From: Enrico Tassi <enrico.tassi@inria.fr>
Date: Sun, 29 Jan 2006 19:54:12 +0000 (+0000)
Subject: chosmetic
X-Git-Tag: make_still_working~7743
X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=30bf48113bb0ea53f5300b3a6a2d129a9f4d73d9;p=helm.git

chosmetic
---

diff --git a/helm/papers/system_T/t.tex b/helm/papers/system_T/t.tex
index 5aece0cd6..9d5622d2b 100644
--- a/helm/papers/system_T/t.tex
+++ b/helm/papers/system_T/t.tex
@@ -172,14 +172,14 @@ informative part of the algorithms.
 say that ax:AX refers to the previous Axioms list...
 
 \[
-  (Proj)\hspace{0.2cm} \Gamma, x:A, \Delta \vdash x:A
+  (Proj)\hspace{0.1cm} \Gamma, x:A, \Delta \vdash x:A
   \hspace{2cm}
-  (Const)\hspace{0.2cm} \Gamma \vdash ax : AX
+  (Const)\hspace{0.1cm} \Gamma \vdash ax : AX
 \]
 
 \[ 
-   (\to_i)\hspace{0.2cm}\frac{\Gamma,x:A \vdash M:Q}{\Gamma \vdash \lambda x:A.M: A \to Q} \hspace{2cm}
-   (\to_e)\hspace{0.2cm}\frac{\Gamma \vdash M: A \to Q \hspace{1cm}\Gamma \vdash N: A}
+   (\to_i)\hspace{0.1cm}\frac{\Gamma,x:A \vdash M:Q}{\Gamma \vdash \lambda x:A.M: A \to Q} \hspace{2cm}
+   (\to_e)\hspace{0.1cm}\frac{\Gamma \vdash M: A \to Q \hspace{1cm}\Gamma \vdash N: A}
     {\Gamma \vdash M N: Q} 
 \]
 
@@ -193,9 +193,9 @@ say that ax:AX refers to the previous Axioms list...
 %\]
 
 \[ 
-   (\forall_i)\hspace{0.2cm}\frac{\Gamma \vdash M:P}{\Gamma \vdash 
+   (\forall_i)\hspace{0.1cm}\frac{\Gamma \vdash M:P}{\Gamma \vdash 
    \lambda x:\N.M: \forall x.P}(*) \hspace{2cm}
-   (\forall_e)\hspace{0.2cm}\frac{\Gamma \vdash M :\forall x.P}{\Gamma \vdash M t: P[t/x]} 
+   (\forall_e)\hspace{0.1cm}\frac{\Gamma \vdash M :\forall x.P}{\Gamma \vdash M t: P[t/x]} 
 \]
 
 
@@ -207,48 +207,78 @@ say that ax:AX refers to the previous Axioms list...
 
 \section{Extraction}
 
-The formulae to types translation function 
-$\sem{\cdot}$ takes in input formulae in HA and returns types in T.
-
-\begin{enumerate}
-\item $\sem{A} = \one$ if A is atomic
-\item $\sem{A \land B} = \sem{A}\times \sem{B}$
-\item $\sem{A \to B} = \sem{A}\to \sem{B}$
-\item $\sem{\forall x:\N.P} = \N \to \sem{P}$
-\item $\sem{\exists x:\N.P} = \N \times \sem{P}$
-\end{enumerate}
-
-definition.
-For any type T of system T $\canonical_T: \one \to T$  is inductively defined as follows:
-\begin{enumerate}
-\item $\canonical_\one = \lambda x:\one.x$
-\item $\canonical_N = \lambda x:\one.0$
-\item $\canonical_{U\times V} = \lambda x:\one.\pair{\canonical_{U} x}{\canonical_{V} x}$
-\item $\canonical_{U\to V} = \lambda x:\one.\lambda \_:U. \canonical_{V} x$
-\end{enumerate}
+The formulae to types translation function $\sem{\cdot}$, see table
+\ref{tab:formulae2types}, takes in input formulae in HA and returns 
+types in T. In table \ref{tab:structproof} we the proofs to terms
+function for structured proofs. Axiom translation is reported in table 
+\ref{tab:axioms}. In table \ref{tab:canonical} we define how the
+canoniac element is formed.
+
+\begin{table}[!h]
+\hrule\vspace{0.1cm}
+\begin{tabular}{p{0.47\textwidth}p{0.47\textwidth}}
+ $\sem{A} = \one$ if A is atomic &
+ $\sem{A \land B} = \sem{A}\times \sem{B}$ \\
+ $\sem{A \to B} = \sem{A}\to \sem{B}$ &
+ $\sem{\forall x:\N.P} = \N \to \sem{P}$ \\
+ $\sem{\exists x:\N.P} = \N \times \sem{P}$ &
+\end{tabular}\vspace{0.1cm}
+\hrule
+\caption{\label{tab:formulae2types}Formulae to types translation}
+\end{table}
+
+\begin{table}[!h]
+\hrule\vspace{0.1cm}
+\begin{tabular}{p{0.47\textwidth}p{0.47\textwidth}}
+ $\semT{M N} = \semT{M} \semT{N}$ &
+ $\semT{\lambda x:A.M} = \lambda x:\sem{A}.\semT{M}$ \\
+ $\semT{\lambda x:\N.M} = \lambda x:\N.\semT{M}$ & 
+ $\semT{M t} = \semT{M} \semT{t}$
+\end{tabular}\vspace{0.1cm}
+\hrule
+\caption{\label{tab:structproof}Structured proofs}
+\end{table}
+
+\begin{table}[!h]
+\hrule\vspace{0.1cm}
+\begin{tabular}{p{0.47\textwidth}p{0.47\textwidth}}
+ $\sem{fst} = \pi_1$&
+ $\sem{snd} = \pi_2$\\
+ $\sem{conj} = \lambda x:\sem{P}.\lambda y:\sem{Q}.\pair{x}{y}$&
+ $\sem{false\_ind} = \canonical_{\sem{Q}}$\\
+ $\sem{discriminate} = \lambda \_:\N.\lambda \_:\one.\star$&
+ $\sem{injS}= \lambda \_:\N. \lambda \_:\N.\lambda \_:\one.\star$\\
+ $\sem{plus\_O} = \sem{times\_O} = \lambda \_:\N.\star$&
+ $\sem{nat\_ind} = R$ \\
+ \multicolumn{2}{l}{
+   $\sem{plus\_S} = \sem{times\_S} = \lambda \_:\N. \lambda \_:\N.\star$
+ }\\
+ \multicolumn{2}{l}{
+   $\sem{ex\_intro} = \lambda x:\N.\lambda f:\sem{P}.\pair{x}{f}$
+ }\\
+ \multicolumn{2}{l}{
+   $\sem{ex\_ind} = 
+   \lambda f:(\N \to \sem{P} \to \sem{Q}).
+   \lambda p:\N\times \sem{P}.f~(fst~p)~(snd~p)$. 
+ }
+\end{tabular}\vspace{0.1cm}
+\hrule
+\caption{\label{tab:axioms}Axioms translation}
+\end{table}
+
+\begin{table}[!h]
+\hrule\vspace{0.1cm}
+\begin{tabular}{p{0.47\textwidth}p{0.47\textwidth}}
+ $\canonical_\one = \lambda x:\one.x$ &
+ $\canonical_N = \lambda x:\one.0$ \\
+ $\canonical_{U\times V} = \lambda x:\one.\pair{\canonical_{U}
+ x}{\canonical_{V} x}$ &
+ $\canonical_{U\to V} = \lambda x:\one.\lambda \_:U. \canonical_{V} x$
+\end{tabular}\vspace{0.1cm}
+\hrule
+\caption{\label{tab:canonical}Canonical element}
+\end{table}
 
-\begin{itemize}
-\item $\sem{nat\_ind} = R$
-\item $\sem{ex\_ind} = (\lambda f:(\N \to \sem{P} \to \sem{Q}).
-\lambda p:\N\times \sem{P}.f (fst \,p) (snd \,p)$. 
-\item $\sem{ex\_intro} = \lambda x:\N.\lambda f:\sem{P}.\pair{x}{f}$
-\item $\sem{fst} = \pi_1$
-\item $\sem{snd} = \pi_2$
-\item $\sem{conj} = \lambda x:\sem{P}.\lambda y:\sem{Q}.\pair{x}{y}$
-\item $\sem{false\_ind} = \canonical_{\sem{Q}}$
-\item $\sem{discriminate} = \lambda \_:\N.\lambda \_:\one.\star$
-\item $\sem{injS}= \lambda \_:\N. \lambda \_:\N.\lambda \_:\one.\star$
-\item $\sem{plus\_O} = \sem{times\_O} = \lambda \_:\N.\star$
-\item $\sem{plus\_S} = \sem{times_S} = \lambda \_:\N. \lambda \_:\N.\star$
-\end{itemize}
-
-In the case of structured proofs:
-\begin{itemize}
-\item $\semT{M N} = \semT{M} \semT{N}$
-\item $\semT{\lambda x:A.M} = \lambda x:\sem{A}.\semT{M}$
-\item $\semT{\lambda x:\N.M} = \lambda x:\N.\semT{M}$
-\item $\semT{M t} = \semT{M} \semT{t}$
-\end{itemize}
 
 \section{Realizability}
 The realizability relation is a relation $f \R P$ where $f: \sem{P}$, and