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-\chapter{Constructive Type Theory}
-\index{Constructive Type Theory|(}
-
-\underscoreoff %this file contains _ in rule names
-
-Martin-L\"of's Constructive Type Theory \cite{martinlof84,nordstrom90} can
-be viewed at many different levels.  It is a formal system that embodies
-the principles of intuitionistic mathematics; it embodies the
-interpretation of propositions as types; it is a vehicle for deriving
-programs from proofs.  
-
-Thompson's book~\cite{thompson91} gives a readable and thorough account of
-Type Theory.  Nuprl is an elaborate implementation~\cite{constable86}.
-{\sc alf} is a more recent tool that allows proof terms to be edited
-directly~\cite{alf}.
-
-Isabelle's original formulation of Type Theory was a kind of sequent
-calculus, following Martin-L\"of~\cite{martinlof84}.  It included rules for
-building the context, namely variable bindings with their types.  A typical
-judgement was
-\[   a(x@1,\ldots,x@n)\in A(x@1,\ldots,x@n) \; 
-    [ x@1\in A@1, x@2\in A@2(x@1), \ldots, x@n\in A@n(x@1,\ldots,x@{n-1}) ]
-\]
-This sequent calculus was not satisfactory because assumptions like
-`suppose $A$ is a type' or `suppose $B(x)$ is a type for all $x$ in $A$'
-could not be formalized.  
-
-The theory~\thydx{CTT} implements Constructive Type Theory, using
-natural deduction.  The judgement above is expressed using $\Forall$ and
-$\Imp$:
-\[ \begin{array}{r@{}l}
-     \Forall x@1\ldots x@n. &
-          \List{x@1\in A@1; 
-                x@2\in A@2(x@1); \cdots \; 
-                x@n\in A@n(x@1,\ldots,x@{n-1})} \Imp \\
-     &  \qquad\qquad a(x@1,\ldots,x@n)\in A(x@1,\ldots,x@n) 
-    \end{array}
-\]
-Assumptions can use all the judgement forms, for instance to express that
-$B$ is a family of types over~$A$:
-\[ \Forall x . x\in A \Imp B(x)\;{\rm type} \]
-To justify the CTT formulation it is probably best to appeal directly to the
-semantic explanations of the rules~\cite{martinlof84}, rather than to the
-rules themselves.  The order of assumptions no longer matters, unlike in
-standard Type Theory.  Contexts, which are typical of many modern type
-theories, are difficult to represent in Isabelle.  In particular, it is
-difficult to enforce that all the variables in a context are distinct.
-\index{assumptions!in CTT}
-
-The theory does not use polymorphism.  Terms in CTT have type~\tydx{i}, the
-type of individuals.  Types in CTT have type~\tydx{t}.
-
-\begin{figure} \tabcolsep=1em  %wider spacing in tables
-\begin{center}
-\begin{tabular}{rrr} 
-  \it name      & \it meta-type         & \it description \\ 
-  \cdx{Type}    & $t \to prop$          & judgement form \\
-  \cdx{Eqtype}  & $[t,t]\to prop$       & judgement form\\
-  \cdx{Elem}    & $[i, t]\to prop$      & judgement form\\
-  \cdx{Eqelem}  & $[i, i, t]\to prop$   & judgement form\\
-  \cdx{Reduce}  & $[i, i]\to prop$      & extra judgement form\\[2ex]
-
-  \cdx{N}       &     $t$               & natural numbers type\\
-  \cdx{0}       &     $i$               & constructor\\
-  \cdx{succ}    & $i\to i$              & constructor\\
-  \cdx{rec}     & $[i,i,[i,i]\to i]\to i$       & eliminator\\[2ex]
-  \cdx{Prod}    & $[t,i\to t]\to t$     & general product type\\
-  \cdx{lambda}  & $(i\to i)\to i$       & constructor\\[2ex]
-  \cdx{Sum}     & $[t, i\to t]\to t$    & general sum type\\
-  \cdx{pair}    & $[i,i]\to i$          & constructor\\
-  \cdx{split}   & $[i,[i,i]\to i]\to i$ & eliminator\\
-  \cdx{fst} \cdx{snd} & $i\to i$        & projections\\[2ex]
-  \cdx{inl} \cdx{inr} & $i\to i$        & constructors for $+$\\
-  \cdx{when}    & $[i,i\to i, i\to i]\to i$    & eliminator for $+$\\[2ex]
-  \cdx{Eq}      & $[t,i,i]\to t$        & equality type\\
-  \cdx{eq}      & $i$                   & constructor\\[2ex]
-  \cdx{F}       & $t$                   & empty type\\
-  \cdx{contr}   & $i\to i$              & eliminator\\[2ex]
-  \cdx{T}       & $t$                   & singleton type\\
-  \cdx{tt}      & $i$                   & constructor
-\end{tabular}
-\end{center}
-\caption{The constants of CTT} \label{ctt-constants}
-\end{figure}
-
-
-CTT supports all of Type Theory apart from list types, well-ordering types,
-and universes.  Universes could be introduced {\em\`a la Tarski}, adding new
-constants as names for types.  The formulation {\em\`a la Russell}, where
-types denote themselves, is only possible if we identify the meta-types~{\tt
-  i} and~{\tt t}.  Most published formulations of well-ordering types have
-difficulties involving extensionality of functions; I suggest that you use
-some other method for defining recursive types.  List types are easy to
-introduce by declaring new rules.
-
-CTT uses the 1982 version of Type Theory, with extensional equality.  The
-computation $a=b\in A$ and the equality $c\in Eq(A,a,b)$ are interchangeable.
-Its rewriting tactics prove theorems of the form $a=b\in A$.  It could be
-modified to have intensional equality, but rewriting tactics would have to
-prove theorems of the form $c\in Eq(A,a,b)$ and the computation rules might
-require a separate simplifier.
-
-
-\begin{figure} \tabcolsep=1em  %wider spacing in tables
-\index{lambda abs@$\lambda$-abstractions!in CTT}
-\begin{center}
-\begin{tabular}{llrrr} 
-  \it symbol &\it name     &\it meta-type & \it priority & \it description \\
-  \sdx{lam} & \cdx{lambda}  & $(i\To o)\To i$ & 10 & $\lambda$-abstraction
-\end{tabular}
-\end{center}
-\subcaption{Binders} 
-
-\begin{center}
-\index{*"` symbol}\index{function applications!in CTT}
-\index{*"+ symbol}
-\begin{tabular}{rrrr} 
-  \it symbol & \it meta-type    & \it priority & \it description \\ 
-  \tt `         & $[i,i]\to i$  & Left 55       & function application\\
-  \tt +         & $[t,t]\to t$  & Right 30      & sum of two types
-\end{tabular}
-\end{center}
-\subcaption{Infixes}
-
-\index{*"* symbol}
-\index{*"-"-"> symbol}
-\begin{center} \tt\frenchspacing
-\begin{tabular}{rrr} 
-  \it external                  & \it internal  & \it standard notation \\ 
-  \sdx{PROD} $x$:$A$ . $B[x]$   &  Prod($A$, $\lambda x. B[x]$) &
-        \rm product $\prod@{x\in A}B[x]$ \\
-  \sdx{SUM} $x$:$A$ . $B[x]$    & Sum($A$, $\lambda x. B[x]$) &
-        \rm sum $\sum@{x\in A}B[x]$ \\
-  $A$ --> $B$     &  Prod($A$, $\lambda x. B$) &
-        \rm function space $A\to B$ \\
-  $A$ * $B$       &  Sum($A$, $\lambda x. B$)  &
-        \rm binary product $A\times B$
-\end{tabular}
-\end{center}
-\subcaption{Translations} 
-
-\index{*"= symbol}
-\begin{center}
-\dquotes
-\[ \begin{array}{rcl}
-prop    & = &  type " type"       \\
-        & | & type " = " type     \\
-        & | & term " : " type        \\
-        & | & term " = " term " : " type 
-\\[2ex]
-type    & = & \hbox{expression of type~$t$} \\
-        & | & "PROD~" id " : " type " . " type  \\
-        & | & "SUM~~" id " : " type " . " type 
-\\[2ex]
-term    & = & \hbox{expression of type~$i$} \\
-        & | & "lam " id~id^* " . " term   \\
-        & | & "< " term " , " term " >"
-\end{array} 
-\]
-\end{center}
-\subcaption{Grammar}
-\caption{Syntax of CTT} \label{ctt-syntax}
-\end{figure}
-
-%%%%\section{Generic Packages}  typedsimp.ML????????????????
-
-
-\section{Syntax}
-The constants are shown in Fig.\ts\ref{ctt-constants}.  The infixes include
-the function application operator (sometimes called `apply'), and the 2-place
-type operators.  Note that meta-level abstraction and application, $\lambda
-x.b$ and $f(a)$, differ from object-level abstraction and application,
-\hbox{\tt lam $x$. $b$} and $b{\tt`}a$.  A CTT function~$f$ is simply an
-individual as far as Isabelle is concerned: its Isabelle type is~$i$, not say
-$i\To i$.
-
-The notation for~CTT (Fig.\ts\ref{ctt-syntax}) is based on that of Nordstr\"om
-et al.~\cite{nordstrom90}.  The empty type is called $F$ and the one-element
-type is $T$; other finite types are built as $T+T+T$, etc.
-
-\index{*SUM symbol}\index{*PROD symbol}
-Quantification is expressed by sums $\sum@{x\in A}B[x]$ and
-products $\prod@{x\in A}B[x]$.  Instead of {\tt Sum($A$,$B$)} and {\tt
-  Prod($A$,$B$)} we may write \hbox{\tt SUM $x$:$A$.\ $B[x]$} and \hbox{\tt
-  PROD $x$:$A$.\ $B[x]$}.  For example, we may write
-\begin{ttbox}
-SUM y:B. PROD x:A. C(x,y)   {\rm for}   Sum(B, \%y. Prod(A, \%x. C(x,y)))
-\end{ttbox}
-The special cases as \hbox{\tt$A$*$B$} and \hbox{\tt$A$-->$B$} abbreviate
-general sums and products over a constant family.\footnote{Unlike normal
-infix operators, {\tt*} and {\tt-->} merely define abbreviations; there are
-no constants~{\tt op~*} and~\hbox{\tt op~-->}.}  Isabelle accepts these
-abbreviations in parsing and uses them whenever possible for printing.
-
-
-\begin{figure} 
-\begin{ttbox}
-\tdx{refl_type}         A type ==> A = A
-\tdx{refl_elem}         a : A ==> a = a : A
-
-\tdx{sym_type}          A = B ==> B = A
-\tdx{sym_elem}          a = b : A ==> b = a : A
-
-\tdx{trans_type}        [| A = B;  B = C |] ==> A = C
-\tdx{trans_elem}        [| a = b : A;  b = c : A |] ==> a = c : A
-
-\tdx{equal_types}       [| a : A;  A = B |] ==> a : B
-\tdx{equal_typesL}      [| a = b : A;  A = B |] ==> a = b : B
-
-\tdx{subst_type}        [| a : A;  !!z. z:A ==> B(z) type |] ==> B(a) type
-\tdx{subst_typeL}       [| a = c : A;  !!z. z:A ==> B(z) = D(z) 
-                  |] ==> B(a) = D(c)
-
-\tdx{subst_elem}        [| a : A;  !!z. z:A ==> b(z):B(z) |] ==> b(a):B(a)
-\tdx{subst_elemL}       [| a = c : A;  !!z. z:A ==> b(z) = d(z) : B(z) 
-                  |] ==> b(a) = d(c) : B(a)
-
-\tdx{refl_red}          Reduce(a,a)
-\tdx{red_if_equal}      a = b : A ==> Reduce(a,b)
-\tdx{trans_red}         [| a = b : A;  Reduce(b,c) |] ==> a = c : A
-\end{ttbox}
-\caption{General equality rules} \label{ctt-equality}
-\end{figure}
-
-
-\begin{figure} 
-\begin{ttbox}
-\tdx{NF}        N type
-
-\tdx{NI0}       0 : N
-\tdx{NI_succ}   a : N ==> succ(a) : N
-\tdx{NI_succL}  a = b : N ==> succ(a) = succ(b) : N
-
-\tdx{NE}        [| p: N;  a: C(0);  
-             !!u v. [| u: N; v: C(u) |] ==> b(u,v): C(succ(u)) 
-          |] ==> rec(p, a, \%u v. b(u,v)) : C(p)
-
-\tdx{NEL}       [| p = q : N;  a = c : C(0);  
-             !!u v. [| u: N; v: C(u) |] ==> b(u,v)=d(u,v): C(succ(u))
-          |] ==> rec(p, a, \%u v. b(u,v)) = rec(q,c,d) : C(p)
-
-\tdx{NC0}       [| a: C(0);  
-             !!u v. [| u: N; v: C(u) |] ==> b(u,v): C(succ(u))
-          |] ==> rec(0, a, \%u v. b(u,v)) = a : C(0)
-
-\tdx{NC_succ}   [| p: N;  a: C(0);  
-             !!u v. [| u: N; v: C(u) |] ==> b(u,v): C(succ(u)) 
-          |] ==> rec(succ(p), a, \%u v. b(u,v)) =
-                 b(p, rec(p, a, \%u v. b(u,v))) : C(succ(p))
-
-\tdx{zero_ne_succ}      [| a: N;  0 = succ(a) : N |] ==> 0: F
-\end{ttbox}
-\caption{Rules for type~$N$} \label{ctt-N}
-\end{figure}
-
-
-\begin{figure} 
-\begin{ttbox}
-\tdx{ProdF}     [| A type; !!x. x:A ==> B(x) type |] ==> PROD x:A. B(x) type
-\tdx{ProdFL}    [| A = C;  !!x. x:A ==> B(x) = D(x) |] ==> 
-          PROD x:A. B(x) = PROD x:C. D(x)
-
-\tdx{ProdI}     [| A type;  !!x. x:A ==> b(x):B(x)
-          |] ==> lam x. b(x) : PROD x:A. B(x)
-\tdx{ProdIL}    [| A type;  !!x. x:A ==> b(x) = c(x) : B(x)
-          |] ==> lam x. b(x) = lam x. c(x) : PROD x:A. B(x)
-
-\tdx{ProdE}     [| p : PROD x:A. B(x);  a : A |] ==> p`a : B(a)
-\tdx{ProdEL}    [| p=q: PROD x:A. B(x);  a=b : A |] ==> p`a = q`b : B(a)
-
-\tdx{ProdC}     [| a : A;  !!x. x:A ==> b(x) : B(x)
-          |] ==> (lam x. b(x)) ` a = b(a) : B(a)
-
-\tdx{ProdC2}    p : PROD x:A. B(x) ==> (lam x. p`x) = p : PROD x:A. B(x)
-\end{ttbox}
-\caption{Rules for the product type $\prod\sb{x\in A}B[x]$} \label{ctt-prod}
-\end{figure}
-
-
-\begin{figure} 
-\begin{ttbox}
-\tdx{SumF}      [| A type;  !!x. x:A ==> B(x) type |] ==> SUM x:A. B(x) type
-\tdx{SumFL}     [| A = C;  !!x. x:A ==> B(x) = D(x) 
-          |] ==> SUM x:A. B(x) = SUM x:C. D(x)
-
-\tdx{SumI}      [| a : A;  b : B(a) |] ==> <a,b> : SUM x:A. B(x)
-\tdx{SumIL}     [| a=c:A;  b=d:B(a) |] ==> <a,b> = <c,d> : SUM x:A. B(x)
-
-\tdx{SumE}      [| p: SUM x:A. B(x);  
-             !!x y. [| x:A; y:B(x) |] ==> c(x,y): C(<x,y>) 
-          |] ==> split(p, \%x y. c(x,y)) : C(p)
-
-\tdx{SumEL}     [| p=q : SUM x:A. B(x); 
-             !!x y. [| x:A; y:B(x) |] ==> c(x,y)=d(x,y): C(<x,y>)
-          |] ==> split(p, \%x y. c(x,y)) = split(q, \%x y. d(x,y)) : C(p)
-
-\tdx{SumC}      [| a: A;  b: B(a);
-             !!x y. [| x:A; y:B(x) |] ==> c(x,y): C(<x,y>)
-          |] ==> split(<a,b>, \%x y. c(x,y)) = c(a,b) : C(<a,b>)
-
-\tdx{fst_def}   fst(a) == split(a, \%x y. x)
-\tdx{snd_def}   snd(a) == split(a, \%x y. y)
-\end{ttbox}
-\caption{Rules for the sum type $\sum\sb{x\in A}B[x]$} \label{ctt-sum}
-\end{figure}
-
-
-\begin{figure} 
-\begin{ttbox}
-\tdx{PlusF}       [| A type;  B type |] ==> A+B type
-\tdx{PlusFL}      [| A = C;  B = D |] ==> A+B = C+D
-
-\tdx{PlusI_inl}   [| a : A;  B type |] ==> inl(a) : A+B
-\tdx{PlusI_inlL}  [| a = c : A;  B type |] ==> inl(a) = inl(c) : A+B
-
-\tdx{PlusI_inr}   [| A type;  b : B |] ==> inr(b) : A+B
-\tdx{PlusI_inrL}  [| A type;  b = d : B |] ==> inr(b) = inr(d) : A+B
-
-\tdx{PlusE}     [| p: A+B;
-             !!x. x:A ==> c(x): C(inl(x));  
-             !!y. y:B ==> d(y): C(inr(y))
-          |] ==> when(p, \%x. c(x), \%y. d(y)) : C(p)
-
-\tdx{PlusEL}    [| p = q : A+B;
-             !!x. x: A ==> c(x) = e(x) : C(inl(x));   
-             !!y. y: B ==> d(y) = f(y) : C(inr(y))
-          |] ==> when(p, \%x. c(x), \%y. d(y)) = 
-                 when(q, \%x. e(x), \%y. f(y)) : C(p)
-
-\tdx{PlusC_inl} [| a: A;
-             !!x. x:A ==> c(x): C(inl(x));  
-             !!y. y:B ==> d(y): C(inr(y))
-          |] ==> when(inl(a), \%x. c(x), \%y. d(y)) = c(a) : C(inl(a))
-
-\tdx{PlusC_inr} [| b: B;
-             !!x. x:A ==> c(x): C(inl(x));  
-             !!y. y:B ==> d(y): C(inr(y))
-          |] ==> when(inr(b), \%x. c(x), \%y. d(y)) = d(b) : C(inr(b))
-\end{ttbox}
-\caption{Rules for the binary sum type $A+B$} \label{ctt-plus}
-\end{figure}
-
-
-\begin{figure} 
-\begin{ttbox}
-\tdx{FF}        F type
-\tdx{FE}        [| p: F;  C type |] ==> contr(p) : C
-\tdx{FEL}       [| p = q : F;  C type |] ==> contr(p) = contr(q) : C
-
-\tdx{TF}        T type
-\tdx{TI}        tt : T
-\tdx{TE}        [| p : T;  c : C(tt) |] ==> c : C(p)
-\tdx{TEL}       [| p = q : T;  c = d : C(tt) |] ==> c = d : C(p)
-\tdx{TC}        p : T ==> p = tt : T)
-\end{ttbox}
-
-\caption{Rules for types $F$ and $T$} \label{ctt-ft}
-\end{figure}
-
-
-\begin{figure} 
-\begin{ttbox}
-\tdx{EqF}       [| A type;  a : A;  b : A |] ==> Eq(A,a,b) type
-\tdx{EqFL}      [| A=B;  a=c: A;  b=d : A |] ==> Eq(A,a,b) = Eq(B,c,d)
-\tdx{EqI}       a = b : A ==> eq : Eq(A,a,b)
-\tdx{EqE}       p : Eq(A,a,b) ==> a = b : A
-\tdx{EqC}       p : Eq(A,a,b) ==> p = eq : Eq(A,a,b)
-\end{ttbox}
-\caption{Rules for the equality type $Eq(A,a,b)$} \label{ctt-eq}
-\end{figure}
-
-
-\begin{figure} 
-\begin{ttbox}
-\tdx{replace_type}    [| B = A;  a : A |] ==> a : B
-\tdx{subst_eqtyparg}  [| a=c : A;  !!z. z:A ==> B(z) type |] ==> B(a)=B(c)
-
-\tdx{subst_prodE}     [| p: Prod(A,B);  a: A;  !!z. z: B(a) ==> c(z): C(z)
-                |] ==> c(p`a): C(p`a)
-
-\tdx{SumIL2}    [| c=a : A;  d=b : B(a) |] ==> <c,d> = <a,b> : Sum(A,B)
-
-\tdx{SumE_fst}  p : Sum(A,B) ==> fst(p) : A
-
-\tdx{SumE_snd}  [| p: Sum(A,B);  A type;  !!x. x:A ==> B(x) type
-          |] ==> snd(p) : B(fst(p))
-\end{ttbox}
-
-\caption{Derived rules for CTT} \label{ctt-derived}
-\end{figure}
-
-
-\section{Rules of inference}
-The rules obey the following naming conventions.  Type formation rules have
-the suffix~{\tt F}\@.  Introduction rules have the suffix~{\tt I}\@.
-Elimination rules have the suffix~{\tt E}\@.  Computation rules, which
-describe the reduction of eliminators, have the suffix~{\tt C}\@.  The
-equality versions of the rules (which permit reductions on subterms) are
-called {\bf long} rules; their names have the suffix~{\tt L}\@.
-Introduction and computation rules are often further suffixed with
-constructor names.
-
-Figure~\ref{ctt-equality} presents the equality rules.  Most of them are
-straightforward: reflexivity, symmetry, transitivity and substitution.  The
-judgement \cdx{Reduce} does not belong to Type Theory proper; it has
-been added to implement rewriting.  The judgement ${\tt Reduce}(a,b)$ holds
-when $a=b:A$ holds.  It also holds when $a$ and $b$ are syntactically
-identical, even if they are ill-typed, because rule {\tt refl_red} does
-not verify that $a$ belongs to $A$.  
-
-The {\tt Reduce} rules do not give rise to new theorems about the standard
-judgements.  The only rule with {\tt Reduce} in a premise is
-{\tt trans_red}, whose other premise ensures that $a$ and~$b$ (and thus
-$c$) are well-typed.
-
-Figure~\ref{ctt-N} presents the rules for~$N$, the type of natural numbers.
-They include \tdx{zero_ne_succ}, which asserts $0\not=n+1$.  This is
-the fourth Peano axiom and cannot be derived without universes \cite[page
-91]{martinlof84}.  
-
-The constant \cdx{rec} constructs proof terms when mathematical
-induction, rule~\tdx{NE}, is applied.  It can also express primitive
-recursion.  Since \cdx{rec} can be applied to higher-order functions,
-it can even express Ackermann's function, which is not primitive recursive
-\cite[page~104]{thompson91}.
-
-Figure~\ref{ctt-prod} shows the rules for general product types, which
-include function types as a special case.  The rules correspond to the
-predicate calculus rules for universal quantifiers and implication.  They
-also permit reasoning about functions, with the rules of a typed
-$\lambda$-calculus.
-
-Figure~\ref{ctt-sum} shows the rules for general sum types, which
-include binary product types as a special case.  The rules correspond to the
-predicate calculus rules for existential quantifiers and conjunction.  They
-also permit reasoning about ordered pairs, with the projections
-\cdx{fst} and~\cdx{snd}.
-
-Figure~\ref{ctt-plus} shows the rules for binary sum types.  They
-correspond to the predicate calculus rules for disjunction.  They also
-permit reasoning about disjoint sums, with the injections \cdx{inl}
-and~\cdx{inr} and case analysis operator~\cdx{when}.
-
-Figure~\ref{ctt-ft} shows the rules for the empty and unit types, $F$
-and~$T$.  They correspond to the predicate calculus rules for absurdity and
-truth.
-
-Figure~\ref{ctt-eq} shows the rules for equality types.  If $a=b\in A$ is
-provable then \cdx{eq} is a canonical element of the type $Eq(A,a,b)$,
-and vice versa.  These rules define extensional equality; the most recent
-versions of Type Theory use intensional equality~\cite{nordstrom90}.
-
-Figure~\ref{ctt-derived} presents the derived rules.  The rule
-\tdx{subst_prodE} is derived from {\tt prodE}, and is easier to use
-in backwards proof.  The rules \tdx{SumE_fst} and \tdx{SumE_snd}
-express the typing of~\cdx{fst} and~\cdx{snd}; together, they are
-roughly equivalent to~{\tt SumE} with the advantage of creating no
-parameters.  Section~\ref{ctt-choice} below demonstrates these rules in a
-proof of the Axiom of Choice.
-
-All the rules are given in $\eta$-expanded form.  For instance, every
-occurrence of $\lambda u\,v. b(u,v)$ could be abbreviated to~$b$ in the
-rules for~$N$.  The expanded form permits Isabelle to preserve bound
-variable names during backward proof.  Names of bound variables in the
-conclusion (here, $u$ and~$v$) are matched with corresponding bound
-variables in the premises.
-
-
-\section{Rule lists}
-The Type Theory tactics provide rewriting, type inference, and logical
-reasoning.  Many proof procedures work by repeatedly resolving certain Type
-Theory rules against a proof state.  CTT defines lists --- each with
-type
-\hbox{\tt thm list} --- of related rules. 
-\begin{ttdescription}
-\item[\ttindexbold{form_rls}] 
-contains formation rules for the types $N$, $\Pi$, $\Sigma$, $+$, $Eq$,
-$F$, and $T$.
-
-\item[\ttindexbold{formL_rls}] 
-contains long formation rules for $\Pi$, $\Sigma$, $+$, and $Eq$.  (For
-other types use \tdx{refl_type}.)
-
-\item[\ttindexbold{intr_rls}] 
-contains introduction rules for the types $N$, $\Pi$, $\Sigma$, $+$, and
-$T$.
-
-\item[\ttindexbold{intrL_rls}] 
-contains long introduction rules for $N$, $\Pi$, $\Sigma$, and $+$.  (For
-$T$ use \tdx{refl_elem}.)
-
-\item[\ttindexbold{elim_rls}] 
-contains elimination rules for the types $N$, $\Pi$, $\Sigma$, $+$, and
-$F$.  The rules for $Eq$ and $T$ are omitted because they involve no
-eliminator.
-
-\item[\ttindexbold{elimL_rls}] 
-contains long elimination rules for $N$, $\Pi$, $\Sigma$, $+$, and $F$.
-
-\item[\ttindexbold{comp_rls}] 
-contains computation rules for the types $N$, $\Pi$, $\Sigma$, and $+$.
-Those for $Eq$ and $T$ involve no eliminator.
-
-\item[\ttindexbold{basic_defs}] 
-contains the definitions of {\tt fst} and {\tt snd}.  
-\end{ttdescription}
-
-
-\section{Tactics for subgoal reordering}
-\begin{ttbox}
-test_assume_tac : int -> tactic
-typechk_tac     : thm list -> tactic
-equal_tac       : thm list -> tactic
-intr_tac        : thm list -> tactic
-\end{ttbox}
-Blind application of CTT rules seldom leads to a proof.  The elimination
-rules, especially, create subgoals containing new unknowns.  These subgoals
-unify with anything, creating a huge search space.  The standard tactic
-\ttindex{filt_resolve_tac}
-(see \iflabelundefined{filt_resolve_tac}{the {\em Reference Manual\/}}%
-        {\S\ref{filt_resolve_tac}})
-%
-fails for goals that are too flexible; so does the CTT tactic {\tt
-  test_assume_tac}.  Used with the tactical \ttindex{REPEAT_FIRST} they
-achieve a simple kind of subgoal reordering: the less flexible subgoals are
-attempted first.  Do some single step proofs, or study the examples below,
-to see why this is necessary.
-\begin{ttdescription}
-\item[\ttindexbold{test_assume_tac} $i$] 
-uses {\tt assume_tac} to solve the subgoal by assumption, but only if
-subgoal~$i$ has the form $a\in A$ and the head of $a$ is not an unknown.
-Otherwise, it fails.
-
-\item[\ttindexbold{typechk_tac} $thms$] 
-uses $thms$ with formation, introduction, and elimination rules to check
-the typing of constructions.  It is designed to solve goals of the form
-$a\in \Var{A}$, where $a$ is rigid and $\Var{A}$ is flexible; thus it
-performs type inference.  The tactic can also solve goals of
-the form $A\;\rm type$.
-
-\item[\ttindexbold{equal_tac} $thms$]
-uses $thms$ with the long introduction and elimination rules to solve goals
-of the form $a=b\in A$, where $a$ is rigid.  It is intended for deriving
-the long rules for defined constants such as the arithmetic operators.  The
-tactic can also perform type-checking.
-
-\item[\ttindexbold{intr_tac} $thms$]
-uses $thms$ with the introduction rules to break down a type.  It is
-designed for goals like $\Var{a}\in A$ where $\Var{a}$ is flexible and $A$
-rigid.  These typically arise when trying to prove a proposition~$A$,
-expressed as a type.
-\end{ttdescription}
-
-
-
-\section{Rewriting tactics}
-\begin{ttbox}
-rew_tac     : thm list -> tactic
-hyp_rew_tac : thm list -> tactic
-\end{ttbox}
-Object-level simplification is accomplished through proof, using the {\tt
-  CTT} equality rules and the built-in rewriting functor
-{\tt TSimpFun}.%
-\footnote{This should not be confused with Isabelle's main simplifier; {\tt
-    TSimpFun} is only useful for CTT and similar logics with type inference
-  rules.  At present it is undocumented.}
-%
-The rewrites include the computation rules and other equations.  The long
-versions of the other rules permit rewriting of subterms and subtypes.
-Also used are transitivity and the extra judgement form \cdx{Reduce}.
-Meta-level simplification handles only definitional equality.
-\begin{ttdescription}
-\item[\ttindexbold{rew_tac} $thms$]
-applies $thms$ and the computation rules as left-to-right rewrites.  It
-solves the goal $a=b\in A$ by rewriting $a$ to $b$.  If $b$ is an unknown
-then it is assigned the rewritten form of~$a$.  All subgoals are rewritten.
-
-\item[\ttindexbold{hyp_rew_tac} $thms$]
-is like {\tt rew_tac}, but includes as rewrites any equations present in
-the assumptions.
-\end{ttdescription}
-
-
-\section{Tactics for logical reasoning}
-Interpreting propositions as types lets CTT express statements of
-intuitionistic logic.  However, Constructive Type Theory is not just another
-syntax for first-order logic.  There are fundamental differences.
-
-\index{assumptions!in CTT}
-Can assumptions be deleted after use?  Not every occurrence of a type
-represents a proposition, and Type Theory assumptions declare variables.
-In first-order logic, $\disj$-elimination with the assumption $P\disj Q$
-creates one subgoal assuming $P$ and another assuming $Q$, and $P\disj Q$
-can be deleted safely.  In Type Theory, $+$-elimination with the assumption
-$z\in A+B$ creates one subgoal assuming $x\in A$ and another assuming $y\in
-B$ (for arbitrary $x$ and $y$).  Deleting $z\in A+B$ when other assumptions
-refer to $z$ may render the subgoal unprovable: arguably,
-meaningless.
-
-Isabelle provides several tactics for predicate calculus reasoning in CTT:
-\begin{ttbox}
-mp_tac       : int -> tactic
-add_mp_tac   : int -> tactic
-safestep_tac : thm list -> int -> tactic
-safe_tac     : thm list -> int -> tactic
-step_tac     : thm list -> int -> tactic
-pc_tac       : thm list -> int -> tactic
-\end{ttbox}
-These are loosely based on the intuitionistic proof procedures
-of~\thydx{FOL}.  For the reasons discussed above, a rule that is safe for
-propositional reasoning may be unsafe for type-checking; thus, some of the
-`safe' tactics are misnamed.
-\begin{ttdescription}
-\item[\ttindexbold{mp_tac} $i$] 
-searches in subgoal~$i$ for assumptions of the form $f\in\Pi(A,B)$ and
-$a\in A$, where~$A$ may be found by unification.  It replaces
-$f\in\Pi(A,B)$ by $z\in B(a)$, where~$z$ is a new parameter.  The tactic
-can produce multiple outcomes for each suitable pair of assumptions.  In
-short, {\tt mp_tac} performs Modus Ponens among the assumptions.
-
-\item[\ttindexbold{add_mp_tac} $i$]
-is like {\tt mp_tac}~$i$ but retains the assumption $f\in\Pi(A,B)$.  It
-avoids information loss but obviously loops if repeated.
-
-\item[\ttindexbold{safestep_tac} $thms$ $i$]
-attacks subgoal~$i$ using formation rules and certain other `safe' rules
-(\tdx{FE}, \tdx{ProdI}, \tdx{SumE}, \tdx{PlusE}), calling
-{\tt mp_tac} when appropriate.  It also uses~$thms$,
-which are typically premises of the rule being derived.
-
-\item[\ttindexbold{safe_tac} $thms$ $i$] attempts to solve subgoal~$i$ by
-  means of backtracking, using {\tt safestep_tac}.
-
-\item[\ttindexbold{step_tac} $thms$ $i$]
-tries to reduce subgoal~$i$ using {\tt safestep_tac}, then tries unsafe
-rules.  It may produce multiple outcomes.
-
-\item[\ttindexbold{pc_tac} $thms$ $i$]
-tries to solve subgoal~$i$ by backtracking, using {\tt step_tac}.
-\end{ttdescription}
-
-
-
-\begin{figure} 
-\index{#+@{\tt\#+} symbol}
-\index{*"- symbol}
-\index{#*@{\tt\#*} symbol}
-\index{*div symbol}
-\index{*mod symbol}
-
-\index{absolute difference}
-\index{"!-"!@{\tt\char124-\char124} symbol}
-%\char124 is vertical bar. We use ! because | stopped working
-
-\begin{constants}
-  \it symbol  & \it meta-type & \it priority & \it description \\ 
-  \tt \#*       & $[i,i]\To i$  &  Left 70      & multiplication \\
-  \tt div       & $[i,i]\To i$  &  Left 70      & division\\
-  \tt mod       & $[i,i]\To i$  &  Left 70      & modulus\\
-  \tt \#+       & $[i,i]\To i$  &  Left 65      & addition\\
-  \tt -         & $[i,i]\To i$  &  Left 65      & subtraction\\
-  \verb'|-|'    & $[i,i]\To i$  &  Left 65      & absolute difference
-\end{constants}
-
-\begin{ttbox}
-\tdx{add_def}           a#+b  == rec(a, b, \%u v. succ(v))  
-\tdx{diff_def}          a-b   == rec(b, a, \%u v. rec(v, 0, \%x y. x))  
-\tdx{absdiff_def}       a|-|b == (a-b) #+ (b-a)  
-\tdx{mult_def}          a#*b  == rec(a, 0, \%u v. b #+ v)  
-
-\tdx{mod_def}           a mod b ==
-                  rec(a, 0, \%u v. rec(succ(v) |-| b, 0, \%x y. succ(v)))
-
-\tdx{div_def}           a div b ==
-                  rec(a, 0, \%u v. rec(succ(u) mod b, succ(v), \%x y. v))
-
-\tdx{add_typing}        [| a:N;  b:N |] ==> a #+ b : N
-\tdx{addC0}             b:N ==> 0 #+ b = b : N
-\tdx{addC_succ}         [| a:N;  b:N |] ==> succ(a) #+ b = succ(a #+ b) : N
-
-\tdx{add_assoc}         [| a:N;  b:N;  c:N |] ==> 
-                  (a #+ b) #+ c = a #+ (b #+ c) : N
-
-\tdx{add_commute}       [| a:N;  b:N |] ==> a #+ b = b #+ a : N
-
-\tdx{mult_typing}       [| a:N;  b:N |] ==> a #* b : N
-\tdx{multC0}            b:N ==> 0 #* b = 0 : N
-\tdx{multC_succ}        [| a:N;  b:N |] ==> succ(a) #* b = b #+ (a#*b) : N
-\tdx{mult_commute}      [| a:N;  b:N |] ==> a #* b = b #* a : N
-
-\tdx{add_mult_dist}     [| a:N;  b:N;  c:N |] ==> 
-                  (a #+ b) #* c = (a #* c) #+ (b #* c) : N
-
-\tdx{mult_assoc}        [| a:N;  b:N;  c:N |] ==> 
-                  (a #* b) #* c = a #* (b #* c) : N
-
-\tdx{diff_typing}       [| a:N;  b:N |] ==> a - b : N
-\tdx{diffC0}            a:N ==> a - 0 = a : N
-\tdx{diff_0_eq_0}       b:N ==> 0 - b = 0 : N
-\tdx{diff_succ_succ}    [| a:N;  b:N |] ==> succ(a) - succ(b) = a - b : N
-\tdx{diff_self_eq_0}    a:N ==> a - a = 0 : N
-\tdx{add_inverse_diff}  [| a:N;  b:N;  b-a=0 : N |] ==> b #+ (a-b) = a : N
-\end{ttbox}
-\caption{The theory of arithmetic} \label{ctt_arith}
-\end{figure}
-
-
-\section{A theory of arithmetic}
-\thydx{Arith} is a theory of elementary arithmetic.  It proves the
-properties of addition, multiplication, subtraction, division, and
-remainder, culminating in the theorem
-\[ a \bmod b + (a/b)\times b = a. \]
-Figure~\ref{ctt_arith} presents the definitions and some of the key
-theorems, including commutative, distributive, and associative laws.
-
-The operators~\verb'#+', \verb'-', \verb'|-|', \verb'#*', \verb'mod'
-and~\verb'div' stand for sum, difference, absolute difference, product,
-remainder and quotient, respectively.  Since Type Theory has only primitive
-recursion, some of their definitions may be obscure.  
-
-The difference~$a-b$ is computed by taking $b$ predecessors of~$a$, where
-the predecessor function is $\lambda v. {\tt rec}(v, 0, \lambda x\,y. x)$.
-
-The remainder $a\bmod b$ counts up to~$a$ in a cyclic fashion, using 0
-as the successor of~$b-1$.  Absolute difference is used to test the
-equality $succ(v)=b$.
-
-The quotient $a/b$ is computed by adding one for every number $x$
-such that $0\leq x \leq a$ and $x\bmod b = 0$.
-
-
-
-\section{The examples directory}
-This directory contains examples and experimental proofs in CTT.
-\begin{ttdescription}
-\item[CTT/ex/typechk.ML]
-contains simple examples of type-checking and type deduction.
-
-\item[CTT/ex/elim.ML]
-contains some examples from Martin-L\"of~\cite{martinlof84}, proved using 
-{\tt pc_tac}.
-
-\item[CTT/ex/equal.ML]
-contains simple examples of rewriting.
-
-\item[CTT/ex/synth.ML]
-demonstrates the use of unknowns with some trivial examples of program
-synthesis. 
-\end{ttdescription}
-
-
-\section{Example: type inference}
-Type inference involves proving a goal of the form $a\in\Var{A}$, where $a$
-is a term and $\Var{A}$ is an unknown standing for its type.  The type,
-initially
-unknown, takes shape in the course of the proof.  Our example is the
-predecessor function on the natural numbers.
-\begin{ttbox}
-Goal "lam n. rec(n, 0, \%x y. x) : ?A";
-{\out Level 0}
-{\out lam n. rec(n,0,\%x y. x) : ?A}
-{\out  1. lam n. rec(n,0,\%x y. x) : ?A}
-\end{ttbox}
-Since the term is a Constructive Type Theory $\lambda$-abstraction (not to
-be confused with a meta-level abstraction), we apply the rule
-\tdx{ProdI}, for $\Pi$-introduction.  This instantiates~$\Var{A}$ to a
-product type of unknown domain and range.
-\begin{ttbox}
-by (resolve_tac [ProdI] 1);
-{\out Level 1}
-{\out lam n. rec(n,0,\%x y. x) : PROD x:?A1. ?B1(x)}
-{\out  1. ?A1 type}
-{\out  2. !!n. n : ?A1 ==> rec(n,0,\%x y. x) : ?B1(n)}
-\end{ttbox}
-Subgoal~1 is too flexible.  It can be solved by instantiating~$\Var{A@1}$
-to any type, but most instantiations will invalidate subgoal~2.  We
-therefore tackle the latter subgoal.  It asks the type of a term beginning
-with {\tt rec}, which can be found by $N$-elimination.%
-\index{*NE theorem}
-\begin{ttbox}
-by (eresolve_tac [NE] 2);
-{\out Level 2}
-{\out lam n. rec(n,0,\%x y. x) : PROD x:N. ?C2(x,x)}
-{\out  1. N type}
-{\out  2. !!n. 0 : ?C2(n,0)}
-{\out  3. !!n x y. [| x : N; y : ?C2(n,x) |] ==> x : ?C2(n,succ(x))}
-\end{ttbox}
-Subgoal~1 is no longer flexible: we now know~$\Var{A@1}$ is the type of
-natural numbers.  However, let us continue proving nontrivial subgoals.
-Subgoal~2 asks, what is the type of~0?\index{*NIO theorem}
-\begin{ttbox}
-by (resolve_tac [NI0] 2);
-{\out Level 3}
-{\out lam n. rec(n,0,\%x y. x) : N --> N}
-{\out  1. N type}
-{\out  2. !!n x y. [| x : N; y : N |] ==> x : N}
-\end{ttbox}
-The type~$\Var{A}$ is now fully determined.  It is the product type
-$\prod@{x\in N}N$, which is shown as the function type $N\to N$ because
-there is no dependence on~$x$.  But we must prove all the subgoals to show
-that the original term is validly typed.  Subgoal~2 is provable by
-assumption and the remaining subgoal falls by $N$-formation.%
-\index{*NF theorem}
-\begin{ttbox}
-by (assume_tac 2);
-{\out Level 4}
-{\out lam n. rec(n,0,\%x y. x) : N --> N}
-{\out  1. N type}
-\ttbreak
-by (resolve_tac [NF] 1);
-{\out Level 5}
-{\out lam n. rec(n,0,\%x y. x) : N --> N}
-{\out No subgoals!}
-\end{ttbox}
-Calling \ttindex{typechk_tac} can prove this theorem in one step.
-
-Even if the original term is ill-typed, one can infer a type for it, but
-unprovable subgoals will be left.  As an exercise, try to prove the
-following invalid goal:
-\begin{ttbox}
-Goal "lam n. rec(n, 0, \%x y. tt) : ?A";
-\end{ttbox}
-
-
-
-\section{An example of logical reasoning}
-Logical reasoning in Type Theory involves proving a goal of the form
-$\Var{a}\in A$, where type $A$ expresses a proposition and $\Var{a}$ stands
-for its proof term, a value of type $A$.  The proof term is initially
-unknown and takes shape during the proof.  
-
-Our example expresses a theorem about quantifiers in a sorted logic:
-\[ \infer{(\ex{x\in A}P(x)) \disj (\ex{x\in A}Q(x))}
-         {\ex{x\in A}P(x)\disj Q(x)} 
-\]
-By the propositions-as-types principle, this is encoded
-using~$\Sigma$ and~$+$ types.  A special case of it expresses a
-distributive law of Type Theory: 
-\[ \infer{(A\times B) + (A\times C)}{A\times(B+C)} \]
-Generalizing this from $\times$ to $\Sigma$, and making the typing
-conditions explicit, yields the rule we must derive:
-\[ \infer{\Var{a} \in (\sum@{x\in A} B(x)) + (\sum@{x\in A} C(x))}
-         {\hbox{$A$ type} &
-          \infer*{\hbox{$B(x)$ type}}{[x\in A]}  &
-          \infer*{\hbox{$C(x)$ type}}{[x\in A]}  &
-          p\in \sum@{x\in A} B(x)+C(x)} 
-\]
-To begin, we bind the rule's premises --- returned by the~{\tt goal}
-command --- to the {\ML} variable~{\tt prems}.
-\begin{ttbox}
-val prems = Goal
-    "[| A type;                       \ttback
-\ttback       !!x. x:A ==> B(x) type;       \ttback
-\ttback       !!x. x:A ==> C(x) type;       \ttback
-\ttback       p: SUM x:A. B(x) + C(x)       \ttback
-\ttback    |] ==>  ?a : (SUM x:A. B(x)) + (SUM x:A. C(x))";
-{\out Level 0}
-{\out ?a : (SUM x:A. B(x)) + (SUM x:A. C(x))}
-{\out  1. ?a : (SUM x:A. B(x)) + (SUM x:A. C(x))}
-\ttbreak
-{\out val prems = ["A type  [A type]",}
-{\out              "?x : A ==> B(?x) type  [!!x. x : A ==> B(x) type]",}
-{\out              "?x : A ==> C(?x) type  [!!x. x : A ==> C(x) type]",}
-{\out              "p : SUM x:A. B(x) + C(x)  [p : SUM x:A. B(x) + C(x)]"]}
-{\out             : thm list}
-\end{ttbox}
-The last premise involves the sum type~$\Sigma$.  Since it is a premise
-rather than the assumption of a goal, it cannot be found by {\tt
-  eresolve_tac}.  We could insert it (and the other atomic premise) by
-calling
-\begin{ttbox}
-cut_facts_tac prems 1;
-\end{ttbox}
-A forward proof step is more straightforward here.  Let us resolve the
-$\Sigma$-elimination rule with the premises using~\ttindex{RL}.  This
-inference yields one result, which we supply to {\tt
-  resolve_tac}.\index{*SumE theorem}
-\begin{ttbox}
-by (resolve_tac (prems RL [SumE]) 1);
-{\out Level 1}
-{\out split(p,?c1) : (SUM x:A. B(x)) + (SUM x:A. C(x))}
-{\out  1. !!x y.}
-{\out        [| x : A; y : B(x) + C(x) |] ==>}
-{\out        ?c1(x,y) : (SUM x:A. B(x)) + (SUM x:A. C(x))}
-\end{ttbox}
-The subgoal has two new parameters, $x$ and~$y$.  In the main goal,
-$\Var{a}$ has been instantiated with a \cdx{split} term.  The
-assumption $y\in B(x) + C(x)$ is eliminated next, causing a case split and
-creating the parameter~$xa$.  This inference also inserts~\cdx{when}
-into the main goal.\index{*PlusE theorem}
-\begin{ttbox}
-by (eresolve_tac [PlusE] 1);
-{\out Level 2}
-{\out split(p,\%x y. when(y,?c2(x,y),?d2(x,y)))}
-{\out : (SUM x:A. B(x)) + (SUM x:A. C(x))}
-{\out  1. !!x y xa.}
-{\out        [| x : A; xa : B(x) |] ==>}
-{\out        ?c2(x,y,xa) : (SUM x:A. B(x)) + (SUM x:A. C(x))}
-\ttbreak
-{\out  2. !!x y ya.}
-{\out        [| x : A; ya : C(x) |] ==>}
-{\out        ?d2(x,y,ya) : (SUM x:A. B(x)) + (SUM x:A. C(x))}
-\end{ttbox}
-To complete the proof object for the main goal, we need to instantiate the
-terms $\Var{c@2}(x,y,xa)$ and $\Var{d@2}(x,y,xa)$.  We attack subgoal~1 by
-a~$+$-introduction rule; since the goal assumes $xa\in B(x)$, we take the left
-injection~(\cdx{inl}).
-\index{*PlusI_inl theorem}
-\begin{ttbox}
-by (resolve_tac [PlusI_inl] 1);
-{\out Level 3}
-{\out split(p,\%x y. when(y,\%xa. inl(?a3(x,y,xa)),?d2(x,y)))}
-{\out : (SUM x:A. B(x)) + (SUM x:A. C(x))}
-{\out  1. !!x y xa. [| x : A; xa : B(x) |] ==> ?a3(x,y,xa) : SUM x:A. B(x)}
-{\out  2. !!x y xa. [| x : A; xa : B(x) |] ==> SUM x:A. C(x) type}
-\ttbreak
-{\out  3. !!x y ya.}
-{\out        [| x : A; ya : C(x) |] ==>}
-{\out        ?d2(x,y,ya) : (SUM x:A. B(x)) + (SUM x:A. C(x))}
-\end{ttbox}
-A new subgoal~2 has appeared, to verify that $\sum@{x\in A}C(x)$ is a type.
-Continuing to work on subgoal~1, we apply the $\Sigma$-introduction rule.
-This instantiates the term $\Var{a@3}(x,y,xa)$; the main goal now contains
-an ordered pair, whose components are two new unknowns.%
-\index{*SumI theorem}
-\begin{ttbox}
-by (resolve_tac [SumI] 1);
-{\out Level 4}
-{\out split(p,\%x y. when(y,\%xa. inl(<?a4(x,y,xa),?b4(x,y,xa)>),?d2(x,y)))}
-{\out : (SUM x:A. B(x)) + (SUM x:A. C(x))}
-{\out  1. !!x y xa. [| x : A; xa : B(x) |] ==> ?a4(x,y,xa) : A}
-{\out  2. !!x y xa. [| x : A; xa : B(x) |] ==> ?b4(x,y,xa) : B(?a4(x,y,xa))}
-{\out  3. !!x y xa. [| x : A; xa : B(x) |] ==> SUM x:A. C(x) type}
-{\out  4. !!x y ya.}
-{\out        [| x : A; ya : C(x) |] ==>}
-{\out        ?d2(x,y,ya) : (SUM x:A. B(x)) + (SUM x:A. C(x))}
-\end{ttbox}
-The two new subgoals both hold by assumption.  Observe how the unknowns
-$\Var{a@4}$ and $\Var{b@4}$ are instantiated throughout the proof state.
-\begin{ttbox}
-by (assume_tac 1);
-{\out Level 5}
-{\out split(p,\%x y. when(y,\%xa. inl(<x,?b4(x,y,xa)>),?d2(x,y)))}
-{\out : (SUM x:A. B(x)) + (SUM x:A. C(x))}
-\ttbreak
-{\out  1. !!x y xa. [| x : A; xa : B(x) |] ==> ?b4(x,y,xa) : B(x)}
-{\out  2. !!x y xa. [| x : A; xa : B(x) |] ==> SUM x:A. C(x) type}
-{\out  3. !!x y ya.}
-{\out        [| x : A; ya : C(x) |] ==>}
-{\out        ?d2(x,y,ya) : (SUM x:A. B(x)) + (SUM x:A. C(x))}
-\ttbreak
-by (assume_tac 1);
-{\out Level 6}
-{\out split(p,\%x y. when(y,\%xa. inl(<x,xa>),?d2(x,y)))}
-{\out : (SUM x:A. B(x)) + (SUM x:A. C(x))}
-{\out  1. !!x y xa. [| x : A; xa : B(x) |] ==> SUM x:A. C(x) type}
-{\out  2. !!x y ya.}
-{\out        [| x : A; ya : C(x) |] ==>}
-{\out        ?d2(x,y,ya) : (SUM x:A. B(x)) + (SUM x:A. C(x))}
-\end{ttbox}
-Subgoal~1 is an example of a well-formedness subgoal~\cite{constable86}.
-Such subgoals are usually trivial; this one yields to
-\ttindex{typechk_tac}, given the current list of premises.
-\begin{ttbox}
-by (typechk_tac prems);
-{\out Level 7}
-{\out split(p,\%x y. when(y,\%xa. inl(<x,xa>),?d2(x,y)))}
-{\out : (SUM x:A. B(x)) + (SUM x:A. C(x))}
-{\out  1. !!x y ya.}
-{\out        [| x : A; ya : C(x) |] ==>}
-{\out        ?d2(x,y,ya) : (SUM x:A. B(x)) + (SUM x:A. C(x))}
-\end{ttbox}
-This subgoal is the other case from the $+$-elimination above, and can be
-proved similarly.  Quicker is to apply \ttindex{pc_tac}.  The main goal
-finally gets a fully instantiated proof object.
-\begin{ttbox}
-by (pc_tac prems 1);
-{\out Level 8}
-{\out split(p,\%x y. when(y,\%xa. inl(<x,xa>),\%y. inr(<x,y>)))}
-{\out : (SUM x:A. B(x)) + (SUM x:A. C(x))}
-{\out No subgoals!}
-\end{ttbox}
-Calling \ttindex{pc_tac} after the first $\Sigma$-elimination above also
-proves this theorem.
-
-
-\section{Example: deriving a currying functional}
-In simply-typed languages such as {\ML}, a currying functional has the type 
-\[ (A\times B \to C) \to (A\to (B\to C)). \]
-Let us generalize this to the dependent types~$\Sigma$ and~$\Pi$.  
-The functional takes a function~$f$ that maps $z:\Sigma(A,B)$
-to~$C(z)$; the resulting function maps $x\in A$ and $y\in B(x)$ to
-$C(\langle x,y\rangle)$.
-
-Formally, there are three typing premises.  $A$ is a type; $B$ is an
-$A$-indexed family of types; $C$ is a family of types indexed by
-$\Sigma(A,B)$.  The goal is expressed using \hbox{\tt PROD f} to ensure
-that the parameter corresponding to the functional's argument is really
-called~$f$; Isabelle echoes the type using \verb|-->| because there is no
-explicit dependence upon~$f$.
-\begin{ttbox}
-val prems = Goal
-    "[| A type; !!x. x:A ==> B(x) type;                           \ttback
-\ttback               !!z. z: (SUM x:A. B(x)) ==> C(z) type             \ttback
-\ttback    |] ==> ?a : PROD f: (PROD z : (SUM x:A . B(x)) . C(z)).      \ttback
-\ttback                     (PROD x:A . PROD y:B(x) . C(<x,y>))";
-\ttbreak
-{\out Level 0}
-{\out ?a : (PROD z:SUM x:A. B(x). C(z)) -->}
-{\out      (PROD x:A. PROD y:B(x). C(<x,y>))}
-{\out  1. ?a : (PROD z:SUM x:A. B(x). C(z)) -->}
-{\out          (PROD x:A. PROD y:B(x). C(<x,y>))}
-\ttbreak
-{\out val prems = ["A type  [A type]",}
-{\out              "?x : A ==> B(?x) type  [!!x. x : A ==> B(x) type]",}
-{\out              "?z : SUM x:A. B(x) ==> C(?z) type}
-{\out               [!!z. z : SUM x:A. B(x) ==> C(z) type]"] : thm list}
-\end{ttbox}
-This is a chance to demonstrate \ttindex{intr_tac}.  Here, the tactic
-repeatedly applies $\Pi$-introduction and proves the rather
-tiresome typing conditions.  
-
-Note that $\Var{a}$ becomes instantiated to three nested
-$\lambda$-abstractions.  It would be easier to read if the bound variable
-names agreed with the parameters in the subgoal.  Isabelle attempts to give
-parameters the same names as corresponding bound variables in the goal, but
-this does not always work.  In any event, the goal is logically correct.
-\begin{ttbox}
-by (intr_tac prems);
-{\out Level 1}
-{\out lam x xa xb. ?b7(x,xa,xb)}
-{\out : (PROD z:SUM x:A. B(x). C(z)) --> (PROD x:A. PROD y:B(x). C(<x,y>))}
-{\out  1. !!f x y.}
-{\out        [| f : PROD z:SUM x:A. B(x). C(z); x : A; y : B(x) |] ==>}
-{\out        ?b7(f,x,y) : C(<x,y>)}
-\end{ttbox}
-Using $\Pi$-elimination, we solve subgoal~1 by applying the function~$f$.
-\index{*ProdE theorem}
-\begin{ttbox}
-by (eresolve_tac [ProdE] 1);
-{\out Level 2}
-{\out lam x xa xb. x ` <xa,xb>}
-{\out : (PROD z:SUM x:A. B(x). C(z)) --> (PROD x:A. PROD y:B(x). C(<x,y>))}
-{\out  1. !!f x y. [| x : A; y : B(x) |] ==> <x,y> : SUM x:A. B(x)}
-\end{ttbox}
-Finally, we verify that the argument's type is suitable for the function
-application.  This is straightforward using introduction rules.
-\index{*intr_tac}
-\begin{ttbox}
-by (intr_tac prems);
-{\out Level 3}
-{\out lam x xa xb. x ` <xa,xb>}
-{\out : (PROD z:SUM x:A. B(x). C(z)) --> (PROD x:A. PROD y:B(x). C(<x,y>))}
-{\out No subgoals!}
-\end{ttbox}
-Calling~\ttindex{pc_tac} would have proved this theorem in one step; it can
-also prove an example by Martin-L\"of, related to $\disj$-elimination
-\cite[page~58]{martinlof84}.
-
-
-\section{Example: proving the Axiom of Choice} \label{ctt-choice}
-Suppose we have a function $h\in \prod@{x\in A}\sum@{y\in B(x)} C(x,y)$,
-which takes $x\in A$ to some $y\in B(x)$ paired with some $z\in C(x,y)$.
-Interpreting propositions as types, this asserts that for all $x\in A$
-there exists $y\in B(x)$ such that $C(x,y)$.  The Axiom of Choice asserts
-that we can construct a function $f\in \prod@{x\in A}B(x)$ such that
-$C(x,f{\tt`}x)$ for all $x\in A$, where the latter property is witnessed by a
-function $g\in \prod@{x\in A}C(x,f{\tt`}x)$.
-
-In principle, the Axiom of Choice is simple to derive in Constructive Type
-Theory.  The following definitions work:
-\begin{eqnarray*}
-    f & \equiv & {\tt fst} \circ h \\
-    g & \equiv & {\tt snd} \circ h
-\end{eqnarray*}
-But a completely formal proof is hard to find.  The rules can be applied in
-countless ways, yielding many higher-order unifiers.  The proof can get
-bogged down in the details.  But with a careful selection of derived rules
-(recall Fig.\ts\ref{ctt-derived}) and the type-checking tactics, we can
-prove the theorem in nine steps.
-\begin{ttbox}
-val prems = Goal
-    "[| A type;  !!x. x:A ==> B(x) type;                    \ttback
-\ttback       !!x y.[| x:A;  y:B(x) |] ==> C(x,y) type            \ttback
-\ttback    |] ==> ?a : PROD h: (PROD x:A. SUM y:B(x). C(x,y)).    \ttback
-\ttback                     (SUM f: (PROD x:A. B(x)). PROD x:A. C(x, f`x))";
-{\out Level 0}
-{\out ?a : (PROD x:A. SUM y:B(x). C(x,y)) -->}
-{\out      (SUM f:PROD x:A. B(x). PROD x:A. C(x,f ` x))}
-{\out  1. ?a : (PROD x:A. SUM y:B(x). C(x,y)) -->}
-{\out          (SUM f:PROD x:A. B(x). PROD x:A. C(x,f ` x))}
-\ttbreak
-{\out val prems = ["A type  [A type]",}
-{\out              "?x : A ==> B(?x) type  [!!x. x : A ==> B(x) type]",}
-{\out              "[| ?x : A; ?y : B(?x) |] ==> C(?x, ?y) type}
-{\out               [!!x y. [| x : A; y : B(x) |] ==> C(x, y) type]"]}
-{\out             : thm list}
-\end{ttbox}
-First, \ttindex{intr_tac} applies introduction rules and performs routine
-type-checking.  This instantiates~$\Var{a}$ to a construction involving
-a $\lambda$-abstraction and an ordered pair.  The pair's components are
-themselves $\lambda$-abstractions and there is a subgoal for each.
-\begin{ttbox}
-by (intr_tac prems);
-{\out Level 1}
-{\out lam x. <lam xa. ?b7(x,xa),lam xa. ?b8(x,xa)>}
-{\out : (PROD x:A. SUM y:B(x). C(x,y)) -->}
-{\out   (SUM f:PROD x:A. B(x). PROD x:A. C(x,f ` x))}
-\ttbreak
-{\out  1. !!h x.}
-{\out        [| h : PROD x:A. SUM y:B(x). C(x,y); x : A |] ==>}
-{\out        ?b7(h,x) : B(x)}
-\ttbreak
-{\out  2. !!h x.}
-{\out        [| h : PROD x:A. SUM y:B(x). C(x,y); x : A |] ==>}
-{\out        ?b8(h,x) : C(x,(lam x. ?b7(h,x)) ` x)}
-\end{ttbox}
-Subgoal~1 asks to find the choice function itself, taking $x\in A$ to some
-$\Var{b@7}(h,x)\in B(x)$.  Subgoal~2 asks, given $x\in A$, for a proof
-object $\Var{b@8}(h,x)$ to witness that the choice function's argument and
-result lie in the relation~$C$.  This latter task will take up most of the
-proof.
-\index{*ProdE theorem}\index{*SumE_fst theorem}\index{*RS}
-\begin{ttbox}
-by (eresolve_tac [ProdE RS SumE_fst] 1);
-{\out Level 2}
-{\out lam x. <lam xa. fst(x ` xa),lam xa. ?b8(x,xa)>}
-{\out : (PROD x:A. SUM y:B(x). C(x,y)) -->}
-{\out   (SUM f:PROD x:A. B(x). PROD x:A. C(x,f ` x))}
-\ttbreak
-{\out  1. !!h x. x : A ==> x : A}
-{\out  2. !!h x.}
-{\out        [| h : PROD x:A. SUM y:B(x). C(x,y); x : A |] ==>}
-{\out        ?b8(h,x) : C(x,(lam x. fst(h ` x)) ` x)}
-\end{ttbox}
-Above, we have composed {\tt fst} with the function~$h$.  Unification
-has deduced that the function must be applied to $x\in A$.  We have our
-choice function.
-\begin{ttbox}
-by (assume_tac 1);
-{\out Level 3}
-{\out lam x. <lam xa. fst(x ` xa),lam xa. ?b8(x,xa)>}
-{\out : (PROD x:A. SUM y:B(x). C(x,y)) -->}
-{\out   (SUM f:PROD x:A. B(x). PROD x:A. C(x,f ` x))}
-{\out  1. !!h x.}
-{\out        [| h : PROD x:A. SUM y:B(x). C(x,y); x : A |] ==>}
-{\out        ?b8(h,x) : C(x,(lam x. fst(h ` x)) ` x)}
-\end{ttbox}
-Before we can compose {\tt snd} with~$h$, the arguments of $C$ must be
-simplified.  The derived rule \tdx{replace_type} lets us replace a type
-by any equivalent type, shown below as the schematic term $\Var{A@{13}}(h,x)$:
-\begin{ttbox}
-by (resolve_tac [replace_type] 1);
-{\out Level 4}
-{\out lam x. <lam xa. fst(x ` xa),lam xa. ?b8(x,xa)>}
-{\out : (PROD x:A. SUM y:B(x). C(x,y)) -->}
-{\out   (SUM f:PROD x:A. B(x). PROD x:A. C(x,f ` x))}
-\ttbreak
-{\out  1. !!h x.}
-{\out        [| h : PROD x:A. SUM y:B(x). C(x,y); x : A |] ==>}
-{\out        C(x,(lam x. fst(h ` x)) ` x) = ?A13(h,x)}
-\ttbreak
-{\out  2. !!h x.}
-{\out        [| h : PROD x:A. SUM y:B(x). C(x,y); x : A |] ==>}
-{\out        ?b8(h,x) : ?A13(h,x)}
-\end{ttbox}
-The derived rule \tdx{subst_eqtyparg} lets us simplify a type's
-argument (by currying, $C(x)$ is a unary type operator):
-\begin{ttbox}
-by (resolve_tac [subst_eqtyparg] 1);
-{\out Level 5}
-{\out lam x. <lam xa. fst(x ` xa),lam xa. ?b8(x,xa)>}
-{\out : (PROD x:A. SUM y:B(x). C(x,y)) -->}
-{\out   (SUM f:PROD x:A. B(x). PROD x:A. C(x,f ` x))}
-\ttbreak
-{\out  1. !!h x.}
-{\out        [| h : PROD x:A. SUM y:B(x). C(x,y); x : A |] ==>}
-{\out        (lam x. fst(h ` x)) ` x = ?c14(h,x) : ?A14(h,x)}
-\ttbreak
-{\out  2. !!h x z.}
-{\out        [| h : PROD x:A. SUM y:B(x). C(x,y); x : A;}
-{\out           z : ?A14(h,x) |] ==>}
-{\out        C(x,z) type}
-\ttbreak
-{\out  3. !!h x.}
-{\out        [| h : PROD x:A. SUM y:B(x). C(x,y); x : A |] ==>}
-{\out        ?b8(h,x) : C(x,?c14(h,x))}
-\end{ttbox}
-Subgoal~1 requires simply $\beta$-contraction, which is the rule
-\tdx{ProdC}.  The term $\Var{c@{14}}(h,x)$ in the last subgoal
-receives the contracted result.
-\begin{ttbox}
-by (resolve_tac [ProdC] 1);
-{\out Level 6}
-{\out lam x. <lam xa. fst(x ` xa),lam xa. ?b8(x,xa)>}
-{\out : (PROD x:A. SUM y:B(x). C(x,y)) -->}
-{\out   (SUM f:PROD x:A. B(x). PROD x:A. C(x,f ` x))}
-\ttbreak
-{\out  1. !!h x.}
-{\out        [| h : PROD x:A. SUM y:B(x). C(x,y); x : A |] ==>}
-{\out        x : ?A15(h,x)}
-\ttbreak
-{\out  2. !!h x xa.}
-{\out        [| h : PROD x:A. SUM y:B(x). C(x,y); x : A;}
-{\out           xa : ?A15(h,x) |] ==>}
-{\out        fst(h ` xa) : ?B15(h,x,xa)}
-\ttbreak
-{\out  3. !!h x z.}
-{\out        [| h : PROD x:A. SUM y:B(x). C(x,y); x : A;}
-{\out           z : ?B15(h,x,x) |] ==>}
-{\out        C(x,z) type}
-\ttbreak
-{\out  4. !!h x.}
-{\out        [| h : PROD x:A. SUM y:B(x). C(x,y); x : A |] ==>}
-{\out        ?b8(h,x) : C(x,fst(h ` x))}
-\end{ttbox}
-Routine type-checking goals proliferate in Constructive Type Theory, but
-\ttindex{typechk_tac} quickly solves them.  Note the inclusion of
-\tdx{SumE_fst} along with the premises.
-\begin{ttbox}
-by (typechk_tac (SumE_fst::prems));
-{\out Level 7}
-{\out lam x. <lam xa. fst(x ` xa),lam xa. ?b8(x,xa)>}
-{\out : (PROD x:A. SUM y:B(x). C(x,y)) -->}
-{\out   (SUM f:PROD x:A. B(x). PROD x:A. C(x,f ` x))}
-\ttbreak
-{\out  1. !!h x.}
-{\out        [| h : PROD x:A. SUM y:B(x). C(x,y); x : A |] ==>}
-{\out        ?b8(h,x) : C(x,fst(h ` x))}
-\end{ttbox}
-We are finally ready to compose {\tt snd} with~$h$.
-\index{*ProdE theorem}\index{*SumE_snd theorem}\index{*RS}
-\begin{ttbox}
-by (eresolve_tac [ProdE RS SumE_snd] 1);
-{\out Level 8}
-{\out lam x. <lam xa. fst(x ` xa),lam xa. snd(x ` xa)>}
-{\out : (PROD x:A. SUM y:B(x). C(x,y)) -->}
-{\out   (SUM f:PROD x:A. B(x). PROD x:A. C(x,f ` x))}
-\ttbreak
-{\out  1. !!h x. x : A ==> x : A}
-{\out  2. !!h x. x : A ==> B(x) type}
-{\out  3. !!h x xa. [| x : A; xa : B(x) |] ==> C(x,xa) type}
-\end{ttbox}
-The proof object has reached its final form.  We call \ttindex{typechk_tac}
-to finish the type-checking.
-\begin{ttbox}
-by (typechk_tac prems);
-{\out Level 9}
-{\out lam x. <lam xa. fst(x ` xa),lam xa. snd(x ` xa)>}
-{\out : (PROD x:A. SUM y:B(x). C(x,y)) -->}
-{\out   (SUM f:PROD x:A. B(x). PROD x:A. C(x,f ` x))}
-{\out No subgoals!}
-\end{ttbox}
-It might be instructive to compare this proof with Martin-L\"of's forward
-proof of the Axiom of Choice \cite[page~50]{martinlof84}.
-
-\index{Constructive Type Theory|)}




