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section\<open>References\<close>
text\<open>
  This theory, we introduce a generic reference. All our typed pointers include such 
  a reference, which allows us to distinguish pointers of the same type, but also to 
  iterate over all pointers in a set.\<close>
theory 
  Ref
  imports
    "HOL-Library.Adhoc_Overloading"
    "../preliminaries/Hiding_Type_Variables"
begin

instantiation sum :: (linorder, linorder) linorder
begin
definition less_eq_sum :: "'a + 'b \<Rightarrow> 'a + 'b \<Rightarrow> bool"
  where
    "less_eq_sum t t' = (case t of
      Inl l \<Rightarrow> (case t' of
        Inl l' \<Rightarrow> l \<le> l'
      | Inr r' \<Rightarrow> True)
    | Inr r \<Rightarrow> (case t' of
        Inl l' \<Rightarrow> False
      | Inr r' \<Rightarrow> r \<le> r'))"
definition less_sum :: "'a + 'b \<Rightarrow> 'a + 'b \<Rightarrow> bool"
  where
    "less_sum t t' \<equiv> t \<le> t' \<and> \<not> t' \<le> t"
instance by(standard) (auto simp add: less_eq_sum_def less_sum_def split: sum.splits)
end

type_synonym ref = nat
consts cast :: 'a

end