--- a/Quot/Nominal/LamEx.thy	Mon Feb 01 13:00:01 2010 +0100
+++ b/Quot/Nominal/LamEx.thy	Mon Feb 01 15:45:40 2010 +0100
@@ -1,5 +1,5 @@
 theory LamEx
-imports "Nominal2_Atoms" "Nominal2_Eqvt" "Nominal2_Supp" "../QuotMain" "Abs"
+imports "Nominal2_Atoms" "Nominal2_Eqvt" "Nominal2_Supp" "../QuotMain" "Abs" "../QuotProd"
 begin
 
 
@@ -26,13 +26,6 @@
 apply(simp)
 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 fresh_plus:
   fixes p q::perm
   shows "\<lbrakk>a \<sharp> p;  a \<sharp> q\<rbrakk> \<Longrightarrow> a \<sharp> (p + q)"
@@ -153,12 +146,13 @@
 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. (rfv t - {atom a} = rfv s - {atom b} \<and> (rfv t - {atom a})\<sharp>* pi \<and> (pi \<bullet> t) \<approx> s)
-       \<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. (rfv t - {atom a} = rfv s - {atom b} \<and> (rfv t - {atom a})\<sharp>* pi \<and> (pi \<bullet> t) \<approx> s)"
+  shows "\<exists>pi. (({atom a}, t) \<approx>gen alpha rfv pi ({atom b}, s))"
 using assms
 apply(erule_tac alpha.cases)
 apply(auto)
@@ -172,11 +166,11 @@
 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(rule conjI)
+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)
@@ -193,24 +187,43 @@
 apply(simp add: a2)
 apply(rule a3)
 apply(rule_tac x="0" in exI)
-apply(simp_all add: fresh_star_def fresh_zero_perm)
+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="- pi" in exI)
-apply(simp)
-apply(simp add: fresh_star_def fresh_minus_perm)
-apply(erule conjE)+
-apply(rotate_tac 3)
-apply(drule_tac pi="- pi" in alpha_eqvt)
-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"
@@ -225,11 +238,13 @@
 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: alpha_gen.simps)
 apply(simp add: fresh_star_plus)
 apply(drule_tac x="- pia \<bullet> sa" in spec)
 apply(drule mp)
@@ -251,89 +266,9 @@
 lemma alpha_rfv:
   shows "t \<approx> s \<Longrightarrow> rfv t = rfv s"
   apply(induct rule: alpha.induct)
-  apply(simp_all)
+  apply(simp_all add: alpha_gen.simps)
   done
 
-inductive
-    alpha2 :: "rlam \<Rightarrow> rlam \<Rightarrow> bool" ("_ \<approx>2 _" [100, 100] 100)
-where
-  a21: "a = b \<Longrightarrow> (rVar a) \<approx>2 (rVar b)"
-| a22: "\<lbrakk>t1 \<approx>2 t2; s1 \<approx>2 s2\<rbrakk> \<Longrightarrow> rApp t1 s1 \<approx>2 rApp t2 s2"
-| a23: "(a = b \<and> t \<approx>2 s) \<or> (a \<noteq> b \<and> ((a \<leftrightarrow> b) \<bullet> t) \<approx>2 s \<and> atom b \<notin> rfv t)\<Longrightarrow> rLam a t \<approx>2 rLam b s"
-
-lemma fv_vars:
-  fixes a::name
-  assumes a1: "\<forall>x \<in> rfv t - {atom a}. pi \<bullet> x = x"
-  shows "(pi \<bullet> t) \<approx>2 ((a \<leftrightarrow> (pi \<bullet> a)) \<bullet> t)"
-using a1
-apply(induct t)
-apply(auto)
-apply(rule a21)
-apply(case_tac "name = a")
-apply(simp)
-apply(simp)
-defer
-apply(rule a22)
-apply(simp)
-apply(simp)
-apply(rule a23)
-apply(case_tac "a = name")
-apply(simp)
-oops
-
-
-lemma 
-  assumes a1: "t \<approx>2 s"
-  shows "t \<approx> s"
-using a1
-apply(induct)
-apply(rule alpha.intros)
-apply(simp)
-apply(rule alpha.intros)
-apply(simp)
-apply(simp)
-apply(rule alpha.intros)
-apply(erule disjE)
-apply(rule_tac x="0" in exI)
-apply(simp add: fresh_star_def fresh_zero_perm)
-apply(erule conjE)+
-apply(drule alpha_rfv)
-apply(simp)
-apply(rule_tac x="(a \<leftrightarrow> b)" in exI)
-apply(simp)
-apply(erule conjE)+
-apply(rule conjI)
-apply(drule alpha_rfv)
-apply(drule sym)
-apply(simp)
-apply(simp add: rfv_eqvt[symmetric])
-defer
-apply(subgoal_tac "atom a \<sharp> (rfv t - {atom a})")
-apply(subgoal_tac "atom b \<sharp> (rfv t - {atom a})")
-
-defer
-sorry
-
-lemma 
-  assumes a1: "t \<approx> s"
-  shows "t \<approx>2 s"
-using a1
-apply(induct)
-apply(rule alpha2.intros)
-apply(simp)
-apply(rule alpha2.intros)
-apply(simp)
-apply(simp)
-apply(clarify)
-apply(rule alpha2.intros)
-apply(frule alpha_rfv)
-apply(rotate_tac 4)
-apply(drule sym)
-apply(simp)
-apply(drule sym)
-apply(simp)
-oops
-
 quotient_type lam = rlam / alpha
   by (rule alpha_equivp)
 
@@ -378,7 +313,7 @@
   apply(rule a3)
   apply(rule_tac x="0" in exI)
   unfolding fresh_star_def 
-  apply(simp add: fresh_star_def fresh_zero_perm)
+  apply(simp add: fresh_star_def fresh_zero_perm alpha_gen.simps)
   apply(simp add: alpha_rfv)
   done
 
@@ -441,10 +376,60 @@
   "\<lbrakk>x = xa; xb = xc\<rbrakk> \<Longrightarrow> App x xb = App xa xc"
   by  (lifting a2)
 
-lemma a3: 
-  "\<lbrakk>\<exists>pi. (fv t - {atom a} = fv s - {atom b} \<and> (fv t - {atom a})\<sharp>* pi \<and> (pi \<bullet> t) = s)\<rbrakk> 
-   \<Longrightarrow> Lam a t = Lam b s"
-  apply  (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)
+sledgehammer
+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: