# [isabelle-dev] Name for derived quotient_def theorem

Florian Haftmann florian.haftmann at informatik.tu-muenchen.de
Wed Mar 28 21:07:54 CEST 2012

```What I had in mind was something as can be found e.g. in
src/HOL/Library/Dlist.thy:

definition empty :: "'a dlist"
where
"empty = Dlist []"

definition insert :: "'a \<Rightarrow> 'a dlist \<Rightarrow> 'a dlist"
where
"insert x dxs = Dlist (List.insert x (list_of_dlist dxs))"

definition remove :: "'a \<Rightarrow> 'a dlist \<Rightarrow> 'a dlist"
where
"remove x dxs = Dlist (remove1 x (list_of_dlist dxs))"

definition map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a dlist
\<Rightarrow> 'b dlist" where
"map f dxs = Dlist (remdups (List.map f (list_of_dlist dxs)))"

definition filter :: "('a \<Rightarrow> bool) \<Rightarrow> 'a dlist
\<Rightarrow> 'a dlist" where
"filter P dxs = Dlist (List.filter P (list_of_dlist dxs))"

lemma list_of_dlist_empty [simp, code abstract]:
"list_of_dlist empty = []"

lemma list_of_dlist_insert [simp, code abstract]:
"list_of_dlist (insert x dxs) = List.insert x (list_of_dlist dxs)"

lemma list_of_dlist_remove [simp, code abstract]:
"list_of_dlist (remove x dxs) = remove1 x (list_of_dlist dxs)"

lemma list_of_dlist_map [simp, code abstract]:
"list_of_dlist (map f dxs) = remdups (List.map f (list_of_dlist dxs))"

lemma list_of_dlist_filter [simp, code abstract]:
"list_of_dlist (filter P dxs) = List.filter P (list_of_dlist dxs)"

Here, the derived equations *are* simplification rules, in the sense
that they simplify from abstract to concrete values.  This pattern
enables convient proofs of abstract equational properties by using
extensionality and then default simplicifcation.  Hence the idea of
».simp«.  Anyway, I do not object to ».rep«, although the disadvantage
is that the corresponding operation will seldom be named »rep«.

Florian

--

PGP available:
http://home.informatik.tu-muenchen.de/haftmann/pgp/florian_haftmann_at_informatik_tu_muenchen_de

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```