--- a/src/HOL/Nominal/Examples/Class.thy Wed Mar 22 14:06:29 2006 +0100
+++ b/src/HOL/Nominal/Examples/Class.thy Wed Mar 22 18:09:35 2006 +0100
@@ -9,19 +9,172 @@
nominal_datatype trm =
Ax "name" "coname"
- | Cut "\<guillemotleft>coname\<guillemotright>trm" "\<guillemotleft>name\<guillemotright>trm" ("Cut <_>._ (_)._" [100,100,100,100] 100)
- | NotR "\<guillemotleft>name\<guillemotright>trm" "coname" ("NotR (_)._ _" [100,100,100] 100)
+ | Cut "\<guillemotleft>coname\<guillemotright>trm" "\<guillemotleft>name\<guillemotright>trm" ("Cut <_>._ [_]._" [100,100,100,100] 100)
+ | NotR "\<guillemotleft>name\<guillemotright>trm" "coname" ("NotR [_]._ _" [100,100,100] 100)
| NotL "\<guillemotleft>coname\<guillemotright>trm" "name" ("NotL <_>._ _" [100,100,100] 100)
| AndR "\<guillemotleft>coname\<guillemotright>trm" "\<guillemotleft>coname\<guillemotright>trm" "coname" ("AndR <_>._ <_>._ _" [100,100,100,100,100] 100)
- | AndL1 "\<guillemotleft>name\<guillemotright>trm" "name" ("AndL1 (_)._ _" [100,100,100] 100)
- | AndL2 "\<guillemotleft>name\<guillemotright>trm" "name" ("AndL2 (_)._ _" [100,100,100] 100)
+ | AndL1 "\<guillemotleft>name\<guillemotright>trm" "name" ("AndL1 [_]._ _" [100,100,100] 100)
+ | AndL2 "\<guillemotleft>name\<guillemotright>trm" "name" ("AndL2 [_]._ _" [100,100,100] 100)
| OrR1 "\<guillemotleft>coname\<guillemotright>trm" "coname" ("OrR1 <_>._ _" [100,100,100] 100)
| OrR2 "\<guillemotleft>coname\<guillemotright>trm" "coname" ("OrR2 <_>._ _" [100,100,100] 100)
- | OrL "\<guillemotleft>name\<guillemotright>trm" "\<guillemotleft>name\<guillemotright>trm" "name" ("OrL (_)._ (_)._ _" [100,100,100,100,100] 100)
- | ImpR "\<guillemotleft>name\<guillemotright>(\<guillemotleft>coname\<guillemotright>trm)" "coname" ("ImpR (_).<_>._ _" [100,100,100,100] 100)
- | ImpL "\<guillemotleft>coname\<guillemotright>trm" "\<guillemotleft>name\<guillemotright>trm" "name" ("ImpL <_>._ (_)._ _" [100,100,100,100,100] 100)
+ | OrL "\<guillemotleft>name\<guillemotright>trm" "\<guillemotleft>name\<guillemotright>trm" "name" ("OrL [_]._ [_]._ _" [100,100,100,100,100] 100)
+ | ImpR "\<guillemotleft>name\<guillemotright>(\<guillemotleft>coname\<guillemotright>trm)" "coname" ("ImpR [_].<_>._ _" [100,100,100,100] 100)
+ | ImpL "\<guillemotleft>coname\<guillemotright>trm" "\<guillemotleft>name\<guillemotright>trm" "name" ("ImpL <_>._ [_]._ _" [100,100,100,100,100] 100)
+
+thm trm_iter_set.intros[no_vars]
-
+lemma it_total:
+ shows "\<exists>r. (t,r) \<in> trm_iter_set f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12"
+ by (induct t rule: trm.induct_weak, auto intro: trm_iter_set.intros)
+
+lemma it_prm_eq:
+ assumes a: "(t,y) \<in> trm_iter_set f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12"
+ and b: "pi1 \<triangleq> pi2"
+ and c: "pi3 \<triangleq> pi4"
+ shows "y pi1 pi3 = y pi2 pi4"
+using a b c
+apply(induct fixing: pi1 pi2 pi3 pi4)
+(* axiom *)
+apply(simp add: pt3[OF pt_name_inst] pt3[OF pt_coname_inst])
+(* Cut *)
+apply(rule_tac f="fresh_fun" in arg_cong)
+apply(simp add: expand_fun_eq)
+apply(rule allI)
+apply(rule_tac f="fresh_fun" in arg_cong)
+apply(simp add: expand_fun_eq)
+apply(rule allI)
+apply(tactic {* DatatypeAux.cong_tac 1 *})+
+apply(simp_all)
+apply(drule meta_spec)+
+apply(drule meta_mp, erule_tac [2] meta_mp)
+apply(rule at_prm_eq_append'[OF at_coname_inst], assumption, assumption)
+apply(drule meta_spec)+
+apply(drule meta_mp, erule_tac [2] meta_mp)
+apply(assumption, rule at_prm_eq_append'[OF at_name_inst], assumption)
+(* NotR *)
+apply(rule_tac f="fresh_fun" in arg_cong)
+apply(simp add: expand_fun_eq)
+apply(rule allI)
+apply(tactic {* DatatypeAux.cong_tac 1 *})+
+apply(simp_all)
+apply(drule meta_spec)+
+apply(drule meta_mp, erule_tac [2] meta_mp)
+apply(assumption, rule at_prm_eq_append'[OF at_name_inst], assumption)
+apply(simp add: pt3[OF pt_name_inst] pt3[OF pt_coname_inst])
+(* NotL *)
+apply(rule_tac f="fresh_fun" in arg_cong)
+apply(simp add: expand_fun_eq)
+apply(rule allI)
+apply(tactic {* DatatypeAux.cong_tac 1 *})+
+apply(simp_all)
+apply(drule meta_spec)+
+apply(drule meta_mp, erule_tac [2] meta_mp)
+apply(rule at_prm_eq_append'[OF at_coname_inst], assumption, assumption)
+apply(simp add: pt3[OF pt_name_inst] pt3[OF pt_coname_inst])
+(* AndR *)
+apply(rule_tac f="fresh_fun" in arg_cong)
+apply(simp add: expand_fun_eq)
+apply(rule allI)
+apply(rule_tac f="fresh_fun" in arg_cong)
+apply(simp add: expand_fun_eq)
+apply(rule allI)
+apply(tactic {* DatatypeAux.cong_tac 1 *})+
+apply(simp_all)
+apply(drule meta_spec)+
+apply(drule meta_mp, erule_tac [2] meta_mp)
+apply(rule at_prm_eq_append'[OF at_coname_inst], assumption, assumption)
+apply(drule meta_spec)+
+apply(drule meta_mp, erule_tac [2] meta_mp)
+apply(rule at_prm_eq_append'[OF at_coname_inst], assumption, assumption)
+apply(simp add: pt3[OF pt_name_inst] pt3[OF pt_coname_inst])
+(* AndL1 *)
+apply(rule_tac f="fresh_fun" in arg_cong)
+apply(simp add: expand_fun_eq)
+apply(rule allI)
+apply(tactic {* DatatypeAux.cong_tac 1 *})+
+apply(simp_all)
+apply(drule meta_spec)+
+apply(drule meta_mp, erule_tac [2] meta_mp)
+apply(assumption, rule at_prm_eq_append'[OF at_name_inst], assumption)
+apply(simp add: pt3[OF pt_name_inst] pt3[OF pt_coname_inst])
+(* AndL1 *)
+apply(rule_tac f="fresh_fun" in arg_cong)
+apply(simp add: expand_fun_eq)
+apply(rule allI)
+apply(tactic {* DatatypeAux.cong_tac 1 *})+
+apply(simp_all)
+apply(drule meta_spec)+
+apply(drule meta_mp, erule_tac [2] meta_mp)
+apply(assumption, rule at_prm_eq_append'[OF at_name_inst], assumption)
+apply(simp add: pt3[OF pt_name_inst] pt3[OF pt_coname_inst])
+(* OrR1 *)
+apply(rule_tac f="fresh_fun" in arg_cong)
+apply(simp add: expand_fun_eq)
+apply(rule allI)
+apply(tactic {* DatatypeAux.cong_tac 1 *})+
+apply(simp_all)
+apply(drule meta_spec)+
+apply(drule meta_mp, erule_tac [2] meta_mp)
+apply(rule at_prm_eq_append'[OF at_coname_inst], assumption, assumption)
+apply(simp add: pt3[OF pt_name_inst] pt3[OF pt_coname_inst])
+(* OrR2 *)
+apply(rule_tac f="fresh_fun" in arg_cong)
+apply(simp add: expand_fun_eq)
+apply(rule allI)
+apply(tactic {* DatatypeAux.cong_tac 1 *})+
+apply(simp_all)
+apply(drule meta_spec)+
+apply(drule meta_mp, erule_tac [2] meta_mp)
+apply(rule at_prm_eq_append'[OF at_coname_inst], assumption, assumption)
+apply(simp add: pt3[OF pt_name_inst] pt3[OF pt_coname_inst])
+(* OrL *)
+apply(rule_tac f="fresh_fun" in arg_cong)
+apply(simp add: expand_fun_eq)
+apply(rule allI)
+apply(rule_tac f="fresh_fun" in arg_cong)
+apply(simp add: expand_fun_eq)
+apply(rule allI)
+apply(tactic {* DatatypeAux.cong_tac 1 *})+
+apply(simp_all)
+apply(drule meta_spec)+
+apply(drule meta_mp, erule_tac [2] meta_mp)
+apply(assumption, rule at_prm_eq_append'[OF at_name_inst], assumption)
+apply(drule meta_spec)+
+apply(drule meta_mp, erule_tac [2] meta_mp)
+apply(assumption, rule at_prm_eq_append'[OF at_name_inst], assumption)
+apply(simp add: pt3[OF pt_name_inst] pt3[OF pt_coname_inst])
+(* ImpR *)
+apply(rule_tac f="fresh_fun" in arg_cong)
+apply(simp add: expand_fun_eq)
+apply(rule allI)
+apply(rule_tac f="fresh_fun" in arg_cong)
+apply(simp add: expand_fun_eq)
+apply(rule allI)
+apply(tactic {* DatatypeAux.cong_tac 1 *})+
+apply(simp_all)
+apply(drule meta_spec)+
+apply(drule meta_mp, erule_tac [2] meta_mp)
+apply(rule at_prm_eq_append'[OF at_coname_inst], assumption)
+apply(rule at_prm_eq_append'[OF at_name_inst], assumption)
+apply(simp add: pt3[OF pt_name_inst] pt3[OF pt_coname_inst])
+(* ImpL *)
+apply(rule_tac f="fresh_fun" in arg_cong)
+apply(simp add: expand_fun_eq)
+apply(rule allI)
+apply(rule_tac f="fresh_fun" in arg_cong)
+apply(simp add: expand_fun_eq)
+apply(rule allI)
+apply(tactic {* DatatypeAux.cong_tac 1 *})+
+apply(simp_all)
+apply(drule meta_spec)+
+apply(drule meta_mp, erule_tac [2] meta_mp)
+apply(rule at_prm_eq_append'[OF at_coname_inst], assumption, assumption)
+apply(drule meta_spec)+
+apply(drule meta_mp, erule_tac [2] meta_mp)
+apply(assumption, rule at_prm_eq_append'[OF at_name_inst], assumption)
+apply(simp add: pt3[OF pt_name_inst] pt3[OF pt_coname_inst])
+done
+
text {* Induction principles *}
thm trm.induct_weak --"weak"