# HG changeset patch
# User urbanc
# Date 1137001591 -3600
# Node ID dde117622dace696123b023d1f06cf8d8ef9eb46
# Parent  9968dc816cda3a778aa24b0346d4b167c4e1e320
updated to new induction principle

diff -r 9968dc816cda -r dde117622dac src/HOL/Nominal/Examples/Class.thy
--- a/src/HOL/Nominal/Examples/Class.thy	Wed Jan 11 18:39:19 2006 +0100
+++ b/src/HOL/Nominal/Examples/Class.thy	Wed Jan 11 18:46:31 2006 +0100
@@ -1,5 +1,5 @@
 
-theory class 
+theory Class 
 imports "../nominal" 
 begin
 
@@ -12,153 +12,7 @@
                  | ImpL "\<guillemotleft>coname\<guillemotright>trm" "\<guillemotleft>name\<guillemotright>trm" "name"("ImpL [_]._ [_]._ _" [100,100,100,100,100] 100)
                  | Cut "\<guillemotleft>coname\<guillemotright>trm" "\<guillemotleft>name\<guillemotright>trm"        ("Cut [_]._ [_]._" [100,100,100,100] 100)
 
-lemma trm_induct_aux:
-  fixes P :: "trm \<Rightarrow> 'a \<Rightarrow> bool"
-  and   f1 :: "'a \<Rightarrow> name set"
-  and   f2 :: "'a \<Rightarrow> coname set"
-  assumes fs1: "\<And>x. finite (f1 x)"
-      and fs2: "\<And>x. finite (f2 x)"
-      and h1: "\<And>k x a. P (Ax x a) k"  
-      and h2: "\<And>k x a t b. x\<notin>f1 k \<Longrightarrow> a\<notin>f2 k \<Longrightarrow> (\<forall>l. P t l) \<Longrightarrow> P (ImpR [x].[a].t b) k" 
-      and h3: "\<And>k a t1 x t2 y. a\<notin>f2 k \<Longrightarrow> x\<notin>f1 k \<Longrightarrow> (\<forall>l. P t1 l) \<Longrightarrow> (\<forall>l. P t2 l) 
-               \<Longrightarrow> P (ImpL [a].t1 [x].t2 y) k" 
-      and h4: "\<And>k a t1 x t2. a\<notin>f2 k \<Longrightarrow> x\<notin>f1 k \<Longrightarrow> (\<forall>l. P t1 l) \<Longrightarrow> (\<forall>l. P t2 l) 
-               \<Longrightarrow> P (Cut [a].t1 [x].t2) k" 
-  shows "\<forall>(pi1::name prm) (pi2::coname prm) k. P (pi1\<bullet>(pi2\<bullet>t)) k"
-proof (induct rule: trm.induct_weak)
-  case (goal1 a)
-  show ?case using h1 by simp
-next
-  case (goal2 x a t b)
-  assume i1: "\<forall>(pi1::name prm)(pi2::coname prm) k. P (pi1\<bullet>(pi2\<bullet>t)) k" 
-  show ?case 
-  proof (intro strip, simp add: abs_perm perm_dj)
-    fix pi1::"name prm" and pi2::"coname prm" and k::"'a"
-    have "\<exists>u::name. u\<sharp>(f1 k,pi1\<bullet>x,pi1\<bullet>(pi2\<bullet>t))"
-      by (rule at_exists_fresh[OF at_name_inst], simp add: supp_prod fs_name1 
-               at_fin_set_supp[OF at_name_inst, OF fs1] fs1)
-    then obtain u::"name" 
-      where f1: "u\<noteq>(pi1\<bullet>x)" and f2: "u\<sharp>(f1 k)" and f3: "u\<sharp>(pi1\<bullet>(pi2\<bullet>t))" 
-      by (auto simp add: fresh_prod at_fresh[OF at_name_inst])
-    have "\<exists>c::coname. c\<sharp>(f2 k,pi2\<bullet>a,pi1\<bullet>(pi2\<bullet>t))"
-      by (rule at_exists_fresh[OF at_coname_inst], simp add: supp_prod fs_coname1 
-               at_fin_set_supp[OF at_coname_inst, OF fs2] fs2)
-    then obtain c::"coname" 
-      where e1: "c\<noteq>(pi2\<bullet>a)" and e2: "c\<sharp>(f2 k)" and e3: "c\<sharp>(pi1\<bullet>(pi2\<bullet>t))" 
-      by (auto simp add: fresh_prod at_fresh[OF at_coname_inst])
-    have g: "ImpR [u].[c].([(u,pi1\<bullet>x)]\<bullet>(pi1\<bullet>([(c,pi2\<bullet>a)]\<bullet>(pi2\<bullet>t)))) (pi2\<bullet>b)
-            =ImpR [(pi1\<bullet>x)].[(pi2\<bullet>a)].(pi1\<bullet>(pi2\<bullet>t)) (pi2\<bullet>b)" using f1 f3 e1 e3
-      apply (auto simp add: ImpR_inject alpha abs_fresh abs_perm perm_dj,
-                  simp add: dj_cp[OF cp_name_coname_inst, OF dj_coname_name])
-      apply(simp add:  pt_fresh_left_ineq[OF pt_name_inst, OF pt_name_inst, 
-                                          OF at_name_inst, OF cp_name_coname_inst] perm_dj)
-      done
-    from i1 have "\<forall>k. P (([(u,pi1\<bullet>x)]@pi1)\<bullet>(([(c,pi2\<bullet>a)]@pi2)\<bullet>t)) k" by force
-    hence i1b: "\<forall>k. P ([(u,pi1\<bullet>x)]\<bullet>(pi1\<bullet>([(c,pi2\<bullet>a)]\<bullet>(pi2\<bullet>t)))) k" 
-      by (simp add: pt_name2[symmetric] pt_coname2[symmetric])
-    with h2 f2 e2 have "P (ImpR [u].[c].([(u,pi1\<bullet>x)]\<bullet>(pi1\<bullet>([(c,pi2\<bullet>a)]\<bullet>(pi2\<bullet>t)))) (pi2\<bullet>b)) k" 
-      by (simp add: fresh_def at_fin_set_supp[OF at_name_inst, OF fs1]
-                                   at_fin_set_supp[OF at_coname_inst, OF fs2])
-    with g show "P (ImpR [(pi1\<bullet>x)].[(pi2\<bullet>a)].(pi1\<bullet>(pi2\<bullet>t)) (pi2\<bullet>b)) k" by simp 
-  qed
-next
-  case (goal3 a t1 x t2 y)
-  assume i1: "\<forall>(pi1::name prm)(pi2::coname prm) k. P (pi1\<bullet>(pi2\<bullet>t1)) k" 
-  and    i2: "\<forall>(pi1::name prm)(pi2::coname prm) k. P (pi1\<bullet>(pi2\<bullet>t2)) k"
-  show ?case
-  proof (intro strip, simp add: abs_perm)
-    fix pi1::"name prm" and pi2::"coname prm" and k::"'a"
-    have "\<exists>u::name. u\<sharp>(f1 k,pi1\<bullet>x,pi1\<bullet>(pi2\<bullet>t2))"
-      by (rule at_exists_fresh[OF at_name_inst], simp add: supp_prod fs_name1 
-               at_fin_set_supp[OF at_name_inst, OF fs1] fs1)
-    then obtain u::"name" 
-      where f1: "u\<noteq>(pi1\<bullet>x)" and f2: "u\<sharp>(f1 k)" and f3: "u\<sharp>(pi1\<bullet>(pi2\<bullet>t2))" 
-      by (auto simp add: fresh_prod at_fresh[OF at_name_inst])
-    have "\<exists>c::coname. c\<sharp>(f2 k,pi2\<bullet>a,pi1\<bullet>(pi2\<bullet>t1))"
-      by (rule at_exists_fresh[OF at_coname_inst], simp add: supp_prod fs_coname1 
-               at_fin_set_supp[OF at_coname_inst, OF fs2] fs2)
-    then obtain c::"coname" 
-      where e1: "c\<noteq>(pi2\<bullet>a)" and e2: "c\<sharp>(f2 k)" and e3: "c\<sharp>(pi1\<bullet>(pi2\<bullet>t1))" 
-      by (auto simp add: fresh_prod at_fresh[OF at_coname_inst])
-    have g: "ImpL [c].([(c,pi2\<bullet>a)]\<bullet>(pi1\<bullet>(pi2\<bullet>t1))) [u].([(u,pi1\<bullet>x)]\<bullet>(pi1\<bullet>(pi2\<bullet>t2))) (pi1\<bullet>y)
-            =ImpL [(pi2\<bullet>a)].(pi1\<bullet>(pi2\<bullet>t1)) [(pi1\<bullet>x)].(pi1\<bullet>(pi2\<bullet>t2)) (pi1\<bullet>y)" using f1 f3 e1 e3
-      by (simp add: ImpL_inject alpha abs_fresh abs_perm perm_dj)
-    from i2 have "\<forall>k. P (([(u,pi1\<bullet>x)]@pi1)\<bullet>(pi2\<bullet>t2)) k" by force
-    hence i2b: "\<forall>k. P ([(u,pi1\<bullet>x)]\<bullet>(pi1\<bullet>(pi2\<bullet>t2))) k" 
-      by (simp add: pt_name2[symmetric])
-    from i1 have "\<forall>k. P (pi1\<bullet>(([(c,pi2\<bullet>a)]@pi2)\<bullet>t1)) k" by force
-    hence i1b: "\<forall>k. P ([(c,pi2\<bullet>a)]\<bullet>(pi1\<bullet>(pi2\<bullet>t1))) k" 
-      by (simp add: pt_coname2[symmetric] dj_cp[OF cp_name_coname_inst, OF dj_coname_name])
-    from h3 f2 e2 i1b i2b 
-    have "P (ImpL [c].([(c,pi2\<bullet>a)]\<bullet>(pi1\<bullet>(pi2\<bullet>t1))) [u].([(u,pi1\<bullet>x)]\<bullet>(pi1\<bullet>(pi2\<bullet>t2))) (pi1\<bullet>y)) k" 
-      by (simp add: fresh_def at_fin_set_supp[OF at_name_inst, OF fs1]
-                                   at_fin_set_supp[OF at_coname_inst, OF fs2])
-    with g show "P (ImpL [(pi2\<bullet>a)].(pi1\<bullet>(pi2\<bullet>t1)) [(pi1\<bullet>x)].(pi1\<bullet>(pi2\<bullet>t2)) (pi1\<bullet>y)) k" by simp 
-  qed
-next
-  case (goal4 a t1 x t2)
-  assume i1: "\<forall>(pi1::name prm)(pi2::coname prm) k. P (pi1\<bullet>(pi2\<bullet>t1)) k" 
-  and    i2: "\<forall>(pi1::name prm)(pi2::coname prm) k. P (pi1\<bullet>(pi2\<bullet>t2)) k"
-  show ?case
-  proof (intro strip, simp add: abs_perm)
-    fix pi1::"name prm" and pi2::"coname prm" and k::"'a"
-    have "\<exists>u::name. u\<sharp>(f1 k,pi1\<bullet>x,pi1\<bullet>(pi2\<bullet>t2))"
-      by (rule at_exists_fresh[OF at_name_inst], simp add: supp_prod fs_name1 
-               at_fin_set_supp[OF at_name_inst, OF fs1] fs1)
-    then obtain u::"name" 
-      where f1: "u\<noteq>(pi1\<bullet>x)" and f2: "u\<sharp>(f1 k)" and f3: "u\<sharp>(pi1\<bullet>(pi2\<bullet>t2))" 
-      by (auto simp add: fresh_prod at_fresh[OF at_name_inst])
-    have "\<exists>c::coname. c\<sharp>(f2 k,pi2\<bullet>a,pi1\<bullet>(pi2\<bullet>t1))"
-      by (rule at_exists_fresh[OF at_coname_inst], simp add: supp_prod fs_coname1 
-               at_fin_set_supp[OF at_coname_inst, OF fs2] fs2)
-    then obtain c::"coname" 
-      where e1: "c\<noteq>(pi2\<bullet>a)" and e2: "c\<sharp>(f2 k)" and e3: "c\<sharp>(pi1\<bullet>(pi2\<bullet>t1))" 
-      by (auto simp add: fresh_prod at_fresh[OF at_coname_inst])
-    have g: "Cut [c].([(c,pi2\<bullet>a)]\<bullet>(pi1\<bullet>(pi2\<bullet>t1))) [u].([(u,pi1\<bullet>x)]\<bullet>(pi1\<bullet>(pi2\<bullet>t2)))
-            =Cut [(pi2\<bullet>a)].(pi1\<bullet>(pi2\<bullet>t1)) [(pi1\<bullet>x)].(pi1\<bullet>(pi2\<bullet>t2))" using f1 f3 e1 e3
-      by (simp add: Cut_inject alpha abs_fresh abs_perm perm_dj)
-    from i2 have "\<forall>k. P (([(u,pi1\<bullet>x)]@pi1)\<bullet>(pi2\<bullet>t2)) k" by force
-    hence i2b: "\<forall>k. P ([(u,pi1\<bullet>x)]\<bullet>(pi1\<bullet>(pi2\<bullet>t2))) k" 
-      by (simp add: pt_name2[symmetric])
-    from i1 have "\<forall>k. P (pi1\<bullet>(([(c,pi2\<bullet>a)]@pi2)\<bullet>t1)) k" by force
-    hence i1b: "\<forall>k. P ([(c,pi2\<bullet>a)]\<bullet>(pi1\<bullet>(pi2\<bullet>t1))) k" 
-      by (simp add: pt_coname2[symmetric] dj_cp[OF cp_name_coname_inst, OF dj_coname_name])
-    from h3 f2 e2 i1b i2b 
-    have "P (Cut [c].([(c,pi2\<bullet>a)]\<bullet>(pi1\<bullet>(pi2\<bullet>t1))) [u].([(u,pi1\<bullet>x)]\<bullet>(pi1\<bullet>(pi2\<bullet>t2)))) k" 
-      by (simp add: fresh_def at_fin_set_supp[OF at_name_inst, OF fs1]
-                                   at_fin_set_supp[OF at_coname_inst, OF fs2])
-    with g show "P (Cut [(pi2\<bullet>a)].(pi1\<bullet>(pi2\<bullet>t1)) [(pi1\<bullet>x)].(pi1\<bullet>(pi2\<bullet>t2))) k" by simp 
-  qed
-qed
+thm trm.induct
+thm trm.inducts
+thm trm.induct'
 
-lemma trm_induct'[case_names Ax ImpR ImpL Cut]: 
-  fixes P :: "trm \<Rightarrow> 'a \<Rightarrow> bool"
-  and   f1 :: "'a \<Rightarrow> name set"
-  and   f2 :: "'a \<Rightarrow> coname set"
-  assumes fs1: "\<And>x. finite (f1 x)"
-      and fs2: "\<And>x. finite (f2 x)"
-      and h1: "\<And>k x a. P (Ax x a) k"  
-      and h2: "\<And>k x a t b. x\<notin>f1 k \<Longrightarrow> a\<notin>f2 k \<Longrightarrow> (\<forall>l. P t l) \<Longrightarrow> P (ImpR [x].[a].t b) k" 
-      and h3: "\<And>k a t1 x t2 y. a\<notin>f2 k \<Longrightarrow> x\<notin>f1 k \<Longrightarrow> (\<forall>l. P t1 l) \<Longrightarrow> (\<forall>l. P t2 l) 
-               \<Longrightarrow> P (ImpL [a].t1 [x].t2 y) k" 
-      and h4: "\<And>k a t1 x t2. a\<notin>f2 k \<Longrightarrow> x\<notin>f1 k \<Longrightarrow> (\<forall>l. P t1 l) \<Longrightarrow> (\<forall>l. P t2 l) 
-               \<Longrightarrow> P (Cut [a].t1 [x].t2) k" 
-  shows  "P t k"
-proof -
-  have "\<forall>(pi1::name prm)(pi2::coname prm) k. P (pi1\<bullet>(pi2\<bullet>t)) k"
-  using fs1 fs2 h1 h2 h3 h4 by (rule trm_induct_aux, auto)
-  hence "P (([]::name prm)\<bullet>(([]::coname prm)\<bullet>t)) k" by blast
-  thus "P t k" by simp
-qed
-
-lemma trm_induct[case_names Ax ImpR ImpL Cut]: 
-  fixes P :: "trm \<Rightarrow> ('a::{fs_name,fs_coname}) \<Rightarrow> bool"
-  assumes h1: "\<And>k x a. P (Ax x a) k"  
-      and h2: "\<And>k x a t b. x\<sharp>k \<Longrightarrow> a\<sharp>k \<Longrightarrow> (\<forall>l. P t l) \<Longrightarrow> P (ImpR [x].[a].t b) k" 
-      and h3: "\<And>k a t1 x t2 y. a\<sharp>k \<Longrightarrow> x\<sharp>k \<Longrightarrow> (\<forall>l. P t1 l) \<Longrightarrow> (\<forall>l. P t2 l) 
-               \<Longrightarrow> P (ImpL [a].t1 [x].t2 y) k" 
-      and h4: "\<And>k a t1 x t2. a\<sharp>k \<Longrightarrow> x\<sharp>k \<Longrightarrow> (\<forall>l. P t1 l) \<Longrightarrow> (\<forall>l. P t2 l) 
-               \<Longrightarrow> P (Cut [a].t1 [x].t2) k" 
-  shows  "P t k"
-by (rule trm_induct'[of "\<lambda>x. ((supp x)::name set)" "\<lambda>x. ((supp x)::coname set)" "P"], 
-    simp_all add: fs_name1 fs_coname1 fresh_def[symmetric], auto intro: h1 h2 h3 h4)