--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Nominal/Abs.thy Mon Mar 15 08:39:23 2010 +0100
@@ -0,0 +1,713 @@
+theory Abs
+imports "Nominal2_Atoms" "Nominal2_Eqvt" "Nominal2_Supp" "../Quotient" "../Quotient_Product"
+begin
+
+(* the next three lemmas that should be in Nominal \<dots>\<dots>must be cleaned *)
+
+
+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 \<and> pi \<bullet> bs = cs"
+
+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 (auto 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(clarify)
+ 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 5)
+ apply(drule_tac pi="- pia" in a)
+ apply(simp)
+ apply(rotate_tac 7)
+ 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)
+ oops
+
+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)
+ unfolding alpha_gen
+ apply(simp)
+ 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
+ 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(simp add: eqvts)
+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 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
+