doc-src/Logics/HOL.tex
 changeset 6119 7e3eb9b4df8e parent 6076 560396301672 child 6141 a6922171b396
     1.1 --- a/doc-src/Logics/HOL.tex	Wed Jan 13 15:18:02 1999 +0100
1.2 +++ b/doc-src/Logics/HOL.tex	Wed Jan 13 16:29:50 1999 +0100
1.3 @@ -1871,15 +1871,19 @@
1.4  itself. Since \texttt{list} is a non-empty datatype, \texttt{datatype t = C (t
1.5    list)} is non-empty as well.
1.6
1.7 +
1.8 +\subsubsection{Freeness of the constructors}
1.9 +
1.10  The datatype constructors are automatically defined as functions of their
1.11  respective type:
1.12  $C^j@i :: [\tau^j@{i,1},\dots,\tau^j@{i,m^j@i}] \To (\alpha@1,\dots,\alpha@h)t@j$
1.13 -These functions have certain {\em freeness} properties.  They are distinct:
1.14 +These functions have certain {\em freeness} properties.  They construct
1.15 +distinct values:
1.16  $1.17 C^j@i~x@1~\dots~x@{m^j@i} \neq C^j@{i'}~y@1~\dots~y@{m^j@{i'}} \qquad 1.18 \mbox{for all}~ i \neq i'. 1.19$
1.20 -and they are injective:
1.21 +The constructor functions are injective:
1.22  $1.23 (C^j@i~x@1~\dots~x@{m^j@i} = C^j@i~y@1~\dots~y@{m^j@i}) = 1.24 (x@1 = y@1 \land \dots \land x@{m^j@i} = y@{m^j@i}) 1.25 @@ -1895,7 +1899,9 @@ 1.26 t@j_ord \, x \neq t@j_ord \, y \Imp x \neq y. 1.27$
1.28
1.29 -\medskip The datatype package also provides structural induction rules.  For
1.30 +\subsubsection{Structural induction}
1.31 +
1.32 +The datatype package also provides structural induction rules.  For
1.33  datatypes without nested recursion, this is of the following form:
1.34  \[
1.35  \infer{P@1~x@1 \wedge \dots \wedge P@n~x@n}
1.36 @@ -2183,7 +2189,7 @@
1.37  {\out No subgoals!}
1.38  \ttbreak
1.39  qed_spec_mp "not_Cons_self";
1.40 -{\out val not_Cons_self = "Cons x xs ~= xs";}
1.41 +{\out val not_Cons_self = "Cons x xs ~= xs" : thm}
1.42  \end{ttbox}
1.43  Because both subgoals could have been proved by \texttt{Asm_simp_tac}
1.44  we could have done that in one step:
1.45 @@ -2266,9 +2272,8 @@
1.46
1.47  Datatypes come with a uniform way of defining functions, {\bf primitive
1.48    recursion}.  In principle, one could introduce primitive recursive functions
1.49 -by asserting their reduction rules as new axioms.  Here is a counter-example
1.50 -(you should not do such things yourself):
1.51 -\begin{ttbox}
1.52 +by asserting their reduction rules as new axioms, but this is not recommended:
1.53 +\begin{ttbox}\slshape
1.54  Append = Main +
1.55  consts app :: ['a list, 'a list] => 'a list
1.56  rules
1.57 @@ -2276,7 +2281,7 @@
1.58     app_Cons  "app (x#xs) ys = x#app xs ys"
1.59  end
1.60  \end{ttbox}
1.61 -But asserting axioms brings the danger of accidentally asserting nonsense, as
1.62 +Asserting axioms brings the danger of accidentally asserting nonsense, as
1.63  in \verb$app [] ys = us$.
1.64
1.65  The \ttindex{primrec} declaration is a safe means of defining primitive
1.66 @@ -2311,24 +2316,21 @@
1.67  calls in $r$ are of the form $f \, \dots \, y@i \, \dots$ for some $i$.  There
1.68  must be at most one reduction rule for each constructor.  The order is
1.69  immaterial.  For missing constructors, the function is defined to return a
1.70 -default value.  Also note that all reduction rules are added to the default
1.71 -simpset.
1.72 -
1.73 +default value.
1.74 +
1.75  If you would like to refer to some rule by name, then you must prefix
1.76  the rule with an identifier.  These identifiers, like those in the
1.77  \texttt{rules} section of a theory, will be visible at the \ML\ level.
1.78
1.79  The primitive recursive function can have infix or mixfix syntax:
1.80  \begin{ttbox}\underscoreon
1.81 -Append = List +
1.82  consts "@"  :: ['a list, 'a list] => 'a list  (infixr 60)
1.83  primrec
1.84     "[] @ ys = ys"
1.85     "(x#xs) @ ys = x#(xs @ ys)"
1.86 -end
1.87  \end{ttbox}
1.88
1.89 -The reduction rules for {\tt\at} become part of the default simpset, which
1.90 +The reduction rules become part of the default simpset, which
1.91  leads to short proof scripts:
1.92  \begin{ttbox}\underscoreon
1.93  Goal "(xs @ ys) @ zs = xs @ (ys @ zs)";