--- a/doc-src/TutorialI/Trie/Trie.thy Mon Oct 09 09:33:45 2000 +0200
+++ b/doc-src/TutorialI/Trie/Trie.thy Mon Oct 09 10:18:21 2000 +0200
@@ -18,8 +18,8 @@
We define two selector functions:
*};
-consts value :: "('a,'v)trie \\<Rightarrow> 'v option"
- alist :: "('a,'v)trie \\<Rightarrow> ('a * ('a,'v)trie)list";
+consts value :: "('a,'v)trie \<Rightarrow> 'v option"
+ alist :: "('a,'v)trie \<Rightarrow> ('a * ('a,'v)trie)list";
primrec "value(Trie ov al) = ov";
primrec "alist(Trie ov al) = al";
@@ -27,7 +27,7 @@
Association lists come with a generic lookup function:
*};
-consts assoc :: "('key * 'val)list \\<Rightarrow> 'key \\<Rightarrow> 'val option";
+consts assoc :: "('key * 'val)list \<Rightarrow> 'key \<Rightarrow> 'val option";
primrec "assoc [] x = None"
"assoc (p#ps) x =
(let (a,b) = p in if a=x then Some b else assoc ps x)";
@@ -39,20 +39,21 @@
recursion on the search string argument:
*};
-consts lookup :: "('a,'v)trie \\<Rightarrow> 'a list \\<Rightarrow> 'v option";
+consts lookup :: "('a,'v)trie \<Rightarrow> 'a list \<Rightarrow> 'v option";
primrec "lookup t [] = value t"
"lookup t (a#as) = (case assoc (alist t) a of
- None \\<Rightarrow> None
- | Some at \\<Rightarrow> lookup at as)";
+ None \<Rightarrow> None
+ | Some at \<Rightarrow> lookup at as)";
text{*
As a first simple property we prove that looking up a string in the empty
-trie @{term"Trie None []"} always returns @{term"None"}. The proof merely
+trie @{term"Trie None []"} always returns @{term None}. The proof merely
distinguishes the two cases whether the search string is empty or not:
*};
lemma [simp]: "lookup (Trie None []) as = None";
-by(case_tac as, simp_all);
+apply(case_tac as, simp_all);
+done
text{*
Things begin to get interesting with the definition of an update function
@@ -60,24 +61,24 @@
associated with that string:
*};
-consts update :: "('a,'v)trie \\<Rightarrow> 'a list \\<Rightarrow> 'v \\<Rightarrow> ('a,'v)trie";
+consts update :: "('a,'v)trie \<Rightarrow> 'a list \<Rightarrow> 'v \<Rightarrow> ('a,'v)trie";
primrec
"update t [] v = Trie (Some v) (alist t)"
"update t (a#as) v =
(let tt = (case assoc (alist t) a of
- None \\<Rightarrow> Trie None [] | Some at \\<Rightarrow> at)
+ None \<Rightarrow> Trie None [] | Some at \<Rightarrow> at)
in Trie (value t) ((a,update tt as v)#alist t))";
text{*\noindent
The base case is obvious. In the recursive case the subtrie
-@{term"tt"} associated with the first letter @{term"a"} is extracted,
+@{term tt} associated with the first letter @{term a} is extracted,
recursively updated, and then placed in front of the association list.
-The old subtrie associated with @{term"a"} is still in the association list
-but no longer accessible via @{term"assoc"}. Clearly, there is room here for
+The old subtrie associated with @{term a} is still in the association list
+but no longer accessible via @{term assoc}. Clearly, there is room here for
optimizations!
-Before we start on any proofs about @{term"update"} we tell the simplifier to
-expand all @{text"let"}s and to split all @{text"case"}-constructs over
+Before we start on any proofs about @{term update} we tell the simplifier to
+expand all @{text let}s and to split all @{text case}-constructs over
options:
*};
@@ -85,23 +86,23 @@
text{*\noindent
The reason becomes clear when looking (probably after a failed proof
-attempt) at the body of @{term"update"}: it contains both
-@{text"let"} and a case distinction over type @{text"option"}.
+attempt) at the body of @{term update}: it contains both
+@{text let} and a case distinction over type @{text option}.
-Our main goal is to prove the correct interaction of @{term"update"} and
-@{term"lookup"}:
+Our main goal is to prove the correct interaction of @{term update} and
+@{term lookup}:
*};
-theorem "\\<forall>t v bs. lookup (update t as v) bs =
+theorem "\<forall>t v bs. lookup (update t as v) bs =
(if as=bs then Some v else lookup t bs)";
txt{*\noindent
-Our plan is to induct on @{term"as"}; hence the remaining variables are
+Our plan is to induct on @{term as}; hence the remaining variables are
quantified. From the definitions it is clear that induction on either
-@{term"as"} or @{term"bs"} is required. The choice of @{term"as"} is merely
-guided by the intuition that simplification of @{term"lookup"} might be easier
-if @{term"update"} has already been simplified, which can only happen if
-@{term"as"} is instantiated.
+@{term as} or @{term bs} is required. The choice of @{term as} is merely
+guided by the intuition that simplification of @{term lookup} might be easier
+if @{term update} has already been simplified, which can only happen if
+@{term as} is instantiated.
The start of the proof is completely conventional:
*};
apply(induct_tac as, auto);
@@ -113,14 +114,15 @@
~2.~\dots~{\isasymLongrightarrow}~lookup~\dots~bs~=~lookup~t~bs\isanewline
~3.~\dots~{\isasymLongrightarrow}~lookup~\dots~bs~=~lookup~t~bs
\end{isabelle}
-Clearly, if we want to make headway we have to instantiate @{term"bs"} as
+Clearly, if we want to make headway we have to instantiate @{term bs} as
well now. It turns out that instead of induction, case distinction
suffices:
*};
-by(case_tac[!] bs, auto);
+apply(case_tac[!] bs, auto);
+done
text{*\noindent
-All methods ending in @{text"tac"} take an optional first argument that
+All methods ending in @{text tac} take an optional first argument that
specifies the range of subgoals they are applied to, where @{text"[!]"} means
all subgoals, i.e.\ @{text"[1-3]"} in our case. Individual subgoal numbers,
e.g. @{text"[2]"} are also allowed.
@@ -128,8 +130,8 @@
This proof may look surprisingly straightforward. However, note that this
comes at a cost: the proof script is unreadable because the intermediate
proof states are invisible, and we rely on the (possibly brittle) magic of
-@{text"auto"} (@{text"simp_all"} will not do---try it) to split the subgoals
-of the induction up in such a way that case distinction on @{term"bs"} makes
+@{text auto} (@{text simp_all} will not do---try it) to split the subgoals
+of the induction up in such a way that case distinction on @{term bs} makes
sense and solves the proof. Part~\ref{Isar} shows you how to write readable
and stable proofs.
*};