diff -r 3d54fcc5f41a -r 6f3b75135638 Quot/Nominal/Terms.thy
--- a/Quot/Nominal/Terms.thy	Thu Feb 18 23:07:28 2010 +0100
+++ b/Quot/Nominal/Terms.thy	Thu Feb 18 23:07:52 2010 +0100
@@ -25,52 +25,71 @@
 where
   "bv1 (BUnit) = {}"
 | "bv1 (BVr x) = {atom x}"
-| "bv1 (BPr bp1 bp2) = (bv1 bp1) \<union> (bv1 bp1)"
-
-local_setup {* define_raw_fv "Terms.rtrm1"
-  [[[[]], [[], []], [[(NONE, 0)], [(NONE, 0)]], [[(NONE, 0)], [], [(SOME @{term bv1}, 0)]]],
-   [[], [[]], [[], []]]] *}
-print_theorems
+| "bv1 (BPr bp1 bp2) = (bv1 bp1) \<union> (bv1 bp2)"
 
 setup {* snd o define_raw_perms ["rtrm1", "bp"] ["Terms.rtrm1", "Terms.bp"] *}
 
-inductive
-  alpha1 :: "rtrm1 \<Rightarrow> rtrm1 \<Rightarrow> bool" ("_ \<approx>1 _" [100, 100] 100)
-where
-  a1: "a = b \<Longrightarrow> (rVr1 a) \<approx>1 (rVr1 b)"
-| a2: "\<lbrakk>t1 \<approx>1 t2; s1 \<approx>1 s2\<rbrakk> \<Longrightarrow> rAp1 t1 s1 \<approx>1 rAp1 t2 s2"
-| a3: "(\<exists>pi. (({atom aa}, t) \<approx>gen alpha1 fv_rtrm1 pi ({atom ab}, s))) \<Longrightarrow> rLm1 aa t \<approx>1 rLm1 ab s"
-| a4: "t1 \<approx>1 t2 \<Longrightarrow> (\<exists>pi. (((bv1 b1), s1) \<approx>gen alpha1 fv_rtrm1 pi ((bv1 b2), s2))) \<Longrightarrow> rLt1 b1 t1 s1 \<approx>1 rLt1 b2 t2 s2"
+local_setup {* snd o define_fv_alpha "Terms.rtrm1"
+  [[[[]], [[], []], [[(NONE, 0)], [(NONE, 0)]], [[(SOME @{term bv1}, 0)], [], [(SOME @{term bv1}, 0)]]],
+   [[], [[]], [[], []]]] *}
+print_theorems
+notation
+  alpha_rtrm1 ("_ \<approx>1 _" [100, 100] 100) and
+  alpha_bp ("_ \<approx>1b _" [100, 100] 100)
+thm alpha_rtrm1_alpha_bp.intros
 
 lemma alpha1_inj:
 "(rVr1 a \<approx>1 rVr1 b) = (a = b)"
 "(rAp1 t1 s1 \<approx>1 rAp1 t2 s2) = (t1 \<approx>1 t2 \<and> s1 \<approx>1 s2)"
-"(rLm1 aa t \<approx>1 rLm1 ab s) = (\<exists>pi. (({atom aa}, t) \<approx>gen alpha1 fv_rtrm1 pi ({atom ab}, s)))"
-"(rLt1 b1 t1 s1 \<approx>1 rLt1 b2 t2 s2) = (t1 \<approx>1 t2 \<and> (\<exists>pi. (((bv1 b1), s1) \<approx>gen alpha1 fv_rtrm1 pi ((bv1 b2), s2))))"
+"(rLm1 aa t \<approx>1 rLm1 ab s) = (\<exists>pi. (({atom aa}, t) \<approx>gen alpha_rtrm1 fv_rtrm1 pi ({atom ab}, s)))"
+"(rLt1 bp rtrm11 rtrm12 \<approx>1 rLt1 bpa rtrm11a rtrm12a) =
+   ((\<exists>pi. (bv1 bp, bp) \<approx>gen alpha_bp fv_bp pi (bv1 bpa, bpa)) \<and> rtrm11 \<approx>1 rtrm11a \<and>
+   (\<exists>pi. (bv1 bp, rtrm12) \<approx>gen alpha_rtrm1 fv_rtrm1 pi (bv1 bpa, rtrm12a)))"
+"alpha_bp BUnit BUnit"
+"(alpha_bp (BVr name) (BVr namea)) = (name = namea)"
+"(alpha_bp (BPr bp1 bp2) (BPr bp1a bp2a)) = (alpha_bp bp1 bp1a \<and> alpha_bp bp2 bp2a)"
 apply -
-apply rule apply (erule alpha1.cases) apply (simp_all add: alpha1.intros)
-apply rule apply (erule alpha1.cases) apply (simp_all add: alpha1.intros)
-apply rule apply (erule alpha1.cases) apply (simp_all add: alpha1.intros)
-apply rule apply (erule alpha1.cases) apply (simp_all add: alpha1.intros)
+apply rule apply (erule alpha_rtrm1.cases) apply (simp_all add: alpha_rtrm1_alpha_bp.intros)
+apply rule apply (erule alpha_rtrm1.cases) apply (simp_all add: alpha_rtrm1_alpha_bp.intros)
+apply rule apply (erule alpha_rtrm1.cases) apply (simp_all add: alpha_rtrm1_alpha_bp.intros)
+apply rule apply (erule alpha_rtrm1.cases) apply (simp_all add: alpha_rtrm1_alpha_bp.intros)
+apply rule apply (erule alpha_bp.cases) apply (simp_all add: alpha_rtrm1_alpha_bp.intros)
+apply rule apply (erule alpha_bp.cases) apply (simp_all add: alpha_rtrm1_alpha_bp.intros)
 done
 
-(* Shouyld we derive it? But bv is given by the user? *)
+lemma alpha_bp_refl: "alpha_bp a a"
+apply induct
+apply (simp_all  add: alpha1_inj)
+done
+
+lemma alpha_bp_eq_eq: "alpha_bp a b = (a = b)"
+apply rule
+apply (induct a b rule: alpha_rtrm1_alpha_bp.inducts(2))
+apply (simp_all add: alpha_bp_refl)
+done
+
+lemma alpha_bp_eq: "alpha_bp = (op =)"
+apply (rule ext)+
+apply (rule alpha_bp_eq_eq)
+done
+
 lemma bv1_eqvt[eqvt]:
   shows "(pi \<bullet> bv1 x) = bv1 (pi \<bullet> x)"
   apply (induct x)
-apply (simp_all add: empty_eqvt insert_eqvt atom_eqvt)
+apply (simp_all add: empty_eqvt insert_eqvt atom_eqvt eqvts)
 done
 
 lemma fv_rtrm1_eqvt[eqvt]:
-  shows "(pi\<bullet>fv_rtrm1 t) = fv_rtrm1 (pi\<bullet>t)"
-  apply (induct t)
+    "(pi\<bullet>fv_rtrm1 t) = fv_rtrm1 (pi\<bullet>t)"
+    "(pi\<bullet>fv_bp b) = fv_bp (pi\<bullet>b)"
+  apply (induct t and b)
   apply (simp_all add: insert_eqvt atom_eqvt empty_eqvt union_eqvt Diff_eqvt bv1_eqvt)
   done
 
-
 lemma alpha1_eqvt:
-  shows "t \<approx>1 s \<Longrightarrow> (pi \<bullet> t) \<approx>1 (pi \<bullet> s)"
-  apply (induct t s rule: alpha1.inducts)
+  "t \<approx>1 s \<Longrightarrow> (pi \<bullet> t) \<approx>1 (pi \<bullet> s)"
+  "alpha_bp a b \<Longrightarrow> alpha_bp (pi \<bullet> a) (pi \<bullet> b)"
+  apply (induct t s and a b rule: alpha_rtrm1_alpha_bp.inducts)
   apply (simp_all add:eqvts alpha1_inj)
   apply (erule exE)
   apply (rule_tac x="pi \<bullet> pia" in exI)
@@ -84,6 +103,8 @@
   apply(simp add: atom_eqvt Diff_eqvt fv_rtrm1_eqvt insert_eqvt empty_eqvt)
   apply(simp add: permute_eqvt[symmetric])
   apply (erule exE)
+  apply (erule exE)
+  apply (rule conjI)
   apply (rule_tac x="pi \<bullet> pia" in exI)
   apply (simp add: alpha_gen)
   apply(erule conjE)+
@@ -94,63 +115,60 @@
   apply(rule_tac ?p1="- pi" in fresh_star_permute_iff[THEN iffD1])
   apply(simp add: atom_eqvt fv_rtrm1_eqvt Diff_eqvt bv1_eqvt)
   apply(simp add: permute_eqvt[symmetric])
+  apply (rule_tac x="pi \<bullet> piaa" in exI)
+  apply (simp add: alpha_gen)
+  apply(erule conjE)+
+  apply(rule conjI)
+  apply(rule_tac ?p1="- pi" in permute_eq_iff[THEN iffD1])
+  apply(simp add: fv_rtrm1_eqvt Diff_eqvt bv1_eqvt)
+  apply(rule conjI)
+  apply(rule_tac ?p1="- pi" in fresh_star_permute_iff[THEN iffD1])
+  apply(simp add: atom_eqvt fv_rtrm1_eqvt Diff_eqvt bv1_eqvt)
+  apply(simp add: permute_eqvt[symmetric])
   done
 
-lemma alpha1_equivp: "equivp alpha1" 
+lemma alpha1_equivp: "equivp alpha_rtrm1" 
   sorry
 
-quotient_type trm1 = rtrm1 / alpha1
+quotient_type trm1 = rtrm1 / alpha_rtrm1
   by (rule alpha1_equivp)
 
-quotient_definition
-  "Vr1 :: name \<Rightarrow> trm1"
-is
-  "rVr1"
-
-quotient_definition
-  "Ap1 :: trm1 \<Rightarrow> trm1 \<Rightarrow> trm1"
-is
-  "rAp1"
-
-quotient_definition
-  "Lm1 :: name \<Rightarrow> trm1 \<Rightarrow> trm1"
-is
-  "rLm1"
-
-quotient_definition
-  "Lt1 :: bp \<Rightarrow> trm1 \<Rightarrow> trm1 \<Rightarrow> trm1"
-is
-  "rLt1"
-
-quotient_definition
-  "fv_trm1 :: trm1 \<Rightarrow> atom set"
-is
-  "fv_rtrm1"
+local_setup {*
+(fn ctxt => ctxt
+ |> snd o (Quotient_Def.quotient_lift_const ("Vr1", @{term rVr1}))
+ |> snd o (Quotient_Def.quotient_lift_const ("Ap1", @{term rAp1}))
+ |> snd o (Quotient_Def.quotient_lift_const ("Lm1", @{term rLm1}))
+ |> snd o (Quotient_Def.quotient_lift_const ("Lt1", @{term rLt1}))
+ |> snd o (Quotient_Def.quotient_lift_const ("fv_trm1", @{term fv_rtrm1})))
+*}
+print_theorems
 
 lemma alpha_rfv1:
   shows "t \<approx>1 s \<Longrightarrow> fv_rtrm1 t = fv_rtrm1 s"
-  apply(induct rule: alpha1.induct)
+  apply(induct rule: alpha_rtrm1_alpha_bp.inducts(1))
   apply(simp_all add: alpha_gen.simps)
   done
 
 lemma [quot_respect]:
- "(op = ===> alpha1) rVr1 rVr1"
- "(alpha1 ===> alpha1 ===> alpha1) rAp1 rAp1"
- "(op = ===> alpha1 ===> alpha1) rLm1 rLm1"
- "(op = ===> alpha1 ===> alpha1 ===> alpha1) rLt1 rLt1"
+ "(op = ===> alpha_rtrm1) rVr1 rVr1"
+ "(alpha_rtrm1 ===> alpha_rtrm1 ===> alpha_rtrm1) rAp1 rAp1"
+ "(op = ===> alpha_rtrm1 ===> alpha_rtrm1) rLm1 rLm1"
+ "(op = ===> alpha_rtrm1 ===> alpha_rtrm1 ===> alpha_rtrm1) rLt1 rLt1"
 apply (auto simp add: alpha1_inj)
 apply (rule_tac x="0" in exI)
 apply (simp add: fresh_star_def fresh_zero_perm alpha_rfv1 alpha_gen)
 apply (rule_tac x="0" in exI)
+apply (simp add: alpha_gen fresh_star_def fresh_zero_perm alpha_rfv1 alpha_bp_eq)
+apply (rule_tac x="0" in exI)
 apply (simp add: alpha_gen fresh_star_def fresh_zero_perm alpha_rfv1)
 done
 
 lemma [quot_respect]:
-  "(op = ===> alpha1 ===> alpha1) permute permute"
+  "(op = ===> alpha_rtrm1 ===> alpha_rtrm1) permute permute"
   by (simp add: alpha1_eqvt)
 
 lemma [quot_respect]:
-  "(alpha1 ===> op =) fv_rtrm1 fv_rtrm1"
+  "(alpha_rtrm1 ===> op =) fv_rtrm1 fv_rtrm1"
   by (simp add: alpha_rfv1)
 
 lemmas trm1_bp_induct = rtrm1_bp.induct[quot_lifted]
@@ -224,6 +242,17 @@
 apply(simp add: supp_Pair supp_atom bp_supp)
 done
 
+lemma fv_eq_bv: "fv_bp bp = bv1 bp"
+apply(induct bp rule: trm1_bp_inducts(2))
+apply(simp_all)
+done
+
+lemma helper: "{b. \<forall>pi. pi \<bullet> (a \<rightleftharpoons> b) \<bullet> bp \<noteq> bp} = {}"
+apply auto
+apply (rule_tac x="(x \<rightleftharpoons> a)" in exI)
+apply auto
+done
+
 lemma supp_fv:
   shows "supp t = fv_trm1 t"
 apply(induct t rule: trm1_bp_inducts(1))
@@ -240,13 +269,13 @@
 apply(simp add: alpha_gen.simps)
 apply(simp add: supp_eqvt[symmetric] fv_trm1_eqvt[symmetric])
 apply(subgoal_tac "supp (Lt1 bp rtrm11 rtrm12) = supp(rtrm11) \<union> supp (Abs (bv1 bp) rtrm12)")
-apply(simp add: supp_Abs fv_trm1)
+apply(simp add: supp_Abs fv_trm1 fv_eq_bv)
 apply(simp (no_asm) add: supp_def)
-apply(simp add: alpha1_INJ)
+apply(simp add: alpha1_INJ alpha_bp_eq)
 apply(simp add: Abs_eq_iff)
 apply(simp add: alpha_gen)
-apply(simp add: supp_eqvt[symmetric] fv_trm1_eqvt[symmetric] bv1_eqvt)
-apply(simp add: Collect_imp_eq Collect_neg_eq)
+apply(simp add: supp_eqvt[symmetric] fv_trm1_eqvt[symmetric] bv1_eqvt fv_eq_bv)
+apply(simp add: Collect_imp_eq Collect_neg_eq fresh_star_def helper)
 done
 
 lemma trm1_supp:
@@ -254,7 +283,7 @@
   "supp (Ap1 t1 t2) = supp t1 \<union> supp t2"
   "supp (Lm1 x t) = (supp t) - {atom x}"
   "supp (Lt1 b t s) = supp t \<union> (supp s - bv1 b)"
-  by (simp_all only: supp_fv fv_trm1)
+by (simp_all add: supp_fv fv_trm1 fv_eq_bv)
 
 lemma trm1_induct_strong:
   assumes "\<And>name b. P b (Vr1 name)"
@@ -280,37 +309,40 @@
 where
   "rbv2 (rAs x t) = {atom x}"
 
-local_setup {* define_raw_fv "Terms.rtrm2"
-  [[[[]], [[], []], [[(NONE, 0)], [(NONE, 0)]], [[], [(SOME @{term rbv2}, 0)]]],
-   [[[(NONE, 0)], []]]] *}
-print_theorems
-
 setup {* snd o define_raw_perms ["rtrm2", "rassign"] ["Terms.rtrm2", "Terms.rassign"] *}
 
-inductive
-  alpha2 :: "rtrm2 \<Rightarrow> rtrm2 \<Rightarrow> bool" ("_ \<approx>2 _" [100, 100] 100)
-and
-  alpha2a :: "rassign \<Rightarrow> rassign \<Rightarrow> bool" ("_ \<approx>2a _" [100, 100] 100)
-where
-  a1: "a = b \<Longrightarrow> (rVr2 a) \<approx>2 (rVr2 b)"
-| a2: "\<lbrakk>t1 \<approx>2 t2; s1 \<approx>2 s2\<rbrakk> \<Longrightarrow> rAp2 t1 s1 \<approx>2 rAp2 t2 s2"
-| a3: "(\<exists>pi. (({atom a}, t) \<approx>gen alpha2 fv_rtrm2 pi ({atom b}, s))) \<Longrightarrow> rLm2 a t \<approx>2 rLm2 b s"
-| a4: "\<lbrakk>\<exists>pi. ((rbv2 bt, t) \<approx>gen alpha2 fv_rtrm2 pi ((rbv2 bs), s));
-        \<exists>pi. ((rbv2 bt, bt) \<approx>gen alpha2a fv_rassign pi (rbv2 bs, bs))\<rbrakk>
-        \<Longrightarrow> rLt2 bt t \<approx>2 rLt2 bs s"
-| a5: "\<lbrakk>a = b; t \<approx>2 s\<rbrakk> \<Longrightarrow> rAs a t \<approx>2a rAs b s" (* This way rbv2 can be lifted *)
+local_setup {* snd o define_fv_alpha "Terms.rtrm2"
+  [[[[]], [[], []], [[(NONE, 0)], [(NONE, 0)]], [[(SOME @{term rbv2}, 0)], [(SOME @{term rbv2}, 0)]]],
+   [[[], []]]] *}
+print_theorems
+
+notation
+  alpha_rtrm2 ("_ \<approx>2 _" [100, 100] 100) and
+  alpha_rassign ("_ \<approx>2b _" [100, 100] 100)
+thm alpha_rtrm2_alpha_rassign.intros
 
 lemma alpha2_equivp:
-  "equivp alpha2"
-  "equivp alpha2a"
+  "equivp alpha_rtrm2"
+  "equivp alpha_rassign"
   sorry
 
 quotient_type
-  trm2 = rtrm2 / alpha2
+  trm2 = rtrm2 / alpha_rtrm2
 and
-  assign = rassign / alpha2a
+  assign = rassign / alpha_rassign
   by (auto intro: alpha2_equivp)
 
+local_setup {*
+(fn ctxt => ctxt
+ |> snd o (Quotient_Def.quotient_lift_const ("Vr2", @{term rVr2}))
+ |> snd o (Quotient_Def.quotient_lift_const ("Ap2", @{term rAp2}))
+ |> snd o (Quotient_Def.quotient_lift_const ("Lm2", @{term rLm2}))
+ |> snd o (Quotient_Def.quotient_lift_const ("Lt2", @{term rLt2}))
+ |> snd o (Quotient_Def.quotient_lift_const ("As", @{term rAs}))
+ |> snd o (Quotient_Def.quotient_lift_const ("fv_trm2", @{term fv_rtrm2}))
+ |> snd o (Quotient_Def.quotient_lift_const ("bv2", @{term rbv2})))
+*}
+print_theorems
 
 
 section {*** lets with many assignments ***}
@@ -331,36 +363,27 @@
   "bv3 ANil = {}"
 | "bv3 (ACons x t as) = {atom x} \<union> (bv3 as)"
 
-local_setup {* define_raw_fv "Terms.trm3"
-  [[[[]], [[], []], [[(NONE, 0)], [(NONE, 0)]], [[], [(SOME @{term bv3}, 0)]]],
-   [[], [[(NONE, 0)], [], []]]] *}
-print_theorems
-
 setup {* snd o define_raw_perms ["rtrm3", "assigns"] ["Terms.trm3", "Terms.assigns"] *}
 
-inductive
-  alpha3 :: "trm3 \<Rightarrow> trm3 \<Rightarrow> bool" ("_ \<approx>3 _" [100, 100] 100)
-and
-  alpha3a :: "assigns \<Rightarrow> assigns \<Rightarrow> bool" ("_ \<approx>3a _" [100, 100] 100)
-where
-  a1: "a = b \<Longrightarrow> (Vr3 a) \<approx>3 (Vr3 b)"
-| a2: "\<lbrakk>t1 \<approx>3 t2; s1 \<approx>3 s2\<rbrakk> \<Longrightarrow> Ap3 t1 s1 \<approx>3 Ap3 t2 s2"
-| a3: "(\<exists>pi. (({atom a}, t) \<approx>gen alpha3 fv_rtrm3 pi ({atom b}, s))) \<Longrightarrow> Lm3 a t \<approx>3 Lm3 b s"
-| a4: "\<lbrakk>\<exists>pi. ((bv3 bt, t) \<approx>gen alpha3 fv_trm3 pi ((bv3 bs), s));
-        \<exists>pi. ((bv3 bt, bt) \<approx>gen alpha3a fv_assign pi (bv3 bs, bs))\<rbrakk>
-        \<Longrightarrow> Lt3 bt t \<approx>3 Lt3 bs s"
-| a5: "ANil \<approx>3a ANil"
-| a6: "\<lbrakk>a = b; t \<approx>3 s; tt \<approx>3a st\<rbrakk> \<Longrightarrow> ACons a t tt \<approx>3a ACons b s st"
+local_setup {* snd o define_fv_alpha "Terms.trm3"
+  [[[[]], [[], []], [[(NONE, 0)], [(NONE, 0)]], [[(SOME @{term bv3}, 0)], [(SOME @{term bv3}, 0)]]],
+   [[], [[], [], []]]] *}
+print_theorems
+
+notation
+  alpha_trm3 ("_ \<approx>3 _" [100, 100] 100) and
+  alpha_assigns ("_ \<approx>3a _" [100, 100] 100)
+thm alpha_trm3_alpha_assigns.intros
 
 lemma alpha3_equivp:
-  "equivp alpha3"
-  "equivp alpha3a"
+  "equivp alpha_trm3"
+  "equivp alpha_assigns"
   sorry
 
 quotient_type
-  qtrm3 = trm3 / alpha3
+  qtrm3 = trm3 / alpha_trm3
 and
-  qassigns = assigns / alpha3a
+  qassigns = assigns / alpha_assigns
   by (auto intro: alpha3_equivp)
 
 
@@ -374,10 +397,6 @@
 
 thm trm4.recs
 
-local_setup {* define_raw_fv "Terms.trm4" [
-  [[[]], [[], []], [[(NONE, 0)], [(NONE, 0)]]], [[], [[], []]]  ] *}
-print_theorems
-
 (* there cannot be a clause for lists, as *)
 (* permutations are  already defined in Nominal (also functions, options, and so on) *)
 setup {* snd o define_raw_perms ["trm4"] ["Terms.trm4"] *}
@@ -393,22 +412,21 @@
 thm permute_trm4_permute_trm4_list.simps
 thm permute_trm4_permute_trm4_list.simps[simplified repaired]
 
-inductive
-    alpha4 :: "trm4 \<Rightarrow> trm4 \<Rightarrow> bool" ("_ \<approx>4 _" [100, 100] 100)
-and alpha4list :: "trm4 list \<Rightarrow> trm4 list \<Rightarrow> bool" ("_ \<approx>4list _" [100, 100] 100) 
-where
-  a1: "a = b \<Longrightarrow> (Vr4 a) \<approx>4 (Vr4 b)"
-| a2: "\<lbrakk>t1 \<approx>4 t2; s1 \<approx>4list s2\<rbrakk> \<Longrightarrow> Ap4 t1 s1 \<approx>4 Ap4 t2 s2"
-| a3: "(\<exists>pi. (({atom a}, t) \<approx>gen alpha4 fv_rtrm4 pi ({atom b}, s))) \<Longrightarrow> Lm4 a t \<approx>4 Lm4 b s"
-| a5: "[] \<approx>4list []"
-| a6: "\<lbrakk>t \<approx>4 s; ts \<approx>4list ss\<rbrakk> \<Longrightarrow> (t#ts) \<approx>4list (s#ss)"
+local_setup {* snd o define_fv_alpha "Terms.trm4" [
+  [[[]], [[], []], [[(NONE, 0)], [(NONE, 0)]]], [[], [[], []]]  ] *}
+print_theorems
 
-lemma alpha4_equivp: "equivp alpha4" sorry
-lemma alpha4list_equivp: "equivp alpha4list" sorry
+notation
+  alpha_trm4 ("_ \<approx>4 _" [100, 100] 100) and
+  alpha_trm4_list ("_ \<approx>4l _" [100, 100] 100)
+thm alpha_trm4_alpha_trm4_list.intros
+
+lemma alpha4_equivp: "equivp alpha_trm4" sorry
+lemma alpha4list_equivp: "equivp alpha_trm4_list" sorry
 
 quotient_type 
-  qtrm4 = trm4 / alpha4 and
-  qtrm4list = "trm4 list" / alpha4list
+  qtrm4 = trm4 / alpha_trm4 and
+  qtrm4list = "trm4 list" / alpha_trm4_list
   by (simp_all add: alpha4_equivp alpha4list_equivp)
 
 
@@ -426,101 +444,70 @@
   "rbv5 rLnil = {}"
 | "rbv5 (rLcons n t ltl) = {atom n} \<union> (rbv5 ltl)"
 
-local_setup {* define_raw_fv "Terms.rtrm5" [
-  [[[]], [[], []], [[(SOME @{term rbv5}, 0)], [(SOME @{term rbv5}, 0)]]], [[], [[(NONE, 0)], [], []]]  ] *}
-print_theorems
 
 setup {* snd o define_raw_perms ["rtrm5", "rlts"] ["Terms.rtrm5", "Terms.rlts"] *}
 print_theorems
 
-inductive
-  alpha5 :: "rtrm5 \<Rightarrow> rtrm5 \<Rightarrow> bool" ("_ \<approx>5 _" [100, 100] 100)
-and
-  alphalts :: "rlts \<Rightarrow> rlts \<Rightarrow> bool" ("_ \<approx>l _" [100, 100] 100)
-where
-  a1: "a = b \<Longrightarrow> (rVr5 a) \<approx>5 (rVr5 b)"
-| a2: "\<lbrakk>t1 \<approx>5 t2; s1 \<approx>5 s2\<rbrakk> \<Longrightarrow> rAp5 t1 s1 \<approx>5 rAp5 t2 s2"
-| a3: "\<lbrakk>\<exists>pi. ((rbv5 l1, t1) \<approx>gen alpha5 fv_rtrm5 pi (rbv5 l2, t2)); 
-        \<exists>pi. ((rbv5 l1, l1) \<approx>gen alphalts fv_rlts pi (rbv5 l2, l2))\<rbrakk>
-        \<Longrightarrow> rLt5 l1 t1 \<approx>5 rLt5 l2 t2"
-| a4: "rLnil \<approx>l rLnil"
-| a5: "ls1 \<approx>l ls2 \<Longrightarrow> t1 \<approx>5 t2 \<Longrightarrow> n1 = n2 \<Longrightarrow> rLcons n1 t1 ls1 \<approx>l rLcons n2 t2 ls2"
+local_setup {* snd o define_fv_alpha "Terms.rtrm5" [
+  [[[]], [[], []], [[(SOME @{term rbv5}, 0)], [(SOME @{term rbv5}, 0)]]], [[], [[], [], []]]  ] *}
+print_theorems
 
-print_theorems
+(* Alternate version with additional binding of name in rlts in rLcons *)
+(*local_setup {* snd o define_fv_alpha "Terms.rtrm5" [
+  [[[]], [[], []], [[(SOME @{term rbv5}, 0)], [(SOME @{term rbv5}, 0)]]], [[], [[(NONE,0)], [], [(NONE,0)]]]  ] *}
+print_theorems*)
+
+notation
+  alpha_rtrm5 ("_ \<approx>5 _" [100, 100] 100) and
+  alpha_rlts ("_ \<approx>l _" [100, 100] 100)
+thm alpha_rtrm5_alpha_rlts.intros
 
 lemma alpha5_inj:
   "((rVr5 a) \<approx>5 (rVr5 b)) = (a = b)"
   "(rAp5 t1 s1 \<approx>5 rAp5 t2 s2) = (t1 \<approx>5 t2 \<and> s1 \<approx>5 s2)"
-  "(rLt5 l1 t1 \<approx>5 rLt5 l2 t2) = ((\<exists>pi. ((rbv5 l1, t1) \<approx>gen alpha5 fv_rtrm5 pi (rbv5 l2, t2))) \<and>
-         (\<exists>pi. ((rbv5 l1, l1) \<approx>gen alphalts fv_rlts pi (rbv5 l2, l2))))"
+  "(rLt5 l1 t1 \<approx>5 rLt5 l2 t2) = ((\<exists>pi. ((rbv5 l1, t1) \<approx>gen alpha_rtrm5 fv_rtrm5 pi (rbv5 l2, t2))) \<and>
+         (\<exists>pi. ((rbv5 l1, l1) \<approx>gen alpha_rlts fv_rlts pi (rbv5 l2, l2))))"
   "rLnil \<approx>l rLnil"
   "(rLcons n1 t1 ls1 \<approx>l rLcons n2 t2 ls2) = (n1 = n2 \<and> ls1 \<approx>l ls2 \<and> t1 \<approx>5 t2)"
 apply -
-apply (simp_all add: alpha5_alphalts.intros)
+apply (simp_all add: alpha_rtrm5_alpha_rlts.intros)
 apply rule
-apply (erule alpha5.cases)
-apply (simp_all add: alpha5_alphalts.intros)
+apply (erule alpha_rtrm5.cases)
+apply (simp_all add: alpha_rtrm5_alpha_rlts.intros)
 apply rule
-apply (erule alpha5.cases)
-apply (simp_all add: alpha5_alphalts.intros)
+apply (erule alpha_rtrm5.cases)
+apply (simp_all add: alpha_rtrm5_alpha_rlts.intros)
 apply rule
-apply (erule alpha5.cases)
-apply (simp_all add: alpha5_alphalts.intros)
+apply (erule alpha_rtrm5.cases)
+apply (simp_all add: alpha_rtrm5_alpha_rlts.intros)
 apply rule
-apply (erule alphalts.cases)
-apply (simp_all add: alpha5_alphalts.intros)
+apply (erule alpha_rlts.cases)
+apply (simp_all add: alpha_rtrm5_alpha_rlts.intros)
 done
 
 lemma alpha5_equivps:
-  shows "equivp alpha5"
-  and   "equivp alphalts"
+  shows "equivp alpha_rtrm5"
+  and   "equivp alpha_rlts"
 sorry
 
 quotient_type
-  trm5 = rtrm5 / alpha5
+  trm5 = rtrm5 / alpha_rtrm5
 and
-  lts = rlts / alphalts
+  lts = rlts / alpha_rlts
   by (auto intro: alpha5_equivps)
 
-quotient_definition
-  "Vr5 :: name \<Rightarrow> trm5"
-is
-  "rVr5"
-
-quotient_definition
-  "Ap5 :: trm5 \<Rightarrow> trm5 \<Rightarrow> trm5"
-is
-  "rAp5"
-
-quotient_definition
-  "Lt5 :: lts \<Rightarrow> trm5 \<Rightarrow> trm5"
-is
-  "rLt5"
-
-quotient_definition
-  "Lnil :: lts"
-is
-  "rLnil"
-
-quotient_definition
-  "Lcons :: name \<Rightarrow> trm5 \<Rightarrow> lts \<Rightarrow> lts"
-is
-  "rLcons"
-
-quotient_definition
-   "fv_trm5 :: trm5 \<Rightarrow> atom set"
-is
-  "fv_rtrm5"
-
-quotient_definition
-   "fv_lts :: lts \<Rightarrow> atom set"
-is
-  "fv_rlts"
-
-quotient_definition
-   "bv5 :: lts \<Rightarrow> atom set"
-is
-  "rbv5"
+local_setup {*
+(fn ctxt => ctxt
+ |> snd o (Quotient_Def.quotient_lift_const ("Vr5", @{term rVr5}))
+ |> snd o (Quotient_Def.quotient_lift_const ("Ap5", @{term rAp5}))
+ |> snd o (Quotient_Def.quotient_lift_const ("Lt5", @{term rLt5}))
+ |> snd o (Quotient_Def.quotient_lift_const ("Lnil", @{term rLnil}))
+ |> snd o (Quotient_Def.quotient_lift_const ("Lcons", @{term rLcons}))
+ |> snd o (Quotient_Def.quotient_lift_const ("fv_trm5", @{term fv_rtrm5}))
+ |> snd o (Quotient_Def.quotient_lift_const ("fv_lts", @{term fv_rlts}))
+ |> snd o (Quotient_Def.quotient_lift_const ("bv5", @{term rbv5})))
+*}
+print_theorems
 
 lemma rbv5_eqvt:
   "pi \<bullet> (rbv5 x) = rbv5 (pi \<bullet> x)"
@@ -537,13 +524,13 @@
 lemma alpha5_eqvt:
   "xa \<approx>5 y \<Longrightarrow> (x \<bullet> xa) \<approx>5 (x \<bullet> y)"
   "xb \<approx>l ya \<Longrightarrow> (x \<bullet> xb) \<approx>l (x \<bullet> ya)"
-  apply(induct rule: alpha5_alphalts.inducts)
+  apply(induct rule: alpha_rtrm5_alpha_rlts.inducts)
   apply (simp_all add: alpha5_inj)
   apply (erule exE)+
   apply(unfold alpha_gen)
   apply (erule conjE)+
   apply (rule conjI)
-  apply (rule_tac x="x \<bullet> pi" in exI)
+  apply (rule_tac x="x \<bullet> pia" in exI)
   apply (rule conjI)
   apply(rule_tac ?p1="- x" in permute_eq_iff[THEN iffD1])
   apply(simp add: atom_eqvt Diff_eqvt insert_eqvt set_eqvt empty_eqvt rbv5_eqvt fv_rtrm5_eqvt)
@@ -552,7 +539,7 @@
   apply(simp add: atom_eqvt Diff_eqvt insert_eqvt set_eqvt empty_eqvt rbv5_eqvt fv_rtrm5_eqvt)
   apply (subst permute_eqvt[symmetric])
   apply (simp)
-  apply (rule_tac x="x \<bullet> pia" in exI)
+  apply (rule_tac x="x \<bullet> pi" in exI)
   apply (rule conjI)
   apply(rule_tac ?p1="- x" in permute_eq_iff[THEN iffD1])
   apply(simp add: atom_eqvt Diff_eqvt insert_eqvt set_eqvt empty_eqvt rbv5_eqvt fv_rlts_eqvt)
@@ -566,27 +553,27 @@
 lemma alpha5_rfv:
   "(t \<approx>5 s \<Longrightarrow> fv_rtrm5 t = fv_rtrm5 s)"
   "(l \<approx>l m \<Longrightarrow> fv_rlts l = fv_rlts m)"
-  apply(induct rule: alpha5_alphalts.inducts)
+  apply(induct rule: alpha_rtrm5_alpha_rlts.inducts)
   apply(simp_all add: alpha_gen)
   done
 
 lemma bv_list_rsp:
   shows "x \<approx>l y \<Longrightarrow> rbv5 x = rbv5 y"
-  apply(induct rule: alpha5_alphalts.inducts(2))
+  apply(induct rule: alpha_rtrm5_alpha_rlts.inducts(2))
   apply(simp_all)
   done
 
 lemma [quot_respect]:
-  "(alphalts ===> op =) fv_rlts fv_rlts"
-  "(alpha5 ===> op =) fv_rtrm5 fv_rtrm5"
-  "(alphalts ===> op =) rbv5 rbv5"
-  "(op = ===> alpha5) rVr5 rVr5"
-  "(alpha5 ===> alpha5 ===> alpha5) rAp5 rAp5"
-  "(alphalts ===> alpha5 ===> alpha5) rLt5 rLt5"
-  "(alphalts ===> alpha5 ===> alpha5) rLt5 rLt5"
-  "(op = ===> alpha5 ===> alphalts ===> alphalts) rLcons rLcons"
-  "(op = ===> alpha5 ===> alpha5) permute permute"
-  "(op = ===> alphalts ===> alphalts) permute permute"
+  "(alpha_rlts ===> op =) fv_rlts fv_rlts"
+  "(alpha_rtrm5 ===> op =) fv_rtrm5 fv_rtrm5"
+  "(alpha_rlts ===> op =) rbv5 rbv5"
+  "(op = ===> alpha_rtrm5) rVr5 rVr5"
+  "(alpha_rtrm5 ===> alpha_rtrm5 ===> alpha_rtrm5) rAp5 rAp5"
+  "(alpha_rlts ===> alpha_rtrm5 ===> alpha_rtrm5) rLt5 rLt5"
+  "(alpha_rlts ===> alpha_rtrm5 ===> alpha_rtrm5) rLt5 rLt5"
+  "(op = ===> alpha_rtrm5 ===> alpha_rlts ===> alpha_rlts) rLcons rLcons"
+  "(op = ===> alpha_rtrm5 ===> alpha_rtrm5) permute permute"
+  "(op = ===> alpha_rlts ===> alpha_rlts) permute permute"
   apply (simp_all add: alpha5_inj alpha5_rfv alpha5_eqvt bv_list_rsp)
   apply (clarify) apply (rule conjI)
   apply (rule_tac x="0" in exI) apply (simp add: fresh_star_def fresh_zero_perm alpha_gen alpha5_rfv)
@@ -597,7 +584,7 @@
   done
 
 lemma
-  shows "(alphalts ===> op =) rbv5 rbv5"
+  shows "(alpha_rlts ===> op =) rbv5 rbv5"
   by (simp add: bv_list_rsp)
 
 lemmas trm5_lts_inducts = rtrm5_rlts.inducts[quot_lifted]
@@ -686,7 +673,7 @@
 lemma distinct_helper:
   shows "\<not>(rVr5 x \<approx>5 rAp5 y z)"
   apply auto
-  apply (erule alpha5.cases)
+  apply (erule alpha_rtrm5.cases)
   apply (simp_all only: rtrm5.distinct)
   done
 
@@ -719,13 +706,15 @@
 | "rbv6 (rLm6 n t) = {atom n} \<union> rbv6 t"
 | "rbv6 (rLt6 l r) = rbv6 l \<union> rbv6 r"
 
-local_setup {* define_raw_fv "Terms.rtrm6" [
-  [[[]], [[(NONE, 0)], [(NONE, 0)]], [[], [(SOME @{term rbv6}, 0)]]]] *}
-print_theorems 
-
 setup {* snd o define_raw_perms ["rtrm6"] ["Terms.rtrm6"] *}
 print_theorems
 
+local_setup {* snd o define_fv_alpha "Terms.rtrm6" [
+  [[[]], [[(NONE, 0)], [(NONE, 0)]], [[(SOME @{term rbv6}, 0)], [(SOME @{term rbv6}, 0)]]]] *}
+notation alpha_rtrm6 ("_ \<approx>6a _" [100, 100] 100)
+(* HERE THE RULES DIFFER *)
+thm alpha_rtrm6.intros
+
 inductive
   alpha6 :: "rtrm6 \<Rightarrow> rtrm6 \<Rightarrow> bool" ("_ \<approx>6 _" [100, 100] 100)
 where
@@ -741,30 +730,15 @@
   trm6 = rtrm6 / alpha6
   by (auto intro: alpha6_equivps)
 
-quotient_definition
-  "Vr6 :: name \<Rightarrow> trm6"
-is
-  "rVr6"
-
-quotient_definition
-  "Lm6 :: name \<Rightarrow> trm6 \<Rightarrow> trm6"
-is
-  "rLm6"
-
-quotient_definition
-  "Lt6 :: trm6 \<Rightarrow> trm6 \<Rightarrow> trm6"
-is
-  "rLt6"
-
-quotient_definition
-   "fv_trm6 :: trm6 \<Rightarrow> atom set"
-is
-  "fv_rtrm6"
-
-quotient_definition
-   "bv6 :: trm6 \<Rightarrow> atom set"
-is
-  "rbv6"
+local_setup {*
+(fn ctxt => ctxt
+ |> snd o (Quotient_Def.quotient_lift_const ("Vr6", @{term rVr6}))
+ |> snd o (Quotient_Def.quotient_lift_const ("Lm6", @{term rLm6}))
+ |> snd o (Quotient_Def.quotient_lift_const ("Lt6", @{term rLt6}))
+ |> snd o (Quotient_Def.quotient_lift_const ("fv_trm6", @{term fv_rtrm6}))
+ |> snd o (Quotient_Def.quotient_lift_const ("bv6", @{term rbv6})))
+*}
+print_theorems
 
 lemma [quot_respect]:
   "(op = ===> alpha6 ===> alpha6) permute permute"
@@ -772,7 +746,6 @@
 sorry
 
 (* Definitely not true , see lemma below *)
-
 lemma [quot_respect]:"(alpha6 ===> op =) rbv6 rbv6"
 apply simp apply clarify
 apply (erule alpha6.induct)
@@ -875,13 +848,16 @@
 | "rbv7 (rLm7 n t) = rbv7 t - {atom n}"
 | "rbv7 (rLt7 l r) = rbv7 l \<union> rbv7 r"
 
-local_setup {* define_raw_fv "Terms.rtrm7" [
-  [[[]], [[(NONE, 0)], [(NONE, 0)]], [[], [(SOME @{term rbv7}, 0)]]]] *}
-print_theorems 
-
 setup {* snd o define_raw_perms ["rtrm7"] ["Terms.rtrm7"] *}
 print_theorems
 
+local_setup {* snd o define_fv_alpha "Terms.rtrm7" [
+  [[[]], [[(NONE, 0)], [(NONE, 0)]], [[(SOME @{term rbv7}, 0)], [(SOME @{term rbv7}, 0)]]]] *}
+notation
+  alpha_rtrm7 ("_ \<approx>7a _" [100, 100] 100)
+(* HERE THE RULES DIFFER *)
+thm alpha_rtrm7.intros
+
 inductive
   alpha7 :: "rtrm7 \<Rightarrow> rtrm7 \<Rightarrow> bool" ("_ \<approx>7 _" [100, 100] 100)
 where
@@ -889,11 +865,6 @@
 | a2: "(\<exists>pi. (({atom a}, t) \<approx>gen alpha7 fv_rtrm7 pi ({atom b}, s))) \<Longrightarrow> rLm7 a t \<approx>7 rLm7 b s"
 | a3: "(\<exists>pi. (((rbv7 t1), s1) \<approx>gen alpha7 fv_rtrm7 pi ((rbv7 t2), s2))) \<Longrightarrow> rLt7 t1 s1 \<approx>7 rLt7 t2 s2"
 
-lemma bvfv7: "rbv7 x = fv_rtrm7 x"
-  apply induct
-  apply simp_all
-done
-
 lemma "(x::name) \<noteq> y \<Longrightarrow> \<not> (alpha7 ===> op =) rbv7 rbv7"
   apply simp
   apply (rule_tac x="rLt7 (rVr7 x) (rVr7 x)" in exI)
@@ -923,13 +894,18 @@
   "rbv8 (Bar0 x) = {}"
 | "rbv8 (Bar1 v x b) = {atom v}"
 
-local_setup {* define_raw_fv "Terms.rfoo8" [
-  [[[]], [[], [(SOME @{term rbv8}, 0)]]], [[[]], [[], [(NONE, 1)], [(NONE, 1)]]]] *}
-print_theorems 
-
 setup {* snd o define_raw_perms ["rfoo8", "rbar8"] ["Terms.rfoo8", "Terms.rbar8"] *}
 print_theorems
 
+local_setup {* snd o define_fv_alpha "Terms.rfoo8" [
+  [[[]], [[(SOME @{term rbv8}, 0)], [(SOME @{term rbv8}, 0)]]], [[[]], [[], [(NONE, 1)], [(NONE, 1)]]]] *}
+notation
+  alpha_rfoo8 ("_ \<approx>f' _" [100, 100] 100) and
+  alpha_rbar8 ("_ \<approx>b' _" [100, 100] 100)
+(* HERE THE RULE DIFFERS *)
+thm alpha_rfoo8_alpha_rbar8.intros
+
+
 inductive
   alpha8f :: "rfoo8 \<Rightarrow> rfoo8 \<Rightarrow> bool" ("_ \<approx>f _" [100, 100] 100)
 and
@@ -958,6 +934,9 @@
   apply simp apply clarify
   apply (erule alpha8f_alpha8b.inducts(1))
   apply (simp_all add: alpha_gen fv_rbar8_rsp_hlp)
+  apply clarify
+  apply (erule alpha8f_alpha8b.inducts(2))
+  apply (simp_all)
 done
 
 
@@ -977,12 +956,17 @@
   "rbv9 (Var9 x) = {}"
 | "rbv9 (Lam9 x b) = {atom x}"
 
-local_setup {* define_raw_fv "Terms.rlam9" [
-  [[[]], [[(NONE, 0)], [(NONE, 0)]]], [[[], [(SOME @{term rbv9}, 0)]]]] *}
+setup {* snd o define_raw_perms ["rlam9", "rbla9"] ["Terms.rlam9", "Terms.rbla9"] *}
 print_theorems
 
-setup {* snd o define_raw_perms ["rlam9", "rbla9"] ["Terms.rlam9", "Terms.rbla9"] *}
-print_theorems
+local_setup {* snd o define_fv_alpha "Terms.rlam9" [
+  [[[]], [[(NONE, 0)], [(NONE, 0)]]], [[[(SOME @{term rbv9}, 0)], [(SOME @{term rbv9}, 0)]]]] *}
+notation
+  alpha_rlam9 ("_ \<approx>9l' _" [100, 100] 100) and
+  alpha_rbla9 ("_ \<approx>9b' _" [100, 100] 100)
+(* HERE THE RULES DIFFER *)
+thm alpha_rlam9_alpha_rbla9.intros
+
 
 inductive
   alpha9l :: "rlam9 \<Rightarrow> rlam9 \<Rightarrow> bool" ("_ \<approx>9l _" [100, 100] 100)
@@ -997,35 +981,16 @@
   lam9 = rlam9 / alpha9l and bla9 = rbla9 / alpha9b
 sorry
 
-quotient_definition
-  "qVar9 :: name \<Rightarrow> lam9"
-is
-  "Var9"
-
-quotient_definition
-  "qLam :: name \<Rightarrow> lam9 \<Rightarrow> lam9"
-is
-  "Lam9"
-
-quotient_definition
-  "qBla9 :: lam9 \<Rightarrow> lam9 \<Rightarrow> bla9"
-is
-  "Bla9"
-
-quotient_definition
-  "fv_lam9 :: lam9 \<Rightarrow> atom set"
-is
-  "fv_rlam9"
-
-quotient_definition
-  "fv_bla9 :: bla9 \<Rightarrow> atom set"
-is
-  "fv_rbla9"
-
-quotient_definition
-  "bv9 :: lam9 \<Rightarrow> atom set"
-is
-  "rbv9"
+local_setup {*
+(fn ctxt => ctxt
+ |> snd o (Quotient_Def.quotient_lift_const ("qVar9", @{term Var9}))
+ |> snd o (Quotient_Def.quotient_lift_const ("qLam9", @{term Lam9}))
+ |> snd o (Quotient_Def.quotient_lift_const ("qBla9", @{term Bla9}))
+ |> snd o (Quotient_Def.quotient_lift_const ("fv_lam9", @{term fv_rlam9}))
+ |> snd o (Quotient_Def.quotient_lift_const ("fv_bla9", @{term fv_rbla9}))
+ |> snd o (Quotient_Def.quotient_lift_const ("bv9", @{term rbv9})))
+*}
+print_theorems
 
 instantiation lam9 and bla9 :: pt
 begin
@@ -1072,14 +1037,11 @@
 setup {* snd o define_raw_perms ["tyS"] ["Terms.tyS"] *}
 print_theorems
 
-abbreviation
-  "atoms xs \<equiv> {atom x| x. x \<in> xs}"
-
 local_setup {* define_raw_fv "Terms.ty" [[[[]], [[], []]]] *}
 print_theorems 
 
 (*
-doesn't work yet
+Doesnot work yet since we do not refer to fv_ty
 local_setup {* define_raw_fv "Terms.tyS" [[[[], []]]] *}
 print_theorems
 *)
@@ -1087,12 +1049,12 @@
 primrec
   fv_tyS
 where 
-  "fv_tyS (All xs T) = (fv_ty T - atoms xs)"
+  "fv_tyS (All xs T) = (fv_ty T - atom ` xs)"
 
 inductive
   alpha_tyS :: "tyS \<Rightarrow> tyS \<Rightarrow> bool" ("_ \<approx>tyS _" [100, 100] 100)
 where
-  a1: "\<exists>pi. ((atoms xs1, T1) \<approx>gen (op =) fv_ty pi (atoms xs2, T2)) 
+  a1: "\<exists>pi. ((atom ` xs1, T1) \<approx>gen (op =) fv_ty pi (atom ` xs2, T2)) 
         \<Longrightarrow> All xs1 T1 \<approx>tyS All xs2 T2"
 
 lemma