(*  Title:      Atoms

    Authors:    Brian Huffman, Christian Urban

    Instantiations of concrete atoms 

*)

theory Atoms

imports Nominal2_Base

begin

section {* @{const nat_of} is an example of a function 

  without finite support *}

lemma not_fresh_nat_of:

  shows "\<not> a \<sharp> nat_of"

unfolding fresh_def supp_def

proof (clarsimp)

  assume "finite {b. (a \<rightleftharpoons> b) \<bullet> nat_of \<noteq> nat_of}"

  hence "finite ({a} \<union> {b. (a \<rightleftharpoons> b) \<bullet> nat_of \<noteq> nat_of})"

    by simp

  then obtain b where

    b1: "b \<noteq> a" and

    b2: "sort_of b = sort_of a" and

    b3: "(a \<rightleftharpoons> b) \<bullet> nat_of = nat_of"

    by (rule obtain_atom) auto

  have "nat_of a = (a \<rightleftharpoons> b) \<bullet> (nat_of a)" by (simp add: permute_nat_def)

  also have "\<dots> = ((a \<rightleftharpoons> b) \<bullet> nat_of) ((a \<rightleftharpoons> b) \<bullet> a)" by (simp add: permute_fun_app_eq)

  also have "\<dots> = nat_of ((a \<rightleftharpoons> b) \<bullet> a)" using b3 by simp

  also have "\<dots> = nat_of b" using b2 by simp

  finally have "nat_of a = nat_of b" by simp

  with b2 have "a = b" by (simp add: atom_components_eq_iff)

  with b1 show "False" by simp

qed

lemma supp_nat_of:

  shows "supp nat_of = UNIV"

  using not_fresh_nat_of [unfolded fresh_def] by auto

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