# HG changeset patch
# User urbanc
# Date 1161205611 -7200
# Node ID ec5531061ed64276b6a59083fd380e201ca17acd
# Parent  c49467a9c1e147bd51334f0b5af8446ac269750d
adapted to Stefan's new inductive package and cleaning up

diff -r c49467a9c1e1 -r ec5531061ed6 src/HOL/Nominal/Examples/Weakening.thy
--- a/src/HOL/Nominal/Examples/Weakening.thy	Wed Oct 18 16:13:03 2006 +0200
+++ b/src/HOL/Nominal/Examples/Weakening.thy	Wed Oct 18 23:06:51 2006 +0200
@@ -1,37 +1,33 @@
 (* $Id$ *)
 
 theory Weakening 
-imports "../Nominal" 
+imports "Nominal" 
 begin
 
-(* WEAKENING EXAMPLE*)
-
-section {* Simply-Typed Lambda-Calculus *}
-(*======================================*)
+section {* Weakening Example for the Simply-Typed Lambda-Calculus *}
+(*================================================================*)
 
 atom_decl name 
 
-nominal_datatype lam = Var "name"
-                     | App "lam" "lam"
-                     | Lam "\<guillemotleft>name\<guillemotright>lam" ("Lam [_]._" [100,100] 100)
+nominal_datatype lam = 
+    Var "name"
+  | App "lam" "lam"
+  | Lam "\<guillemotleft>name\<guillemotright>lam" ("Lam [_]._" [100,100] 100)
 
 nominal_datatype ty =
     TVar "nat"
   | TArr "ty" "ty" (infix "\<rightarrow>" 200)
 
-lemma perm_ty[simp]:
+lemma [simp]:
   fixes pi ::"name prm"
   and   \<tau>  ::"ty"
   shows "pi\<bullet>\<tau> = \<tau>"
-by (induct \<tau> rule: ty.induct_weak, simp_all add: perm_nat_def)  
+by (induct \<tau> rule: ty.induct_weak)
+   (simp_all add: perm_nat_def)  
 
-(* valid contexts *)
-consts
-  ctxts :: "((name\<times>ty) list) set" 
+text {* valid contexts *}
+inductive2
   valid :: "(name\<times>ty) list \<Rightarrow> bool"
-translations
-  "valid \<Gamma>" \<rightleftharpoons> "\<Gamma> \<in> ctxts"  
-inductive ctxts
 intros
 v1[intro]: "valid []"
 v2[intro]: "\<lbrakk>valid \<Gamma>;a\<sharp>\<Gamma>\<rbrakk>\<Longrightarrow> valid ((a,\<sigma>)#\<Gamma>)"
@@ -41,108 +37,100 @@
   assumes a: "valid \<Gamma>"
   shows   "valid (pi\<bullet>\<Gamma>)"
 using a
-apply(induct)
-apply(auto simp add: fresh_bij)
-done
+by (induct)
+   (auto simp add: fresh_bij)
 
-(* typing judgements *)
-consts
-  typing :: "(((name\<times>ty) list)\<times>lam\<times>ty) set" 
-syntax
-  "_typing_judge" :: "(name\<times>ty) list\<Rightarrow>lam\<Rightarrow>ty\<Rightarrow>bool" (" _ \<turnstile> _ : _ " [80,80,80] 80) 
-translations
-  "\<Gamma> \<turnstile> t : \<tau>" \<rightleftharpoons> "(\<Gamma>,t,\<tau>) \<in> typing"  
-
-inductive typing
+text{* typing judgements *}
+inductive2
+  typing :: "(name\<times>ty) list\<Rightarrow>lam\<Rightarrow>ty\<Rightarrow>bool" (" _ \<turnstile> _ : _ " [80,80,80] 80) 
 intros
-t1[intro]: "\<lbrakk>valid \<Gamma>; (a,\<tau>)\<in>set \<Gamma>\<rbrakk>\<Longrightarrow> \<Gamma> \<turnstile> Var a : \<tau>"
-t2[intro]: "\<lbrakk>\<Gamma> \<turnstile> t1 : \<tau>\<rightarrow>\<sigma>; \<Gamma> \<turnstile> t2 : \<tau>\<rbrakk>\<Longrightarrow> \<Gamma> \<turnstile> App t1 t2 : \<sigma>"
-t3[intro]: "\<lbrakk>a\<sharp>\<Gamma>;((a,\<tau>)#\<Gamma>) \<turnstile> t : \<sigma>\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> Lam [a].t : \<tau>\<rightarrow>\<sigma>"
+t_Var[intro]: "\<lbrakk>valid \<Gamma>; (a,\<tau>)\<in>set \<Gamma>\<rbrakk>\<Longrightarrow> \<Gamma> \<turnstile> Var a : \<tau>"
+t_App[intro]: "\<lbrakk>\<Gamma> \<turnstile> t1 : \<tau>\<rightarrow>\<sigma>; \<Gamma> \<turnstile> t2 : \<tau>\<rbrakk>\<Longrightarrow> \<Gamma> \<turnstile> App t1 t2 : \<sigma>"
+t_Lam[intro]: "\<lbrakk>a\<sharp>\<Gamma>;((a,\<tau>)#\<Gamma>) \<turnstile> t : \<sigma>\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> Lam [a].t : \<tau>\<rightarrow>\<sigma>"
 
 lemma eqvt_typing: 
-  fixes  \<Gamma> :: "(name\<times>ty) list"
-  and    t :: "lam"
-  and    \<tau> :: "ty"
-  and    pi:: "name prm"
+  fixes pi:: "name prm"
   assumes a: "\<Gamma> \<turnstile> t : \<tau>"
   shows "(pi\<bullet>\<Gamma>) \<turnstile> (pi\<bullet>t) : \<tau>"
 using a
 proof (induct)
-  case (t1 \<Gamma> \<tau> a)
+  case (t_Var \<Gamma> a \<tau>)
   have "valid (pi\<bullet>\<Gamma>)" by (rule eqvt_valid)
   moreover
   have "(pi\<bullet>(a,\<tau>))\<in>((pi::name prm)\<bullet>set \<Gamma>)" by (rule pt_set_bij2[OF pt_name_inst, OF at_name_inst])
   ultimately show "(pi\<bullet>\<Gamma>) \<turnstile> ((pi::name prm)\<bullet>Var a) : \<tau>"
     using typing.intros by (force simp add: pt_list_set_pi[OF pt_name_inst, symmetric])
 next 
-  case (t3 \<Gamma> \<sigma> \<tau> a t)
+  case (t_Lam a \<Gamma> \<tau> t \<sigma>)
   moreover have "(pi\<bullet>a)\<sharp>(pi\<bullet>\<Gamma>)" by (simp add: fresh_bij)
   ultimately show "(pi\<bullet>\<Gamma>) \<turnstile> (pi\<bullet>Lam [a].t) :\<tau>\<rightarrow>\<sigma>" by force 
 qed (auto)
 
-lemma typing_induct[consumes 1, case_names t1 t2 t3]:
+text {* the strong induction principle needs to be derived manually *}
+
+lemma typing_induct[consumes 1, case_names t_Var t_App t_Lam]:
   fixes  P :: "'a::fs_name\<Rightarrow>(name\<times>ty) list \<Rightarrow> lam \<Rightarrow> ty \<Rightarrow>bool"
   and    \<Gamma> :: "(name\<times>ty) list"
   and    t :: "lam"
   and    \<tau> :: "ty"
   and    x :: "'a::fs_name"
   assumes a: "\<Gamma> \<turnstile> t : \<tau>"
-  and a1:    "\<And>\<Gamma> (a::name) \<tau> x. valid \<Gamma> \<Longrightarrow> (a,\<tau>) \<in> set \<Gamma> \<Longrightarrow> P x \<Gamma> (Var a) \<tau>"
+  and a1:    "\<And>\<Gamma> a \<tau> x. \<lbrakk>valid \<Gamma>; (a,\<tau>) \<in> set \<Gamma>\<rbrakk> \<Longrightarrow> P x \<Gamma> (Var a) \<tau>"
   and a2:    "\<And>\<Gamma> \<tau> \<sigma> t1 t2 x. 
-              \<Gamma> \<turnstile> t1 : \<tau>\<rightarrow>\<sigma> \<Longrightarrow> (\<And>z. P z \<Gamma> t1 (\<tau>\<rightarrow>\<sigma>)) \<Longrightarrow> \<Gamma> \<turnstile> t2 : \<tau> \<Longrightarrow> (\<And>z. P z \<Gamma> t2 \<tau>)
+              \<lbrakk>\<Gamma> \<turnstile> t1 : \<tau>\<rightarrow>\<sigma>; (\<And>z. P z \<Gamma> t1 (\<tau>\<rightarrow>\<sigma>)); \<Gamma> \<turnstile> t2 : \<tau>; (\<And>z. P z \<Gamma> t2 \<tau>)\<rbrakk>
               \<Longrightarrow> P x \<Gamma> (App t1 t2) \<sigma>"
-  and a3:    "\<And>a \<Gamma> \<tau> \<sigma> t x. a\<sharp>x \<Longrightarrow> a\<sharp>\<Gamma> \<Longrightarrow> ((a,\<tau>) # \<Gamma>) \<turnstile> t : \<sigma> \<Longrightarrow> (\<And>z. P z ((a,\<tau>)#\<Gamma>) t \<sigma>)
+  and a3:    "\<And>a \<Gamma> \<tau> \<sigma> t x. \<lbrakk>a\<sharp>x; a\<sharp>\<Gamma>; ((a,\<tau>)#\<Gamma>) \<turnstile> t : \<sigma>; (\<And>z. P z ((a,\<tau>)#\<Gamma>) t \<sigma>)\<rbrakk>
               \<Longrightarrow> P x \<Gamma> (Lam [a].t) (\<tau>\<rightarrow>\<sigma>)"
   shows "P x \<Gamma> t \<tau>"
 proof -
   from a have "\<And>(pi::name prm) x. P x (pi\<bullet>\<Gamma>) (pi\<bullet>t) \<tau>"
   proof (induct)
-    case (t1 \<Gamma> \<tau> a)
-    have j1: "valid \<Gamma>" by fact
-    have j2: "(a,\<tau>)\<in>set \<Gamma>" by fact
-    from j1 have j3: "valid (pi\<bullet>\<Gamma>)" by (rule eqvt_valid)
-    from j2 have "pi\<bullet>(a,\<tau>)\<in>pi\<bullet>(set \<Gamma>)" by (simp only: pt_set_bij[OF pt_name_inst, OF at_name_inst])  
-    hence j4: "(pi\<bullet>a,\<tau>)\<in>set (pi\<bullet>\<Gamma>)" by (simp add: pt_list_set_pi[OF pt_name_inst])
-    show "P x (pi\<bullet>\<Gamma>) (pi\<bullet>(Var a)) \<tau>" using a1 j3 j4 by simp
+    case (t_Var \<Gamma> a \<tau>)
+    have "valid \<Gamma>" by fact
+    then have "valid (pi\<bullet>\<Gamma>)" by (rule eqvt_valid)
+    moreover
+    have "(a,\<tau>)\<in>set \<Gamma>" by fact
+    then have "pi\<bullet>(a,\<tau>)\<in>pi\<bullet>(set \<Gamma>)" by (simp only: pt_set_bij[OF pt_name_inst, OF at_name_inst])  
+    then have "(pi\<bullet>a,\<tau>)\<in>set (pi\<bullet>\<Gamma>)" by (simp add: pt_list_set_pi[OF pt_name_inst])
+    ultimately show "P x (pi\<bullet>\<Gamma>) (pi\<bullet>(Var a)) \<tau>" using a1 by simp
   next
-    case (t2 \<Gamma> \<sigma> \<tau> t1 t2)
-    thus ?case using a2 by (simp, blast intro: eqvt_typing)
+    case (t_App \<Gamma> t1 \<tau> \<sigma> t2)
+    thus "P x (pi\<bullet>\<Gamma>) (pi\<bullet>(App t1 t2)) \<sigma>" using a2 by (simp, blast intro: eqvt_typing)
   next
-    case (t3 \<Gamma> \<sigma> \<tau> a t)
+    case (t_Lam a \<Gamma> \<tau> t \<sigma>)
     have k1: "a\<sharp>\<Gamma>" by fact
     have k2: "((a,\<tau>)#\<Gamma>)\<turnstile>t:\<sigma>" by fact
     have k3: "\<And>(pi::name prm) (x::'a::fs_name). P x (pi \<bullet>((a,\<tau>)#\<Gamma>)) (pi\<bullet>t) \<sigma>" by fact
     have f: "\<exists>c::name. c\<sharp>(pi\<bullet>a,pi\<bullet>t,pi\<bullet>\<Gamma>,x)"
-      by (rule at_exists_fresh[OF at_name_inst], simp add: fs_name1)
+      by (rule exists_fresh, simp add: fs_name1)
     then obtain c::"name" 
       where f1: "c\<noteq>(pi\<bullet>a)" and f2: "c\<sharp>x" and f3: "c\<sharp>(pi\<bullet>t)" and f4: "c\<sharp>(pi\<bullet>\<Gamma>)"
-      by (force simp add: fresh_prod at_fresh[OF at_name_inst])
-    from k1 have k1a: "(pi\<bullet>a)\<sharp>(pi\<bullet>\<Gamma>)" 
-      by (simp add: pt_fresh_left[OF pt_name_inst, OF at_name_inst] 
-                    pt_rev_pi[OF pt_name_inst, OF at_name_inst])
+      by (force simp add: fresh_prod fresh_atm)
+    from k1 have k1a: "(pi\<bullet>a)\<sharp>(pi\<bullet>\<Gamma>)"  by (simp add: fresh_bij)
     have l1: "(([(c,pi\<bullet>a)]@pi)\<bullet>\<Gamma>) = (pi\<bullet>\<Gamma>)" using f4 k1a 
-      by (simp only: pt2[OF pt_name_inst], rule pt_fresh_fresh[OF pt_name_inst, OF at_name_inst])
+      by (simp only: pt_name2, rule perm_fresh_fresh)
     have "\<And>x. P x (([(c,pi\<bullet>a)]@pi)\<bullet>((a,\<tau>)#\<Gamma>)) (([(c,pi\<bullet>a)]@pi)\<bullet>t) \<sigma>" using k3 by force
     hence l2: "\<And>x. P x ((c, \<tau>)#(pi\<bullet>\<Gamma>)) (([(c,pi\<bullet>a)]@pi)\<bullet>t) \<sigma>" using f1 l1
-      by (force simp add: pt2[OF pt_name_inst]  at_calc[OF at_name_inst])
+      by (force simp add: pt_name2  calc_atm)
     have "(([(c,pi\<bullet>a)]@pi)\<bullet>((a,\<tau>)#\<Gamma>)) \<turnstile> (([(c,pi\<bullet>a)]@pi)\<bullet>t) : \<sigma>" using k2 by (rule eqvt_typing)
     hence l3: "((c, \<tau>)#(pi\<bullet>\<Gamma>)) \<turnstile> (([(c,pi\<bullet>a)]@pi)\<bullet>t) : \<sigma>" using l1 f1 
-      by (force simp add: pt2[OF pt_name_inst]  at_calc[OF at_name_inst])
+      by (force simp add: pt_name2  calc_atm)
     have l4: "P x (pi\<bullet>\<Gamma>) (Lam [c].(([(c,pi\<bullet>a)]@pi)\<bullet>t)) (\<tau> \<rightarrow> \<sigma>)" using f2 f4 l2 l3 a3 by auto
     have alpha: "(Lam [c].([(c,pi\<bullet>a)]\<bullet>(pi\<bullet>t))) = (Lam [(pi\<bullet>a)].(pi\<bullet>t))" using f1 f3
       by (simp add: lam.inject alpha)
-    show "P x (pi\<bullet>\<Gamma>) (pi\<bullet>(Lam [a].t)) (\<tau> \<rightarrow> \<sigma>)" using l4 alpha 
-      by (simp only: pt2[OF pt_name_inst], simp)
+    show "P x (pi\<bullet>\<Gamma>) (pi\<bullet>(Lam [a].t)) (\<tau> \<rightarrow> \<sigma>)" using l4 alpha by (simp only: pt_name2, simp)
   qed
   hence "P x (([]::name prm)\<bullet>\<Gamma>) (([]::name prm)\<bullet>t) \<tau>" by blast
   thus "P x \<Gamma> t \<tau>" by simp
 qed
 
-(* Now it comes: The Weakening Lemma *)
+text {* definition of a subcontext *}
 
-constdefs
+abbreviation
   "sub" :: "(name\<times>ty) list \<Rightarrow> (name\<times>ty) list \<Rightarrow> bool" (" _ \<lless> _ " [80,80] 80)
-  "\<Gamma>1 \<lless> \<Gamma>2 \<equiv> \<forall>a \<sigma>. (a,\<sigma>)\<in>set \<Gamma>1 \<longrightarrow>  (a,\<sigma>)\<in>set \<Gamma>2"
+  "\<Gamma>1 \<lless> \<Gamma>2 \<equiv> \<forall>a \<sigma>. (a,\<sigma>)\<in>set \<Gamma>1 \<longrightarrow> (a,\<sigma>)\<in>set \<Gamma>2"
+
+text {* Now it comes: The Weakening Lemma *}
 
 lemma weakening_version1: 
   assumes a: "\<Gamma>1 \<turnstile> t : \<sigma>" 
@@ -151,11 +139,8 @@
   shows "\<Gamma>2 \<turnstile> t:\<sigma>"
 using a b c
 apply(nominal_induct \<Gamma>1 t \<sigma> avoiding: \<Gamma>2 rule: typing_induct)
-apply(auto simp add: sub_def)
-(* FIXME: this was completely automatic before the *)
-(* change to meta-connectives :o(                  *)
-apply(atomize)
-apply(auto)
+apply(auto | atomize)+
+(* FIXME: meta-quantifiers seem to not ba as "automatic" as object-quantifiers *)
 done
 
 lemma weakening_version2: 
@@ -167,24 +152,26 @@
   and     c: "\<Gamma>1 \<lless> \<Gamma>2"
   shows "\<Gamma>2 \<turnstile> t:\<sigma>"
 using a b c
-proof (nominal_induct \<Gamma>1 t \<sigma> avoiding: \<Gamma>2 rule: typing_induct, auto)
-  case (t1 \<Gamma>1 a \<tau>)  (* variable case *)
-  have "\<Gamma>1 \<lless> \<Gamma>2" 
-  and  "valid \<Gamma>2" 
-  and  "(a,\<tau>)\<in> set \<Gamma>1" by fact+
-  thus "\<Gamma>2 \<turnstile> Var a : \<tau>" by (force simp add: sub_def)
+proof (nominal_induct \<Gamma>1 t \<sigma> avoiding: \<Gamma>2 rule: typing_induct)
+  case (t_Var \<Gamma>1 a \<tau>)  (* variable case *)
+  have "\<Gamma>1 \<lless> \<Gamma>2" by fact 
+  moreover  
+  have "valid \<Gamma>2" by fact 
+  moreover 
+  have "(a,\<tau>)\<in> set \<Gamma>1" by fact
+  ultimately show "\<Gamma>2 \<turnstile> Var a : \<tau>" by auto
 next
-  case (t3 a \<Gamma>1 \<tau> \<sigma> t) (* lambda case *)
-  have a1: "\<Gamma>1 \<lless> \<Gamma>2" by fact
-  have a2: "valid \<Gamma>2" by fact
-  have a3: "a\<sharp>\<Gamma>2" by fact
-  have ih: "\<And>\<Gamma>3. valid \<Gamma>3 \<Longrightarrow> ((a,\<tau>)#\<Gamma>1) \<lless> \<Gamma>3 \<Longrightarrow>  \<Gamma>3 \<turnstile> t:\<sigma>" by fact
-  have "((a,\<tau>)#\<Gamma>1) \<lless> ((a,\<tau>)#\<Gamma>2)" using a1 by (simp add: sub_def)
+  case (t_Lam a \<Gamma>1 \<tau> \<sigma> t) (* lambda case *)
+  have vc: "a\<sharp>\<Gamma>2" by fact (* variable convention *)
+  have ih: "\<And>\<Gamma>3. \<lbrakk>valid \<Gamma>3; ((a,\<tau>)#\<Gamma>1) \<lless> \<Gamma>3\<rbrakk> \<Longrightarrow>  \<Gamma>3 \<turnstile> t:\<sigma>" by fact
+  have "\<Gamma>1 \<lless> \<Gamma>2" by fact
+  then have "((a,\<tau>)#\<Gamma>1) \<lless> ((a,\<tau>)#\<Gamma>2)" by simp
   moreover
-  have "valid ((a,\<tau>)#\<Gamma>2)" using a2 a3 v2 by force
-  ultimately have "((a,\<tau>)#\<Gamma>2) \<turnstile> t:\<sigma>" using ih by force
-  with a3 show "\<Gamma>2 \<turnstile> (Lam [a].t) : \<tau> \<rightarrow> \<sigma>" by force
-qed
+  have "valid \<Gamma>2" by fact
+  then have "valid ((a,\<tau>)#\<Gamma>2)" using vc v2 by simp
+  ultimately have "((a,\<tau>)#\<Gamma>2) \<turnstile> t:\<sigma>" using ih by simp
+  with vc show "\<Gamma>2 \<turnstile> (Lam [a].t) : \<tau> \<rightarrow> \<sigma>" by auto
+qed (auto)
 
 lemma weakening_version3: 
   assumes a: "\<Gamma>1 \<turnstile> t:\<sigma>"
@@ -193,44 +180,43 @@
   shows "\<Gamma>2 \<turnstile> t:\<sigma>"
 using a b c
 proof (nominal_induct \<Gamma>1 t \<sigma> avoiding: \<Gamma>2 rule: typing_induct)
-  case (t3 a \<Gamma>1 \<tau> \<sigma> t) (* lambda case *)
-  have fc: "a\<sharp>\<Gamma>2" by fact
-  have ih: "\<And>\<Gamma>3. valid \<Gamma>3 \<Longrightarrow> ((a,\<tau>)#\<Gamma>1) \<lless> \<Gamma>3  \<Longrightarrow>  \<Gamma>3 \<turnstile> t:\<sigma>" by fact 
-  have a1: "\<Gamma>1 \<lless> \<Gamma>2" by fact
-  have a2: "valid \<Gamma>2" by fact
-  have "((a,\<tau>)#\<Gamma>1) \<lless> ((a,\<tau>)#\<Gamma>2)" using a1 sub_def by simp 
+  case (t_Lam a \<Gamma>1 \<tau> \<sigma> t) (* lambda case *)
+  have vc: "a\<sharp>\<Gamma>2" by fact (* variable convention *)
+  have ih: "\<And>\<Gamma>3. \<lbrakk>valid \<Gamma>3; ((a,\<tau>)#\<Gamma>1) \<lless> \<Gamma>3\<rbrakk> \<Longrightarrow>  \<Gamma>3 \<turnstile> t:\<sigma>" by fact
+  have "\<Gamma>1 \<lless> \<Gamma>2" by fact
+  then have "((a,\<tau>)#\<Gamma>1) \<lless> ((a,\<tau>)#\<Gamma>2)" by simp
   moreover
-  have "valid ((a,\<tau>)#\<Gamma>2)" using a2 fc by force
-  ultimately have "((a,\<tau>)#\<Gamma>2) \<turnstile> t:\<sigma>" using ih by simp 
-  with fc show "\<Gamma>2 \<turnstile> (Lam [a].t) : \<tau> \<rightarrow> \<sigma>" by force
-qed (auto simp add: sub_def) (* app and var case *)
+  have "valid \<Gamma>2" by fact
+  then have "valid ((a,\<tau>)#\<Gamma>2)" using vc v2 by simp
+  ultimately have "((a,\<tau>)#\<Gamma>2) \<turnstile> t:\<sigma>" using ih by simp
+  with vc show "\<Gamma>2 \<turnstile> (Lam [a].t) : \<tau> \<rightarrow> \<sigma>" by auto
+qed (auto) (* app and var case *)
 
-text{* The original induction principle for typing 
-       is not strong enough - so the simple proof fails *}
+text{* The original induction principle for the typing relation
+       is not strong enough - even this simple lemma fails      *}
 lemma weakening_too_weak: 
   assumes a: "\<Gamma>1 \<turnstile> t:\<sigma>"
   and     b: "valid \<Gamma>2" 
   and     c: "\<Gamma>1 \<lless> \<Gamma>2"
   shows "\<Gamma>2 \<turnstile> t:\<sigma>"
-
-thm typing.induct[no_vars]
-
 using a b c
 proof (induct arbitrary: \<Gamma>2)
-  case (t1 \<Gamma>1 \<tau> a) (* variable case *)
-  have "\<Gamma>1 \<lless> \<Gamma>2" 
-  and  "valid \<Gamma>2"
-  and  "(a,\<tau>) \<in> (set \<Gamma>1)" by fact+ 
-  thus "\<Gamma>2 \<turnstile> Var a : \<tau>" by (force simp add: sub_def)
+  case (t_Var \<Gamma>1 a \<tau>) (* variable case *)
+  have "\<Gamma>1 \<lless> \<Gamma>2" by fact
+  moreover
+  have "valid \<Gamma>2" by fact
+  moreover
+  have "(a,\<tau>) \<in> (set \<Gamma>1)" by fact 
+  ultimately show "\<Gamma>2 \<turnstile> Var a : \<tau>" by auto
 next
-  case (t3 \<Gamma>1 \<sigma> \<tau> a t) (* lambda case *)
+  case (t_Lam a \<Gamma>1 \<tau> t \<sigma>) (* lambda case *)
   (* all assumption in this case*)
   have a0: "a\<sharp>\<Gamma>1" by fact
   have a1: "((a,\<tau>)#\<Gamma>1) \<turnstile> t : \<sigma>" by fact
   have a2: "\<Gamma>1 \<lless> \<Gamma>2" by fact
   have a3: "valid \<Gamma>2" by fact
-  have ih: "\<And>\<Gamma>3. valid \<Gamma>3 \<Longrightarrow> ((a,\<tau>)#\<Gamma>1) \<lless> \<Gamma>3  \<Longrightarrow>  \<Gamma>3 \<turnstile> t:\<sigma>" by fact
-  have "((a,\<tau>)#\<Gamma>1) \<lless> ((a,\<tau>)#\<Gamma>2)" using a2 by (simp add: sub_def)
+  have ih: "\<And>\<Gamma>3. \<lbrakk>valid \<Gamma>3; ((a,\<tau>)#\<Gamma>1) \<lless> \<Gamma>3\<rbrakk>  \<Longrightarrow>  \<Gamma>3 \<turnstile> t:\<sigma>" by fact
+  have "((a,\<tau>)#\<Gamma>1) \<lless> ((a,\<tau>)#\<Gamma>2)" using a2 by simp
   moreover
   have "valid ((a,\<tau>)#\<Gamma>2)" using v2 (* fails *) 
     oops