Theory Atoms

(*  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 "¬ a ♯ nat_of"
unfolding fresh_def supp_def
proof (clarsimp)
  assume "finite {b. (a ⇌ b) ∙ nat_of ≠ nat_of}"
  hence "finite ({a} ∪ {b. (a ⇌ b) ∙ nat_of ≠ nat_of})"
    by simp
  then obtain b where
    b1: "b ≠ a" and
    b2: "sort_of b = sort_of a" and
    b3: "(a ⇌ b) ∙ nat_of = nat_of"
    by (rule obtain_atom) auto
  have "nat_of a = (a ⇌ b) ∙ (nat_of a)" by (simp add: permute_nat_def)
  also have "… = ((a ⇌ b) ∙ nat_of) ((a ⇌ b) ∙ a)" by (simp add: permute_fun_app_eq)
  also have "… = nat_of ((a ⇌ b) ∙ a)" using b3 by simp
  also have "… = 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 ‹at›.›

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

instantiation name :: at
begin

definition
  "p ∙ a = Abs_name (p ∙ 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'' []"
  by (simp add: atom_name_def Rep_name[simplified])

text ‹Custom syntax for concrete atoms of type at›

term "a:::name"


section ‹Automatic instantiation of class ‹at›.›

atom_decl name2

lemma 
  "sort_of (atom (a::name2)) ≠ sort_of (atom (b::name))"
by (simp add: sort_of_atom_name)


text ‹example swappings›
lemma
  fixes a b::"atom"
  assumes "sort_of a = sort_of b"
  shows "(a ⇌ b) ∙ (a, b) = (b, a)"
using assms
by simp

lemma
  fixes a b::"name2"
  shows "(a ↔ b) ∙ (a, b) = (b, a)"
by simp

section ‹An example for multiple-sort atoms›

datatype ty =
  TVar string
| Fun ty ty ("_ → _")

primrec
  sort_of_ty::"ty ⇒ 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 ⟷ x = y"
apply(induct x arbitrary: y)
apply(case_tac [!] y)
apply(simp_all)
done

declare sort_of_ty.simps [simp del]

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

instantiation var :: at_base
begin

definition
  "p ∙ a = Abs_var (p ∙ 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 ⇒ ty ⇒ 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 ⟷ x = y ∧ 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 "α ≠ β" 
  shows "(Var x α ↔ Var y α) ∙ (Var x α, Var x β) = (Var y α, Var x β)"
  using assms by simp

text ‹Projecting out the type component of a variable.›

definition
  ty_of :: "var ⇒ 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 ∙ x) = ty_of x"
  unfolding ty_of_def
  unfolding atom_eqvt [symmetric]
  by (simp only: sort_of_permute)


section ‹Tests with subtyping and automatic coercions›

declare [[coercion_enabled]]

atom_decl var1
atom_decl var2

declare [[coercion "atom::var1⇒atom"]]

declare [[coercion "atom::var2⇒atom"]]

lemma
  fixes a::"var1" and b::"var2"
  shows "a ♯ t ∧ b ♯ t"
oops


(* does not yet work: it needs image as
   coercion map *)

lemma
  fixes as::"var1 set"
  shows "atom ` as ♯* t"

oops

end