(*  Title:      Atoms
    Authors:    Brian Huffman, Christian Urban

    Instantiations of concrete atoms 
*)
theory Atoms
imports Nominal2_Atoms
begin

section {* Manual instantiation of class @{text at}. *}

typedef (open) name = "{a. sort_of a = Sort ''name'' []}"
by (rule exists_eq_simple_sort)

instantiation name :: at
begin

definition
  "p \<bullet> a = Abs_name (p \<bullet> Rep_name a)"

definition
  "atom a = Rep_name a"

instance
apply (rule at_class)
apply (rule type_definition_name)
apply (rule atom_name_def)
apply (rule permute_name_def)
done

end

lemma sort_of_atom_name: 
  shows "sort_of (atom (a::name)) = Sort ''name'' []"
  unfolding atom_name_def using Rep_name by simp

text {* Custom syntax for concrete atoms of type at *}

term "a:::name"

text {* 
  a:::name stands for (atom a) with a being of concrete atom 
  type name. The above lemma can therefore also be stated as

  "sort_of (a:::name) = Sort ''name'' []"

  This does not work for multi-sorted atoms.
*}


section {* Automatic instantiation of class @{text at}. *}

atom_decl name2

lemma sort_of_atom_name2:
  "sort_of (atom (a::name2)) = Sort ''Atoms.name2'' []"
unfolding atom_name2_def 
using Rep_name2 
by simp

text {* example swappings *}
lemma
  fixes a b::"atom"
  assumes "sort_of a = sort_of b"
  shows "(a \<rightleftharpoons> b) \<bullet> (a, b) = (b, a)"
using assms
by simp

lemma
  fixes a b::"name2"
  shows "(a \<leftrightarrow> b) \<bullet> (a, b) = (b, a)"
by simp

section {* An example for multiple-sort atoms *}

datatype ty =
  TVar string
| Fun ty ty ("_ \<rightarrow> _")

primrec
  sort_of_ty::"ty \<Rightarrow> atom_sort"
where
  "sort_of_ty (TVar s) = Sort ''TVar'' [Sort s []]"
| "sort_of_ty (Fun ty1 ty2) = Sort ''Fun'' [sort_of_ty ty1, sort_of_ty ty2]"

lemma sort_of_ty_eq_iff:
  shows "sort_of_ty x = sort_of_ty y \<longleftrightarrow> x = y"
apply(induct x arbitrary: y)
apply(case_tac [!] y)
apply(simp_all)
done

declare sort_of_ty.simps [simp del]

typedef (open) var = "{a. sort_of a \<in> range sort_of_ty}"
  by (rule_tac x="Atom (sort_of_ty x) y" in exI, simp)

instantiation var :: at_base
begin

definition
  "p \<bullet> a = Abs_var (p \<bullet> Rep_var a)"

definition
  "atom a = Rep_var a"

instance
apply (rule at_base_class)
apply (rule type_definition_var)
apply (rule atom_var_def)
apply (rule permute_var_def)
done

end

text {* Constructor for variables. *}

definition
  Var :: "nat \<Rightarrow> ty \<Rightarrow> var"
where
  "Var x t = Abs_var (Atom (sort_of_ty t) x)"

lemma Var_eq_iff [simp]:
  shows "Var x s = Var y t \<longleftrightarrow> x = y \<and> s = t"
  unfolding Var_def
  by (auto simp add: Abs_var_inject sort_of_ty_eq_iff)

lemma sort_of_atom_var [simp]:
  "sort_of (atom (Var n ty)) = sort_of_ty ty"
  unfolding atom_var_def Var_def
  by (simp add: Abs_var_inverse)

lemma 
  assumes "\<alpha> \<noteq> \<beta>" 
  shows "(Var x \<alpha> \<leftrightarrow> Var y \<alpha>) \<bullet> (Var x \<alpha>, Var x \<beta>) = (Var y \<alpha>, Var x \<beta>)"
  using assms by simp

text {* Projecting out the type component of a variable. *}

definition
  ty_of :: "var \<Rightarrow> ty"
where
  "ty_of x = inv sort_of_ty (sort_of (atom x))"

text {*
  Functions @{term Var}/@{term ty_of} satisfy many of the same
  properties as @{term Atom}/@{term sort_of}.
*}

lemma ty_of_Var [simp]:
  shows "ty_of (Var x t) = t"
  unfolding ty_of_def
  unfolding sort_of_atom_var
  apply (rule inv_f_f)
  apply (simp add: inj_on_def sort_of_ty_eq_iff)
  done

lemma ty_of_permute [simp]:
  shows "ty_of (p \<bullet> x) = ty_of x"
  unfolding ty_of_def
  unfolding atom_eqvt [symmetric]
  by simp

end