Theory Ref

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section‹References›
text‹
  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.›
theory 
  Ref
  imports
    "HOL-Library.Adhoc_Overloading"
    "../preliminaries/Hiding_Type_Variables"
begin

instantiation sum :: (linorder, linorder) linorder
begin
definition less_eq_sum :: "'a + 'b ⇒ 'a + 'b ⇒ bool"
  where
    "less_eq_sum t t' = (case t of
      Inl l ⇒ (case t' of
        Inl l' ⇒ l ≤ l'
      | Inr r' ⇒ True)
    | Inr r ⇒ (case t' of
        Inl l' ⇒ False
      | Inr r' ⇒ r ≤ r'))"
definition less_sum :: "'a + 'b ⇒ 'a + 'b ⇒ bool"
  where
    "less_sum t t' ≡ t ≤ t' ∧ ¬ t' ≤ 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