doc-src/Intro/foundations.tex
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%% $Id$
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\part{Foundations} 
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This Part presents Isabelle's logical foundations in detail:
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representing logical syntax in the typed $\lambda$-calculus; expressing
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inference rules in Isabelle's meta-logic; combining rules by resolution.
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Readers wishing to use Isabelle immediately may prefer to skip straight to
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Part~II, using this Part (via the index) for reference only.
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\begin{figure} 
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\begin{eqnarray*}
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  \neg P   & \hbox{abbreviates} & P\imp\bot \\
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  P\bimp Q & \hbox{abbreviates} & (P\imp Q) \conj (Q\imp P)
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\end{eqnarray*}
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\vskip 4ex
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\(\begin{array}{c@{\qquad\qquad}c}
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  \infer[({\conj}I)]{P\conj Q}{P & Q}  &
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  \infer[({\conj}E1)]{P}{P\conj Q} \qquad 
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  \infer[({\conj}E2)]{Q}{P\conj Q} \\[4ex]
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  \infer[({\disj}I1)]{P\disj Q}{P} \qquad 
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  \infer[({\disj}I2)]{P\disj Q}{Q} &
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  \infer[({\disj}E)]{R}{P\disj Q & \infer*{R}{[P]} & \infer*{R}{[Q]}}\\[4ex]
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  \infer[({\imp}I)]{P\imp Q}{\infer*{Q}{[P]}} &
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  \infer[({\imp}E)]{Q}{P\imp Q & P}  \\[4ex]
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  &
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  \infer[({\bot}E)]{P}{\bot}\\[4ex]
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  \infer[({\forall}I)*]{\forall x.P}{P} &
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  \infer[({\forall}E)]{P[t/x]}{\forall x.P} \\[3ex]
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  \infer[({\exists}I)]{\exists x.P}{P[t/x]} &
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  \infer[({\exists}E)*]{Q}{{\exists x.P} & \infer*{Q}{[P]} } \\[3ex]
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  {t=t} \,(refl)   &  \vcenter{\infer[(subst)]{P[u/x]}{t=u & P[t/x]}} 
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\end{array} \)
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\bigskip\bigskip
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*{\em Eigenvariable conditions\/}:
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$\forall I$: provided $x$ is not free in the assumptions
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$\exists E$: provided $x$ is not free in $Q$ or any assumption except $P$
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\caption{Intuitionistic first-order logic} \label{fol-fig}
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\end{figure}
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\section{Formalizing logical syntax in Isabelle}
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\index{Isabelle!formalizing syntax|bold}
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Figure~\ref{fol-fig} presents intuitionistic first-order logic,
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including equality and the natural numbers.  Let us see how to formalize
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this logic in Isabelle, illustrating the main features of Isabelle's
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polymorphic meta-logic.
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Isabelle represents syntax using the typed $\lambda$-calculus.  We declare
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a type for each syntactic category of the logic.  We declare a constant for
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each symbol of the logic, giving each $n$-ary operation an $n$-argument
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curried function type.  Most importantly, $\lambda$-abstraction represents
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variable binding in quantifiers.
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\index{$\To$|bold}\index{types}
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Isabelle has \ML-style type constructors such as~$(\alpha)list$, where
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$(bool)list$ is the type of lists of booleans.  Function types have the
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form $\sigma\To\tau$, where $\sigma$ and $\tau$ are types.  Functions
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taking $n$~arguments require curried function types; we may abbreviate
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\[  \sigma@1\To (\cdots \sigma@n\To \tau\cdots)  \quad \hbox{as} \quad
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   [\sigma@1, \ldots, \sigma@n] \To \tau. $$ 
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The syntax for terms is summarised below.  Note that function application is
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written $t(u)$ rather than the usual $t\,u$.
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\[ 
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\begin{array}{ll}
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  t :: \sigma   & \hbox{type constraint, on a term or variable} \\
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  \lambda x.t   & \hbox{abstraction} \\
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  \lambda x@1\ldots x@n.t
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        & \hbox{curried abstraction, $\lambda x@1. \ldots \lambda x@n.t$} \\
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  t(u)          & \hbox{application} \\
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  t (u@1, \ldots, u@n) & \hbox{curried application, $t(u@1)\ldots(u@n)$} 
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\end{array}
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\]
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\subsection{Simple types and constants}
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\index{types!simple|bold}
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The syntactic categories of our logic (Figure~\ref{fol-fig}) are 
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{\bf formulae} and {\bf terms}.  Formulae denote truth
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values, so (following logical tradition) we call their type~$o$.  Terms can
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be constructed using~0 and~$Suc$, requiring a type~$nat$ of natural
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numbers.  Later, we shall see how to admit terms of other types.
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After declaring the types~$o$ and~$nat$, we may declare constants for the
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symbols of our logic.  Since $\bot$ denotes a truth value (falsity) and 0
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denotes a number, we put \begin{eqnarray*}
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  \bot  & :: & o \\
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  0     & :: & nat.
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\end{eqnarray*}
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If a symbol requires operands, the corresponding constant must have a
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function type.  In our logic, the successor function
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($Suc$) is from natural numbers to natural numbers, negation ($\neg$) is a
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function from truth values to truth values, and the binary connectives are
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curried functions taking two truth values as arguments: 
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\begin{eqnarray*}
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  Suc    & :: & nat\To nat  \\
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  {\neg} & :: & o\To o      \\
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  \conj,\disj,\imp,\bimp  & :: & [o,o]\To o 
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\end{eqnarray*}
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Isabelle allows us to declare the binary connectives as infixes, with
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appropriate precedences, so that we write $P\conj Q\disj R$ instead of
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$\disj(\conj(P,Q), R)$.
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\subsection{Polymorphic types and constants} \label{polymorphic}
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\index{types!polymorphic|bold}
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Which type should we assign to the equality symbol?  If we tried
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$[nat,nat]\To o$, then equality would be restricted to the natural
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numbers; we would have to declare different equality symbols for each
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type.  Isabelle's type system is polymorphic, so we could declare
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\begin{eqnarray*}
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  {=}  & :: & [\alpha,\alpha]\To o.
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\end{eqnarray*}
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But this is also wrong.  The declaration is too polymorphic; $\alpha$
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ranges over all types, including~$o$ and $nat\To nat$.  Thus, it admits
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$\bot=\neg(\bot)$ and $Suc=Suc$ as formulae, which is acceptable in
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higher-order logic but not in first-order logic.
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Isabelle's \bfindex{classes} control polymorphism.  Each type variable
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belongs to a class, which denotes a set of types.  Classes are partially
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ordered by the subclass relation, which is essentially the subset relation
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on the sets of types.  They closely resemble the classes of the functional
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language Haskell~\cite{haskell-tutorial,haskell-report}.  Nipkow and
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Prehofer discuss type inference for type classes~\cite{nipkow-prehofer}.
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Isabelle provides the built-in class $logic$, which consists of the logical
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types: the ones we want to reason about.  Let us declare a class $term$, to
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consist of all legal types of terms in our logic.  The subclass structure
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is now $term\le logic$.
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We declare $nat$ to be in class $term$.  Type variables of class~$term$
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should resemble Standard~\ML's equality type variables, which range over
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those types that possess an equality test.  Thus, we declare the equality
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constant by
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\begin{eqnarray*}
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  {=}  & :: & [\alpha{::}term,\alpha]\To o 
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\end{eqnarray*}
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We give function types and~$o$ the class $logic$ rather than~$term$, since
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they are not legal types for terms.  We may introduce new types of class
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$term$ --- for instance, type $string$ or $real$ --- at any time.  We can
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even declare type constructors such as $(\alpha)list$, and state that type
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$(\sigma)list$ belongs to class~$term$ provided $\sigma$ does;  equality
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applies to lists of natural numbers but not to lists of formulae.  We may
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summarize this paragraph by a set of {\bf arity declarations} for type
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constructors: 
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\index{$\To$|bold}\index{arities!declaring}
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\begin{eqnarray*}
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  o     & :: & logic \\
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  {\To} & :: & (logic,logic)logic \\
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  nat, string, real     & :: & term \\
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  list  & :: & (term)term
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\end{eqnarray*}
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Higher-order logic, where equality does apply to truth values and
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functions, would require different arity declarations, namely ${o::term}$
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and ${{\To}::(term,term)term}$.  The class system can also handle
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overloading.\index{overloading|bold} We could declare $arith$ to be the
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subclass of $term$ consisting of the `arithmetic' types, such as~$nat$.
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Then we could declare the operators
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\begin{eqnarray*}
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  {+},{-},{\times},{/}  & :: & [\alpha{::}arith,\alpha]\To \alpha
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\end{eqnarray*}
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If we declare new types $real$ and $complex$ of class $arith$, then we
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effectively have three sets of operators:
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\begin{eqnarray*}
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  {+},{-},{\times},{/}  & :: & [nat,nat]\To nat \\
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  {+},{-},{\times},{/}  & :: & [real,real]\To real \\
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  {+},{-},{\times},{/}  & :: & [complex,complex]\To complex 
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\end{eqnarray*}
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Isabelle will regard these as distinct constants, each of which can be defined
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separately.  We could even introduce the type $(\alpha)vector$, make
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$(\sigma)vector$ belong to $arith$ provided $\sigma$ is in $arith$, and define
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\begin{eqnarray*}
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  {+}  & :: & [(\sigma)vector,(\sigma)vector]\To (\sigma)vector 
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\end{eqnarray*}
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in terms of ${+} :: [\sigma,\sigma]\To \sigma$.
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Although we have pretended so far that a type variable belongs to only one
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class --- Isabelle's concrete syntax helps to uphold this illusion --- it
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may in fact belong to any finite number of classes.  For example suppose
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that we had declared yet another class $ord \le term$, the class of all
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`ordered' types, and a constant
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\begin{eqnarray*}
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  {\le}  & :: & [\alpha{::}ord,\alpha]\To o.
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\end{eqnarray*}
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In this context the variable $x$ in $x \le (x+x)$ will be assigned type
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$\alpha{::}\{arith,ord\}$, i.e.\ $\alpha$ belongs to both $arith$ and $ord$.
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Semantically the set $\{arith,ord\}$ should be understood
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as the intersection of the sets of types represented by $arith$ and $ord$.
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Such intersections of classes are called \bfindex{sorts}.  The empty
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intersection of classes, $\{\}$, contains all types and is thus the
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{\bf universal sort}.
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The type checker handles overloading, assigning each term a unique type.  For
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this to be possible, the class and type declarations must satisfy certain
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technical constraints~\cite{nipkow-prehofer}.
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\subsection{Higher types and quantifiers}
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\index{types!higher|bold}
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Quantifiers are regarded as operations upon functions.  Ignoring polymorphism
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for the moment, consider the formula $\forall x. P(x)$, where $x$ ranges
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over type~$nat$.  This is true if $P(x)$ is true for all~$x$.  Abstracting
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$P(x)$ into a function, this is the same as saying that $\lambda x.P(x)$
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returns true for all arguments.  Thus, the universal quantifier can be
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represented by a constant
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\begin{eqnarray*}
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  \forall  & :: & (nat\To o) \To o,
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\end{eqnarray*}
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which is essentially an infinitary truth table.  The representation of $\forall
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x. P(x)$ is $\forall(\lambda x. P(x))$.  
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The existential quantifier is treated
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in the same way.  Other binding operators are also easily handled; for
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instance, the summation operator $\Sigma@{k=i}^j f(k)$ can be represented as
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$\Sigma(i,j,\lambda k.f(k))$, where
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\begin{eqnarray*}
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  \Sigma  & :: & [nat,nat, nat\To nat] \To nat.
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\end{eqnarray*}
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Quantifiers may be polymorphic.  We may define $\forall$ and~$\exists$ over
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all legal types of terms, not just the natural numbers, and
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allow summations over all arithmetic types:
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\begin{eqnarray*}
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   \forall,\exists      & :: & (\alpha{::}term\To o) \To o \\
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   \Sigma               & :: & [nat,nat, nat\To \alpha{::}arith] \To \alpha
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\end{eqnarray*}
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Observe that the index variables still have type $nat$, while the values
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being summed may belong to any arithmetic type.
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\section{Formalizing logical rules in Isabelle}
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\index{meta-logic|bold}
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\index{Isabelle!formalizing rules|bold}
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\index{$\Imp$|bold}\index{$\Forall$|bold}\index{$\equiv$|bold}
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\index{implication!meta-level|bold}
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\index{quantifiers!meta-level|bold}
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\index{equality!meta-level|bold}
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Object-logics are formalized by extending Isabelle's meta-logic, which is
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intuitionistic higher-order logic.  The meta-level connectives are {\bf
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implication}, the {\bf universal quantifier}, and {\bf equality}.
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\begin{itemize}
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  \item The implication \(\phi\Imp \psi\) means `\(\phi\) implies
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\(\psi\)', and expresses logical {\bf entailment}.  
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  \item The quantification \(\Forall x.\phi\) means `\(\phi\) is true for
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all $x$', and expresses {\bf generality} in rules and axiom schemes. 
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\item The equality \(a\equiv b\) means `$a$ equals $b$', for expressing
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  \bfindex{definitions} (see~\S\ref{definitions}).  Equalities left over
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  from the unification process, so called \bfindex{flex-flex equations},
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  are written $a\qeq b$.  The two equality symbols have the same logical
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  meaning. 
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\end{itemize}
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The syntax of the meta-logic is formalized in precisely in the same manner
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as object-logics, using the typed $\lambda$-calculus.  Analogous to
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type~$o$ above, there is a built-in type $prop$ of meta-level truth values.
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Meta-level formulae will have this type.  Type $prop$ belongs to
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class~$logic$; also, $\sigma\To\tau$ belongs to $logic$ provided $\sigma$
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and $\tau$ do.  Here are the types of the built-in connectives:
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\begin{eqnarray*}
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  \Imp     & :: & [prop,prop]\To prop \\
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  \Forall  & :: & (\alpha{::}logic\To prop) \To prop \\
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  {\equiv} & :: & [\alpha{::}\{\},\alpha]\To prop \\
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  \qeq & :: & [\alpha{::}\{\},\alpha]\To prop c
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\end{eqnarray*}
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The restricted polymorphism in $\Forall$ excludes certain types, those used
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just for parsing. 
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In our formalization of first-order logic, we declared a type~$o$ of
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object-level truth values, rather than using~$prop$ for this purpose.  If
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we declared the object-level connectives to have types such as
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${\neg}::prop\To prop$, then these connectives would be applicable to
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meta-level formulae.  Keeping $prop$ and $o$ as separate types maintains
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the distinction between the meta-level and the object-level.  To formalize
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the inference rules, we shall need to relate the two levels; accordingly,
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we declare the constant
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\index{Trueprop@{$Trueprop$}}
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\begin{eqnarray*}
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  Trueprop & :: & o\To prop.
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\end{eqnarray*}
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We may regard $Trueprop$ as a meta-level predicate, reading $Trueprop(P)$ as
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`$P$ is true at the object-level.'  Put another way, $Trueprop$ is a coercion
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from $o$ to $prop$.
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\subsection{Expressing propositional rules}
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\index{rules!propositional|bold}
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We shall illustrate the use of the meta-logic by formalizing the rules of
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Figure~\ref{fol-fig}.  Each object-level rule is expressed as a meta-level
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axiom. 
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One of the simplest rules is $(\conj E1)$.  Making
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everything explicit, its formalization in the meta-logic is
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$$ \Forall P\;Q. Trueprop(P\conj Q) \Imp Trueprop(P).   \eqno(\conj E1) $$
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This may look formidable, but it has an obvious reading: for all object-level
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truth values $P$ and~$Q$, if $P\conj Q$ is true then so is~$P$.  The
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reading is correct because the meta-logic has simple models, where
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types denote sets and $\Forall$ really means `for all.'
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\index{Trueprop@{$Trueprop$}}
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Isabelle adopts notational conventions to ease the writing of rules.  We may
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hide the occurrences of $Trueprop$ by making it an implicit coercion.
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Outer universal quantifiers may be dropped.  Finally, the nested implication
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\[  \phi@1\Imp(\cdots \phi@n\Imp\psi\cdots) \]
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may be abbreviated as $\List{\phi@1; \ldots; \phi@n} \Imp \psi$, which
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formalizes a rule of $n$~premises.
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Using these conventions, the conjunction rules become the following axioms.
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These fully specify the properties of~$\conj$:
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$$ \List{P; Q} \Imp P\conj Q                 \eqno(\conj I) $$
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$$ P\conj Q \Imp P  \qquad  P\conj Q \Imp Q  \eqno(\conj E1,2) $$
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\noindent
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Next, consider the disjunction rules.  The discharge of assumption in
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$(\disj E)$ is expressed  using $\Imp$:
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$$ P \Imp P\disj Q  \qquad  Q \Imp P\disj Q  \eqno(\disj I1,2) $$
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$$ \List{P\disj Q; P\Imp R; Q\Imp R} \Imp R  \eqno(\disj E) $$
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\noindent
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To understand this treatment of assumptions\index{assumptions} in natural
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deduction, look at implication.  The rule $({\imp}I)$ is the classic
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example of natural deduction: to prove that $P\imp Q$ is true, assume $P$
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is true and show that $Q$ must then be true.  More concisely, if $P$
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implies $Q$ (at the meta-level), then $P\imp Q$ is true (at the
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object-level).  Showing the coercion explicitly, this is formalized as
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\[ (Trueprop(P)\Imp Trueprop(Q)) \Imp Trueprop(P\imp Q). \]
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The rule $({\imp}E)$ is straightforward; hiding $Trueprop$, the axioms to
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specify $\imp$ are 
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$$ (P \Imp Q)  \Imp  P\imp Q   \eqno({\imp}I) $$
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$$ \List{P\imp Q; P}  \Imp Q.  \eqno({\imp}E) $$
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\noindent
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Finally, the intuitionistic contradiction rule is formalized as the axiom
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$$ \bot \Imp P.   \eqno(\bot E) $$
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\begin{warn}
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Earlier versions of Isabelle, and certain
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papers~\cite{paulson89,paulson700}, use $\List{P}$ to mean $Trueprop(P)$.
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\index{Trueprop@{$Trueprop$}}
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\end{warn}
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\subsection{Quantifier rules and substitution}
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\index{rules!quantifier|bold}\index{substitution|bold}
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\index{variables!bound}
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Isabelle expresses variable binding using $\lambda$-abstraction; for instance,
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$\forall x.P$ is formalized as $\forall(\lambda x.P)$.  Recall that $F(t)$
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is Isabelle's syntax for application of the function~$F$ to the argument~$t$;
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it is not a meta-notation for substitution.  On the other hand, a substitution
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will take place if $F$ has the form $\lambda x.P$;  Isabelle transforms
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$(\lambda x.P)(t)$ to~$P[t/x]$ by $\beta$-conversion.  Thus, we can express
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inference rules that involve substitution for bound variables.
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\index{parameters|bold}\index{eigenvariables|see{parameters}}
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A logic may attach provisos to certain of its rules, especially quantifier
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rules.  We cannot hope to formalize arbitrary provisos.  Fortunately, those
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typical of quantifier rules always have the same form, namely `$x$ not free in
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\ldots {\it (some set of formulae)},' where $x$ is a variable (called a {\bf
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parameter} or {\bf eigenvariable}) in some premise.  Isabelle treats
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provisos using~$\Forall$, its inbuilt notion of `for all'.
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\index{$\Forall$}
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The purpose of the proviso `$x$ not free in \ldots' is
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to ensure that the premise may not make assumptions about the value of~$x$,
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and therefore holds for all~$x$.  We formalize $(\forall I)$ by
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\[ \left(\Forall x. Trueprop(P(x))\right) \Imp Trueprop(\forall x.P(x)). \]
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This means, `if $P(x)$ is true for all~$x$, then $\forall x.P(x)$ is true.'
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The $\forall E$ rule exploits $\beta$-conversion.  Hiding $Trueprop$, the
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$\forall$ axioms are
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$$ \left(\Forall x. P(x)\right)  \Imp  \forall x.P(x)   \eqno(\forall I) $$
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$$ \forall x.P(x)  \Imp P(t).  \eqno(\forall E)$$
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\noindent
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We have defined the object-level universal quantifier~($\forall$)
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using~$\Forall$.  But we do not require meta-level counterparts of all the
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connectives of the object-logic!  Consider the existential quantifier: 
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$$ P(t)  \Imp  \exists x.P(x)  \eqno(\exists I)$$
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$$ \List{\exists x.P(x);\; \Forall x. P(x)\Imp Q} \Imp Q  \eqno(\exists E) $$
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   386
Let us verify $(\exists E)$ semantically.  Suppose that the premises
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   387
hold; since $\exists x.P(x)$ is true, we may choose $a$ such that $P(a)$ is
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   388
true.  Instantiating $\Forall x. P(x)\Imp Q$ with $a$ yields $P(a)\Imp Q$, and
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   389
we obtain the desired conclusion, $Q$.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   390
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lcp
parents:
diff changeset
   391
The treatment of substitution deserves mention.  The rule
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   392
\[ \infer{P[u/t]}{t=u & P} \]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   393
would be hard to formalize in Isabelle.  It calls for replacing~$t$ by $u$
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   394
throughout~$P$, which cannot be expressed using $\beta$-conversion.  Our
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   395
rule~$(subst)$ uses the occurrences of~$x$ in~$P$ as a template for
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   396
substitution, inferring $P[u/x]$ from~$P[t/x]$.  When we formalize this as
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   397
an axiom, the template becomes a function variable:
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   398
$$ \List{t=u; P(t)} \Imp P(u).  \eqno(subst)$$
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   399
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lcp
parents:
diff changeset
   400
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lcp
parents:
diff changeset
   401
\subsection{Signatures and theories}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   402
\index{signatures|bold}\index{theories|bold}
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lcp
parents:
diff changeset
   403
A {\bf signature} contains the information necessary for type checking,
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   404
parsing and pretty printing.  It specifies classes and their
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   405
relationships; types, with their arities, and constants, with
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   406
their types.  It also contains syntax rules, specified using mixfix
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   407
declarations.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   408
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lcp
parents:
diff changeset
   409
Two signatures can be merged provided their specifications are compatible ---
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   410
they must not, for example, assign different types to the same constant.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   411
Under similar conditions, a signature can be extended.  Signatures are
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   412
managed internally by Isabelle; users seldom encounter them.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   413
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lcp
parents:
diff changeset
   414
A {\bf theory} consists of a signature plus a collection of axioms.  The
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   415
{\bf pure} theory contains only the meta-logic.  Theories can be combined
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   416
provided their signatures are compatible.  A theory definition extends an
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   417
existing theory with further signature specifications --- classes, types,
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   418
constants and mixfix declarations --- plus a list of axioms, expressed as
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   419
strings to be parsed.  A theory can formalize a small piece of mathematics,
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   420
such as lists and their operations, or an entire logic.  A mathematical
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   421
development typically involves many theories in a hierarchy.  For example,
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   422
the pure theory could be extended to form a theory for
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   423
Figure~\ref{fol-fig}; this could be extended in two separate ways to form a
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   424
theory for natural numbers and a theory for lists; the union of these two
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   425
could be extended into a theory defining the length of a list:
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   426
\[ \bf
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   427
\begin{array}{c@{}c@{}c@{}c@{}c}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   428
     {}   &     {} & \hbox{Length} &  {}   &     {}   \\
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   429
     {}   &     {}   &  \uparrow &     {}   &     {}   \\
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   430
     {}   &     {} &\hbox{Nat}+\hbox{List}&  {}   &     {}   \\
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   431
     {}   & \nearrow &     {}    & \nwarrow &     {}   \\
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lcp
parents:
diff changeset
   432
 \hbox{Nat} &   {}   &     {}    &     {}   & \hbox{List} \\
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   433
     {}   & \nwarrow &     {}    & \nearrow &     {}   \\
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   434
     {}   &     {}   &\hbox{FOL} &     {}   &     {}   \\
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   435
     {}   &     {}   &  \uparrow &     {}   &     {}   \\
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   436
     {}   &     {}   &\hbox{Pure}&     {}  &     {}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   437
\end{array}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   438
\]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   439
Each Isabelle proof typically works within a single theory, which is
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   440
associated with the proof state.  However, many different theories may
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   441
coexist at the same time, and you may work in each of these during a single
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   442
session.  
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   443
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   444
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   445
\section{Proof construction in Isabelle}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   446
\index{Isabelle!proof construction in|bold}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   447
There is a one-to-one correspondence between meta-level proofs and
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   448
object-level proofs~\cite{paulson89}.  To each use of a meta-level axiom,
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   449
such as $(\forall I)$, there is a use of the corresponding object-level
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   450
rule.  Object-level assumptions and parameters have meta-level
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   451
counterparts.  The meta-level formalization is {\bf faithful}, admitting no
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   452
incorrect object-level inferences, and {\bf adequate}, admitting all
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   453
correct object-level inferences.  These properties must be demonstrated
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   454
separately for each object-logic.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   455
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   456
The meta-logic is defined by a collection of inference rules, including
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   457
equational rules for the $\lambda$-calculus, and logical rules.  The rules
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   458
for~$\Imp$ and~$\Forall$ resemble those for~$\imp$ and~$\forall$ in
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   459
Figure~\ref{fol-fig}.  Proofs performed using the primitive meta-rules
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   460
would be lengthy; Isabelle proofs normally use certain derived rules.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   461
{\bf Resolution}, in particular, is convenient for backward proof.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   462
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   463
Unification is central to theorem proving.  It supports quantifier
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   464
reasoning by allowing certain `unknown' terms to be instantiated later,
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   465
possibly in stages.  When proving that the time required to sort $n$
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   466
integers is proportional to~$n^2$, we need not state the constant of
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   467
proportionality; when proving that a hardware adder will deliver the sum of
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   468
its inputs, we need not state how many clock ticks will be required.  Such
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   469
quantities often emerge from the proof.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   470
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   471
\index{variables!schematic|see{unknowns}}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   472
Isabelle provides {\bf schematic variables}, or \bfindex{unknowns}, for
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   473
unification.  Logically, unknowns are free variables.  Pragmatically, they
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   474
may be instantiated during a proof, while ordinary variables remain fixed.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   475
Unknowns are written with a ?\ prefix and are frequently subscripted:
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   476
$\Var{a}$, $\Var{a@1}$, $\Var{a@2}$, \ldots, $\Var{P}$, $\Var{P@1}$, \ldots.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   477
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   478
Recall that an inference rule of the form
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   479
\[ \infer{\phi}{\phi@1 & \ldots & \phi@n} \]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   480
is formalized in Isabelle's meta-logic as the axiom
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   481
$\List{\phi@1; \ldots; \phi@n} \Imp \phi$.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   482
Such axioms resemble {\sc Prolog}'s Horn clauses, and can be combined by
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   483
resolution --- Isabelle's principal proof method.  Resolution yields both
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   484
forward and backward proof.  Backward proof works by unifying a goal with
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   485
the conclusion of a rule, whose premises become new subgoals.  Forward proof
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   486
works by unifying theorems with the premises of a rule, deriving a new theorem.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   487
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   488
Isabelle axioms will require an extended notion of resolution, because
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   489
they differ from Horn clauses in two major respects:
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   490
\begin{itemize}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   491
  \item They are written in the typed $\lambda$-calculus, and therefore must be
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   492
resolved using higher-order unification.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   493
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   494
  \item Horn clauses are composed of atomic formulae.  Any formula of the form
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   495
$Trueprop(\cdots)$ is atomic, but axioms such as ${\imp}I$ and $\forall I$
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   496
contain non-atomic formulae.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   497
\index{Trueprop@{$Trueprop$}}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   498
\end{itemize}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   499
Isabelle should not be confused with classical resolution theorem provers
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   500
such as Otter~\cite{wos-bledsoe}.  At the meta-level, Isabelle proves
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   501
theorems in their positive form, not by refutation.  However, an
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   502
object-logic that includes a contradiction rule may employ a refutation
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   503
proof procedure.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   504
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   505
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   506
\subsection{Higher-order unification}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   507
\index{unification!higher-order|bold}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   508
Unification is equation solving.  The solution of $f(\Var{x},c) \qeq
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   509
f(d,\Var{y})$ is $\Var{x}\equiv d$ and $\Var{y}\equiv c$.  {\bf
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   510
Higher-order unification} is equation solving for typed $\lambda$-terms.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   511
To handle $\beta$-conversion, it must reduce $(\lambda x.t)u$ to $t[u/x]$.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   512
That is easy --- in the typed $\lambda$-calculus, all reduction sequences
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   513
terminate at a normal form.  But it must guess the unknown
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   514
function~$\Var{f}$ in order to solve the equation
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   515
\begin{equation} \label{hou-eqn}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   516
 \Var{f}(t) \qeq g(u@1,\ldots,u@k).
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   517
\end{equation}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   518
Huet's~\cite{huet75} search procedure solves equations by imitation and
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   519
projection.  {\bf Imitation}\index{imitation|bold} makes~$\Var{f}$ apply
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   520
leading symbol (if a constant) of the right-hand side.  To solve
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   521
equation~(\ref{hou-eqn}), it guesses
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   522
\[ \Var{f} \equiv \lambda x. g(\Var{h@1}(x),\ldots,\Var{h@k}(x)), \]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   523
where $\Var{h@1}$, \ldots, $\Var{h@k}$ are new unknowns.  Assuming there are no
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   524
other occurrences of~$\Var{f}$, equation~(\ref{hou-eqn}) simplifies to the
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   525
set of equations
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   526
\[ \Var{h@1}(t)\qeq u@1 \quad\ldots\quad \Var{h@k}(t)\qeq u@k. \]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   527
If the procedure solves these equations, instantiating $\Var{h@1}$, \ldots,
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   528
$\Var{h@k}$, then it yields an instantiation for~$\Var{f}$.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   529
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   530
\index{projection|bold}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   531
{\bf Projection} makes $\Var{f}$ apply one of its arguments.  To solve
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   532
equation~(\ref{hou-eqn}), if $t$ expects~$m$ arguments and delivers a
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   533
result of suitable type, it guesses
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   534
\[ \Var{f} \equiv \lambda x. x(\Var{h@1}(x),\ldots,\Var{h@m}(x)), \]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   535
where $\Var{h@1}$, \ldots, $\Var{h@m}$ are new unknowns.  Assuming there are no
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   536
other occurrences of~$\Var{f}$, equation~(\ref{hou-eqn}) simplifies to the 
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   537
equation 
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   538
\[ t(\Var{h@1}(t),\ldots,\Var{h@m}(t)) \qeq g(u@1,\ldots,u@k). $$ 
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   539
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   540
\begin{warn}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   541
Huet's unification procedure is complete.  Isabelle's polymorphic version,
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   542
which solves for type unknowns as well as for term unknowns, is incomplete.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   543
The problem is that projection requires type information.  In
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   544
equation~(\ref{hou-eqn}), if the type of~$t$ is unknown, then projections
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   545
are possible for all~$m\geq0$, and the types of the $\Var{h@i}$ will be
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   546
similarly unconstrained.  Therefore, Isabelle never attempts such
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   547
projections, and may fail to find unifiers where a type unknown turns out
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   548
to be a function type.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   549
\end{warn}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   550
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   551
\index{unknowns!of function type|bold}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   552
Given $\Var{f}(t@1,\ldots,t@n)\qeq u$, Huet's procedure could make up to
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   553
$n+1$ guesses.  The search tree and set of unifiers may be infinite.  But
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   554
higher-order unification can work effectively, provided you are careful
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   555
with {\bf function unknowns}:
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   556
\begin{itemize}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   557
  \item Equations with no function unknowns are solved using first-order
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   558
unification, extended to treat bound variables.  For example, $\lambda x.x
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   559
\qeq \lambda x.\Var{y}$ has no solution because $\Var{y}\equiv x$ would
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   560
capture the free variable~$x$.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   561
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   562
  \item An occurrence of the term $\Var{f}(x,y,z)$, where the arguments are
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   563
distinct bound variables, causes no difficulties.  Its projections can only
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   564
match the corresponding variables.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   565
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   566
  \item Even an equation such as $\Var{f}(a)\qeq a+a$ is all right.  It has
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   567
four solutions, but Isabelle evaluates them lazily, trying projection before
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   568
imitation. The first solution is usually the one desired:
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   569
\[ \Var{f}\equiv \lambda x. x+x \quad
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   570
   \Var{f}\equiv \lambda x. a+x \quad
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   571
   \Var{f}\equiv \lambda x. x+a \quad
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   572
   \Var{f}\equiv \lambda x. a+a \]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   573
  \item  Equations such as $\Var{f}(\Var{x},\Var{y})\qeq t$ and
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   574
$\Var{f}(\Var{g}(x))\qeq t$ admit vast numbers of unifiers, and must be
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   575
avoided. 
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   576
\end{itemize}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   577
In problematic cases, you may have to instantiate some unknowns before
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   578
invoking unification. 
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   579
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   580
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   581
\subsection{Joining rules by resolution} \label{joining}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   582
\index{resolution|bold}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   583
Let $\List{\psi@1; \ldots; \psi@m} \Imp \psi$ and $\List{\phi@1; \ldots;
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   584
\phi@n} \Imp \phi$ be two Isabelle theorems, representing object-level rules. 
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   585
Choosing some~$i$ from~1 to~$n$, suppose that $\psi$ and $\phi@i$ have a
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   586
higher-order unifier.  Writing $Xs$ for the application of substitution~$s$ to
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   587
expression~$X$, this means there is some~$s$ such that $\psi s\equiv \phi@i s$.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   588
By resolution, we may conclude
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   589
\[ (\List{\phi@1; \ldots; \phi@{i-1}; \psi@1; \ldots; \psi@m;
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   590
          \phi@{i+1}; \ldots; \phi@n} \Imp \phi)s.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   591
\]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   592
The substitution~$s$ may instantiate unknowns in both rules.  In short,
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   593
resolution is the following rule:
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   594
\[ \infer[(\psi s\equiv \phi@i s)]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   595
         {(\List{\phi@1; \ldots; \phi@{i-1}; \psi@1; \ldots; \psi@m;
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   596
          \phi@{i+1}; \ldots; \phi@n} \Imp \phi)s}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   597
         {\List{\psi@1; \ldots; \psi@m} \Imp \psi & &
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   598
          \List{\phi@1; \ldots; \phi@n} \Imp \phi}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   599
\]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   600
It operates at the meta-level, on Isabelle theorems, and is justified by
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   601
the properties of $\Imp$ and~$\Forall$.  It takes the number~$i$ (for
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   602
$1\leq i\leq n$) as a parameter and may yield infinitely many conclusions,
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   603
one for each unifier of $\psi$ with $\phi@i$.  Isabelle returns these
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   604
conclusions as a sequence (lazy list).
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   605
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   606
Resolution expects the rules to have no outer quantifiers~($\Forall$).  It
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   607
may rename or instantiate any schematic variables, but leaves free
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   608
variables unchanged.  When constructing a theory, Isabelle puts the rules
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   609
into a standard form containing no free variables; for instance, $({\imp}E)$
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   610
becomes
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   611
\[ \List{\Var{P}\imp \Var{Q}; \Var{P}}  \Imp \Var{Q}. 
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   612
\]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   613
When resolving two rules, the unknowns in the first rule are renamed, by
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   614
subscripting, to make them distinct from the unknowns in the second rule.  To
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   615
resolve $({\imp}E)$ with itself, the first copy of the rule would become
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   616
\[ \List{\Var{P@1}\imp \Var{Q@1}; \Var{P@1}}  \Imp \Var{Q@1}. \]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   617
Resolving this with $({\imp}E)$ in the first premise, unifying $\Var{Q@1}$ with
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   618
$\Var{P}\imp \Var{Q}$, is the meta-level inference
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   619
\[ \infer{\List{\Var{P@1}\imp (\Var{P}\imp \Var{Q}); \Var{P@1}; \Var{P}} 
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   620
           \Imp\Var{Q}.}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   621
         {\List{\Var{P@1}\imp \Var{Q@1}; \Var{P@1}}  \Imp \Var{Q@1} & &
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   622
          \List{\Var{P}\imp \Var{Q}; \Var{P}}  \Imp \Var{Q}}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   623
\]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   624
Renaming the unknowns in the resolvent, we have derived the
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   625
object-level rule
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   626
\[ \infer{Q.}{R\imp (P\imp Q)  &  R  &  P}  \]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   627
\index{rules!derived}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   628
Joining rules in this fashion is a simple way of proving theorems.  The
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   629
derived rules are conservative extensions of the object-logic, and may permit
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   630
simpler proofs.  Let us consider another example.  Suppose we have the axiom
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   631
$$ \forall x\,y. Suc(x)=Suc(y)\imp x=y. \eqno (inject) $$
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   632
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   633
\noindent 
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   634
The standard form of $(\forall E)$ is
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   635
$\forall x.\Var{P}(x)  \Imp \Var{P}(\Var{t})$.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   636
Resolving $(inject)$ with $(\forall E)$ replaces $\Var{P}$ by
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   637
$\lambda x. \forall y. Suc(x)=Suc(y)\imp x=y$ and leaves $\Var{t}$
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   638
unchanged, yielding  
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   639
\[ \forall y. Suc(\Var{t})=Suc(y)\imp \Var{t}=y. \]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   640
Resolving this with $(\forall E)$ puts a subscript on~$\Var{t}$
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   641
and yields
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   642
\[ Suc(\Var{t@1})=Suc(\Var{t})\imp \Var{t@1}=\Var{t}. \]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   643
Resolving this with $({\imp}E)$ increases the subscripts and yields
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   644
\[ Suc(\Var{t@2})=Suc(\Var{t@1})\Imp \Var{t@2}=\Var{t@1}. 
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   645
\]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   646
We have derived the rule
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   647
\[ \infer{m=n,}{Suc(m)=Suc(n)} \]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   648
which goes directly from $Suc(m)=Suc(n)$ to $m=n$.  It is handy for simplifying
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   649
an equation like $Suc(Suc(Suc(m)))=Suc(Suc(Suc(0)))$.  
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   650
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   651
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   652
\subsection{Lifting a rule into a context}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   653
The rules $({\imp}I)$ and $(\forall I)$ may seem unsuitable for
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   654
resolution.  They have non-atomic premises, namely $P\Imp Q$ and $\Forall
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   655
x.P(x)$, while the conclusions of all the rules are atomic (they have the form
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   656
$Trueprop(\cdots)$).  Isabelle gets round the problem through a meta-inference
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   657
called \bfindex{lifting}.  Let us consider how to construct proofs such as
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   658
\[ \infer[({\imp}I)]{P\imp(Q\imp R)}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   659
         {\infer[({\imp}I)]{Q\imp R}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   660
                        {\infer*{R}{[P] & [Q]}}}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   661
   \qquad
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   662
   \infer[(\forall I)]{\forall x\,y.P(x,y)}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   663
         {\infer[(\forall I)]{\forall y.P(x,y)}{P(x,y)}}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   664
\]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   665
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   666
\subsubsection{Lifting over assumptions}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   667
\index{lifting!over assumptions|bold}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   668
Lifting over $\theta\Imp{}$ is the following meta-inference rule:
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   669
\[ \infer{\List{\theta\Imp\phi@1; \ldots; \theta\Imp\phi@n} \Imp
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   670
          (\theta \Imp \phi)}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   671
         {\List{\phi@1; \ldots; \phi@n} \Imp \phi} \]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   672
This is clearly sound: if $\List{\phi@1; \ldots; \phi@n} \Imp \phi$ is true and
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   673
$\theta\Imp\phi@1$, \ldots, $\theta\Imp\phi@n$, $\theta$ are all true then
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   674
$\phi$ must be true.  Iterated lifting over a series of meta-formulae
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   675
$\theta@k$, \ldots, $\theta@1$ yields an object-rule whose conclusion is
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   676
$\List{\theta@1; \ldots; \theta@k} \Imp \phi$.  Typically the $\theta@i$ are
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   677
the assumptions in a natural deduction proof; lifting copies them into a rule's
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   678
premises and conclusion.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   679
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   680
When resolving two rules, Isabelle lifts the first one if necessary.  The
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   681
standard form of $({\imp}I)$ is
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   682
\[ (\Var{P} \Imp \Var{Q})  \Imp  \Var{P}\imp \Var{Q}.   \]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   683
To resolve this rule with itself, Isabelle modifies one copy as follows: it
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   684
renames the unknowns to $\Var{P@1}$ and $\Var{Q@1}$, then lifts the rule over
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   685
$\Var{P}\Imp{}$:
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   686
\[ \List{\List{\Var{P}; \Var{P@1}} \Imp \Var{Q@1}; \Var{P}} 
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   687
   \Imp  \Var{P@1}\imp \Var{Q@1}.   \]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   688
Unifying $\Var{P}\Imp \Var{P@1}\imp\Var{Q@1}$ with $\Var{P} \Imp
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   689
\Var{Q}$ instantiates $\Var{Q}$ to ${\Var{P@1}\imp\Var{Q@1}}$.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   690
Resolution yields
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   691
\[ (\List{\Var{P}; \Var{P@1}} \Imp \Var{Q@1}) \Imp
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   692
\Var{P}\imp(\Var{P@1}\imp\Var{Q@1}).   \]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   693
This represents the derived rule
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   694
\[ \infer{P\imp(Q\imp R).}{\infer*{R}{[P,Q]}} \]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   695
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   696
\subsubsection{Lifting over parameters}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   697
\index{lifting!over parameters|bold}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   698
An analogous form of lifting handles premises of the form $\Forall x\ldots\,$. 
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   699
Here, lifting prefixes an object-rule's premises and conclusion with $\Forall
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   700
x$.  At the same time, lifting introduces a dependence upon~$x$.  It replaces
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   701
each unknown $\Var{a}$ in the rule by $\Var{a'}(x)$, where $\Var{a'}$ is a new
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   702
unknown (by subscripting) of suitable type --- necessarily a function type.  In
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   703
short, lifting is the meta-inference
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   704
\[ \infer{\List{\Forall x.\phi@1^x; \ldots; \Forall x.\phi@n^x} 
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   705
           \Imp \Forall x.\phi^x,}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   706
         {\List{\phi@1; \ldots; \phi@n} \Imp \phi} \]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   707
where $\phi^x$ stands for the result of lifting unknowns over~$x$ in $\phi$. 
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   708
It is not hard to verify that this meta-inference is sound.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   709
For example, $(\disj I)$ might be lifted to
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   710
\[ (\Forall x.\Var{P@1}(x)) \Imp (\Forall x. \Var{P@1}(x)\disj \Var{Q@1}(x))\]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   711
and $(\forall I)$ to
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   712
\[ (\Forall x\,y.\Var{P@1}(x,y)) \Imp (\Forall x. \forall y.\Var{P@1}(x,y)). \]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   713
Isabelle has renamed a bound variable in $(\forall I)$ from $x$ to~$y$,
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   714
avoiding a clash.  Resolving the above with $(\forall I)$ is the meta-inference
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   715
\[ \infer{\Forall x\,y.\Var{P@1}(x,y)) \Imp \forall x\,y.\Var{P@1}(x,y)) }
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   716
         {(\Forall x\,y.\Var{P@1}(x,y)) \Imp 
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   717
               (\Forall x. \forall y.\Var{P@1}(x,y))  &
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   718
          (\Forall x.\Var{P}(x)) \Imp (\forall x.\Var{P}(x))} \]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   719
Here, $\Var{P}$ is replaced by $\lambda x.\forall y.\Var{P@1}(x,y)$; the
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   720
resolvent expresses the derived rule
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   721
\[ \vcenter{ \infer{\forall x\,y.Q(x,y)}{Q(x,y)} }
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   722
   \quad\hbox{provided $x$, $y$ not free in the assumptions} 
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   723
\] 
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   724
I discuss lifting and parameters at length elsewhere~\cite{paulson89}.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   725
Miller goes into even greater detail~\cite{miller-jsc}.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   726
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   727
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   728
\section{Backward proof by resolution}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   729
\index{resolution!in backward proof}\index{proof!backward|bold}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   730
Resolution is convenient for deriving simple rules and for reasoning
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   731
forward from facts.  It can also support backward proof, where we start
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   732
with a goal and refine it to progressively simpler subgoals until all have
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   733
been solved.  {\sc lcf} (and its descendants {\sc hol} and Nuprl) provide
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   734
tactics and tacticals, which constitute a high-level language for
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   735
expressing proof searches.  \bfindex{Tactics} perform primitive refinements
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   736
while \bfindex{tacticals} combine tactics.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   737
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   738
\index{LCF}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   739
Isabelle's tactics and tacticals work differently from {\sc lcf}'s.  An
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   740
Isabelle rule is {\bf bidirectional}: there is no distinction between
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   741
inputs and outputs.  {\sc lcf} has a separate tactic for each rule;
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   742
Isabelle performs refinement by any rule in a uniform fashion, using
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   743
resolution.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   744
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   745
\index{subgoals|bold}\index{main goal|bold}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   746
Isabelle works with meta-level theorems of the form
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   747
\( \List{\phi@1; \ldots; \phi@n} \Imp \phi \).
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   748
We have viewed this as the {\bf rule} with premises
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   749
$\phi@1$,~\ldots,~$\phi@n$ and conclusion~$\phi$.  It can also be viewed as
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   750
the \bfindex{proof state} with subgoals $\phi@1$,~\ldots,~$\phi@n$ and main
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   751
goal~$\phi$.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   752
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   753
To prove the formula~$\phi$, take $\phi\Imp \phi$ as the initial proof
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   754
state.  This assertion is, trivially, a theorem.  At a later stage in the
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   755
backward proof, a typical proof state is $\List{\phi@1; \ldots; \phi@n}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   756
\Imp \phi$.  This proof state is a theorem, guaranteeing that the subgoals
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   757
$\phi@1$,~\ldots,~$\phi@n$ imply~$\phi$.  If $m=0$ then we have
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   758
proved~$\phi$ outright.  If $\phi$ contains unknowns, they may become
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   759
instantiated during the proof; a proof state may be $\List{\phi@1; \ldots;
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   760
\phi@n} \Imp \phi'$, where $\phi'$ is an instance of~$\phi$.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   761
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   762
\subsection{Refinement by resolution}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   763
\index{refinement|bold}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   764
To refine subgoal~$i$ of a proof state by a rule, perform the following
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   765
resolution: 
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   766
\[ \infer{\hbox{new proof state}}{\hbox{rule} & & \hbox{proof state}} \]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   767
If the rule is $\List{\psi'@1; \ldots; \psi'@m} \Imp \psi'$ after lifting
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   768
over subgoal~$i$, and the proof state is $\List{\phi@1; \ldots; \phi@n}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   769
\Imp \phi$, then the new proof state is (for~$1\leq i\leq n$)
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   770
\[ (\List{\phi@1; \ldots; \phi@{i-1}; \psi'@1;
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   771
          \ldots; \psi'@m; \phi@{i+1}; \ldots; \phi@n} \Imp \phi)s.  \]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   772
Substitution~$s$ unifies $\psi'$ with~$\phi@i$.  In the proof state,
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   773
subgoal~$i$ is replaced by $m$ new subgoals, the rule's instantiated premises.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   774
If some of the rule's unknowns are left un-instantiated, they become new
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   775
unknowns in the proof state.  Refinement by~$(\exists I)$, namely
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   776
\[ \Var{P}(\Var{t}) \Imp \exists x. \Var{P}(x), \]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   777
inserts a new unknown derived from~$\Var{t}$ by subscripting and lifting.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   778
We do not have to specify an `existential witness' when
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   779
applying~$(\exists I)$.  Further resolutions may instantiate unknowns in
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   780
the proof state.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   781
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   782
\subsection{Proof by assumption}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   783
\index{proof!by assumption|bold}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   784
In the course of a natural deduction proof, parameters $x@1$, \ldots,~$x@l$ and
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   785
assumptions $\theta@1$, \ldots, $\theta@k$ accumulate, forming a context for
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   786
each subgoal.  Repeated lifting steps can lift a rule into any context.  To
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   787
aid readability, Isabelle puts contexts into a normal form, gathering the
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   788
parameters at the front:
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   789
\begin{equation} \label{context-eqn}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   790
\Forall x@1 \ldots x@l. \List{\theta@1; \ldots; \theta@k}\Imp\theta. 
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   791
\end{equation}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   792
Under the usual reading of the connectives, this expresses that $\theta$
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   793
follows from $\theta@1$,~\ldots~$\theta@k$ for arbitrary
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   794
$x@1$,~\ldots,~$x@l$.  It is trivially true if $\theta$ equals any of
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   795
$\theta@1$,~\ldots~$\theta@k$, or is unifiable with any of them.  This
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   796
models proof by assumption in natural deduction.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   797
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   798
Isabelle automates the meta-inference for proof by assumption.  Its arguments
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   799
are the meta-theorem $\List{\phi@1; \ldots; \phi@n} \Imp \phi$, and some~$i$
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   800
from~1 to~$n$, where $\phi@i$ has the form~(\ref{context-eqn}).  Its results
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   801
are meta-theorems of the form
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   802
\[ (\List{\phi@1; \ldots; \phi@{i-1}; \phi@{i+1}; \phi@n} \Imp \phi)s \]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   803
for each $s$ and~$j$ such that $s$ unifies $\lambda x@1 \ldots x@l. \theta@j$
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   804
with $\lambda x@1 \ldots x@l. \theta$.  Isabelle supplies the parameters
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   805
$x@1$,~\ldots,~$x@l$ to higher-order unification as bound variables, which
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   806
regards them as unique constants with a limited scope --- this enforces
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   807
parameter provisos~\cite{paulson89}.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   808
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   809
The premise represents a proof state with~$n$ subgoals, of which the~$i$th is
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   810
to be solved by assumption.  Isabelle searches the subgoal's context for an
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   811
assumption, say $\theta@j$, that can solve it by unification.  For each
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   812
unifier, the meta-inference returns an instantiated proof state from which the
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   813
$i$th subgoal has been removed.  Isabelle searches for a unifying assumption;
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   814
for readability and robustness, proofs do not refer to assumptions by number.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   815
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   816
Consider the proof state $(\List{P(a); P(b)} \Imp P(\Var{x})) \Imp Q(\Var{x})$.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   817
Proof by assumption (with $i=1$, the only possibility) yields two results:
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   818
\begin{itemize}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   819
  \item $Q(a)$, instantiating $\Var{x}\equiv a$
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   820
  \item $Q(b)$, instantiating $\Var{x}\equiv b$
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   821
\end{itemize}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   822
Here, proof by assumption affects the main goal.  It could also affect
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   823
other subgoals.  Consider the effect of having the
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   824
additional subgoal ${\List{P(b); P(c)} \Imp P(\Var{x})}$.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   825
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   826
\subsection{A propositional proof} \label{prop-proof}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   827
\index{examples!propositional}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   828
Our first example avoids quantifiers.  Given the main goal $P\disj P\imp
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   829
P$, Isabelle creates the initial state
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   830
\[ (P\disj P\imp P)\Imp (P\disj P\imp P). \]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   831
Bear in mind that every proof state we derive will be a meta-theorem,
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   832
expressing that the subgoals imply the main goal.  The first step is to refine
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   833
subgoal~1 by (${\imp}I)$, creating a new state where $P\disj P$ is an
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   834
assumption: 
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   835
\[ (P\disj P\Imp P)\Imp (P\disj P\imp P) \]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   836
The next step is $(\disj E)$, which replaces subgoal~1 by three new subgoals. 
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   837
Because of lifting, each subgoal contains a copy of the context --- the
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   838
assumption $P\disj P$.  (In fact, this assumption is now redundant; we shall
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   839
shortly see how to get rid of it!)  The new proof state is the following
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   840
meta-theorem, laid out for clarity:
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   841
\[ \begin{array}{l@{}l@{\qquad\qquad}l} 
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   842
  \lbrakk\;& P\disj P\Imp \Var{P@1}\disj\Var{Q@1}; & \hbox{(subgoal 1)} \\
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   843
           & \List{P\disj P; \Var{P@1}} \Imp P;    & \hbox{(subgoal 2)} \\
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   844
           & \List{P\disj P; \Var{Q@1}} \Imp P     & \hbox{(subgoal 3)} \\
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   845
  \rbrakk\;& \Imp (P\disj P\imp P)                 & \hbox{(main goal)}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   846
   \end{array} 
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   847
\]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   848
Notice the unknowns in the proof state.  Because we have applied $(\disj E)$,
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   849
we must prove some disjunction, $\Var{P@1}\disj\Var{Q@1}$.  Of course,
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   850
subgoal~1 is provable by assumption.  This instantiates both $\Var{P@1}$ and
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   851
$\Var{Q@1}$ to~$P$ throughout the proof state:
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   852
\[ \begin{array}{l@{}l@{\qquad\qquad}l} 
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   853
    \lbrakk\;& \List{P\disj P; P} \Imp P; & \hbox{(subgoal 1)} \\
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   854
             & \List{P\disj P; P} \Imp P  & \hbox{(subgoal 2)} \\
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   855
    \rbrakk\;& \Imp (P\disj P\imp P)      & \hbox{(main goal)}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   856
   \end{array} \]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   857
Both of the remaining subgoals can be proved by assumption.  After two such
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   858
steps, the proof state is simply the meta-theorem $P\disj P\imp P$.  This is
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   859
our desired result.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   860
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   861
\subsection{A quantifier proof}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   862
\index{examples!with quantifiers}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   863
To illustrate quantifiers and $\Forall$-lifting, let us prove
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   864
$(\exists x.P(f(x)))\imp(\exists x.P(x))$.  The initial proof
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   865
state is the trivial meta-theorem 
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   866
\[ (\exists x.P(f(x)))\imp(\exists x.P(x)) \Imp 
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   867
   (\exists x.P(f(x)))\imp(\exists x.P(x)). \]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   868
As above, the first step is refinement by (${\imp}I)$: 
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   869
\[ (\exists x.P(f(x))\Imp \exists x.P(x)) \Imp 
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   870
   (\exists x.P(f(x)))\imp(\exists x.P(x)) 
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   871
\]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   872
The next step is $(\exists E)$, which replaces subgoal~1 by two new subgoals.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   873
Both have the assumption $\exists x.P(f(x))$.  The new proof
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   874
state is the meta-theorem  
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   875
\[ \begin{array}{l@{}l@{\qquad\qquad}l} 
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   876
   \lbrakk\;& \exists x.P(f(x)) \Imp \exists x.\Var{P@1}(x); & \hbox{(subgoal 1)} \\
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   877
            & \Forall x.\List{\exists x.P(f(x)); \Var{P@1}(x)} \Imp 
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   878
                       \exists x.P(x)  & \hbox{(subgoal 2)} \\
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   879
    \rbrakk\;& \Imp (\exists x.P(f(x)))\imp(\exists x.P(x))  & \hbox{(main goal)}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   880
   \end{array} 
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   881
\]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   882
The unknown $\Var{P@1}$ appears in both subgoals.  Because we have applied
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   883
$(\exists E)$, we must prove $\exists x.\Var{P@1}(x)$, where $\Var{P@1}(x)$ may
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   884
become any formula possibly containing~$x$.  Proving subgoal~1 by assumption
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   885
instantiates $\Var{P@1}$ to~$\lambda x.P(f(x))$:  
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   886
\[ \left(\Forall x.\List{\exists x.P(f(x)); P(f(x))} \Imp 
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   887
         \exists x.P(x)\right) 
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   888
   \Imp (\exists x.P(f(x)))\imp(\exists x.P(x)) 
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   889
\]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   890
The next step is refinement by $(\exists I)$.  The rule is lifted into the
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   891
context of the parameter~$x$ and the assumption $P(f(x))$.  This ensures that
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   892
the context is copied to the subgoal and allows the existential witness to
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   893
depend upon~$x$: 
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   894
\[ \left(\Forall x.\List{\exists x.P(f(x)); P(f(x))} \Imp 
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   895
         P(\Var{x@2}(x))\right) 
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   896
   \Imp (\exists x.P(f(x)))\imp(\exists x.P(x)) 
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   897
\]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   898
The existential witness, $\Var{x@2}(x)$, consists of an unknown
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   899
applied to a parameter.  Proof by assumption unifies $\lambda x.P(f(x))$ 
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   900
with $\lambda x.P(\Var{x@2}(x))$, instantiating $\Var{x@2}$ to $f$.  The final
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   901
proof state contains no subgoals: $(\exists x.P(f(x)))\imp(\exists x.P(x))$.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   902
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   903
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   904
\subsection{Tactics and tacticals}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   905
\index{tactics|bold}\index{tacticals|bold}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   906
{\bf Tactics} perform backward proof.  Isabelle tactics differ from those
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   907
of {\sc lcf}, {\sc hol} and Nuprl by operating on entire proof states,
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   908
rather than on individual subgoals.  An Isabelle tactic is a function that
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   909
takes a proof state and returns a sequence (lazy list) of possible
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   910
successor states.  Sequences are coded in ML as functions, a standard
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   911
technique~\cite{paulson91}.  Isabelle represents proof states by theorems.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   912
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   913
Basic tactics execute the meta-rules described above, operating on a
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   914
given subgoal.  The {\bf resolution tactics} take a list of rules and
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   915
return next states for each combination of rule and unifier.  The {\bf
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   916
assumption tactic} examines the subgoal's assumptions and returns next
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   917
states for each combination of assumption and unifier.  Lazy lists are
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   918
essential because higher-order resolution may return infinitely many
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   919
unifiers.  If there are no matching rules or assumptions then no next
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   920
states are generated; a tactic application that returns an empty list is
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   921
said to {\bf fail}.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   922
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   923
Sequences realize their full potential with {\bf tacticals} --- operators
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   924
for combining tactics.  Depth-first search, breadth-first search and
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   925
best-first search (where a heuristic function selects the best state to
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   926
explore) return their outcomes as a sequence.  Isabelle provides such
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   927
procedures in the form of tacticals.  Simpler procedures can be expressed
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   928
directly using the basic tacticals {\it THEN}, {\it ORELSE} and {\it REPEAT}:
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   929
\begin{description}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   930
\item[$tac1\;THEN\;tac2$] is a tactic for sequential composition.  Applied
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   931
to a proof state, it returns all states reachable in two steps by applying
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   932
$tac1$ followed by~$tac2$.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   933
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   934
\item[$tac1\;ORELSE\;tac2$] is a choice tactic.  Applied to a state, it
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   935
tries~$tac1$ and returns the result if non-empty; otherwise, it uses~$tac2$.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   936
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   937
\item[$REPEAT\;tac$] is a repetition tactic.  Applied to a state, it
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   938
returns all states reachable by applying~$tac$ as long as possible (until
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   939
it would fail).  $REPEAT$ can be expressed in a few lines of \ML{} using
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   940
{\it THEN} and~{\it ORELSE}.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   941
\end{description}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   942
For instance, this tactic repeatedly applies $tac1$ and~$tac2$, giving
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   943
$tac1$ priority:
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   944
\[ REPEAT(tac1\;ORELSE\;tac2) \]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   945
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   946
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   947
\section{Variations on resolution}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   948
In principle, resolution and proof by assumption suffice to prove all
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   949
theorems.  However, specialized forms of resolution are helpful for working
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   950
with elimination rules.  Elim-resolution applies an elimination rule to an
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   951
assumption; destruct-resolution is similar, but applies a rule in a forward
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   952
style.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   953
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   954
The last part of the section shows how the techniques for proving theorems
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   955
can also serve to derive rules.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   956
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   957
\subsection{Elim-resolution}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   958
\index{elim-resolution|bold}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   959
Consider proving the theorem $((R\disj R)\disj R)\disj R\imp R$.  By
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   960
$({\imp}I)$, we prove $R$ from the assumption $((R\disj R)\disj R)\disj R$.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   961
Applying $(\disj E)$ to this assumption yields two subgoals, one that
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   962
assumes~$R$ (and is therefore trivial) and one that assumes $(R\disj
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   963
R)\disj R$.  This subgoal admits another application of $(\disj E)$.  Since
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   964
natural deduction never discards assumptions, we eventually generate a
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   965
subgoal containing much that is redundant:
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   966
\[ \List{((R\disj R)\disj R)\disj R; (R\disj R)\disj R; R\disj R; R} \Imp R. \]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   967
In general, using $(\disj E)$ on the assumption $P\disj Q$ creates two new
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   968
subgoals with the additional assumption $P$ or~$Q$.  In these subgoals,
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   969
$P\disj Q$ is redundant.  It wastes space; worse, it could make a theorem
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   970
prover repeatedly apply~$(\disj E)$, looping.  Other elimination rules pose
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   971
similar problems.  In first-order logic, only universally quantified
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   972
assumptions are sometimes needed more than once --- say, to prove
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   973
$P(f(f(a)))$ from the assumptions $\forall x.P(x)\imp P(f(x))$ and~$P(a)$.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   974
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   975
Many logics can be formulated as sequent calculi that delete redundant
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   976
assumptions after use.  The rule $(\disj E)$ might become
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   977
\[ \infer[\disj\hbox{-left}]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   978
         {\Gamma,P\disj Q,\Delta \turn \Theta}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   979
         {\Gamma,P,\Delta \turn \Theta && \Gamma,Q,\Delta \turn \Theta}  \] 
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   980
In backward proof, a goal containing $P\disj Q$ on the left of the~$\turn$
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   981
(that is, as an assumption) splits into two subgoals, replacing $P\disj Q$
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   982
by $P$ or~$Q$.  But the sequent calculus, with its explicit handling of
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   983
assumptions, can be tiresome to use.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   984
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   985
Elim-resolution is Isabelle's way of getting sequent calculus behaviour
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   986
from natural deduction rules.  It lets an elimination rule consume an
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   987
assumption.  Elim-resolution takes a rule $\List{\psi@1; \ldots; \psi@m}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   988
\Imp \psi$ a proof state $\List{\phi@1; \ldots; \phi@n} \Imp \phi$, and a
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   989
subgoal number~$i$.  The rule must have at least one premise, thus $m>0$.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   990
Write the rule's lifted form as $\List{\psi'@1; \ldots; \psi'@m} \Imp
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   991
\psi'$.  Ordinary resolution would attempt to reduce~$\phi@i$,
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   992
replacing subgoal~$i$ by $m$ new ones.  Elim-resolution tries {\bf
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   993
simultaneously} to reduce~$\phi@i$ and to solve~$\psi'@1$ by assumption; it
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   994
returns a sequence of next states.  Each of these replaces subgoal~$i$ by
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   995
instances of $\psi'@2$, \ldots, $\psi'@m$ from which the selected
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   996
assumption has been deleted.  Suppose $\phi@i$ has the parameter~$x$ and
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   997
assumptions $\theta@1$,~\ldots,~$\theta@k$.  Then $\psi'@1$, the rule's first
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   998
premise after lifting, will be
216d6ed87399 Initial revision
lcp
parents:
diff changeset
   999
\( \Forall x. \List{\theta@1; \ldots; \theta@k}\Imp \psi^{x}@1 \).
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1000
Elim-resolution tries to unify simultaneously $\psi'\qeq\phi@i$ and
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1001
$\lambda x. \theta@j \qeq \lambda x. \psi^{x}@1$, for $j=1$,~\ldots,~$k$.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1002
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1003
Let us redo the example from~\S\ref{prop-proof}.  The elimination rule
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1004
is~$(\disj E)$,
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1005
\[ \List{\Var{P}\disj \Var{Q};\; \Var{P}\Imp \Var{R};\; \Var{Q}\Imp \Var{R}}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1006
      \Imp \Var{R}  \]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1007
and the proof state is $(P\disj P\Imp P)\Imp (P\disj P\imp P)$.  The
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1008
lifted rule would be
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1009
\[ \begin{array}{l@{}l}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1010
  \lbrakk\;& P\disj P \Imp \Var{P@1}\disj\Var{Q@1}; \\
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1011
           & \List{P\disj P ;\; \Var{P@1}} \Imp \Var{R@1};    \\
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1012
           & \List{P\disj P ;\; \Var{Q@1}} \Imp \Var{R@1}     \\
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1013
  \rbrakk\;& \Imp \Var{R@1}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1014
  \end{array} 
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1015
\]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1016
Unification would take the simultaneous equations
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1017
$P\disj P \qeq \Var{P@1}\disj\Var{Q@1}$ and $\Var{R@1} \qeq P$, yielding
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1018
$\Var{P@1}\equiv\Var{Q@1}\equiv\Var{R@1} \equiv P$.  The new proof state
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1019
would be simply
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1020
\[ \List{P \Imp P;\; P \Imp P} \Imp (P\disj P\imp P). 
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1021
\]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1022
Elim-resolution's simultaneous unification gives better control
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1023
than ordinary resolution.  Recall the substitution rule:
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1024
$$ \List{\Var{t}=\Var{u}; \Var{P}(\Var{t})} \Imp \Var{P}(\Var{u}) 
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1025
   \eqno(subst) $$
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1026
Unsuitable for ordinary resolution because $\Var{P}(\Var{u})$ admits many
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1027
unifiers, $(subst)$ works well with elim-resolution.  It deletes some
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1028
assumption of the form $x=y$ and replaces every~$y$ by~$x$ in the subgoal
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1029
formula.  The simultaneous unification instantiates $\Var{u}$ to~$y$; if
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1030
$y$ is not an unknown, then $\Var{P}(y)$ can easily be unified with another
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1031
formula.  
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1032
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1033
In logical parlance, the premise containing the connective to be eliminated
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1034
is called the \bfindex{major premise}.  Elim-resolution expects the major
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1035
premise to come first.  The order of the premises is significant in
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1036
Isabelle.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1037
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1038
\subsection{Destruction rules} \label{destruct}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1039
\index{destruction rules|bold}\index{proof!forward}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1040
Looking back to Figure~\ref{fol-fig}, notice that there are two kinds of
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1041
elimination rule.  The rules $({\conj}E1)$, $({\conj}E2)$, $({\imp}E)$ and
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1042
$({\forall}E)$ extract the conclusion from the major premise.  In Isabelle
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1043
parlance, such rules are called \bfindex{destruction rules}; they are readable
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1044
and easy to use in forward proof.  The rules $({\disj}E)$, $({\bot}E)$ and
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1045
$({\exists}E)$ work by discharging assumptions; they support backward proof
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1046
in a style reminiscent of the sequent calculus.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1047
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1048
The latter style is the most general form of elimination rule.  In natural
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1049
deduction, there is no way to recast $({\disj}E)$, $({\bot}E)$ or
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1050
$({\exists}E)$ as destruction rules.  But we can write general elimination
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1051
rules for $\conj$, $\imp$ and~$\forall$:
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1052
\[
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1053
\infer{R}{P\conj Q & \infer*{R}{[P,Q]}} \qquad
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1054
\infer{R}{P\imp Q & P & \infer*{R}{[Q]}}  \qquad
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1055
\infer{Q}{\forall x.P & \infer*{Q}{[P[t/x]]}} 
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1056
\]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1057
Because they are concise, destruction rules are simpler to derive than the
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1058
corresponding elimination rules.  To facilitate their use in backward
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1059
proof, Isabelle provides a means of transforming a destruction rule such as
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1060
\[ \infer[\quad\hbox{to the elimination rule}\quad]{Q}{P@1 & \ldots & P@m} 
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1061
   \infer{R.}{P@1 & \ldots & P@m & \infer*{R}{[Q]}} 
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1062
\]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1063
\index{destruct-resolution|bold}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1064
{\bf Destruct-resolution} combines this transformation with
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1065
elim-resolution.  It applies a destruction rule to some assumption of a
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1066
subgoal.  Given the rule above, it replaces the assumption~$P@1$ by~$Q$,
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1067
with new subgoals of showing instances of $P@2$, \ldots,~$P@m$.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1068
Destruct-resolution works forward from a subgoal's assumptions.  Ordinary
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1069
resolution performs forward reasoning from theorems, as illustrated
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1070
in~\S\ref{joining}. 
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1071
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1072
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1073
\subsection{Deriving rules by resolution}  \label{deriving}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1074
\index{rules!derived|bold}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1075
The meta-logic, itself a form of the predicate calculus, is defined by a
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1076
system of natural deduction rules.  Each theorem may depend upon
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1077
meta-assumptions.  The theorem that~$\phi$ follows from the assumptions
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1078
$\phi@1$, \ldots, $\phi@n$ is written
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1079
\[ \phi \quad [\phi@1,\ldots,\phi@n]. \]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1080
A more conventional notation might be $\phi@1,\ldots,\phi@n \turn \phi$,
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1081
but Isabelle's notation is more readable with large formulae.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1082
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1083
Meta-level natural deduction provides a convenient mechanism for deriving
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1084
new object-level rules.  To derive the rule
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1085
\[ \infer{\phi,}{\theta@1 & \ldots & \theta@k} \]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1086
assume the premises $\theta@1$,~\ldots,~$\theta@k$ at the
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1087
meta-level.  Then prove $\phi$, possibly using these assumptions.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1088
Starting with a proof state $\phi\Imp\phi$, assumptions may accumulate,
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1089
reaching a final state such as
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1090
\[ \phi \quad [\theta@1,\ldots,\theta@k]. \]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1091
The meta-rule for $\Imp$ introduction discharges an assumption.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1092
Discharging them in the order $\theta@k,\ldots,\theta@1$ yields the
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1093
meta-theorem $\List{\theta@1; \ldots; \theta@k} \Imp \phi$, with no
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1094
assumptions.  This represents the desired rule.
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1095
Let us derive the general $\conj$ elimination rule:
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1096
$$ \infer{R}{P\conj Q & \infer*{R}{[P,Q]}}  \eqno(\conj E)$$
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1097
We assume $P\conj Q$ and $\List{P;Q}\Imp R$, and commence backward proof in
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1098
the state $R\Imp R$.  Resolving this with the second assumption yields the
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1099
state 
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1100
\[ \phantom{\List{P\conj Q;\; P\conj Q}}
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1101
   \llap{$\List{P;Q}$}\Imp R \quad [\,\List{P;Q}\Imp R\,]. \]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1102
Resolving subgoals~1 and~2 with $({\conj}E1)$ and $({\conj}E2)$,
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1103
respectively, yields the state
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1104
\[ \List{P\conj Q;\; P\conj Q}\Imp R \quad [\,\List{P;Q}\Imp R\,]. \]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1105
Resolving both subgoals with the assumption $P\conj Q$ yields
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1106
\[ R \quad [\, \List{P;Q}\Imp R, P\conj Q \,]. \]
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1107
The proof may use the meta-assumptions in any order, and as often as
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1108
necessary; when finished, we discharge them in the correct order to
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1109
obtain the desired form:
216d6ed87399 Initial revision
lcp
parents:
diff changeset
  1110
\[ \List{P\conj Q;\; \List{P;Q}\Imp R} \Imp R \]
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lcp
parents:
diff changeset
  1111
We have derived the rule using free variables, which prevents their
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parents:
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  1112
premature instantiation during the proof; we may now replace them by
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parents:
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  1113
schematic variables.
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parents:
diff changeset
  1114
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lcp
parents:
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  1115
\begin{warn}
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lcp
parents:
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  1116
Schematic variables are not allowed in (meta) assumptions because they would
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lcp
parents:
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  1117
complicate lifting.  Assumptions remain fixed throughout a proof.
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parents:
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  1118
\end{warn}
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diff changeset
  1119
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  1120
% Local Variables: 
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  1121
% mode: latex
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  1122
% TeX-master: t
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  1123
% End: