)$ such that $$0=v(t_0,x_0)-\varphi(t_0,x_0)=\underset{(t,x)\in[0,T)\times \R^d}{\min}(v-\varphi)(t,x),$$ we have $$-\partial_tv(t_0,x_0)-\hat h(t_0,x_0,\varphi(t_0,x_0),D\varphi(t_0,x_0),\Kc \varphi(t_0,x_0,\cdot),D^0\varphi(t_0,x_0),\varphi(t_0,x_0+\cdot))\