--- a/Nominal/Abs.thy	Sat Mar 20 10:12:09 2010 +0100
+++ b/Nominal/Abs.thy	Sat Mar 20 18:16:26 2010 +0100
@@ -377,128 +377,6 @@
 apply(simp add: eqvts)
 done
 
-lemma perm_zero:
-  assumes a: "\<forall>x::atom. p \<bullet> x = x"
-  shows "p = 0"
-using a
-by (simp add: expand_perm_eq)
-
-fun
-  add_perm 
-where
-  "add_perm [] = 0"
-| "add_perm ((a, b) # xs) = (a \<rightleftharpoons> b) + add_perm xs"
-
-fun
-  elem_perm
-where
-  "elem_perm [] = {}"
-| "elem_perm ((a, b) # xs) = {a, b} \<union> elem_perm xs"
-
-
-lemma add_perm_apend:
-  shows "add_perm (xs @ ys) = add_perm xs + add_perm ys"
-apply(induct xs arbitrary: ys)
-apply(auto simp add: add_assoc)
-done
-
-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(drule perm_zero)
-apply(simp)
-apply(rule_tac x="[]" in exI)
-apply(simp add: supp_Nil)
-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_apend)
-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 left_minus minus_unique 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
-
-lemma tt0:
-  fixes p::perm
-  shows "(supp x) \<sharp>* p \<Longrightarrow> \<forall>a \<in> supp p. a \<sharp> x"
-apply(auto simp add: fresh_star_def supp_perm fresh_def)
-done
-
-lemma uu0:
-  shows "(\<forall>a \<in> elem_perm xs. a \<sharp> x) \<Longrightarrow> (add_perm xs \<bullet> x) = x"
-apply(induct xs rule: add_perm.induct)
-apply(simp)
-apply(simp add: swap_fresh_fresh)
-done
-
-lemma yy0:
-  fixes xs::"(atom \<times> atom) list"
-  shows "supp xs = elem_perm xs"
-apply(induct xs rule: elem_perm.induct)
-apply(auto simp add: supp_Nil supp_Cons supp_Pair supp_atom)
-done
-
-lemma tt1:
-  shows "(supp x) \<sharp>* p \<Longrightarrow> p \<bullet> x = x"
-apply(drule tt0)
-apply(subgoal_tac "\<exists>xs. p = add_perm xs \<and> supp xs \<subseteq> supp p")
-prefer 2
-apply(rule perm_list_exists)
-apply(erule exE)
-apply(simp only: yy0)
-apply(rule uu0)
-apply(auto)
-done
-
 
 lemma perm_induct_test:
   fixes P :: "perm => bool"
@@ -509,10 +387,6 @@
   shows "P p"
 using fin
 apply(induct F\<equiv>"supp p" arbitrary: p rule: finite_induct)
-apply(simp add: supp_perm)
-apply(drule perm_zero)
-apply(simp add: zero)
-apply(rotate_tac 3)
 oops
 
 lemma ii:
@@ -618,13 +492,13 @@
 apply(case_tac "a \<notin> supp x")
 apply(simp)
 apply(subgoal_tac "supp x \<sharp>* p")
-apply(drule tt1)
+apply(drule supp_perm_eq)
 apply(simp)
 apply(simp)
 apply(simp)
 apply(case_tac "a \<notin> supp y")
 apply(simp)
-apply(drule tt1)
+apply(drule supp_perm_eq)
 apply(clarify)
 apply(simp (no_asm_use))
 apply(simp)
@@ -635,7 +509,7 @@
 apply(simp)
 apply(case_tac "a \<sharp> p")
 apply(subgoal_tac "supp y \<sharp>* p")
-apply(drule tt1)
+apply(drule supp_perm_eq)
 apply(clarify)
 apply(simp (no_asm_use))
 apply(metis)

--- a/Nominal/Abs_res.thy	Sat Mar 20 10:12:09 2010 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,723 +0,0 @@
-theory Abs_res
-imports "Nominal2_Atoms" "Nominal2_Eqvt" "Nominal2_Supp" "../Quotient" "../Quotient_Product"
-begin
-
-fun
-  alpha_gen 
-where
-  alpha_gen[simp del]:
-  "alpha_gen (bs, x) R f pi (cs, y) \<longleftrightarrow> f x - bs = f y - cs \<and> (f x - bs) \<sharp>* pi \<and> R (pi \<bullet> x) y"
-
-notation
-  alpha_gen ("_ \<approx>gen _ _ _ _" [100, 100, 100, 100, 100] 100)
-
-lemma [mono]: "R1 \<le> R2 \<Longrightarrow> alpha_gen x R1 \<le> alpha_gen x R2"
-  by (cases x) (auto simp add: le_fun_def le_bool_def alpha_gen.simps)
-
-lemma alpha_gen_refl:
-  assumes a: "R x x"
-  shows "(bs, x) \<approx>gen R f 0 (bs, x)"
-  using a by (simp add: alpha_gen fresh_star_def fresh_zero_perm)
-
-lemma alpha_gen_sym:
-  assumes a: "(bs, x) \<approx>gen R f p (cs, y)"
-  and     b: "R (p \<bullet> x) y \<Longrightarrow> R (- p \<bullet> y) x"
-  shows "(cs, y) \<approx>gen R f (- p) (bs, x)"
-  using a b by (simp add: alpha_gen fresh_star_def fresh_def supp_minus_perm)
-
-lemma alpha_gen_trans:
-  assumes a: "(bs, x) \<approx>gen R f p1 (cs, y)"
-  and     b: "(cs, y) \<approx>gen R f p2 (ds, z)"
-  and     c: "\<lbrakk>R (p1 \<bullet> x) y; R (p2 \<bullet> y) z\<rbrakk> \<Longrightarrow> R ((p2 + p1) \<bullet> x) z"
-  shows "(bs, x) \<approx>gen R f (p2 + p1) (ds, z)"
-  using a b c using supp_plus_perm
-  apply(simp add: alpha_gen fresh_star_def fresh_def)
-  apply(blast)
-  done
-
-lemma alpha_gen_eqvt:
-  assumes a: "(bs, x) \<approx>gen R f q (cs, y)"
-  and     b: "R (q \<bullet> x) y \<Longrightarrow> R (p \<bullet> (q \<bullet> x)) (p \<bullet> y)"
-  and     c: "p \<bullet> (f x) = f (p \<bullet> x)"
-  and     d: "p \<bullet> (f y) = f (p \<bullet> y)"
-  shows "(p \<bullet> bs, p \<bullet> x) \<approx>gen R f (p \<bullet> q) (p \<bullet> cs, p \<bullet> y)"
-  using a b
-  apply(simp add: alpha_gen c[symmetric] d[symmetric] Diff_eqvt[symmetric])
-  apply(simp add: permute_eqvt[symmetric])
-  apply(simp add: fresh_star_permute_iff)
-  apply(clarsimp)
-  done
-
-lemma alpha_gen_compose_sym:
-  fixes pi
-  assumes b: "(aa, t) \<approx>gen (\<lambda>x1 x2. R x1 x2 \<and> R x2 x1) f pi (ab, s)"
-  and a: "\<And>pi t s. (R t s \<Longrightarrow> R (pi \<bullet> t) (pi \<bullet> s))"
-  shows "(ab, s) \<approx>gen R f (- pi) (aa, t)"
-  using b apply -
-  apply(simp add: alpha_gen.simps)
-  apply(erule conjE)+
-  apply(rule conjI)
-  apply(simp add: fresh_star_def fresh_minus_perm)
-  apply(subgoal_tac "R (- pi \<bullet> s) ((- pi) \<bullet> (pi \<bullet> t))")
-  apply simp
-  apply(rule a)
-  apply assumption
-  done
-
-lemma alpha_gen_compose_trans:
-  fixes pi pia
-  assumes b: "(aa, t) \<approx>gen (\<lambda>x1 x2. R x1 x2 \<and> (\<forall>x. R x2 x \<longrightarrow> R x1 x)) f pi (ab, ta)"
-  and c: "(ab, ta) \<approx>gen R f pia (ac, sa)"
-  and a: "\<And>pi t s. (R t s \<Longrightarrow> R (pi \<bullet> t) (pi \<bullet> s))"
-  shows "(aa, t) \<approx>gen R f (pia + pi) (ac, sa)"
-  using b c apply -
-  apply(simp add: alpha_gen.simps)
-  apply(erule conjE)+
-  apply(simp add: fresh_star_plus)
-  apply(drule_tac x="- pia \<bullet> sa" in spec)
-  apply(drule mp)
-  apply(rotate_tac 4)
-  apply(drule_tac pi="- pia" in a)
-  apply(simp)
-  apply(rotate_tac 6)
-  apply(drule_tac pi="pia" in a)
-  apply(simp)
-  done
-
-lemma alpha_gen_compose_eqvt:
-  fixes  pia
-  assumes b: "(g d, t) \<approx>gen (\<lambda>x1 x2. R x1 x2 \<and> R (pi \<bullet> x1) (pi \<bullet> x2)) f pia (g e, s)"
-  and     c: "\<And>y. pi \<bullet> (g y) = g (pi \<bullet> y)"
-  and     a: "\<And>x. pi \<bullet> (f x) = f (pi \<bullet> x)"
-  shows  "(g (pi \<bullet> d), pi \<bullet> t) \<approx>gen R f (pi \<bullet> pia) (g (pi \<bullet> e), pi \<bullet> s)"
-  using b
-  apply -
-  apply(simp add: alpha_gen.simps)
-  apply(erule conjE)+
-  apply(rule conjI)
-  apply(rule_tac ?p1="- pi" in permute_eq_iff[THEN iffD1])
-  apply(simp add: a[symmetric] atom_eqvt Diff_eqvt insert_eqvt set_eqvt empty_eqvt c[symmetric])
-  apply(rule conjI)
-  apply(rule_tac ?p1="- pi" in fresh_star_permute_iff[THEN iffD1])
-  apply(simp add: a[symmetric] atom_eqvt Diff_eqvt insert_eqvt set_eqvt empty_eqvt c[symmetric])
-  apply(subst permute_eqvt[symmetric])
-  apply(simp)
-  done
-
-fun
-  alpha_abs 
-where
-  "alpha_abs (bs, x) (cs, y) = (\<exists>p. (bs, x) \<approx>gen (op=) supp p (cs, y))"
-
-notation
-  alpha_abs ("_ \<approx>abs _")
-
-lemma alpha_abs_swap:
-  assumes a1: "a \<notin> (supp x) - bs"
-  and     a2: "b \<notin> (supp x) - bs"
-  shows "(bs, x) \<approx>abs ((a \<rightleftharpoons> b) \<bullet> bs, (a \<rightleftharpoons> b) \<bullet> x)"
-  apply(simp)
-  apply(rule_tac x="(a \<rightleftharpoons> b)" in exI)
-  apply(simp add: alpha_gen)
-  apply(simp add: supp_eqvt[symmetric] Diff_eqvt[symmetric])
-  apply(simp add: swap_set_not_in[OF a1 a2])
-  apply(subgoal_tac "supp (a \<rightleftharpoons> b) \<subseteq> {a, b}")
-  using a1 a2
-  apply(simp add: fresh_star_def fresh_def)
-  apply(blast)
-  apply(simp add: supp_swap)
-  done
-
-lemma alpha_gen_swap_fun:
-  assumes f_eqvt: "\<And>pi. (pi \<bullet> (f x)) = f (pi \<bullet> x)"
-  assumes a1: "a \<notin> (f x) - bs"
-  and     a2: "b \<notin> (f x) - bs"
-  shows "\<exists>pi. (bs, x) \<approx>gen (op=) f pi ((a \<rightleftharpoons> b) \<bullet> bs, (a \<rightleftharpoons> b) \<bullet> x)"
-  apply(rule_tac x="(a \<rightleftharpoons> b)" in exI)
-  apply(simp add: alpha_gen)
-  apply(simp add: f_eqvt[symmetric] Diff_eqvt[symmetric])
-  apply(simp add: swap_set_not_in[OF a1 a2])
-  apply(subgoal_tac "supp (a \<rightleftharpoons> b) \<subseteq> {a, b}")
-  using a1 a2
-  apply(simp add: fresh_star_def fresh_def)
-  apply(blast)
-  apply(simp add: supp_swap)
-  done
-
-
-fun
-  supp_abs_fun
-where
-  "supp_abs_fun (bs, x) = (supp x) - bs"
-
-lemma supp_abs_fun_lemma:
-  assumes a: "x \<approx>abs y" 
-  shows "supp_abs_fun x = supp_abs_fun y"
-  using a
-  apply(induct rule: alpha_abs.induct)
-  apply(simp add: alpha_gen)
-  done
-  
-quotient_type 'a abs = "(atom set \<times> 'a::pt)" / "alpha_abs"
-  apply(rule equivpI)
-  unfolding reflp_def symp_def transp_def
-  apply(simp_all)
-  (* refl *)
-  apply(clarify)
-  apply(rule exI)
-  apply(rule alpha_gen_refl)
-  apply(simp)
-  (* symm *)
-  apply(clarify)
-  apply(rule exI)
-  apply(rule alpha_gen_sym)
-  apply(assumption)
-  apply(clarsimp)
-  (* trans *)
-  apply(clarify)
-  apply(rule exI)
-  apply(rule alpha_gen_trans)
-  apply(assumption)
-  apply(assumption)
-  apply(simp)
-  done
-
-quotient_definition
-  "Abs::atom set \<Rightarrow> ('a::pt) \<Rightarrow> 'a abs"
-is
-  "Pair::atom set \<Rightarrow> ('a::pt) \<Rightarrow> (atom set \<times> 'a)"
-
-lemma [quot_respect]:
-  shows "((op =) ===> (op =) ===> alpha_abs) Pair Pair"
-  apply(clarsimp)
-  apply(rule exI)
-  apply(rule alpha_gen_refl)
-  apply(simp)
-  done
-
-lemma [quot_respect]:
-  shows "((op =) ===> alpha_abs ===> alpha_abs) permute permute"
-  apply(clarsimp)
-  apply(rule exI)
-  apply(rule alpha_gen_eqvt)
-  apply(assumption)
-  apply(simp_all add: supp_eqvt)
-  done
-
-lemma [quot_respect]:
-  shows "(alpha_abs ===> (op =)) supp_abs_fun supp_abs_fun"
-  apply(simp add: supp_abs_fun_lemma)
-  done
-
-lemma abs_induct:
-  "\<lbrakk>\<And>as (x::'a::pt). P (Abs as x)\<rbrakk> \<Longrightarrow> P t"
-  apply(lifting prod.induct[where 'a="atom set" and 'b="'a"])
-  done
-
-(* TEST case *)
-lemmas abs_induct2 = prod.induct[where 'a="atom set" and 'b="'a::pt", quot_lifted]
-thm abs_induct abs_induct2
-
-instantiation abs :: (pt) pt
-begin
-
-quotient_definition
-  "permute_abs::perm \<Rightarrow> ('a::pt abs) \<Rightarrow> 'a abs"
-is
-  "permute:: perm \<Rightarrow> (atom set \<times> 'a::pt) \<Rightarrow> (atom set \<times> 'a::pt)"
-
-lemma permute_ABS [simp]:
-  fixes x::"'a::pt"  (* ??? has to be 'a \<dots> 'b does not work *)
-  shows "(p \<bullet> (Abs as x)) = Abs (p \<bullet> as) (p \<bullet> x)"
-  by (lifting permute_prod.simps(1)[where 'a="atom set" and 'b="'a"])
-
-instance
-  apply(default)
-  apply(induct_tac [!] x rule: abs_induct)
-  apply(simp_all)
-  done
-
-end
-
-quotient_definition
-  "supp_Abs_fun :: ('a::pt) abs \<Rightarrow> atom \<Rightarrow> bool"
-is
-  "supp_abs_fun"
-
-lemma supp_Abs_fun_simp:
-  shows "supp_Abs_fun (Abs bs x) = (supp x) - bs"
-  by (lifting supp_abs_fun.simps(1))
-
-lemma supp_Abs_fun_eqvt [eqvt]:
-  shows "(p \<bullet> supp_Abs_fun x) = supp_Abs_fun (p \<bullet> x)"
-  apply(induct_tac x rule: abs_induct)
-  apply(simp add: supp_Abs_fun_simp supp_eqvt Diff_eqvt)
-  done
-
-lemma supp_Abs_fun_fresh:
-  shows "a \<sharp> Abs bs x \<Longrightarrow> a \<sharp> supp_Abs_fun (Abs bs x)"
-  apply(rule fresh_fun_eqvt_app)
-  apply(simp add: eqvts_raw)
-  apply(simp)
-  done
-
-lemma Abs_swap:
-  assumes a1: "a \<notin> (supp x) - bs"
-  and     a2: "b \<notin> (supp x) - bs"
-  shows "(Abs bs x) = (Abs ((a \<rightleftharpoons> b) \<bullet> bs) ((a \<rightleftharpoons> b) \<bullet> x))"
-  using a1 a2 by (lifting alpha_abs_swap)
-
-lemma Abs_supports:
-  shows "((supp x) - as) supports (Abs as x)"
-  unfolding supports_def
-  apply(clarify)
-  apply(simp (no_asm))
-  apply(subst Abs_swap[symmetric])
-  apply(simp_all)
-  done
-
-lemma supp_Abs_subset1:
-  fixes x::"'a::fs"
-  shows "(supp x) - as \<subseteq> supp (Abs as x)"
-  apply(simp add: supp_conv_fresh)
-  apply(auto)
-  apply(drule_tac supp_Abs_fun_fresh)
-  apply(simp only: supp_Abs_fun_simp)
-  apply(simp add: fresh_def)
-  apply(simp add: supp_finite_atom_set finite_supp)
-  done
-
-lemma supp_Abs_subset2:
-  fixes x::"'a::fs"
-  shows "supp (Abs as x) \<subseteq> (supp x) - as"
-  apply(rule supp_is_subset)
-  apply(rule Abs_supports)
-  apply(simp add: finite_supp)
-  done
-
-lemma supp_Abs:
-  fixes x::"'a::fs"
-  shows "supp (Abs as x) = (supp x) - as"
-  apply(rule_tac subset_antisym)
-  apply(rule supp_Abs_subset2)
-  apply(rule supp_Abs_subset1)
-  done
-
-instance abs :: (fs) fs
-  apply(default)
-  apply(induct_tac x rule: abs_induct)
-  apply(simp add: supp_Abs)
-  apply(simp add: finite_supp)
-  done
-
-lemma Abs_fresh_iff:
-  fixes x::"'a::fs"
-  shows "a \<sharp> Abs bs x \<longleftrightarrow> a \<in> bs \<or> (a \<notin> bs \<and> a \<sharp> x)"
-  apply(simp add: fresh_def)
-  apply(simp add: supp_Abs)
-  apply(auto)
-  done
-
-lemma Abs_eq_iff:
-  shows "Abs bs x = Abs cs y \<longleftrightarrow> (\<exists>p. (bs, x) \<approx>gen (op =) supp p (cs, y))"
-  by (lifting alpha_abs.simps(1))
-
-
-
-(* 
-  below is a construction site for showing that in the
-  single-binder case, the old and new alpha equivalence 
-  coincide
-*)
-
-fun
-  alpha1
-where
-  "alpha1 (a, x) (b, y) \<longleftrightarrow> (a = b \<and> x = y) \<or> (a \<noteq> b \<and> x = (a \<rightleftharpoons> b) \<bullet> y \<and> a \<sharp> y)"
-
-notation 
-  alpha1 ("_ \<approx>abs1 _")
-
-fun
-  alpha2
-where
-  "alpha2 (a, x) (b, y) \<longleftrightarrow> (\<exists>c. c \<sharp> (a,b,x,y) \<and> ((c \<rightleftharpoons> a) \<bullet> x) = ((c \<rightleftharpoons> b) \<bullet> y))"
-
-notation 
-  alpha2 ("_ \<approx>abs2 _")
-
-lemma qq:
-  fixes S::"atom set"
-  assumes a: "supp p \<inter> S = {}"
-  shows "p \<bullet> S = S"
-using a
-apply(simp add: supp_perm permute_set_eq)
-apply(auto)
-apply(simp only: disjoint_iff_not_equal)
-apply(simp)
-apply (metis permute_atom_def_raw)
-apply(rule_tac x="(- p) \<bullet> x" in exI)
-apply(simp)
-apply(simp only: disjoint_iff_not_equal)
-apply(simp)
-apply(metis permute_minus_cancel)
-done
-
-lemma alpha_old_new:
-  assumes a: "(a, x) \<approx>abs1 (b, y)" "sort_of a = sort_of b"
-  shows "({a}, x) \<approx>abs ({b}, y)"
-using a
-apply(simp)
-apply(erule disjE)
-apply(simp)
-apply(rule exI)
-apply(rule alpha_gen_refl)
-apply(simp)
-apply(rule_tac x="(a \<rightleftharpoons> b)" in  exI)
-apply(simp add: alpha_gen)
-apply(simp add: fresh_def)
-apply(rule conjI)
-apply(rule_tac ?p1="(a \<rightleftharpoons> b)" in  permute_eq_iff[THEN iffD1])
-apply(rule trans)
-apply(simp add: Diff_eqvt supp_eqvt)
-apply(subst swap_set_not_in)
-back
-apply(simp)
-apply(simp)
-apply(simp add: permute_set_eq)
-apply(rule_tac ?p1="(a \<rightleftharpoons> b)" in fresh_star_permute_iff[THEN iffD1])
-apply(simp add: permute_self)
-apply(simp add: Diff_eqvt supp_eqvt)
-apply(simp add: permute_set_eq)
-apply(subgoal_tac "supp (a \<rightleftharpoons> b) \<subseteq> {a, b}")
-apply(simp add: fresh_star_def fresh_def)
-apply(blast)
-apply(simp add: supp_swap)
-done
-
-lemma perm_zero:
-  assumes a: "\<forall>x::atom. p \<bullet> x = x"
-  shows "p = 0"
-using a
-by (simp add: expand_perm_eq)
-
-fun
-  add_perm 
-where
-  "add_perm [] = 0"
-| "add_perm ((a, b) # xs) = (a \<rightleftharpoons> b) + add_perm xs"
-
-fun
-  elem_perm
-where
-  "elem_perm [] = {}"
-| "elem_perm ((a, b) # xs) = {a, b} \<union> elem_perm xs"
-
-
-lemma add_perm_apend:
-  shows "add_perm (xs @ ys) = add_perm xs + add_perm ys"
-apply(induct xs arbitrary: ys)
-apply(auto simp add: add_assoc)
-done
-
-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(drule perm_zero)
-apply(simp)
-apply(rule_tac x="[]" in exI)
-apply(simp add: supp_Nil)
-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_apend)
-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 left_minus minus_unique 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
-
-lemma tt0:
-  fixes p::perm
-  shows "(supp x) \<sharp>* p \<Longrightarrow> \<forall>a \<in> supp p. a \<sharp> x"
-apply(auto simp add: fresh_star_def supp_perm fresh_def)
-done
-
-lemma uu0:
-  shows "(\<forall>a \<in> elem_perm xs. a \<sharp> x) \<Longrightarrow> (add_perm xs \<bullet> x) = x"
-apply(induct xs rule: add_perm.induct)
-apply(simp)
-apply(simp add: swap_fresh_fresh)
-done
-
-lemma yy0:
-  fixes xs::"(atom \<times> atom) list"
-  shows "supp xs = elem_perm xs"
-apply(induct xs rule: elem_perm.induct)
-apply(auto simp add: supp_Nil supp_Cons supp_Pair supp_atom)
-done
-
-lemma tt1:
-  shows "(supp x) \<sharp>* p \<Longrightarrow> p \<bullet> x = x"
-apply(drule tt0)
-apply(subgoal_tac "\<exists>xs. p = add_perm xs \<and> supp xs \<subseteq> supp p")
-prefer 2
-apply(rule perm_list_exists)
-apply(erule exE)
-apply(simp only: yy0)
-apply(rule uu0)
-apply(auto)
-done
-
-
-lemma perm_induct_test:
-  fixes P :: "perm => bool"
-  assumes fin: "finite (supp p)" 
-  assumes zero: "P 0"
-  assumes swap: "\<And>a b. \<lbrakk>sort_of a = sort_of b; a \<noteq> b\<rbrakk> \<Longrightarrow> P (a \<rightleftharpoons> b)"
-  assumes plus: "\<And>p1 p2. \<lbrakk>supp p1 \<inter> supp p2 = {}; P p1; P p2\<rbrakk> \<Longrightarrow> P (p1 + p2)"
-  shows "P p"
-using fin
-apply(induct F\<equiv>"supp p" arbitrary: p rule: finite_induct)
-apply(simp add: supp_perm)
-apply(drule perm_zero)
-apply(simp add: zero)
-apply(rotate_tac 3)
-oops
-lemma tt:
-  "(supp x) \<sharp>* p \<Longrightarrow> p \<bullet> x = x"
-oops
-
-lemma yy:
-  assumes "S1 - {x} = S2 - {x}" "x \<in> S1" "x \<in> S2"
-  shows "S1 = S2"
-using assms
-apply (metis insert_Diff_single insert_absorb)
-done
-
-lemma permute_boolI:
-  fixes P::"bool"
-  shows "p \<bullet> P \<Longrightarrow> P"
-apply(simp add: permute_bool_def)
-done
-
-lemma permute_boolE:
-  fixes P::"bool"
-  shows "P \<Longrightarrow> p \<bullet> P"
-apply(simp add: permute_bool_def)
-done
-
-lemma kk:
-  assumes a: "p \<bullet> x = y"
-  shows "\<forall>a \<in> supp x. (p \<bullet> a) \<in> supp y"
-using a
-apply(auto)
-apply(rule_tac p="- p" in permute_boolI)
-apply(simp add: mem_eqvt supp_eqvt)
-done
-
-lemma ww:
-  assumes "a \<notin> supp x" "b \<in> supp x" "a \<noteq> b" "sort_of a = sort_of b"
-  shows "((a \<rightleftharpoons> b) \<bullet> x) \<noteq> x"
-apply(subgoal_tac "(supp x) supports x")
-apply(simp add: supports_def)
-using assms
-apply -
-apply(drule_tac x="a" in spec)
-defer
-apply(rule supp_supports)
-apply(auto)
-apply(rotate_tac 1)
-apply(drule_tac p="(a \<rightleftharpoons> b)" in permute_boolE)
-apply(simp add: mem_eqvt supp_eqvt)
-done
-
-lemma alpha_abs_sym:
-  assumes a: "({a}, x) \<approx>abs ({b}, y)"
-  shows "({b}, y) \<approx>abs ({a}, x)"
-using a
-apply(simp)
-apply(erule exE)
-apply(rule_tac x="- p" in exI)
-apply(simp add: alpha_gen)
-apply(simp add: fresh_star_def fresh_minus_perm)
-apply (metis permute_minus_cancel(2))
-done
-
-lemma alpha_abs_trans:
-  assumes a: "({a1}, x1) \<approx>abs ({a2}, x2)"
-  assumes b: "({a2}, x2) \<approx>abs ({a3}, x3)"
-  shows "({a1}, x1) \<approx>abs ({a3}, x3)"
-using a b
-apply(simp)
-apply(erule exE)+
-apply(rule_tac x="pa + p" in exI)
-apply(simp add: alpha_gen)
-apply(simp add: fresh_star_def fresh_plus_perm)
-done
-
-lemma alpha_equal:
-  assumes a: "({a}, x) \<approx>abs ({a}, y)" 
-  shows "(a, x) \<approx>abs1 (a, y)"
-using a
-apply(simp)
-apply(erule exE)
-apply(simp add: alpha_gen)
-apply(erule conjE)+
-apply(case_tac "a \<notin> supp x")
-apply(simp)
-apply(subgoal_tac "supp x \<sharp>* p")
-apply(drule tt1)
-apply(simp)
-apply(simp)
-apply(simp)
-apply(case_tac "a \<notin> supp y")
-apply(simp)
-apply(drule tt1)
-apply(clarify)
-apply(simp (no_asm_use))
-apply(simp)
-apply(simp)
-apply(drule yy)
-apply(simp)
-apply(simp)
-apply(simp)
-apply(case_tac "a \<sharp> p")
-apply(subgoal_tac "supp y \<sharp>* p")
-apply(drule tt1)
-apply(clarify)
-apply(simp (no_asm_use))
-apply(metis)
-apply(auto simp add: fresh_star_def)[1]
-apply(frule_tac kk)
-apply(drule_tac x="a" in bspec)
-apply(simp)
-apply(simp add: fresh_def)
-apply(simp add: supp_perm)
-apply(subgoal_tac "((p \<bullet> a) \<sharp> p)")
-apply(simp add: fresh_def supp_perm)
-apply(simp add: fresh_star_def)
-done
-
-lemma alpha_unequal:
-  assumes a: "({a}, x) \<approx>abs ({b}, y)" "sort_of a = sort_of b" "a \<noteq> b"
-  shows "(a, x) \<approx>abs1 (b, y)"
-using a
-apply -
-apply(subgoal_tac "a \<notin> supp x - {a}")
-apply(subgoal_tac "b \<notin> supp x - {a}")
-defer
-apply(simp add: alpha_gen)
-apply(simp)
-apply(drule_tac alpha_abs_swap)
-apply(assumption)
-apply(simp only: insert_eqvt empty_eqvt swap_atom_simps)
-apply(drule alpha_abs_sym)
-apply(rotate_tac 4)
-apply(drule_tac alpha_abs_trans)
-apply(assumption)
-apply(drule alpha_equal)
-apply(simp)
-apply(rule_tac p="(a \<rightleftharpoons> b)" in permute_boolI)
-apply(simp add: fresh_eqvt)
-apply(simp add: fresh_def)
-done
-
-lemma alpha_new_old:
-  assumes a: "({a}, x) \<approx>abs ({b}, y)" "sort_of a = sort_of b" 
-  shows "(a, x) \<approx>abs1 (b, y)"
-using a
-apply(case_tac "a=b")
-apply(simp only: alpha_equal)
-apply(drule alpha_unequal)
-apply(simp)
-apply(simp)
-apply(simp)
-done
-
-fun
-  distinct_perms 
-where
-  "distinct_perms [] = True"
-| "distinct_perms (p # ps) = (supp p \<inter> supp ps = {} \<and> distinct_perms ps)"
-
-(* support of concrete atom sets *)
-
-lemma atom_eqvt_raw:
-  fixes p::"perm"
-  shows "(p \<bullet> atom) = atom"
-by (simp add: expand_fun_eq permute_fun_def atom_eqvt)
-
-lemma atom_image_cong:
-  shows "(atom ` X = atom ` Y) = (X = Y)"
-apply(rule inj_image_eq_iff)
-apply(simp add: inj_on_def)
-done
-
-lemma supp_atom_image:
-  fixes as::"'a::at_base set"
-  shows "supp (atom ` as) = supp as"
-apply(simp add: supp_def)
-apply(simp add: image_eqvt)
-apply(simp add: atom_eqvt_raw)
-apply(simp add: atom_image_cong)
-done
-
-lemma swap_atom_image_fresh: "\<lbrakk>a \<sharp> atom ` (fn :: ('a :: at_base set)); b \<sharp> atom ` fn\<rbrakk> \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> fn = fn"
-  apply (simp add: fresh_def)
-  apply (simp add: supp_atom_image)
-  apply (fold fresh_def)
-  apply (simp add: swap_fresh_fresh)
-  done
-
-
-end
-

--- a/Nominal/Nominal2_Supp.thy	Sat Mar 20 10:12:09 2010 +0100
+++ b/Nominal/Nominal2_Supp.thy	Sat Mar 20 18:16:26 2010 +0100
@@ -372,23 +372,115 @@
 done
 
 
-(*
-lemma supp_infinite:
-  fixes S::"atom set"
-  assumes asm: "finite (UNIV - S)"
-  shows "(supp S) = (UNIV - S)"
-apply(rule finite_supp_unique)
-apply(auto simp add: supports_def permute_set_eq swap_atom)[1]
-apply(rule asm)
-apply(auto simp add: permute_set_eq swap_atom)[1]
+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 left_minus minus_unique 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
 
-lemma supp_infinite_coinfinite:
-  fixes S::"atom set"
-  assumes asm1: "infinite S"
-  and     asm2: "infinite (UNIV-S)"
-  shows "(supp S) = (UNIV::atom set)"
-*)
+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
 
 end
\ No newline at end of file