),Z) -> mark#(Z) mark#(U11(X)) -> mark#(X) mark#(U12(X)) -> mark#(X) a____#(__(X,Y),Z) -> mark#(Y) a____#(__(X,Y),Z) -> a____#(mark(Y),mark(Z)) a____#(__(X,Y),Z) -> mark#(X) a____#(__(X,Y),Z) -> a____#(mark(X),a____(mark(Y),mark(Z))) a____#(X,nil()) -> mark#(X) a____#(nil(),X) -> mark#(X) TRS: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) a____(X,nil()) -> mark(X) a____(nil(),X) -> mark(X) a__U11(tt()) -> a__U12(tt()) a__U12(tt()) -> tt() a__isNePal(__(I,__(P,I))) -> a__U11(tt()) mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) mark(U11(X)) -> a__U11(mark(X)) mark(U12(X)) -> a__U12(mark(X)) mark(isNePal(X)) -> a__isNePal(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() a____(X1,X2) -> __(X1,X2) a__U11(X) -> U11(X) a__U12(X) -> U12(X) a__isNePal(X) -> isNePal(X) Arctic Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0 + 0, [a____#](x0, x1) = x0 + x1 + 0, [isNePal](x0) = 2x0 + 1, [U12](x0) = x0 + 0, [U11](x0) = x0, [a__isNePal](x0) = 2x0 + 1, [a__U12](x0) = x0 + 0, [a__U11](x0) = x0, [tt] = 0, [nil] = 0, [mark](x0) = x0, [a____](x0, x1) = x0 + x1 + 0, [__](x0, x1) = x0 + x1 + 0 orientation: mark#(isNePal(X)) = 2X + 1 >= X + 0 = mark#(X) mark#(__(X1,X2)) = X1 + X2 + 0 >= X2 + 0 = mark#(X2) mark#(__(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(__(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a____#(mark(X1),mark(X2)) a____#(__(X,Y),Z) = X + Y + Z + 0 >= Z + 0 = mark#(Z) mark#(U11(X)) = X + 0 >= X + 0 = mark#(X) mark#(U12(X)) = X + 0 >= X + 0 = mark#(X) a____#(__(X,Y),Z) = X + Y + Z + 0 >= Y + 0 = mark#(Y) a____#(__(X,Y),Z) = X + Y + Z + 0 >= Y + Z + 0 = a____#(mark(Y),mark(Z)) a____#(__(X,Y),Z) = X + Y + Z + 0 >= X + 0 = mark#(X) a____#(__(X,Y),Z) = X + Y + Z + 0 >= X + Y + Z + 0 = a____#(mark(X),a____(mark(Y),mark(Z))) a____#(X,nil()) = X + 0 >= X + 0 = mark#(X) a____#(nil(),X) = X + 0 >= X + 0 = mark#(X) a____(__(X,Y),Z) = X + Y + Z + 0 >= X + Y + Z + 0 = a____(mark(X),a____(mark(Y),mark(Z))) a____(X,nil()) = X + 0 >= X = mark(X) a____(nil(),X) = X + 0 >= X = mark(X) a__U11(tt()) = 0 >= 0 = a__U12(tt()) a__U12(tt()) = 0 >= 0 = tt() a__isNePal(__(I,__(P,I))) = 2I + 2P + 2 >= 0 = a__U11(tt()) mark(__(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a____(mark(X1),mark(X2)) mark(U11(X)) = X >= X = a__U11(mark(X)) mark(U12(X)) = X + 0 >= X + 0 = a__U12(mark(X)) mark(isNePal(X)) = 2X + 1 >= 2X + 1 = a__isNePal(mark(X)) mark(nil()) = 0 >= 0 = nil() mark(tt()) = 0 >= 0 = tt() a____(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = __(X1,X2) a__U11(X) = X >= X = U11(X) a__U12(X) = X + 0 >= X + 0 = U12(X) a__isNePal(X) = 2X + 1 >= 2X + 1 = isNePal(X) problem: DPs: mark#(__(X1,X2)) -> mark#(X2) mark#(__(X1,X2)) -> mark#(X1) mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) a____#(__(X,Y),Z) -> mark#(Z) mark#(U11(X)) -> mark#(X) mark#(U12(X)) -> mark#(X) a____#(__(X,Y),Z) -> mark#(Y) a____#(__(X,Y),Z) -> a____#(mark(Y),mark(Z)) a____#(__(X,Y),Z) -> mark#(X) a____#(__(X,Y),Z) -> a____#(mark(X),a____(mark(Y),mark(Z))) a____#(X,nil()) -> mark#(X) a____#(nil(),X) -> mark#(X) TRS: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) a____(X,nil()) -> mark(X) a____(nil(),X) -> mark(X) a__U11(tt()) -> a__U12(tt()) a__U12(tt()) -> tt() a__isNePal(__(I,__(P,I))) -> a__U11(tt()) mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) mark(U11(X)) -> a__U11(mark(X)) mark(U12(X)) -> a__U12(mark(X)) mark(isNePal(X)) -> a__isNePal(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() a____(X1,X2) -> __(X1,X2) a__U11(X) -> U11(X) a__U12(X) -> U12(X) a__isNePal(X) -> isNePal(X) Arctic Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [a____#](x0, x1) = x0 + x1, [isNePal](x0) = 5, [U12](x0) = x0 + 0, [U11](x0) = 4x0 + 2, [a__isNePal](x0) = 5, [a__U12](x0) = x0 + 0, [a__U11](x0) = 4x0 + 2, [tt] = 1, [nil] = 2, [mark](x0) = x0, [a____](x0, x1) = x0 + x1 + 0, [__](x0, x1) = x0 + x1 + 0 orientation: mark#(__(X1,X2)) = X1 + X2 + 0 >= X2 = mark#(X2) mark#(__(X1,X2)) = X1 + X2 + 0 >= X1 = mark#(X1) mark#(__(X1,X2)) = X1 + X2 + 0 >= X1 + X2 = a____#(mark(X1),mark(X2)) a____#(__(X,Y),Z) = X + Y + Z + 0 >= Z = mark#(Z) mark#(U11(X)) = 4X + 2 >= X = mark#(X) mark#(U12(X)) = X + 0 >= X = mark#(X) a____#(__(X,Y),Z) = X + Y + Z + 0 >= Y = mark#(Y) a____#(__(X,Y),Z) = X + Y + Z + 0 >= Y + Z = a____#(mark(Y),mark(Z)) a____#(__(X,Y),Z) = X + Y + Z + 0 >= X = mark#(X) a____#(__(X,Y),Z) = X + Y + Z + 0 >= X + Y + Z + 0 = a____#(mark(X),a____(mark(Y),mark(Z))) a____#(X,nil()) = X + 2 >= X = mark#(X) a____#(nil(),X) = X + 2 >= X = mark#(X) a____(__(X,Y),Z) = X + Y + Z + 0 >= X + Y + Z + 0 = a____(mark(X),a____(mark(Y),mark(Z))) a____(X,nil()) = X + 2 >= X = mark(X) a____(nil(),X) = X + 2 >= X = mark(X) a__U11(tt()) = 5 >= 1 = a__U12(tt()) a__U12(tt()) = 1 >= 1 = tt() a__isNePal(__(I,__(P,I))) = 5 >= 5 = a__U11(tt()) mark(__(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a____(mark(X1),mark(X2)) mark(U11(X)) = 4X + 2 >= 4X + 2 = a__U11(mark(X)) mark(U12(X)) = X + 0 >= X + 0 = a__U12(mark(X)) mark(isNePal(X)) = 5 >= 5 = a__isNePal(mark(X)) mark(nil()) = 2 >= 2 = nil() mark(tt()) = 1 >= 1 = tt() a____(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = __(X1,X2) a__U11(X) = 4X + 2 >= 4X + 2 = U11(X) a__U12(X) = X + 0 >= X + 0 = U12(X) a__isNePal(X) = 5 >= 5 = isNePal(X) problem: DPs: mark#(__(X1,X2)) -> mark#(X2) mark#(__(X1,X2)) -> mark#(X1) mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) a____#(__(X,Y),Z) -> mark#(Z) mark#(U12(X)) -> mark#(X) a____#(__(X,Y),Z) -> mark#(Y) a____#(__(X,Y),Z) -> a____#(mark(Y),mark(Z)) a____#(__(X,Y),Z) -> mark#(X) a____#(__(X,Y),Z) -> a____#(mark(X),a____(mark(Y),mark(Z))) a____#(X,nil()) -> mark#(X) a____#(nil(),X) -> mark#(X) TRS: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) a____(X,nil()) -> mark(X) a____(nil(),X) -> mark(X) a__U11(tt()) -> a__U12(tt()) a__U12(tt()) -> tt() a__isNePal(__(I,__(P,I))) -> a__U11(tt()) mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) mark(U11(X)) -> a__U11(mark(X)) mark(U12(X)) -> a__U12(mark(X)) mark(isNePal(X)) -> a__isNePal(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() a____(X1,X2) -> __(X1,X2) a__U11(X) -> U11(X) a__U12(X) -> U12(X) a__isNePal(X) -> isNePal(X) Arctic Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0 + 1, [a____#](x0, x1) = 1x0 + x1 + 1, [isNePal](x0) = x0, [U12](x0) = x0, [U11](x0) = 0, [a__isNePal](x0) = x0 + 0, [a__U12](x0) = x0 + 0, [a__U11](x0) = 0, [tt] = 0, [nil] = 0, [mark](x0) = x0 + 0, [a____](x0, x1) = 1x0 + x1 + 1, [__](x0, x1) = 1x0 + x1 + 1 orientation: mark#(__(X1,X2)) = 1X1 + X2 + 1 >= X2 + 1 = mark#(X2) mark#(__(X1,X2)) = 1X1 + X2 + 1 >= X1 + 1 = mark#(X1) mark#(__(X1,X2)) = 1X1 + X2 + 1 >= 1X1 + X2 + 1 = a____#(mark(X1),mark(X2)) a____#(__(X,Y),Z) = 2X + 1Y + Z + 2 >= Z + 1 = mark#(Z) mark#(U12(X)) = X + 1 >= X + 1 = mark#(X) a____#(__(X,Y),Z) = 2X + 1Y + Z + 2 >= Y + 1 = mark#(Y) a____#(__(X,Y),Z) = 2X + 1Y + Z + 2 >= 1Y + Z + 1 = a____#(mark(Y),mark(Z)) a____#(__(X,Y),Z) = 2X + 1Y + Z + 2 >= X + 1 = mark#(X) a____#(__(X,Y),Z) = 2X + 1Y + Z + 2 >= 1X + 1Y + Z + 1 = a____#(mark(X),a____(mark(Y),mark(Z))) a____#(X,nil()) = 1X + 1 >= X + 1 = mark#(X) a____#(nil(),X) = X + 1 >= X + 1 = mark#(X) a____(__(X,Y),Z) = 2X + 1Y + Z + 2 >= 1X + 1Y + Z + 1 = a____(mark(X),a____(mark(Y),mark(Z))) a____(X,nil()) = 1X + 1 >= X + 0 = mark(X) a____(nil(),X) = X + 1 >= X + 0 = mark(X) a__U11(tt()) = 0 >= 0 = a__U12(tt()) a__U12(tt()) = 0 >= 0 = tt() a__isNePal(__(I,__(P,I))) = 1I + 1P + 1 >= 0 = a__U11(tt()) mark(__(X1,X2)) = 1X1 + X2 + 1 >= 1X1 + X2 + 1 = a____(mark(X1),mark(X2)) mark(U11(X)) = 0 >= 0 = a__U11(mark(X)) mark(U12(X)) = X + 0 >= X + 0 = a__U12(mark(X)) mark(isNePal(X)) = X + 0 >= X + 0 = a__isNePal(mark(X)) mark(nil()) = 0 >= 0 = nil() mark(tt()) = 0 >= 0 = tt() a____(X1,X2) = 1X1 + X2 + 1 >= 1X1 + X2 + 1 = __(X1,X2) a__U11(X) = 0 >= 0 = U11(X) a__U12(X) = X + 0 >= X = U12(X) a__isNePal(X) = X + 0 >= X = isNePal(X) problem: DPs: mark#(__(X1,X2)) -> mark#(X2) mark#(__(X1,X2)) -> mark#(X1) mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) a____#(__(X,Y),Z) -> mark#(Z) mark#(U12(X)) -> mark#(X) a____#(__(X,Y),Z) -> a____#(mark(Y),mark(Z)) a____#(__(X,Y),Z) -> a____#(mark(X),a____(mark(Y),mark(Z))) a____#(X,nil()) -> mark#(X) a____#(nil(),X) -> mark#(X) TRS: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) a____(X,nil()) -> mark(X) a____(nil(),X) -> mark(X) a__U11(tt()) -> a__U12(tt()) a__U12(tt()) -> tt() a__isNePal(__(I,__(P,I))) -> a__U11(tt()) mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) mark(U11(X)) -> a__U11(mark(X)) mark(U12(X)) -> a__U12(mark(X)) mark(isNePal(X)) -> a__isNePal(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() a____(X1,X2) -> __(X1,X2) a__U11(X) -> U11(X) a__U12(X) -> U12(X) a__isNePal(X) -> isNePal(X) Arctic Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [a____#](x0, x1) = x0 + x1, [isNePal](x0) = 2x0, [U12](x0) = 2x0, [U11](x0) = 2x0 + 0, [a__isNePal](x0) = 2x0, [a__U12](x0) = 2x0, [a__U11](x0) = 2x0 + 0, [tt] = 0, [nil] = 7, [mark](x0) = x0, [a____](x0, x1) = x0 + x1 + 0, [__](x0, x1) = x0 + x1 + 0 orientation: mark#(__(X1,X2)) = X1 + X2 + 0 >= X2 = mark#(X2) mark#(__(X1,X2)) = X1 + X2 + 0 >= X1 = mark#(X1) mark#(__(X1,X2)) = X1 + X2 + 0 >= X1 + X2 = a____#(mark(X1),mark(X2)) a____#(__(X,Y),Z) = X + Y + Z + 0 >= Z = mark#(Z) mark#(U12(X)) = 2X >= X = mark#(X) a____#(__(X,Y),Z) = X + Y + Z + 0 >= Y + Z = a____#(mark(Y),mark(Z)) a____#(__(X,Y),Z) = X + Y + Z + 0 >= X + Y + Z + 0 = a____#(mark(X),a____(mark(Y),mark(Z))) a____#(X,nil()) = X + 7 >= X = mark#(X) a____#(nil(),X) = X + 7 >= X = mark#(X) a____(__(X,Y),Z) = X + Y + Z + 0 >= X + Y + Z + 0 = a____(mark(X),a____(mark(Y),mark(Z))) a____(X,nil()) = X + 7 >= X = mark(X) a____(nil(),X) = X + 7 >= X = mark(X) a__U11(tt()) = 2 >= 2 = a__U12(tt()) a__U12(tt()) = 2 >= 0 = tt() a__isNePal(__(I,__(P,I))) = 2I + 2P + 2 >= 2 = a__U11(tt()) mark(__(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a____(mark(X1),mark(X2)) mark(U11(X)) = 2X + 0 >= 2X + 0 = a__U11(mark(X)) mark(U12(X)) = 2X >= 2X = a__U12(mark(X)) mark(isNePal(X)) = 2X >= 2X = a__isNePal(mark(X)) mark(nil()) = 7 >= 7 = nil() mark(tt()) = 0 >= 0 = tt() a____(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = __(X1,X2) a__U11(X) = 2X + 0 >= 2X + 0 = U11(X) a__U12(X) = 2X >= 2X = U12(X) a__isNePal(X) = 2X >= 2X = isNePal(X) problem: DPs: mark#(__(X1,X2)) -> mark#(X2) mark#(__(X1,X2)) -> mark#(X1) mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) a____#(__(X,Y),Z) -> mark#(Z) a____#(__(X,Y),Z) -> a____#(mark(Y),mark(Z)) a____#(__(X,Y),Z) -> a____#(mark(X),a____(mark(Y),mark(Z))) a____#(X,nil()) -> mark#(X) a____#(nil(),X) -> mark#(X) TRS: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) a____(X,nil()) -> mark(X) a____(nil(),X) -> mark(X) a__U11(tt()) -> a__U12(tt()) a__U12(tt()) -> tt() a__isNePal(__(I,__(P,I))) -> a__U11(tt()) mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) mark(U11(X)) -> a__U11(mark(X)) mark(U12(X)) -> a__U12(mark(X)) mark(isNePal(X)) -> a__isNePal(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() a____(X1,X2) -> __(X1,X2) a__U11(X) -> U11(X) a__U12(X) -> U12(X) a__isNePal(X) -> isNePal(X) Arctic Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = 2x0 + 0, [a____#](x0, x1) = 4x0 + 2x1, [isNePal](x0) = 0, [U12](x0) = 0, [U11](x0) = x0, [a__isNePal](x0) = 0, [a__U12](x0) = 0, [a__U11](x0) = x0, [tt] = 0, [nil] = 0, [mark](x0) = x0, [a____](x0, x1) = 2x0 + x1 + 2, [__](x0, x1) = 2x0 + x1 + 2 orientation: mark#(__(X1,X2)) = 4X1 + 2X2 + 4 >= 2X2 + 0 = mark#(X2) mark#(__(X1,X2)) = 4X1 + 2X2 + 4 >= 2X1 + 0 = mark#(X1) mark#(__(X1,X2)) = 4X1 + 2X2 + 4 >= 4X1 + 2X2 = a____#(mark(X1),mark(X2)) a____#(__(X,Y),Z) = 6X + 4Y + 2Z + 6 >= 2Z + 0 = mark#(Z) a____#(__(X,Y),Z) = 6X + 4Y + 2Z + 6 >= 4Y + 2Z = a____#(mark(Y),mark(Z)) a____#(__(X,Y),Z) = 6X + 4Y + 2Z + 6 >= 4X + 4Y + 2Z + 4 = a____#(mark(X),a____(mark(Y),mark(Z))) a____#(X,nil()) = 4X + 2 >= 2X + 0 = mark#(X) a____#(nil(),X) = 2X + 4 >= 2X + 0 = mark#(X) a____(__(X,Y),Z) = 4X + 2Y + Z + 4 >= 2X + 2Y + Z + 2 = a____(mark(X),a____(mark(Y),mark(Z))) a____(X,nil()) = 2X + 2 >= X = mark(X) a____(nil(),X) = X + 2 >= X = mark(X) a__U11(tt()) = 0 >= 0 = a__U12(tt()) a__U12(tt()) = 0 >= 0 = tt() a__isNePal(__(I,__(P,I))) = 0 >= 0 = a__U11(tt()) mark(__(X1,X2)) = 2X1 + X2 + 2 >= 2X1 + X2 + 2 = a____(mark(X1),mark(X2)) mark(U11(X)) = X >= X = a__U11(mark(X)) mark(U12(X)) = 0 >= 0 = a__U12(mark(X)) mark(isNePal(X)) = 0 >= 0 = a__isNePal(mark(X)) mark(nil()) = 0 >= 0 = nil() mark(tt()) = 0 >= 0 = tt() a____(X1,X2) = 2X1 + X2 + 2 >= 2X1 + X2 + 2 = __(X1,X2) a__U11(X) = X >= X = U11(X) a__U12(X) = 0 >= 0 = U12(X) a__isNePal(X) = 0 >= 0 = isNePal(X) problem: DPs: mark#(__(X1,X2)) -> mark#(X2) mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) a____#(__(X,Y),Z) -> mark#(Z) a____#(__(X,Y),Z) -> a____#(mark(Y),mark(Z)) a____#(__(X,Y),Z) -> a____#(mark(X),a____(mark(Y),mark(Z))) a____#(nil(),X) -> mark#(X) TRS: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) a____(X,nil()) -> mark(X) a____(nil(),X) -> mark(X) a__U11(tt()) -> a__U12(tt()) a__U12(tt()) -> tt() a__isNePal(__(I,__(P,I))) -> a__U11(tt()) mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) mark(U11(X)) -> a__U11(mark(X)) mark(U12(X)) -> a__U12(mark(X)) mark(isNePal(X)) -> a__isNePal(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() a____(X1,X2) -> __(X1,X2) a__U11(X) -> U11(X) a__U12(X) -> U12(X) a__isNePal(X) -> isNePal(X) KBO Processor: argument filtering: pi(__) = [0,1] pi(a____) = [0,1] pi(mark) = [0] pi(nil) = [] pi(tt) = [] pi(a__U11) = [0] pi(a__U12) = [] pi(a__isNePal) = 0 pi(U11) = [0] pi(U12) = [] pi(isNePal) = 0 pi(a____#) = [0,1] pi(mark#) = 0 weight function: w0 = 1 w(isNePal) = w(U12) = w(U11) = w(a__isNePal) = w(a__U12) = w( a__U11) = w(tt) = w(nil) = w(a____) = w(__) = 1 w(mark#) = w(a____#) = w(mark) = 0 precedence: mark > a__U12 ~ a__U11 > a____ > mark# ~ a____# ~ isNePal ~ U12 ~ U11 ~ a__isNePal ~ tt ~ nil ~ __ problem: DPs: TRS: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) a____(X,nil()) -> mark(X) a____(nil(),X) -> mark(X) a__U11(tt()) -> a__U12(tt()) a__U12(tt()) -> tt() a__isNePal(__(I,__(P,I))) -> a__U11(tt()) mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) mark(U11(X)) -> a__U11(mark(X)) mark(U12(X)) -> a__U12(mark(X)) mark(isNePal(X)) -> a__isNePal(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() a____(X1,X2) -> __(X1,X2) a__U11(X) -> U11(X) a__U12(X) -> U12(X) a__isNePal(X) -> isNePal(X) Qed