Define recognizers for the following languages:
$L_{0} = \{ w \in \{0,1\}^* \mid \forall w'.\ \forall a \in \{0,1\}.\ w = w' a \Rightarrow a \neq 0 \}$ $L_1 = \{ w \in \{0,1\}^* \mid \forall x,y.\ w = x 0 y \Rightarrow |y|_0 = 0 \}$ $L_2 = \{ w \in \{0,1,2\}^* \mid \forall x,y.\ w = x 1 y \Rightarrow |y|_0 = 0 \}$ $L_3 = \{ w \mid w = x 0 y \Rightarrow \exists z .\ y = 1 1 z \}$ $L_4 = \{ w \in \{0,1\}^* \mid |w|_0 \geq |w|_1 \}$ $L_5 = \{ w \in \{0,1\}^* \mid |w|_0 = |w|_1 \}$ $L_6 = \{ 0^n 1^n \mid n \geq 0 \}$ $L_7 = \{ 0^n 1 0^n \mid n \geq 0 \}$ $L_8 = \{ w \in \{0,1,2\}^* \mid \forall x,y.\ w = x 1 y \Rightarrow |y|_2 = |y|_0 \}$ $L_9 = \{ w \mid \exists w_0,w_1 .\ w = w_0 w_1 \land |w_0|_0 = |w_1|_1 \land |w_0|_1 = 0 = |w_1|_0 \}$