# HG changeset patch
# User Christian Urban <urbanc@in.tum.de>
# Date 1270409968 -7200
# Node ID c34347ec7ab3d10b99769739ec1ca26d0d27fdf2
# Parent  c0eac04ae3b467f361eba434b7bbff95e39c372a
separated general nominal theory into separate folder

diff -r c0eac04ae3b4 -r c34347ec7ab3 IsaMakefile
--- a/IsaMakefile	Sat Apr 03 22:31:11 2010 +0200
+++ b/IsaMakefile	Sun Apr 04 21:39:28 2010 +0200
@@ -36,7 +36,7 @@
 
 pearl: $(LOG)/HOL-Pearl.gz
 
-$(LOG)/HOL-Pearl.gz: Nominal/Nominal*.thy Pearl/ROOT.ML Pearl/document/root.* Pearl/*.thy
+$(LOG)/HOL-Pearl.gz: Nominal-General/Nominal*.thy Pearl/ROOT.ML Pearl/document/root.* Pearl/*.thy
 	@$(USEDIR) -D generated HOL Pearl
 	$(ISABELLE_TOOL) document -o pdf Pearl/generated
 	@cp Pearl/document.pdf pearl.pdf
diff -r c0eac04ae3b4 -r c34347ec7ab3 Nominal-General/Atoms.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Nominal-General/Atoms.thy	Sun Apr 04 21:39:28 2010 +0200
@@ -0,0 +1,164 @@
+(*  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
diff -r c0eac04ae3b4 -r c34347ec7ab3 Nominal-General/Nominal2_Atoms.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Nominal-General/Nominal2_Atoms.thy	Sun Apr 04 21:39:28 2010 +0200
@@ -0,0 +1,255 @@
+(*  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)"
+
+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" => "CONST 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
diff -r c0eac04ae3b4 -r c34347ec7ab3 Nominal-General/Nominal2_Base.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Nominal-General/Nominal2_Base.thy	Sun Apr 04 21:39:28 2010 +0200
@@ -0,0 +1,1062 @@
+(*  Title:      Nominal2_Base
+    Authors:    Brian Huffman, Christian Urban
+
+    Basic definitions and lemma infrastructure for 
+    Nominal Isabelle. 
+*)
+theory Nominal2_Base
+imports Main Infinite_Set
+begin
+
+section {* Atoms and Sorts *}
+
+text {* A simple implementation for atom_sorts is strings. *}
+(* types atom_sort = string *)
+
+text {* To deal with Church-like binding we use trees of  
+  strings as sorts. *}
+
+datatype atom_sort = Sort "string" "atom_sort list"
+
+datatype atom = Atom atom_sort nat
+
+
+text {* Basic projection function. *}
+
+primrec
+  sort_of :: "atom \<Rightarrow> atom_sort"
+where
+  "sort_of (Atom s i) = s"
+
+
+text {* There are infinitely many atoms of each sort. *}
+lemma INFM_sort_of_eq: 
+  shows "INFM a. sort_of a = s"
+proof -
+  have "INFM i. sort_of (Atom s i) = s" by simp
+  moreover have "inj (Atom s)" by (simp add: inj_on_def)
+  ultimately show "INFM a. sort_of a = s" by (rule INFM_inj)
+qed
+
+lemma infinite_sort_of_eq:
+  shows "infinite {a. sort_of a = s}"
+  using INFM_sort_of_eq unfolding INFM_iff_infinite .
+
+lemma atom_infinite [simp]: 
+  shows "infinite (UNIV :: atom set)"
+  using subset_UNIV infinite_sort_of_eq
+  by (rule infinite_super)
+
+lemma obtain_atom:
+  fixes X :: "atom set"
+  assumes X: "finite X"
+  obtains a where "a \<notin> X" "sort_of a = s"
+proof -
+  from X have "MOST a. a \<notin> X"
+    unfolding MOST_iff_cofinite by simp
+  with INFM_sort_of_eq
+  have "INFM a. sort_of a = s \<and> a \<notin> X"
+    by (rule INFM_conjI)
+  then obtain a where "a \<notin> X" "sort_of a = s"
+    by (auto elim: INFM_E)
+  then show ?thesis ..
+qed
+
+section {* Sort-Respecting Permutations *}
+
+typedef perm =
+  "{f. bij f \<and> finite {a. f a \<noteq> a} \<and> (\<forall>a. sort_of (f a) = sort_of a)}"
+proof
+  show "id \<in> ?perm" by simp
+qed
+
+lemma permI:
+  assumes "bij f" and "MOST x. f x = x" and "\<And>a. sort_of (f a) = sort_of a"
+  shows "f \<in> perm"
+  using assms unfolding perm_def MOST_iff_cofinite by simp
+
+lemma perm_is_bij: "f \<in> perm \<Longrightarrow> bij f"
+  unfolding perm_def by simp
+
+lemma perm_is_finite: "f \<in> perm \<Longrightarrow> finite {a. f a \<noteq> a}"
+  unfolding perm_def by simp
+
+lemma perm_is_sort_respecting: "f \<in> perm \<Longrightarrow> sort_of (f a) = sort_of a"
+  unfolding perm_def by simp
+
+lemma perm_MOST: "f \<in> perm \<Longrightarrow> MOST x. f x = x"
+  unfolding perm_def MOST_iff_cofinite by simp
+
+lemma perm_id: "id \<in> perm"
+  unfolding perm_def by simp
+
+lemma perm_comp:
+  assumes f: "f \<in> perm" and g: "g \<in> perm"
+  shows "(f \<circ> g) \<in> perm"
+apply (rule permI)
+apply (rule bij_comp)
+apply (rule perm_is_bij [OF g])
+apply (rule perm_is_bij [OF f])
+apply (rule MOST_rev_mp [OF perm_MOST [OF g]])
+apply (rule MOST_rev_mp [OF perm_MOST [OF f]])
+apply (simp)
+apply (simp add: perm_is_sort_respecting [OF f])
+apply (simp add: perm_is_sort_respecting [OF g])
+done
+
+lemma perm_inv:
+  assumes f: "f \<in> perm"
+  shows "(inv f) \<in> perm"
+apply (rule permI)
+apply (rule bij_imp_bij_inv)
+apply (rule perm_is_bij [OF f])
+apply (rule MOST_mono [OF perm_MOST [OF f]])
+apply (erule subst, rule inv_f_f)
+apply (rule bij_is_inj [OF perm_is_bij [OF f]])
+apply (rule perm_is_sort_respecting [OF f, THEN sym, THEN trans])
+apply (simp add: surj_f_inv_f [OF bij_is_surj [OF perm_is_bij [OF f]]])
+done
+
+lemma bij_Rep_perm: "bij (Rep_perm p)"
+  using Rep_perm [of p] unfolding perm_def by simp
+
+lemma finite_Rep_perm: "finite {a. Rep_perm p a \<noteq> a}"
+  using Rep_perm [of p] unfolding perm_def by simp
+
+lemma sort_of_Rep_perm: "sort_of (Rep_perm p a) = sort_of a"
+  using Rep_perm [of p] unfolding perm_def by simp
+
+lemma Rep_perm_ext:
+  "Rep_perm p1 = Rep_perm p2 \<Longrightarrow> p1 = p2"
+  by (simp add: expand_fun_eq Rep_perm_inject [symmetric])
+
+
+subsection {* Permutations form a group *}
+
+instantiation perm :: group_add
+begin
+
+definition
+  "0 = Abs_perm id"
+
+definition
+  "- p = Abs_perm (inv (Rep_perm p))"
+
+definition
+  "p + q = Abs_perm (Rep_perm p \<circ> Rep_perm q)"
+
+definition
+  "(p1::perm) - p2 = p1 + - p2"
+
+lemma Rep_perm_0: "Rep_perm 0 = id"
+  unfolding zero_perm_def
+  by (simp add: Abs_perm_inverse perm_id)
+
+lemma Rep_perm_add:
+  "Rep_perm (p1 + p2) = Rep_perm p1 \<circ> Rep_perm p2"
+  unfolding plus_perm_def
+  by (simp add: Abs_perm_inverse perm_comp Rep_perm)
+
+lemma Rep_perm_uminus:
+  "Rep_perm (- p) = inv (Rep_perm p)"
+  unfolding uminus_perm_def
+  by (simp add: Abs_perm_inverse perm_inv Rep_perm)
+
+instance
+apply default
+unfolding Rep_perm_inject [symmetric]
+unfolding minus_perm_def
+unfolding Rep_perm_add
+unfolding Rep_perm_uminus
+unfolding Rep_perm_0
+by (simp_all add: o_assoc inv_o_cancel [OF bij_is_inj [OF bij_Rep_perm]])
+
+end
+
+
+section {* Implementation of swappings *}
+
+definition
+  swap :: "atom \<Rightarrow> atom \<Rightarrow> perm" ("'(_ \<rightleftharpoons> _')")
+where
+  "(a \<rightleftharpoons> b) =
+    Abs_perm (if sort_of a = sort_of b 
+              then (\<lambda>c. if a = c then b else if b = c then a else c) 
+              else id)"
+
+lemma Rep_perm_swap:
+  "Rep_perm (a \<rightleftharpoons> b) =
+    (if sort_of a = sort_of b 
+     then (\<lambda>c. if a = c then b else if b = c then a else c)
+     else id)"
+unfolding swap_def
+apply (rule Abs_perm_inverse)
+apply (rule permI)
+apply (auto simp add: bij_def inj_on_def surj_def)[1]
+apply (rule MOST_rev_mp [OF MOST_neq(1) [of a]])
+apply (rule MOST_rev_mp [OF MOST_neq(1) [of b]])
+apply (simp)
+apply (simp)
+done
+
+lemmas Rep_perm_simps =
+  Rep_perm_0
+  Rep_perm_add
+  Rep_perm_uminus
+  Rep_perm_swap
+
+lemma swap_different_sorts [simp]:
+  "sort_of a \<noteq> sort_of b \<Longrightarrow> (a \<rightleftharpoons> b) = 0"
+  by (rule Rep_perm_ext) (simp add: Rep_perm_simps)
+
+lemma swap_cancel:
+  "(a \<rightleftharpoons> b) + (a \<rightleftharpoons> b) = 0"
+by (rule Rep_perm_ext) 
+   (simp add: Rep_perm_simps expand_fun_eq)
+
+lemma swap_self [simp]:
+  "(a \<rightleftharpoons> a) = 0"
+  by (rule Rep_perm_ext, simp add: Rep_perm_simps expand_fun_eq)
+
+lemma minus_swap [simp]:
+  "- (a \<rightleftharpoons> b) = (a \<rightleftharpoons> b)"
+  by (rule minus_unique [OF swap_cancel])
+
+lemma swap_commute:
+  "(a \<rightleftharpoons> b) = (b \<rightleftharpoons> a)"
+  by (rule Rep_perm_ext)
+     (simp add: Rep_perm_swap expand_fun_eq)
+
+lemma swap_triple:
+  assumes "a \<noteq> b" and "c \<noteq> b"
+  assumes "sort_of a = sort_of b" "sort_of b = sort_of c"
+  shows "(a \<rightleftharpoons> c) + (b \<rightleftharpoons> c) + (a \<rightleftharpoons> c) = (a \<rightleftharpoons> b)"
+  using assms
+  by (rule_tac Rep_perm_ext)
+     (auto simp add: Rep_perm_simps expand_fun_eq)
+
+
+section {* Permutation Types *}
+
+text {*
+  Infix syntax for @{text permute} has higher precedence than
+  addition, but lower than unary minus.
+*}
+
+class pt =
+  fixes permute :: "perm \<Rightarrow> 'a \<Rightarrow> 'a" ("_ \<bullet> _" [76, 75] 75)
+  assumes permute_zero [simp]: "0 \<bullet> x = x"
+  assumes permute_plus [simp]: "(p + q) \<bullet> x = p \<bullet> (q \<bullet> x)"
+begin
+
+lemma permute_diff [simp]:
+  shows "(p - q) \<bullet> x = p \<bullet> - q \<bullet> x"
+  unfolding diff_minus by simp
+
+lemma permute_minus_cancel [simp]:
+  shows "p \<bullet> - p \<bullet> x = x"
+  and   "- p \<bullet> p \<bullet> x = x"
+  unfolding permute_plus [symmetric] by simp_all
+
+lemma permute_swap_cancel [simp]:
+  shows "(a \<rightleftharpoons> b) \<bullet> (a \<rightleftharpoons> b) \<bullet> x = x"
+  unfolding permute_plus [symmetric]
+  by (simp add: swap_cancel)
+
+lemma permute_swap_cancel2 [simp]:
+  shows "(a \<rightleftharpoons> b) \<bullet> (b \<rightleftharpoons> a) \<bullet> x = x"
+  unfolding permute_plus [symmetric]
+  by (simp add: swap_commute)
+
+lemma inj_permute [simp]: 
+  shows "inj (permute p)"
+  by (rule inj_on_inverseI)
+     (rule permute_minus_cancel)
+
+lemma surj_permute [simp]: 
+  shows "surj (permute p)"
+  by (rule surjI, rule permute_minus_cancel)
+
+lemma bij_permute [simp]: 
+  shows "bij (permute p)"
+  by (rule bijI [OF inj_permute surj_permute])
+
+lemma inv_permute: 
+  shows "inv (permute p) = permute (- p)"
+  by (rule inv_equality) (simp_all)
+
+lemma permute_minus: 
+  shows "permute (- p) = inv (permute p)"
+  by (simp add: inv_permute)
+
+lemma permute_eq_iff [simp]: 
+  shows "p \<bullet> x = p \<bullet> y \<longleftrightarrow> x = y"
+  by (rule inj_permute [THEN inj_eq])
+
+end
+
+subsection {* Permutations for atoms *}
+
+instantiation atom :: pt
+begin
+
+definition
+  "p \<bullet> a = Rep_perm p a"
+
+instance 
+apply(default)
+apply(simp_all add: permute_atom_def Rep_perm_simps)
+done
+
+end
+
+lemma sort_of_permute [simp]:
+  shows "sort_of (p \<bullet> a) = sort_of a"
+  unfolding permute_atom_def by (rule sort_of_Rep_perm)
+
+lemma swap_atom:
+  shows "(a \<rightleftharpoons> b) \<bullet> c =
+           (if sort_of a = sort_of b
+            then (if c = a then b else if c = b then a else c) else c)"
+  unfolding permute_atom_def
+  by (simp add: Rep_perm_swap)
+
+lemma swap_atom_simps [simp]:
+  "sort_of a = sort_of b \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> a = b"
+  "sort_of a = sort_of b \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> b = a"
+  "c \<noteq> a \<Longrightarrow> c \<noteq> b \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> c = c"
+  unfolding swap_atom by simp_all
+
+lemma expand_perm_eq:
+  fixes p q :: "perm"
+  shows "p = q \<longleftrightarrow> (\<forall>a::atom. p \<bullet> a = q \<bullet> a)"
+  unfolding permute_atom_def
+  by (metis Rep_perm_ext ext)
+
+
+subsection {* Permutations for permutations *}
+
+instantiation perm :: pt
+begin
+
+definition
+  "p \<bullet> q = p + q - p"
+
+instance
+apply default
+apply (simp add: permute_perm_def)
+apply (simp add: permute_perm_def diff_minus minus_add add_assoc)
+done
+
+end
+
+lemma permute_self: "p \<bullet> p = p"
+unfolding permute_perm_def by (simp add: diff_minus add_assoc)
+
+lemma permute_eqvt:
+  shows "p \<bullet> (q \<bullet> x) = (p \<bullet> q) \<bullet> (p \<bullet> x)"
+  unfolding permute_perm_def by simp
+
+lemma zero_perm_eqvt:
+  shows "p \<bullet> (0::perm) = 0"
+  unfolding permute_perm_def by simp
+
+lemma add_perm_eqvt:
+  fixes p p1 p2 :: perm
+  shows "p \<bullet> (p1 + p2) = p \<bullet> p1 + p \<bullet> p2"
+  unfolding permute_perm_def
+  by (simp add: expand_perm_eq)
+
+lemma swap_eqvt:
+  shows "p \<bullet> (a \<rightleftharpoons> b) = (p \<bullet> a \<rightleftharpoons> p \<bullet> b)"
+  unfolding permute_perm_def
+  by (auto simp add: swap_atom expand_perm_eq)
+
+
+subsection {* Permutations for functions *}
+
+instantiation "fun" :: (pt, pt) pt
+begin
+
+definition
+  "p \<bullet> f = (\<lambda>x. p \<bullet> (f (- p \<bullet> x)))"
+
+instance
+apply default
+apply (simp add: permute_fun_def)
+apply (simp add: permute_fun_def minus_add)
+done
+
+end
+
+lemma permute_fun_app_eq:
+  shows "p \<bullet> (f x) = (p \<bullet> f) (p \<bullet> x)"
+unfolding permute_fun_def by simp
+
+
+subsection {* Permutations for booleans *}
+
+instantiation bool :: pt
+begin
+
+definition "p \<bullet> (b::bool) = b"
+
+instance
+apply(default) 
+apply(simp_all add: permute_bool_def)
+done
+
+end
+
+lemma Not_eqvt:
+  shows "p \<bullet> (\<not> A) = (\<not> (p \<bullet> A))"
+by (simp add: permute_bool_def)
+
+lemma permute_boolE:
+  fixes P::"bool"
+  shows "p \<bullet> P \<Longrightarrow> P"
+  by (simp add: permute_bool_def)
+
+lemma permute_boolI:
+  fixes P::"bool"
+  shows "P \<Longrightarrow> p \<bullet> P"
+  by(simp add: permute_bool_def)
+
+subsection {* Permutations for sets *}
+
+lemma permute_set_eq:
+  fixes x::"'a::pt"
+  and   p::"perm"
+  shows "(p \<bullet> X) = {p \<bullet> x | x. x \<in> X}"
+  apply(auto simp add: permute_fun_def permute_bool_def mem_def)
+  apply(rule_tac x="- p \<bullet> x" in exI)
+  apply(simp)
+  done
+
+lemma permute_set_eq_image:
+  shows "p \<bullet> X = permute p ` X"
+unfolding permute_set_eq by auto
+
+lemma permute_set_eq_vimage:
+  shows "p \<bullet> X = permute (- p) -` X"
+unfolding permute_fun_def permute_bool_def
+unfolding vimage_def Collect_def mem_def ..
+
+lemma swap_set_not_in:
+  assumes a: "a \<notin> S" "b \<notin> S"
+  shows "(a \<rightleftharpoons> b) \<bullet> S = S"
+  using a by (auto simp add: permute_set_eq swap_atom)
+
+lemma swap_set_in:
+  assumes a: "a \<in> S" "b \<notin> S" "sort_of a = sort_of b"
+  shows "(a \<rightleftharpoons> b) \<bullet> S \<noteq> S"
+  using a by (auto simp add: permute_set_eq swap_atom)
+
+
+subsection {* Permutations for units *}
+
+instantiation unit :: pt
+begin
+
+definition "p \<bullet> (u::unit) = u"
+
+instance proof
+qed (simp_all add: permute_unit_def)
+
+end
+
+
+subsection {* Permutations for products *}
+
+instantiation "*" :: (pt, pt) pt
+begin
+
+primrec 
+  permute_prod 
+where
+  Pair_eqvt: "p \<bullet> (x, y) = (p \<bullet> x, p \<bullet> y)"
+
+instance
+by default auto
+
+end
+
+subsection {* Permutations for sums *}
+
+instantiation "+" :: (pt, pt) pt
+begin
+
+primrec 
+  permute_sum 
+where
+  "p \<bullet> (Inl x) = Inl (p \<bullet> x)"
+| "p \<bullet> (Inr y) = Inr (p \<bullet> y)"
+
+instance proof
+qed (case_tac [!] x, simp_all)
+
+end
+
+subsection {* Permutations for lists *}
+
+instantiation list :: (pt) pt
+begin
+
+primrec 
+  permute_list 
+where
+  "p \<bullet> [] = []"
+| "p \<bullet> (x # xs) = p \<bullet> x # p \<bullet> xs"
+
+instance proof
+qed (induct_tac [!] x, simp_all)
+
+end
+
+subsection {* Permutations for options *}
+
+instantiation option :: (pt) pt
+begin
+
+primrec 
+  permute_option 
+where
+  "p \<bullet> None = None"
+| "p \<bullet> (Some x) = Some (p \<bullet> x)"
+
+instance proof
+qed (induct_tac [!] x, simp_all)
+
+end
+
+subsection {* Permutations for @{typ char}, @{typ nat}, and @{typ int} *}
+
+instantiation char :: pt
+begin
+
+definition "p \<bullet> (c::char) = c"
+
+instance proof
+qed (simp_all add: permute_char_def)
+
+end
+
+instantiation nat :: pt
+begin
+
+definition "p \<bullet> (n::nat) = n"
+
+instance proof
+qed (simp_all add: permute_nat_def)
+
+end
+
+instantiation int :: pt
+begin
+
+definition "p \<bullet> (i::int) = i"
+
+instance proof
+qed (simp_all add: permute_int_def)
+
+end
+
+
+section {* Pure types *}
+
+text {* Pure types will have always empty support. *}
+
+class pure = pt +
+  assumes permute_pure: "p \<bullet> x = x"
+
+text {* Types @{typ unit} and @{typ bool} are pure. *}
+
+instance unit :: pure
+proof qed (rule permute_unit_def)
+
+instance bool :: pure
+proof qed (rule permute_bool_def)
+
+text {* Other type constructors preserve purity. *}
+
+instance "fun" :: (pure, pure) pure
+by default (simp add: permute_fun_def permute_pure)
+
+instance "*" :: (pure, pure) pure
+by default (induct_tac x, simp add: permute_pure)
+
+instance "+" :: (pure, pure) pure
+by default (induct_tac x, simp_all add: permute_pure)
+
+instance list :: (pure) pure
+by default (induct_tac x, simp_all add: permute_pure)
+
+instance option :: (pure) pure
+by default (induct_tac x, simp_all add: permute_pure)
+
+
+subsection {* Types @{typ char}, @{typ nat}, and @{typ int} *}
+
+instance char :: pure
+proof qed (rule permute_char_def)
+
+instance nat :: pure
+proof qed (rule permute_nat_def)
+
+instance int :: pure
+proof qed (rule permute_int_def)
+
+
+subsection {* Supp, Freshness and Supports *}
+
+context pt
+begin
+
+definition
+  supp :: "'a \<Rightarrow> atom set"
+where
+  "supp x = {a. infinite {b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x}}"
+
+end
+
+definition
+  fresh :: "atom \<Rightarrow> 'a::pt \<Rightarrow> bool" ("_ \<sharp> _" [55, 55] 55)
+where   
+  "a \<sharp> x \<equiv> a \<notin> supp x"
+
+lemma supp_conv_fresh: 
+  shows "supp x = {a. \<not> a \<sharp> x}"
+  unfolding fresh_def by simp
+
+lemma swap_rel_trans:
+  assumes "sort_of a = sort_of b"
+  assumes "sort_of b = sort_of c"
+  assumes "(a \<rightleftharpoons> c) \<bullet> x = x"
+  assumes "(b \<rightleftharpoons> c) \<bullet> x = x"
+  shows "(a \<rightleftharpoons> b) \<bullet> x = x"
+proof (cases)
+  assume "a = b \<or> c = b"
+  with assms show "(a \<rightleftharpoons> b) \<bullet> x = x" by auto
+next
+  assume *: "\<not> (a = b \<or> c = b)"
+  have "((a \<rightleftharpoons> c) + (b \<rightleftharpoons> c) + (a \<rightleftharpoons> c)) \<bullet> x = x"
+    using assms by simp
+  also have "(a \<rightleftharpoons> c) + (b \<rightleftharpoons> c) + (a \<rightleftharpoons> c) = (a \<rightleftharpoons> b)"
+    using assms * by (simp add: swap_triple)
+  finally show "(a \<rightleftharpoons> b) \<bullet> x = x" .
+qed
+
+lemma swap_fresh_fresh:
+  assumes a: "a \<sharp> x" 
+  and     b: "b \<sharp> x"
+  shows "(a \<rightleftharpoons> b) \<bullet> x = x"
+proof (cases)
+  assume asm: "sort_of a = sort_of b" 
+  have "finite {c. (a \<rightleftharpoons> c) \<bullet> x \<noteq> x}" "finite {c. (b \<rightleftharpoons> c) \<bullet> x \<noteq> x}" 
+    using a b unfolding fresh_def supp_def by simp_all
+  then have "finite ({c. (a \<rightleftharpoons> c) \<bullet> x \<noteq> x} \<union> {c. (b \<rightleftharpoons> c) \<bullet> x \<noteq> x})" by simp
+  then obtain c 
+    where "(a \<rightleftharpoons> c) \<bullet> x = x" "(b \<rightleftharpoons> c) \<bullet> x = x" "sort_of c = sort_of b"
+    by (rule obtain_atom) (auto)
+  then show "(a \<rightleftharpoons> b) \<bullet> x = x" using asm by (rule_tac swap_rel_trans) (simp_all)
+next
+  assume "sort_of a \<noteq> sort_of b"
+  then show "(a \<rightleftharpoons> b) \<bullet> x = x" by simp
+qed
+
+
+subsection {* supp and fresh are equivariant *}
+
+lemma finite_Collect_bij:
+  assumes a: "bij f"
+  shows "finite {x. P (f x)} = finite {x. P x}"
+by (metis a finite_vimage_iff vimage_Collect_eq)
+
+lemma fresh_permute_iff:
+  shows "(p \<bullet> a) \<sharp> (p \<bullet> x) \<longleftrightarrow> a \<sharp> x"
+proof -
+  have "(p \<bullet> a) \<sharp> (p \<bullet> x) \<longleftrightarrow> finite {b. (p \<bullet> a \<rightleftharpoons> b) \<bullet> p \<bullet> x \<noteq> p \<bullet> x}"
+    unfolding fresh_def supp_def by simp
+  also have "\<dots> \<longleftrightarrow> finite {b. (p \<bullet> a \<rightleftharpoons> p \<bullet> b) \<bullet> p \<bullet> x \<noteq> p \<bullet> x}"
+    using bij_permute by (rule finite_Collect_bij [symmetric])
+  also have "\<dots> \<longleftrightarrow> finite {b. p \<bullet> (a \<rightleftharpoons> b) \<bullet> x \<noteq> p \<bullet> x}"
+    by (simp only: permute_eqvt [of p] swap_eqvt)
+  also have "\<dots> \<longleftrightarrow> finite {b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x}"
+    by (simp only: permute_eq_iff)
+  also have "\<dots> \<longleftrightarrow> a \<sharp> x"
+    unfolding fresh_def supp_def by simp
+  finally show ?thesis .
+qed
+
+lemma fresh_eqvt:
+  shows "p \<bullet> (a \<sharp> x) = (p \<bullet> a) \<sharp> (p \<bullet> x)"
+  by (simp add: permute_bool_def fresh_permute_iff)
+
+lemma supp_eqvt:
+  fixes  p  :: "perm"
+  and    x   :: "'a::pt"
+  shows "p \<bullet> (supp x) = supp (p \<bullet> x)"
+  unfolding supp_conv_fresh
+  unfolding permute_fun_def Collect_def
+  by (simp add: Not_eqvt fresh_eqvt)
+
+subsection {* supports *}
+
+definition
+  supports :: "atom set \<Rightarrow> 'a::pt \<Rightarrow> bool" (infixl "supports" 80)
+where  
+  "S supports x \<equiv> \<forall>a b. (a \<notin> S \<and> b \<notin> S \<longrightarrow> (a \<rightleftharpoons> b) \<bullet> x = x)"
+
+lemma supp_is_subset:
+  fixes S :: "atom set"
+  and   x :: "'a::pt"
+  assumes a1: "S supports x"
+  and     a2: "finite S"
+  shows "(supp x) \<subseteq> S"
+proof (rule ccontr)
+  assume "\<not>(supp x \<subseteq> S)"
+  then obtain a where b1: "a \<in> supp x" and b2: "a \<notin> S" by auto
+  from a1 b2 have "\<forall>b. b \<notin> S \<longrightarrow> (a \<rightleftharpoons> b) \<bullet> x = x" by (unfold supports_def) (auto)
+  hence "{b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x} \<subseteq> S" by auto
+  with a2 have "finite {b. (a \<rightleftharpoons> b)\<bullet>x \<noteq> x}" by (simp add: finite_subset)
+  then have "a \<notin> (supp x)" unfolding supp_def by simp
+  with b1 show False by simp
+qed
+
+lemma supports_finite:
+  fixes S :: "atom set"
+  and   x :: "'a::pt"
+  assumes a1: "S supports x"
+  and     a2: "finite S"
+  shows "finite (supp x)"
+proof -
+  have "(supp x) \<subseteq> S" using a1 a2 by (rule supp_is_subset)
+  then show "finite (supp x)" using a2 by (simp add: finite_subset)
+qed
+
+lemma supp_supports:
+  fixes x :: "'a::pt"
+  shows "(supp x) supports x"
+proof (unfold supports_def, intro strip)
+  fix a b
+  assume "a \<notin> (supp x) \<and> b \<notin> (supp x)"
+  then have "a \<sharp> x" and "b \<sharp> x" by (simp_all add: fresh_def)
+  then show "(a \<rightleftharpoons> b) \<bullet> x = x" by (rule swap_fresh_fresh)
+qed
+
+lemma supp_is_least_supports:
+  fixes S :: "atom set"
+  and   x :: "'a::pt"
+  assumes  a1: "S supports x"
+  and      a2: "finite S"
+  and      a3: "\<And>S'. finite S' \<Longrightarrow> (S' supports x) \<Longrightarrow> S \<subseteq> S'"
+  shows "(supp x) = S"
+proof (rule equalityI)
+  show "(supp x) \<subseteq> S" using a1 a2 by (rule supp_is_subset)
+  with a2 have fin: "finite (supp x)" by (rule rev_finite_subset)
+  have "(supp x) supports x" by (rule supp_supports)
+  with fin a3 show "S \<subseteq> supp x" by blast
+qed
+
+lemma subsetCI: 
+  shows "(\<And>x. x \<in> A \<Longrightarrow> x \<notin> B \<Longrightarrow> False) \<Longrightarrow> A \<subseteq> B"
+  by auto
+
+lemma finite_supp_unique:
+  assumes a1: "S supports x"
+  assumes a2: "finite S"
+  assumes a3: "\<And>a b. \<lbrakk>a \<in> S; b \<notin> S; sort_of a = sort_of b\<rbrakk> \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> x \<noteq> x"
+  shows "(supp x) = S"
+  using a1 a2
+proof (rule supp_is_least_supports)
+  fix S'
+  assume "finite S'" and "S' supports x"
+  show "S \<subseteq> S'"
+  proof (rule subsetCI)
+    fix a
+    assume "a \<in> S" and "a \<notin> S'"
+    have "finite (S \<union> S')"
+      using `finite S` `finite S'` by simp
+    then obtain b where "b \<notin> S \<union> S'" and "sort_of b = sort_of a"
+      by (rule obtain_atom)
+    then have "b \<notin> S" and "b \<notin> S'"  and "sort_of a = sort_of b"
+      by simp_all
+    then have "(a \<rightleftharpoons> b) \<bullet> x = x"
+      using `a \<notin> S'` `S' supports x` by (simp add: supports_def)
+    moreover have "(a \<rightleftharpoons> b) \<bullet> x \<noteq> x"
+      using `a \<in> S` `b \<notin> S` `sort_of a = sort_of b`
+      by (rule a3)
+    ultimately show "False" by simp
+  qed
+qed
+
+section {* Finitely-supported types *}
+
+class fs = pt +
+  assumes finite_supp: "finite (supp x)"
+
+lemma pure_supp: 
+  shows "supp (x::'a::pure) = {}"
+  unfolding supp_def by (simp add: permute_pure)
+
+lemma pure_fresh:
+  fixes x::"'a::pure"
+  shows "a \<sharp> x"
+  unfolding fresh_def by (simp add: pure_supp)
+
+instance pure < fs
+by default (simp add: pure_supp)
+
+
+subsection  {* Type @{typ atom} is finitely-supported. *}
+
+lemma supp_atom:
+  shows "supp a = {a}"
+apply (rule finite_supp_unique)
+apply (clarsimp simp add: supports_def)
+apply simp
+apply simp
+done
+
+lemma fresh_atom: 
+  shows "a \<sharp> b \<longleftrightarrow> a \<noteq> b"
+  unfolding fresh_def supp_atom by simp
+
+instance atom :: fs
+by default (simp add: supp_atom)
+
+
+section {* Type @{typ perm} is finitely-supported. *}
+
+lemma perm_swap_eq:
+  shows "(a \<rightleftharpoons> b) \<bullet> p = p \<longleftrightarrow> (p \<bullet> (a \<rightleftharpoons> b)) = (a \<rightleftharpoons> b)"
+unfolding permute_perm_def
+by (metis add_diff_cancel minus_perm_def)
+
+lemma supports_perm: 
+  shows "{a. p \<bullet> a \<noteq> a} supports p"
+  unfolding supports_def
+  by (simp add: perm_swap_eq swap_eqvt)
+
+lemma finite_perm_lemma: 
+  shows "finite {a::atom. p \<bullet> a \<noteq> a}"
+  using finite_Rep_perm [of p]
+  unfolding permute_atom_def .
+
+lemma supp_perm:
+  shows "supp p = {a. p \<bullet> a \<noteq> a}"
+apply (rule finite_supp_unique)
+apply (rule supports_perm)
+apply (rule finite_perm_lemma)
+apply (simp add: perm_swap_eq swap_eqvt)
+apply (auto simp add: expand_perm_eq swap_atom)
+done
+
+lemma fresh_perm: 
+  shows "a \<sharp> p \<longleftrightarrow> p \<bullet> a = a"
+unfolding fresh_def by (simp add: supp_perm)
+
+lemma supp_swap:
+  shows "supp (a \<rightleftharpoons> b) = (if a = b \<or> sort_of a \<noteq> sort_of b then {} else {a, b})"
+  by (auto simp add: supp_perm swap_atom)
+
+lemma fresh_zero_perm: 
+  shows "a \<sharp> (0::perm)"
+  unfolding fresh_perm by simp
+
+lemma supp_zero_perm: 
+  shows "supp (0::perm) = {}"
+  unfolding supp_perm by simp
+
+lemma fresh_plus_perm:
+  fixes p q::perm
+  assumes "a \<sharp> p" "a \<sharp> q"
+  shows "a \<sharp> (p + q)"
+  using assms
+  unfolding fresh_def
+  by (auto simp add: supp_perm)
+
+lemma supp_plus_perm:
+  fixes p q::perm
+  shows "supp (p + q) \<subseteq> supp p \<union> supp q"
+  by (auto simp add: supp_perm)
+
+lemma fresh_minus_perm:
+  fixes p::perm
+  shows "a \<sharp> (- p) \<longleftrightarrow> a \<sharp> p"
+  unfolding fresh_def
+  apply(auto simp add: supp_perm)
+  apply(metis permute_minus_cancel)+
+  done
+
+lemma supp_minus_perm:
+  fixes p::perm
+  shows "supp (- p) = supp p"
+  unfolding supp_conv_fresh
+  by (simp add: fresh_minus_perm)
+
+instance perm :: fs
+by default (simp add: supp_perm finite_perm_lemma)
+
+lemma plus_perm_eq:
+  fixes p q::"perm"
+  assumes asm: "supp p \<inter>  supp q = {}"
+  shows "p + q  = q + p"
+unfolding expand_perm_eq
+proof
+  fix a::"atom"
+  show "(p + q) \<bullet> a = (q + p) \<bullet> a"
+  proof -
+    { assume "a \<notin> supp p" "a \<notin> supp q"
+      then have "(p + q) \<bullet> a = (q + p) \<bullet> a" 
+	by (simp add: supp_perm)
+    }
+    moreover
+    { assume a: "a \<in> supp p" "a \<notin> supp q"
+      then have "p \<bullet> a \<in> supp p" by (simp add: supp_perm)
+      then have "p \<bullet> a \<notin> supp q" using asm by auto
+      with a have "(p + q) \<bullet> a = (q + p) \<bullet> a" 
+	by (simp add: supp_perm)
+    }
+    moreover
+    { assume a: "a \<notin> supp p" "a \<in> supp q"
+      then have "q \<bullet> a \<in> supp q" by (simp add: supp_perm)
+      then have "q \<bullet> a \<notin> supp p" using asm by auto 
+      with a have "(p + q) \<bullet> a = (q + p) \<bullet> a" 
+	by (simp add: supp_perm)
+    }
+    ultimately show "(p + q) \<bullet> a = (q + p) \<bullet> a" 
+      using asm by blast
+  qed
+qed
+
+section {* Finite Support instances for other types *}
+
+subsection {* Type @{typ "'a \<times> 'b"} is finitely-supported. *}
+
+lemma supp_Pair: 
+  shows "supp (x, y) = supp x \<union> supp y"
+  by (simp add: supp_def Collect_imp_eq Collect_neg_eq)
+
+lemma fresh_Pair: 
+  shows "a \<sharp> (x, y) \<longleftrightarrow> a \<sharp> x \<and> a \<sharp> y"
+  by (simp add: fresh_def supp_Pair)
+
+instance "*" :: (fs, fs) fs
+apply default
+apply (induct_tac x)
+apply (simp add: supp_Pair finite_supp)
+done
+
+subsection {* Type @{typ "'a + 'b"} is finitely supported *}
+
+lemma supp_Inl: 
+  shows "supp (Inl x) = supp x"
+  by (simp add: supp_def)
+
+lemma supp_Inr: 
+  shows "supp (Inr x) = supp x"
+  by (simp add: supp_def)
+
+lemma fresh_Inl: 
+  shows "a \<sharp> Inl x \<longleftrightarrow> a \<sharp> x"
+  by (simp add: fresh_def supp_Inl)
+
+lemma fresh_Inr: 
+  shows "a \<sharp> Inr y \<longleftrightarrow> a \<sharp> y"
+  by (simp add: fresh_def supp_Inr)
+
+instance "+" :: (fs, fs) fs
+apply default
+apply (induct_tac x)
+apply (simp_all add: supp_Inl supp_Inr finite_supp)
+done
+
+subsection {* Type @{typ "'a option"} is finitely supported *}
+
+lemma supp_None: 
+  shows "supp None = {}"
+by (simp add: supp_def)
+
+lemma supp_Some: 
+  shows "supp (Some x) = supp x"
+  by (simp add: supp_def)
+
+lemma fresh_None: 
+  shows "a \<sharp> None"
+  by (simp add: fresh_def supp_None)
+
+lemma fresh_Some: 
+  shows "a \<sharp> Some x \<longleftrightarrow> a \<sharp> x"
+  by (simp add: fresh_def supp_Some)
+
+instance option :: (fs) fs
+apply default
+apply (induct_tac x)
+apply (simp_all add: supp_None supp_Some finite_supp)
+done
+
+subsubsection {* Type @{typ "'a list"} is finitely supported *}
+
+lemma supp_Nil: 
+  shows "supp [] = {}"
+  by (simp add: supp_def)
+
+lemma supp_Cons: 
+  shows "supp (x # xs) = supp x \<union> supp xs"
+by (simp add: supp_def Collect_imp_eq Collect_neg_eq)
+
+lemma fresh_Nil: 
+  shows "a \<sharp> []"
+  by (simp add: fresh_def supp_Nil)
+
+lemma fresh_Cons: 
+  shows "a \<sharp> (x # xs) \<longleftrightarrow> a \<sharp> x \<and> a \<sharp> xs"
+  by (simp add: fresh_def supp_Cons)
+
+instance list :: (fs) fs
+apply default
+apply (induct_tac x)
+apply (simp_all add: supp_Nil supp_Cons finite_supp)
+done
+
+section {* Support and freshness for applications *}
+
+lemma supp_fun_app:
+  shows "supp (f x) \<subseteq> (supp f) \<union> (supp x)"
+proof (rule subsetCI)
+  fix a::"atom"
+  assume a: "a \<in> supp (f x)"
+  assume b: "a \<notin> supp f \<union> supp x"
+  then have "finite {b. (a \<rightleftharpoons> b) \<bullet> f \<noteq> f}" "finite {b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x}" 
+    unfolding supp_def by auto
+  then have "finite ({b. (a \<rightleftharpoons> b) \<bullet> f \<noteq> f} \<union> {b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x})" by simp
+  moreover 
+  have "{b. ((a \<rightleftharpoons> b) \<bullet> f) ((a \<rightleftharpoons> b) \<bullet> x) \<noteq> f x} \<subseteq> ({b. (a \<rightleftharpoons> b) \<bullet> f \<noteq> f} \<union> {b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x})"
+    by auto
+  ultimately have "finite {b. ((a \<rightleftharpoons> b) \<bullet> f) ((a \<rightleftharpoons> b) \<bullet> x) \<noteq> f x}"
+    using finite_subset by auto
+  then have "a \<notin> supp (f x)" unfolding supp_def
+    by (simp add: permute_fun_app_eq)
+  with a show "False" by simp
+qed
+    
+lemma fresh_fun_app:
+  shows "a \<sharp> (f, x) \<Longrightarrow> a \<sharp> f x"
+unfolding fresh_def
+using supp_fun_app
+by (auto simp add: supp_Pair)
+
+lemma fresh_fun_eqvt_app:
+  assumes a: "\<forall>p. p \<bullet> f = f"
+  shows "a \<sharp> x \<Longrightarrow> a \<sharp> f x"
+proof -
+  from a have b: "supp f = {}"
+    unfolding supp_def by simp
+  show "a \<sharp> x \<Longrightarrow> a \<sharp> f x"
+    unfolding fresh_def
+    using supp_fun_app b
+    by auto
+qed
+
+end
diff -r c0eac04ae3b4 -r c34347ec7ab3 Nominal-General/Nominal2_Eqvt.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Nominal-General/Nominal2_Eqvt.thy	Sun Apr 04 21:39:28 2010 +0200
@@ -0,0 +1,305 @@
+(*  Title:      Nominal2_Eqvt
+    Authors:    Brian Huffman, Christian Urban
+
+    Equivariance, Supp and Fresh Lemmas for Operators. 
+    (Contains most, but not all such lemmas.)
+*)
+theory Nominal2_Eqvt
+imports Nominal2_Base Nominal2_Atoms
+uses ("nominal_thmdecls.ML")
+     ("nominal_permeq.ML")
+begin
+
+section {* Logical Operators *}
+
+lemma eq_eqvt:
+  shows "p \<bullet> (x = y) \<longleftrightarrow> (p \<bullet> x) = (p \<bullet> y)"
+  unfolding permute_eq_iff permute_bool_def ..
+
+lemma if_eqvt:
+  shows "p \<bullet> (if b then x else y) = (if p \<bullet> b then p \<bullet> x else p \<bullet> y)"
+  by (simp add: permute_fun_def permute_bool_def)
+
+lemma True_eqvt:
+  shows "p \<bullet> True = True"
+  unfolding permute_bool_def ..
+
+lemma False_eqvt:
+  shows "p \<bullet> False = False"
+  unfolding permute_bool_def ..
+
+lemma imp_eqvt:
+  shows "p \<bullet> (A \<longrightarrow> B) = ((p \<bullet> A) \<longrightarrow> (p \<bullet> B))"
+  by (simp add: permute_bool_def)
+
+lemma conj_eqvt:
+  shows "p \<bullet> (A \<and> B) = ((p \<bullet> A) \<and> (p \<bullet> B))"
+  by (simp add: permute_bool_def)
+
+lemma disj_eqvt:
+  shows "p \<bullet> (A \<or> B) = ((p \<bullet> A) \<or> (p \<bullet> B))"
+  by (simp add: permute_bool_def)
+
+lemma Not_eqvt:
+  shows "p \<bullet> (\<not> A) = (\<not> (p \<bullet> A))"
+  by (simp add: permute_bool_def)
+
+lemma all_eqvt:
+  shows "p \<bullet> (\<forall>x. P x) = (\<forall>x. (p \<bullet> P) x)"
+  unfolding permute_fun_def permute_bool_def
+  by (auto, drule_tac x="p \<bullet> x" in spec, simp)
+
+lemma all_eqvt2:
+  shows "p \<bullet> (\<forall>x. P x) = (\<forall>x. p \<bullet> P (- p \<bullet> x))"
+  unfolding permute_fun_def permute_bool_def
+  by (auto, drule_tac x="p \<bullet> x" in spec, simp)
+
+lemma ex_eqvt:
+  shows "p \<bullet> (\<exists>x. P x) = (\<exists>x. (p \<bullet> P) x)"
+  unfolding permute_fun_def permute_bool_def
+  by (auto, rule_tac x="p \<bullet> x" in exI, simp)
+
+lemma ex_eqvt2:
+  shows "p \<bullet> (\<exists>x. P x) = (\<exists>x. p \<bullet> P (- p \<bullet> x))"
+  unfolding permute_fun_def permute_bool_def
+  by (auto, rule_tac x="p \<bullet> x" in exI, simp)
+
+lemma ex1_eqvt:
+  shows "p \<bullet> (\<exists>!x. P x) = (\<exists>!x. (p \<bullet> P) x)"
+  unfolding Ex1_def 
+  by (simp add: ex_eqvt permute_fun_def conj_eqvt all_eqvt imp_eqvt eq_eqvt)
+
+lemma ex1_eqvt2:
+  shows "p \<bullet> (\<exists>!x. P x) = (\<exists>!x. p \<bullet> P (- p \<bullet> x))"
+  unfolding Ex1_def ex_eqvt2 conj_eqvt all_eqvt2 imp_eqvt eq_eqvt
+  by simp
+
+lemma the_eqvt:
+  assumes unique: "\<exists>!x. P x"
+  shows "(p \<bullet> (THE x. P x)) = (THE x. p \<bullet> P (- p \<bullet> x))"
+  apply(rule the1_equality [symmetric])
+  apply(simp add: ex1_eqvt2[symmetric])
+  apply(simp add: permute_bool_def unique)
+  apply(simp add: permute_bool_def)
+  apply(rule theI'[OF unique])
+  done
+
+section {* Set Operations *}
+
+lemma mem_permute_iff:
+  shows "(p \<bullet> x) \<in> (p \<bullet> X) \<longleftrightarrow> x \<in> X"
+unfolding mem_def permute_fun_def permute_bool_def
+by simp
+
+lemma mem_eqvt:
+  shows "p \<bullet> (x \<in> A) \<longleftrightarrow> (p \<bullet> x) \<in> (p \<bullet> A)"
+  unfolding mem_permute_iff permute_bool_def by simp
+
+lemma not_mem_eqvt:
+  shows "p \<bullet> (x \<notin> A) \<longleftrightarrow> (p \<bullet> x) \<notin> (p \<bullet> A)"
+  unfolding mem_def permute_fun_def by (simp add: Not_eqvt)
+
+lemma Collect_eqvt:
+  shows "p \<bullet> {x. P x} = {x. (p \<bullet> P) x}"
+  unfolding Collect_def permute_fun_def ..
+
+lemma Collect_eqvt2:
+  shows "p \<bullet> {x. P x} = {x. p \<bullet> (P (-p \<bullet> x))}"
+  unfolding Collect_def permute_fun_def ..
+
+lemma empty_eqvt:
+  shows "p \<bullet> {} = {}"
+  unfolding empty_def Collect_eqvt2 False_eqvt ..
+
+lemma supp_set_empty:
+  shows "supp {} = {}"
+  by (simp add: supp_def empty_eqvt)
+
+lemma fresh_set_empty:
+  shows "a \<sharp> {}"
+  by (simp add: fresh_def supp_set_empty)
+
+lemma UNIV_eqvt:
+  shows "p \<bullet> UNIV = UNIV"
+  unfolding UNIV_def Collect_eqvt2 True_eqvt ..
+
+lemma union_eqvt:
+  shows "p \<bullet> (A \<union> B) = (p \<bullet> A) \<union> (p \<bullet> B)"
+  unfolding Un_def Collect_eqvt2 disj_eqvt mem_eqvt by simp
+
+lemma inter_eqvt:
+  shows "p \<bullet> (A \<inter> B) = (p \<bullet> A) \<inter> (p \<bullet> B)"
+  unfolding Int_def Collect_eqvt2 conj_eqvt mem_eqvt by simp
+
+lemma Diff_eqvt:
+  fixes A B :: "'a::pt set"
+  shows "p \<bullet> (A - B) = p \<bullet> A - p \<bullet> B"
+  unfolding set_diff_eq Collect_eqvt2 conj_eqvt Not_eqvt mem_eqvt by simp
+
+lemma Compl_eqvt:
+  fixes A :: "'a::pt set"
+  shows "p \<bullet> (- A) = - (p \<bullet> A)"
+  unfolding Compl_eq_Diff_UNIV Diff_eqvt UNIV_eqvt ..
+
+lemma insert_eqvt:
+  shows "p \<bullet> (insert x A) = insert (p \<bullet> x) (p \<bullet> A)"
+  unfolding permute_set_eq_image image_insert ..
+
+lemma vimage_eqvt:
+  shows "p \<bullet> (f -` A) = (p \<bullet> f) -` (p \<bullet> A)"
+  unfolding vimage_def permute_fun_def [where f=f]
+  unfolding Collect_eqvt2 mem_eqvt ..
+
+lemma image_eqvt:
+  shows "p \<bullet> (f ` A) = (p \<bullet> f) ` (p \<bullet> A)"
+  unfolding permute_set_eq_image
+  unfolding permute_fun_def [where f=f]
+  by (simp add: image_image)
+
+lemma finite_permute_iff:
+  shows "finite (p \<bullet> A) \<longleftrightarrow> finite A"
+  unfolding permute_set_eq_vimage
+  using bij_permute by (rule finite_vimage_iff)
+
+lemma finite_eqvt:
+  shows "p \<bullet> finite A = finite (p \<bullet> A)"
+  unfolding finite_permute_iff permute_bool_def ..
+
+
+section {* List Operations *}
+
+lemma append_eqvt:
+  shows "p \<bullet> (xs @ ys) = (p \<bullet> xs) @ (p \<bullet> ys)"
+  by (induct xs) auto
+
+lemma supp_append:
+  shows "supp (xs @ ys) = supp xs \<union> supp ys"
+  by (induct xs) (auto simp add: supp_Nil supp_Cons)
+
+lemma fresh_append:
+  shows "a \<sharp> (xs @ ys) \<longleftrightarrow> a \<sharp> xs \<and> a \<sharp> ys"
+  by (induct xs) (simp_all add: fresh_Nil fresh_Cons)
+
+lemma rev_eqvt:
+  shows "p \<bullet> (rev xs) = rev (p \<bullet> xs)"
+  by (induct xs) (simp_all add: append_eqvt)
+
+lemma supp_rev:
+  shows "supp (rev xs) = supp xs"
+  by (induct xs) (auto simp add: supp_append supp_Cons supp_Nil)
+
+lemma fresh_rev:
+  shows "a \<sharp> rev xs \<longleftrightarrow> a \<sharp> xs"
+  by (induct xs) (auto simp add: fresh_append fresh_Cons fresh_Nil)
+
+lemma set_eqvt:
+  shows "p \<bullet> (set xs) = set (p \<bullet> xs)"
+  by (induct xs) (simp_all add: empty_eqvt insert_eqvt)
+
+(* needs finite support premise
+lemma supp_set:
+  fixes x :: "'a::pt"
+  shows "supp (set xs) = supp xs"
+*)
+
+
+section {* Product Operations *}
+
+lemma fst_eqvt:
+  "p \<bullet> (fst x) = fst (p \<bullet> x)"
+ by (cases x) simp
+
+lemma snd_eqvt:
+  "p \<bullet> (snd x) = snd (p \<bullet> x)"
+ by (cases x) simp
+
+section {* Units *}
+
+lemma supp_unit:
+  shows "supp () = {}"
+  by (simp add: supp_def)
+
+lemma fresh_unit:
+  shows "a \<sharp> ()"
+  by (simp add: fresh_def supp_unit)
+
+section {* Equivariance automation *}
+
+text {* Setup of the theorem attributes @{text eqvt} and @{text eqvt_force} *}
+
+use "nominal_thmdecls.ML"
+setup "Nominal_ThmDecls.setup"
+
+lemmas [eqvt] = 
+  (* connectives *)
+  eq_eqvt if_eqvt imp_eqvt disj_eqvt conj_eqvt Not_eqvt 
+  True_eqvt False_eqvt ex_eqvt all_eqvt ex1_eqvt
+  imp_eqvt [folded induct_implies_def]
+
+  (* nominal *)
+  (*permute_eqvt commented out since it loops *)
+  supp_eqvt fresh_eqvt
+  permute_pure
+
+  (* datatypes *)
+  permute_prod.simps append_eqvt rev_eqvt set_eqvt
+  fst_eqvt snd_eqvt Pair_eqvt
+
+  (* sets *)
+  empty_eqvt UNIV_eqvt union_eqvt inter_eqvt mem_eqvt
+  Diff_eqvt Compl_eqvt insert_eqvt Collect_eqvt image_eqvt
+
+  atom_eqvt add_perm_eqvt
+
+thm eqvts
+thm eqvts_raw
+
+text {* helper lemmas for the eqvt_tac *}
+
+definition
+  "unpermute p = permute (- p)"
+
+lemma eqvt_apply:
+  fixes f :: "'a::pt \<Rightarrow> 'b::pt" 
+  and x :: "'a::pt"
+  shows "p \<bullet> (f x) \<equiv> (p \<bullet> f) (p \<bullet> x)"
+  unfolding permute_fun_def by simp
+
+lemma eqvt_lambda:
+  fixes f :: "'a::pt \<Rightarrow> 'b::pt"
+  shows "p \<bullet> (\<lambda>x. f x) \<equiv> (\<lambda>x. p \<bullet> (f (unpermute p x)))"
+  unfolding permute_fun_def unpermute_def by simp
+
+lemma eqvt_bound:
+  shows "p \<bullet> unpermute p x \<equiv> x"
+  unfolding unpermute_def by simp
+
+use "nominal_permeq.ML"
+
+
+lemma "p \<bullet> (A \<longrightarrow> B = C)"
+apply (tactic {* Nominal_Permeq.eqvt_tac @{context} 1 *}) 
+oops
+
+lemma "p \<bullet> (\<lambda>(x::'a::pt). A \<longrightarrow> (B::'a \<Rightarrow> bool) x = C) = foo"
+apply (tactic {* Nominal_Permeq.eqvt_tac @{context} 1 *})
+oops
+
+lemma "p \<bullet> (\<lambda>x y. \<exists>z. x = z \<and> x = y \<longrightarrow> z \<noteq> x) = foo"
+apply (tactic {* Nominal_Permeq.eqvt_tac @{context} 1 *})
+oops
+
+lemma "p \<bullet> (\<lambda>f x. f (g (f x))) = foo"
+apply (tactic {* Nominal_Permeq.eqvt_tac @{context} 1 *})
+oops
+
+lemma "p \<bullet> (\<lambda>q. q \<bullet> (r \<bullet> x)) = foo"
+apply (tactic {* Nominal_Permeq.eqvt_tac @{context} 1 *})
+oops
+
+lemma "p \<bullet> (q \<bullet> r \<bullet> x) = foo"
+apply (tactic {* Nominal_Permeq.eqvt_tac @{context} 1 *})
+oops
+
+
+end
diff -r c0eac04ae3b4 -r c34347ec7ab3 Nominal-General/Nominal2_Supp.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Nominal-General/Nominal2_Supp.thy	Sun Apr 04 21:39:28 2010 +0200
@@ -0,0 +1,501 @@
+(*  Title:      Nominal2_Supp
+    Authors:    Brian Huffman, Christian Urban
+
+    Supplementary Lemmas and Definitions for 
+    Nominal Isabelle. 
+*)
+theory Nominal2_Supp
+imports Nominal2_Base Nominal2_Eqvt Nominal2_Atoms
+begin
+
+
+section {* Fresh-Star *}
+
+text {* The fresh-star generalisation of fresh is used in strong
+  induction principles. *}
+
+definition 
+  fresh_star :: "atom set \<Rightarrow> 'a::pt \<Rightarrow> bool" ("_ \<sharp>* _" [80,80] 80)
+where 
+  "as \<sharp>* x \<equiv> \<forall>a \<in> as. a \<sharp> x"
+
+lemma fresh_star_prod:
+  fixes as::"atom set"
+  shows "as \<sharp>* (x, y) = (as \<sharp>* x \<and> as \<sharp>* y)"
+  by (auto simp add: fresh_star_def fresh_Pair)
+
+lemma fresh_star_union:
+  shows "(as \<union> bs) \<sharp>* x = (as \<sharp>* x \<and> bs \<sharp>* x)"
+  by (auto simp add: fresh_star_def)
+
+lemma fresh_star_insert:
+  shows "(insert a as) \<sharp>* x = (a \<sharp> x \<and> as \<sharp>* x)"
+  by (auto simp add: fresh_star_def)
+
+lemma fresh_star_Un_elim:
+  "((as \<union> bs) \<sharp>* x \<Longrightarrow> PROP C) \<equiv> (as \<sharp>* x \<Longrightarrow> bs \<sharp>* x \<Longrightarrow> PROP C)"
+  unfolding fresh_star_def
+  apply(rule)
+  apply(erule meta_mp)
+  apply(auto)
+  done
+
+lemma fresh_star_insert_elim:
+  "(insert a as \<sharp>* x \<Longrightarrow> PROP C) \<equiv> (a \<sharp> x \<Longrightarrow> as \<sharp>* x \<Longrightarrow> PROP C)"
+  unfolding fresh_star_def
+  by rule (simp_all add: fresh_star_def)
+
+lemma fresh_star_empty_elim:
+  "({} \<sharp>* x \<Longrightarrow> PROP C) \<equiv> PROP C"
+  by (simp add: fresh_star_def)
+
+lemma fresh_star_unit_elim: 
+  shows "(a \<sharp>* () \<Longrightarrow> PROP C) \<equiv> PROP C"
+  by (simp add: fresh_star_def fresh_unit) 
+
+lemma fresh_star_prod_elim: 
+  shows "(a \<sharp>* (x, y) \<Longrightarrow> PROP C) \<equiv> (a \<sharp>* x \<Longrightarrow> a \<sharp>* y \<Longrightarrow> PROP C)"
+  by (rule, simp_all add: fresh_star_prod)
+
+lemma fresh_star_plus:
+  fixes p q::perm
+  shows "\<lbrakk>a \<sharp>* p;  a \<sharp>* q\<rbrakk> \<Longrightarrow> a \<sharp>* (p + q)"
+  unfolding fresh_star_def
+  by (simp add: fresh_plus_perm)
+
+lemma fresh_star_permute_iff:
+  shows "(p \<bullet> a) \<sharp>* (p \<bullet> x) \<longleftrightarrow> a \<sharp>* x"
+  unfolding fresh_star_def
+  by (metis mem_permute_iff permute_minus_cancel fresh_permute_iff)
+
+lemma fresh_star_eqvt:
+  shows "(p \<bullet> (as \<sharp>* x)) = (p \<bullet> as) \<sharp>* (p \<bullet> x)"
+unfolding fresh_star_def
+unfolding Ball_def
+apply(simp add: all_eqvt)
+apply(subst permute_fun_def)
+apply(simp add: imp_eqvt fresh_eqvt mem_eqvt)
+done
+
+section {* Avoiding of atom sets *}
+
+text {* 
+  For every set of atoms, there is another set of atoms
+  avoiding a finitely supported c and there is a permutation
+  which 'translates' between both sets.
+*}
+
+lemma at_set_avoiding_aux:
+  fixes Xs::"atom set"
+  and   As::"atom set"
+  assumes b: "Xs \<subseteq> As"
+  and     c: "finite As"
+  shows "\<exists>p. (p \<bullet> Xs) \<inter> As = {} \<and> (supp p) \<subseteq> (Xs \<union> (p \<bullet> Xs))"
+proof -
+  from b c have "finite Xs" by (rule finite_subset)
+  then show ?thesis using b
+  proof (induct rule: finite_subset_induct)
+    case empty
+    have "0 \<bullet> {} \<inter> As = {}" by simp
+    moreover
+    have "supp (0::perm) \<subseteq> {} \<union> 0 \<bullet> {}" by (simp add: supp_zero_perm)
+    ultimately show ?case by blast
+  next
+    case (insert x Xs)
+    then obtain p where
+      p1: "(p \<bullet> Xs) \<inter> As = {}" and 
+      p2: "supp p \<subseteq> (Xs \<union> (p \<bullet> Xs))" by blast
+    from `x \<in> As` p1 have "x \<notin> p \<bullet> Xs" by fast
+    with `x \<notin> Xs` p2 have "x \<notin> supp p" by fast
+    hence px: "p \<bullet> x = x" unfolding supp_perm by simp
+    have "finite (As \<union> p \<bullet> Xs)"
+      using `finite As` `finite Xs`
+      by (simp add: permute_set_eq_image)
+    then obtain y where "y \<notin> (As \<union> p \<bullet> Xs)" "sort_of y = sort_of x"
+      by (rule obtain_atom)
+    hence y: "y \<notin> As" "y \<notin> p \<bullet> Xs" "sort_of y = sort_of x"
+      by simp_all
+    let ?q = "(x \<rightleftharpoons> y) + p"
+    have q: "?q \<bullet> insert x Xs = insert y (p \<bullet> Xs)"
+      unfolding insert_eqvt
+      using `p \<bullet> x = x` `sort_of y = sort_of x`
+      using `x \<notin> p \<bullet> Xs` `y \<notin> p \<bullet> Xs`
+      by (simp add: swap_atom swap_set_not_in)
+    have "?q \<bullet> insert x Xs \<inter> As = {}"
+      using `y \<notin> As` `p \<bullet> Xs \<inter> As = {}`
+      unfolding q by simp
+    moreover
+    have "supp ?q \<subseteq> insert x Xs \<union> ?q \<bullet> insert x Xs"
+      using p2 unfolding q
+      apply (intro subset_trans [OF supp_plus_perm])
+      apply (auto simp add: supp_swap)
+      done
+    ultimately show ?case by blast
+  qed
+qed
+
+lemma at_set_avoiding:
+  assumes a: "finite Xs"
+  and     b: "finite (supp c)"
+  obtains p::"perm" where "(p \<bullet> Xs)\<sharp>*c" and "(supp p) \<subseteq> (Xs \<union> (p \<bullet> Xs))"
+  using a b at_set_avoiding_aux [where Xs="Xs" and As="Xs \<union> supp c"]
+  unfolding fresh_star_def fresh_def by blast
+
+
+section {* The freshness lemma according to Andrew Pitts *}
+
+lemma fresh_conv_MOST: 
+  shows "a \<sharp> x \<longleftrightarrow> (MOST b. (a \<rightleftharpoons> b) \<bullet> x = x)"
+  unfolding fresh_def supp_def MOST_iff_cofinite by simp
+
+lemma fresh_apply:
+  assumes "a \<sharp> f" and "a \<sharp> x" 
+  shows "a \<sharp> f x"
+  using assms unfolding fresh_conv_MOST
+  unfolding permute_fun_app_eq [where f=f]
+  by (elim MOST_rev_mp, simp)
+
+lemma freshness_lemma:
+  fixes h :: "'a::at \<Rightarrow> 'b::pt"
+  assumes a: "\<exists>a. atom a \<sharp> (h, h a)"
+  shows  "\<exists>x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x"
+proof -
+  from a obtain b where a1: "atom b \<sharp> h" and a2: "atom b \<sharp> h b"
+    by (auto simp add: fresh_Pair)
+  show "\<exists>x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x"
+  proof (intro exI allI impI)
+    fix a :: 'a
+    assume a3: "atom a \<sharp> h"
+    show "h a = h b"
+    proof (cases "a = b")
+      assume "a = b"
+      thus "h a = h b" by simp
+    next
+      assume "a \<noteq> b"
+      hence "atom a \<sharp> b" by (simp add: fresh_at_base)
+      with a3 have "atom a \<sharp> h b" by (rule fresh_apply)
+      with a2 have d1: "(atom b \<rightleftharpoons> atom a) \<bullet> (h b) = (h b)"
+        by (rule swap_fresh_fresh)
+      from a1 a3 have d2: "(atom b \<rightleftharpoons> atom a) \<bullet> h = h"
+        by (rule swap_fresh_fresh)
+      from d1 have "h b = (atom b \<rightleftharpoons> atom a) \<bullet> (h b)" by simp
+      also have "\<dots> = ((atom b \<rightleftharpoons> atom a) \<bullet> h) ((atom b \<rightleftharpoons> atom a) \<bullet> b)"
+        by (rule permute_fun_app_eq)
+      also have "\<dots> = h a"
+        using d2 by simp
+      finally show "h a = h b"  by simp
+    qed
+  qed
+qed
+
+lemma freshness_lemma_unique:
+  fixes h :: "'a::at \<Rightarrow> 'b::pt"
+  assumes a: "\<exists>a. atom a \<sharp> (h, h a)"
+  shows "\<exists>!x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x"
+proof (rule ex_ex1I)
+  from a show "\<exists>x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x"
+    by (rule freshness_lemma)
+next
+  fix x y
+  assume x: "\<forall>a. atom a \<sharp> h \<longrightarrow> h a = x"
+  assume y: "\<forall>a. atom a \<sharp> h \<longrightarrow> h a = y"
+  from a x y show "x = y"
+    by (auto simp add: fresh_Pair)
+qed
+
+text {* packaging the freshness lemma into a function *}
+
+definition
+  fresh_fun :: "('a::at \<Rightarrow> 'b::pt) \<Rightarrow> 'b"
+where
+  "fresh_fun h = (THE x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x)"
+
+lemma fresh_fun_app:
+  fixes h :: "'a::at \<Rightarrow> 'b::pt"
+  assumes a: "\<exists>a. atom a \<sharp> (h, h a)"
+  assumes b: "atom a \<sharp> h"
+  shows "fresh_fun h = h a"
+unfolding fresh_fun_def
+proof (rule the_equality)
+  show "\<forall>a'. atom a' \<sharp> h \<longrightarrow> h a' = h a"
+  proof (intro strip)
+    fix a':: 'a
+    assume c: "atom a' \<sharp> h"
+    from a have "\<exists>x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x" by (rule freshness_lemma)
+    with b c show "h a' = h a" by auto
+  qed
+next
+  fix fr :: 'b
+  assume "\<forall>a. atom a \<sharp> h \<longrightarrow> h a = fr"
+  with b show "fr = h a" by auto
+qed
+
+lemma fresh_fun_app':
+  fixes h :: "'a::at \<Rightarrow> 'b::pt"
+  assumes a: "atom a \<sharp> h" "atom a \<sharp> h a"
+  shows "fresh_fun h = h a"
+  apply (rule fresh_fun_app)
+  apply (auto simp add: fresh_Pair intro: a)
+  done
+
+lemma fresh_fun_eqvt:
+  fixes h :: "'a::at \<Rightarrow> 'b::pt"
+  assumes a: "\<exists>a. atom a \<sharp> (h, h a)"
+  shows "p \<bullet> (fresh_fun h) = fresh_fun (p \<bullet> h)"
+  using a
+  apply (clarsimp simp add: fresh_Pair)
+  apply (subst fresh_fun_app', assumption+)
+  apply (drule fresh_permute_iff [where p=p, THEN iffD2])
+  apply (drule fresh_permute_iff [where p=p, THEN iffD2])
+  apply (simp add: atom_eqvt permute_fun_app_eq [where f=h])
+  apply (erule (1) fresh_fun_app' [symmetric])
+  done
+
+lemma fresh_fun_supports:
+  fixes h :: "'a::at \<Rightarrow> 'b::pt"
+  assumes a: "\<exists>a. atom a \<sharp> (h, h a)"
+  shows "(supp h) supports (fresh_fun h)"
+  apply (simp add: supports_def fresh_def [symmetric])
+  apply (simp add: fresh_fun_eqvt [OF a] swap_fresh_fresh)
+  done
+
+notation fresh_fun (binder "FRESH " 10)
+
+lemma FRESH_f_iff:
+  fixes P :: "'a::at \<Rightarrow> 'b::pure"
+  fixes f :: "'b \<Rightarrow> 'c::pure"
+  assumes P: "finite (supp P)"
+  shows "(FRESH x. f (P x)) = f (FRESH x. P x)"
+proof -
+  obtain a::'a where "atom a \<notin> supp P"
+    using P by (rule obtain_at_base)
+  hence "atom a \<sharp> P"
+    by (simp add: fresh_def)
+  show "(FRESH x. f (P x)) = f (FRESH x. P x)"
+    apply (subst fresh_fun_app' [where a=a, OF _ pure_fresh])
+    apply (cut_tac `atom a \<sharp> P`)
+    apply (simp add: fresh_conv_MOST)
+    apply (elim MOST_rev_mp, rule MOST_I, clarify)
+    apply (simp add: permute_fun_def permute_pure expand_fun_eq)
+    apply (subst fresh_fun_app' [where a=a, OF `atom a \<sharp> P` pure_fresh])
+    apply (rule refl)
+    done
+qed
+
+lemma FRESH_binop_iff:
+  fixes P :: "'a::at \<Rightarrow> 'b::pure"
+  fixes Q :: "'a::at \<Rightarrow> 'c::pure"
+  fixes binop :: "'b \<Rightarrow> 'c \<Rightarrow> 'd::pure"
+  assumes P: "finite (supp P)" 
+  and     Q: "finite (supp Q)"
+  shows "(FRESH x. binop (P x) (Q x)) = binop (FRESH x. P x) (FRESH x. Q x)"
+proof -
+  from assms have "finite (supp P \<union> supp Q)" by simp
+  then obtain a::'a where "atom a \<notin> (supp P \<union> supp Q)"
+    by (rule obtain_at_base)
+  hence "atom a \<sharp> P" and "atom a \<sharp> Q"
+    by (simp_all add: fresh_def)
+  show ?thesis
+    apply (subst fresh_fun_app' [where a=a, OF _ pure_fresh])
+    apply (cut_tac `atom a \<sharp> P` `atom a \<sharp> Q`)
+    apply (simp add: fresh_conv_MOST)
+    apply (elim MOST_rev_mp, rule MOST_I, clarify)
+    apply (simp add: permute_fun_def permute_pure expand_fun_eq)
+    apply (subst fresh_fun_app' [where a=a, OF `atom a \<sharp> P` pure_fresh])
+    apply (subst fresh_fun_app' [where a=a, OF `atom a \<sharp> Q` pure_fresh])
+    apply (rule refl)
+    done
+qed
+
+lemma FRESH_conj_iff:
+  fixes P Q :: "'a::at \<Rightarrow> bool"
+  assumes P: "finite (supp P)" and Q: "finite (supp Q)"
+  shows "(FRESH x. P x \<and> Q x) \<longleftrightarrow> (FRESH x. P x) \<and> (FRESH x. Q x)"
+using P Q by (rule FRESH_binop_iff)
+
+lemma FRESH_disj_iff:
+  fixes P Q :: "'a::at \<Rightarrow> bool"
+  assumes P: "finite (supp P)" and Q: "finite (supp Q)"
+  shows "(FRESH x. P x \<or> Q x) \<longleftrightarrow> (FRESH x. P x) \<or> (FRESH x. Q x)"
+using P Q by (rule FRESH_binop_iff)
+
+
+section {* An example of a function without finite support *}
+
+primrec
+  nat_of :: "atom \<Rightarrow> nat"
+where
+  "nat_of (Atom s n) = n"
+
+lemma atom_eq_iff:
+  fixes a b :: atom
+  shows "a = b \<longleftrightarrow> sort_of a = sort_of b \<and> nat_of a = nat_of b"
+  by (induct a, induct b, simp)
+
+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_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 {* Support for sets of atoms *}
+
+lemma supp_finite_atom_set:
+  fixes S::"atom set"
+  assumes "finite S"
+  shows "supp S = S"
+  apply(rule finite_supp_unique)
+  apply(simp add: supports_def)
+  apply(simp add: swap_set_not_in)
+  apply(rule assms)
+  apply(simp add: swap_set_in)
+done
+
+
+section {* transpositions of permutations *}
+
+fun
+  add_perm 
+where
+  "add_perm [] = 0"
+| "add_perm ((a, b) # xs) = (a \<rightleftharpoons> b) + add_perm xs"
+
+lemma add_perm_append:
+  shows "add_perm (xs @ ys) = add_perm xs + add_perm ys"
+by (induct xs arbitrary: ys)
+   (auto simp add: add_assoc)
+
+lemma perm_list_exists:
+  fixes p::perm
+  shows "\<exists>xs. p = add_perm xs \<and> supp xs \<subseteq> supp p"
+apply(induct p taking: "\<lambda>p::perm. card (supp p)" rule: measure_induct)
+apply(rename_tac p)
+apply(case_tac "supp p = {}")
+apply(simp)
+apply(simp add: supp_perm)
+apply(rule_tac x="[]" in exI)
+apply(simp add: supp_Nil)
+apply(simp add: expand_perm_eq)
+apply(subgoal_tac "\<exists>x. x \<in> supp p")
+defer
+apply(auto)[1]
+apply(erule exE)
+apply(drule_tac x="p + (((- p) \<bullet> x) \<rightleftharpoons> x)" in spec)
+apply(drule mp)
+defer
+apply(erule exE)
+apply(rule_tac x="xs @ [((- p) \<bullet> x, x)]" in exI)
+apply(simp add: add_perm_append)
+apply(erule conjE)
+apply(drule sym)
+apply(simp)
+apply(simp add: expand_perm_eq)
+apply(simp add: supp_append)
+apply(simp add: supp_perm supp_Cons supp_Nil supp_Pair supp_atom)
+apply(rule conjI)
+prefer 2
+apply(auto)[1]
+apply (metis permute_atom_def_raw permute_minus_cancel(2))
+defer
+apply(rule psubset_card_mono)
+apply(simp add: finite_supp)
+apply(rule psubsetI)
+defer
+apply(subgoal_tac "x \<notin> supp (p + (- p \<bullet> x \<rightleftharpoons> x))")
+apply(blast)
+apply(simp add: supp_perm)
+defer
+apply(auto simp add: supp_perm)[1]
+apply(case_tac "x = xa")
+apply(simp)
+apply(case_tac "((- p) \<bullet> x) = xa")
+apply(simp)
+apply(case_tac "sort_of xa = sort_of x")
+apply(simp)
+apply(auto)[1]
+apply(simp)
+apply(simp)
+apply(subgoal_tac "{a. p \<bullet> (- p \<bullet> x \<rightleftharpoons> x) \<bullet> a \<noteq> a} \<subseteq> {a. p \<bullet> a \<noteq> a}")
+apply(blast)
+apply(auto simp add: supp_perm)[1]
+apply(case_tac "x = xa")
+apply(simp)
+apply(case_tac "((- p) \<bullet> x) = xa")
+apply(simp)
+apply(case_tac "sort_of xa = sort_of x")
+apply(simp)
+apply(auto)[1]
+apply(simp)
+apply(simp)
+done
+
+section {* Lemma about support and permutations *}
+
+lemma supp_perm_eq:
+  assumes a: "(supp x) \<sharp>* p"
+  shows "p \<bullet> x = x"
+proof -
+  obtain xs where eq: "p = add_perm xs" and sub: "supp xs \<subseteq> supp p"
+    using perm_list_exists by blast
+  from a have "\<forall>a \<in> supp p. a \<sharp> x"
+    by (auto simp add: fresh_star_def fresh_def supp_perm)
+  with eq sub have "\<forall>a \<in> supp xs. a \<sharp> x" by auto
+  then have "add_perm xs \<bullet> x = x" 
+    by (induct xs rule: add_perm.induct)
+       (simp_all add: supp_Cons supp_Pair supp_atom swap_fresh_fresh)
+  then show "p \<bullet> x = x" using eq by simp
+qed
+
+section {* at_set_avoiding2 *}
+
+lemma at_set_avoiding2:
+  assumes "finite xs"
+  and     "finite (supp c)" "finite (supp x)"
+  and     "xs \<sharp>* x"
+  shows "\<exists>p. (p \<bullet> xs) \<sharp>* c \<and> supp x \<sharp>* p"
+using assms
+apply(erule_tac c="(c, x)" in at_set_avoiding)
+apply(simp add: supp_Pair)
+apply(rule_tac x="p" in exI)
+apply(simp add: fresh_star_prod)
+apply(subgoal_tac "\<forall>a \<in> supp p. a \<sharp> x")
+apply(auto simp add: fresh_star_def fresh_def supp_perm)[1]
+apply(auto simp add: fresh_star_def fresh_def)
+done
+
+lemma at_set_avoiding2_atom:
+  assumes "finite (supp c)" "finite (supp x)"
+  and     b: "xa \<sharp> x"
+  shows "\<exists>p. (p \<bullet> xa) \<sharp> c \<and> supp x \<sharp>* p"
+proof -
+  have a: "{xa} \<sharp>* x" unfolding fresh_star_def by (simp add: b)
+  obtain p where p1: "(p \<bullet> {xa}) \<sharp>* c" and p2: "supp x \<sharp>* p"
+    using at_set_avoiding2[of "{xa}" "c" "x"] assms a by blast
+  have c: "(p \<bullet> xa) \<sharp> c" using p1
+    unfolding fresh_star_def Ball_def 
+    by (erule_tac x="p \<bullet> xa" in allE) (simp add: eqvts)
+  hence "p \<bullet> xa \<sharp> c \<and> supp x \<sharp>* p" using p2 by blast
+  then show ?thesis by blast
+qed
+
+end
diff -r c0eac04ae3b4 -r c34347ec7ab3 Nominal-General/ROOT.ML
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Nominal-General/ROOT.ML	Sun Apr 04 21:39:28 2010 +0200
@@ -0,0 +1,7 @@
+
+no_document use_thys
+   ["Nominal2_Base",
+    "Nominal2_Eqvt",
+    "Nominal2_Atoms",
+    "Nominal2_Supp",
+    "Atoms"];
diff -r c0eac04ae3b4 -r c34347ec7ab3 Nominal-General/nominal_atoms.ML
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Nominal-General/nominal_atoms.ML	Sun Apr 04 21:39:28 2010 +0200
@@ -0,0 +1,94 @@
+(*  Title:      nominal_atoms/ML
+    Authors:    Brian Huffman, Christian Urban
+
+    Command for defining concrete atom types. 
+    
+    At the moment, only single-sorted atom types
+    are supported. 
+*)
+
+signature ATOM_DECL =
+sig
+  val add_atom_decl: (binding * (binding option)) -> theory -> theory
+end;
+
+structure Atom_Decl :> ATOM_DECL =
+struct
+
+val atomT = @{typ atom};
+val permT = @{typ perm};
+
+val sort_of_const = @{term sort_of};
+fun atom_const T = Const (@{const_name atom}, T --> atomT);
+fun permute_const T = Const (@{const_name permute}, permT --> T --> T);
+
+fun mk_sort_of t = sort_of_const $ t;
+fun mk_atom t = atom_const (fastype_of t) $ t;
+fun mk_permute (p, t) = permute_const (fastype_of t) $ p $ t;
+
+fun atom_decl_set (str : string) : term =
+  let
+    val a = Free ("a", atomT);
+    val s = Const (@{const_name "Sort"}, @{typ "string => atom_sort list => atom_sort"})
+              $ HOLogic.mk_string str $ HOLogic.nil_const @{typ "atom_sort"};
+  in
+    HOLogic.mk_Collect ("a", atomT, HOLogic.mk_eq (mk_sort_of a, s))
+  end
+
+fun add_atom_decl (name : binding, arg : binding option) (thy : theory) =
+  let
+    val _ = Theory.requires thy "Nominal2_Atoms" "nominal logic";
+    val str = Sign.full_name thy name;
+
+    (* typedef *)
+    val set = atom_decl_set str;
+    val tac = rtac @{thm exists_eq_simple_sort} 1;
+    val ((full_tname, info  as ({Rep_name, Abs_name, ...}, {type_definition, ...})), thy) =
+      Typedef.add_typedef_global false NONE (name, [], NoSyn) set NONE tac thy;
+
+    (* definition of atom and permute *)
+    val newT = #abs_type (fst info);
+    val RepC = Const (Rep_name, newT --> atomT);
+    val AbsC = Const (Abs_name, atomT --> newT);
+    val a = Free ("a", newT);
+    val p = Free ("p", permT);
+    val atom_eqn =
+      HOLogic.mk_Trueprop (HOLogic.mk_eq (mk_atom a, RepC $ a));
+    val permute_eqn =
+      HOLogic.mk_Trueprop (HOLogic.mk_eq
+        (mk_permute (p, a), AbsC $ (mk_permute (p, RepC $ a))));
+    val atom_def_name =
+      Binding.prefix_name "atom_" (Binding.suffix_name "_def" name);
+    val permute_def_name =
+      Binding.prefix_name "permute_" (Binding.suffix_name "_def" name);
+
+    (* at class instance *)
+    val lthy =
+      Theory_Target.instantiation ([full_tname], [], @{sort at}) thy;
+    val ((_, (_, permute_ldef)), lthy) =
+      Specification.definition (NONE, ((permute_def_name, []), permute_eqn)) lthy;
+    val ((_, (_, atom_ldef)), lthy) =
+      Specification.definition (NONE, ((atom_def_name, []), atom_eqn)) lthy;
+    val ctxt_thy = ProofContext.init (ProofContext.theory_of lthy);
+    val permute_def = singleton (ProofContext.export lthy ctxt_thy) permute_ldef;
+    val atom_def = singleton (ProofContext.export lthy ctxt_thy) atom_ldef;
+    val class_thm = @{thm at_class} OF [type_definition, atom_def, permute_def];
+    val thy = lthy
+      |> Class.prove_instantiation_instance (K (Tactic.rtac class_thm 1))
+      |> Local_Theory.exit_global;
+  in
+    thy
+  end;
+
+(** outer syntax **)
+
+local structure P = OuterParse and K = OuterKeyword in
+
+val _ =
+  OuterSyntax.command "atom_decl" "declaration of a concrete atom type" K.thy_decl
+    ((P.binding -- Scan.option (Args.parens (P.binding))) >>
+      (Toplevel.print oo (Toplevel.theory o add_atom_decl)));
+
+end;
+
+end;
diff -r c0eac04ae3b4 -r c34347ec7ab3 Nominal-General/nominal_permeq.ML
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Nominal-General/nominal_permeq.ML	Sun Apr 04 21:39:28 2010 +0200
@@ -0,0 +1,71 @@
+(*  Title:      nominal_thmdecls.ML
+    Author:     Brian Huffman, Christian Urban
+*)
+
+signature NOMINAL_PERMEQ =
+sig
+  val eqvt_tac: Proof.context -> int -> tactic 
+
+end;
+
+(* TODO:
+
+ - provide a method interface with the usual add and del options
+
+ - print a warning if for a constant no eqvt lemma is stored
+
+ - there seems to be too much simplified in case of multiple 
+   permutations, like
+
+      p o q o r o x 
+
+   we usually only want the outermost permutation to "float" in
+*)
+
+
+structure Nominal_Permeq: NOMINAL_PERMEQ =
+struct
+
+local
+
+fun eqvt_apply_conv ctxt ct =
+  case (term_of ct) of
+    (Const (@{const_name "permute"}, _) $ _ $ (_ $ _)) =>
+      let
+        val (perm, t) = Thm.dest_comb ct
+        val (_, p) = Thm.dest_comb perm
+        val (f, x) = Thm.dest_comb t
+        val a = ctyp_of_term x;
+        val b = ctyp_of_term t;
+        val ty_insts = map SOME [b, a]
+        val term_insts = map SOME [p, f, x]
+      in
+        Drule.instantiate' ty_insts term_insts @{thm eqvt_apply}
+      end
+  | _ => Conv.no_conv ct
+
+fun eqvt_lambda_conv ctxt ct =
+  case (term_of ct) of
+    (Const (@{const_name "permute"}, _) $ _ $ Abs _) =>
+      Conv.rewr_conv @{thm eqvt_lambda} ct
+  | _ => Conv.no_conv ct
+
+in
+
+fun eqvt_conv ctxt ct =
+  Conv.first_conv
+    [ Conv.rewr_conv @{thm eqvt_bound},
+      eqvt_apply_conv ctxt
+        then_conv Conv.comb_conv (eqvt_conv ctxt),
+      eqvt_lambda_conv ctxt
+        then_conv Conv.abs_conv (fn (v, ctxt) => eqvt_conv ctxt) ctxt,
+      More_Conv.rewrs_conv (Nominal_ThmDecls.get_eqvts_raw_thms ctxt),
+      Conv.all_conv
+    ] ct
+
+fun eqvt_tac ctxt = 
+  CONVERSION (More_Conv.bottom_conv (fn ctxt => eqvt_conv ctxt) ctxt)
+
+end
+
+end; (* structure *)
\ No newline at end of file
diff -r c0eac04ae3b4 -r c34347ec7ab3 Nominal-General/nominal_thmdecls.ML
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Nominal-General/nominal_thmdecls.ML	Sun Apr 04 21:39:28 2010 +0200
@@ -0,0 +1,134 @@
+(*  Title:      nominal_thmdecls.ML
+    Author:     Christian Urban
+
+  Infrastructure for the lemma collection "eqvts".
+
+  Provides the attributes [eqvt] and [eqvt_raw], and the theorem
+  lists eqvts and eqvts_raw. The first attribute will store the 
+  theorem in the eqvts list and also in the eqvts_raw list. For 
+  the latter the theorem is expected to be of the form
+
+    p o (c x1 x2 ...) = c (p o x1) (p o x2) ...
+
+  and it is stored in the form
+
+    p o c == c
+
+  The [eqvt_raw] attribute just adds the theorem to eqvts_raw.
+
+  TODO:
+
+   - deal with eqvt-lemmas of the form 
+
+       c x1 x2 ... ==> c (p o x1) (p o x2) ..
+*)
+
+signature NOMINAL_THMDECLS =
+sig
+  val eqvt_add: attribute
+  val eqvt_del: attribute
+  val eqvt_raw_add: attribute
+  val eqvt_raw_del: attribute
+  val setup: theory -> theory
+  val get_eqvts_thms: Proof.context -> thm list
+  val get_eqvts_raw_thms: Proof.context -> thm list
+
+end;
+
+structure Nominal_ThmDecls: NOMINAL_THMDECLS =
+struct
+
+
+structure EqvtData = Generic_Data
+( type T = thm Item_Net.T;
+  val empty = Thm.full_rules;
+  val extend = I;
+  val merge = Item_Net.merge );
+
+structure EqvtRawData = Generic_Data
+( type T = thm Item_Net.T;
+  val empty = Thm.full_rules;
+  val extend = I;
+  val merge = Item_Net.merge );
+
+val eqvts = Item_Net.content o EqvtData.get;
+val eqvts_raw = Item_Net.content o EqvtRawData.get;
+
+val get_eqvts_thms = eqvts o Context.Proof; 
+val get_eqvts_raw_thms = eqvts_raw o Context.Proof; 
+
+val add_thm = EqvtData.map o Item_Net.update;
+val del_thm = EqvtData.map o Item_Net.remove;
+
+val add_raw_thm = EqvtRawData.map o Item_Net.update;
+val del_raw_thm = EqvtRawData.map o Item_Net.remove;
+
+fun dest_perm (Const (@{const_name "permute"}, _) $ p $ t) = (p, t)
+  | dest_perm t = raise TERM("dest_perm", [t])
+
+fun mk_perm p trm =
+let
+  val ty = fastype_of trm
+in
+  Const (@{const_name "permute"}, @{typ "perm"} --> ty --> ty) $ p $ trm
+end
+
+fun eqvt_transform_tac thm = REPEAT o FIRST' 
+  [CHANGED o simp_tac (HOL_basic_ss addsimps @{thms permute_minus_cancel}),
+   rtac (thm RS @{thm trans}),
+   rtac @{thm trans[OF permute_fun_def]} THEN' rtac @{thm ext}]
+
+(* transform equations into the required form *)
+fun transform_eq ctxt thm lhs rhs = 
+let
+  val (p, t) = dest_perm lhs
+  val (c, args) = strip_comb t
+  val (c', args') = strip_comb rhs 
+  val eargs = map Envir.eta_contract args 
+  val eargs' = map Envir.eta_contract args'
+  val p_str = fst (fst (dest_Var p))
+  val goal = HOLogic.mk_Trueprop (HOLogic.mk_eq (mk_perm p c, c))
+in
+  if c <> c' 
+    then error "eqvt lemma is not of the right form (constants do not agree)"
+  else if eargs' <> map (mk_perm p) eargs 
+    then error "eqvt lemma is not of the right form (arguments do not agree)"
+  else if args = [] 
+    then thm
+  else Goal.prove ctxt [p_str] [] goal
+    (fn _ => eqvt_transform_tac thm 1)
+end
+
+fun transform addel_fun thm context = 
+let
+  val ctxt = Context.proof_of context
+  val trm = HOLogic.dest_Trueprop (prop_of thm)
+in
+  case trm of
+    Const (@{const_name "op ="}, _) $ lhs $ rhs =>
+      let 
+        val thm' = transform_eq ctxt thm lhs rhs RS @{thm eq_reflection}
+      in
+        addel_fun thm' context
+      end
+  | _ => raise (error "only (op=) case implemented yet")
+end 
+
+val eqvt_add = Thm.declaration_attribute (fn thm => (add_thm thm) o (transform add_raw_thm thm));
+val eqvt_del = Thm.declaration_attribute (fn thm => (del_thm thm) o (transform del_raw_thm thm));
+
+val eqvt_raw_add = Thm.declaration_attribute add_raw_thm;
+val eqvt_raw_del = Thm.declaration_attribute del_raw_thm;
+
+val setup =
+  Attrib.setup @{binding "eqvt"} (Attrib.add_del eqvt_add eqvt_del) 
+    (cat_lines ["declaration of equivariance lemmas - they will automtically be",  
+       "brought into the form p o c = c"]) #>
+  Attrib.setup @{binding "eqvt_raw"} (Attrib.add_del eqvt_raw_add eqvt_raw_del) 
+    (cat_lines ["declaration of equivariance lemmas - no",
+       "transformation is performed"]) #>
+  PureThy.add_thms_dynamic (@{binding "eqvts"}, eqvts) #>
+  PureThy.add_thms_dynamic (@{binding "eqvts_raw"}, eqvts_raw);
+
+
+end;
diff -r c0eac04ae3b4 -r c34347ec7ab3 Nominal/Atoms.thy
--- a/Nominal/Atoms.thy	Sat Apr 03 22:31:11 2010 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,164 +0,0 @@
-(*  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
diff -r c0eac04ae3b4 -r c34347ec7ab3 Nominal/Ex/ExLet.thy
--- a/Nominal/Ex/ExLet.thy	Sat Apr 03 22:31:11 2010 +0200
+++ b/Nominal/Ex/ExLet.thy	Sun Apr 04 21:39:28 2010 +0200
@@ -167,6 +167,7 @@
     apply(subgoal_tac "\<exists>q. (q \<bullet> set (bn (p \<bullet> lts))) \<sharp>* c \<and> supp (Abs_lst (bn (p \<bullet> lts)) (p \<bullet> trm)) \<sharp>* q")
     apply(erule exE)
     apply(erule conjE)
+    thm Lt_subst
     apply(subst Lt_subst)
     apply assumption
     apply(rule a4)
diff -r c0eac04ae3b4 -r c34347ec7ab3 Nominal/Fv.thy
--- a/Nominal/Fv.thy	Sat Apr 03 22:31:11 2010 +0200
+++ b/Nominal/Fv.thy	Sun Apr 04 21:39:28 2010 +0200
@@ -1,5 +1,5 @@
 theory Fv
-imports "Nominal2_Atoms" "Abs" "Perm" "Rsp" "Nominal2_FSet"
+imports "../Nominal-General/Nominal2_Atoms" "Abs" "Perm" "Rsp" "Nominal2_FSet"
 begin
 
 (* The bindings data structure:
diff -r c0eac04ae3b4 -r c34347ec7ab3 Nominal/Lift.thy
--- a/Nominal/Lift.thy	Sat Apr 03 22:31:11 2010 +0200
+++ b/Nominal/Lift.thy	Sun Apr 04 21:39:28 2010 +0200
@@ -1,5 +1,8 @@
 theory Lift
-imports "Nominal2_Atoms" "Nominal2_Eqvt" "Nominal2_Supp" "Abs" "Perm" "Fv" "Rsp"
+imports "../Nominal-General/Nominal2_Atoms" 
+        "../Nominal-General/Nominal2_Eqvt" 
+        "../Nominal_General/Nominal2_Supp" 
+        "Abs" "Perm" "Fv" "Rsp"
 begin
 
 
diff -r c0eac04ae3b4 -r c34347ec7ab3 Nominal/Nominal2_Atoms.thy
--- a/Nominal/Nominal2_Atoms.thy	Sat Apr 03 22:31:11 2010 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,255 +0,0 @@
-(*  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)"
-
-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" => "CONST 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
diff -r c0eac04ae3b4 -r c34347ec7ab3 Nominal/Nominal2_Base.thy
--- a/Nominal/Nominal2_Base.thy	Sat Apr 03 22:31:11 2010 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,1062 +0,0 @@
-(*  Title:      Nominal2_Base
-    Authors:    Brian Huffman, Christian Urban
-
-    Basic definitions and lemma infrastructure for 
-    Nominal Isabelle. 
-*)
-theory Nominal2_Base
-imports Main Infinite_Set
-begin
-
-section {* Atoms and Sorts *}
-
-text {* A simple implementation for atom_sorts is strings. *}
-(* types atom_sort = string *)
-
-text {* To deal with Church-like binding we use trees of  
-  strings as sorts. *}
-
-datatype atom_sort = Sort "string" "atom_sort list"
-
-datatype atom = Atom atom_sort nat
-
-
-text {* Basic projection function. *}
-
-primrec
-  sort_of :: "atom \<Rightarrow> atom_sort"
-where
-  "sort_of (Atom s i) = s"
-
-
-text {* There are infinitely many atoms of each sort. *}
-lemma INFM_sort_of_eq: 
-  shows "INFM a. sort_of a = s"
-proof -
-  have "INFM i. sort_of (Atom s i) = s" by simp
-  moreover have "inj (Atom s)" by (simp add: inj_on_def)
-  ultimately show "INFM a. sort_of a = s" by (rule INFM_inj)
-qed
-
-lemma infinite_sort_of_eq:
-  shows "infinite {a. sort_of a = s}"
-  using INFM_sort_of_eq unfolding INFM_iff_infinite .
-
-lemma atom_infinite [simp]: 
-  shows "infinite (UNIV :: atom set)"
-  using subset_UNIV infinite_sort_of_eq
-  by (rule infinite_super)
-
-lemma obtain_atom:
-  fixes X :: "atom set"
-  assumes X: "finite X"
-  obtains a where "a \<notin> X" "sort_of a = s"
-proof -
-  from X have "MOST a. a \<notin> X"
-    unfolding MOST_iff_cofinite by simp
-  with INFM_sort_of_eq
-  have "INFM a. sort_of a = s \<and> a \<notin> X"
-    by (rule INFM_conjI)
-  then obtain a where "a \<notin> X" "sort_of a = s"
-    by (auto elim: INFM_E)
-  then show ?thesis ..
-qed
-
-section {* Sort-Respecting Permutations *}
-
-typedef perm =
-  "{f. bij f \<and> finite {a. f a \<noteq> a} \<and> (\<forall>a. sort_of (f a) = sort_of a)}"
-proof
-  show "id \<in> ?perm" by simp
-qed
-
-lemma permI:
-  assumes "bij f" and "MOST x. f x = x" and "\<And>a. sort_of (f a) = sort_of a"
-  shows "f \<in> perm"
-  using assms unfolding perm_def MOST_iff_cofinite by simp
-
-lemma perm_is_bij: "f \<in> perm \<Longrightarrow> bij f"
-  unfolding perm_def by simp
-
-lemma perm_is_finite: "f \<in> perm \<Longrightarrow> finite {a. f a \<noteq> a}"
-  unfolding perm_def by simp
-
-lemma perm_is_sort_respecting: "f \<in> perm \<Longrightarrow> sort_of (f a) = sort_of a"
-  unfolding perm_def by simp
-
-lemma perm_MOST: "f \<in> perm \<Longrightarrow> MOST x. f x = x"
-  unfolding perm_def MOST_iff_cofinite by simp
-
-lemma perm_id: "id \<in> perm"
-  unfolding perm_def by simp
-
-lemma perm_comp:
-  assumes f: "f \<in> perm" and g: "g \<in> perm"
-  shows "(f \<circ> g) \<in> perm"
-apply (rule permI)
-apply (rule bij_comp)
-apply (rule perm_is_bij [OF g])
-apply (rule perm_is_bij [OF f])
-apply (rule MOST_rev_mp [OF perm_MOST [OF g]])
-apply (rule MOST_rev_mp [OF perm_MOST [OF f]])
-apply (simp)
-apply (simp add: perm_is_sort_respecting [OF f])
-apply (simp add: perm_is_sort_respecting [OF g])
-done
-
-lemma perm_inv:
-  assumes f: "f \<in> perm"
-  shows "(inv f) \<in> perm"
-apply (rule permI)
-apply (rule bij_imp_bij_inv)
-apply (rule perm_is_bij [OF f])
-apply (rule MOST_mono [OF perm_MOST [OF f]])
-apply (erule subst, rule inv_f_f)
-apply (rule bij_is_inj [OF perm_is_bij [OF f]])
-apply (rule perm_is_sort_respecting [OF f, THEN sym, THEN trans])
-apply (simp add: surj_f_inv_f [OF bij_is_surj [OF perm_is_bij [OF f]]])
-done
-
-lemma bij_Rep_perm: "bij (Rep_perm p)"
-  using Rep_perm [of p] unfolding perm_def by simp
-
-lemma finite_Rep_perm: "finite {a. Rep_perm p a \<noteq> a}"
-  using Rep_perm [of p] unfolding perm_def by simp
-
-lemma sort_of_Rep_perm: "sort_of (Rep_perm p a) = sort_of a"
-  using Rep_perm [of p] unfolding perm_def by simp
-
-lemma Rep_perm_ext:
-  "Rep_perm p1 = Rep_perm p2 \<Longrightarrow> p1 = p2"
-  by (simp add: expand_fun_eq Rep_perm_inject [symmetric])
-
-
-subsection {* Permutations form a group *}
-
-instantiation perm :: group_add
-begin
-
-definition
-  "0 = Abs_perm id"
-
-definition
-  "- p = Abs_perm (inv (Rep_perm p))"
-
-definition
-  "p + q = Abs_perm (Rep_perm p \<circ> Rep_perm q)"
-
-definition
-  "(p1::perm) - p2 = p1 + - p2"
-
-lemma Rep_perm_0: "Rep_perm 0 = id"
-  unfolding zero_perm_def
-  by (simp add: Abs_perm_inverse perm_id)
-
-lemma Rep_perm_add:
-  "Rep_perm (p1 + p2) = Rep_perm p1 \<circ> Rep_perm p2"
-  unfolding plus_perm_def
-  by (simp add: Abs_perm_inverse perm_comp Rep_perm)
-
-lemma Rep_perm_uminus:
-  "Rep_perm (- p) = inv (Rep_perm p)"
-  unfolding uminus_perm_def
-  by (simp add: Abs_perm_inverse perm_inv Rep_perm)
-
-instance
-apply default
-unfolding Rep_perm_inject [symmetric]
-unfolding minus_perm_def
-unfolding Rep_perm_add
-unfolding Rep_perm_uminus
-unfolding Rep_perm_0
-by (simp_all add: o_assoc inv_o_cancel [OF bij_is_inj [OF bij_Rep_perm]])
-
-end
-
-
-section {* Implementation of swappings *}
-
-definition
-  swap :: "atom \<Rightarrow> atom \<Rightarrow> perm" ("'(_ \<rightleftharpoons> _')")
-where
-  "(a \<rightleftharpoons> b) =
-    Abs_perm (if sort_of a = sort_of b 
-              then (\<lambda>c. if a = c then b else if b = c then a else c) 
-              else id)"
-
-lemma Rep_perm_swap:
-  "Rep_perm (a \<rightleftharpoons> b) =
-    (if sort_of a = sort_of b 
-     then (\<lambda>c. if a = c then b else if b = c then a else c)
-     else id)"
-unfolding swap_def
-apply (rule Abs_perm_inverse)
-apply (rule permI)
-apply (auto simp add: bij_def inj_on_def surj_def)[1]
-apply (rule MOST_rev_mp [OF MOST_neq(1) [of a]])
-apply (rule MOST_rev_mp [OF MOST_neq(1) [of b]])
-apply (simp)
-apply (simp)
-done
-
-lemmas Rep_perm_simps =
-  Rep_perm_0
-  Rep_perm_add
-  Rep_perm_uminus
-  Rep_perm_swap
-
-lemma swap_different_sorts [simp]:
-  "sort_of a \<noteq> sort_of b \<Longrightarrow> (a \<rightleftharpoons> b) = 0"
-  by (rule Rep_perm_ext) (simp add: Rep_perm_simps)
-
-lemma swap_cancel:
-  "(a \<rightleftharpoons> b) + (a \<rightleftharpoons> b) = 0"
-by (rule Rep_perm_ext) 
-   (simp add: Rep_perm_simps expand_fun_eq)
-
-lemma swap_self [simp]:
-  "(a \<rightleftharpoons> a) = 0"
-  by (rule Rep_perm_ext, simp add: Rep_perm_simps expand_fun_eq)
-
-lemma minus_swap [simp]:
-  "- (a \<rightleftharpoons> b) = (a \<rightleftharpoons> b)"
-  by (rule minus_unique [OF swap_cancel])
-
-lemma swap_commute:
-  "(a \<rightleftharpoons> b) = (b \<rightleftharpoons> a)"
-  by (rule Rep_perm_ext)
-     (simp add: Rep_perm_swap expand_fun_eq)
-
-lemma swap_triple:
-  assumes "a \<noteq> b" and "c \<noteq> b"
-  assumes "sort_of a = sort_of b" "sort_of b = sort_of c"
-  shows "(a \<rightleftharpoons> c) + (b \<rightleftharpoons> c) + (a \<rightleftharpoons> c) = (a \<rightleftharpoons> b)"
-  using assms
-  by (rule_tac Rep_perm_ext)
-     (auto simp add: Rep_perm_simps expand_fun_eq)
-
-
-section {* Permutation Types *}
-
-text {*
-  Infix syntax for @{text permute} has higher precedence than
-  addition, but lower than unary minus.
-*}
-
-class pt =
-  fixes permute :: "perm \<Rightarrow> 'a \<Rightarrow> 'a" ("_ \<bullet> _" [76, 75] 75)
-  assumes permute_zero [simp]: "0 \<bullet> x = x"
-  assumes permute_plus [simp]: "(p + q) \<bullet> x = p \<bullet> (q \<bullet> x)"
-begin
-
-lemma permute_diff [simp]:
-  shows "(p - q) \<bullet> x = p \<bullet> - q \<bullet> x"
-  unfolding diff_minus by simp
-
-lemma permute_minus_cancel [simp]:
-  shows "p \<bullet> - p \<bullet> x = x"
-  and   "- p \<bullet> p \<bullet> x = x"
-  unfolding permute_plus [symmetric] by simp_all
-
-lemma permute_swap_cancel [simp]:
-  shows "(a \<rightleftharpoons> b) \<bullet> (a \<rightleftharpoons> b) \<bullet> x = x"
-  unfolding permute_plus [symmetric]
-  by (simp add: swap_cancel)
-
-lemma permute_swap_cancel2 [simp]:
-  shows "(a \<rightleftharpoons> b) \<bullet> (b \<rightleftharpoons> a) \<bullet> x = x"
-  unfolding permute_plus [symmetric]
-  by (simp add: swap_commute)
-
-lemma inj_permute [simp]: 
-  shows "inj (permute p)"
-  by (rule inj_on_inverseI)
-     (rule permute_minus_cancel)
-
-lemma surj_permute [simp]: 
-  shows "surj (permute p)"
-  by (rule surjI, rule permute_minus_cancel)
-
-lemma bij_permute [simp]: 
-  shows "bij (permute p)"
-  by (rule bijI [OF inj_permute surj_permute])
-
-lemma inv_permute: 
-  shows "inv (permute p) = permute (- p)"
-  by (rule inv_equality) (simp_all)
-
-lemma permute_minus: 
-  shows "permute (- p) = inv (permute p)"
-  by (simp add: inv_permute)
-
-lemma permute_eq_iff [simp]: 
-  shows "p \<bullet> x = p \<bullet> y \<longleftrightarrow> x = y"
-  by (rule inj_permute [THEN inj_eq])
-
-end
-
-subsection {* Permutations for atoms *}
-
-instantiation atom :: pt
-begin
-
-definition
-  "p \<bullet> a = Rep_perm p a"
-
-instance 
-apply(default)
-apply(simp_all add: permute_atom_def Rep_perm_simps)
-done
-
-end
-
-lemma sort_of_permute [simp]:
-  shows "sort_of (p \<bullet> a) = sort_of a"
-  unfolding permute_atom_def by (rule sort_of_Rep_perm)
-
-lemma swap_atom:
-  shows "(a \<rightleftharpoons> b) \<bullet> c =
-           (if sort_of a = sort_of b
-            then (if c = a then b else if c = b then a else c) else c)"
-  unfolding permute_atom_def
-  by (simp add: Rep_perm_swap)
-
-lemma swap_atom_simps [simp]:
-  "sort_of a = sort_of b \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> a = b"
-  "sort_of a = sort_of b \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> b = a"
-  "c \<noteq> a \<Longrightarrow> c \<noteq> b \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> c = c"
-  unfolding swap_atom by simp_all
-
-lemma expand_perm_eq:
-  fixes p q :: "perm"
-  shows "p = q \<longleftrightarrow> (\<forall>a::atom. p \<bullet> a = q \<bullet> a)"
-  unfolding permute_atom_def
-  by (metis Rep_perm_ext ext)
-
-
-subsection {* Permutations for permutations *}
-
-instantiation perm :: pt
-begin
-
-definition
-  "p \<bullet> q = p + q - p"
-
-instance
-apply default
-apply (simp add: permute_perm_def)
-apply (simp add: permute_perm_def diff_minus minus_add add_assoc)
-done
-
-end
-
-lemma permute_self: "p \<bullet> p = p"
-unfolding permute_perm_def by (simp add: diff_minus add_assoc)
-
-lemma permute_eqvt:
-  shows "p \<bullet> (q \<bullet> x) = (p \<bullet> q) \<bullet> (p \<bullet> x)"
-  unfolding permute_perm_def by simp
-
-lemma zero_perm_eqvt:
-  shows "p \<bullet> (0::perm) = 0"
-  unfolding permute_perm_def by simp
-
-lemma add_perm_eqvt:
-  fixes p p1 p2 :: perm
-  shows "p \<bullet> (p1 + p2) = p \<bullet> p1 + p \<bullet> p2"
-  unfolding permute_perm_def
-  by (simp add: expand_perm_eq)
-
-lemma swap_eqvt:
-  shows "p \<bullet> (a \<rightleftharpoons> b) = (p \<bullet> a \<rightleftharpoons> p \<bullet> b)"
-  unfolding permute_perm_def
-  by (auto simp add: swap_atom expand_perm_eq)
-
-
-subsection {* Permutations for functions *}
-
-instantiation "fun" :: (pt, pt) pt
-begin
-
-definition
-  "p \<bullet> f = (\<lambda>x. p \<bullet> (f (- p \<bullet> x)))"
-
-instance
-apply default
-apply (simp add: permute_fun_def)
-apply (simp add: permute_fun_def minus_add)
-done
-
-end
-
-lemma permute_fun_app_eq:
-  shows "p \<bullet> (f x) = (p \<bullet> f) (p \<bullet> x)"
-unfolding permute_fun_def by simp
-
-
-subsection {* Permutations for booleans *}
-
-instantiation bool :: pt
-begin
-
-definition "p \<bullet> (b::bool) = b"
-
-instance
-apply(default) 
-apply(simp_all add: permute_bool_def)
-done
-
-end
-
-lemma Not_eqvt:
-  shows "p \<bullet> (\<not> A) = (\<not> (p \<bullet> A))"
-by (simp add: permute_bool_def)
-
-lemma permute_boolE:
-  fixes P::"bool"
-  shows "p \<bullet> P \<Longrightarrow> P"
-  by (simp add: permute_bool_def)
-
-lemma permute_boolI:
-  fixes P::"bool"
-  shows "P \<Longrightarrow> p \<bullet> P"
-  by(simp add: permute_bool_def)
-
-subsection {* Permutations for sets *}
-
-lemma permute_set_eq:
-  fixes x::"'a::pt"
-  and   p::"perm"
-  shows "(p \<bullet> X) = {p \<bullet> x | x. x \<in> X}"
-  apply(auto simp add: permute_fun_def permute_bool_def mem_def)
-  apply(rule_tac x="- p \<bullet> x" in exI)
-  apply(simp)
-  done
-
-lemma permute_set_eq_image:
-  shows "p \<bullet> X = permute p ` X"
-unfolding permute_set_eq by auto
-
-lemma permute_set_eq_vimage:
-  shows "p \<bullet> X = permute (- p) -` X"
-unfolding permute_fun_def permute_bool_def
-unfolding vimage_def Collect_def mem_def ..
-
-lemma swap_set_not_in:
-  assumes a: "a \<notin> S" "b \<notin> S"
-  shows "(a \<rightleftharpoons> b) \<bullet> S = S"
-  using a by (auto simp add: permute_set_eq swap_atom)
-
-lemma swap_set_in:
-  assumes a: "a \<in> S" "b \<notin> S" "sort_of a = sort_of b"
-  shows "(a \<rightleftharpoons> b) \<bullet> S \<noteq> S"
-  using a by (auto simp add: permute_set_eq swap_atom)
-
-
-subsection {* Permutations for units *}
-
-instantiation unit :: pt
-begin
-
-definition "p \<bullet> (u::unit) = u"
-
-instance proof
-qed (simp_all add: permute_unit_def)
-
-end
-
-
-subsection {* Permutations for products *}
-
-instantiation "*" :: (pt, pt) pt
-begin
-
-primrec 
-  permute_prod 
-where
-  Pair_eqvt: "p \<bullet> (x, y) = (p \<bullet> x, p \<bullet> y)"
-
-instance
-by default auto
-
-end
-
-subsection {* Permutations for sums *}
-
-instantiation "+" :: (pt, pt) pt
-begin
-
-primrec 
-  permute_sum 
-where
-  "p \<bullet> (Inl x) = Inl (p \<bullet> x)"
-| "p \<bullet> (Inr y) = Inr (p \<bullet> y)"
-
-instance proof
-qed (case_tac [!] x, simp_all)
-
-end
-
-subsection {* Permutations for lists *}
-
-instantiation list :: (pt) pt
-begin
-
-primrec 
-  permute_list 
-where
-  "p \<bullet> [] = []"
-| "p \<bullet> (x # xs) = p \<bullet> x # p \<bullet> xs"
-
-instance proof
-qed (induct_tac [!] x, simp_all)
-
-end
-
-subsection {* Permutations for options *}
-
-instantiation option :: (pt) pt
-begin
-
-primrec 
-  permute_option 
-where
-  "p \<bullet> None = None"
-| "p \<bullet> (Some x) = Some (p \<bullet> x)"
-
-instance proof
-qed (induct_tac [!] x, simp_all)
-
-end
-
-subsection {* Permutations for @{typ char}, @{typ nat}, and @{typ int} *}
-
-instantiation char :: pt
-begin
-
-definition "p \<bullet> (c::char) = c"
-
-instance proof
-qed (simp_all add: permute_char_def)
-
-end
-
-instantiation nat :: pt
-begin
-
-definition "p \<bullet> (n::nat) = n"
-
-instance proof
-qed (simp_all add: permute_nat_def)
-
-end
-
-instantiation int :: pt
-begin
-
-definition "p \<bullet> (i::int) = i"
-
-instance proof
-qed (simp_all add: permute_int_def)
-
-end
-
-
-section {* Pure types *}
-
-text {* Pure types will have always empty support. *}
-
-class pure = pt +
-  assumes permute_pure: "p \<bullet> x = x"
-
-text {* Types @{typ unit} and @{typ bool} are pure. *}
-
-instance unit :: pure
-proof qed (rule permute_unit_def)
-
-instance bool :: pure
-proof qed (rule permute_bool_def)
-
-text {* Other type constructors preserve purity. *}
-
-instance "fun" :: (pure, pure) pure
-by default (simp add: permute_fun_def permute_pure)
-
-instance "*" :: (pure, pure) pure
-by default (induct_tac x, simp add: permute_pure)
-
-instance "+" :: (pure, pure) pure
-by default (induct_tac x, simp_all add: permute_pure)
-
-instance list :: (pure) pure
-by default (induct_tac x, simp_all add: permute_pure)
-
-instance option :: (pure) pure
-by default (induct_tac x, simp_all add: permute_pure)
-
-
-subsection {* Types @{typ char}, @{typ nat}, and @{typ int} *}
-
-instance char :: pure
-proof qed (rule permute_char_def)
-
-instance nat :: pure
-proof qed (rule permute_nat_def)
-
-instance int :: pure
-proof qed (rule permute_int_def)
-
-
-subsection {* Supp, Freshness and Supports *}
-
-context pt
-begin
-
-definition
-  supp :: "'a \<Rightarrow> atom set"
-where
-  "supp x = {a. infinite {b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x}}"
-
-end
-
-definition
-  fresh :: "atom \<Rightarrow> 'a::pt \<Rightarrow> bool" ("_ \<sharp> _" [55, 55] 55)
-where   
-  "a \<sharp> x \<equiv> a \<notin> supp x"
-
-lemma supp_conv_fresh: 
-  shows "supp x = {a. \<not> a \<sharp> x}"
-  unfolding fresh_def by simp
-
-lemma swap_rel_trans:
-  assumes "sort_of a = sort_of b"
-  assumes "sort_of b = sort_of c"
-  assumes "(a \<rightleftharpoons> c) \<bullet> x = x"
-  assumes "(b \<rightleftharpoons> c) \<bullet> x = x"
-  shows "(a \<rightleftharpoons> b) \<bullet> x = x"
-proof (cases)
-  assume "a = b \<or> c = b"
-  with assms show "(a \<rightleftharpoons> b) \<bullet> x = x" by auto
-next
-  assume *: "\<not> (a = b \<or> c = b)"
-  have "((a \<rightleftharpoons> c) + (b \<rightleftharpoons> c) + (a \<rightleftharpoons> c)) \<bullet> x = x"
-    using assms by simp
-  also have "(a \<rightleftharpoons> c) + (b \<rightleftharpoons> c) + (a \<rightleftharpoons> c) = (a \<rightleftharpoons> b)"
-    using assms * by (simp add: swap_triple)
-  finally show "(a \<rightleftharpoons> b) \<bullet> x = x" .
-qed
-
-lemma swap_fresh_fresh:
-  assumes a: "a \<sharp> x" 
-  and     b: "b \<sharp> x"
-  shows "(a \<rightleftharpoons> b) \<bullet> x = x"
-proof (cases)
-  assume asm: "sort_of a = sort_of b" 
-  have "finite {c. (a \<rightleftharpoons> c) \<bullet> x \<noteq> x}" "finite {c. (b \<rightleftharpoons> c) \<bullet> x \<noteq> x}" 
-    using a b unfolding fresh_def supp_def by simp_all
-  then have "finite ({c. (a \<rightleftharpoons> c) \<bullet> x \<noteq> x} \<union> {c. (b \<rightleftharpoons> c) \<bullet> x \<noteq> x})" by simp
-  then obtain c 
-    where "(a \<rightleftharpoons> c) \<bullet> x = x" "(b \<rightleftharpoons> c) \<bullet> x = x" "sort_of c = sort_of b"
-    by (rule obtain_atom) (auto)
-  then show "(a \<rightleftharpoons> b) \<bullet> x = x" using asm by (rule_tac swap_rel_trans) (simp_all)
-next
-  assume "sort_of a \<noteq> sort_of b"
-  then show "(a \<rightleftharpoons> b) \<bullet> x = x" by simp
-qed
-
-
-subsection {* supp and fresh are equivariant *}
-
-lemma finite_Collect_bij:
-  assumes a: "bij f"
-  shows "finite {x. P (f x)} = finite {x. P x}"
-by (metis a finite_vimage_iff vimage_Collect_eq)
-
-lemma fresh_permute_iff:
-  shows "(p \<bullet> a) \<sharp> (p \<bullet> x) \<longleftrightarrow> a \<sharp> x"
-proof -
-  have "(p \<bullet> a) \<sharp> (p \<bullet> x) \<longleftrightarrow> finite {b. (p \<bullet> a \<rightleftharpoons> b) \<bullet> p \<bullet> x \<noteq> p \<bullet> x}"
-    unfolding fresh_def supp_def by simp
-  also have "\<dots> \<longleftrightarrow> finite {b. (p \<bullet> a \<rightleftharpoons> p \<bullet> b) \<bullet> p \<bullet> x \<noteq> p \<bullet> x}"
-    using bij_permute by (rule finite_Collect_bij [symmetric])
-  also have "\<dots> \<longleftrightarrow> finite {b. p \<bullet> (a \<rightleftharpoons> b) \<bullet> x \<noteq> p \<bullet> x}"
-    by (simp only: permute_eqvt [of p] swap_eqvt)
-  also have "\<dots> \<longleftrightarrow> finite {b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x}"
-    by (simp only: permute_eq_iff)
-  also have "\<dots> \<longleftrightarrow> a \<sharp> x"
-    unfolding fresh_def supp_def by simp
-  finally show ?thesis .
-qed
-
-lemma fresh_eqvt:
-  shows "p \<bullet> (a \<sharp> x) = (p \<bullet> a) \<sharp> (p \<bullet> x)"
-  by (simp add: permute_bool_def fresh_permute_iff)
-
-lemma supp_eqvt:
-  fixes  p  :: "perm"
-  and    x   :: "'a::pt"
-  shows "p \<bullet> (supp x) = supp (p \<bullet> x)"
-  unfolding supp_conv_fresh
-  unfolding permute_fun_def Collect_def
-  by (simp add: Not_eqvt fresh_eqvt)
-
-subsection {* supports *}
-
-definition
-  supports :: "atom set \<Rightarrow> 'a::pt \<Rightarrow> bool" (infixl "supports" 80)
-where  
-  "S supports x \<equiv> \<forall>a b. (a \<notin> S \<and> b \<notin> S \<longrightarrow> (a \<rightleftharpoons> b) \<bullet> x = x)"
-
-lemma supp_is_subset:
-  fixes S :: "atom set"
-  and   x :: "'a::pt"
-  assumes a1: "S supports x"
-  and     a2: "finite S"
-  shows "(supp x) \<subseteq> S"
-proof (rule ccontr)
-  assume "\<not>(supp x \<subseteq> S)"
-  then obtain a where b1: "a \<in> supp x" and b2: "a \<notin> S" by auto
-  from a1 b2 have "\<forall>b. b \<notin> S \<longrightarrow> (a \<rightleftharpoons> b) \<bullet> x = x" by (unfold supports_def) (auto)
-  hence "{b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x} \<subseteq> S" by auto
-  with a2 have "finite {b. (a \<rightleftharpoons> b)\<bullet>x \<noteq> x}" by (simp add: finite_subset)
-  then have "a \<notin> (supp x)" unfolding supp_def by simp
-  with b1 show False by simp
-qed
-
-lemma supports_finite:
-  fixes S :: "atom set"
-  and   x :: "'a::pt"
-  assumes a1: "S supports x"
-  and     a2: "finite S"
-  shows "finite (supp x)"
-proof -
-  have "(supp x) \<subseteq> S" using a1 a2 by (rule supp_is_subset)
-  then show "finite (supp x)" using a2 by (simp add: finite_subset)
-qed
-
-lemma supp_supports:
-  fixes x :: "'a::pt"
-  shows "(supp x) supports x"
-proof (unfold supports_def, intro strip)
-  fix a b
-  assume "a \<notin> (supp x) \<and> b \<notin> (supp x)"
-  then have "a \<sharp> x" and "b \<sharp> x" by (simp_all add: fresh_def)
-  then show "(a \<rightleftharpoons> b) \<bullet> x = x" by (rule swap_fresh_fresh)
-qed
-
-lemma supp_is_least_supports:
-  fixes S :: "atom set"
-  and   x :: "'a::pt"
-  assumes  a1: "S supports x"
-  and      a2: "finite S"
-  and      a3: "\<And>S'. finite S' \<Longrightarrow> (S' supports x) \<Longrightarrow> S \<subseteq> S'"
-  shows "(supp x) = S"
-proof (rule equalityI)
-  show "(supp x) \<subseteq> S" using a1 a2 by (rule supp_is_subset)
-  with a2 have fin: "finite (supp x)" by (rule rev_finite_subset)
-  have "(supp x) supports x" by (rule supp_supports)
-  with fin a3 show "S \<subseteq> supp x" by blast
-qed
-
-lemma subsetCI: 
-  shows "(\<And>x. x \<in> A \<Longrightarrow> x \<notin> B \<Longrightarrow> False) \<Longrightarrow> A \<subseteq> B"
-  by auto
-
-lemma finite_supp_unique:
-  assumes a1: "S supports x"
-  assumes a2: "finite S"
-  assumes a3: "\<And>a b. \<lbrakk>a \<in> S; b \<notin> S; sort_of a = sort_of b\<rbrakk> \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> x \<noteq> x"
-  shows "(supp x) = S"
-  using a1 a2
-proof (rule supp_is_least_supports)
-  fix S'
-  assume "finite S'" and "S' supports x"
-  show "S \<subseteq> S'"
-  proof (rule subsetCI)
-    fix a
-    assume "a \<in> S" and "a \<notin> S'"
-    have "finite (S \<union> S')"
-      using `finite S` `finite S'` by simp
-    then obtain b where "b \<notin> S \<union> S'" and "sort_of b = sort_of a"
-      by (rule obtain_atom)
-    then have "b \<notin> S" and "b \<notin> S'"  and "sort_of a = sort_of b"
-      by simp_all
-    then have "(a \<rightleftharpoons> b) \<bullet> x = x"
-      using `a \<notin> S'` `S' supports x` by (simp add: supports_def)
-    moreover have "(a \<rightleftharpoons> b) \<bullet> x \<noteq> x"
-      using `a \<in> S` `b \<notin> S` `sort_of a = sort_of b`
-      by (rule a3)
-    ultimately show "False" by simp
-  qed
-qed
-
-section {* Finitely-supported types *}
-
-class fs = pt +
-  assumes finite_supp: "finite (supp x)"
-
-lemma pure_supp: 
-  shows "supp (x::'a::pure) = {}"
-  unfolding supp_def by (simp add: permute_pure)
-
-lemma pure_fresh:
-  fixes x::"'a::pure"
-  shows "a \<sharp> x"
-  unfolding fresh_def by (simp add: pure_supp)
-
-instance pure < fs
-by default (simp add: pure_supp)
-
-
-subsection  {* Type @{typ atom} is finitely-supported. *}
-
-lemma supp_atom:
-  shows "supp a = {a}"
-apply (rule finite_supp_unique)
-apply (clarsimp simp add: supports_def)
-apply simp
-apply simp
-done
-
-lemma fresh_atom: 
-  shows "a \<sharp> b \<longleftrightarrow> a \<noteq> b"
-  unfolding fresh_def supp_atom by simp
-
-instance atom :: fs
-by default (simp add: supp_atom)
-
-
-section {* Type @{typ perm} is finitely-supported. *}
-
-lemma perm_swap_eq:
-  shows "(a \<rightleftharpoons> b) \<bullet> p = p \<longleftrightarrow> (p \<bullet> (a \<rightleftharpoons> b)) = (a \<rightleftharpoons> b)"
-unfolding permute_perm_def
-by (metis add_diff_cancel minus_perm_def)
-
-lemma supports_perm: 
-  shows "{a. p \<bullet> a \<noteq> a} supports p"
-  unfolding supports_def
-  by (simp add: perm_swap_eq swap_eqvt)
-
-lemma finite_perm_lemma: 
-  shows "finite {a::atom. p \<bullet> a \<noteq> a}"
-  using finite_Rep_perm [of p]
-  unfolding permute_atom_def .
-
-lemma supp_perm:
-  shows "supp p = {a. p \<bullet> a \<noteq> a}"
-apply (rule finite_supp_unique)
-apply (rule supports_perm)
-apply (rule finite_perm_lemma)
-apply (simp add: perm_swap_eq swap_eqvt)
-apply (auto simp add: expand_perm_eq swap_atom)
-done
-
-lemma fresh_perm: 
-  shows "a \<sharp> p \<longleftrightarrow> p \<bullet> a = a"
-unfolding fresh_def by (simp add: supp_perm)
-
-lemma supp_swap:
-  shows "supp (a \<rightleftharpoons> b) = (if a = b \<or> sort_of a \<noteq> sort_of b then {} else {a, b})"
-  by (auto simp add: supp_perm swap_atom)
-
-lemma fresh_zero_perm: 
-  shows "a \<sharp> (0::perm)"
-  unfolding fresh_perm by simp
-
-lemma supp_zero_perm: 
-  shows "supp (0::perm) = {}"
-  unfolding supp_perm by simp
-
-lemma fresh_plus_perm:
-  fixes p q::perm
-  assumes "a \<sharp> p" "a \<sharp> q"
-  shows "a \<sharp> (p + q)"
-  using assms
-  unfolding fresh_def
-  by (auto simp add: supp_perm)
-
-lemma supp_plus_perm:
-  fixes p q::perm
-  shows "supp (p + q) \<subseteq> supp p \<union> supp q"
-  by (auto simp add: supp_perm)
-
-lemma fresh_minus_perm:
-  fixes p::perm
-  shows "a \<sharp> (- p) \<longleftrightarrow> a \<sharp> p"
-  unfolding fresh_def
-  apply(auto simp add: supp_perm)
-  apply(metis permute_minus_cancel)+
-  done
-
-lemma supp_minus_perm:
-  fixes p::perm
-  shows "supp (- p) = supp p"
-  unfolding supp_conv_fresh
-  by (simp add: fresh_minus_perm)
-
-instance perm :: fs
-by default (simp add: supp_perm finite_perm_lemma)
-
-lemma plus_perm_eq:
-  fixes p q::"perm"
-  assumes asm: "supp p \<inter>  supp q = {}"
-  shows "p + q  = q + p"
-unfolding expand_perm_eq
-proof
-  fix a::"atom"
-  show "(p + q) \<bullet> a = (q + p) \<bullet> a"
-  proof -
-    { assume "a \<notin> supp p" "a \<notin> supp q"
-      then have "(p + q) \<bullet> a = (q + p) \<bullet> a" 
-	by (simp add: supp_perm)
-    }
-    moreover
-    { assume a: "a \<in> supp p" "a \<notin> supp q"
-      then have "p \<bullet> a \<in> supp p" by (simp add: supp_perm)
-      then have "p \<bullet> a \<notin> supp q" using asm by auto
-      with a have "(p + q) \<bullet> a = (q + p) \<bullet> a" 
-	by (simp add: supp_perm)
-    }
-    moreover
-    { assume a: "a \<notin> supp p" "a \<in> supp q"
-      then have "q \<bullet> a \<in> supp q" by (simp add: supp_perm)
-      then have "q \<bullet> a \<notin> supp p" using asm by auto 
-      with a have "(p + q) \<bullet> a = (q + p) \<bullet> a" 
-	by (simp add: supp_perm)
-    }
-    ultimately show "(p + q) \<bullet> a = (q + p) \<bullet> a" 
-      using asm by blast
-  qed
-qed
-
-section {* Finite Support instances for other types *}
-
-subsection {* Type @{typ "'a \<times> 'b"} is finitely-supported. *}
-
-lemma supp_Pair: 
-  shows "supp (x, y) = supp x \<union> supp y"
-  by (simp add: supp_def Collect_imp_eq Collect_neg_eq)
-
-lemma fresh_Pair: 
-  shows "a \<sharp> (x, y) \<longleftrightarrow> a \<sharp> x \<and> a \<sharp> y"
-  by (simp add: fresh_def supp_Pair)
-
-instance "*" :: (fs, fs) fs
-apply default
-apply (induct_tac x)
-apply (simp add: supp_Pair finite_supp)
-done
-
-subsection {* Type @{typ "'a + 'b"} is finitely supported *}
-
-lemma supp_Inl: 
-  shows "supp (Inl x) = supp x"
-  by (simp add: supp_def)
-
-lemma supp_Inr: 
-  shows "supp (Inr x) = supp x"
-  by (simp add: supp_def)
-
-lemma fresh_Inl: 
-  shows "a \<sharp> Inl x \<longleftrightarrow> a \<sharp> x"
-  by (simp add: fresh_def supp_Inl)
-
-lemma fresh_Inr: 
-  shows "a \<sharp> Inr y \<longleftrightarrow> a \<sharp> y"
-  by (simp add: fresh_def supp_Inr)
-
-instance "+" :: (fs, fs) fs
-apply default
-apply (induct_tac x)
-apply (simp_all add: supp_Inl supp_Inr finite_supp)
-done
-
-subsection {* Type @{typ "'a option"} is finitely supported *}
-
-lemma supp_None: 
-  shows "supp None = {}"
-by (simp add: supp_def)
-
-lemma supp_Some: 
-  shows "supp (Some x) = supp x"
-  by (simp add: supp_def)
-
-lemma fresh_None: 
-  shows "a \<sharp> None"
-  by (simp add: fresh_def supp_None)
-
-lemma fresh_Some: 
-  shows "a \<sharp> Some x \<longleftrightarrow> a \<sharp> x"
-  by (simp add: fresh_def supp_Some)
-
-instance option :: (fs) fs
-apply default
-apply (induct_tac x)
-apply (simp_all add: supp_None supp_Some finite_supp)
-done
-
-subsubsection {* Type @{typ "'a list"} is finitely supported *}
-
-lemma supp_Nil: 
-  shows "supp [] = {}"
-  by (simp add: supp_def)
-
-lemma supp_Cons: 
-  shows "supp (x # xs) = supp x \<union> supp xs"
-by (simp add: supp_def Collect_imp_eq Collect_neg_eq)
-
-lemma fresh_Nil: 
-  shows "a \<sharp> []"
-  by (simp add: fresh_def supp_Nil)
-
-lemma fresh_Cons: 
-  shows "a \<sharp> (x # xs) \<longleftrightarrow> a \<sharp> x \<and> a \<sharp> xs"
-  by (simp add: fresh_def supp_Cons)
-
-instance list :: (fs) fs
-apply default
-apply (induct_tac x)
-apply (simp_all add: supp_Nil supp_Cons finite_supp)
-done
-
-section {* Support and freshness for applications *}
-
-lemma supp_fun_app:
-  shows "supp (f x) \<subseteq> (supp f) \<union> (supp x)"
-proof (rule subsetCI)
-  fix a::"atom"
-  assume a: "a \<in> supp (f x)"
-  assume b: "a \<notin> supp f \<union> supp x"
-  then have "finite {b. (a \<rightleftharpoons> b) \<bullet> f \<noteq> f}" "finite {b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x}" 
-    unfolding supp_def by auto
-  then have "finite ({b. (a \<rightleftharpoons> b) \<bullet> f \<noteq> f} \<union> {b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x})" by simp
-  moreover 
-  have "{b. ((a \<rightleftharpoons> b) \<bullet> f) ((a \<rightleftharpoons> b) \<bullet> x) \<noteq> f x} \<subseteq> ({b. (a \<rightleftharpoons> b) \<bullet> f \<noteq> f} \<union> {b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x})"
-    by auto
-  ultimately have "finite {b. ((a \<rightleftharpoons> b) \<bullet> f) ((a \<rightleftharpoons> b) \<bullet> x) \<noteq> f x}"
-    using finite_subset by auto
-  then have "a \<notin> supp (f x)" unfolding supp_def
-    by (simp add: permute_fun_app_eq)
-  with a show "False" by simp
-qed
-    
-lemma fresh_fun_app:
-  shows "a \<sharp> (f, x) \<Longrightarrow> a \<sharp> f x"
-unfolding fresh_def
-using supp_fun_app
-by (auto simp add: supp_Pair)
-
-lemma fresh_fun_eqvt_app:
-  assumes a: "\<forall>p. p \<bullet> f = f"
-  shows "a \<sharp> x \<Longrightarrow> a \<sharp> f x"
-proof -
-  from a have b: "supp f = {}"
-    unfolding supp_def by simp
-  show "a \<sharp> x \<Longrightarrow> a \<sharp> f x"
-    unfolding fresh_def
-    using supp_fun_app b
-    by auto
-qed
-
-end
diff -r c0eac04ae3b4 -r c34347ec7ab3 Nominal/Nominal2_Eqvt.thy
--- a/Nominal/Nominal2_Eqvt.thy	Sat Apr 03 22:31:11 2010 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,305 +0,0 @@
-(*  Title:      Nominal2_Eqvt
-    Authors:    Brian Huffman, Christian Urban
-
-    Equivariance, Supp and Fresh Lemmas for Operators. 
-    (Contains most, but not all such lemmas.)
-*)
-theory Nominal2_Eqvt
-imports Nominal2_Base Nominal2_Atoms
-uses ("nominal_thmdecls.ML")
-     ("nominal_permeq.ML")
-begin
-
-section {* Logical Operators *}
-
-lemma eq_eqvt:
-  shows "p \<bullet> (x = y) \<longleftrightarrow> (p \<bullet> x) = (p \<bullet> y)"
-  unfolding permute_eq_iff permute_bool_def ..
-
-lemma if_eqvt:
-  shows "p \<bullet> (if b then x else y) = (if p \<bullet> b then p \<bullet> x else p \<bullet> y)"
-  by (simp add: permute_fun_def permute_bool_def)
-
-lemma True_eqvt:
-  shows "p \<bullet> True = True"
-  unfolding permute_bool_def ..
-
-lemma False_eqvt:
-  shows "p \<bullet> False = False"
-  unfolding permute_bool_def ..
-
-lemma imp_eqvt:
-  shows "p \<bullet> (A \<longrightarrow> B) = ((p \<bullet> A) \<longrightarrow> (p \<bullet> B))"
-  by (simp add: permute_bool_def)
-
-lemma conj_eqvt:
-  shows "p \<bullet> (A \<and> B) = ((p \<bullet> A) \<and> (p \<bullet> B))"
-  by (simp add: permute_bool_def)
-
-lemma disj_eqvt:
-  shows "p \<bullet> (A \<or> B) = ((p \<bullet> A) \<or> (p \<bullet> B))"
-  by (simp add: permute_bool_def)
-
-lemma Not_eqvt:
-  shows "p \<bullet> (\<not> A) = (\<not> (p \<bullet> A))"
-  by (simp add: permute_bool_def)
-
-lemma all_eqvt:
-  shows "p \<bullet> (\<forall>x. P x) = (\<forall>x. (p \<bullet> P) x)"
-  unfolding permute_fun_def permute_bool_def
-  by (auto, drule_tac x="p \<bullet> x" in spec, simp)
-
-lemma all_eqvt2:
-  shows "p \<bullet> (\<forall>x. P x) = (\<forall>x. p \<bullet> P (- p \<bullet> x))"
-  unfolding permute_fun_def permute_bool_def
-  by (auto, drule_tac x="p \<bullet> x" in spec, simp)
-
-lemma ex_eqvt:
-  shows "p \<bullet> (\<exists>x. P x) = (\<exists>x. (p \<bullet> P) x)"
-  unfolding permute_fun_def permute_bool_def
-  by (auto, rule_tac x="p \<bullet> x" in exI, simp)
-
-lemma ex_eqvt2:
-  shows "p \<bullet> (\<exists>x. P x) = (\<exists>x. p \<bullet> P (- p \<bullet> x))"
-  unfolding permute_fun_def permute_bool_def
-  by (auto, rule_tac x="p \<bullet> x" in exI, simp)
-
-lemma ex1_eqvt:
-  shows "p \<bullet> (\<exists>!x. P x) = (\<exists>!x. (p \<bullet> P) x)"
-  unfolding Ex1_def 
-  by (simp add: ex_eqvt permute_fun_def conj_eqvt all_eqvt imp_eqvt eq_eqvt)
-
-lemma ex1_eqvt2:
-  shows "p \<bullet> (\<exists>!x. P x) = (\<exists>!x. p \<bullet> P (- p \<bullet> x))"
-  unfolding Ex1_def ex_eqvt2 conj_eqvt all_eqvt2 imp_eqvt eq_eqvt
-  by simp
-
-lemma the_eqvt:
-  assumes unique: "\<exists>!x. P x"
-  shows "(p \<bullet> (THE x. P x)) = (THE x. p \<bullet> P (- p \<bullet> x))"
-  apply(rule the1_equality [symmetric])
-  apply(simp add: ex1_eqvt2[symmetric])
-  apply(simp add: permute_bool_def unique)
-  apply(simp add: permute_bool_def)
-  apply(rule theI'[OF unique])
-  done
-
-section {* Set Operations *}
-
-lemma mem_permute_iff:
-  shows "(p \<bullet> x) \<in> (p \<bullet> X) \<longleftrightarrow> x \<in> X"
-unfolding mem_def permute_fun_def permute_bool_def
-by simp
-
-lemma mem_eqvt:
-  shows "p \<bullet> (x \<in> A) \<longleftrightarrow> (p \<bullet> x) \<in> (p \<bullet> A)"
-  unfolding mem_permute_iff permute_bool_def by simp
-
-lemma not_mem_eqvt:
-  shows "p \<bullet> (x \<notin> A) \<longleftrightarrow> (p \<bullet> x) \<notin> (p \<bullet> A)"
-  unfolding mem_def permute_fun_def by (simp add: Not_eqvt)
-
-lemma Collect_eqvt:
-  shows "p \<bullet> {x. P x} = {x. (p \<bullet> P) x}"
-  unfolding Collect_def permute_fun_def ..
-
-lemma Collect_eqvt2:
-  shows "p \<bullet> {x. P x} = {x. p \<bullet> (P (-p \<bullet> x))}"
-  unfolding Collect_def permute_fun_def ..
-
-lemma empty_eqvt:
-  shows "p \<bullet> {} = {}"
-  unfolding empty_def Collect_eqvt2 False_eqvt ..
-
-lemma supp_set_empty:
-  shows "supp {} = {}"
-  by (simp add: supp_def empty_eqvt)
-
-lemma fresh_set_empty:
-  shows "a \<sharp> {}"
-  by (simp add: fresh_def supp_set_empty)
-
-lemma UNIV_eqvt:
-  shows "p \<bullet> UNIV = UNIV"
-  unfolding UNIV_def Collect_eqvt2 True_eqvt ..
-
-lemma union_eqvt:
-  shows "p \<bullet> (A \<union> B) = (p \<bullet> A) \<union> (p \<bullet> B)"
-  unfolding Un_def Collect_eqvt2 disj_eqvt mem_eqvt by simp
-
-lemma inter_eqvt:
-  shows "p \<bullet> (A \<inter> B) = (p \<bullet> A) \<inter> (p \<bullet> B)"
-  unfolding Int_def Collect_eqvt2 conj_eqvt mem_eqvt by simp
-
-lemma Diff_eqvt:
-  fixes A B :: "'a::pt set"
-  shows "p \<bullet> (A - B) = p \<bullet> A - p \<bullet> B"
-  unfolding set_diff_eq Collect_eqvt2 conj_eqvt Not_eqvt mem_eqvt by simp
-
-lemma Compl_eqvt:
-  fixes A :: "'a::pt set"
-  shows "p \<bullet> (- A) = - (p \<bullet> A)"
-  unfolding Compl_eq_Diff_UNIV Diff_eqvt UNIV_eqvt ..
-
-lemma insert_eqvt:
-  shows "p \<bullet> (insert x A) = insert (p \<bullet> x) (p \<bullet> A)"
-  unfolding permute_set_eq_image image_insert ..
-
-lemma vimage_eqvt:
-  shows "p \<bullet> (f -` A) = (p \<bullet> f) -` (p \<bullet> A)"
-  unfolding vimage_def permute_fun_def [where f=f]
-  unfolding Collect_eqvt2 mem_eqvt ..
-
-lemma image_eqvt:
-  shows "p \<bullet> (f ` A) = (p \<bullet> f) ` (p \<bullet> A)"
-  unfolding permute_set_eq_image
-  unfolding permute_fun_def [where f=f]
-  by (simp add: image_image)
-
-lemma finite_permute_iff:
-  shows "finite (p \<bullet> A) \<longleftrightarrow> finite A"
-  unfolding permute_set_eq_vimage
-  using bij_permute by (rule finite_vimage_iff)
-
-lemma finite_eqvt:
-  shows "p \<bullet> finite A = finite (p \<bullet> A)"
-  unfolding finite_permute_iff permute_bool_def ..
-
-
-section {* List Operations *}
-
-lemma append_eqvt:
-  shows "p \<bullet> (xs @ ys) = (p \<bullet> xs) @ (p \<bullet> ys)"
-  by (induct xs) auto
-
-lemma supp_append:
-  shows "supp (xs @ ys) = supp xs \<union> supp ys"
-  by (induct xs) (auto simp add: supp_Nil supp_Cons)
-
-lemma fresh_append:
-  shows "a \<sharp> (xs @ ys) \<longleftrightarrow> a \<sharp> xs \<and> a \<sharp> ys"
-  by (induct xs) (simp_all add: fresh_Nil fresh_Cons)
-
-lemma rev_eqvt:
-  shows "p \<bullet> (rev xs) = rev (p \<bullet> xs)"
-  by (induct xs) (simp_all add: append_eqvt)
-
-lemma supp_rev:
-  shows "supp (rev xs) = supp xs"
-  by (induct xs) (auto simp add: supp_append supp_Cons supp_Nil)
-
-lemma fresh_rev:
-  shows "a \<sharp> rev xs \<longleftrightarrow> a \<sharp> xs"
-  by (induct xs) (auto simp add: fresh_append fresh_Cons fresh_Nil)
-
-lemma set_eqvt:
-  shows "p \<bullet> (set xs) = set (p \<bullet> xs)"
-  by (induct xs) (simp_all add: empty_eqvt insert_eqvt)
-
-(* needs finite support premise
-lemma supp_set:
-  fixes x :: "'a::pt"
-  shows "supp (set xs) = supp xs"
-*)
-
-
-section {* Product Operations *}
-
-lemma fst_eqvt:
-  "p \<bullet> (fst x) = fst (p \<bullet> x)"
- by (cases x) simp
-
-lemma snd_eqvt:
-  "p \<bullet> (snd x) = snd (p \<bullet> x)"
- by (cases x) simp
-
-section {* Units *}
-
-lemma supp_unit:
-  shows "supp () = {}"
-  by (simp add: supp_def)
-
-lemma fresh_unit:
-  shows "a \<sharp> ()"
-  by (simp add: fresh_def supp_unit)
-
-section {* Equivariance automation *}
-
-text {* Setup of the theorem attributes @{text eqvt} and @{text eqvt_force} *}
-
-use "nominal_thmdecls.ML"
-setup "Nominal_ThmDecls.setup"
-
-lemmas [eqvt] = 
-  (* connectives *)
-  eq_eqvt if_eqvt imp_eqvt disj_eqvt conj_eqvt Not_eqvt 
-  True_eqvt False_eqvt ex_eqvt all_eqvt ex1_eqvt
-  imp_eqvt [folded induct_implies_def]
-
-  (* nominal *)
-  (*permute_eqvt commented out since it loops *)
-  supp_eqvt fresh_eqvt
-  permute_pure
-
-  (* datatypes *)
-  permute_prod.simps append_eqvt rev_eqvt set_eqvt
-  fst_eqvt snd_eqvt Pair_eqvt
-
-  (* sets *)
-  empty_eqvt UNIV_eqvt union_eqvt inter_eqvt mem_eqvt
-  Diff_eqvt Compl_eqvt insert_eqvt Collect_eqvt image_eqvt
-
-  atom_eqvt add_perm_eqvt
-
-thm eqvts
-thm eqvts_raw
-
-text {* helper lemmas for the eqvt_tac *}
-
-definition
-  "unpermute p = permute (- p)"
-
-lemma eqvt_apply:
-  fixes f :: "'a::pt \<Rightarrow> 'b::pt" 
-  and x :: "'a::pt"
-  shows "p \<bullet> (f x) \<equiv> (p \<bullet> f) (p \<bullet> x)"
-  unfolding permute_fun_def by simp
-
-lemma eqvt_lambda:
-  fixes f :: "'a::pt \<Rightarrow> 'b::pt"
-  shows "p \<bullet> (\<lambda>x. f x) \<equiv> (\<lambda>x. p \<bullet> (f (unpermute p x)))"
-  unfolding permute_fun_def unpermute_def by simp
-
-lemma eqvt_bound:
-  shows "p \<bullet> unpermute p x \<equiv> x"
-  unfolding unpermute_def by simp
-
-use "nominal_permeq.ML"
-
-
-lemma "p \<bullet> (A \<longrightarrow> B = C)"
-apply (tactic {* Nominal_Permeq.eqvt_tac @{context} 1 *}) 
-oops
-
-lemma "p \<bullet> (\<lambda>(x::'a::pt). A \<longrightarrow> (B::'a \<Rightarrow> bool) x = C) = foo"
-apply (tactic {* Nominal_Permeq.eqvt_tac @{context} 1 *})
-oops
-
-lemma "p \<bullet> (\<lambda>x y. \<exists>z. x = z \<and> x = y \<longrightarrow> z \<noteq> x) = foo"
-apply (tactic {* Nominal_Permeq.eqvt_tac @{context} 1 *})
-oops
-
-lemma "p \<bullet> (\<lambda>f x. f (g (f x))) = foo"
-apply (tactic {* Nominal_Permeq.eqvt_tac @{context} 1 *})
-oops
-
-lemma "p \<bullet> (\<lambda>q. q \<bullet> (r \<bullet> x)) = foo"
-apply (tactic {* Nominal_Permeq.eqvt_tac @{context} 1 *})
-oops
-
-lemma "p \<bullet> (q \<bullet> r \<bullet> x) = foo"
-apply (tactic {* Nominal_Permeq.eqvt_tac @{context} 1 *})
-oops
-
-
-end
diff -r c0eac04ae3b4 -r c34347ec7ab3 Nominal/Nominal2_FSet.thy
--- a/Nominal/Nominal2_FSet.thy	Sat Apr 03 22:31:11 2010 +0200
+++ b/Nominal/Nominal2_FSet.thy	Sun Apr 04 21:39:28 2010 +0200
@@ -1,5 +1,5 @@
 theory Nominal2_FSet
-imports FSet Nominal2_Supp
+imports FSet "../Nominal-General/Nominal2_Supp"
 begin
 
 lemma permute_rsp_fset[quot_respect]:
diff -r c0eac04ae3b4 -r c34347ec7ab3 Nominal/Nominal2_Supp.thy
--- a/Nominal/Nominal2_Supp.thy	Sat Apr 03 22:31:11 2010 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,501 +0,0 @@
-(*  Title:      Nominal2_Supp
-    Authors:    Brian Huffman, Christian Urban
-
-    Supplementary Lemmas and Definitions for 
-    Nominal Isabelle. 
-*)
-theory Nominal2_Supp
-imports Nominal2_Base Nominal2_Eqvt Nominal2_Atoms
-begin
-
-
-section {* Fresh-Star *}
-
-text {* The fresh-star generalisation of fresh is used in strong
-  induction principles. *}
-
-definition 
-  fresh_star :: "atom set \<Rightarrow> 'a::pt \<Rightarrow> bool" ("_ \<sharp>* _" [80,80] 80)
-where 
-  "as \<sharp>* x \<equiv> \<forall>a \<in> as. a \<sharp> x"
-
-lemma fresh_star_prod:
-  fixes as::"atom set"
-  shows "as \<sharp>* (x, y) = (as \<sharp>* x \<and> as \<sharp>* y)"
-  by (auto simp add: fresh_star_def fresh_Pair)
-
-lemma fresh_star_union:
-  shows "(as \<union> bs) \<sharp>* x = (as \<sharp>* x \<and> bs \<sharp>* x)"
-  by (auto simp add: fresh_star_def)
-
-lemma fresh_star_insert:
-  shows "(insert a as) \<sharp>* x = (a \<sharp> x \<and> as \<sharp>* x)"
-  by (auto simp add: fresh_star_def)
-
-lemma fresh_star_Un_elim:
-  "((as \<union> bs) \<sharp>* x \<Longrightarrow> PROP C) \<equiv> (as \<sharp>* x \<Longrightarrow> bs \<sharp>* x \<Longrightarrow> PROP C)"
-  unfolding fresh_star_def
-  apply(rule)
-  apply(erule meta_mp)
-  apply(auto)
-  done
-
-lemma fresh_star_insert_elim:
-  "(insert a as \<sharp>* x \<Longrightarrow> PROP C) \<equiv> (a \<sharp> x \<Longrightarrow> as \<sharp>* x \<Longrightarrow> PROP C)"
-  unfolding fresh_star_def
-  by rule (simp_all add: fresh_star_def)
-
-lemma fresh_star_empty_elim:
-  "({} \<sharp>* x \<Longrightarrow> PROP C) \<equiv> PROP C"
-  by (simp add: fresh_star_def)
-
-lemma fresh_star_unit_elim: 
-  shows "(a \<sharp>* () \<Longrightarrow> PROP C) \<equiv> PROP C"
-  by (simp add: fresh_star_def fresh_unit) 
-
-lemma fresh_star_prod_elim: 
-  shows "(a \<sharp>* (x, y) \<Longrightarrow> PROP C) \<equiv> (a \<sharp>* x \<Longrightarrow> a \<sharp>* y \<Longrightarrow> PROP C)"
-  by (rule, simp_all add: fresh_star_prod)
-
-lemma fresh_star_plus:
-  fixes p q::perm
-  shows "\<lbrakk>a \<sharp>* p;  a \<sharp>* q\<rbrakk> \<Longrightarrow> a \<sharp>* (p + q)"
-  unfolding fresh_star_def
-  by (simp add: fresh_plus_perm)
-
-lemma fresh_star_permute_iff:
-  shows "(p \<bullet> a) \<sharp>* (p \<bullet> x) \<longleftrightarrow> a \<sharp>* x"
-  unfolding fresh_star_def
-  by (metis mem_permute_iff permute_minus_cancel fresh_permute_iff)
-
-lemma fresh_star_eqvt:
-  shows "(p \<bullet> (as \<sharp>* x)) = (p \<bullet> as) \<sharp>* (p \<bullet> x)"
-unfolding fresh_star_def
-unfolding Ball_def
-apply(simp add: all_eqvt)
-apply(subst permute_fun_def)
-apply(simp add: imp_eqvt fresh_eqvt mem_eqvt)
-done
-
-section {* Avoiding of atom sets *}
-
-text {* 
-  For every set of atoms, there is another set of atoms
-  avoiding a finitely supported c and there is a permutation
-  which 'translates' between both sets.
-*}
-
-lemma at_set_avoiding_aux:
-  fixes Xs::"atom set"
-  and   As::"atom set"
-  assumes b: "Xs \<subseteq> As"
-  and     c: "finite As"
-  shows "\<exists>p. (p \<bullet> Xs) \<inter> As = {} \<and> (supp p) \<subseteq> (Xs \<union> (p \<bullet> Xs))"
-proof -
-  from b c have "finite Xs" by (rule finite_subset)
-  then show ?thesis using b
-  proof (induct rule: finite_subset_induct)
-    case empty
-    have "0 \<bullet> {} \<inter> As = {}" by simp
-    moreover
-    have "supp (0::perm) \<subseteq> {} \<union> 0 \<bullet> {}" by (simp add: supp_zero_perm)
-    ultimately show ?case by blast
-  next
-    case (insert x Xs)
-    then obtain p where
-      p1: "(p \<bullet> Xs) \<inter> As = {}" and 
-      p2: "supp p \<subseteq> (Xs \<union> (p \<bullet> Xs))" by blast
-    from `x \<in> As` p1 have "x \<notin> p \<bullet> Xs" by fast
-    with `x \<notin> Xs` p2 have "x \<notin> supp p" by fast
-    hence px: "p \<bullet> x = x" unfolding supp_perm by simp
-    have "finite (As \<union> p \<bullet> Xs)"
-      using `finite As` `finite Xs`
-      by (simp add: permute_set_eq_image)
-    then obtain y where "y \<notin> (As \<union> p \<bullet> Xs)" "sort_of y = sort_of x"
-      by (rule obtain_atom)
-    hence y: "y \<notin> As" "y \<notin> p \<bullet> Xs" "sort_of y = sort_of x"
-      by simp_all
-    let ?q = "(x \<rightleftharpoons> y) + p"
-    have q: "?q \<bullet> insert x Xs = insert y (p \<bullet> Xs)"
-      unfolding insert_eqvt
-      using `p \<bullet> x = x` `sort_of y = sort_of x`
-      using `x \<notin> p \<bullet> Xs` `y \<notin> p \<bullet> Xs`
-      by (simp add: swap_atom swap_set_not_in)
-    have "?q \<bullet> insert x Xs \<inter> As = {}"
-      using `y \<notin> As` `p \<bullet> Xs \<inter> As = {}`
-      unfolding q by simp
-    moreover
-    have "supp ?q \<subseteq> insert x Xs \<union> ?q \<bullet> insert x Xs"
-      using p2 unfolding q
-      apply (intro subset_trans [OF supp_plus_perm])
-      apply (auto simp add: supp_swap)
-      done
-    ultimately show ?case by blast
-  qed
-qed
-
-lemma at_set_avoiding:
-  assumes a: "finite Xs"
-  and     b: "finite (supp c)"
-  obtains p::"perm" where "(p \<bullet> Xs)\<sharp>*c" and "(supp p) \<subseteq> (Xs \<union> (p \<bullet> Xs))"
-  using a b at_set_avoiding_aux [where Xs="Xs" and As="Xs \<union> supp c"]
-  unfolding fresh_star_def fresh_def by blast
-
-
-section {* The freshness lemma according to Andrew Pitts *}
-
-lemma fresh_conv_MOST: 
-  shows "a \<sharp> x \<longleftrightarrow> (MOST b. (a \<rightleftharpoons> b) \<bullet> x = x)"
-  unfolding fresh_def supp_def MOST_iff_cofinite by simp
-
-lemma fresh_apply:
-  assumes "a \<sharp> f" and "a \<sharp> x" 
-  shows "a \<sharp> f x"
-  using assms unfolding fresh_conv_MOST
-  unfolding permute_fun_app_eq [where f=f]
-  by (elim MOST_rev_mp, simp)
-
-lemma freshness_lemma:
-  fixes h :: "'a::at \<Rightarrow> 'b::pt"
-  assumes a: "\<exists>a. atom a \<sharp> (h, h a)"
-  shows  "\<exists>x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x"
-proof -
-  from a obtain b where a1: "atom b \<sharp> h" and a2: "atom b \<sharp> h b"
-    by (auto simp add: fresh_Pair)
-  show "\<exists>x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x"
-  proof (intro exI allI impI)
-    fix a :: 'a
-    assume a3: "atom a \<sharp> h"
-    show "h a = h b"
-    proof (cases "a = b")
-      assume "a = b"
-      thus "h a = h b" by simp
-    next
-      assume "a \<noteq> b"
-      hence "atom a \<sharp> b" by (simp add: fresh_at_base)
-      with a3 have "atom a \<sharp> h b" by (rule fresh_apply)
-      with a2 have d1: "(atom b \<rightleftharpoons> atom a) \<bullet> (h b) = (h b)"
-        by (rule swap_fresh_fresh)
-      from a1 a3 have d2: "(atom b \<rightleftharpoons> atom a) \<bullet> h = h"
-        by (rule swap_fresh_fresh)
-      from d1 have "h b = (atom b \<rightleftharpoons> atom a) \<bullet> (h b)" by simp
-      also have "\<dots> = ((atom b \<rightleftharpoons> atom a) \<bullet> h) ((atom b \<rightleftharpoons> atom a) \<bullet> b)"
-        by (rule permute_fun_app_eq)
-      also have "\<dots> = h a"
-        using d2 by simp
-      finally show "h a = h b"  by simp
-    qed
-  qed
-qed
-
-lemma freshness_lemma_unique:
-  fixes h :: "'a::at \<Rightarrow> 'b::pt"
-  assumes a: "\<exists>a. atom a \<sharp> (h, h a)"
-  shows "\<exists>!x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x"
-proof (rule ex_ex1I)
-  from a show "\<exists>x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x"
-    by (rule freshness_lemma)
-next
-  fix x y
-  assume x: "\<forall>a. atom a \<sharp> h \<longrightarrow> h a = x"
-  assume y: "\<forall>a. atom a \<sharp> h \<longrightarrow> h a = y"
-  from a x y show "x = y"
-    by (auto simp add: fresh_Pair)
-qed
-
-text {* packaging the freshness lemma into a function *}
-
-definition
-  fresh_fun :: "('a::at \<Rightarrow> 'b::pt) \<Rightarrow> 'b"
-where
-  "fresh_fun h = (THE x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x)"
-
-lemma fresh_fun_app:
-  fixes h :: "'a::at \<Rightarrow> 'b::pt"
-  assumes a: "\<exists>a. atom a \<sharp> (h, h a)"
-  assumes b: "atom a \<sharp> h"
-  shows "fresh_fun h = h a"
-unfolding fresh_fun_def
-proof (rule the_equality)
-  show "\<forall>a'. atom a' \<sharp> h \<longrightarrow> h a' = h a"
-  proof (intro strip)
-    fix a':: 'a
-    assume c: "atom a' \<sharp> h"
-    from a have "\<exists>x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x" by (rule freshness_lemma)
-    with b c show "h a' = h a" by auto
-  qed
-next
-  fix fr :: 'b
-  assume "\<forall>a. atom a \<sharp> h \<longrightarrow> h a = fr"
-  with b show "fr = h a" by auto
-qed
-
-lemma fresh_fun_app':
-  fixes h :: "'a::at \<Rightarrow> 'b::pt"
-  assumes a: "atom a \<sharp> h" "atom a \<sharp> h a"
-  shows "fresh_fun h = h a"
-  apply (rule fresh_fun_app)
-  apply (auto simp add: fresh_Pair intro: a)
-  done
-
-lemma fresh_fun_eqvt:
-  fixes h :: "'a::at \<Rightarrow> 'b::pt"
-  assumes a: "\<exists>a. atom a \<sharp> (h, h a)"
-  shows "p \<bullet> (fresh_fun h) = fresh_fun (p \<bullet> h)"
-  using a
-  apply (clarsimp simp add: fresh_Pair)
-  apply (subst fresh_fun_app', assumption+)
-  apply (drule fresh_permute_iff [where p=p, THEN iffD2])
-  apply (drule fresh_permute_iff [where p=p, THEN iffD2])
-  apply (simp add: atom_eqvt permute_fun_app_eq [where f=h])
-  apply (erule (1) fresh_fun_app' [symmetric])
-  done
-
-lemma fresh_fun_supports:
-  fixes h :: "'a::at \<Rightarrow> 'b::pt"
-  assumes a: "\<exists>a. atom a \<sharp> (h, h a)"
-  shows "(supp h) supports (fresh_fun h)"
-  apply (simp add: supports_def fresh_def [symmetric])
-  apply (simp add: fresh_fun_eqvt [OF a] swap_fresh_fresh)
-  done
-
-notation fresh_fun (binder "FRESH " 10)
-
-lemma FRESH_f_iff:
-  fixes P :: "'a::at \<Rightarrow> 'b::pure"
-  fixes f :: "'b \<Rightarrow> 'c::pure"
-  assumes P: "finite (supp P)"
-  shows "(FRESH x. f (P x)) = f (FRESH x. P x)"
-proof -
-  obtain a::'a where "atom a \<notin> supp P"
-    using P by (rule obtain_at_base)
-  hence "atom a \<sharp> P"
-    by (simp add: fresh_def)
-  show "(FRESH x. f (P x)) = f (FRESH x. P x)"
-    apply (subst fresh_fun_app' [where a=a, OF _ pure_fresh])
-    apply (cut_tac `atom a \<sharp> P`)
-    apply (simp add: fresh_conv_MOST)
-    apply (elim MOST_rev_mp, rule MOST_I, clarify)
-    apply (simp add: permute_fun_def permute_pure expand_fun_eq)
-    apply (subst fresh_fun_app' [where a=a, OF `atom a \<sharp> P` pure_fresh])
-    apply (rule refl)
-    done
-qed
-
-lemma FRESH_binop_iff:
-  fixes P :: "'a::at \<Rightarrow> 'b::pure"
-  fixes Q :: "'a::at \<Rightarrow> 'c::pure"
-  fixes binop :: "'b \<Rightarrow> 'c \<Rightarrow> 'd::pure"
-  assumes P: "finite (supp P)" 
-  and     Q: "finite (supp Q)"
-  shows "(FRESH x. binop (P x) (Q x)) = binop (FRESH x. P x) (FRESH x. Q x)"
-proof -
-  from assms have "finite (supp P \<union> supp Q)" by simp
-  then obtain a::'a where "atom a \<notin> (supp P \<union> supp Q)"
-    by (rule obtain_at_base)
-  hence "atom a \<sharp> P" and "atom a \<sharp> Q"
-    by (simp_all add: fresh_def)
-  show ?thesis
-    apply (subst fresh_fun_app' [where a=a, OF _ pure_fresh])
-    apply (cut_tac `atom a \<sharp> P` `atom a \<sharp> Q`)
-    apply (simp add: fresh_conv_MOST)
-    apply (elim MOST_rev_mp, rule MOST_I, clarify)
-    apply (simp add: permute_fun_def permute_pure expand_fun_eq)
-    apply (subst fresh_fun_app' [where a=a, OF `atom a \<sharp> P` pure_fresh])
-    apply (subst fresh_fun_app' [where a=a, OF `atom a \<sharp> Q` pure_fresh])
-    apply (rule refl)
-    done
-qed
-
-lemma FRESH_conj_iff:
-  fixes P Q :: "'a::at \<Rightarrow> bool"
-  assumes P: "finite (supp P)" and Q: "finite (supp Q)"
-  shows "(FRESH x. P x \<and> Q x) \<longleftrightarrow> (FRESH x. P x) \<and> (FRESH x. Q x)"
-using P Q by (rule FRESH_binop_iff)
-
-lemma FRESH_disj_iff:
-  fixes P Q :: "'a::at \<Rightarrow> bool"
-  assumes P: "finite (supp P)" and Q: "finite (supp Q)"
-  shows "(FRESH x. P x \<or> Q x) \<longleftrightarrow> (FRESH x. P x) \<or> (FRESH x. Q x)"
-using P Q by (rule FRESH_binop_iff)
-
-
-section {* An example of a function without finite support *}
-
-primrec
-  nat_of :: "atom \<Rightarrow> nat"
-where
-  "nat_of (Atom s n) = n"
-
-lemma atom_eq_iff:
-  fixes a b :: atom
-  shows "a = b \<longleftrightarrow> sort_of a = sort_of b \<and> nat_of a = nat_of b"
-  by (induct a, induct b, simp)
-
-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_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 {* Support for sets of atoms *}
-
-lemma supp_finite_atom_set:
-  fixes S::"atom set"
-  assumes "finite S"
-  shows "supp S = S"
-  apply(rule finite_supp_unique)
-  apply(simp add: supports_def)
-  apply(simp add: swap_set_not_in)
-  apply(rule assms)
-  apply(simp add: swap_set_in)
-done
-
-
-section {* transpositions of permutations *}
-
-fun
-  add_perm 
-where
-  "add_perm [] = 0"
-| "add_perm ((a, b) # xs) = (a \<rightleftharpoons> b) + add_perm xs"
-
-lemma add_perm_append:
-  shows "add_perm (xs @ ys) = add_perm xs + add_perm ys"
-by (induct xs arbitrary: ys)
-   (auto simp add: add_assoc)
-
-lemma perm_list_exists:
-  fixes p::perm
-  shows "\<exists>xs. p = add_perm xs \<and> supp xs \<subseteq> supp p"
-apply(induct p taking: "\<lambda>p::perm. card (supp p)" rule: measure_induct)
-apply(rename_tac p)
-apply(case_tac "supp p = {}")
-apply(simp)
-apply(simp add: supp_perm)
-apply(rule_tac x="[]" in exI)
-apply(simp add: supp_Nil)
-apply(simp add: expand_perm_eq)
-apply(subgoal_tac "\<exists>x. x \<in> supp p")
-defer
-apply(auto)[1]
-apply(erule exE)
-apply(drule_tac x="p + (((- p) \<bullet> x) \<rightleftharpoons> x)" in spec)
-apply(drule mp)
-defer
-apply(erule exE)
-apply(rule_tac x="xs @ [((- p) \<bullet> x, x)]" in exI)
-apply(simp add: add_perm_append)
-apply(erule conjE)
-apply(drule sym)
-apply(simp)
-apply(simp add: expand_perm_eq)
-apply(simp add: supp_append)
-apply(simp add: supp_perm supp_Cons supp_Nil supp_Pair supp_atom)
-apply(rule conjI)
-prefer 2
-apply(auto)[1]
-apply (metis permute_atom_def_raw permute_minus_cancel(2))
-defer
-apply(rule psubset_card_mono)
-apply(simp add: finite_supp)
-apply(rule psubsetI)
-defer
-apply(subgoal_tac "x \<notin> supp (p + (- p \<bullet> x \<rightleftharpoons> x))")
-apply(blast)
-apply(simp add: supp_perm)
-defer
-apply(auto simp add: supp_perm)[1]
-apply(case_tac "x = xa")
-apply(simp)
-apply(case_tac "((- p) \<bullet> x) = xa")
-apply(simp)
-apply(case_tac "sort_of xa = sort_of x")
-apply(simp)
-apply(auto)[1]
-apply(simp)
-apply(simp)
-apply(subgoal_tac "{a. p \<bullet> (- p \<bullet> x \<rightleftharpoons> x) \<bullet> a \<noteq> a} \<subseteq> {a. p \<bullet> a \<noteq> a}")
-apply(blast)
-apply(auto simp add: supp_perm)[1]
-apply(case_tac "x = xa")
-apply(simp)
-apply(case_tac "((- p) \<bullet> x) = xa")
-apply(simp)
-apply(case_tac "sort_of xa = sort_of x")
-apply(simp)
-apply(auto)[1]
-apply(simp)
-apply(simp)
-done
-
-section {* Lemma about support and permutations *}
-
-lemma supp_perm_eq:
-  assumes a: "(supp x) \<sharp>* p"
-  shows "p \<bullet> x = x"
-proof -
-  obtain xs where eq: "p = add_perm xs" and sub: "supp xs \<subseteq> supp p"
-    using perm_list_exists by blast
-  from a have "\<forall>a \<in> supp p. a \<sharp> x"
-    by (auto simp add: fresh_star_def fresh_def supp_perm)
-  with eq sub have "\<forall>a \<in> supp xs. a \<sharp> x" by auto
-  then have "add_perm xs \<bullet> x = x" 
-    by (induct xs rule: add_perm.induct)
-       (simp_all add: supp_Cons supp_Pair supp_atom swap_fresh_fresh)
-  then show "p \<bullet> x = x" using eq by simp
-qed
-
-section {* at_set_avoiding2 *}
-
-lemma at_set_avoiding2:
-  assumes "finite xs"
-  and     "finite (supp c)" "finite (supp x)"
-  and     "xs \<sharp>* x"
-  shows "\<exists>p. (p \<bullet> xs) \<sharp>* c \<and> supp x \<sharp>* p"
-using assms
-apply(erule_tac c="(c, x)" in at_set_avoiding)
-apply(simp add: supp_Pair)
-apply(rule_tac x="p" in exI)
-apply(simp add: fresh_star_prod)
-apply(subgoal_tac "\<forall>a \<in> supp p. a \<sharp> x")
-apply(auto simp add: fresh_star_def fresh_def supp_perm)[1]
-apply(auto simp add: fresh_star_def fresh_def)
-done
-
-lemma at_set_avoiding2_atom:
-  assumes "finite (supp c)" "finite (supp x)"
-  and     b: "xa \<sharp> x"
-  shows "\<exists>p. (p \<bullet> xa) \<sharp> c \<and> supp x \<sharp>* p"
-proof -
-  have a: "{xa} \<sharp>* x" unfolding fresh_star_def by (simp add: b)
-  obtain p where p1: "(p \<bullet> {xa}) \<sharp>* c" and p2: "supp x \<sharp>* p"
-    using at_set_avoiding2[of "{xa}" "c" "x"] assms a by blast
-  have c: "(p \<bullet> xa) \<sharp> c" using p1
-    unfolding fresh_star_def Ball_def 
-    by (erule_tac x="p \<bullet> xa" in allE) (simp add: eqvts)
-  hence "p \<bullet> xa \<sharp> c \<and> supp x \<sharp>* p" using p2 by blast
-  then show ?thesis by blast
-qed
-
-end
diff -r c0eac04ae3b4 -r c34347ec7ab3 Nominal/Parser.thy
--- a/Nominal/Parser.thy	Sat Apr 03 22:31:11 2010 +0200
+++ b/Nominal/Parser.thy	Sun Apr 04 21:39:28 2010 +0200
@@ -1,5 +1,7 @@
 theory Parser
-imports "Nominal2_Atoms" "Nominal2_Eqvt" "Nominal2_Supp" "Perm" "Fv" "Rsp" "Lift"
+imports "../Nominal-General/Nominal2_Atoms" 
+        "../Nominal-General/Nominal2_Eqvt" 
+        "../Nominal-General/Nominal2_Supp" "Perm" "Fv" "Rsp" "Lift"
 begin
 
 section{* Interface for nominal_datatype *}
diff -r c0eac04ae3b4 -r c34347ec7ab3 Nominal/Perm.thy
--- a/Nominal/Perm.thy	Sat Apr 03 22:31:11 2010 +0200
+++ b/Nominal/Perm.thy	Sun Apr 04 21:39:28 2010 +0200
@@ -1,5 +1,5 @@
 theory Perm
-imports "Nominal2_Atoms"
+imports "../Nominal-General/Nominal2_Atoms"
 begin
 
 ML {*
diff -r c0eac04ae3b4 -r c34347ec7ab3 Nominal/ROOT.ML
--- a/Nominal/ROOT.ML	Sat Apr 03 22:31:11 2010 +0200
+++ b/Nominal/ROOT.ML	Sun Apr 04 21:39:28 2010 +0200
@@ -1,11 +1,7 @@
 quick_and_dirty := true;
 
 no_document use_thys
-   ["Nominal2_Base",
-    "Nominal2_Eqvt",
-    "Nominal2_Atoms",
-    "Nominal2_Supp",
-    "Ex/ExLam",
+   ["Ex/ExLam",
     "Ex/ExLF",
     "Ex/Ex1",
     "Ex/Ex1rec",
diff -r c0eac04ae3b4 -r c34347ec7ab3 Nominal/nominal_atoms.ML
--- a/Nominal/nominal_atoms.ML	Sat Apr 03 22:31:11 2010 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,94 +0,0 @@
-(*  Title:      nominal_atoms/ML
-    Authors:    Brian Huffman, Christian Urban
-
-    Command for defining concrete atom types. 
-    
-    At the moment, only single-sorted atom types
-    are supported. 
-*)
-
-signature ATOM_DECL =
-sig
-  val add_atom_decl: (binding * (binding option)) -> theory -> theory
-end;
-
-structure Atom_Decl :> ATOM_DECL =
-struct
-
-val atomT = @{typ atom};
-val permT = @{typ perm};
-
-val sort_of_const = @{term sort_of};
-fun atom_const T = Const (@{const_name atom}, T --> atomT);
-fun permute_const T = Const (@{const_name permute}, permT --> T --> T);
-
-fun mk_sort_of t = sort_of_const $ t;
-fun mk_atom t = atom_const (fastype_of t) $ t;
-fun mk_permute (p, t) = permute_const (fastype_of t) $ p $ t;
-
-fun atom_decl_set (str : string) : term =
-  let
-    val a = Free ("a", atomT);
-    val s = Const (@{const_name "Sort"}, @{typ "string => atom_sort list => atom_sort"})
-              $ HOLogic.mk_string str $ HOLogic.nil_const @{typ "atom_sort"};
-  in
-    HOLogic.mk_Collect ("a", atomT, HOLogic.mk_eq (mk_sort_of a, s))
-  end
-
-fun add_atom_decl (name : binding, arg : binding option) (thy : theory) =
-  let
-    val _ = Theory.requires thy "Nominal2_Atoms" "nominal logic";
-    val str = Sign.full_name thy name;
-
-    (* typedef *)
-    val set = atom_decl_set str;
-    val tac = rtac @{thm exists_eq_simple_sort} 1;
-    val ((full_tname, info  as ({Rep_name, Abs_name, ...}, {type_definition, ...})), thy) =
-      Typedef.add_typedef_global false NONE (name, [], NoSyn) set NONE tac thy;
-
-    (* definition of atom and permute *)
-    val newT = #abs_type (fst info);
-    val RepC = Const (Rep_name, newT --> atomT);
-    val AbsC = Const (Abs_name, atomT --> newT);
-    val a = Free ("a", newT);
-    val p = Free ("p", permT);
-    val atom_eqn =
-      HOLogic.mk_Trueprop (HOLogic.mk_eq (mk_atom a, RepC $ a));
-    val permute_eqn =
-      HOLogic.mk_Trueprop (HOLogic.mk_eq
-        (mk_permute (p, a), AbsC $ (mk_permute (p, RepC $ a))));
-    val atom_def_name =
-      Binding.prefix_name "atom_" (Binding.suffix_name "_def" name);
-    val permute_def_name =
-      Binding.prefix_name "permute_" (Binding.suffix_name "_def" name);
-
-    (* at class instance *)
-    val lthy =
-      Theory_Target.instantiation ([full_tname], [], @{sort at}) thy;
-    val ((_, (_, permute_ldef)), lthy) =
-      Specification.definition (NONE, ((permute_def_name, []), permute_eqn)) lthy;
-    val ((_, (_, atom_ldef)), lthy) =
-      Specification.definition (NONE, ((atom_def_name, []), atom_eqn)) lthy;
-    val ctxt_thy = ProofContext.init (ProofContext.theory_of lthy);
-    val permute_def = singleton (ProofContext.export lthy ctxt_thy) permute_ldef;
-    val atom_def = singleton (ProofContext.export lthy ctxt_thy) atom_ldef;
-    val class_thm = @{thm at_class} OF [type_definition, atom_def, permute_def];
-    val thy = lthy
-      |> Class.prove_instantiation_instance (K (Tactic.rtac class_thm 1))
-      |> Local_Theory.exit_global;
-  in
-    thy
-  end;
-
-(** outer syntax **)
-
-local structure P = OuterParse and K = OuterKeyword in
-
-val _ =
-  OuterSyntax.command "atom_decl" "declaration of a concrete atom type" K.thy_decl
-    ((P.binding -- Scan.option (Args.parens (P.binding))) >>
-      (Toplevel.print oo (Toplevel.theory o add_atom_decl)));
-
-end;
-
-end;
diff -r c0eac04ae3b4 -r c34347ec7ab3 Nominal/nominal_permeq.ML
--- a/Nominal/nominal_permeq.ML	Sat Apr 03 22:31:11 2010 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,71 +0,0 @@
-(*  Title:      nominal_thmdecls.ML
-    Author:     Brian Huffman, Christian Urban
-*)
-
-signature NOMINAL_PERMEQ =
-sig
-  val eqvt_tac: Proof.context -> int -> tactic 
-
-end;
-
-(* TODO:
-
- - provide a method interface with the usual add and del options
-
- - print a warning if for a constant no eqvt lemma is stored
-
- - there seems to be too much simplified in case of multiple 
-   permutations, like
-
-      p o q o r o x 
-
-   we usually only want the outermost permutation to "float" in
-*)
-
-
-structure Nominal_Permeq: NOMINAL_PERMEQ =
-struct
-
-local
-
-fun eqvt_apply_conv ctxt ct =
-  case (term_of ct) of
-    (Const (@{const_name "permute"}, _) $ _ $ (_ $ _)) =>
-      let
-        val (perm, t) = Thm.dest_comb ct
-        val (_, p) = Thm.dest_comb perm
-        val (f, x) = Thm.dest_comb t
-        val a = ctyp_of_term x;
-        val b = ctyp_of_term t;
-        val ty_insts = map SOME [b, a]
-        val term_insts = map SOME [p, f, x]
-      in
-        Drule.instantiate' ty_insts term_insts @{thm eqvt_apply}
-      end
-  | _ => Conv.no_conv ct
-
-fun eqvt_lambda_conv ctxt ct =
-  case (term_of ct) of
-    (Const (@{const_name "permute"}, _) $ _ $ Abs _) =>
-      Conv.rewr_conv @{thm eqvt_lambda} ct
-  | _ => Conv.no_conv ct
-
-in
-
-fun eqvt_conv ctxt ct =
-  Conv.first_conv
-    [ Conv.rewr_conv @{thm eqvt_bound},
-      eqvt_apply_conv ctxt
-        then_conv Conv.comb_conv (eqvt_conv ctxt),
-      eqvt_lambda_conv ctxt
-        then_conv Conv.abs_conv (fn (v, ctxt) => eqvt_conv ctxt) ctxt,
-      More_Conv.rewrs_conv (Nominal_ThmDecls.get_eqvts_raw_thms ctxt),
-      Conv.all_conv
-    ] ct
-
-fun eqvt_tac ctxt = 
-  CONVERSION (More_Conv.bottom_conv (fn ctxt => eqvt_conv ctxt) ctxt)
-
-end
-
-end; (* structure *)
\ No newline at end of file
diff -r c0eac04ae3b4 -r c34347ec7ab3 Nominal/nominal_thmdecls.ML
--- a/Nominal/nominal_thmdecls.ML	Sat Apr 03 22:31:11 2010 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,134 +0,0 @@
-(*  Title:      nominal_thmdecls.ML
-    Author:     Christian Urban
-
-  Infrastructure for the lemma collection "eqvts".
-
-  Provides the attributes [eqvt] and [eqvt_raw], and the theorem
-  lists eqvts and eqvts_raw. The first attribute will store the 
-  theorem in the eqvts list and also in the eqvts_raw list. For 
-  the latter the theorem is expected to be of the form
-
-    p o (c x1 x2 ...) = c (p o x1) (p o x2) ...
-
-  and it is stored in the form
-
-    p o c == c
-
-  The [eqvt_raw] attribute just adds the theorem to eqvts_raw.
-
-  TODO:
-
-   - deal with eqvt-lemmas of the form 
-
-       c x1 x2 ... ==> c (p o x1) (p o x2) ..
-*)
-
-signature NOMINAL_THMDECLS =
-sig
-  val eqvt_add: attribute
-  val eqvt_del: attribute
-  val eqvt_raw_add: attribute
-  val eqvt_raw_del: attribute
-  val setup: theory -> theory
-  val get_eqvts_thms: Proof.context -> thm list
-  val get_eqvts_raw_thms: Proof.context -> thm list
-
-end;
-
-structure Nominal_ThmDecls: NOMINAL_THMDECLS =
-struct
-
-
-structure EqvtData = Generic_Data
-( type T = thm Item_Net.T;
-  val empty = Thm.full_rules;
-  val extend = I;
-  val merge = Item_Net.merge );
-
-structure EqvtRawData = Generic_Data
-( type T = thm Item_Net.T;
-  val empty = Thm.full_rules;
-  val extend = I;
-  val merge = Item_Net.merge );
-
-val eqvts = Item_Net.content o EqvtData.get;
-val eqvts_raw = Item_Net.content o EqvtRawData.get;
-
-val get_eqvts_thms = eqvts o Context.Proof; 
-val get_eqvts_raw_thms = eqvts_raw o Context.Proof; 
-
-val add_thm = EqvtData.map o Item_Net.update;
-val del_thm = EqvtData.map o Item_Net.remove;
-
-val add_raw_thm = EqvtRawData.map o Item_Net.update;
-val del_raw_thm = EqvtRawData.map o Item_Net.remove;
-
-fun dest_perm (Const (@{const_name "permute"}, _) $ p $ t) = (p, t)
-  | dest_perm t = raise TERM("dest_perm", [t])
-
-fun mk_perm p trm =
-let
-  val ty = fastype_of trm
-in
-  Const (@{const_name "permute"}, @{typ "perm"} --> ty --> ty) $ p $ trm
-end
-
-fun eqvt_transform_tac thm = REPEAT o FIRST' 
-  [CHANGED o simp_tac (HOL_basic_ss addsimps @{thms permute_minus_cancel}),
-   rtac (thm RS @{thm trans}),
-   rtac @{thm trans[OF permute_fun_def]} THEN' rtac @{thm ext}]
-
-(* transform equations into the required form *)
-fun transform_eq ctxt thm lhs rhs = 
-let
-  val (p, t) = dest_perm lhs
-  val (c, args) = strip_comb t
-  val (c', args') = strip_comb rhs 
-  val eargs = map Envir.eta_contract args 
-  val eargs' = map Envir.eta_contract args'
-  val p_str = fst (fst (dest_Var p))
-  val goal = HOLogic.mk_Trueprop (HOLogic.mk_eq (mk_perm p c, c))
-in
-  if c <> c' 
-    then error "eqvt lemma is not of the right form (constants do not agree)"
-  else if eargs' <> map (mk_perm p) eargs 
-    then error "eqvt lemma is not of the right form (arguments do not agree)"
-  else if args = [] 
-    then thm
-  else Goal.prove ctxt [p_str] [] goal
-    (fn _ => eqvt_transform_tac thm 1)
-end
-
-fun transform addel_fun thm context = 
-let
-  val ctxt = Context.proof_of context
-  val trm = HOLogic.dest_Trueprop (prop_of thm)
-in
-  case trm of
-    Const (@{const_name "op ="}, _) $ lhs $ rhs =>
-      let 
-        val thm' = transform_eq ctxt thm lhs rhs RS @{thm eq_reflection}
-      in
-        addel_fun thm' context
-      end
-  | _ => raise (error "only (op=) case implemented yet")
-end 
-
-val eqvt_add = Thm.declaration_attribute (fn thm => (add_thm thm) o (transform add_raw_thm thm));
-val eqvt_del = Thm.declaration_attribute (fn thm => (del_thm thm) o (transform del_raw_thm thm));
-
-val eqvt_raw_add = Thm.declaration_attribute add_raw_thm;
-val eqvt_raw_del = Thm.declaration_attribute del_raw_thm;
-
-val setup =
-  Attrib.setup @{binding "eqvt"} (Attrib.add_del eqvt_add eqvt_del) 
-    (cat_lines ["declaration of equivariance lemmas - they will automtically be",  
-       "brought into the form p o c = c"]) #>
-  Attrib.setup @{binding "eqvt_raw"} (Attrib.add_del eqvt_raw_add eqvt_raw_del) 
-    (cat_lines ["declaration of equivariance lemmas - no",
-       "transformation is performed"]) #>
-  PureThy.add_thms_dynamic (@{binding "eqvts"}, eqvts) #>
-  PureThy.add_thms_dynamic (@{binding "eqvts_raw"}, eqvts_raw);
-
-
-end;
diff -r c0eac04ae3b4 -r c34347ec7ab3 Pearl/Paper.thy
--- a/Pearl/Paper.thy	Sat Apr 03 22:31:11 2010 +0200
+++ b/Pearl/Paper.thy	Sun Apr 04 21:39:28 2010 +0200
@@ -1,9 +1,9 @@
 (*<*)
 theory Paper
-imports "../Nominal/Nominal2_Base" 
-        "../Nominal/Nominal2_Atoms" 
-        "../Nominal/Nominal2_Eqvt" 
-        "../Nominal/Atoms" 
+imports "../Nominal-General/Nominal2_Base" 
+        "../Nominal-General/Nominal2_Atoms" 
+        "../Nominal-General/Nominal2_Eqvt" 
+        "../Nominal-General/Atoms" 
         "LaTeXsugar"
 begin
 
diff -r c0eac04ae3b4 -r c34347ec7ab3 Pearl/ROOT.ML
--- a/Pearl/ROOT.ML	Sat Apr 03 22:31:11 2010 +0200
+++ b/Pearl/ROOT.ML	Sun Apr 04 21:39:28 2010 +0200
@@ -1,7 +1,7 @@
-no_document use_thys ["../Nominal/Nominal2_Base", 
-                      "../Nominal/Nominal2_Atoms", 
-                      "../Nominal/Nominal2_Eqvt",
-                      "../Nominal/Atoms", 
+no_document use_thys ["../Nominal-General/Nominal2_Base", 
+                      "../Nominal-General/Nominal2_Atoms", 
+                      "../Nominal-General/Nominal2_Eqvt",
+                      "../Nominal-General/Atoms", 
                       "LaTeXsugar"];
 
 use_thys ["Paper"];
\ No newline at end of file
diff -r c0eac04ae3b4 -r c34347ec7ab3 README
--- a/README	Sat Apr 03 22:31:11 2010 +0200
+++ b/README	Sun Apr 04 21:39:28 2010 +0200
@@ -3,16 +3,18 @@
 
 Subdirectories:
 
-Attic       ... old version of the quotient package (is now 
-                part of the Isabelle distribution)
+Attic         ... old version of the quotient package (is now 
+                  part of the Isabelle distribution)
 
-Literature  ... some relevant papers bout binders and
-                Core-Haskell
+Literature    ... some relevant papers about binders and
+                  Core-Haskell
+
+Nominal-General . implementation of the abstract nominal theory
 
-Nominal     ... main files for Nominal Isabelle
+Nominal       ... main files for new Nominal Isabelle
 
-Nominal/Ex  ... examples for new implementation
+Nominal/Ex    ... examples for new implementation
 
-Paper       ... submitted
+Paper         ... submitted to ICFP
 
-Pearl       ... paper accepted at ITP 
\ No newline at end of file
+Pearl         ... paper accepted at ITP 
\ No newline at end of file