diff -r 1dd314a00b0c -r 4239a0784e5f Quot/Nominal/LamEx2.thy
--- a/Quot/Nominal/LamEx2.thy	Mon Feb 01 15:57:37 2010 +0100
+++ b/Quot/Nominal/LamEx2.thy	Mon Feb 01 16:05:59 2010 +0100
@@ -1,7 +1,58 @@
 theory LamEx
-imports Nominal "../QuotMain" "../QuotList"
+imports "Nominal2_Atoms" "Nominal2_Eqvt" "Nominal2_Supp" "../QuotMain" "Abs" "../QuotProd"
 begin
 
+
+(* lemmas that should be in Nominal \<dots>\<dots>must be cleaned *)
+lemma in_permute_iff:
+  shows "(p \<bullet> x) \<in> (p \<bullet> X) \<longleftrightarrow> x \<in> X"
+apply(unfold mem_def permute_fun_def)[1]
+apply(simp add: permute_bool_def) 
+done
+
+lemma fresh_star_permute_iff:
+  shows "(p \<bullet> a) \<sharp>* (p \<bullet> x) \<longleftrightarrow> a \<sharp>* x"
+apply(simp add: fresh_star_def)
+apply(auto)
+apply(drule_tac x="p \<bullet> xa" in bspec)
+apply(unfold mem_def permute_fun_def)[1] 
+apply(simp add: eqvts)
+apply(simp add: fresh_permute_iff)
+apply(rule_tac ?p1="- p" in fresh_permute_iff[THEN iffD1])
+apply(simp)
+apply(drule_tac x="- p \<bullet> xa" in bspec)
+apply(rule_tac ?p1="p" in in_permute_iff[THEN iffD1])
+apply(simp)
+apply(simp)
+done
+
+lemma fresh_plus:
+  fixes p q::perm
+  shows "\<lbrakk>a \<sharp> p;  a \<sharp> q\<rbrakk> \<Longrightarrow> a \<sharp> (p + q)"
+unfolding fresh_def
+using supp_plus_perm
+apply(auto)
+done
+
+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)
+
+lemma supp_finite_set:
+  fixes S::"atom set"
+  assumes "finite S"
+  shows "supp S = S"
+  apply(rule finite_supp_unique)
+  apply(simp add: supports_def)
+  apply(auto simp add: permute_set_eq swap_atom)[1]
+  apply(metis)
+  apply(rule assms)
+  apply(auto simp add: permute_set_eq swap_atom)[1]
+done
+  
+
 atom_decl name
 
 datatype rlam =
@@ -10,57 +61,84 @@
 | rLam "name" "rlam"
 
 fun
-  rfv :: "rlam \<Rightarrow> name set"
+  rfv :: "rlam \<Rightarrow> atom set"
 where
-  rfv_var: "rfv (rVar a) = {a}"
+  rfv_var: "rfv (rVar a) = {atom a}"
 | rfv_app: "rfv (rApp t1 t2) = (rfv t1) \<union> (rfv t2)"
-| rfv_lam: "rfv (rLam a t) = (rfv t) - {a}"
+| rfv_lam: "rfv (rLam a t) = (rfv t) - {atom a}"
 
-overloading
-  perm_rlam \<equiv> "perm :: 'x prm \<Rightarrow> rlam \<Rightarrow> rlam"   (unchecked)
+instantiation rlam :: pt
 begin
 
-fun
-  perm_rlam
+primrec
+  permute_rlam
 where
-  "perm_rlam pi (rVar a) = rVar (pi \<bullet> a)"
-| "perm_rlam pi (rApp t1 t2) = rApp (perm_rlam pi t1) (perm_rlam pi t2)"
-| "perm_rlam pi (rLam a t) = rLam (pi \<bullet> a) (perm_rlam pi t)"
+  "permute_rlam pi (rVar a) = rVar (pi \<bullet> a)"
+| "permute_rlam pi (rApp t1 t2) = rApp (permute_rlam pi t1) (permute_rlam pi t2)"
+| "permute_rlam pi (rLam a t) = rLam (pi \<bullet> a) (permute_rlam pi t)"
+
+instance
+apply default
+apply(induct_tac [!] x)
+apply(simp_all)
+done
 
 end
 
-declare perm_rlam.simps[eqvt]
+instantiation rlam :: fs
+begin
 
-instance rlam::pt_name
-  apply(default)
-  apply(induct_tac [!] x rule: rlam.induct)
-  apply(simp_all add: pt_name2 pt_name3)
-  done
+lemma neg_conj:
+  "\<not>(P \<and> Q) \<longleftrightarrow> (\<not>P) \<or> (\<not>Q)"
+  by simp
+
+lemma infinite_Un:
+  "infinite (S \<union> T) \<longleftrightarrow> infinite S \<or> infinite T"
+  by simp
 
-instance rlam::fs_name
-  apply(default)
-  apply(induct_tac [!] x rule: rlam.induct)
-  apply(simp add: supp_def)
-  apply(fold supp_def)
-  apply(simp add: supp_atm)
-  apply(simp add: supp_def Collect_imp_eq Collect_neg_eq)
-  apply(simp add: supp_def)
-  apply(simp add: supp_def Collect_imp_eq Collect_neg_eq[symmetric])
-  apply(fold supp_def)
-  apply(simp add: supp_atm)
-  done
+instance
+apply default
+apply(induct_tac x)
+(* var case *)
+apply(simp add: supp_def)
+apply(fold supp_def)[1]
+apply(simp add: supp_at_base)
+(* app case *)
+apply(simp only: supp_def)
+apply(simp only: permute_rlam.simps) 
+apply(simp only: rlam.inject) 
+apply(simp only: neg_conj)
+apply(simp only: Collect_disj_eq)
+apply(simp only: infinite_Un)
+apply(simp only: Collect_disj_eq)
+apply(simp)
+(* lam case *)
+apply(simp only: supp_def)
+apply(simp only: permute_rlam.simps) 
+apply(simp only: rlam.inject) 
+apply(simp only: neg_conj)
+apply(simp only: Collect_disj_eq)
+apply(simp only: infinite_Un)
+apply(simp only: Collect_disj_eq)
+apply(simp)
+apply(fold supp_def)[1]
+apply(simp add: supp_at_base)
+done
 
-declare set_diff_eqvt[eqvt]
+end
+
+
+(* for the eqvt proof of the alpha-equivalence *)
+declare permute_rlam.simps[eqvt]
 
 lemma rfv_eqvt[eqvt]:
-  fixes pi::"name prm"
   shows "(pi\<bullet>rfv t) = rfv (pi\<bullet>t)"
 apply(induct t)
 apply(simp_all)
-apply(simp add: perm_set_eq)
+apply(simp add: permute_set_eq atom_eqvt)
 apply(simp add: union_eqvt)
-apply(simp add: set_diff_eqvt)
-apply(simp add: perm_set_eq)
+apply(simp add: Diff_eqvt)
+apply(simp add: permute_set_eq atom_eqvt)
 done
 
 inductive
@@ -68,35 +146,37 @@
 where
   a1: "a = b \<Longrightarrow> (rVar a) \<approx> (rVar b)"
 | a2: "\<lbrakk>t1 \<approx> t2; s1 \<approx> s2\<rbrakk> \<Longrightarrow> rApp t1 s1 \<approx> rApp t2 s2"
-| a3: "\<exists>pi::name prm. (rfv t - {a} = rfv s - {b} \<and> (rfv t - {a})\<sharp>* pi \<and> (pi \<bullet> t) \<approx> s \<and> (pi \<bullet> a) = b)
-       \<Longrightarrow> rLam a t \<approx> rLam b s"
+| a3: "\<exists>pi. (({atom a}, t) \<approx>gen alpha rfv pi ({atom b}, s)) \<Longrightarrow> rLam a t \<approx> rLam b s"
 
-
+thm alpha.induct
 
+lemma a3_inverse:
+  assumes "rLam a t \<approx> rLam b s"
+  shows "\<exists>pi. (({atom a}, t) \<approx>gen alpha rfv pi ({atom b}, s))"
+using assms
+apply(erule_tac alpha.cases)
+apply(auto)
+done
 
-(* should be automatic with new version of eqvt-machinery *)
+text {* should be automatic with new version of eqvt-machinery *}
 lemma alpha_eqvt:
-  fixes pi::"name prm"
   shows "t \<approx> s \<Longrightarrow> (pi \<bullet> t) \<approx> (pi \<bullet> s)"
 apply(induct rule: alpha.induct)
 apply(simp add: a1)
 apply(simp add: a2)
 apply(simp)
 apply(rule a3)
-apply(erule conjE)
 apply(erule exE)
-apply(erule conjE)
 apply(rule_tac x="pi \<bullet> pia" in exI)
+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: eqvts atom_eqvt)
 apply(rule conjI)
-apply(rule_tac pi1="rev pi" in perm_bij[THEN iffD1])
-apply(perm_simp add: eqvts)
-apply(rule conjI)
-apply(rule_tac pi1="rev pi" in pt_fresh_star_bij(1)[OF pt_name_inst at_name_inst, THEN iffD1])
-apply(perm_simp add: eqvts)
-apply(rule conjI)
-apply(subst perm_compose[symmetric])
-apply(simp)
-apply(subst perm_compose[symmetric])
+apply(rule_tac ?p1="- pi" in fresh_star_permute_iff[THEN iffD1])
+apply(simp add: eqvts atom_eqvt)
+apply(subst permute_eqvt[symmetric])
 apply(simp)
 done
 
@@ -106,28 +186,44 @@
 apply(simp add: a1)
 apply(simp add: a2)
 apply(rule a3)
-apply(rule_tac x="[]" in exI)
-apply(simp_all add: fresh_star_def fresh_list_nil)
+apply(rule_tac x="0" in exI)
+apply(rule alpha_gen_refl)
+apply(assumption)
 done
 
+lemma fresh_minus_perm:
+  fixes p::perm
+  shows "a \<sharp> (- p) \<longleftrightarrow> a \<sharp> p"
+  apply(simp add: fresh_def)
+  apply(simp only: supp_minus_perm)
+  done
+
+lemma alpha_gen_atom_sym:
+  assumes a: "\<And>pi t s. (R t s \<Longrightarrow> R (pi \<bullet> t) (pi \<bullet> s))"
+  shows "\<exists>pi. ({atom a}, t) \<approx>gen \<lambda>x1 x2. R x1 x2 \<and> R x2 x1 f pi ({atom b}, s) \<Longrightarrow>
+       \<exists>pi. ({atom b}, s) \<approx>gen R f pi ({atom a}, t)"
+  apply(erule exE)
+  apply(rule_tac x="- pi" in exI)
+  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_sym:
   shows "t \<approx> s \<Longrightarrow> s \<approx> t"
-apply(induct rule: alpha.induct)
-apply(simp add: a1)
-apply(simp add: a2)
-apply(rule a3)
-apply(erule exE)
-apply(rule_tac x="rev pi" in exI)
-apply(simp)
-apply(simp add: fresh_star_def fresh_list_rev)
-apply(rule conjI)
-apply(erule conjE)+
-apply(rotate_tac 3)
-apply(drule_tac pi="rev pi" in alpha_eqvt)
-apply(perm_simp)
-apply(rule pt_bij2[OF pt_name_inst at_name_inst])
-apply(simp)
-done
+  apply(induct rule: alpha.induct)
+  apply(simp add: a1)
+  apply(simp add: a2)
+  apply(rule a3)
+  apply(rule alpha_gen_atom_sym)
+  apply(rule alpha_eqvt)
+  apply(assumption)+
+  done
 
 lemma alpha_trans:
   shows "t1 \<approx> t2 \<Longrightarrow> t2 \<approx> t3 \<Longrightarrow> t1 \<approx> t3"
@@ -142,42 +238,40 @@
 apply(rotate_tac 1)
 apply(erule alpha.cases)
 apply(simp_all)
+apply(simp add: alpha_gen.simps)
 apply(erule conjE)+
 apply(erule exE)+
 apply(erule conjE)+
 apply(rule a3)
-apply(rule_tac x="pia @ pi" in exI)
-apply(simp add: fresh_star_def fresh_list_append)
-apply(simp add: pt_name2)
-apply(drule_tac x="rev pia \<bullet> sa" in spec)
+apply(rule_tac x="pia + pi" in exI)
+apply(simp add: alpha_gen.simps)
+apply(simp add: fresh_star_plus)
+apply(drule_tac x="- pia \<bullet> sa" in spec)
 apply(drule mp)
-apply(rotate_tac 8)
-apply(drule_tac pi="rev pia" in alpha_eqvt)
-apply(perm_simp)
-apply(rotate_tac 11)
+apply(rotate_tac 7)
+apply(drule_tac pi="- pia" in alpha_eqvt)
+apply(simp)
+apply(rotate_tac 9)
 apply(drule_tac pi="pia" in alpha_eqvt)
-apply(perm_simp)
+apply(simp)
 done
 
 lemma alpha_equivp:
   shows "equivp alpha"
-apply(rule equivpI)
-unfolding reflp_def symp_def transp_def
-apply(auto intro: alpha_refl alpha_sym alpha_trans)
-done
+  apply(rule equivpI)
+  unfolding reflp_def symp_def transp_def
+  apply(auto intro: alpha_refl alpha_sym alpha_trans)
+  done
 
 lemma alpha_rfv:
   shows "t \<approx> s \<Longrightarrow> rfv t = rfv s"
-apply(induct rule: alpha.induct)
-apply(simp)
-apply(simp)
-apply(simp)
-done
+  apply(induct rule: alpha.induct)
+  apply(simp_all add: alpha_gen.simps)
+  done
 
 quotient_type lam = rlam / alpha
   by (rule alpha_equivp)
 
-
 quotient_definition
   "Var :: name \<Rightarrow> lam"
 as
@@ -194,48 +288,32 @@
   "rLam"
 
 quotient_definition
-  "fv :: lam \<Rightarrow> name set"
+  "fv :: lam \<Rightarrow> atom set"
 as
   "rfv"
 
-(* definition of overloaded permutation function *)
-(* for the lifted type lam                       *)
-overloading
-  perm_lam \<equiv> "perm :: 'x prm \<Rightarrow> lam \<Rightarrow> lam"   (unchecked)
-begin
-
-quotient_definition
-  "perm_lam :: 'x prm \<Rightarrow> lam \<Rightarrow> lam"
-as
-  "perm::'x prm \<Rightarrow> rlam \<Rightarrow> rlam"
-
-end
-
 lemma perm_rsp[quot_respect]:
-  "(op = ===> alpha ===> alpha) op \<bullet> op \<bullet>"
+  "(op = ===> alpha ===> alpha) permute permute"
   apply(auto)
-  (* this is propably true if some type conditions are imposed ;o) *)
-  sorry
-
-lemma fresh_rsp:
-  "(op = ===> alpha ===> op =) fresh fresh"
-  apply(auto)
-  (* this is probably only true if some type conditions are imposed *)
-  sorry
+  apply(rule alpha_eqvt)
+  apply(simp)
+  done
 
 lemma rVar_rsp[quot_respect]:
   "(op = ===> alpha) rVar rVar"
   by (auto intro: a1)
 
-lemma rApp_rsp[quot_respect]: "(alpha ===> alpha ===> alpha) rApp rApp"
+lemma rApp_rsp[quot_respect]: 
+  "(alpha ===> alpha ===> alpha) rApp rApp"
   by (auto intro: a2)
 
-lemma rLam_rsp[quot_respect]: "(op = ===> alpha ===> alpha) rLam rLam"
+lemma rLam_rsp[quot_respect]: 
+  "(op = ===> alpha ===> alpha) rLam rLam"
   apply(auto)
   apply(rule a3)
-  apply(rule_tac x="[]" in exI)
-  unfolding fresh_star_def
-  apply(simp add: fresh_list_nil)
+  apply(rule_tac x="0" in exI)
+  unfolding fresh_star_def 
+  apply(simp add: fresh_star_def fresh_zero_perm alpha_gen.simps)
   apply(simp add: alpha_rfv)
   done
 
@@ -244,6 +322,7 @@
 apply(simp add: alpha_rfv)
 done
 
+
 section {* lifted theorems *}
 
 lemma lam_induct:
@@ -251,29 +330,43 @@
     \<And>lam1 lam2. \<lbrakk>P lam1; P lam2\<rbrakk> \<Longrightarrow> P (App lam1 lam2);
     \<And>name lam. P lam \<Longrightarrow> P (Lam name lam)\<rbrakk> 
     \<Longrightarrow> P lam"
-  by (lifting rlam.induct)
+  apply (lifting rlam.induct)
+  done
+
+instantiation lam :: pt
+begin
 
-lemma perm_lam [simp]:
-  fixes pi::"'a prm"
+quotient_definition
+  "permute_lam :: perm \<Rightarrow> lam \<Rightarrow> lam"
+as
+  "permute :: perm \<Rightarrow> rlam \<Rightarrow> rlam"
+
+lemma permute_lam [simp]:
   shows "pi \<bullet> Var a = Var (pi \<bullet> a)"
   and   "pi \<bullet> App t1 t2 = App (pi \<bullet> t1) (pi \<bullet> t2)"
   and   "pi \<bullet> Lam a t = Lam (pi \<bullet> a) (pi \<bullet> t)"
-apply(lifting perm_rlam.simps)
+apply(lifting permute_rlam.simps)
 done
 
-instance lam::pt_name
-apply(default)
+instance
+apply default
 apply(induct_tac [!] x rule: lam_induct)
-apply(simp_all add: pt_name2 pt_name3)
+apply(simp_all)
 done
 
+end
+
 lemma fv_lam [simp]: 
-  shows "fv (Var a) = {a}"
+  shows "fv (Var a) = {atom a}"
   and   "fv (App t1 t2) = fv t1 \<union> fv t2"
-  and   "fv (Lam a t) = fv t - {a}"
+  and   "fv (Lam a t) = fv t - {atom a}"
 apply(lifting rfv_var rfv_app rfv_lam)
 done
 
+lemma fv_eqvt:
+  shows "(p \<bullet> fv t) = fv (p \<bullet> t)"
+apply(lifting rfv_eqvt)
+done
 
 lemma a1: 
   "a = b \<Longrightarrow> Var a = Var b"
@@ -283,32 +376,97 @@
   "\<lbrakk>x = xa; xb = xc\<rbrakk> \<Longrightarrow> App x xb = App xa xc"
   by  (lifting a2)
 
-lemma a3: 
-  "\<lbrakk>\<exists>pi::name prm. (fv t - {a} = fv s - {b} \<and> (fv t - {a})\<sharp>* pi \<and> (pi \<bullet> t) = s \<and> (pi \<bullet> a) = b)\<rbrakk> 
-   \<Longrightarrow> Lam a t = Lam b s"
-  by  (lifting a3)
+lemma alpha_gen_rsp_pre:
+  assumes a5: "\<And>t s. R t s \<Longrightarrow> R (pi \<bullet> t) (pi \<bullet> s)"
+  and     a1: "R s1 t1"
+  and     a2: "R s2 t2"
+  and     a3: "\<And>a b c d. R a b \<Longrightarrow> R c d \<Longrightarrow> R1 a c = R2 b d"
+  and     a4: "\<And>x y. R x y \<Longrightarrow> fv1 x = fv2 y"
+  shows   "(a, s1) \<approx>gen R1 fv1 pi (b, s2) = (a, t1) \<approx>gen R2 fv2 pi (b, t2)"
+apply (simp add: alpha_gen.simps)
+apply (simp only: a4[symmetric, OF a1] a4[symmetric, OF a2])
+apply auto
+apply (subst a3[symmetric])
+apply (rule a5)
+apply (rule a1)
+apply (rule a2)
+apply (assumption)
+apply (subst a3)
+apply (rule a5)
+apply (rule a1)
+apply (rule a2)
+apply (assumption)
+done
+
+lemma [quot_respect]: "(prod_rel op = alpha ===>
+           (alpha ===> alpha ===> op =) ===> (alpha ===> op =) ===> op = ===> prod_rel op = alpha ===> op =)
+           alpha_gen alpha_gen"
+apply simp
+apply clarify
+apply (rule alpha_gen_rsp_pre[of "alpha",OF alpha_eqvt])
+apply auto
+done
+
+lemma pi_rep: "pi \<bullet> (rep_lam x) = rep_lam (pi \<bullet> x)"
+apply (unfold rep_lam_def)
+sorry
+
+lemma [quot_preserve]: "(prod_fun id rep_lam --->
+           (abs_lam ---> abs_lam ---> id) ---> (abs_lam ---> id) ---> id ---> (prod_fun id rep_lam) ---> id)
+           alpha_gen = alpha_gen"
+apply (simp add: expand_fun_eq)
+apply (simp add: alpha_gen.simps)
+apply (simp add: pi_rep)
+apply (simp only: Quotient_abs_rep[OF Quotient_lam])
+apply auto
+done
+
+lemma alpha_prs [quot_preserve]: "(rep_lam ---> rep_lam ---> id) alpha = (op =)"
+apply (simp add: expand_fun_eq)
+sorry
+
+
+lemma a3:
+  "\<exists>pi. ({atom a}, t) \<approx>gen (op =) fv pi ({atom b}, s) \<Longrightarrow> Lam a t = Lam b s"
+  apply (lifting a3)
+  done
+
+lemma a3_inv:
+  assumes "Lam a t = Lam b s"
+  shows "\<exists>pi. (fv t - {atom a} = fv s - {atom b} \<and> (fv t - {atom a})\<sharp>* pi \<and> (pi \<bullet> t) = s)"
+using assms
+apply(lifting a3_inverse)
+done
 
 lemma alpha_cases: 
   "\<lbrakk>a1 = a2; \<And>a b. \<lbrakk>a1 = Var a; a2 = Var b; a = b\<rbrakk> \<Longrightarrow> P;
     \<And>x xa xb xc. \<lbrakk>a1 = App x xb; a2 = App xa xc; x = xa; xb = xc\<rbrakk> \<Longrightarrow> P;
     \<And>t a s b. \<lbrakk>a1 = Lam a t; a2 = Lam b s; 
-         \<exists>pi::name prm. fv t - {a} = fv s - {b} \<and> (fv t - {a}) \<sharp>* pi \<and> (pi \<bullet> t) = s \<and> pi \<bullet> a = b\<rbrakk> \<Longrightarrow> P\<rbrakk>
+         \<exists>pi. fv t - {atom a} = fv s - {atom b} \<and> (fv t - {atom a}) \<sharp>* pi \<and> (pi \<bullet> t) = s\<rbrakk> 
+   \<Longrightarrow> P\<rbrakk>
     \<Longrightarrow> P"
   by (lifting alpha.cases)
 
+(* not sure whether needed *)
 lemma alpha_induct: 
   "\<lbrakk>qx = qxa; \<And>a b. a = b \<Longrightarrow> qxb (Var a) (Var b);
     \<And>x xa xb xc. \<lbrakk>x = xa; qxb x xa; xb = xc; qxb xb xc\<rbrakk> \<Longrightarrow> qxb (App x xb) (App xa xc);
      \<And>t a s b.
-        \<lbrakk>\<exists>pi::name prm. fv t - {a} = fv s - {b} \<and>
-         (fv t - {a}) \<sharp>* pi \<and> ((pi \<bullet> t) = s \<and> qxb (pi \<bullet> t) s) \<and> pi \<bullet> a = b\<rbrakk> \<Longrightarrow> qxb (Lam a t) (Lam b s)\<rbrakk>
+        \<lbrakk>\<exists>pi. fv t - {atom a} = fv s - {atom b} \<and>
+         (fv t - {atom a}) \<sharp>* pi \<and> ((pi \<bullet> t) = s \<and> qxb (pi \<bullet> t) s)\<rbrakk> 
+     \<Longrightarrow> qxb (Lam a t) (Lam b s)\<rbrakk>
     \<Longrightarrow> qxb qx qxa"
   by (lifting alpha.induct)
 
+(* should they lift automatically *)
 lemma lam_inject [simp]: 
   shows "(Var a = Var b) = (a = b)"
   and   "(App t1 t2 = App s1 s2) = (t1 = s1 \<and> t2 = s2)"
 apply(lifting rlam.inject(1) rlam.inject(2))
+apply(regularize)
+prefer 2
+apply(regularize)
+prefer 2
 apply(auto)
 apply(drule alpha.cases)
 apply(simp_all)
@@ -321,6 +479,16 @@
 apply(simp_all)
 done
 
+lemma Lam_pseudo_inject:
+  shows "(Lam a t = Lam b s) = 
+      (\<exists>pi. (fv t - {atom a} = fv s - {atom b} \<and> (fv t - {atom a})\<sharp>* pi \<and> (pi \<bullet> t) = s))"
+apply(rule iffI)
+apply(rule a3_inv)
+apply(assumption)
+apply(rule a3)
+apply(assumption)
+done
+
 lemma rlam_distinct:
   shows "\<not>(rVar nam \<approx> rApp rlam1' rlam2')"
   and   "\<not>(rApp rlam1' rlam2' \<approx> rVar nam)"
@@ -329,18 +497,18 @@
   and   "\<not>(rApp rlam1 rlam2 \<approx> rLam nam' rlam')"
   and   "\<not>(rLam nam' rlam' \<approx> rApp rlam1 rlam2)"
 apply auto
-apply(erule alpha.cases)
-apply simp_all
-apply(erule alpha.cases)
-apply simp_all
-apply(erule alpha.cases)
-apply simp_all
-apply(erule alpha.cases)
-apply simp_all
-apply(erule alpha.cases)
-apply simp_all
-apply(erule alpha.cases)
-apply simp_all
+apply (erule alpha.cases)
+apply (simp_all only: rlam.distinct)
+apply (erule alpha.cases)
+apply (simp_all only: rlam.distinct)
+apply (erule alpha.cases)
+apply (simp_all only: rlam.distinct)
+apply (erule alpha.cases)
+apply (simp_all only: rlam.distinct)
+apply (erule alpha.cases)
+apply (simp_all only: rlam.distinct)
+apply (erule alpha.cases)
+apply (simp_all only: rlam.distinct)
 done
 
 lemma lam_distinct[simp]:
@@ -354,49 +522,88 @@
 done
 
 lemma var_supp1:
-  shows "(supp (Var a)) = ((supp a)::name set)"
-  by (simp add: supp_def)
+  shows "(supp (Var a)) = (supp a)"
+  apply (simp add: supp_def)
+  done
 
 lemma var_supp:
-  shows "(supp (Var a)) = {a::name}"
-  using var_supp1 by (simp add: supp_atm)
+  shows "(supp (Var a)) = {a:::name}"
+  using var_supp1 by (simp add: supp_at_base)
 
 lemma app_supp:
-  shows "supp (App t1 t2) = (supp t1) \<union> ((supp t2)::name set)"
-apply(simp only: perm_lam supp_def lam_inject)
+  shows "supp (App t1 t2) = (supp t1) \<union> (supp t2)"
+apply(simp only: supp_def lam_inject)
 apply(simp add: Collect_imp_eq Collect_neg_eq)
 done
 
-lemma lam_supp:
-  shows "supp (Lam x t) = ((supp ([x].t))::name set)"
-apply(simp add: supp_def)
-apply(simp add: abs_perm)
-sorry
+(* supp for lam *)
+lemma lam_supp1:
+  shows "(supp (atom x, t)) supports (Lam x t) "
+apply(simp add: supports_def)
+apply(fold fresh_def)
+apply(simp add: fresh_Pair swap_fresh_fresh)
+apply(clarify)
+apply(subst swap_at_base_simps(3))
+apply(simp_all add: fresh_atom)
+done
 
+lemma lam_fsupp1:
+  assumes a: "finite (supp t)"
+  shows "finite (supp (Lam x t))"
+apply(rule supports_finite)
+apply(rule lam_supp1)
+apply(simp add: a supp_Pair supp_atom)
+done
 
-instance lam::fs_name
+instance lam :: fs
 apply(default)
 apply(induct_tac x rule: lam_induct)
 apply(simp add: var_supp)
 apply(simp add: app_supp)
-apply(simp add: lam_supp abs_supp)
+apply(simp add: lam_fsupp1)
+done
+
+lemma supp_fv:
+  shows "supp t = fv t"
+apply(induct t rule: lam_induct)
+apply(simp add: var_supp)
+apply(simp add: app_supp)
+apply(subgoal_tac "supp (Lam name lam) = supp (Abst {atom name} lam)")
+apply(simp add: supp_Abst)
+apply(simp (no_asm) add: supp_def permute_set_eq atom_eqvt)
+apply(simp add: Lam_pseudo_inject)
+apply(simp add: abs_eq alpha_gen)
+apply(simp add: supp_eqvt[symmetric] fv_eqvt[symmetric])
+done
+
+lemma lam_supp2:
+  shows "supp (Lam x t) = supp (Abst {atom x} t)"
+apply(simp add: supp_def permute_set_eq atom_eqvt)
+apply(simp add: Lam_pseudo_inject)
+apply(simp add: abs_eq supp_fv alpha_gen)
+done
+
+lemma lam_supp:
+  shows "supp (Lam x t) = ((supp t) - {atom x})"
+apply(simp add: lam_supp2)
+apply(simp add: supp_Abst)
 done
 
 lemma fresh_lam:
-  "(a \<sharp> Lam b t) \<longleftrightarrow> (a = b) \<or> (a \<noteq> b \<and> a \<sharp> t)"
+  "(atom a \<sharp> Lam b t) \<longleftrightarrow> (a = b) \<or> (a \<noteq> b \<and> atom a \<sharp> t)"
 apply(simp add: fresh_def)
-apply(simp add: lam_supp abs_supp)
+apply(simp add: lam_supp)
 apply(auto)
 done
 
 lemma lam_induct_strong:
-  fixes a::"'a::fs_name"
+  fixes a::"'a::fs"
   assumes a1: "\<And>name b. P b (Var name)"
   and     a2: "\<And>lam1 lam2 b. \<lbrakk>\<And>c. P c lam1; \<And>c. P c lam2\<rbrakk> \<Longrightarrow> P b (App lam1 lam2)"
-  and     a3: "\<And>name lam b. \<lbrakk>\<And>c. P c lam; name \<sharp> b\<rbrakk> \<Longrightarrow> P b (Lam name lam)"
+  and     a3: "\<And>name lam b. \<lbrakk>\<And>c. P c lam; (atom name) \<sharp> b\<rbrakk> \<Longrightarrow> P b (Lam name lam)"
   shows "P a lam"
 proof -
-  have "\<And>(pi::name prm) a. P a (pi \<bullet> lam)" 
+  have "\<And>pi a. P a (pi \<bullet> lam)" 
   proof (induct lam rule: lam_induct)
     case (1 name pi)
     show "P a (pi \<bullet> Var name)"
@@ -405,8 +612,8 @@
       done
   next
     case (2 lam1 lam2 pi)
-    have b1: "\<And>(pi::name prm) a. P a (pi \<bullet> lam1)" by fact
-    have b2: "\<And>(pi::name prm) a. P a (pi \<bullet> lam2)" by fact
+    have b1: "\<And>pi a. P a (pi \<bullet> lam1)" by fact
+    have b2: "\<And>pi a. P a (pi \<bullet> lam2)" by fact
     show "P a (pi \<bullet> App lam1 lam2)"
       apply (simp)
       apply (rule a2)
@@ -415,217 +622,43 @@
       done
   next
     case (3 name lam pi a)
-    have b: "\<And>(pi::name prm) a. P a (pi \<bullet> lam)" by fact
-    obtain c::name where fr: "c\<sharp>(a, pi\<bullet>name, pi\<bullet>lam)"
-      apply(rule exists_fresh[of "(a, pi\<bullet>name, pi\<bullet>lam)"])
-      apply(simp_all add: fs_name1)
-      done
-    from b fr have p: "P a (Lam c (([(c, pi\<bullet>name)]@pi)\<bullet>lam))" 
+    have b: "\<And>pi a. P a (pi \<bullet> lam)" by fact
+    obtain c::name where fr: "atom c\<sharp>(a, pi\<bullet>name, pi\<bullet>lam)"
+      apply(rule obtain_atom)
+      apply(auto)
+      sorry
+    from b fr have p: "P a (Lam c (((c \<leftrightarrow> (pi \<bullet> name)) + pi)\<bullet>lam))" 
       apply -
       apply(rule a3)
       apply(blast)
-      apply(simp)
+      apply(simp add: fresh_Pair)
       done
-    have eq: "[(c, pi\<bullet>name)] \<bullet> Lam (pi \<bullet> name) (pi \<bullet> lam) = Lam (pi \<bullet> name) (pi \<bullet> lam)"
-      apply(rule perm_fresh_fresh)
+    have eq: "(atom c \<rightleftharpoons> atom (pi\<bullet>name)) \<bullet> Lam (pi \<bullet> name) (pi \<bullet> lam) = Lam (pi \<bullet> name) (pi \<bullet> lam)"
+      apply(rule swap_fresh_fresh)
       using fr
-      apply(simp add: fresh_lam)
-      apply(simp add: fresh_lam)
+      apply(simp add: fresh_lam fresh_Pair)
+      apply(simp add: fresh_lam fresh_Pair)
       done
     show "P a (pi \<bullet> Lam name lam)" 
       apply (simp)
       apply(subst eq[symmetric])
       using p
-      apply(simp only: perm_lam pt_name2 swap_simps)
+      apply(simp only: permute_lam)
+      apply(simp add: flip_def)
       done
   qed
-  then have "P a (([]::name prm) \<bullet> lam)" by blast
+  then have "P a (0 \<bullet> lam)" by blast
   then show "P a lam" by simp 
 qed
 
 
 lemma var_fresh:
   fixes a::"name"
-  shows "(a \<sharp> (Var b)) = (a \<sharp> b)"
+  shows "(atom a \<sharp> (Var b)) = (atom a \<sharp> b)"
   apply(simp add: fresh_def)
   apply(simp add: var_supp1)
   done
 
-(* lemma hom_reg: *)
-
-lemma rlam_rec_eqvt:
-  fixes pi::"name prm"
-  and   f1::"name \<Rightarrow> ('a::pt_name)"
-  shows "(pi\<bullet>rlam_rec f1 f2 f3 t) = rlam_rec (pi\<bullet>f1) (pi\<bullet>f2) (pi\<bullet>f3) (pi\<bullet>t)"
-apply(induct t)
-apply(simp_all)
-apply(simp add: perm_fun_def)
-apply(perm_simp)
-apply(subst pt_fun_app_eq[OF pt_name_inst at_name_inst])
-back
-apply(subst pt_fun_app_eq[OF pt_name_inst at_name_inst])
-apply(subst pt_fun_app_eq[OF pt_name_inst at_name_inst])
-apply(subst pt_fun_app_eq[OF pt_name_inst at_name_inst])
-apply(simp)
-apply(subst pt_fun_app_eq[OF pt_name_inst at_name_inst])
-back
-apply(subst pt_fun_app_eq[OF pt_name_inst at_name_inst])
-apply(subst pt_fun_app_eq[OF pt_name_inst at_name_inst])
-apply(simp)
-done
- 
-
-lemma rlam_rec_respects:
-  assumes f1: "f_var \<in> Respects (op= ===> op=)"
-  and     f2: "f_app \<in> Respects (alpha ===> alpha ===> op= ===> op= ===> op=)"
-  and     f3: "f_lam \<in> Respects (op= ===> alpha ===> op= ===> op=)"
-  shows "rlam_rec f_var f_app f_lam \<in> Respects (alpha ===> op =)"
-apply(simp add: mem_def)
-apply(simp add: Respects_def)
-apply(rule allI)
-apply(rule allI)
-apply(rule impI)
-apply(erule alpha.induct)
-apply(simp)
-apply(simp)
-using f2
-apply(simp add: mem_def)
-apply(simp add: Respects_def)
-using f3[simplified mem_def Respects_def]
-apply(simp)
-apply(case_tac "a=b")
-apply(clarify)
-apply(simp)
-(* probably true *)
-sorry
-
-function
-  term1_hom :: "(name \<Rightarrow> 'a) \<Rightarrow>
-                (rlam \<Rightarrow> rlam \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow>
-                ((name \<Rightarrow> rlam) \<Rightarrow> (name \<Rightarrow> 'a) \<Rightarrow> 'a) \<Rightarrow> rlam \<Rightarrow> 'a"
-where
-  "term1_hom var app abs' (rVar x) = (var x)"
-| "term1_hom var app abs' (rApp t u) =
-     app t u (term1_hom var app abs' t) (term1_hom var app abs' u)"
-| "term1_hom var app abs' (rLam x u) =
-     abs' (\<lambda>y. [(x, y)] \<bullet> u) (\<lambda>y. term1_hom var app abs' ([(x, y)] \<bullet> u))"
-apply(pat_completeness)
-apply(auto)
-done
-
-lemma pi_size:
-  fixes pi::"name prm"
-  and   t::"rlam"
-  shows "size (pi \<bullet> t) = size t"
-apply(induct t)
-apply(auto)
-done
-
-termination term1_hom
-  apply(relation "measure (\<lambda>(f1, f2, f3, t). size t)")
-apply(auto simp add: pi_size)
-done
-
-lemma lam_exhaust:
-  "\<lbrakk>\<And>name. y = Var name \<Longrightarrow> P; \<And>rlam1 rlam2. y = App rlam1 rlam2 \<Longrightarrow> P; \<And>name rlam. y = Lam name rlam \<Longrightarrow> P\<rbrakk>
-    \<Longrightarrow> P"
-apply(lifting rlam.exhaust)
-done
-
-(* THIS IS NOT TRUE, but it lets prove the existence of the hom function *)
-lemma lam_inject':
-  "(Lam a x = Lam b y) = ((\<lambda>c. [(a, c)] \<bullet> x) = (\<lambda>c. [(b, c)] \<bullet> y))"
-sorry
-
-function
-  hom :: "(name \<Rightarrow> 'a) \<Rightarrow>
-                (lam \<Rightarrow> lam \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow>
-                ((name \<Rightarrow> lam) \<Rightarrow> (name \<Rightarrow> 'a) \<Rightarrow> 'a) \<Rightarrow> lam \<Rightarrow> 'a"
-where
-  "hom f_var f_app f_lam (Var x) = f_var x"
-| "hom f_var f_app f_lam (App l r) = f_app l r (hom f_var f_app f_lam l) (hom f_var f_app f_lam r)"
-| "hom f_var f_app f_lam (Lam a x) = f_lam (\<lambda>b. ([(a,b)] \<bullet> x)) (\<lambda>b. hom f_var f_app f_lam ([(a,b)] \<bullet> x))"
-defer
-apply(simp_all add: lam_inject') (* inject, distinct *)
-apply(tactic {* Cong_Tac.cong_tac @{thm cong} 1 *})
-apply(rule refl)
-apply(rule ext)
-apply(tactic {* Cong_Tac.cong_tac @{thm cong} 1 *})
-apply simp_all
-apply(erule conjE)+
-apply(rule_tac x="b" in cong)
-apply simp_all
-apply auto
-apply(rule_tac y="b" in lam_exhaust)
-apply simp_all
-apply auto
-apply meson
-apply(simp_all add: lam_inject')
-apply metis
-done
-
-termination hom
-  apply -
-(*
-ML_prf {* Size.size_thms @{theory} "LamEx.lam" *}
-*)
-sorry
-
-thm hom.simps
-
-lemma term1_hom_rsp:
-  "\<lbrakk>(alpha ===> alpha ===> op =) f_app f_app; ((op = ===> alpha) ===> op =) f_lam f_lam\<rbrakk>
-       \<Longrightarrow> (alpha ===> op =) (term1_hom f_var f_app f_lam) (term1_hom f_var f_app f_lam)"
-apply(simp)
-apply(rule allI)+
-apply(rule impI)
-apply(erule alpha.induct)
-apply(auto)[1]
-apply(auto)[1]
-apply(simp)
-apply(erule conjE)+
-apply(erule exE)+
-apply(erule conjE)+
-apply(clarify)
-sorry
-
-lemma hom: "
-\<forall>f_var. \<forall>f_app \<in> Respects(alpha ===> alpha ===> op =).
-\<forall>f_lam \<in> Respects((op = ===> alpha) ===> op =).
-\<exists>hom\<in>Respects (alpha ===> op =). 
-    ((\<forall>x. hom (rVar x) = f_var x) \<and>
-     (\<forall>l r. hom (rApp l r) = f_app l r (hom l) (hom r)) \<and>
-     (\<forall>x a. hom (rLam a x) = f_lam (\<lambda>b. ([(a,b)]\<bullet> x)) (\<lambda>b. hom ([(a,b)] \<bullet> x))))"
-apply(rule allI)
-apply(rule ballI)+
-apply(rule_tac x="term1_hom f_var f_app f_lam" in bexI)
-apply(simp_all)
-apply(simp only: in_respects)
-apply(rule term1_hom_rsp)
-apply(assumption)+
-done
-
-lemma hom':
-"\<exists>hom.
-  ((\<forall>x. hom (Var x) = f_var x) \<and>
-   (\<forall>l r. hom (App l r) = f_app l r (hom l) (hom r)) \<and>
-   (\<forall>x a. hom (Lam a x) = f_lam (\<lambda>b. ([(a,b)] \<bullet> x)) (\<lambda>b. hom ([(a,b)] \<bullet> x))))"
-apply (lifting hom)
-done
-
-(* test test
-lemma raw_hom_correct: 
-  assumes f1: "f_var \<in> Respects (op= ===> op=)"
-  and     f2: "f_app \<in> Respects (alpha ===> alpha ===> op= ===> op= ===> op=)"
-  and     f3: "f_lam \<in> Respects ((op= ===> alpha) ===> (op= ===> op=) ===> op=)"
-  shows "\<exists>!hom\<in>Respects (alpha ===> op =). 
-    ((\<forall>x. hom (rVar x) = f_var x) \<and>
-     (\<forall>l r. hom (rApp l r) = f_app l r (hom l) (hom r)) \<and>
-     (\<forall>x a. hom (rLam a x) = f_lam (\<lambda>b. ([(a,b)]\<bullet> x)) (\<lambda>b. hom ([(a,b)] \<bullet> x))))"
-unfolding Bex1_def
-apply(rule ex1I)
-sorry
-*)
 
 
 end