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__and(tt(),X) -> mark(X) a__isNePal(__(I,__(P,I))) -> tt() mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNePal(X)) -> a__isNePal(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() a____(X1,X2) -> __(X1,X2) a__and(X1,X2) -> and(X1,X2) 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__and#(tt(),X) -> mark#(X) mark#(__(X1,X2)) -> mark#(X2) mark#(__(X1,X2)) -> mark#(X1) mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) 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__and(tt(),X) -> mark(X) a__isNePal(__(I,__(P,I))) -> tt() mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNePal(X)) -> a__isNePal(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() a____(X1,X2) -> __(X1,X2) a__and(X1,X2) -> and(X1,X2) 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__and#(tt(),X) -> mark#(X) mark#(__(X1,X2)) -> mark#(X2) mark#(__(X1,X2)) -> mark#(X1) mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) 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__and(tt(),X) -> mark(X) a__isNePal(__(I,__(P,I))) -> tt() mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNePal(X)) -> a__isNePal(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() a____(X1,X2) -> __(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNePal(X) -> isNePal(X) graph: a__and#(tt(),X) -> mark#(X) -> mark#(isNePal(X)) -> a__isNePal#(mark(X)) a__and#(tt(),X) -> mark#(X) -> mark#(isNePal(X)) -> mark#(X) a__and#(tt(),X) -> mark#(X) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) a__and#(tt(),X) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X1) a__and#(tt(),X) -> mark#(X) -> mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) a__and#(tt(),X) -> mark#(X) -> mark#(__(X1,X2)) -> mark#(X1) a__and#(tt(),X) -> mark#(X) -> mark#(__(X1,X2)) -> mark#(X2) 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#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNePal(X)) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X1) 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#(and(X1,X2)) -> a__and#(mark(X1),X2) -> a__and#(tt(),X) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(isNePal(X)) -> a__isNePal#(mark(X)) mark#(and(X1,X2)) -> mark#(X1) -> mark#(isNePal(X)) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(and(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) mark#(and(X1,X2)) -> mark#(X1) -> mark#(__(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> 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#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(__(X1,X2)) -> mark#(X2) -> mark#(and(X1,X2)) -> mark#(X1) 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#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(__(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) 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#(and(X1,X2)) -> a__and#(mark(X1),X2) a____#(nil(),X) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X1) 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#(and(X1,X2)) -> a__and#(mark(X1),X2) a____#(__(X,Y),Z) -> mark#(Z) -> mark#(and(X1,X2)) -> mark#(X1) 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#(and(X1,X2)) -> a__and#(mark(X1),X2) a____#(__(X,Y),Z) -> mark#(Y) -> mark#(and(X1,X2)) -> mark#(X1) 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#(and(X1,X2)) -> a__and#(mark(X1),X2) a____#(__(X,Y),Z) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X1) 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#(and(X1,X2)) -> a__and#(mark(X1),X2) a____#(X,nil()) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X1) 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: 14 #arcs: 92/225 DPs: a__and#(tt(),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#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(isNePal(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__and(tt(),X) -> mark(X) a__isNePal(__(I,__(P,I))) -> tt() mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNePal(X)) -> a__isNePal(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() a____(X1,X2) -> __(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNePal(X) -> isNePal(X) Arctic Interpretation Processor: dimension: 1 interpretation: [a__and#](x0, x1) = x1 + 0, [mark#](x0) = x0 + 0, [a____#](x0, x1) = x0 + x1, [isNePal](x0) = 2x0 + 1, [and](x0, x1) = x0 + x1, [a__isNePal](x0) = 2x0 + 1, [a__and](x0, x1) = x0 + x1, [tt] = 1, [nil] = 1, [mark](x0) = x0, [a____](x0, x1) = x0 + x1 + 0, [__](x0, x1) = x0 + x1 + 0 orientation: a__and#(tt(),X) = X + 0 >= 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 = a____#(mark(X1),mark(X2)) a____#(__(X,Y),Z) = X + Y + Z + 0 >= Z + 0 = mark#(Z) mark#(and(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 + 0 >= X2 + 0 = a__and#(mark(X1),X2) mark#(isNePal(X)) = 2X + 1 >= 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 = 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 + 1 >= X + 0 = mark#(X) a____#(nil(),X) = X + 1 >= 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 + 1 >= X = mark(X) a____(nil(),X) = X + 1 >= X = mark(X) a__and(tt(),X) = X + 1 >= X = mark(X) a__isNePal(__(I,__(P,I))) = 2I + 2P + 2 >= 1 = tt() mark(__(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a____(mark(X1),mark(X2)) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = a__and(mark(X1),X2) mark(isNePal(X)) = 2X + 1 >= 2X + 1 = a__isNePal(mark(X)) mark(nil()) = 1 >= 1 = nil() mark(tt()) = 1 >= 1 = tt() a____(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = __(X1,X2) a__and(X1,X2) = X1 + X2 >= X1 + X2 = and(X1,X2) a__isNePal(X) = 2X + 1 >= 2X + 1 = isNePal(X) problem: DPs: a__and#(tt(),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#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) 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__and(tt(),X) -> mark(X) a__isNePal(__(I,__(P,I))) -> tt() mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNePal(X)) -> a__isNePal(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() a____(X1,X2) -> __(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNePal(X) -> isNePal(X) Arctic Interpretation Processor: dimension: 1 interpretation: [a__and#](x0, x1) = x0 + x1 + 0, [mark#](x0) = x0, [a____#](x0, x1) = 1x0 + x1, [isNePal](x0) = 0, [and](x0, x1) = x0 + x1 + 0, [a__isNePal](x0) = 0, [a__and](x0, x1) = x0 + x1 + 0, [tt] = 0, [nil] = 0, [mark](x0) = x0, [a____](x0, x1) = 1x0 + x1, [__](x0, x1) = 1x0 + x1 orientation: a__and#(tt(),X) = X + 0 >= X = mark#(X) mark#(__(X1,X2)) = 1X1 + X2 >= X2 = mark#(X2) mark#(__(X1,X2)) = 1X1 + X2 >= X1 = mark#(X1) mark#(__(X1,X2)) = 1X1 + X2 >= 1X1 + X2 = a____#(mark(X1),mark(X2)) a____#(__(X,Y),Z) = 2X + 1Y + Z >= Z = mark#(Z) mark#(and(X1,X2)) = X1 + X2 + 0 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__and#(mark(X1),X2) a____#(__(X,Y),Z) = 2X + 1Y + Z >= Y = mark#(Y) a____#(__(X,Y),Z) = 2X + 1Y + Z >= 1Y + Z = a____#(mark(Y),mark(Z)) a____#(__(X,Y),Z) = 2X + 1Y + Z >= X = mark#(X) a____#(__(X,Y),Z) = 2X + 1Y + Z >= 1X + 1Y + Z = a____#(mark(X),a____(mark(Y),mark(Z))) a____#(X,nil()) = 1X + 0 >= X = mark#(X) a____#(nil(),X) = X + 1 >= X = mark#(X) a____(__(X,Y),Z) = 2X + 1Y + Z >= 1X + 1Y + Z = a____(mark(X),a____(mark(Y),mark(Z))) a____(X,nil()) = 1X + 0 >= X = mark(X) a____(nil(),X) = X + 1 >= X = mark(X) a__and(tt(),X) = X + 0 >= X = mark(X) a__isNePal(__(I,__(P,I))) = 0 >= 0 = tt() mark(__(X1,X2)) = 1X1 + X2 >= 1X1 + X2 = a____(mark(X1),mark(X2)) mark(and(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__and(mark(X1),X2) mark(isNePal(X)) = 0 >= 0 = a__isNePal(mark(X)) mark(nil()) = 0 >= 0 = nil() mark(tt()) = 0 >= 0 = tt() a____(X1,X2) = 1X1 + X2 >= 1X1 + X2 = __(X1,X2) a__and(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = and(X1,X2) a__isNePal(X) = 0 >= 0 = isNePal(X) problem: DPs: a__and#(tt(),X) -> mark#(X) mark#(__(X1,X2)) -> mark#(X2) mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) a____#(__(X,Y),Z) -> mark#(Z) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) 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__and(tt(),X) -> mark(X) a__isNePal(__(I,__(P,I))) -> tt() mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNePal(X)) -> a__isNePal(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() a____(X1,X2) -> __(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNePal(X) -> isNePal(X) Arctic Interpretation Processor: dimension: 1 interpretation: [a__and#](x0, x1) = x1 + 5, [mark#](x0) = x0, [a____#](x0, x1) = x0 + x1 + 0, [isNePal](x0) = x0, [and](x0, x1) = 7x0 + x1 + 5, [a__isNePal](x0) = x0, [a__and](x0, x1) = 7x0 + x1 + 5, [tt] = 0, [nil] = 0, [mark](x0) = x0, [a____](x0, x1) = x0 + x1 + 0, [__](x0, x1) = x0 + x1 + 0 orientation: a__and#(tt(),X) = X + 5 >= X = mark#(X) mark#(__(X1,X2)) = X1 + X2 + 0 >= X2 = mark#(X2) mark#(__(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a____#(mark(X1),mark(X2)) a____#(__(X,Y),Z) = X + Y + Z + 0 >= Z = mark#(Z) mark#(and(X1,X2)) = 7X1 + X2 + 5 >= X1 = mark#(X1) mark#(and(X1,X2)) = 7X1 + X2 + 5 >= X2 + 5 = a__and#(mark(X1),X2) 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 + Y + Z + 0 = a____#(mark(X),a____(mark(Y),mark(Z))) a____#(nil(),X) = X + 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 + 0 >= X = mark(X) a____(nil(),X) = X + 0 >= X = mark(X) a__and(tt(),X) = X + 7 >= X = mark(X) a__isNePal(__(I,__(P,I))) = I + P + 0 >= 0 = tt() mark(__(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a____(mark(X1),mark(X2)) mark(and(X1,X2)) = 7X1 + X2 + 5 >= 7X1 + X2 + 5 = a__and(mark(X1),X2) mark(isNePal(X)) = X >= X = 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__and(X1,X2) = 7X1 + X2 + 5 >= 7X1 + X2 + 5 = and(X1,X2) a__isNePal(X) = X >= X = isNePal(X) problem: DPs: a__and#(tt(),X) -> mark#(X) mark#(__(X1,X2)) -> mark#(X2) mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) a____#(__(X,Y),Z) -> mark#(Z) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) 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__and(tt(),X) -> mark(X) a__isNePal(__(I,__(P,I))) -> tt() mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNePal(X)) -> a__isNePal(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() a____(X1,X2) -> __(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNePal(X) -> isNePal(X) Arctic Interpretation Processor: dimension: 1 interpretation: [a__and#](x0, x1) = x0 + 4x1 + 1, [mark#](x0) = x0, [a____#](x0, x1) = x0 + x1 + 0, [isNePal](x0) = 4, [and](x0, x1) = x0 + 4x1 + 1, [a__isNePal](x0) = 4, [a__and](x0, x1) = x0 + 4x1 + 1, [tt] = 4, [nil] = 4, [mark](x0) = x0 + 0, [a____](x0, x1) = x0 + x1 + 0, [__](x0, x1) = x0 + x1 + 0 orientation: a__and#(tt(),X) = 4X + 4 >= X = mark#(X) mark#(__(X1,X2)) = X1 + X2 + 0 >= X2 = mark#(X2) mark#(__(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a____#(mark(X1),mark(X2)) a____#(__(X,Y),Z) = X + Y + Z + 0 >= Z = mark#(Z) mark#(and(X1,X2)) = X1 + 4X2 + 1 >= X1 + 4X2 + 1 = a__and#(mark(X1),X2) 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 + Y + Z + 0 = a____#(mark(X),a____(mark(Y),mark(Z))) a____#(nil(),X) = X + 4 >= 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 + 4 >= X + 0 = mark(X) a____(nil(),X) = X + 4 >= X + 0 = mark(X) a__and(tt(),X) = 4X + 4 >= X + 0 = mark(X) a__isNePal(__(I,__(P,I))) = 4 >= 4 = tt() mark(__(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a____(mark(X1),mark(X2)) mark(and(X1,X2)) = X1 + 4X2 + 1 >= X1 + 4X2 + 1 = a__and(mark(X1),X2) mark(isNePal(X)) = 4 >= 4 = a__isNePal(mark(X)) mark(nil()) = 4 >= 4 = nil() mark(tt()) = 4 >= 4 = tt() a____(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = __(X1,X2) a__and(X1,X2) = X1 + 4X2 + 1 >= X1 + 4X2 + 1 = and(X1,X2) a__isNePal(X) = 4 >= 4 = isNePal(X) problem: DPs: mark#(__(X1,X2)) -> mark#(X2) mark#(__(X1,X2)) -> a____#(mark(X1),mark(X2)) a____#(__(X,Y),Z) -> mark#(Z) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) 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__and(tt(),X) -> mark(X) a__isNePal(__(I,__(P,I))) -> tt() mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNePal(X)) -> a__isNePal(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() a____(X1,X2) -> __(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNePal(X) -> isNePal(X) SCC Processor: #sccs: 1 #rules: 6 #arcs: 82/49 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__and(tt(),X) -> mark(X) a__isNePal(__(I,__(P,I))) -> tt() mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNePal(X)) -> a__isNePal(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() a____(X1,X2) -> __(X1,X2) a__and(X1,X2) -> and(X1,X2) 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__and) = [0,1] pi(a__isNePal) = [] pi(and) = [0,1] pi(isNePal) = [] pi(a____#) = [0,1] pi(mark#) = 0 weight function: w0 = 1 w(isNePal) = w(and) = w(a__isNePal) = w(a__and) = w(tt) = w( nil) = 1 w(mark#) = w(a____#) = w(mark) = w(a____) = w(__) = 0 precedence: mark > a____ > a__isNePal ~ a__and ~ __ > mark# ~ a____# ~ isNePal ~ and ~ 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__and(tt(),X) -> mark(X) a__isNePal(__(I,__(P,I))) -> tt() mark(__(X1,X2)) -> a____(mark(X1),mark(X2)) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(isNePal(X)) -> a__isNePal(mark(X)) mark(nil()) -> nil() mark(tt()) -> tt() a____(X1,X2) -> __(X1,X2) a__and(X1,X2) -> and(X1,X2) a__isNePal(X) -> isNePal(X) Qed