diff -r dd50de393330 -r 3dcc34f18bfa src/HOL/Nominal/Examples/Weakening.thy
--- a/src/HOL/Nominal/Examples/Weakening.thy	Wed Nov 30 12:28:47 2005 +0100
+++ b/src/HOL/Nominal/Examples/Weakening.thy	Wed Nov 30 14:27:09 2005 +0100
@@ -96,35 +96,35 @@
 
 
 lemma typing_induct_weak[THEN spec, case_names t1 t2 t3]:
-  fixes  P :: "(name\<times>ty) list \<Rightarrow> lam \<Rightarrow> ty \<Rightarrow>'a\<Rightarrow>bool"
+  fixes  P :: "'a\<Rightarrow>(name\<times>ty) list \<Rightarrow> lam \<Rightarrow> ty \<Rightarrow>bool"
   and    \<Gamma> :: "(name\<times>ty) list"
   and    t :: "lam"
   and    \<tau> :: "ty"
   assumes a: "\<Gamma> \<turnstile> t : \<tau>"
-  and a1:    "\<And>x \<Gamma> (a::name) \<tau>. valid \<Gamma> \<Longrightarrow> (a,\<tau>) \<in> set \<Gamma> \<Longrightarrow> P \<Gamma> (Var a) \<tau> x"
-  and a2:    "\<And>x \<Gamma> \<tau> \<sigma> t1 t2. 
-              \<Gamma> \<turnstile> t1 : \<tau>\<rightarrow>\<sigma> \<Longrightarrow> (\<forall>z. P \<Gamma> t1 (\<tau>\<rightarrow>\<sigma>) z) \<Longrightarrow> \<Gamma> \<turnstile> t2 : \<tau> \<Longrightarrow> (\<forall>z. P \<Gamma> t2 \<tau> z)
-              \<Longrightarrow> P \<Gamma> (App t1 t2) \<sigma> x"
-  and a3:    "\<And>x (a::name) \<Gamma> \<tau> \<sigma> t. 
-              a\<sharp>\<Gamma> \<Longrightarrow> ((a,\<tau>) # \<Gamma>) \<turnstile> t : \<sigma> \<Longrightarrow> (\<forall>z. P ((a,\<tau>)#\<Gamma>) t \<sigma> z)
-              \<Longrightarrow> P \<Gamma> (Lam [a].t) (\<tau>\<rightarrow>\<sigma>) x"
-  shows "\<forall>x. P \<Gamma> t \<tau> x"
+  and a1:    "\<And>\<Gamma> (a::name) \<tau> x. valid \<Gamma> \<Longrightarrow> (a,\<tau>) \<in> set \<Gamma> \<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>)
+              \<Longrightarrow> P x \<Gamma> (App t1 t2) \<sigma>"
+  and a3:    "\<And>(a::name) \<Gamma> \<tau> \<sigma> t x. 
+              a\<sharp>\<Gamma> \<Longrightarrow> ((a,\<tau>) # \<Gamma>) \<turnstile> t : \<sigma> \<Longrightarrow> (\<And>z. P z ((a,\<tau>)#\<Gamma>) t \<sigma>)
+              \<Longrightarrow> P x \<Gamma> (Lam [a].t) (\<tau>\<rightarrow>\<sigma>)"
+  shows "\<forall>x. P x \<Gamma> t \<tau>"
 using a by (induct, simp_all add: a1 a2 a3)
 
 lemma typing_induct_aux[rule_format]:
-  fixes  P :: "(name\<times>ty) list \<Rightarrow> lam \<Rightarrow> ty \<Rightarrow>'a::fs_name\<Rightarrow>bool"
+  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"
   assumes a: "\<Gamma> \<turnstile> t : \<tau>"
-  and a1:    "\<And>x \<Gamma> (a::name) \<tau>. valid \<Gamma> \<Longrightarrow> (a,\<tau>) \<in> set \<Gamma> \<Longrightarrow> P \<Gamma> (Var a) \<tau> x"
-  and a2:    "\<And>x \<Gamma> \<tau> \<sigma> t1 t2. 
-              \<Gamma> \<turnstile> t1 : \<tau>\<rightarrow>\<sigma> \<Longrightarrow> (\<And>z. P \<Gamma> t1 (\<tau>\<rightarrow>\<sigma>) z) \<Longrightarrow> \<Gamma> \<turnstile> t2 : \<tau> \<Longrightarrow> (\<And>z. P \<Gamma> t2 \<tau> z)
-              \<Longrightarrow> P \<Gamma> (App t1 t2) \<sigma> x"
-  and a3:    "\<And>x (a::name) \<Gamma> \<tau> \<sigma> t. 
-              a\<sharp>x \<Longrightarrow> a\<sharp>\<Gamma> \<Longrightarrow> ((a,\<tau>) # \<Gamma>) \<turnstile> t : \<sigma> \<Longrightarrow> (\<forall>z. P ((a,\<tau>)#\<Gamma>) t \<sigma> z)
-              \<Longrightarrow> P \<Gamma> (Lam [a].t) (\<tau>\<rightarrow>\<sigma>) x"
-  shows "\<forall>(pi::name prm) (x::'a::fs_name). P (pi\<bullet>\<Gamma>) (pi\<bullet>t) \<tau> x"
+  and a1:    "\<And>\<Gamma> (a::name) \<tau> x. valid \<Gamma> \<Longrightarrow> (a,\<tau>) \<in> set \<Gamma> \<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>)
+              \<Longrightarrow> P x \<Gamma> (App t1 t2) \<sigma>"
+  and a3:    "\<And>(a::name) \<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>)
+              \<Longrightarrow> P x \<Gamma> (Lam [a].t) (\<tau>\<rightarrow>\<sigma>)"
+  shows "\<forall>(pi::name prm) (x::'a::fs_name). P x (pi\<bullet>\<Gamma>) (pi\<bullet>t) \<tau>"
 using a
 proof (induct)
   case (t1 \<Gamma> \<tau> a)
@@ -136,7 +136,7 @@
     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 (pi\<bullet>\<Gamma>) (Var (pi\<bullet>a)) \<tau> x" using a1 j3 j4 by force
+    show "P x (pi\<bullet>\<Gamma>) (Var (pi\<bullet>a)) \<tau>" using a1 j3 j4 by force
   qed
 next
   case (t2 \<Gamma> \<sigma> \<tau> t1 t2)
@@ -145,7 +145,7 @@
   case (t3 \<Gamma> \<sigma> \<tau> a t)
   have k1: "a\<sharp>\<Gamma>" by fact
   have k2: "((a,\<tau>)#\<Gamma>)\<turnstile>t:\<sigma>" by fact
-  have k3: "\<forall>(pi::name prm) (x::'a::fs_name). P (pi \<bullet> ((a,\<tau>)#\<Gamma>)) (pi\<bullet>t) \<sigma> x" by fact
+  have k3: "\<forall>(pi::name prm) (x::'a::fs_name). P x (pi \<bullet>((a,\<tau>)#\<Gamma>)) (pi\<bullet>t) \<sigma>" by fact
   show ?case
   proof (intro strip, simp)
     fix pi::"name prm" and x::"'a::fs_name"
@@ -159,36 +159,36 @@
                     pt_rev_pi[OF pt_name_inst, OF at_name_inst])
     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])
-    have "\<forall>x. P (([(c,pi\<bullet>a)]@pi)\<bullet>((a,\<tau>)#\<Gamma>)) (([(c,pi\<bullet>a)]@pi)\<bullet>t) \<sigma> x" using k3 by force
-    hence l2: "\<forall>x. P ((c, \<tau>)#(pi\<bullet>\<Gamma>)) (([(c,pi\<bullet>a)]@pi)\<bullet>t) \<sigma> x" using f1 l1
+    have "\<forall>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: "\<forall>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] split: if_splits)
     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] split: if_splits)
-    have l4: "P (pi\<bullet>\<Gamma>) (Lam [c].(([(c,pi\<bullet>a)]@pi)\<bullet>t)) (\<tau> \<rightarrow> \<sigma>) x" using f2 f4 l2 l3 a3 by auto
+    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 (pi\<bullet>\<Gamma>) (Lam [(pi\<bullet>a)].(pi\<bullet>t)) (\<tau> \<rightarrow> \<sigma>) x" using l4 alpha 
+    show "P x (pi\<bullet>\<Gamma>) (Lam [(pi\<bullet>a)].(pi\<bullet>t)) (\<tau> \<rightarrow> \<sigma>)" using l4 alpha 
       by (simp only: pt2[OF pt_name_inst])
   qed
 qed
 
-lemma typing_induct[case_names t1 t2 t3]:
-  fixes  P :: "(name\<times>ty) list \<Rightarrow> lam \<Rightarrow> ty \<Rightarrow>'a::fs_name\<Rightarrow>bool"
+lemma typing_induct[consumes 1, case_names t1 t2 t3]:
+  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>x \<Gamma> (a::name) \<tau>. valid \<Gamma> \<Longrightarrow> (a,\<tau>) \<in> set \<Gamma> \<Longrightarrow> P \<Gamma> (Var a) \<tau> x"
-  and a2:    "\<And>x \<Gamma> \<tau> \<sigma> t1 t2. 
-              \<Gamma> \<turnstile> t1 : \<tau>\<rightarrow>\<sigma> \<Longrightarrow> (\<forall>z. P \<Gamma> t1 (\<tau>\<rightarrow>\<sigma>) z) \<Longrightarrow> \<Gamma> \<turnstile> t2 : \<tau> \<Longrightarrow> (\<forall>z. P \<Gamma> t2 \<tau> z)
-              \<Longrightarrow> P \<Gamma> (App t1 t2) \<sigma> x"
-  and a3:    "\<And>x (a::name) \<Gamma> \<tau> \<sigma> t. 
-              a\<sharp>x \<Longrightarrow> a\<sharp>\<Gamma> \<Longrightarrow> ((a,\<tau>) # \<Gamma>) \<turnstile> t : \<sigma> \<Longrightarrow> (\<forall>z. P ((a,\<tau>)#\<Gamma>) t \<sigma> z)
-              \<Longrightarrow> P \<Gamma> (Lam [a].t) (\<tau>\<rightarrow>\<sigma>) x"
-  shows "P \<Gamma> t \<tau> x"
-using a a1 a2 a3 typing_induct_aux[of "\<Gamma>" "t" "\<tau>" "P" "[]" "x", simplified] by force
+  and a1:    "\<And>\<Gamma> (a::name) \<tau> x. valid \<Gamma> \<Longrightarrow> (a,\<tau>) \<in> set \<Gamma> \<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>)
+              \<Longrightarrow> P x \<Gamma> (App t1 t2) \<sigma>"
+  and a3:    "\<And>(a::name) \<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>)
+              \<Longrightarrow> P x \<Gamma> (Lam [a].t) (\<tau>\<rightarrow>\<sigma>)"
+  shows "P x \<Gamma> t \<tau>"
+using a a1 a2 a3 typing_induct_aux[of "\<Gamma>" "t" "\<tau>" "P" "x" "[]", simplified] by force
 
 
 (* Now it comes: The Weakening Lemma *)
@@ -198,11 +198,15 @@
   "\<Gamma>1 \<lless> \<Gamma>2 \<equiv> \<forall>a \<sigma>. (a,\<sigma>)\<in>set \<Gamma>1 \<longrightarrow>  (a,\<sigma>)\<in>set \<Gamma>2"
 
 lemma weakening_version1[rule_format]: 
-  assumes a: "\<Gamma>1 \<turnstile> t : \<sigma>"
-  shows "valid \<Gamma>2 \<longrightarrow> \<Gamma>1 \<lless> \<Gamma>2 \<longrightarrow> \<Gamma>2 \<turnstile> t:\<sigma>"
-using a
-apply(nominal_induct \<Gamma>1 t \<sigma> rule: typing_induct)
+  assumes "\<Gamma>1 \<turnstile> t : \<sigma>" 
+  and     "valid \<Gamma>2" 
+  and     "\<Gamma>1 \<lless> \<Gamma>2"
+  shows "\<Gamma>2 \<turnstile> t:\<sigma>"
+using prems
+apply(nominal_induct \<Gamma>1 t \<sigma> fresh: \<Gamma>2 rule: typing_induct)
 apply(auto simp add: sub_def)
+apply(atomize) (* FIXME: earlier this was completely automatic :o(  *)
+apply(auto)
 done
 
 lemma weakening_version2[rule_format]: 
@@ -211,19 +215,19 @@
   and   \<tau> ::"ty"
   assumes a: "\<Gamma>1 \<turnstile> t:\<sigma>"
   shows "valid \<Gamma>2 \<longrightarrow> \<Gamma>1 \<lless> \<Gamma>2 \<longrightarrow> \<Gamma>2 \<turnstile> t:\<sigma>"
-using a
-proof (nominal_induct \<Gamma>1 t \<sigma> rule: typing_induct, auto)
-  case (t1 \<Gamma>2 \<Gamma>1 a \<tau>)  (* variable case *)
+using prems
+proof (nominal_induct \<Gamma>1 t \<sigma> fresh: \<Gamma>2 rule: typing_induct, auto)
+  case (t1 \<Gamma>1 a \<tau> \<Gamma>2)  (* variable case *)
   assume "\<Gamma>1 \<lless> \<Gamma>2" 
   and    "valid \<Gamma>2" 
   and    "(a,\<tau>)\<in> set \<Gamma>1" 
   thus  "\<Gamma>2 \<turnstile> Var a : \<tau>" by (force simp add: sub_def)
 next
-  case (t3 \<Gamma>2 a \<Gamma>1 \<tau> \<sigma> t) (* lambda case *)
+  case (t3 a \<Gamma>1 \<tau> \<sigma> t \<Gamma>2) (* lambda case *)
   assume a1: "\<Gamma>1 \<lless> \<Gamma>2"
   and    a2: "valid \<Gamma>2"
   and    a3: "a\<sharp>\<Gamma>2"
-  have i: "\<forall>\<Gamma>3. valid \<Gamma>3 \<longrightarrow> ((a,\<tau>)#\<Gamma>1) \<lless> \<Gamma>3 \<longrightarrow>  \<Gamma>3 \<turnstile> t:\<sigma>" by fact
+  have i: "\<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)
   moreover
   have "valid ((a,\<tau>)#\<Gamma>2)" using a2 a3 v2 by force
@@ -235,19 +239,19 @@
   fixes \<Gamma>1::"(name\<times>ty) list"
   and   t ::"lam"
   and   \<tau> ::"ty"
-  assumes a: "\<Gamma>1 \<turnstile> t:\<sigma>"
+  assumes "\<Gamma>1 \<turnstile> t:\<sigma>"
   shows "valid \<Gamma>2 \<longrightarrow> \<Gamma>1 \<lless> \<Gamma>2 \<longrightarrow> \<Gamma>2 \<turnstile> t:\<sigma>"
-using a
-proof (nominal_induct \<Gamma>1 t \<sigma> rule: typing_induct)
-  case (t1 \<Gamma>2 \<Gamma>1 a \<tau>)  (* variable case *)
+using prems
+proof (nominal_induct \<Gamma>1 t \<sigma> fresh: \<Gamma>2 rule: typing_induct)
+  case (t1 \<Gamma>1 a \<tau> \<Gamma>2)  (* variable case *)
   thus "valid \<Gamma>2 \<longrightarrow> \<Gamma>1 \<lless> \<Gamma>2 \<longrightarrow> \<Gamma>2 \<turnstile> Var a : \<tau>" by (force simp add: sub_def)
 next 
-  case (t2 \<Gamma>2 \<Gamma>1 \<tau> \<sigma> t1 t2)  (* variable case *)
+  case (t2 \<Gamma>1 \<tau> \<sigma> t1 t2 \<Gamma>2)  (* variable case *)
   thus "valid \<Gamma>2 \<longrightarrow> \<Gamma>1 \<lless> \<Gamma>2 \<longrightarrow> \<Gamma>2 \<turnstile> App t1 t2 : \<sigma>" by force
 next
-  case (t3 \<Gamma>2 a \<Gamma>1 \<tau> \<sigma> t) (* lambda case *)
-  have a3: "a\<sharp>\<Gamma>2" 
-  and  i: "\<forall>\<Gamma>3. valid \<Gamma>3 \<longrightarrow> ((a,\<tau>)#\<Gamma>1) \<lless> \<Gamma>3 \<longrightarrow>  \<Gamma>3 \<turnstile> t:\<sigma>" by fact
+  case (t3 a \<Gamma>1 \<tau> \<sigma> t \<Gamma>2) (* lambda case *)
+  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
   show "valid \<Gamma>2 \<longrightarrow> \<Gamma>1 \<lless> \<Gamma>2 \<longrightarrow> \<Gamma>2 \<turnstile> (Lam [a].t) : \<tau> \<rightarrow> \<sigma>"
     proof (intro strip)
       assume a1: "\<Gamma>1 \<lless> \<Gamma>2"
@@ -255,19 +259,19 @@
       have "((a,\<tau>)#\<Gamma>1) \<lless> ((a,\<tau>)#\<Gamma>2)" using a1 by (simp add: sub_def)
       moreover
       have "valid ((a,\<tau>)#\<Gamma>2)" using a2 a3 v2 by force
-      ultimately have "((a,\<tau>)#\<Gamma>2) \<turnstile> t:\<sigma>" using i 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
 qed
 
 lemma weakening_version4[rule_format]: 
-  assumes a: "\<Gamma>1 \<turnstile> t:\<sigma>"
+  assumes "\<Gamma>1 \<turnstile> t:\<sigma>"
   shows "valid \<Gamma>2 \<longrightarrow> \<Gamma>1 \<lless> \<Gamma>2 \<longrightarrow> \<Gamma>2 \<turnstile> t:\<sigma>"
-using a
-proof (nominal_induct \<Gamma>1 t \<sigma> rule: typing_induct)
-  case (t3 \<Gamma>2 a \<Gamma>1 \<tau> \<sigma> t) (* lambda case *)
-  have fc: "a\<sharp>\<Gamma>2" 
-  and ih: "\<forall>\<Gamma>3. valid \<Gamma>3 \<longrightarrow> ((a,\<tau>)#\<Gamma>1) \<lless> \<Gamma>3  \<longrightarrow>  \<Gamma>3 \<turnstile> t:\<sigma>" by fact 
+using prems
+proof (nominal_induct \<Gamma>1 t \<sigma> fresh: \<Gamma>2 rule: typing_induct)
+  case (t3 a \<Gamma>1 \<tau> \<sigma> t \<Gamma>2) (* 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 
   show "valid \<Gamma>2 \<longrightarrow> \<Gamma>1 \<lless> \<Gamma>2 \<longrightarrow> \<Gamma>2 \<turnstile> (Lam [a].t) : \<tau> \<rightarrow> \<sigma>"
   proof (intro strip)
     assume a1: "\<Gamma>1 \<lless> \<Gamma>2"
@@ -284,17 +288,17 @@
 (* original induction principle is not strong *)
 (* enough - so the simple proof fails         *)
 lemma weakening_too_weak[rule_format]: 
-  assumes a: "\<Gamma>1 \<turnstile> t:\<sigma>"
+  assumes "\<Gamma>1 \<turnstile> t:\<sigma>"
   shows "valid \<Gamma>2 \<longrightarrow> \<Gamma>1 \<lless> \<Gamma>2 \<longrightarrow> \<Gamma>2 \<turnstile> t:\<sigma>"
-using a
-proof (nominal_induct \<Gamma>1 t \<sigma> rule: typing_induct_weak, auto)
-  case (t1 \<Gamma>2 \<Gamma>1 a \<tau>)  (* variable case *)
+using prems
+proof (induct, auto)
+  case (t1 \<Gamma>1 \<tau> a) (* variable case *)
   assume "\<Gamma>1 \<lless> \<Gamma>2"
   and    "valid \<Gamma>2"
-  and    "(a,\<tau>)\<in> set \<Gamma>1" 
+  and    "(a,\<tau>) \<in> (set \<Gamma>1)" 
   thus "\<Gamma>2 \<turnstile> Var a : \<tau>" by (force simp add: sub_def)
 next
-  case (t3 \<Gamma>2 a \<Gamma>1 \<tau> \<sigma> t) (* lambda case *)
+  case (t3 \<Gamma>1 \<tau> \<sigma> a t) (* lambda case *)
   assume a1: "\<Gamma>1 \<lless> \<Gamma>2"
   and    a2: "valid \<Gamma>2"
   and    i: "\<forall>\<Gamma>3. valid \<Gamma>3 \<longrightarrow> ((a,\<tau>)#\<Gamma>1) \<lless> \<Gamma>3  \<longrightarrow>  \<Gamma>3 \<turnstile> t:\<sigma>"