Spahn rules of type theories

• Bart jacobs, Categorical logic and type theory

p.121-123 lists the following as the basic rules of simple type theory:

(1) identity

$$\frac{}{x:X\vdash x:X}$$

(2) function symbol

$$\frac{M_1:s_1\;\;\dots \;\; M_n:s_n}{F(M_1,\dots,M_n:s_{n+1})}$$

for $F:s_1,\dots,s_n\to s_{n+1}$ in $\Sigma$

(3) weakening

$$\frac{x_1:s_1,\;\dots\;x_n:s_n\vdash M:t}{x_1:s_1,\dots,x_n:s_n,x_{n+1},s_{n+1}\vdash M:t}$$

(4) contraction

$$\frac{\Gamma, x_n: s, x_{n+1}: s\vdash M:t}{\Gamma, x_n: s\vdash M[x_n / x_{n+1}]: t}$$

(5) exchange

$$\frac{\Gamma, x_i:s_i, x_{i+1}:s_{i+1},\Delta\vdash M: t}{\Gamma, x_i:s_{i+1}, x_{i+1}: s_i,\Delta\vdash M[x_i/x_{it1},x_{i+1}/x_i]:t}$$

where we omitted (except in (4) and (5)) the $\Gamma \vdash$ which we could prefix to every typing judgement.

From these rule one can derive a further important rule:

(6) substitution

$$\frac{x_n:s\vdash M:t\;\;\; N:s}{M[N/x_n]:t}$$