(*  Title:      Nominal2_Atoms
    Authors:    Brian Huffman, Christian Urban

    Definitions for concrete atom types. 
*)
theory Nominal2_Atoms
imports Nominal2_Base
uses ("nominal_atoms.ML")
begin

section {* Concrete atom types *}

text {*
  Class @{text at_base} allows types containing multiple sorts of atoms.
  Class @{text at} only allows types with a single sort.
*}

class at_base = pt +
  fixes atom :: "'a \<Rightarrow> atom"
  assumes atom_eq_iff [simp]: "atom a = atom b \<longleftrightarrow> a = b"
  assumes atom_eqvt: "p \<bullet> (atom a) = atom (p \<bullet> a)"

class at = at_base +
  assumes sort_of_atom_eq [simp]: "sort_of (atom a) = sort_of (atom b)"

instance at < at_base ..

lemma supp_at_base: 
  fixes a::"'a::at_base"
  shows "supp a = {atom a}"
  by (simp add: supp_atom [symmetric] supp_def atom_eqvt)

lemma fresh_at_base: 
  shows "a \<sharp> b \<longleftrightarrow> a \<noteq> atom b"
  unfolding fresh_def by (simp add: supp_at_base)

instance at_base < fs
proof qed (simp add: supp_at_base)

lemma at_base_infinite [simp]:
  shows "infinite (UNIV :: 'a::at_base set)" (is "infinite ?U")
proof
  obtain a :: 'a where "True" by auto
  assume "finite ?U"
  hence "finite (atom ` ?U)"
    by (rule finite_imageI)
  then obtain b where b: "b \<notin> atom ` ?U" "sort_of b = sort_of (atom a)"
    by (rule obtain_atom)
  from b(2) have "b = atom ((atom a \<rightleftharpoons> b) \<bullet> a)"
    unfolding atom_eqvt [symmetric]
    by (simp add: swap_atom)
  hence "b \<in> atom ` ?U" by simp
  with b(1) show "False" by simp
qed

lemma swap_at_base_simps [simp]:
  fixes x y::"'a::at_base"
  shows "sort_of (atom x) = sort_of (atom y) \<Longrightarrow> (atom x \<rightleftharpoons> atom y) \<bullet> x = y"
  and   "sort_of (atom x) = sort_of (atom y) \<Longrightarrow> (atom x \<rightleftharpoons> atom y) \<bullet> y = x"
  and   "atom x \<noteq> a \<Longrightarrow> atom x \<noteq> b \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> x = x"
  unfolding atom_eq_iff [symmetric]
  unfolding atom_eqvt [symmetric]
  by simp_all

lemma obtain_at_base:
  assumes X: "finite X"
  obtains a::"'a::at_base" where "atom a \<notin> X"
proof -
  have "inj (atom :: 'a \<Rightarrow> atom)"
    by (simp add: inj_on_def)
  with X have "finite (atom -` X :: 'a set)"
    by (rule finite_vimageI)
  with at_base_infinite have "atom -` X \<noteq> (UNIV :: 'a set)"
    by auto
  then obtain a :: 'a where "atom a \<notin> X"
    by auto
  thus ?thesis ..
qed


section {* A swapping operation for concrete atoms *}
  
definition
  flip :: "'a::at_base \<Rightarrow> 'a \<Rightarrow> perm" ("'(_ \<leftrightarrow> _')")
where
  "(a \<leftrightarrow> b) = (atom a \<rightleftharpoons> atom b)"

lemma flip_self [simp]: "(a \<leftrightarrow> a) = 0"
  unfolding flip_def by (rule swap_self)

lemma flip_commute: "(a \<leftrightarrow> b) = (b \<leftrightarrow> a)"
  unfolding flip_def by (rule swap_commute)

lemma minus_flip [simp]: "- (a \<leftrightarrow> b) = (a \<leftrightarrow> b)"
  unfolding flip_def by (rule minus_swap)

lemma add_flip_cancel: "(a \<leftrightarrow> b) + (a \<leftrightarrow> b) = 0"
  unfolding flip_def by (rule swap_cancel)

lemma permute_flip_cancel [simp]: "(a \<leftrightarrow> b) \<bullet> (a \<leftrightarrow> b) \<bullet> x = x"
  unfolding permute_plus [symmetric] add_flip_cancel by simp

lemma permute_flip_cancel2 [simp]: "(a \<leftrightarrow> b) \<bullet> (b \<leftrightarrow> a) \<bullet> x = x"
  by (simp add: flip_commute)

lemma flip_eqvt: 
  fixes a b c::"'a::at_base"
  shows "p \<bullet> (a \<leftrightarrow> b) = (p \<bullet> a \<leftrightarrow> p \<bullet> b)"
  unfolding flip_def
  by (simp add: swap_eqvt atom_eqvt)

lemma flip_at_base_simps [simp]:
  shows "sort_of (atom a) = sort_of (atom b) \<Longrightarrow> (a \<leftrightarrow> b) \<bullet> a = b"
  and   "sort_of (atom a) = sort_of (atom b) \<Longrightarrow> (a \<leftrightarrow> b) \<bullet> b = a"
  and   "\<lbrakk>a \<noteq> c; b \<noteq> c\<rbrakk> \<Longrightarrow> (a \<leftrightarrow> b) \<bullet> c = c"
  and   "sort_of (atom a) \<noteq> sort_of (atom b) \<Longrightarrow> (a \<leftrightarrow> b) \<bullet> x = x"
  unfolding flip_def
  unfolding atom_eq_iff [symmetric]
  unfolding atom_eqvt [symmetric]
  by simp_all

text {* the following two lemmas do not hold for at_base, 
  only for single sort atoms from at *}

lemma permute_flip_at:
  fixes a b c::"'a::at"
  shows "(a \<leftrightarrow> b) \<bullet> c = (if c = a then b else if c = b then a else c)"
  unfolding flip_def
  apply (rule atom_eq_iff [THEN iffD1])
  apply (subst atom_eqvt [symmetric])
  apply (simp add: swap_atom)
  done

lemma flip_at_simps [simp]:
  fixes a b::"'a::at"
  shows "(a \<leftrightarrow> b) \<bullet> a = b" 
  and   "(a \<leftrightarrow> b) \<bullet> b = a"
  unfolding permute_flip_at by simp_all

lemma flip_fresh_fresh:
  fixes a b::"'a::at_base"
  assumes "atom a \<sharp> x" "atom b \<sharp> x"
  shows "(a \<leftrightarrow> b) \<bullet> x = x"
using assms
by (simp add: flip_def swap_fresh_fresh)

subsection {* Syntax for coercing at-elements to the atom-type *}

(*
syntax
  "_atom_constrain" :: "logic \<Rightarrow> type \<Rightarrow> logic" ("_:::_" [4, 0] 3)

translations
  "_atom_constrain a t" => "atom (_constrain a t)"
*)

subsection {* A lemma for proving instances of class @{text at}. *}

setup {* Sign.add_const_constraint (@{const_name "permute"}, NONE) *}
setup {* Sign.add_const_constraint (@{const_name "atom"}, NONE) *}

text {*
  New atom types are defined as subtypes of @{typ atom}.
*}

lemma exists_eq_simple_sort: 
  shows "\<exists>a. a \<in> {a. sort_of a = s}"
  by (rule_tac x="Atom s 0" in exI, simp)

lemma exists_eq_sort: 
  shows "\<exists>a. a \<in> {a. sort_of a \<in> range sort_fun}"
  by (rule_tac x="Atom (sort_fun x) y" in exI, simp)

lemma at_base_class:
  fixes sort_fun :: "'b \<Rightarrow>atom_sort"
  fixes Rep :: "'a \<Rightarrow> atom" and Abs :: "atom \<Rightarrow> 'a"
  assumes type: "type_definition Rep Abs {a. sort_of a \<in> range sort_fun}"
  assumes atom_def: "\<And>a. atom a = Rep a"
  assumes permute_def: "\<And>p a. p \<bullet> a = Abs (p \<bullet> Rep a)"
  shows "OFCLASS('a, at_base_class)"
proof
  interpret type_definition Rep Abs "{a. sort_of a \<in> range sort_fun}" by (rule type)
  have sort_of_Rep: "\<And>a. sort_of (Rep a) \<in> range sort_fun" using Rep by simp
  fix a b :: 'a and p p1 p2 :: perm
  show "0 \<bullet> a = a"
    unfolding permute_def by (simp add: Rep_inverse)
  show "(p1 + p2) \<bullet> a = p1 \<bullet> p2 \<bullet> a"
    unfolding permute_def by (simp add: Abs_inverse sort_of_Rep)
  show "atom a = atom b \<longleftrightarrow> a = b"
    unfolding atom_def by (simp add: Rep_inject)
  show "p \<bullet> atom a = atom (p \<bullet> a)"
    unfolding permute_def atom_def by (simp add: Abs_inverse sort_of_Rep)
qed

(*
lemma at_class:
  fixes s :: atom_sort
  fixes Rep :: "'a \<Rightarrow> atom" and Abs :: "atom \<Rightarrow> 'a"
  assumes type: "type_definition Rep Abs {a. sort_of a \<in> range (\<lambda>x::unit. s)}"
  assumes atom_def: "\<And>a. atom a = Rep a"
  assumes permute_def: "\<And>p a. p \<bullet> a = Abs (p \<bullet> Rep a)"
  shows "OFCLASS('a, at_class)"
proof
  interpret type_definition Rep Abs "{a. sort_of a \<in> range (\<lambda>x::unit. s)}" by (rule type)
  have sort_of_Rep: "\<And>a. sort_of (Rep a) = s" using Rep by (simp add: image_def)
  fix a b :: 'a and p p1 p2 :: perm
  show "0 \<bullet> a = a"
    unfolding permute_def by (simp add: Rep_inverse)
  show "(p1 + p2) \<bullet> a = p1 \<bullet> p2 \<bullet> a"
    unfolding permute_def by (simp add: Abs_inverse sort_of_Rep)
  show "sort_of (atom a) = sort_of (atom b)"
    unfolding atom_def by (simp add: sort_of_Rep)
  show "atom a = atom b \<longleftrightarrow> a = b"
    unfolding atom_def by (simp add: Rep_inject)
  show "p \<bullet> atom a = atom (p \<bullet> a)"
    unfolding permute_def atom_def by (simp add: Abs_inverse sort_of_Rep)
qed
*)

lemma at_class:
  fixes s :: atom_sort
  fixes Rep :: "'a \<Rightarrow> atom" and Abs :: "atom \<Rightarrow> 'a"
  assumes type: "type_definition Rep Abs {a. sort_of a = s}"
  assumes atom_def: "\<And>a. atom a = Rep a"
  assumes permute_def: "\<And>p a. p \<bullet> a = Abs (p \<bullet> Rep a)"
  shows "OFCLASS('a, at_class)"
proof
  interpret type_definition Rep Abs "{a. sort_of a = s}" by (rule type)
  have sort_of_Rep: "\<And>a. sort_of (Rep a) = s" using Rep by (simp add: image_def)
  fix a b :: 'a and p p1 p2 :: perm
  show "0 \<bullet> a = a"
    unfolding permute_def by (simp add: Rep_inverse)
  show "(p1 + p2) \<bullet> a = p1 \<bullet> p2 \<bullet> a"
    unfolding permute_def by (simp add: Abs_inverse sort_of_Rep)
  show "sort_of (atom a) = sort_of (atom b)"
    unfolding atom_def by (simp add: sort_of_Rep)
  show "atom a = atom b \<longleftrightarrow> a = b"
    unfolding atom_def by (simp add: Rep_inject)
  show "p \<bullet> atom a = atom (p \<bullet> a)"
    unfolding permute_def atom_def by (simp add: Abs_inverse sort_of_Rep)
qed

setup {* Sign.add_const_constraint
  (@{const_name "permute"}, SOME @{typ "perm \<Rightarrow> 'a::pt \<Rightarrow> 'a"}) *}
setup {* Sign.add_const_constraint
  (@{const_name "atom"}, SOME @{typ "'a::at_base \<Rightarrow> atom"}) *}


section {* Automation for creating concrete atom types *}

text {* at the moment only single-sort concrete atoms are supported *}

use "nominal_atoms.ML"




end