# HG changeset patch
# User urbanc
# Date 1175012902 -7200
# Node ID 62c76754da32598e5bd12fa1b4e88247e7da7376
# Parent  7b9f346ac3662cc09e49ff502486b6159e6571d8
adapted to new nominal_inductive infrastructure

diff -r 7b9f346ac366 -r 62c76754da32 src/HOL/Nominal/Examples/Weakening.thy
--- a/src/HOL/Nominal/Examples/Weakening.thy	Tue Mar 27 17:57:42 2007 +0200
+++ b/src/HOL/Nominal/Examples/Weakening.thy	Tue Mar 27 18:28:22 2007 +0200
@@ -18,113 +18,43 @@
     TVar "nat"
   | TArr "ty" "ty" (infix "\<rightarrow>" 200)
 
-lemma ty_perm[simp]:
-  fixes pi ::"name prm"
-  and   T  ::"ty"
-  shows "pi\<bullet>T = T"
-by (induct T rule: ty.induct_weak)
-   (simp_all add: perm_nat_def)  
+lemma ty_fresh:
+  fixes x::"name"
+  and   T::"ty"
+  shows "x\<sharp>T"
+by (nominal_induct T rule: ty.induct)
+   (auto simp add: fresh_nat)
 
 text {* valid contexts *}
+
 inductive2
   valid :: "(name\<times>ty) list \<Rightarrow> bool"
 where
     v1[intro]: "valid []"
   | v2[intro]: "\<lbrakk>valid \<Gamma>;a\<sharp>\<Gamma>\<rbrakk>\<Longrightarrow> valid ((a,\<sigma>)#\<Gamma>)"
 
-lemma eqvt_valid[eqvt]:
-  fixes   pi:: "name prm"
-  assumes a: "valid \<Gamma>"
-  shows   "valid (pi\<bullet>\<Gamma>)"
-using a
-by (induct) (auto simp add: fresh_bij)
+equivariance valid
 
 text{* typing judgements *}
 inductive2
   typing :: "(name\<times>ty) list\<Rightarrow>lam\<Rightarrow>ty\<Rightarrow>bool" (" _ \<turnstile> _ : _ " [80,80,80] 80) 
 where
-    t_Var[intro]: "\<lbrakk>valid \<Gamma>; (a,T)\<in>set \<Gamma>\<rbrakk>\<Longrightarrow> \<Gamma> \<turnstile> Var a : T"
+    t_Var[intro]: "\<lbrakk>valid \<Gamma>; (x,T)\<in>set \<Gamma>\<rbrakk>\<Longrightarrow> \<Gamma> \<turnstile> Var x : T"
   | t_App[intro]: "\<lbrakk>\<Gamma> \<turnstile> t1 : T1\<rightarrow>T2; \<Gamma> \<turnstile> t2 : T1\<rbrakk>\<Longrightarrow> \<Gamma> \<turnstile> App t1 t2 : T2"
-  | t_Lam[intro]: "\<lbrakk>a\<sharp>\<Gamma>;((a,T1)#\<Gamma>) \<turnstile> t : T2\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> Lam [a].t : T1\<rightarrow>T2"
-
-lemma eqvt_typing[eqvt]: 
-  fixes pi:: "name prm"
-  assumes a: "\<Gamma> \<turnstile> t : T"
-  shows "(pi\<bullet>\<Gamma>) \<turnstile> (pi\<bullet>t) : (pi\<bullet>T)"
-using a
-proof (induct)
-  case (t_Var \<Gamma> a T)
-  have "valid (pi\<bullet>\<Gamma>)" by (rule eqvt_valid)
-  moreover
-  have "(pi\<bullet>(a,T))\<in>(pi\<bullet>set \<Gamma>)" by (rule pt_set_bij2[OF pt_name_inst, OF at_name_inst])
-  ultimately show "(pi\<bullet>\<Gamma>) \<turnstile> (pi\<bullet>Var a) : (pi\<bullet>T)"
-    using typing.intros by (force simp add: set_eqvt)
-next 
-  case (t_Lam a \<Gamma> T1 t T2)
-  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) :(pi\<bullet>T1\<rightarrow>T2)" by force 
-qed (auto)
-
-text {* the strong induction principle needs to be derived manually *}
+  | t_Lam[intro]: "\<lbrakk>x\<sharp>\<Gamma>;((x,T1)#\<Gamma>) \<turnstile> t : T2\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> Lam [x].t : T1\<rightarrow>T2"
 
-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    T :: "ty"
-  and    x :: "'a::fs_name"
-  assumes a: "\<Gamma> \<turnstile> t : T"
-  and a1:    "\<And>\<Gamma> a T x. \<lbrakk>valid \<Gamma>; (a,T) \<in> set \<Gamma>\<rbrakk> \<Longrightarrow> P x \<Gamma> (Var a) T"
-  and a2:    "\<And>\<Gamma> T1 T2 t1 t2 x. \<lbrakk>\<And>z. P z \<Gamma> t1 (T1\<rightarrow>T2); \<And>z. P z \<Gamma> t2 T1\<rbrakk>
-              \<Longrightarrow> P x \<Gamma> (App t1 t2) T2"
-  and a3:    "\<And>a \<Gamma> T1 T2 t x. \<lbrakk>a\<sharp>x; a\<sharp>\<Gamma>; \<And>z. P z ((a,T1)#\<Gamma>) t T2\<rbrakk>
-              \<Longrightarrow> P x \<Gamma> (Lam [a].t) (T1\<rightarrow>T2)"
-  shows "P x \<Gamma> t T"
-proof -
-  from a have "\<And>(pi::name prm) x. P x (pi\<bullet>\<Gamma>) (pi\<bullet>t) (pi\<bullet>T)"
-  proof (induct)
-    case (t_Var \<Gamma> a T)
-    have "valid \<Gamma>" by fact
-    then have "valid (pi\<bullet>\<Gamma>)" by (rule eqvt)
-    moreover
-    have "(a,T)\<in>set \<Gamma>" by fact
-    then have "pi\<bullet>(a,T)\<in>pi\<bullet>(set \<Gamma>)" by (simp only: pt_set_bij[OF pt_name_inst, OF at_name_inst])  
-    then have "(pi\<bullet>a,T)\<in>set (pi\<bullet>\<Gamma>)" by (simp add: set_eqvt)
-    ultimately show "P x (pi\<bullet>\<Gamma>) (pi\<bullet>(Var a)) (pi\<bullet>T)" using a1 by simp
-  next
-    case (t_App \<Gamma> t1 T1 T2 t2)
-    thus "P x (pi\<bullet>\<Gamma>) (pi\<bullet>(App t1 t2)) (pi\<bullet>T2)" using a2 
-      by (simp only: eqvt) (blast)
-  next
-    case (t_Lam a \<Gamma> T1 t T2)
-    obtain c::"name" where fs: "c\<sharp>(pi\<bullet>a,pi\<bullet>t,pi\<bullet>\<Gamma>,x)" by (rule exists_fresh[OF fs_name1])
-    let ?sw="[(pi\<bullet>a,c)]"
-    let ?pi'="?sw@pi"
-    have f1: "a\<sharp>\<Gamma>" by fact
-    have f2: "(pi\<bullet>a)\<sharp>(pi\<bullet>\<Gamma>)" using f1 by (simp add: fresh_bij)
-    have f3: "c\<sharp>?pi'\<bullet>\<Gamma>" using f1 by (auto simp add: pt_name2 fresh_left calc_atm perm_pi_simp)
-    have ih1: "\<And>x. P x (?pi'\<bullet>((a,T1)#\<Gamma>)) (?pi'\<bullet>t) (?pi'\<bullet>T2)" by fact
-    then have "\<And>x. P x ((c,T1)#(?pi'\<bullet>\<Gamma>)) (?pi'\<bullet>t) (?pi'\<bullet>T2)" by (simp add: calc_atm)
-    then have "P x (?pi'\<bullet>\<Gamma>) (Lam [c].(?pi'\<bullet>t)) (T1\<rightarrow>T2)" using a3 f3 fs by simp
-    then have "P x (?sw\<bullet>pi\<bullet>\<Gamma>) (?sw\<bullet>(Lam [(pi\<bullet>a)].(pi\<bullet>t))) (T1\<rightarrow>T2)" 
-      by (simp del: append_Cons add: calc_atm pt_name2)
-    moreover have "(?sw\<bullet>(pi\<bullet>\<Gamma>)) = (pi\<bullet>\<Gamma>)" 
-      by (rule perm_fresh_fresh) (simp_all add: fs f2)
-    moreover have "(?sw\<bullet>(Lam [(pi\<bullet>a)].(pi\<bullet>t))) = Lam [(pi\<bullet>a)].(pi\<bullet>t)" 
-      by (rule perm_fresh_fresh) (simp_all add: fs f2 abs_fresh)
-    ultimately show "P x (pi\<bullet>\<Gamma>) (pi\<bullet>(Lam [a].t)) (pi\<bullet>T1\<rightarrow>T2)" by (simp)
-  qed
-  hence "P x (([]::name prm)\<bullet>\<Gamma>) (([]::name prm)\<bullet>t) (([]::name prm)\<bullet>T)" by blast
-  thus "P x \<Gamma> t T" by simp
-qed
+(* automatically deriving the strong induction principle *)
+nominal_inductive typing
+  by (simp_all add: abs_fresh ty_fresh)
 
 text {* definition of a subcontext *}
 
 abbreviation
-  "sub" :: "(name\<times>ty) list \<Rightarrow> (name\<times>ty) list \<Rightarrow> bool" (" _ \<lless> _ " [80,80] 80) where
-  "\<Gamma>1 \<lless> \<Gamma>2 \<equiv> \<forall>a \<sigma>. (a,\<sigma>)\<in>set \<Gamma>1 \<longrightarrow> (a,\<sigma>)\<in>set \<Gamma>2"
+  "sub_context" :: "(name\<times>ty) list \<Rightarrow> (name\<times>ty) list \<Rightarrow> bool" (" _ \<lless> _ " [80,80] 80) 
+where
+  "\<Gamma>1 \<lless> \<Gamma>2 \<equiv> \<forall>x T. (x,T)\<in>set \<Gamma>1 \<longrightarrow> (x,T)\<in>set \<Gamma>2"
 
-text {* now it comes: The Weakening Lemma *}
+text {* Now it comes: The Weakening Lemma *}
 
 lemma weakening_version1: 
   assumes a: "\<Gamma>1 \<turnstile> t : T" 
@@ -132,9 +62,8 @@
   and     c: "\<Gamma>1 \<lless> \<Gamma>2"
   shows "\<Gamma>2 \<turnstile> t : T"
 using a b c
-by (nominal_induct \<Gamma>1 t T avoiding: \<Gamma>2 rule: typing_induct)
+by (nominal_induct \<Gamma>1 t T avoiding: \<Gamma>2 rule: typing.strong_induct)
    (auto | atomize)+
-(* FIXME: meta-quantifiers seem to be not as "automatic" as object-quantifiers *)
 
 lemma weakening_version2: 
   fixes \<Gamma>1::"(name\<times>ty) list"
@@ -145,25 +74,25 @@
   and     c: "\<Gamma>1 \<lless> \<Gamma>2"
   shows "\<Gamma>2 \<turnstile> t:T"
 using a b c
-proof (nominal_induct \<Gamma>1 t T avoiding: \<Gamma>2 rule: typing_induct)
-  case (t_Var \<Gamma>1 a T)  (* variable case *)
+proof (nominal_induct \<Gamma>1 t T avoiding: \<Gamma>2 rule: typing.strong_induct)
+  case (t_Var \<Gamma>1 x T)  (* variable case *)
   have "\<Gamma>1 \<lless> \<Gamma>2" by fact 
   moreover  
   have "valid \<Gamma>2" by fact 
   moreover 
-  have "(a,T)\<in> set \<Gamma>1" by fact
-  ultimately show "\<Gamma>2 \<turnstile> Var a : T" by auto
+  have "(x,T)\<in> set \<Gamma>1" by fact
+  ultimately show "\<Gamma>2 \<turnstile> Var x : T" by auto
 next
-  case (t_Lam a \<Gamma>1 T1 T2 t) (* lambda case *)
-  have vc: "a\<sharp>\<Gamma>2" by fact (* variable convention *)
-  have ih: "\<And>\<Gamma>3. \<lbrakk>valid \<Gamma>3; ((a,T1)#\<Gamma>1) \<lless> \<Gamma>3\<rbrakk> \<Longrightarrow>  \<Gamma>3 \<turnstile> t:T2" by fact
+  case (t_Lam x \<Gamma>1 T1 t T2) (* lambda case *)
+  have vc: "x\<sharp>\<Gamma>2" by fact   (* variable convention *)
+  have ih: "\<And>\<Gamma>3. \<lbrakk>valid \<Gamma>3; ((x,T1)#\<Gamma>1) \<lless> \<Gamma>3\<rbrakk> \<Longrightarrow>  \<Gamma>3 \<turnstile> t:T2" by fact
   have "\<Gamma>1 \<lless> \<Gamma>2" by fact
-  then have "((a,T1)#\<Gamma>1) \<lless> ((a,T1)#\<Gamma>2)" by simp
+  then have "((x,T1)#\<Gamma>1) \<lless> ((x,T1)#\<Gamma>2)" by simp
   moreover
   have "valid \<Gamma>2" by fact
-  then have "valid ((a,T1)#\<Gamma>2)" using vc by (simp add: v2)
-  ultimately have "((a,T1)#\<Gamma>2) \<turnstile> t:T2" using ih by simp
-  with vc show "\<Gamma>2 \<turnstile> (Lam [a].t) : T1\<rightarrow>T2" by auto
+  then have "valid ((x,T1)#\<Gamma>2)" using vc by (simp add: v2)
+  ultimately have "((x,T1)#\<Gamma>2) \<turnstile> t:T2" using ih by simp
+  with vc show "\<Gamma>2 \<turnstile> (Lam [x].t) : T1\<rightarrow>T2" by auto
 qed (auto) (* app case *)
 
 lemma weakening_version3: 
@@ -172,17 +101,17 @@
   and     c: "\<Gamma>1 \<lless> \<Gamma>2"
   shows "\<Gamma>2 \<turnstile> t : T"
 using a b c
-proof (nominal_induct \<Gamma>1 t T avoiding: \<Gamma>2 rule: typing_induct)
-  case (t_Lam a \<Gamma>1 T1 T2 t) (* lambda case *)
-  have vc: "a\<sharp>\<Gamma>2" by fact (* variable convention *)
-  have ih: "\<And>\<Gamma>3. \<lbrakk>valid \<Gamma>3; ((a,T1)#\<Gamma>1) \<lless> \<Gamma>3\<rbrakk> \<Longrightarrow>  \<Gamma>3 \<turnstile> t : T2" by fact
+proof (nominal_induct \<Gamma>1 t T avoiding: \<Gamma>2 rule: typing.strong_induct)
+  case (t_Lam x \<Gamma>1 T1 t T2) (* lambda case *)
+  have vc: "x\<sharp>\<Gamma>2" by fact (* variable convention *)
+  have ih: "\<And>\<Gamma>3. \<lbrakk>valid \<Gamma>3; ((x,T1)#\<Gamma>1) \<lless> \<Gamma>3\<rbrakk> \<Longrightarrow>  \<Gamma>3 \<turnstile> t : T2" by fact
   have "\<Gamma>1 \<lless> \<Gamma>2" by fact
-  then have "((a,T1)#\<Gamma>1) \<lless> ((a,T1)#\<Gamma>2)" by simp
+  then have "((x,T1)#\<Gamma>1) \<lless> ((x,T1)#\<Gamma>2)" by simp
   moreover
   have "valid \<Gamma>2" by fact
-  then have "valid ((a,T1)#\<Gamma>2)" using vc by (simp add: v2)
-  ultimately have "((a,T1)#\<Gamma>2) \<turnstile> t : T2" using ih by simp
-  with vc show "\<Gamma>2 \<turnstile> (Lam [a].t) : T1 \<rightarrow> T2" by auto
+  then have "valid ((x,T1)#\<Gamma>2)" using vc by (simp add: v2)
+  ultimately have "((x,T1)#\<Gamma>2) \<turnstile> t : T2" using ih by simp
+  with vc show "\<Gamma>2 \<turnstile> (Lam [x].t) : T1 \<rightarrow> T2" by auto
 qed (auto) (* app and var case *)
 
 text{* The original induction principle for the typing relation
@@ -194,24 +123,24 @@
   shows "\<Gamma>2 \<turnstile> t : T"
 using a b c
 proof (induct arbitrary: \<Gamma>2)
-  case (t_Var \<Gamma>1 a T) (* variable case *)
+  case (t_Var \<Gamma>1 x T) (* variable case *)
   have "\<Gamma>1 \<lless> \<Gamma>2" by fact
   moreover
   have "valid \<Gamma>2" by fact
   moreover
-  have "(a,T) \<in> (set \<Gamma>1)" by fact 
-  ultimately show "\<Gamma>2 \<turnstile> Var a : T" by auto
+  have "(x,T) \<in> (set \<Gamma>1)" by fact 
+  ultimately show "\<Gamma>2 \<turnstile> Var x : T" by auto
 next
-  case (t_Lam a \<Gamma>1 T1 t T2) (* lambda case *)
+  case (t_Lam x \<Gamma>1 T1 t T2) (* lambda case *)
   (* all assumptions available in this case*)
-  have a0: "a\<sharp>\<Gamma>1" by fact
-  have a1: "((a,T1)#\<Gamma>1) \<turnstile> t : T2" by fact
+  have a0: "x\<sharp>\<Gamma>1" by fact
+  have a1: "((x,T1)#\<Gamma>1) \<turnstile> t : T2" by fact
   have a2: "\<Gamma>1 \<lless> \<Gamma>2" by fact
   have a3: "valid \<Gamma>2" by fact
-  have ih: "\<And>\<Gamma>3. \<lbrakk>valid \<Gamma>3; ((a,T1)#\<Gamma>1) \<lless> \<Gamma>3\<rbrakk>  \<Longrightarrow>  \<Gamma>3 \<turnstile> t : T2" by fact
-  have "((a,T1)#\<Gamma>1) \<lless> ((a,T1)#\<Gamma>2)" using a2 by simp
+  have ih: "\<And>\<Gamma>3. \<lbrakk>valid \<Gamma>3; ((x,T1)#\<Gamma>1) \<lless> \<Gamma>3\<rbrakk>  \<Longrightarrow>  \<Gamma>3 \<turnstile> t : T2" by fact
+  have "((x,T1)#\<Gamma>1) \<lless> ((x,T1)#\<Gamma>2)" using a2 by simp
   moreover
-  have "valid ((a,T1)#\<Gamma>2)" using v2 (* fails *) 
+  have "valid ((x,T1)#\<Gamma>2)" using v2 (* fails *) 
     oops
 
 end
\ No newline at end of file