\documentclass[a4paper]{article}
\pagestyle{headings}
%\usepackage{graphicx}
-\usepackage{amssymb,amsmath,mathrsfs}
+\usepackage{amssymb,amsmath,mathrsfs,stmaryrd,amsthm}
%\usepackage{hyperref}
%\usepackage{picins}
-\newcommand{\sem}[1]{[\![ #1 ]\!]}
-\newcommand{\R}{\;\mathscr{R}\;}
+\newcommand{\semT}[1]{\ensuremath{\llbracket #1 \rrbracket}}
+\newcommand{\sem}[1]{\llbracket \ensuremath{#1} \rrbracket}
+\newcommand{\pair}[2]{<\!#1,#2\!>}
+\newcommand{\canonical}{\bot}
+\newcommand{\R}{~\mathscr{R}~}
\newcommand{\N}{\,\mathbb{N}\,}
+\newcommand{\B}{\,\mathbb{B}\,}
\newcommand{\NT}{\,\mathbb{N}\,}
\newcommand{\NH}{\,\mathbb{N}\,}
\renewcommand{\star}{\ast}
-\newcommand{\one}{\mathbb{1}}
-\renewcommand{\times}{\cdot}
-\title{A gentle approach to program extraction and realizability}
+\renewcommand{\vec}{\overrightarrow}
+\newcommand{\one}{{\bf 1}}
+\newcommand{\mult}{\cdot}
+\newcommand{\ind}{Ind(X)}
+\newcommand{\indP}{Ind(\vec{P}~|~X)}
+\newcommand{\Xind}{\ensuremath{X_{ind}}}
+\newcommand{\XindP}{\ensuremath{X_{ind}}}
+\renewcommand{\|}{\ensuremath{\quad | \quad}}
+\newcommand{\triUP}{\ensuremath{\Delta}}
+\newcommand{\triDOWN}{\ensuremath{\nabla}}
+\newcommand{\Rx}{\ensuremath{R_X}}
+
+\newtheorem{thm}{Theorem}[subsection]
+
+\title{Modified Realizability and Inductive Types}
\author{...}
...
\end{abstract}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Introduction}
+The characterization of the provable recursive functions of
+Peano Arithmetic as the terms of system T is a well known
+result of G\"odel \cite{Godel58,Godel90}. Although several authors acknowledge
+that the functional interpretation of the Dialectica paper
+is not among the major achievements of the author (see e.g. \cite{Girard87}),
+the result has been extensively investigated and there is a wide
+literature on the
+topic (see e.g. \cite{Troelstra,HS86,Girard87}, just to mention textbooks,
+and the bibliography therein).
+
+A different, more neglected, but for many respects much more
+direct relation between Peano (or Heyting) Arithmetics and
+G\"odel System T is provided
+by the so called {\em modified realizability}. Modified realizability
+was first introduced by Kreisel in \cite{Kreisel59} - although it will take you
+a bit of effort to recognize it in the few lines of paragraph 3.52 -
+and later in \cite{Kreisel62} under the name of generalized realizability.
+The name of modified realizability seems to be due to Troelstra
+\cite{Troelstra}
+- who contested Kreisel's name but unfortunately failed in proposing
+a valid alternative; we shall reluctantly adopt this latter name
+to avoid further confusion. Modified realizability is a typed variant of
+realizability, essentially providing interpretations
+of $HA^{\omega}$ into itself: each theorem is realized by a typed function
+of system T, that also gives the actual computational content extracted
+from the proof.
+In spite of the simplicity and the elegance of the proof, it is extremely
+difficult to find a modern discussion of this result; the most recent
+exposition we are aware of is in the encyclopedic work by
+Troelstra \cite{Troelstra} (pp.213-229) going back to thirty years ago.
+Even modern introductory books
+to Type Theory and Proof Theory devoting much space to system T
+such as \cite{GLT} and \cite{TS} surprisingly leave out this simple and
+illuminating result. Both the previous textbooks
+prefer to focus on higher order arithmetics and its relation with
+Girard's System $F$ \cite{Girard86}, but the technical complexity and
+the didactical value of the two proofs is not comparable: when you
+prove that the Induction Principle is realized by the recursor $R$
+of system $T$ you catch a sudden gleam of understanding in the
+students eyes; usually, the same does not happen when you show, say,
+that the ``forgetful'' interpretation of the higher order predicate defining
+the natural numbers is the system $F$ encoding
+$\forall X.(X\to X) \to X \to X$ of $\N$.
+Moreover, after a first period of enthusiasm, the impredicative
+encoding of inductive types in Logical Frameworks has shown several
+problems and limitations (see e.g. \cite{Werner} pp.24-25) mostly
+solved by assuming inductive types as a primitive logical notion
+(leading e.g. form the Calculus of Constructions to the Calculus
+of Inductive Constructions - CIC). Even the extraction algorithm of
+CIC, strictly based on realizability principles, and in a first time
+still oriented towards System F \cite{Paulin87,Paulin89} has been
+recently rewritten \cite{Letouzey04}
+to take advantage of concrete types and pattern matching of ML-like
+languages. Unfortunately, systems like the Calculus of Inductive
+Constructions are so complex, from the logical point of view, to
+substantially prevent a really neat theoretical exposition (at present,
+it does not
+even exists a truly complete consistency proofs covering all aspects
+of such systems); moreover, not everybody may be interested in all the features
+offered by these frameworks, from polymorphism to types depending on
+proofs. Our program is to restart the analysis of logical systems with
+primitive inductive types in a smooth way, starting form first order
+logic and adding little by little small bits of logical power.
+This paper is the first step in this direction.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{G\"odel system T}
+We shall use a variant of system T with three atomic types $\N$ (natural
+numbers), $\B$ (booleans) and $\one$ (a terminal object), and two binary
+type constructors $\times$ (product) and $\to$ (arrow type).
+
+The terms of the language comprise the usual simply typed lambda terms
+with explicit pairs, plus the following additional constants:
+\begin{itemize}
+\item $*:\one$,
+\item $true: \B$, $false:\B$, $D:A\to A \to \B \to A$
+\item $O:\N$, $S:\N \to \N$, $R:A \to (A \to \N \to A) \to \N \to A$,
+\end{itemize}
+Redexes comprise $\beta$-reduction
+\[(\beta)~~ \lambda x:U.M ~ N \leadsto M[N/x]\]
+projections
+
+\[(\pi_1)~~fst \pair{M}{N} \leadsto M\\ \hspace{.6cm} (\pi_2)~~ snd \pair{M}{N}
+\leadsto N \]
+and the following type specific reductions:
+\[(D_{true})~~\\D~M~N~ true \leadsto M \hspace{.6cm}
+ (D_{false})~~ D~M~N~false \leadsto N \]
+\[(R_0)~~\\R~M~F~ 0 \leadsto M \hspace{.6cm}
+ (R_S)~~ R~M~F~(S~n) \leadsto F~n~(R~M~F~n) \]
+\[(*)~~ M \leadsto * \]
+where (*) holds for any $M$ of type $\one$.
+
+Note that using the well known isomorpshims
+$\one \to A \cong A$, $A \to \one \cong \one$
+and $A \times \one \cong A \cong \one\times A$ (see \cite{AL91}, pp.231-239)
+we may always get rid of $\one$ (apart the trivial case).
+The terminal object does not play a major role in our treatment, but
+it allows to extract better algorithms. In particular we use it
+to realize atomic proposition, and stripping out the terminal object using
+the above isomorphisms gives a simple way of just keeping the truly
+informative part of the algorithms.
+
+
+
\section{Heyting's arithmetics}
{\bf Axioms}
\begin{itemize}
-
\item $nat\_ind: P(0) \to (\forall x.P(x) \to P(S(x))) \to \forall x.P(x)$
\item $ex\_ind: (\forall x.P(x) \to Q) \to \exists x.P(x) \to Q$
\item $ex\_intro: \forall x.(P \to \exists x.P)$
\item $injS: \forall x,y.S(x) = S(y) \to x=y$
\item $plus\_O:\forall x.x+0=x$
\item $plus\_S:\forall x,y.x+S(y)=S(x+y)$
-\item $times\_O:\forall x.x\times0=0$
-\item $times\_S:\forall x,y.x\times S(y)=x+(x\times y)$
+\item $times\_O:\forall x.x\mult0=0$
+\item $times\_S:\forall x,y.x\mult S(y)=x+(x\mult y)$
\end{itemize}
\noindent
{\bf Inference Rules}
+say that ax:AX refers to the previous Axioms list...
+
+\[
+ (Proj)\hspace{0.2cm} \Gamma, x:A, \Delta \vdash x:A
+ \hspace{2cm}
+ (Const)\hspace{0.2cm} \Gamma \vdash ax : AX
+\]
+
\[
- (\to_i)\frac{\Gamma,x:A \vdash M:Q}{\Gamma \vdash \lambda x:A.M: A \to Q} \hspace{2cm}
- (\to_e)\frac{\Gamma \vdash M: A \to Q \hspace{1cm}\Gamma \vdash N: A}
+ (\to_i)\hspace{0.2cm}\frac{\Gamma,x:A \vdash M:Q}{\Gamma \vdash \lambda x:A.M: A \to Q} \hspace{2cm}
+ (\to_e)\hspace{0.2cm}\frac{\Gamma \vdash M: A \to Q \hspace{1cm}\Gamma \vdash N: A}
{\Gamma \vdash M N: Q}
\]
%\[
% (\land_i)\frac{\Gamma \vdash M:A \hspace{1cm}\Gamma \vdash N:B}
-% {\Gamma \vdash <M,N> : A \land B}
+% {\Gamma \vdash \pair{M}{N} : A \land B}
%\hspace{2cm}
% (\land_{el})\frac{\Gamma \vdash A \land B}{\Gamma \vdash A}
%\hspace{2cm}
%\]
\[
- (\forall_i)\frac{\Gamma \vdash M:P}{\Gamma \vdash
+ (\forall_i)\hspace{0.2cm}\frac{\Gamma \vdash M:P}{\Gamma \vdash
\lambda x:\N.M: \forall x.P}(*) \hspace{2cm}
- (\forall_e)\frac{\Gamma \vdash M :\forall x.P}{\Gamma \vdash M t: P[t/x]}
+ (\forall_e)\hspace{0.2cm}\frac{\Gamma \vdash M :\forall x.P}{\Gamma \vdash M t: P[t/x]}
\]
\section{Extraction}
-\begin{itemize}
-\item $\sem{A} = 1$ if A is atomic
+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 $\sem{\forall x.P} = \N \to \sem{P}$
-\item $\sem{\exists x.P} = \N \times \sem{P}$
-\end{itemize}
+\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 $\bot_T: 1 \to T$ is inuctively defined as follows:
+For any type T of system T $\canonical_T: \one \to T$ is inductively defined as follows:
\begin{enumerate}
-\item $\bot_1 = \lambda x:1.x$
-\item $\bot_N = \lambda x:1.0$
-\item $\bot_{U\times V} = \lambda x:1.<\bot_{U} x,\bot_{V} x>$
-\item $\bot_{U\to V} = \lambda x:1.\lambda \_:U. \bot_{V} x$
+\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}
\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 $\sem{ex\_intro} = \lambda x:\N.\lambda f:\sem{P}.<x,f>$
+\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 $\sem{conj} = \lambda x:\sem{P}.\lambda y:\sem{Q}.<x,y>$
-\item $\sem{false\_ind} = \bot_{\sem{Q}}$
-\item $\sem{discriminate} = \lambda \_:\N.\lambda \_:1.\star$
-\item $\sem{injS}= \lambda \_:\N. \lambda \_:\N.\lambda \_:1.\star$
+\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 $\sem{M N} = \sem{M} \sem{N}$
-\item $\sem{\lambda x:A.M} = \lambda x:\sem{A}.\sem{M}$
-\item $\sem{\lambda x:\N.M} = \lambda x:\N.\sem{M}$
-\item $\sem{M t} = \sem{M} \sem{t}$
+\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 $f: \sem{P}$.
+The realizability relation is a relation $f \R P$ where $f: \sem{P}$, and
+$P$ is a closed formula.
In particular:
\begin{itemize}
\item $\neg (\star \R \bot)$
\item $* \R (t_1=t_2)$ iff $t_1=t_2$ is true ...
-\item $<f,g> \R (P\land Q)$ iff $f \R P$ and $g \R Q$
+\item $\pair{f}{g} \R (P\land Q)$ iff $f \R P$ and $g \R Q$
\item $f \R (P\to Q)$ iff for any $m$ such that $m \R P$, $(f \,m) \R Q$
\item $f \R (\forall x.P)$ iff for any natural number $n$ $(f n) \R P[\underline{n}/x]$
-\item $<n,g> \R (\exists x.P)$ iff $g \R P[\underline{n}/x]$
+\item $\pair{n}{g}\R (\exists x.P)$ iff $g \R P[\underline{n}/x]$
\end{itemize}
+%We need to generalize the notion of realizability to sequents.
+%Given a sequent $B_1, \ldots, B_n \vdash A$ with free variables in
+%$\vec{x} = x_1,\ldots, x_m$, we say that $f \R B1, \ldots, B_n \vdash A$ iff
+%forall natural numbers $n_1, \ldots, n_m$,
+%if forall $i \in {1,\ldots,n}$
+%$m_i \R B_i[\vec{\underline{n}}/\vec{x}]$ then
+%$$f <m_1, \ldots, m_n> \R A[\vec{\underline{n}}/\vec{x}]$$.
+%
+\noindent
+We need to generalize the notion of realizability to sequents.\\
+Let $\vec{x} = FV_{\N}( B_1, \ldots, B_n, P)$ a vector of variables of type
+$\N$ that occur free in $B_1, \ldots, B_n, P$. Let $\vec{b:B}$ the vector
+$b_1:B_1, \ldots, b_n:B_n$.\\
+We say that $f \R B_1, \ldots, B_n \vdash A:P$ iff
+$$\lambda \vec{x:\N}. \lambda \vec{b:B}.f \R
+\forall \vec{x}. B_1 \to \ldots \to B_n \to P$$
+Note that $\forall \vec{x}. B_1 \to \ldots \to B_n \to P$ is a closed formula,
+so we can use the previous definition of realizability on it.
\noindent
We proceed to prove that all axioms $ax:Ax$ are realized by $\sem{ax}$.
\to Q) \to (\exists x:(P x)) \to Q$$ Following the definition of $\R$ we have
to prove that given\\ $f~\R~\forall~x:((P~x)~\to~Q)$ and
$p~\R~\exists~x:(P~x)$, then $\underline{ex\_ind}~f~p \R Q$.\\
- $p$ is a couple $<n_p,g_p>$ such that $g_p \R P[\underline{n_p}/x]$, while
+ $p$ is a couple $\pair{n_p}{g_p}$ such that $g_p \R P[\underline{n_p}/x]$, while
$f$ is a function such that forall $n$ and for all $m \R P[\underline{n}/x]$
then $f~n~m \R Q$ (note that $x$ is not free in $Q$ so $[\underline{n}/x]$
affects only $P$).\\
- Exapanding the definition of $\underline{ex\_ind}$, $fst$
+ Expanding the definition of $\underline{ex\_ind}$, $fst$
and $snd$ we obtain $f~n_p~g_p$ that we know is in relation $\R$ with $Q$
since $g_p \R P[\underline{n_p}/x]$.
\item $ex\_intro$.
We must prove that
- $$\lambda x:\N.\lambda f:\sem{P}.<x,f> \R \forall x.(P\to\exists x.P(x)$$
- that leads to prove that foreach n
+ $$\lambda x:\N.\lambda f:\sem{P}.\pair{x}{f} \R \forall x.(P\to\exists x.P(x)$$
+ that leads to prove that for each n
$\underline{ex\_into}~n \R (P\to\exists x.P(x))[\underline{n}/x]$.\\
Evaluating the substitution we have
$\underline{ex\_into}~n \R (P[\underline{n}/x]\to\exists x.P(x))$.\\
Again by definition of $\R$ we have to prove that given a
$m \R P[\underline{n}/x]$ then $\underline{ex\_into}~n~m \R \exists x.P(x)$.
Expanding the definition of $\underline{ex\_intro}$ we have
- $<n,m> \R \exists x.P(x)$ that is true since $m \R P[\underline{n}/x]$.
+ $\pair{n}{m} \R \exists x.P(x)$ that is true since $m \R P[\underline{n}/x]$.
\item $fst$.
- Wehave to prove that $\pi_1 \R P \land Q \to P$, that is equal to proving
+ We have to prove that $\pi_1 \R P \land Q \to P$, that is equal to proving
that for each $m \R P \land Q$ then $\pi_1~m \R P$ .
- $m$ must be a couple $<f_m,g_m>$ such that $f_m \R P$ and $g_m \R Q$.
+ $m$ must be a couple $\pair{f_m}{g_m}$ such that $f_m \R P$ and $g_m \R Q$.
So we conclude that $\pi_1~m$ reduces to $f_m$ that is in relation $\R$
with $P$.
\item $conj$.
We have to prove that
- $$\lambda x:\sem{P}. \lambda y:\sem{Q}.<x,y> \R P \to Q \to P \land Q$$
+ $$\lambda x:\sem{P}. \lambda y:\sem{Q}.\pair{x}{y}\R P \to Q \to P \land Q$$
Following the definition of $\R$ we have to show that
- foreach $m \R P$ and foreach $n \R Q$ then
- $(\lambda x:\sem{P}. \lambda y:\sem{Q}.<x,y>)~m~n \R P \land Q$.\\
- This is the same of $<m,n> \R P \land Q$ that is verified since
+ for each $m \R P$ and for each $n \R Q$ then
+ $(\lambda x:\sem{P}. \lambda y:\sem{Q}.\pair{x}{y})~m~n \R P \land Q$.\\
+ This is the same of $\pair{m}{n} \R P \land Q$ that is verified since
$m \R P$ and $n \R Q$.
\item $injS$.
We have to prove that for each $n_1$ and $n_2$\\
- $\lambda \_:\N. \lambda \_:\N.\lambda \_:1.*~n_1~n_2 \R
+ $\lambda \_:\N. \lambda \_:\N.\lambda \_:\one.*~n_1~n_2 \R
(S(x)=S(y)\to x=y)[n_1/x][n_2/y]$.\\
We assume that $m \R S(n_1)=S(n_2)$ and we have to show that
- $\lambda \_:\N. \lambda \_:\N.\lambda \_:1.*~n_1~n_2~m$ that reduces to
+ $\lambda \_:\N. \lambda \_:\N.\lambda \_:\one.*~n_1~n_2~m$ that reduces to
$*$ is in relation $\R$ with $n_1=n_2$. Since in the standard model of
natural numbers $S(n_1)=S(n_2)$ implies $n_1=n_2$ we have that
$* \R n_1=n_2$.
\item $times\_O$.
Since in the standard model for natural numbers $0$ is the absorbing element
- for multiplication $\lambda \_:\N.\star \R \forall x.x \times 0 = x$.
+ for multiplication $\lambda \_:\N.\star \R \forall x.x \mult 0 = 0$.
\item $times\_S$.
In the standard model of natural numbers the multiplications of two
equal to the second number. So
$$\lambda \_:\N. \lambda \_:\N. \star \R \forall x,y.x+S(y)=S(x+y)$$.
-
\end{itemize}
+
+\noindent
+{\bf example}\\
+Let us prove the following principle of well founded induction:
+\[(\forall m.(\forall p. p < m \to P~p) \to P~m) \to \forall n.P~n\]
+In the following proof we shall make use of proof-terms, since we finally
+wish to extract the computational content; we leave to reader the easy
+check that the proof object describes the usual and natural proof
+of the statement.
+
+We assume to have already proved the following lemmas (having trivial
+realizers):\\
+\[L : \forall p, q.p < q \to q \le 0 \to \bot\]
+\[M : \forall p,q,n.p < q \to q \le (S n) \to p \le n \]
+Let us assume $h : \forall m.(\forall p. p < m \to P~p) \to P~m$.
+We prove by induction on $n$ that $\forall q. q \le n \to P~q$.
+For $n=0$, we get a proof of $P ~q$ by
+\[ B \equiv \lambda q.\lambda h_0:q \le 0. h ~q~
+(\lambda p.\lambda k:p < q. false\_ind ~(L~p~q~k~h_0)) \]
+In the inductive case, we must prove that, for any $n$,
+\[(\forall q. q \le n \to P~q) \to (\forall q. q \le S n \to P~q)\]
+Assume $h_1: \forall q. q \le n \to P q$ and
+$h_2: q \le S ~n$. Let us prove $\forall p. p < q \to P~p$.
+If $h_3: p < q$ then $(M~ p~ q~ n~ h_3~ h_2): p \le n$, hence
+$h_1 ~p ~ (M~ p~ q~ n~ h_3~ h_2): P~p$.\\
+In conclusion, the proof of the
+inductive case is
+\[I \equiv \lambda n.\lambda h_1:\forall q. q \le n \to P~ q.\lambda q.\lambda h_2:q \le S n.
+h ~ q ~ (\lambda p.\lambda h_3:p < q.h_1 ~p~ (M~ p~ q~ n~ h_3~ h_2)) \]
+(where $h$ is free in I).
+The full proof is
+\[ \lambda h: \forall m.(\forall p. p < m \to P~p) \to P~m.\lambda m.
+nat\_ind ~B ~ I ~m~m~ (le\_n ~ m) \]
+where $le\_n$ is a proof that $\forall n. n \le n$, and the free $P$ in the definition of $nat_{ind}$ is instantiated with $\forall m.m \le m \to P~m$.\\
+Form the previous proof,after stripping terminal objects,
+and a bit of eta-contraction to make
+the term more readable, we extract the following term (types are omitted):
+
+\[R' \equiv \lambda f.\lambda m.
+R~ (\lambda n.f ~n~ (\lambda q.*))~
+(\lambda n\lambda g\lambda q.f ~q~g)~m ~m\]
+
+The intuition of this operator is the following: supose to
+have a recursive definition $h q = F[h]$ where $q:\N$ and
+$F[h]: A$. This defines a functional
+$f: \lambda q.\lambda g.F[g]: N\to(N\to A) \to A$, such that
+(morally) $h$ is the fixpoint of $f$. For instance,
+in the case of the fibonacci function, $f$ is
+\[fibo \equiv \lambda q. \lambda g.
+if~ q = 0~then~ 1~ else~ if~ q = 1~ then~ 1~ else~ g (q-1)+g (q-2)\]
+
+So $f$ build a new
+approximation of $h$ from the previous approximation $h$ taken
+as input. $R'$ precisely computes the mth-approximation starting
+from a dummy function $(\lambda q.*_A)$. Alternatively,
+you may look at $g$ as the ``history'' (curse of values) of $h$
+for all values less or equal to $q$; then $f$ extend $g$ to
+$q+1$.
+
+Let's compute for example
+\begin{eqnarray}
+R'~fibo~2 & \leadsto &
+ R~ (\lambda n.fibo ~n~ (\lambda q.*))~
+ (\lambda n\lambda g\lambda q.fibo ~q~g)~2 ~2\nonumber\\
+& \leadsto &
+ (\lambda n\lambda g\lambda q.fibo ~q~g)~1~
+ (R~
+ (\lambda n.fibo ~n~ (\lambda q.*))~
+ (\lambda n\lambda g\lambda q.fibo ~q~g)~1)~
+ 2 \nonumber\\
+& \leadsto &
+ \lambda q.fibo ~q~
+ (R~
+ (\lambda n.fibo ~n~ (\lambda q.*))~
+ (\lambda n\lambda g\lambda q.fibo ~q~g)~1)~
+ 2 \nonumber\\
+& \leadsto &
+ \lambda q.fibo ~q~
+ ((\lambda n\lambda g\lambda q.fibo ~q~g)~0~
+ (R~
+ (\lambda n.fibo ~n~ (\lambda q.*))~
+ (\lambda n\lambda g\lambda q.fibo ~q~g)~0))~
+ 2 \nonumber\\
+& \leadsto &
+ \lambda q.fibo ~q~
+ (\lambda q.fibo ~q~
+ (R~
+ (\lambda n.fibo ~n~ (\lambda q.*))~
+ (\lambda n\lambda g\lambda q.fibo ~q~g)~0)
+ )2 \nonumber\\
+& \leadsto &
+ \lambda q.fibo ~q~
+ (\lambda q.fibo ~q~
+ (\lambda n.fibo ~n~ (\lambda q.*)))2
+ \nonumber\\
+& \leadsto &
+ fibo~2~(\lambda q.fibo ~q~ (\lambda n.fibo ~n~ (\lambda q.*))) \nonumber\\
+& \leadsto &
+ (\lambda q.fibo ~q~ (\lambda n.fibo ~n~ (\lambda q.*))) 1 +
+ (\lambda q.fibo ~q~ (\lambda n.fibo ~n~ (\lambda q.*))) 0 \nonumber\\
+& \leadsto &
+ fibo ~1~ (\lambda n.fibo ~n~ (\lambda q.*)) +
+ fibo ~0~ (\lambda n.fibo ~n~ (\lambda q.*)) \nonumber\\
+& \leadsto &
+ 1 + 1 \nonumber
+\end{eqnarray}
+Note that the second argument of $fibo$ is always a method to calculate all the prvious values of $fibo$. DA CAPIRE (per me) come mai $\lambda n$ non viene usata...
+CAPITA CON csc:
+
+n non serve perche' c'e' una relazione logica di n con q,
+in particolare $q <= Sn$ ... quindi $q < n$ (lemma M)...
+e quindi posso usare come history $< n$ una history $< q$.
+il $\lambda h2$ essendo $[[q <= Sn]]$ = 1 viene scartata.
+
+se si spiega come array viene decente... forse. lunedi' provo a scrivere
+meglio.
+
+\section{Inductive types}
+The notation we will use is similar to the one used in
+\cite{Werner} and \cite{Paulin89} but we prefer
+giving a label to each constructor and use that label instead of the
+longer $Constr(n,\ind\{\ldots\})$ to indicate the $n^{th}$ constructor.
+We adopt the vector notation to make things more readable.
+$\vec{m}$ has to be intended as $m_1~\ldots~m_n$ where $n$ may
+be equal to 0 (we use $m_1~\vec{m}$ when we want to give a
+name to the first $m$ and assert $n>0$). If the vector notation is
+used inside an arrow type it has a slightly different meaning,
+$A \to \vec{B} \to C$ is a shortcut for
+$A \to B_1 \to \ldots \to B_n \to C$.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Extensions to the logic framework}
+To talk about arbitrary inductive types (and not hard coded natural numbers) we
+have to extend a bit our framework.
+
+First we admit quantification over inductive types $T$, thus $\forall x:T.A$
+and $\exists x:T.A$ are allowed. Then rules 4 and 5 of the $\sem{\cdot}$
+definition are replaced by $\sem{\forall x:T.P} = T \to \sem{P}$ and
+$\sem{\exists x:T.P} = T \times \sem{P}$.
+
+For each inductive type we will describe the formation rules and the
+corresponding induction principle schema.
+
+Symmetrically we have to extend System T with arbitrary inductive types and
+we will see how theyr recursors are defined in the following sections.
+
+The definition of $\R$ is modified substituting each occurrence of $\N$ with
+a generic inductive type $T$.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Type definition}
+$$\ind\{c_1:C(X); \ldots ; c_n:C(X)\}$$
+$$C(X) ::= X \| T \to C(X) \| X \to C(X)$$
+In the second case we mean $T \neq X$.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Induction principle}
+The induction principle for an inductive type $X$ and a predicate $Q$
+is a constant with the following type
+$$\Xind:\vec{\triUP\{C(X), c\}} \to \forall t:X.Q(t)$$
+$\triUP$ takes a constructor type $C(X)$ and a term $c$ (initially $c$ is a
+constructor of X, and $c:C(X)$) and is defined by recursion as follows:
+\begin{eqnarray}
+\triUP\{X, c\} & = & Q(c) \nonumber\\
+\triUP\{T \to C(X), c\} & = &
+ \forall m:T.\triUP\{C(X),c~m\} \nonumber\\
+\triUP\{X \to C(X), c\} & = &
+ \forall t:X.Q(t) \to \triUP\{C(X), c~t\} \nonumber
+\end{eqnarray}
+
+%%%%%%%%%%%%%%%%%%%%%
+\subsection{Recursor}
+\subsubsection{Type}
+The type of the recursor $\Rx$ on an inductive type $X$ is
+$$\Rx : \vec{\square\{C(X)\}} \to X \to \alpha$$
+$\square$ is defined by recursion on the constructor type $C(X)$.
+\begin{eqnarray}
+\square\{X\} & = & \alpha \nonumber \\
+\square\{T \to C(X)\} & = & T \to \square\{C(X)\}\nonumber \\
+\square\{X \to C(X)\} & = & X \to \alpha \to \square\{C(X)\}\nonumber
+\end{eqnarray}
+\subsubsection{Reduction rules}
+We say that
+$$\Rx~\vec{f}~(c_i~\vec{m}) \leadsto
+\triDOWN\{C(X)_i, f_i, \vec{m}\}$$
+$\triDOWN$ takes a constructor type $C(X)$, a term $f$
+(of type $\square\{C(X)\}$) and is defined by recursion as follows:
+\begin{eqnarray}
+\triDOWN\{X, f, \} & = & f\nonumber \\
+\triDOWN\{T \to C(X), f, m_1~\vec{m}\} & = &
+ \triDOWN\{C(X), f~m_1, \vec{m}\}\nonumber \\
+\triDOWN\{X \to C(X), f, m_1~\vec{m}\} & = &
+ \triDOWN\{C(X), f~m_1~(\Rx~\vec{f}~m_1),
+ \vec{m}\}\nonumber
+\end{eqnarray}
+We assume $\Rx~\vec{f}~(c_i~\vec{m})$ is well typed, so in the first case we
+can omit $\vec{m}$ since it is an empty sequence.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Realizability of the induction principle}
+Once we have inductive types and their induction principle we want to show that
+the recursor $\Rx$ realizes $\Xind$, that is that $\Rx$ has type
+$\sem{\Xind}$ and is in relation $\R$ with $\Xind$.
+
+\begin{thm}$\Rx : \sem{\Xind}$\end{thm}
+\begin{proof}
+We have to compare the definition of $\square$ and $\triUP$
+since they play the same role in constructing respectively the types of
+$\Rx$ and
+$\Xind$. If we assume $\alpha = \sem{Q}$ and we apply the $\sem{\cdot}$
+function to each right side of the $\triUP$ definition we obtain
+exactly $\square$. The last two elements of the arrows $\Rx$ and
+$\Xind$ are again the same up to $\sem{\cdot}$.
+\end{proof}
+
+\begin{thm}$\Rx\R \Xind$\end{thm}
+\begin{proof}
+To prove that $\Rx\R \Xind$ we must assume that for each $i$ index
+of a constructor of $X$, $f_i \R \triUP\{C(X)_i, c_i\}$ and we
+have to prove that for each $t:X$
+$$\Rx~\vec{f}~t \R Q(t)$$
+\noindent
+We proceed by induction on the structure of $t$.
+\\
+The base case is when the
+type of the head constructor of $t$ has no recursive arguments (i.e. the type
+is generated using only the first two rules $C(X)$), so
+$(\Rx~\vec{f}~(c_i~\vec{m}))$ reduces in one step to $(f_i~\vec{m})$. $f_i$
+realizes $\triUP\{C(X)_i, c_i\}$ by assumption and since we are in the base
+case $\triUP\{C(X)_i, c_i\}$ is of the form $\vec{\forall t:T}.Q(c_i~\vec{t})$.
+Thus $f_i~\vec{m} \R Q(c_i~\vec{m})$.
+\\
+In the induction step we have as induction hypothesis that for each recursive
+argument $t_i$ of the head constructor $c_i$, $r_i\equiv
+\Rx~\vec{f}~t_i \R Q(t_i)$. By the third rule of $\triDOWN$ we obtain the reduct
+$f_i~\vec{m}~\vec{t~r}$ (here we write first all the non recursive arguments,
+then all the recursive one. In general they can be mixed and the proof is
+exactly the same but the notation is really heavier). We know by hypothesis
+that $f_i \R \triUP\{C(X)_i, c_i\} \equiv \vec{\forall m:T}.\vec{\forall
+t:X.Q(t)} \to Q(c_i~\vec{m}~\vec{t})$, thus $f_i~\vec{m}~\vec{t~r} \R
+Q(c_i~\vec{m}~\vec{t})$.
+\end{proof}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Improoving inductive types}
+It is possible to parametrize inductive types over other inductive types
+without much difficulties since the type $T$ in $C(X)$ is free. Both the
+recursor and the induction principle are schemas, parametric over $T$.
+
+Possiamo anche definire $X_{\vec{P}}\equiv Ind(P|X)={c_i : C(P|X)}$ e poi
+fare variare $T$ su $\vec{P}$, ma non ottengo niente di meglio.
+
+Credo anche che quantificare su eventuali variabili di tipo non cambi niente
+visto che non abbiamo funzioni.
+
+Se ammettiamo che i tipi dipendano da termini di tipo induttivo
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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projects / helm.git / blobdiff