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: Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [isNePal](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [U12](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [U11](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [a__isNePal](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [a__U12](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [a__U11](x0) = [0 0 0]x0 [0 0 0] , [0] [tt] = [0] [0], [1] [nil] = [0] [0], [1 0 0] [0] [mark](x0) = [1 0 0]x0 + [0] [0 0 0] [1], [1 0 0] [1 0 0] [0] [a____](x0, x1) = [1 0 0]x0 + [1 0 0]x1 + [0] [0 0 0] [0 0 0] [1], [1 0 0] [1 0 0] [__](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] orientation: [1 0 0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [0] a____(__(X,Y),Z) = [1 0 0]X + [1 0 0]Y + [1 0 0]Z + [0] >= [1 0 0]X + [1 0 0]Y + [1 0 0]Z + [0] = a____(mark(X),a____(mark(Y),mark(Z))) [0 0 0] [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [0 0 0] [1] [1 0 0] [1] [1 0 0] [0] a____(X,nil()) = [1 0 0]X + [1] >= [1 0 0]X + [0] = mark(X) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1] [1 0 0] [0] a____(nil(),X) = [1 0 0]X + [1] >= [1 0 0]X + [0] = mark(X) [0 0 0] [1] [0 0 0] [1] [0] [0] a__U11(tt()) = [0] >= [0] = a__U12(tt()) [0] [0] [0] [0] a__U12(tt()) = [0] >= [0] = tt() [0] [0] [2 0 0] [1 0 0] [0] a__isNePal(__(I,__(P,I))) = [0 0 0]I + [0 0 0]P >= [0] = a__U11(tt()) [0 0 0] [0 0 0] [0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] mark(__(X1,X2)) = [1 0 0]X1 + [1 0 0]X2 + [0] >= [1 0 0]X1 + [1 0 0]X2 + [0] = a____(mark(X1),mark(X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 0] [0] [1 0 0] mark(U11(X)) = [1 0 0]X + [0] >= [0 0 0]X = a__U11(mark(X)) [0 0 0] [1] [0 0 0] [1 0 0] [0] [1 0 0] [0] mark(U12(X)) = [1 0 0]X + [0] >= [0 0 0]X + [0] = a__U12(mark(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] mark(isNePal(X)) = [1 0 0]X + [0] >= [0 0 0]X = a__isNePal(mark(X)) [0 0 0] [1] [0 0 0] [1] [1] mark(nil()) = [1] >= [0] = nil() [1] [0] [0] [0] mark(tt()) = [0] >= [0] = tt() [1] [0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] a____(X1,X2) = [1 0 0]X1 + [1 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 = __(X1,X2) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1 0 0] [1 0 0] a__U11(X) = [0 0 0]X >= [0 0 0]X = U11(X) [0 0 0] [0 0 0] [1 0 0] [1 0 0] a__U12(X) = [0 0 0]X >= [0 0 0]X = U12(X) [0 0 1] [0 0 0] [1 0 0] [1 0 0] a__isNePal(X) = [0 0 0]X >= [0 0 0]X = isNePal(X) [0 0 0] [0 0 0] problem: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) 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) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [isNePal](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [1] [U12](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [1 0 0] [1] [U11](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [a__isNePal](x0) = x0 , [1 0 0] [1] [a__U12](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [1 0 0] [1] [a__U11](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [0] [tt] = [0] [0], [0] [nil] = [0] [0], [1 0 0] [mark](x0) = [0 1 0]x0 [1 1 0] , [1 1 0] [1 1 0] [0] [a____](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [1] [1 1 0] [0 1 0] [0], [1 1 0] [1 1 0] [0] [__](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [1] [0 0 0] [0 1 0] [0] orientation: [1 1 0] [1 1 0] [1 1 0] [1] [1 1 0] [1 1 0] [1 1 0] [1] a____(__(X,Y),Z) = [0 0 0]X + [0 0 0]Y + [0 0 0]Z + [1] >= [0 0 0]X + [0 0 0]Y + [0 0 0]Z + [1] = a____(mark(X),a____(mark(Y),mark(Z))) [1 1 0] [1 1 0] [0 1 0] [1] [1 1 0] [0 0 0] [0 0 0] [1] [1] [1] a__U11(tt()) = [0] >= [0] = a__U12(tt()) [0] [0] [1] [0] a__U12(tt()) = [0] >= [0] = tt() [0] [0] [2 2 0] [1 1 0] [1] [1] a__isNePal(__(I,__(P,I))) = [0 0 0]I + [0 0 0]P + [1] >= [0] = a__U11(tt()) [0 0 0] [0 0 0] [1] [0] [1 1 0] [1 1 0] [0] [1 1 0] [1 1 0] [0] mark(__(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = a____(mark(X1),mark(X2)) [1 1 0] [1 1 0] [1] [1 1 0] [0 1 0] [0] [1 0 0] [1] [1 0 0] [1] mark(U11(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = a__U11(mark(X)) [1 0 0] [1] [0 0 0] [0] [1 0 0] [1] [1 0 0] [1] mark(U12(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = a__U12(mark(X)) [1 0 0] [1] [0 0 0] [0] [1 0 0] [1 0 0] mark(isNePal(X)) = [0 1 0]X >= [0 1 0]X = a__isNePal(mark(X)) [1 1 0] [1 1 0] [0] [0] mark(nil()) = [0] >= [0] = nil() [0] [0] [0] [0] mark(tt()) = [0] >= [0] = tt() [0] [0] [1 1 0] [1 1 0] [0] [1 1 0] [1 1 0] [0] a____(X1,X2) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = __(X1,X2) [1 1 0] [0 1 0] [0] [0 0 0] [0 1 0] [0] [1 0 0] [1] [1 0 0] [1] a__U11(X) = [0 0 0]X + [0] >= [0 0 0]X + [0] = U11(X) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1] [1 0 0] [1] a__U12(X) = [0 0 0]X + [0] >= [0 0 0]X + [0] = U12(X) [0 0 0] [0] [0 0 0] [0] [1 0 0] a__isNePal(X) = X >= [0 1 0]X = isNePal(X) [0 0 0] problem: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) a__U11(tt()) -> a__U12(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) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [isNePal](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [U12](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [U11](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [a__isNePal](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [a__U12](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [a__U11](x0) = [0 0 0]x0 [0 0 0] , [0] [tt] = [0] [0], [0] [nil] = [0] [0], [0] [mark](x0) = x0 + [0] [1], [1 1 0] [1 1 0] [0] [a____](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [1] [1 1 0] [0 0 1] [0], [1 1 0] [1 1 0] [0] [__](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [1] [1 1 0] [0 0 1] [0] orientation: [1 1 0] [1 1 0] [1 1 0] [1] [1 1 0] [1 1 0] [1 1 0] [1] a____(__(X,Y),Z) = [0 0 0]X + [0 0 0]Y + [0 0 0]Z + [1] >= [0 0 0]X + [0 0 0]Y + [0 0 0]Z + [1] = a____(mark(X),a____(mark(Y),mark(Z))) [1 1 0] [1 1 0] [0 0 1] [1] [1 1 0] [1 1 0] [0 0 1] [1] [0] [0] a__U11(tt()) = [0] >= [0] = a__U12(tt()) [0] [0] [2 2 0] [1 1 0] [1] [0] a__isNePal(__(I,__(P,I))) = [0 0 0]I + [0 0 0]P + [0] >= [0] = a__U11(tt()) [0 0 0] [0 0 0] [0] [0] [1 1 0] [1 1 0] [0] [1 1 0] [1 1 0] [0] mark(__(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = a____(mark(X1),mark(X2)) [1 1 0] [0 0 1] [1] [1 1 0] [0 0 1] [1] [1 0 0] [0] [1 0 0] mark(U11(X)) = [0 0 0]X + [0] >= [0 0 0]X = a__U11(mark(X)) [0 0 0] [1] [0 0 0] [1 0 0] [0] [1 0 0] mark(U12(X)) = [0 0 0]X + [0] >= [0 0 0]X = a__U12(mark(X)) [0 0 0] [1] [0 0 0] [1 0 0] [0] [1 0 0] mark(isNePal(X)) = [0 0 0]X + [0] >= [0 0 0]X = a__isNePal(mark(X)) [0 0 0] [1] [0 0 0] [0] [0] mark(nil()) = [0] >= [0] = nil() [1] [0] [0] [0] mark(tt()) = [0] >= [0] = tt() [1] [0] [1 1 0] [1 1 0] [0] [1 1 0] [1 1 0] [0] a____(X1,X2) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = __(X1,X2) [1 1 0] [0 0 1] [0] [1 1 0] [0 0 1] [0] [1 0 0] [1 0 0] a__U11(X) = [0 0 0]X >= [0 0 0]X = U11(X) [0 0 0] [0 0 0] [1 0 0] [1 0 0] a__U12(X) = [0 0 0]X >= [0 0 0]X = U12(X) [0 0 0] [0 0 0] [1 0 0] [1 0 0] a__isNePal(X) = [0 0 0]X >= [0 0 0]X = isNePal(X) [0 0 0] [0 0 0] problem: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) a__U11(tt()) -> a__U12(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) Matrix Interpretation Processor: dim=1 interpretation: [isNePal](x0) = x0 + 4, [U12](x0) = x0, [U11](x0) = 2x0 + 4, [a__isNePal](x0) = x0 + 4, [a__U12](x0) = x0, [a__U11](x0) = 2x0 + 4, [tt] = 4, [nil] = 2, [mark](x0) = x0, [a____](x0, x1) = x0 + x1 + 4, [__](x0, x1) = x0 + x1 + 4 orientation: a____(__(X,Y),Z) = X + Y + Z + 8 >= X + Y + Z + 8 = a____(mark(X),a____(mark(Y),mark(Z))) a__U11(tt()) = 12 >= 4 = a__U12(tt()) mark(__(X1,X2)) = X1 + X2 + 4 >= X1 + X2 + 4 = a____(mark(X1),mark(X2)) mark(U11(X)) = 2X + 4 >= 2X + 4 = a__U11(mark(X)) mark(U12(X)) = X >= X = a__U12(mark(X)) mark(isNePal(X)) = X + 4 >= X + 4 = a__isNePal(mark(X)) mark(nil()) = 2 >= 2 = nil() mark(tt()) = 4 >= 4 = tt() a____(X1,X2) = X1 + X2 + 4 >= X1 + X2 + 4 = __(X1,X2) a__U11(X) = 2X + 4 >= 2X + 4 = U11(X) a__U12(X) = X >= X = U12(X) a__isNePal(X) = X + 4 >= X + 4 = isNePal(X) problem: a____(__(X,Y),Z) -> a____(mark(X),a____(mark(Y),mark(Z))) 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: weight function: w0 = 1 w(isNePal) = w(U12) = w(U11) = w(a__isNePal) = w(a__U12) = w(a__U11) = w( tt) = w(nil) = 1 w(mark) = w(a____) = w(__) = 0 precedence: mark > a__isNePal ~ a__U12 ~ a__U11 ~ a____ > isNePal ~ U12 ~ U11 ~ tt ~ nil ~ __ problem: Qed