--- a/src/HOL/Nominal/Examples/Weakening.thy Thu Oct 26 16:08:40 2006 +0200
+++ b/src/HOL/Nominal/Examples/Weakening.thy Mon Oct 30 13:07:51 2006 +0100
@@ -124,6 +124,72 @@
thus "P x \<Gamma> t \<tau>" by simp
qed
+lemma typing_induct_test[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 \<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.
+ \<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. \<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 (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 (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 (t_Lam a \<Gamma> \<tau> t \<sigma> pi x)
+ have p1: "((a,\<tau>)#\<Gamma>)\<turnstile>t:\<sigma>" by fact
+ have ih1: "\<And>(pi::name prm) x. P x (pi\<bullet>((a,\<tau>)#\<Gamma>)) (pi\<bullet>t) \<sigma>" by fact
+ have f: "a\<sharp>\<Gamma>" by fact
+ then have f': "(pi\<bullet>a)\<sharp>(pi\<bullet>\<Gamma>)" by (simp add: fresh_bij)
+ have "\<exists>c::name. c\<sharp>(pi\<bullet>a,pi\<bullet>t,pi\<bullet>\<Gamma>,x)"
+ by (rule exists_fresh, simp add: fs_name1)
+ then obtain c::"name"
+ where fs: "c\<noteq>(pi\<bullet>a)" "c\<sharp>x" "c\<sharp>(pi\<bullet>t)" "c\<sharp>(pi\<bullet>\<Gamma>)"
+ by (force simp add: fresh_prod fresh_atm)
+ let ?pi'="[(pi\<bullet>a,c)]@pi"
+ have eq: "((pi\<bullet>a,c)#pi)\<bullet>a = c" by (simp add: calc_atm)
+ have p1': "(?pi'\<bullet>((a,\<tau>)#\<Gamma>))\<turnstile>(?pi'\<bullet>t):\<sigma>" using p1 by (simp only: eqvt_typing)
+ have ih1': "\<And>x. P x (?pi'\<bullet>((a,\<tau>)#\<Gamma>)) (?pi'\<bullet>t) \<sigma>" using ih1 by simp
+ have "P x (?pi'\<bullet>\<Gamma>) (?pi'\<bullet>(Lam [a].t)) (\<tau>\<rightarrow>\<sigma>)" using f f' fs p1' ih1' eq
+ apply -
+ apply(simp del: append_Cons)
+ apply(rule a3)
+ apply(simp_all add: fresh_left calc_atm pt_name2)
+ done
+ then have "P x ([(pi\<bullet>a,c)]\<bullet>(pi\<bullet>\<Gamma>)) ([(pi\<bullet>a,c)]\<bullet>(Lam [(pi\<bullet>a)].(pi\<bullet>t))) (\<tau>\<rightarrow>\<sigma>)"
+ by (simp del: append_Cons add: pt_name2)
+ then show "P x (pi\<bullet>\<Gamma>) (pi\<bullet>(Lam [a].t)) (\<tau> \<rightarrow> \<sigma>)" using f f' fs
+ apply -
+ apply(subgoal_tac "c\<sharp>Lam [(pi\<bullet>a)].(pi\<bullet>t)")
+ apply(subgoal_tac "(pi\<bullet>a)\<sharp>Lam [(pi\<bullet>a)].(pi\<bullet>t)")
+ apply(simp only: perm_fresh_fresh)
+ apply(simp)
+ apply(simp add: abs_fresh)
+ apply(simp add: abs_fresh)
+ done
+ qed
+ hence "P x (([]::name prm)\<bullet>\<Gamma>) (([]::name prm)\<bullet>t) \<tau>" by blast
+ thus "P x \<Gamma> t \<tau>" by simp
+qed
+
+
text {* definition of a subcontext *}
abbreviation