Spahn rules of type theories (Rev #2, changes)

Showing changes from revision #1 to #2: Added | ~~Removed~~ | ~~Chan~~ged

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

(1) identity

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

(2) function symbol

\frac{M_{1} : {x s}_{1} \dots M_{n} : {x s}_{n}}{F (M_{1}, \dots, M_{n} : {x s}_{n + 1})}

~~\frac{M_1:x_1\;\;\dots~~ \frac{M_1:s_1\;\;\dots \;\; ~~M_n:x_n}{F(M_1,\dots,M_n:x_{n+1}}~~ M_n:s_n}{F(M_1,\dots,M_n:s_{n+1})}

for $F : {x s}_{1}, \dots, {x s}_{n} \to {x s}_{n + 1}$ ~~F:x_1,\dots,x_n\to~~ F:s_1,\dots,s_n\to ~~x_{n+1}~~ 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

~~where we omitted the $\Gamma \vdash$ which we could prefix to every typing judgement.~~

\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}

Revision on February 26, 2013 at 02:53:29 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.