PN.2 formula workbench

$$K(x,y,z)=(x\odot y)\odot z-x\odot(y\odot z)=r_L-r_R$$

$$\Delta_S=|S(r_L)-S(r_R)|,\quad\Delta_D=|\mathfrak D(b_L)-\mathfrak D(b_R)|,\quad\Delta_R=\|K\|$$

$$e_X=\max_{b\in\{L,R\}}d_X(\widehat x,x_b)$$

$$\Delta_S(u)\ge c\|K(u)\|,\qquad c>0,\quad u\in\Omega$$

$$e_Se_D\ge\frac{\Delta_S\Delta_D}{4}$$

$$e_Re_D\ge\frac{\|K\|\Delta_D}{4}$$

$$e_Se_D\ge\frac{c\|K\|\Delta_D}{4}$$

$$\tau=\operatorname{TV}(P_L,P_R)=\frac12\int|p-q|\,d\mu,\quad\mu=P_L+P_R$$

$$\nu(\mathcal Y)=\int\min(p,q)\,d\mu=1-\tau$$

$$\mathcal R_X=\max_{b\in\{L,R\}}\mathbb E_b\,d_X(\widehat X,x_b)$$

$$\mathcal R_S\ge\frac{\Delta_S}{2}(1-\tau)$$

$$\mathcal R_D\ge\frac{\Delta_D}{2}(1-\tau)$$

$$\mathcal R_R\ge\frac{\|K\|}{2}(1-\tau)$$

$$\mathcal R_S\mathcal R_D\ge\frac{\Delta_S\Delta_D}{4}(1-\tau)^2$$

$$\mathcal R_R\mathcal R_D\ge\frac{\|K\|\Delta_D}{4}(1-\tau)^2$$

$$\mathcal R_S\mathcal R_D\ge\frac{c\|K\|\Delta_D}{4}(1-\tau)^2$$