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+%% $Id$
+\chapter{Constructive Type Theory}
+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. The logic is complex and many authors have attempted
+to simplify it. Thompson~\cite{thompson91} is a readable and thorough
+account of the theory.
+
+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 directory~\ttindexbold{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.
+
+The theory has the {\ML} identifier \ttindexbold{CTT.thy}. It does not
+use polymorphism. Terms in {\CTT} have type~$i$, the type of individuals.
+Types in {\CTT} have type~$t$.
+
+{\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~$i$ and~$o$. 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 second simplifier.
+
+
+\begin{figure} \tabcolsep=1em %wider spacing in tables
+\begin{center}
+\begin{tabular}{rrr}
+ \it symbol & \it meta-type & \it description \\
+ \idx{Type} & $t \to prop$ & judgement form \\
+ \idx{Eqtype} & $[t,t]\to prop$ & judgement form\\
+ \idx{Elem} & $[i, t]\to prop$ & judgement form\\
+ \idx{Eqelem} & $[i, i, t]\to prop$ & judgement form\\
+ \idx{Reduce} & $[i, i]\to prop$ & extra judgement form\\[2ex]
+
+ \idx{N} & $t$ & natural numbers type\\
+ \idx{0} & $i$ & constructor\\
+ \idx{succ} & $i\to i$ & constructor\\
+ \idx{rec} & $[i,i,[i,i]\to i]\to i$ & eliminator\\[2ex]
+ \idx{Prod} & $[t,i\to t]\to t$ & general product type\\
+ \idx{lambda} & $(i\to i)\to i$ & constructor\\[2ex]
+ \idx{Sum} & $[t, i\to t]\to t$ & general sum type\\
+ \idx{pair} & $[i,i]\to i$ & constructor\\
+ \idx{split} & $[i,[i,i]\to i]\to i$ & eliminator\\
+ \idx{fst} snd & $i\to i$ & projections\\[2ex]
+ \idx{inl} inr & $i\to i$ & constructors for $+$\\
+ \idx{when} & $[i,i\to i, i\to i]\to i$ & eliminator for $+$\\[2ex]
+ \idx{Eq} & $[t,i,i]\to t$ & equality type\\
+ \idx{eq} & $i$ & constructor\\[2ex]
+ \idx{F} & $t$ & empty type\\
+ \idx{contr} & $i\to i$ & eliminator\\[2ex]
+ \idx{T} & $t$ & singleton type\\
+ \idx{tt} & $i$ & constructor
+\end{tabular}
+\end{center}
+\caption{The constants of {\CTT}} \label{ctt-constants}
+\end{figure}
+
+
+\begin{figure} \tabcolsep=1em %wider spacing in tables
+\begin{center}
+\begin{tabular}{llrrr}
+ \it symbol &\it name &\it meta-type & \it precedence & \it description \\
+ \idx{lam} & \idx{lambda} & $(i\To o)\To i$ & 10 & $\lambda$-abstraction
+\end{tabular}
+\end{center}
+\subcaption{Binders}
+
+\begin{center}
+\indexbold{*"`}
+\indexbold{*"+}
+\begin{tabular}{rrrr}
+ \it symbol & \it meta-type & \it precedence & \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}
+
+\indexbold{*"*}
+\indexbold{*"-"-">}
+\begin{center} \tt\frenchspacing
+\begin{tabular}{rrr}
+ \it external & \it internal & \it standard notation \\
+ \idx{PROD} $x$:$A$ . $B[x]$ & Prod($A$, $\lambda x.B[x]$) &
+ \rm product $\prod@{x\in A}B[x]$ \\
+ \idx{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}
+
+\indexbold{*"=}
+\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 Figure~\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 $i\To i$.
+
+\indexbold{*F}\indexbold{*T}\indexbold{*SUM}\indexbold{*PROD}
+The empty type is called $F$ and the one-element type is $T$; other finite
+sets are built as $T+T+T$, etc. The notation for~{\CTT}
+(Figure~\ref{ctt-syntax}) is based on that of Nordstr\"om et
+al.~\cite{nordstrom90}. We can 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}
+\idx{refl_type} A type ==> A = A
+\idx{refl_elem} a : A ==> a = a : A
+
+\idx{sym_type} A = B ==> B = A
+\idx{sym_elem} a = b : A ==> b = a : A
+
+\idx{trans_type} [| A = B; B = C |] ==> A = C
+\idx{trans_elem} [| a = b : A; b = c : A |] ==> a = c : A
+
+\idx{equal_types} [| a : A; A = B |] ==> a : B
+\idx{equal_typesL} [| a = b : A; A = B |] ==> a = b : B
+
+\idx{subst_type} [| a : A; !!z. z:A ==> B(z) type |] ==> B(a) type
+\idx{subst_typeL} [| a = c : A; !!z. z:A ==> B(z) = D(z)
+ |] ==> B(a) = D(c)
+
+\idx{subst_elem} [| a : A; !!z. z:A ==> b(z):B(z) |] ==> b(a):B(a)
+\idx{subst_elemL} [| a = c : A; !!z. z:A ==> b(z) = d(z) : B(z)
+ |] ==> b(a) = d(c) : B(a)
+
+\idx{refl_red} Reduce(a,a)
+\idx{red_if_equal} a = b : A ==> Reduce(a,b)
+\idx{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}
+\idx{NF} N type
+
+\idx{NI0} 0 : N
+\idx{NI_succ} a : N ==> succ(a) : N
+\idx{NI_succL} a = b : N ==> succ(a) = succ(b) : N
+
+\idx{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)
+
+\idx{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)
+
+\idx{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)
+
+\idx{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))
+
+\idx{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}
+\idx{ProdF} [| A type; !!x. x:A ==> B(x) type |] ==> PROD x:A.B(x) type
+\idx{ProdFL} [| A = C; !!x. x:A ==> B(x) = D(x) |] ==>
+ PROD x:A.B(x) = PROD x:C.D(x)
+
+\idx{ProdI} [| A type; !!x. x:A ==> b(x):B(x)
+ |] ==> lam x.b(x) : PROD x:A.B(x)
+\idx{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)
+
+\idx{ProdE} [| p : PROD x:A.B(x); a : A |] ==> p`a : B(a)
+\idx{ProdEL} [| p=q: PROD x:A.B(x); a=b : A |] ==> p`a = q`b : B(a)
+
+\idx{ProdC} [| a : A; !!x. x:A ==> b(x) : B(x)
+ |] ==> (lam x.b(x)) ` a = b(a) : B(a)
+
+\idx{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 $\prod@{x\in A}B[x]$} \label{ctt-prod}
+\end{figure}
+
+
+\begin{figure}
+\begin{ttbox}
+\idx{SumF} [| A type; !!x. x:A ==> B(x) type |] ==> SUM x:A.B(x) type
+\idx{SumFL} [| A = C; !!x. x:A ==> B(x) = D(x)
+ |] ==> SUM x:A.B(x) = SUM x:C.D(x)
+
+\idx{SumI} [| a : A; b : B(a) |] ==> : SUM x:A.B(x)
+\idx{SumIL} [| a=c:A; b=d:B(a) |] ==> = : SUM x:A.B(x)
+
+\idx{SumE} [| p: SUM x:A.B(x);
+ !!x y. [| x:A; y:B(x) |] ==> c(x,y): C()
+ |] ==> split(p, %x y.c(x,y)) : C(p)
+
+\idx{SumEL} [| p=q : SUM x:A.B(x);
+ !!x y. [| x:A; y:B(x) |] ==> c(x,y)=d(x,y): C()
+ |] ==> split(p, %x y.c(x,y)) = split(q, %x y.d(x,y)) : C(p)
+
+\idx{SumC} [| a: A; b: B(a);
+ !!x y. [| x:A; y:B(x) |] ==> c(x,y): C()
+ |] ==> split(, %x y.c(x,y)) = c(a,b) : C()
+
+\idx{fst_def} fst(a) == split(a, %x y.x)
+\idx{snd_def} snd(a) == split(a, %x y.y)
+\end{ttbox}
+\caption{Rules for the sum type $\sum@{x\in A}B[x]$} \label{ctt-sum}
+\end{figure}
+
+
+\begin{figure}
+\begin{ttbox}
+\idx{PlusF} [| A type; B type |] ==> A+B type
+\idx{PlusFL} [| A = C; B = D |] ==> A+B = C+D
+
+\idx{PlusI_inl} [| a : A; B type |] ==> inl(a) : A+B
+\idx{PlusI_inlL} [| a = c : A; B type |] ==> inl(a) = inl(c) : A+B
+
+\idx{PlusI_inr} [| A type; b : B |] ==> inr(b) : A+B
+\idx{PlusI_inrL} [| A type; b = d : B |] ==> inr(b) = inr(d) : A+B
+
+\idx{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)
+
+\idx{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)
+
+\idx{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))
+
+\idx{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}
+\idx{EqF} [| A type; a : A; b : A |] ==> Eq(A,a,b) type
+\idx{EqFL} [| A=B; a=c: A; b=d : A |] ==> Eq(A,a,b) = Eq(B,c,d)
+\idx{EqI} a = b : A ==> eq : Eq(A,a,b)
+\idx{EqE} p : Eq(A,a,b) ==> a = b : A
+\idx{EqC} p : Eq(A,a,b) ==> p = eq : Eq(A,a,b)
+\end{ttbox}
+\subcaption{The equality type $Eq(A,a,b)$}
+
+\begin{ttbox}
+\idx{FF} F type
+\idx{FE} [| p: F; C type |] ==> contr(p) : C
+\idx{FEL} [| p = q : F; C type |] ==> contr(p) = contr(q) : C
+\end{ttbox}
+\subcaption{The empty type $F$}
+
+\begin{ttbox}
+\idx{TF} T type
+\idx{TI} tt : T
+\idx{TE} [| p : T; c : C(tt) |] ==> c : C(p)
+\idx{TEL} [| p = q : T; c = d : C(tt) |] ==> c = d : C(p)
+\idx{TC} p : T ==> p = tt : T)
+\end{ttbox}
+\subcaption{The unit type $T$}
+
+\caption{Rules for other {\CTT} types} \label{ctt-others}
+\end{figure}
+
+
+
+\begin{figure}
+\begin{ttbox}
+\idx{replace_type} [| B = A; a : A |] ==> a : B
+\idx{subst_eqtyparg} [| a=c : A; !!z. z:A ==> B(z) type |] ==> B(a)=B(c)
+
+\idx{subst_prodE} [| p: Prod(A,B); a: A; !!z. z: B(a) ==> c(z): C(z)
+ |] ==> c(p`a): C(p`a)
+
+\idx{SumIL2} [| c=a : A; d=b : B(a) |] ==> = : Sum(A,B)
+
+\idx{SumE_fst} p : Sum(A,B) ==> fst(p) : A
+
+\idx{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 {\em long} rules; their names have the suffix~{\tt L}\@.
+Introduction and computation rules often are further suffixed with
+constructor names.
+
+Figures~\ref{ctt-equality}--\ref{ctt-others} shows the rules. Those
+for~$N$ include \ttindex{zero_ne_succ}, $0\not=n+1$: the fourth Peano axiom
+cannot be derived without universes \cite[page 91]{martinlof84}.
+Figure~\ref{ctt-sum} shows the rules for general sums, which include binary
+products as a special case, with the projections \ttindex{fst}
+and~\ttindex{snd}.
+
+The extra judgement \ttindex{Reduce} is used 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 \ttindex{refl_red} does not verify that $a$ belongs to $A$. These
+rules do not give rise to new theorems about the standard judgements ---
+note that the only rule that makes use of {\tt Reduce} is \ttindex{trans_red},
+whose first premise ensures that $a$ and $b$ (and thus $c$) are well-typed.
+
+Derived rules are shown in Figure~\ref{ctt-derived}. The rule
+\ttindex{subst_prodE} is derived from \ttindex{prodE}, and is easier to
+use in backwards proof. The rules \ttindex{SumE_fst} and
+\ttindex{SumE_snd} express the typing of~\ttindex{fst} and~\ttindex{snd};
+together, they are roughly equivalent to~\ttindex{SumE} with the advantage
+of creating no parameters. These rules are demonstrated in a proof of the
+Axiom of Choice~(\S\ref{ctt-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~$N$. This 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{description}
+\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 \ttindex{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 \ttindex{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 \ttindex{fst} and \ttindex{snd}.
+\end{description}
+
+
+\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, causing an undirectional search. The standard tactic
+\ttindex{filt_resolve_tac} (see the {\em Reference Manual}) can reject
+overly flexible goals; 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{description}
+\item[\ttindexbold{test_assume_tac} $i$]
+uses \ttindex{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 Hindley-Milner 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{description}
+
+
+
+\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
+\ttindex{TSimpFun}.\footnote{This should not be confused with {\tt
+SimpFun}, which is the main rewriting functor; {\tt TSimpFun} is only
+useful for {\CTT} and similar logics with type inference rules.}
+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 \ttindex{Reduce}.
+Meta-level simplification handles only definitional equality.
+\begin{description}
+\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{description}
+
+
+\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. A key question: 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. 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 $z\in A+B$ may render the subgoals
+unprovable if other assumptions refer to $z$. Some people might argue that
+such subgoals are not even meaningful.
+\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~\ttindex{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{description}
+\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 $f\in\Pi(A,B)$.
+
+\item[\ttindexbold{safestep_tac} $thms$ $i$]
+attacks subgoal~$i$ using formation rules and certain other `safe' rules
+(\ttindex{FE}, \ttindex{ProdI}, \ttindex{SumE}, \ttindex{PlusE}), calling
+{\tt mp_tac} when appropriate. It also uses~$thms$,
+which are typically premises of the rule being derived.
+
+\item[\ttindexbold{safe_tac} $thms$ $i$]
+tries to solve subgoal~$i$ by 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{description}
+
+
+
+\begin{figure}
+\begin{ttbox}
+\idx{add_def} a#+b == rec(a, b, %u v.succ(v))
+\idx{diff_def} a-b == rec(b, a, %u v.rec(v, 0, %x y.x))
+\idx{absdiff_def} a|-|b == (a-b) #+ (b-a)
+\idx{mult_def} a#*b == rec(a, 0, %u v. b #+ v)
+
+\idx{mod_def} a//b == rec(a, 0,
+ %u v. rec(succ(v) |-| b, 0, %x y.succ(v)))
+
+\idx{quo_def} a/b == rec(a, 0,
+ %u v. rec(succ(u) // b, succ(v), %x y.v))
+\end{ttbox}
+\subcaption{Definitions of the operators}
+
+\begin{ttbox}
+\idx{add_typing} [| a:N; b:N |] ==> a #+ b : N
+\idx{addC0} b:N ==> 0 #+ b = b : N
+\idx{addC_succ} [| a:N; b:N |] ==> succ(a) #+ b = succ(a #+ b) : N
+
+\idx{add_assoc} [| a:N; b:N; c:N |] ==>
+ (a #+ b) #+ c = a #+ (b #+ c) : N
+
+\idx{add_commute} [| a:N; b:N |] ==> a #+ b = b #+ a : N
+
+\idx{mult_typing} [| a:N; b:N |] ==> a #* b : N
+\idx{multC0} b:N ==> 0 #* b = 0 : N
+\idx{multC_succ} [| a:N; b:N |] ==> succ(a) #* b = b #+ (a#*b) : N
+\idx{mult_commute} [| a:N; b:N |] ==> a #* b = b #* a : N
+
+\idx{add_mult_dist} [| a:N; b:N; c:N |] ==>
+ (a #+ b) #* c = (a #* c) #+ (b #* c) : N
+
+\idx{mult_assoc} [| a:N; b:N; c:N |] ==>
+ (a #* b) #* c = a #* (b #* c) : N
+
+\idx{diff_typing} [| a:N; b:N |] ==> a - b : N
+\idx{diffC0} a:N ==> a - 0 = a : N
+\idx{diff_0_eq_0} b:N ==> 0 - b = 0 : N
+\idx{diff_succ_succ} [| a:N; b:N |] ==> succ(a) - succ(b) = a - b : N
+\idx{diff_self_eq_0} a:N ==> a - a = 0 : N
+\idx{add_inverse_diff} [| a:N; b:N; b-a=0 : N |] ==> b #+ (a-b) = a : N
+\end{ttbox}
+\subcaption{Some theorems of arithmetic}
+\caption{The theory of arithmetic} \label{ctt-arith}
+\end{figure}
+
+
+\section{A theory of arithmetic}
+{\CTT} contains 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
+theory has the {\ML} identifier \ttindexbold{arith.thy}. All proofs are on
+the file \ttindexbold{CTT/arith.ML}.
+
+The operators~\verb'#+', \verb'-', \verb'|-|', \verb'#*', \verb'//'
+and~\verb'/' 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 $a//b$ 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 $x//b = 0$.
+
+
+
+\section{The examples directory}
+This directory contains examples and experimental proofs in {\CTT}.
+\begin{description}
+\item[\ttindexbold{CTT/ex/typechk.ML}]
+contains simple examples of type checking and type deduction.
+
+\item[\ttindexbold{CTT/ex/elim.ML}]
+contains some examples from Martin-L\"of~\cite{martinlof84}, proved using
+{\tt pc_tac}.
+
+\item[\ttindexbold{CTT/ex/equal.ML}]
+contains simple examples of rewriting.
+
+\item[\ttindexbold{CTT/ex/synth.ML}]
+demonstrates the use of unknowns with some trivial examples of program
+synthesis.
+\end{description}
+
+
+\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 CTT.thy "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
+\ttindex{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 can be solved by instantiating~$\Var{A@1}$ to any type, but this
+could 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}
+\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}
+We now know~$\Var{A@1}$ is the type of natural numbers. However, let us
+continue with subgoal~2. What is the type of~0?\index{*NIO}
+\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 determined. It is $\prod@{n\in N}N$, which is
+equivalent to $N\to N$. 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}
+\begin{ttbox}
+by (assume_tac 2);
+{\out Level 4}
+{\out lam n. rec(n,0,%x y. x) : N --> N}
+{\out 1. N type}
+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.
+
+
+\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 $\Var{a}$ is an
+unknown standing
+for its proof term: a value of type $A$. This term is initially unknown, as
+with type inference, and takes shape during the proof. Our example
+expresses, by propositions-as-types, a theorem about quantifiers in a
+sorted logic:
+\[ \infer{(\ex{x\in A}P(x)) \disj (\ex{x\in A}Q(x))}
+ {\ex{x\in A}P(x)\disj Q(x)}
+\]
+It it related to 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
+\[ \infer{\Var{a} \in (\sum@{x\in A} B(x)) + (\sum@{x\in A} C(x))}
+ {\hbox{$A$ type} &
+ \infer*{\hbox{$B(x)$ type}}{[x\in A]} &
+ \infer*{\hbox{$C(x)$ type}}{[x\in A]} &
+ p\in \sum@{x\in A} B(x)+C(x)}
+\]
+To derive this rule, we bind its premises --- returned by~\ttindex{goal}
+--- to the {\ML} variable~{\tt prems}.
+\begin{ttbox}
+val prems = goal CTT.thy
+ "[| 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))}
+\end{ttbox}
+One of the premises involves summation ($\Sigma$). Since it is a premise
+rather than the assumption of a goal, it cannot be found by
+\ttindex{eresolve_tac}. We could insert it by calling
+\hbox{\tt \ttindex{cut_facts_tac} prems 1}. Instead, let us resolve the
+$\Sigma$-elimination rule with the premises; this yields one result, which
+we supply to \ttindex{resolve_tac}.\index{*SumE}\index{*RL}
+\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. In the main goal, $\Var{a}$ has been
+instantiated with a \ttindex{split} term. The assumption $y\in B(x) + C(x)$ is
+eliminated next, causing a case split and a new parameter. The main goal
+now contains~\ttindex{when}.
+\index{*PlusE}
+\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))}
+{\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
+introduction of~$+$; since it assumes $xa\in B(x)$, we take the left
+injection~(\ttindex{inl}).
+\index{*PlusI_inl}
+\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}
+{\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 has appeared, to verify that $\sum@{x\in A}C(x)$ is a type.
+Continuing with subgoal~1, we apply $\Sigma$-introduction. The main goal
+now contains an ordered pair.
+\index{*SumI}
+\begin{ttbox}
+by (resolve_tac [SumI] 1);
+{\out Level 4}
+{\out split(p,%x y. when(y,%xa. inl(),?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(),?d2(x,y)))}
+{\out : (SUM x:A. B(x)) + (SUM x:A. C(x))}
+{\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))}
+by (assume_tac 1);
+{\out Level 6}
+{\out split(p,%x y. when(y,%xa. inl(),?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 just type checking. It yields to \ttindex{typechk_tac},
+supplied with the current list of premises.
+\begin{ttbox}
+by (typechk_tac prems);
+{\out Level 7}
+{\out split(p,%x y. when(y,%xa. inl(),?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}
+The other case is similar. Let us prove it by \ttindex{pc_tac}, and note
+the final proof object.
+\begin{ttbox}
+by (pc_tac prems 1);
+{\out Level 8}
+{\out split(p,%x y. when(y,%xa. inl(),%y. inr()))}
+{\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~$\Sigma$ and~$\Pi$. The argument of the
+functional is a function that maps $z:\Sigma(A,B)$ to~$C(z)$; the resulting
+function maps $x\in A$ and $y\in B(x)$ to $C(\langle x,y\rangle)$. Here
+$B$ is a family over~$A$, while $C$ is a family over $\Sigma(A,B)$.
+\begin{ttbox}
+val prems = goal CTT.thy
+ "[| A type; !!x. x:A ==> B(x) type; \ttback
+\ttback !!z. z: (SUM x:A. B(x)) ==> C(z) type |] \ttback
+\ttback ==> ?a : (PROD z : (SUM x:A . B(x)) . C(z)) \ttback
+\ttback --> (PROD x:A . PROD y:B(x) . C())";
+{\out Level 0}
+{\out ?a : (PROD z:SUM x:A. B(x). C(z)) --> (PROD x:A. PROD y:B(x). C())}
+{\out 1. ?a : (PROD z:SUM x:A. B(x). C(z)) -->}
+{\out (PROD x:A. PROD y:B(x). C())}
+\end{ttbox}
+This is an opportunity to demonstrate \ttindex{intr_tac}. Here, the tactic
+repeatedly applies $\Pi$-introduction, automatically proving the rather
+tiresome typing conditions. Note that $\Var{a}$ becomes instantiated to
+three nested $\lambda$-abstractions.
+\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())}
+{\out 1. !!uu x y.}
+{\out [| uu : PROD z:SUM x:A. B(x). C(z); x : A; y : B(x) |] ==>}
+{\out ?b7(uu,x,y) : C()}
+\end{ttbox}
+Using $\Pi$-elimination, we solve subgoal~1 by applying the function~$uu$.
+\index{*ProdE}
+\begin{ttbox}
+by (eresolve_tac [ProdE] 1);
+{\out Level 2}
+{\out lam x xa xb. x ` }
+{\out : (PROD z:SUM x:A. B(x). C(z)) --> (PROD x:A. PROD y:B(x). C())}
+{\out 1. !!uu x y. [| x : A; y : B(x) |] ==> : SUM x:A. B(x)}
+\end{ttbox}
+Finally, we exhibit a suitable argument 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 ` }
+{\out : (PROD z:SUM x:A. B(x). C(z)) --> (PROD x:A. PROD y:B(x). C())}
+{\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 \cite[page~50]{martinlof84}. 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. Many of the rules can be
+applied in a multiplicity of ways, yielding a large number of higher-order
+unifiers. The proof can get bogged down in the details. But with a
+careful selection of derived rules (recall Figure~\ref{ctt-derived}) and
+the type checking tactics, we can prove the theorem in nine steps.
+\begin{ttbox}
+val prems = goal CTT.thy
+ "[| 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 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))}
+\end{ttbox}
+First, \ttindex{intr_tac} applies introduction rules and performs routine
+type checking. This instantiates~$\Var{a}$ to a construction involving
+three $\lambda$-abstractions and an ordered pair.
+\begin{ttbox}
+by (intr_tac prems);
+{\out Level 1}
+{\out lam x. }
+{\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. !!uu x.}
+{\out [| uu : PROD x:A. SUM y:B(x). C(x,y); x : A |] ==>}
+{\out ?b7(uu,x) : B(x)}
+{\out 2. !!uu x.}
+{\out [| uu : PROD x:A. SUM y:B(x). C(x,y); x : A |] ==>}
+{\out ?b8(uu,x) : C(x,(lam x. ?b7(uu,x)) ` x)}
+\end{ttbox}
+Subgoal~1 asks to find the choice function itself, taking $x\in A$ to some
+$\Var{b@7}(uu,x)\in B(x)$. Subgoal~2 asks, given $x\in A$, for a proof
+object $\Var{b@8}(uu,x)$ to witness that the choice function's argument
+and result lie in the relation~$C$.
+\index{*ProdE}\index{*SumE_fst}\index{*RS}
+\begin{ttbox}
+by (eresolve_tac [ProdE RS SumE_fst] 1);
+{\out Level 2}
+{\out lam x. }
+{\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. !!uu x. x : A ==> x : A}
+{\out 2. !!uu x.}
+{\out [| uu : PROD x:A. SUM y:B(x). C(x,y); x : A |] ==>}
+{\out ?b8(uu,x) : C(x,(lam x. fst(uu ` x)) ` x)}
+\end{ttbox}
+Above, we have composed \ttindex{fst} with the function~$h$ (named~$uu$ in
+the assumptions). Unification has deduced that the function must be
+applied to $x\in A$.
+\begin{ttbox}
+by (assume_tac 1);
+{\out Level 3}
+{\out lam x. }
+{\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. !!uu x.}
+{\out [| uu : PROD x:A. SUM y:B(x). C(x,y); x : A |] ==>}
+{\out ?b8(uu,x) : C(x,(lam x. fst(uu ` x)) ` x)}
+\end{ttbox}
+Before we can compose \ttindex{snd} with~$h$, the arguments of $C$ must be
+simplified. The derived rule \ttindex{replace_type} lets us replace a type
+by any equivalent type:
+\begin{ttbox}
+by (resolve_tac [replace_type] 1);
+{\out Level 4}
+{\out lam x. }
+{\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. !!uu x.}
+{\out [| uu : PROD x:A. SUM y:B(x). C(x,y); x : A |] ==>}
+{\out C(x,(lam x. fst(uu ` x)) ` x) = ?A13(uu,x)}
+{\out 2. !!uu x.}
+{\out [| uu : PROD x:A. SUM y:B(x). C(x,y); x : A |] ==>}
+{\out ?b8(uu,x) : ?A13(uu,x)}
+\end{ttbox}
+The derived rule \ttindex{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. }
+{\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. !!uu x.}
+{\out [| uu : PROD x:A. SUM y:B(x). C(x,y); x : A |] ==>}
+{\out (lam x. fst(uu ` x)) ` x = ?c14(uu,x) : ?A14(uu,x)}
+{\out 2. !!uu x z.}
+{\out [| uu : PROD x:A. SUM y:B(x). C(x,y); x : A;}
+{\out z : ?A14(uu,x) |] ==>}
+{\out C(x,z) type}
+{\out 3. !!uu x.}
+{\out [| uu : PROD x:A. SUM y:B(x). C(x,y); x : A |] ==>}
+{\out ?b8(uu,x) : C(x,?c14(uu,x))}
+\end{ttbox}
+The rule \ttindex{ProdC} is simply $\beta$-reduction. The term
+$\Var{c@{14}}(uu,x)$ receives the simplified form, $f{\tt`}x$.
+\begin{ttbox}
+by (resolve_tac [ProdC] 1);
+{\out Level 6}
+{\out lam x. }
+{\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. !!uu x.}
+{\out [| uu : PROD x:A. SUM y:B(x). C(x,y); x : A |] ==> x : ?A15(uu,x)}
+{\out 2. !!uu x xa.}
+{\out [| uu : PROD x:A. SUM y:B(x). C(x,y); x : A;}
+{\out xa : ?A15(uu,x) |] ==>}
+{\out fst(uu ` xa) : ?B15(uu,x,xa)}
+{\out 3. !!uu x z.}
+{\out [| uu : PROD x:A. SUM y:B(x). C(x,y); x : A;}
+{\out z : ?B15(uu,x,x) |] ==>}
+{\out C(x,z) type}
+{\out 4. !!uu x.}
+{\out [| uu : PROD x:A. SUM y:B(x). C(x,y); x : A |] ==>}
+{\out ?b8(uu,x) : C(x,fst(uu ` x))}
+\end{ttbox}
+Routine type checking goals proliferate in Constructive Type Theory, but
+\ttindex{typechk_tac} quickly solves them. Note the inclusion of
+\ttindex{SumE_fst}.
+\begin{ttbox}
+by (typechk_tac (SumE_fst::prems));
+{\out Level 7}
+{\out lam x. }
+{\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. !!uu x.}
+{\out [| uu : PROD x:A. SUM y:B(x). C(x,y); x : A |] ==>}
+{\out ?b8(uu,x) : C(x,fst(uu ` x))}
+\end{ttbox}
+We are finally ready to compose \ttindex{snd} with~$h$.
+\index{*ProdE}\index{*SumE_snd}\index{*RS}
+\begin{ttbox}
+by (eresolve_tac [ProdE RS SumE_snd] 1);
+{\out Level 8}
+{\out lam x. }
+{\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. !!uu x. x : A ==> x : A}
+{\out 2. !!uu x. x : A ==> B(x) type}
+{\out 3. !!uu 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. }
+{\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}