author nipkow Mon, 29 Sep 2014 22:09:05 +0200 changeset 58486 f62a887c3ae7 parent 58483 d5f24630c104 child 58487 1ae4ab94ea63
tuned
 src/Doc/Prog_Prove/Isar.thy file | annotate | diff | comparison | revisions src/Doc/Prog_Prove/Types_and_funs.thy file | annotate | diff | comparison | revisions
--- a/src/Doc/Prog_Prove/Isar.thy	Mon Sep 29 21:34:48 2014 +0200
+++ b/src/Doc/Prog_Prove/Isar.thy	Mon Sep 29 22:09:05 2014 +0200
@@ -546,7 +546,7 @@

Sometimes one would like to prove some lemma locally within a proof,
a lemma that shares the current context of assumptions but that
-has its own assumptions and is generalized over its locally fixed
+has its own assumptions and is generalised over its locally fixed
variables at the end. This is what a \concept{raw proof block} does:
\begin{quote}\index{$IMP053@@{text"{ ... }"} (proof block)} @{text"{"} \isacom{fix} @{text"x\<^sub>1 \<dots> x\<^sub>n"}\\ @@ -1031,7 +1031,7 @@ \isacom{lemma} @{text[source]"I x y z \<Longrightarrow> x = r \<Longrightarrow> y = s \<Longrightarrow> z = t \<Longrightarrow> \<dots>"} \end{isabelle} Standard rule induction will work fine now, provided the free variables in -@{text r}, @{text s}, @{text t} are generalized via @{text"arbitrary"}. +@{text r}, @{text s}, @{text t} are generalised via @{text"arbitrary"}. However, induction can do the above transformation for us, behind the curtains, so we never need to see the expanded version of the lemma. This is what we need to write: --- a/src/Doc/Prog_Prove/Types_and_funs.thy Mon Sep 29 21:34:48 2014 +0200 +++ b/src/Doc/Prog_Prove/Types_and_funs.thy Mon Sep 29 22:09:05 2014 +0200 @@ -249,7 +249,7 @@ if the function is defined by recursion on argument number$i\$.}
\end{quote}
The key heuristic, and the main point of this section, is to
-\emph{generalize the goal before induction}.
+\emph{generalise the goal before induction}.
The reason is simple: if the goal is
too specific, the induction hypothesis is too weak to allow the induction
step to go through. Let us illustrate the idea with an example.
@@ -294,10 +294,10 @@
argument,~@{term"[]"}, prevents it from rewriting the conclusion.
This example suggests a heuristic:
\begin{quote}
-\emph{Generalize goals for induction by replacing constants by variables.}
+\emph{Generalise goals for induction by replacing constants by variables.}
\end{quote}
Of course one cannot do this na\"{\i}vely: @{prop"itrev xs ys = rev xs"} is
-just not true.  The correct generalization is
+just not true.  The correct generalisation is
*};
(*<*)oops;(*>*)
lemma "itrev xs ys = rev xs @ ys"
@@ -305,7 +305,7 @@
txt{*
If @{text ys} is replaced by @{term"[]"}, the right-hand side simplifies to
@{term"rev xs"}, as required.
-In this instance it was easy to guess the right generalization.
+In this instance it was easy to guess the right generalisation.
Other situations can require a good deal of creativity.

Although we now have two variables, only @{text xs} is suitable for
@@ -313,12 +313,12 @@
not there:
@{subgoals[display,margin=65]}
The induction hypothesis is still too weak, but this time it takes no
-intuition to generalize: the problem is that the @{text ys}
+intuition to generalise: the problem is that the @{text ys}
in the induction hypothesis is fixed,
but the induction hypothesis needs to be applied with
@{term"a # ys"} instead of @{text ys}. Hence we prove the theorem
for all @{text ys} instead of a fixed one. We can instruct induction
-to perform this generalization for us by adding @{text "arbitrary: ys"}\index{arbitrary@@{text"arbitrary:"}}.
+to perform this generalisation for us by adding @{text "arbitrary: ys"}\index{arbitrary@@{text"arbitrary:"}}.
*}
(*<*)oops
lemma "itrev xs ys = rev xs @ ys"
@@ -334,12 +334,12 @@
done

text{*
-This leads to another heuristic for generalization:
+This leads to another heuristic for generalisation:
\begin{quote}
-\emph{Generalize induction by generalizing all free
+\emph{Generalise induction by generalising all free
variables\\ {\em(except the induction variable itself)}.}
\end{quote}
-Generalization is best performed with @{text"arbitrary: y\<^sub>1 \<dots> y\<^sub>k"}.
+Generalisation is best performed with @{text"arbitrary: y\<^sub>1 \<dots> y\<^sub>k"}.
This heuristic prevents trivial failures like the one above.
However, it should not be applied blindly.
It is not always required, and the additional quantifiers can complicate