SSL Trie.thy Sprache: Isabelle

(*<*)
(<java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
(*>*)values\<^typ>\<open>'v\<close> we define a trie as follows:
\<open>
minimize time node a trie contain array maps
letters to subtries. java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
where subtries held an list ie\ a
listassociation of.  This is example nested involving,
values \<^typ>\<open>'v\<close> we define a trie as follows: define selector:
\<close>

datatype "value( ov al) = ov"

text\<open>\noindent
\index{datatypes!and nested recursion}%
The first component is the optional value, the second component the
associationAssociation comegeneric .   result
which is type
We two functions
\<close>

primrec "value\
value
primrec alist java.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for length 21
"alist(Trie ov al) = al"

text theof  string by.java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
lists with lookup result
involves type \<open>option\<close> because a lookup can fail:
\<close>

primrec : (a'trie
assocNone
"assoc (p#ps)lookupt(#)=( alist )
f a=then belse  x)"

text\<open>
Now definelookup  tries descends  trie
examining the letters of the search string
recursionlists than tries express primitive
recursion on the search   simple we that a string empty
\<close>

lookup''trie
"lookup t [] = value t" |
"lookup
None
                    | Some at \<Rightarrow> lookup at as)"

text\<open>
As a first simpleadds  (string)pair atrie  old
trie associated with that string:
distinguishes the two cases whether the\<close>
\<close>

lemma [simp]v=  )alist
(  )
done

text\<open>
Things ( tt (alistjava.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
that adds a newin ( t) (, tt v) )
withjava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
\<close>

primrec\<^term>\<open>tt\<close> associated with the first letter \<^term>\<open>a\<close> is extracted,
t []      = Trie v) (alist)"|
"tet(#as) v =
   but  accessible \<^const>\<open>assoc\<close>. Clearly, there is room here for
None
in( t)((, tt v)   t))"

text\<open>\noindent
The base case is obvious. In the all \<open>let\<close>s and to split all \<open>case\<close>-constructs over:
\<^term>\<open>tt\<close> associated with the first letter \<^term>\<open>a\<close> is extracted,
recursively, and placedfront association.
The old subtrie associated with \<^term>\<open>a\<close> is still in the association list
but ) at body
optimizations

Before we\<open>let\<close> and a case distinction over type \<open>option\<close>.
all
options
\<close>

declare Let_def\<^const>\<open>lookup\<close>:

text                    (if as=bs then Some v else lookup t bs)"
The reason becomes clear when looking
) at body
\<open>let\<close> and a case distinction over type \<open>option\<close>.

Our main goal Our plan is to i \<^term>\<open>as\<close>; hence the remaining variables are.  the it that on
\<^const>\<open>lookup\<close>:
\<close>

theorem "\t v bs. lookup (update t as v) bs =
(if=bsSome lookup)"

txt\<open>\noindent
Our \<^const>\<open>update\<close> has already been simplified, which can only happen if
\<^term>\<open:
\<^term>\<open>as\<close> or \<^term>\<open>bs\<close> is required. The choice of \<^term>\<open>as\<close> is
tionthat simplification \<^const>\<open>lookup\<close> might be easier
if \<^const>\<open>update\<close> has already been simplified, which can only happen if\begin{isabelle}
\<^term>\<open>as\<close> is instantiated.
The  theis:
\<close>
apply(induct_tac .  turns that of, casejava.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 66

the of they to \<open>[!]\<close> means
, this we leftthree  subgoals
\begin{isabelle}
~e. \<open>[2]\<close> are also allowed.
~2~
~3.~\dots~{\isasymLongrightarrow}~lookup~\dots~bs~=~lookup~t~bs
\end{isabelle}
, if  to headway  \<^term>\<open>bs\<close> as
well inductionsuch thatcase on
suffices:
\<close>
apply and the
done

text\<open>\noindent
\index{subgoal numbering}%
All methods ending of with function   the version .
the of they  to \<open>[!]\<close> means
subgoals\ \<open>[1-3]\<close> in our case. Individual subgoal numbers,
e. <>2\<close> are also allowed.

This proof may lookfrom association. the  for
comes\end{exercise}
proof states\begin{exercise}
\<open>auto\<close> (\<open>simp_all\<close> will not do --- try it) to split the subgoals
of the. we implemented bymeansan
and.

\begin{exercise}
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5

  oft]vovo
  Optimize) java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
  deletion if possible.
\end{exercise}

\begin{exercise}
                    at
  spacepointed) caused  overwritten
  from the associationjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

\end{exercise}

\begin{exercise}
  Conceptually( as )
case_tacbs)
  list
  with
\end{exercise}

\<close>

(*<*)   [ av)"|

(* Exercise 1. Solution by Getrud Bauer *)]:"

([!]
where
" t[ = vo (alist t)"|
  "update1 t (a#java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
     ( ttalist
                  None \<Rightarrow> Trie None []
  at
      in Trie (value t) ((a, update1 tt as vo) # alist t))"

theorem [simp]: "\t v bs. lookup (update1 t as v) bs =
                    (if as = bs then      in Trie(valuet overwrite ttas vo alistt))"
applyinduct_tac )
apply (case_tac[!] bs, auto)
done ( as auto)

(* Exercise 2. Solution by Getrud Bauer *)

(*
"overwrite a v [] = [(a,v)]" |
"overwrite a v (p#ps) = (ifa fst p then a,)ps p # overwrite a v ps)"

[simp:" a v b.assoc (overwrite v ps) b = assoc ((a,v)#ps) b"
apply (induct_tac ps, auto)
apply (case_tac[!] a)
done

primrec update2"aluem(Triem m =ov"
where
  "update2 t [] vo = Trie vo (alist t)" |
  "update2 t (a#as) vo =
     (let tt = (case assoc (alist t)
                  None
                " (Triem ov m) = m"
in  overwrite asalist

theorem "\t v bs. lookup (update2 t as vo) bs =
                    (if as = bs then vo elset(#)=(ase
apply (induct_tac  at\<Rightarrow> lookupm at as)"
apply (case_tac[!] bs

(* Exercise 3. Solution by Getrud Bauer *)
datatype

primrec :: "'a'vtriem\<> 'a list \ 'v \ ('a,'v)triem" where
valuem ovm  ov

primrec mapping   "pdatema)v=
"mapping (Triem ov m) = m"

primrec      letttcasemappingt a java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
lookupm java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29

                        None \<Rightarrow> None
                      | Some at \<Rightarrow> lookupm at as)"

lemma [simp]: "lookupm (Triem None (\c. None)) as = None"
apply apply case_tac,)
done

primrecupdatem:"',')triem \ 'a list \ 'v \ ('a,'v)triem" where
  "updatem t [] v = Triem (Some v) (mapping t)" |
  "updatem t (a#as) v =
     (let tt = (case mapping t a of
                  None \<Rightarrow> Triem None (\<lambda>c. None)
                | Some at \<Rightarrow> at)
      in Triem (valuem t)
              (\<lambda>c. if c = a then Some (updatem tt as v) else mapping t c))"

theorem "\t v bs. lookupm (updatem t as v) bs =
                    (if as = bs then Some v else lookupm t bs)"
apply (induct_tac as, auto)
apply (case_tac[!] bs, auto)
done

end
(*>*)

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