Definition. Typing rules for PCF [dt-003N]

\dfrac {n \in \mathbb {N}}{\Gamma \vdash n : \mathsf {nat}}\quad \dfrac {x : \tau \in \Gamma }{\Gamma \vdash x : \tau }\quad \dfrac {\Gamma , x : \tau _1 \vdash e : \tau _2 }{ \Gamma \vdash \lambda x : \tau _1.\ e : \tau _1 \rightarrow \tau _2}\quad \dfrac {\Gamma \vdash e_1 : \tau _1 \rightarrow \tau _2 \quad \Gamma \vdash e_2 : \tau _1}{\Gamma \vdash e_1\ e_2 : \tau _2}\\[2em] \dfrac { }{ \Gamma \vdash \mathsf {succ} : \mathsf {nat} \rightarrow \mathsf {nat}}\quad \dfrac { }{ \Gamma \vdash \mathsf {pred} : \mathsf {nat} \rightarrow \mathsf {nat}}\quad \dfrac { \Gamma \vdash e_1 : \mathsf {nat} \quad \Gamma \vdash e_2 : \tau \quad \Gamma \vdash e_3 : \tau }{ \Gamma \vdash \mathsf {ifz}\ e_1\ \mathsf {then}\ e_2\ \mathsf {else}\ e_3 : \tau }\quad \dfrac { \Gamma , x : \tau \vdash e : \tau }{\Gamma \vdash \mathsf {fix}\ x : \tau .\ e : \tau }