wneuper/isa: src/HOL/Nominal/Examples/Class2.thy@74bf65a1910a (annotated)

wenzelm@36278	1	theory Class2
wenzelm@36278	2	imports Class1
wenzelm@36278	3	begin
wenzelm@36278	4
wenzelm@36278	5	text {* Reduction *}
wenzelm@36278	6
wenzelm@36278	7	lemma fin_not_Cut:
wenzelm@36278	8	assumes a: "fin M x"
wenzelm@36278	9	shows "\<not>(\<exists>a M' x N'. M = Cut <a>.M' (x).N')"
wenzelm@36278	10	using a
wenzelm@36278	11	by (induct) (auto)
wenzelm@36278	12
wenzelm@36278	13	lemma fresh_not_fin:
wenzelm@36278	14	assumes a: "x\<sharp>M"
wenzelm@36278	15	shows "\<not>fin M x"
wenzelm@36278	16	proof -
wenzelm@36278	17	have "fin M x \<Longrightarrow> x\<sharp>M \<Longrightarrow> False" by (induct rule: fin.induct) (auto simp add: abs_fresh fresh_atm)
wenzelm@36278	18	with a show "\<not>fin M x" by blast
wenzelm@36278	19	qed
wenzelm@36278	20
wenzelm@36278	21	lemma fresh_not_fic:
wenzelm@36278	22	assumes a: "a\<sharp>M"
wenzelm@36278	23	shows "\<not>fic M a"
wenzelm@36278	24	proof -
wenzelm@36278	25	have "fic M a \<Longrightarrow> a\<sharp>M \<Longrightarrow> False" by (induct rule: fic.induct) (auto simp add: abs_fresh fresh_atm)
wenzelm@36278	26	with a show "\<not>fic M a" by blast
wenzelm@36278	27	qed
wenzelm@36278	28
wenzelm@36278	29	lemma c_redu_subst1:
wenzelm@54152	30	assumes a: "M \<longrightarrow>\<^sub>c M'" "c\<sharp>M" "y\<sharp>P"
wenzelm@54152	31	shows "M{y:=<c>.P} \<longrightarrow>\<^sub>c M'{y:=<c>.P}"
wenzelm@36278	32	using a
wenzelm@36278	33	proof(nominal_induct avoiding: y c P rule: c_redu.strong_induct)
wenzelm@36278	34	case (left M a N x)
wenzelm@36278	35	then show ?case
wenzelm@36278	36	apply -
wenzelm@36278	37	apply(simp)
wenzelm@36278	38	apply(rule conjI)
wenzelm@36278	39	apply(force)
wenzelm@36278	40	apply(auto)
wenzelm@36278	41	apply(subgoal_tac "M{a:=(x).N}{y:=<c>.P} = M{y:=<c>.P}{a:=(x).(N{y:=<c>.P})}")(A)
wenzelm@36278	42	apply(simp)
wenzelm@36278	43	apply(rule c_redu.intros)
wenzelm@36278	44	apply(rule not_fic_subst1)
wenzelm@36278	45	apply(simp)
wenzelm@36278	46	apply(simp add: subst_fresh)
wenzelm@36278	47	apply(simp add: subst_fresh)
wenzelm@36278	48	apply(simp add: abs_fresh fresh_atm)
wenzelm@36278	49	apply(rule subst_subst2)
wenzelm@36278	50	apply(simp add: fresh_prod fresh_atm)
wenzelm@36278	51	apply(simp add: fresh_prod fresh_atm)
wenzelm@36278	52	apply(simp add: fresh_prod fresh_atm)
wenzelm@36278	53	apply(simp)
wenzelm@36278	54	done
wenzelm@36278	55	next
wenzelm@36278	56	case (right N x a M)
wenzelm@36278	57	then show ?case
wenzelm@36278	58	apply -
wenzelm@36278	59	apply(simp)
wenzelm@36278	60	apply(rule conjI)
wenzelm@36278	61	(* case M = Ax y a *)
wenzelm@36278	62	apply(rule impI)
wenzelm@36278	63	apply(subgoal_tac "N{x:=<a>.Ax y a}{y:=<c>.P} = N{y:=<c>.P}{x:=<c>.P}")
wenzelm@36278	64	apply(simp)
wenzelm@36278	65	apply(rule c_redu.right)
wenzelm@36278	66	apply(rule not_fin_subst2)
wenzelm@36278	67	apply(simp)
wenzelm@36278	68	apply(rule subst_fresh)
wenzelm@36278	69	apply(simp add: abs_fresh)
wenzelm@36278	70	apply(simp add: abs_fresh)
wenzelm@36278	71	apply(rule sym)
wenzelm@36278	72	apply(rule interesting_subst1')
wenzelm@36278	73	apply(simp add: fresh_atm)
wenzelm@36278	74	apply(simp)
wenzelm@36278	75	apply(simp)
wenzelm@36278	76	(* case M \<noteq> Ax y a*)
wenzelm@36278	77	apply(rule impI)
wenzelm@36278	78	apply(subgoal_tac "N{x:=<a>.M}{y:=<c>.P} = N{y:=<c>.P}{x:=<a>.(M{y:=<c>.P})}")
wenzelm@36278	79	apply(simp)
wenzelm@36278	80	apply(rule c_redu.right)
wenzelm@36278	81	apply(rule not_fin_subst2)
wenzelm@36278	82	apply(simp)
wenzelm@36278	83	apply(simp add: subst_fresh)
wenzelm@36278	84	apply(simp add: subst_fresh)
wenzelm@36278	85	apply(simp add: abs_fresh fresh_atm)
wenzelm@36278	86	apply(rule subst_subst3)
wenzelm@36278	87	apply(simp_all add: fresh_atm fresh_prod)
wenzelm@36278	88	done
wenzelm@36278	89	qed
wenzelm@36278	90
wenzelm@36278	91	lemma c_redu_subst2:
wenzelm@54152	92	assumes a: "M \<longrightarrow>\<^sub>c M'" "c\<sharp>P" "y\<sharp>M"
wenzelm@54152	93	shows "M{c:=(y).P} \<longrightarrow>\<^sub>c M'{c:=(y).P}"
wenzelm@36278	94	using a
wenzelm@36278	95	proof(nominal_induct avoiding: y c P rule: c_redu.strong_induct)
wenzelm@36278	96	case (right N x a M)
wenzelm@36278	97	then show ?case
wenzelm@36278	98	apply -
wenzelm@36278	99	apply(simp)
wenzelm@36278	100	apply(rule conjI)
wenzelm@36278	101	apply(force)
wenzelm@36278	102	apply(auto)
wenzelm@36278	103	apply(subgoal_tac "N{x:=<a>.M}{c:=(y).P} = N{c:=(y).P}{x:=<a>.(M{c:=(y).P})}")(A)
wenzelm@36278	104	apply(simp)
wenzelm@36278	105	apply(rule c_redu.intros)
wenzelm@36278	106	apply(rule not_fin_subst1)
wenzelm@36278	107	apply(simp)
wenzelm@36278	108	apply(simp add: subst_fresh)
wenzelm@36278	109	apply(simp add: subst_fresh)
wenzelm@36278	110	apply(simp add: abs_fresh fresh_atm)
wenzelm@36278	111	apply(rule subst_subst1)
wenzelm@36278	112	apply(simp add: fresh_prod fresh_atm)
wenzelm@36278	113	apply(simp add: fresh_prod fresh_atm)
wenzelm@36278	114	apply(simp add: fresh_prod fresh_atm)
wenzelm@36278	115	apply(simp)
wenzelm@36278	116	done
wenzelm@36278	117	next
wenzelm@36278	118	case (left M a N x)
wenzelm@36278	119	then show ?case
wenzelm@36278	120	apply -
wenzelm@36278	121	apply(simp)
wenzelm@36278	122	apply(rule conjI)
wenzelm@36278	123	(* case N = Ax x c *)
wenzelm@36278	124	apply(rule impI)
wenzelm@36278	125	apply(subgoal_tac "M{a:=(x).Ax x c}{c:=(y).P} = M{c:=(y).P}{a:=(y).P}")
wenzelm@36278	126	apply(simp)
wenzelm@36278	127	apply(rule c_redu.left)
wenzelm@36278	128	apply(rule not_fic_subst2)
wenzelm@36278	129	apply(simp)
wenzelm@36278	130	apply(simp)
wenzelm@36278	131	apply(rule subst_fresh)
wenzelm@36278	132	apply(simp add: abs_fresh)
wenzelm@36278	133	apply(rule sym)
wenzelm@36278	134	apply(rule interesting_subst2')
wenzelm@36278	135	apply(simp add: fresh_atm)
wenzelm@36278	136	apply(simp)
wenzelm@36278	137	apply(simp)
wenzelm@36278	138	(* case M \<noteq> Ax y a*)
wenzelm@36278	139	apply(rule impI)
wenzelm@36278	140	apply(subgoal_tac "M{a:=(x).N}{c:=(y).P} = M{c:=(y).P}{a:=(x).(N{c:=(y).P})}")
wenzelm@36278	141	apply(simp)
wenzelm@36278	142	apply(rule c_redu.left)
wenzelm@36278	143	apply(rule not_fic_subst2)
wenzelm@36278	144	apply(simp)
wenzelm@36278	145	apply(simp add: subst_fresh)
wenzelm@36278	146	apply(simp add: subst_fresh)
wenzelm@36278	147	apply(simp add: abs_fresh fresh_atm)
wenzelm@36278	148	apply(rule subst_subst4)
wenzelm@36278	149	apply(simp add: fresh_prod fresh_atm)
wenzelm@36278	150	apply(simp add: fresh_prod fresh_atm)
wenzelm@36278	151	apply(simp add: fresh_prod fresh_atm)
wenzelm@36278	152	apply(simp add: fresh_prod fresh_atm)
wenzelm@36278	153	apply(simp)
wenzelm@36278	154	done
wenzelm@36278	155	qed
wenzelm@36278	156
wenzelm@36278	157	lemma c_redu_subst1':
wenzelm@54152	158	assumes a: "M \<longrightarrow>\<^sub>c M'"
wenzelm@54152	159	shows "M{y:=<c>.P} \<longrightarrow>\<^sub>c M'{y:=<c>.P}"
wenzelm@36278	160	using a
wenzelm@36278	161	proof -
wenzelm@36278	162	obtain y'::"name" where fs1: "y'\<sharp>(M,M',P,P,y)" by (rule exists_fresh(1), rule fin_supp, blast)
wenzelm@36278	163	obtain c'::"coname" where fs2: "c'\<sharp>(M,M',P,P,c)" by (rule exists_fresh(2), rule fin_supp, blast)
wenzelm@36278	164	have "M{y:=<c>.P} = ([(y',y)]\<bullet>M){y':=<c'>.([(c',c)]\<bullet>P)}" using fs1 fs2
wenzelm@36278	165	apply -
wenzelm@36278	166	apply(rule trans)
wenzelm@36278	167	apply(rule_tac y="y'" in subst_rename(3))
wenzelm@36278	168	apply(simp)
wenzelm@36278	169	apply(rule subst_rename(4))
wenzelm@36278	170	apply(simp)
wenzelm@36278	171	done
wenzelm@54152	172	also have "\<dots> \<longrightarrow>\<^sub>c ([(y',y)]\<bullet>M'){y':=<c'>.([(c',c)]\<bullet>P)}" using fs1 fs2
wenzelm@36278	173	apply -
wenzelm@36278	174	apply(rule c_redu_subst1)
wenzelm@36278	175	apply(simp add: c_redu.eqvt a)
wenzelm@36278	176	apply(simp_all add: fresh_left calc_atm fresh_prod)
wenzelm@36278	177	done
wenzelm@36278	178	also have "\<dots> = M'{y:=<c>.P}" using fs1 fs2
wenzelm@36278	179	apply -
wenzelm@36278	180	apply(rule sym)
wenzelm@36278	181	apply(rule trans)
wenzelm@36278	182	apply(rule_tac y="y'" in subst_rename(3))
wenzelm@36278	183	apply(simp)
wenzelm@36278	184	apply(rule subst_rename(4))
wenzelm@36278	185	apply(simp)
wenzelm@36278	186	done
wenzelm@36278	187	finally show ?thesis by simp
wenzelm@36278	188	qed
wenzelm@36278	189
wenzelm@36278	190	lemma c_redu_subst2':
wenzelm@54152	191	assumes a: "M \<longrightarrow>\<^sub>c M'"
wenzelm@54152	192	shows "M{c:=(y).P} \<longrightarrow>\<^sub>c M'{c:=(y).P}"
wenzelm@36278	193	using a
wenzelm@36278	194	proof -
wenzelm@36278	195	obtain y'::"name" where fs1: "y'\<sharp>(M,M',P,P,y)" by (rule exists_fresh(1), rule fin_supp, blast)
wenzelm@36278	196	obtain c'::"coname" where fs2: "c'\<sharp>(M,M',P,P,c)" by (rule exists_fresh(2), rule fin_supp, blast)
wenzelm@36278	197	have "M{c:=(y).P} = ([(c',c)]\<bullet>M){c':=(y').([(y',y)]\<bullet>P)}" using fs1 fs2
wenzelm@36278	198	apply -
wenzelm@36278	199	apply(rule trans)
wenzelm@36278	200	apply(rule_tac c="c'" in subst_rename(1))
wenzelm@36278	201	apply(simp)
wenzelm@36278	202	apply(rule subst_rename(2))
wenzelm@36278	203	apply(simp)
wenzelm@36278	204	done
wenzelm@54152	205	also have "\<dots> \<longrightarrow>\<^sub>c ([(c',c)]\<bullet>M'){c':=(y').([(y',y)]\<bullet>P)}" using fs1 fs2
wenzelm@36278	206	apply -
wenzelm@36278	207	apply(rule c_redu_subst2)
wenzelm@36278	208	apply(simp add: c_redu.eqvt a)
wenzelm@36278	209	apply(simp_all add: fresh_left calc_atm fresh_prod)
wenzelm@36278	210	done
wenzelm@36278	211	also have "\<dots> = M'{c:=(y).P}" using fs1 fs2
wenzelm@36278	212	apply -
wenzelm@36278	213	apply(rule sym)
wenzelm@36278	214	apply(rule trans)
wenzelm@36278	215	apply(rule_tac c="c'" in subst_rename(1))
wenzelm@36278	216	apply(simp)
wenzelm@36278	217	apply(rule subst_rename(2))
wenzelm@36278	218	apply(simp)
wenzelm@36278	219	done
wenzelm@36278	220
wenzelm@36278	221	finally show ?thesis by simp
wenzelm@36278	222	qed
wenzelm@36278	223
wenzelm@36278	224	lemma aux1:
wenzelm@54152	225	assumes a: "M = M'" "M' \<longrightarrow>\<^sub>l M''"
wenzelm@54152	226	shows "M \<longrightarrow>\<^sub>l M''"
wenzelm@36278	227	using a by simp
wenzelm@36278	228
wenzelm@36278	229	lemma aux2:
wenzelm@54152	230	assumes a: "M \<longrightarrow>\<^sub>l M'" "M' = M''"
wenzelm@54152	231	shows "M \<longrightarrow>\<^sub>l M''"
wenzelm@36278	232	using a by simp
wenzelm@36278	233
wenzelm@36278	234	lemma aux3:
wenzelm@54152	235	assumes a: "M = M'" "M' \<longrightarrow>\<^sub>a* M''"
wenzelm@54152	236	shows "M \<longrightarrow>\<^sub>a* M''"
wenzelm@36278	237	using a by simp
wenzelm@36278	238
wenzelm@36278	239	lemma aux4:
wenzelm@36278	240	assumes a: "M = M'"
wenzelm@54152	241	shows "M \<longrightarrow>\<^sub>a* M'"
wenzelm@36278	242	using a by blast
wenzelm@36278	243
wenzelm@36278	244	lemma l_redu_subst1:
wenzelm@54152	245	assumes a: "M \<longrightarrow>\<^sub>l M'"
wenzelm@54152	246	shows "M{y:=<c>.P} \<longrightarrow>\<^sub>a* M'{y:=<c>.P}"
wenzelm@36278	247	using a
wenzelm@36278	248	proof(nominal_induct M M' avoiding: y c P rule: l_redu.strong_induct)
wenzelm@36278	249	case LAxR
wenzelm@36278	250	then show ?case
wenzelm@36278	251	apply -
wenzelm@36278	252	apply(rule aux3)
wenzelm@36278	253	apply(rule better_Cut_substn)
wenzelm@36278	254	apply(simp add: abs_fresh)
wenzelm@36278	255	apply(simp)
wenzelm@36278	256	apply(simp add: fresh_atm)
wenzelm@36278	257	apply(auto)
wenzelm@36278	258	apply(rule aux4)
wenzelm@36278	259	apply(simp add: trm.inject alpha calc_atm fresh_atm)
wenzelm@36278	260	apply(rule a_star_trans)
wenzelm@36278	261	apply(rule a_starI)
wenzelm@36278	262	apply(rule al_redu)
wenzelm@36278	263	apply(rule l_redu.intros)
wenzelm@36278	264	apply(simp add: subst_fresh)
wenzelm@36278	265	apply(simp add: fresh_atm)
wenzelm@36278	266	apply(rule fic_subst2)
wenzelm@36278	267	apply(simp_all)
wenzelm@36278	268	apply(rule aux4)
wenzelm@36278	269	apply(rule subst_comm')
wenzelm@36278	270	apply(simp_all)
wenzelm@36278	271	done
wenzelm@36278	272	next
wenzelm@36278	273	case LAxL
wenzelm@36278	274	then show ?case
wenzelm@36278	275	apply -
wenzelm@36278	276	apply(rule aux3)
wenzelm@36278	277	apply(rule better_Cut_substn)
wenzelm@36278	278	apply(simp add: abs_fresh)
wenzelm@36278	279	apply(simp)
wenzelm@36278	280	apply(simp add: trm.inject fresh_atm)
wenzelm@36278	281	apply(auto)
wenzelm@36278	282	apply(rule aux4)
wenzelm@36278	283	apply(rule sym)
wenzelm@36278	284	apply(rule fin_substn_nrename)
wenzelm@36278	285	apply(simp_all)
wenzelm@36278	286	apply(rule a_starI)
wenzelm@36278	287	apply(rule al_redu)
wenzelm@36278	288	apply(rule aux2)
wenzelm@36278	289	apply(rule l_redu.intros)
wenzelm@36278	290	apply(simp add: subst_fresh)
wenzelm@36278	291	apply(simp add: fresh_atm)
wenzelm@36278	292	apply(rule fin_subst1)
wenzelm@36278	293	apply(simp_all)
wenzelm@36278	294	apply(rule subst_comm')
wenzelm@36278	295	apply(simp_all)
wenzelm@36278	296	done
wenzelm@36278	297	next
wenzelm@36278	298	case (LNot v M N u a b)
wenzelm@36278	299	then show ?case
wenzelm@36278	300	proof -
wenzelm@36278	301	{ assume asm: "N\<noteq>Ax y b"
wenzelm@36278	302	have "(Cut <a>.NotR (u).M a (v).NotL <b>.N v){y:=<c>.P} =
wenzelm@42764	303	(Cut <a>.NotR (u).(M{y:=<c>.P}) a (v).NotL <b>.(N{y:=<c>.P}) v)" using LNot
wenzelm@36278	304	by (simp add: subst_fresh abs_fresh fresh_atm)
wenzelm@54152	305	also have "\<dots> \<longrightarrow>\<^sub>l (Cut <b>.(N{y:=<c>.P}) (u).(M{y:=<c>.P}))" using LNot
wenzelm@36278	306	by (auto intro: l_redu.intros simp add: subst_fresh)
wenzelm@42764	307	also have "\<dots> = (Cut <b>.N (u).M){y:=<c>.P}" using LNot asm
wenzelm@36278	308	by (simp add: subst_fresh abs_fresh fresh_atm)
wenzelm@36278	309	finally have ?thesis by auto
wenzelm@36278	310	}
wenzelm@36278	311	moreover
wenzelm@36278	312	{ assume asm: "N=Ax y b"
wenzelm@36278	313	have "(Cut <a>.NotR (u).M a (v).NotL <b>.N v){y:=<c>.P} =
wenzelm@42764	314	(Cut <a>.NotR (u).(M{y:=<c>.P}) a (v).NotL <b>.(N{y:=<c>.P}) v)" using LNot
wenzelm@36278	315	by (simp add: subst_fresh abs_fresh fresh_atm)
wenzelm@54152	316	also have "\<dots> \<longrightarrow>\<^sub>a* (Cut <b>.(N{y:=<c>.P}) (u).(M{y:=<c>.P}))" using LNot
wenzelm@36278	317	apply -
wenzelm@36278	318	apply(rule a_starI)
wenzelm@36278	319	apply(rule al_redu)
wenzelm@36278	320	apply(auto intro: l_redu.intros simp add: subst_fresh abs_fresh)
wenzelm@36278	321	done
wenzelm@42764	322	also have "\<dots> = (Cut <b>.(Cut <c>.P (y).Ax y b) (u).(M{y:=<c>.P}))" using LNot asm
wenzelm@36278	323	by simp
wenzelm@54152	324	also have "\<dots> \<longrightarrow>\<^sub>a* (Cut <b>.(P[c\<turnstile>c>b]) (u).(M{y:=<c>.P}))"
wenzelm@36278	325	proof (cases "fic P c")
wenzelm@36278	326	case True
wenzelm@36278	327	assume "fic P c"
wenzelm@42764	328	then show ?thesis using LNot
wenzelm@36278	329	apply -
wenzelm@36278	330	apply(rule a_starI)
wenzelm@36278	331	apply(rule better_CutL_intro)
wenzelm@36278	332	apply(rule al_redu)
wenzelm@36278	333	apply(rule better_LAxR_intro)
wenzelm@36278	334	apply(simp)
wenzelm@36278	335	done
wenzelm@36278	336	next
wenzelm@36278	337	case False
wenzelm@36278	338	assume "\<not>fic P c"
wenzelm@36278	339	then show ?thesis
wenzelm@36278	340	apply -
wenzelm@36278	341	apply(rule a_star_CutL)
wenzelm@36278	342	apply(rule a_star_trans)
wenzelm@36278	343	apply(rule a_starI)
wenzelm@36278	344	apply(rule ac_redu)
wenzelm@36278	345	apply(rule better_left)
wenzelm@36278	346	apply(simp)
wenzelm@36278	347	apply(simp add: subst_with_ax2)
wenzelm@36278	348	done
wenzelm@36278	349	qed
wenzelm@42764	350	also have "\<dots> = (Cut <b>.N (u).M){y:=<c>.P}" using LNot asm
wenzelm@36278	351	apply -
wenzelm@36278	352	apply(auto simp add: subst_fresh abs_fresh)
wenzelm@36278	353	apply(simp add: trm.inject)
wenzelm@36278	354	apply(simp add: alpha fresh_atm)
wenzelm@36278	355	apply(rule sym)
wenzelm@36278	356	apply(rule crename_swap)
wenzelm@36278	357	apply(simp)
wenzelm@36278	358	done
wenzelm@54152	359	finally have "(Cut <a>.NotR (u).M a (v).NotL <b>.N v){y:=<c>.P} \<longrightarrow>\<^sub>a* (Cut <b>.N (u).M){y:=<c>.P}"
wenzelm@36278	360	by simp
wenzelm@36278	361	}
wenzelm@36278	362	ultimately show ?thesis by blast
wenzelm@36278	363	qed
wenzelm@36278	364	next
wenzelm@36278	365	case (LAnd1 b a1 M1 a2 M2 N z u)
wenzelm@36278	366	then show ?case
wenzelm@36278	367	proof -
wenzelm@36278	368	{ assume asm: "M1\<noteq>Ax y a1"
wenzelm@36278	369	have "(Cut <b>.AndR <a1>.M1 <a2>.M2 b (z).AndL1 (u).N z){y:=<c>.P} =
wenzelm@36278	370	Cut <b>.AndR <a1>.(M1{y:=<c>.P}) <a2>.(M2{y:=<c>.P}) b (z).AndL1 (u).(N{y:=<c>.P}) z"
wenzelm@42764	371	using LAnd1 by (simp add: subst_fresh abs_fresh fresh_atm)
wenzelm@54152	372	also have "\<dots> \<longrightarrow>\<^sub>a* Cut <a1>.(M1{y:=<c>.P}) (u).(N{y:=<c>.P})"
wenzelm@42764	373	using LAnd1
wenzelm@36278	374	apply -
wenzelm@36278	375	apply(rule a_starI)
wenzelm@36278	376	apply(rule al_redu)
wenzelm@36278	377	apply(auto intro: l_redu.intros simp add: subst_fresh abs_fresh)
wenzelm@36278	378	done
wenzelm@42764	379	also have "\<dots> = (Cut <a1>.M1 (u).N){y:=<c>.P}" using LAnd1 asm
wenzelm@36278	380	by (simp add: subst_fresh abs_fresh fresh_atm)
wenzelm@36278	381	finally
wenzelm@54152	382	have "(Cut <b>.AndR <a1>.M1 <a2>.M2 b (z).AndL1 (u).N z){y:=<c>.P} \<longrightarrow>\<^sub>a* (Cut <a1>.M1 (u).N){y:=<c>.P}"
wenzelm@36278	383	by simp
wenzelm@36278	384	}
wenzelm@36278	385	moreover
wenzelm@36278	386	{ assume asm: "M1=Ax y a1"
wenzelm@36278	387	have "(Cut <b>.AndR <a1>.M1 <a2>.M2 b (z).AndL1 (u).N z){y:=<c>.P} =
wenzelm@36278	388	Cut <b>.AndR <a1>.(M1{y:=<c>.P}) <a2>.(M2{y:=<c>.P}) b (z).AndL1 (u).(N{y:=<c>.P}) z"
wenzelm@42764	389	using LAnd1 by (simp add: subst_fresh abs_fresh fresh_atm)
wenzelm@54152	390	also have "\<dots> \<longrightarrow>\<^sub>a* Cut <a1>.(M1{y:=<c>.P}) (u).(N{y:=<c>.P})"
wenzelm@42764	391	using LAnd1
wenzelm@36278	392	apply -
wenzelm@36278	393	apply(rule a_starI)
wenzelm@36278	394	apply(rule al_redu)
wenzelm@36278	395	apply(auto intro: l_redu.intros simp add: subst_fresh abs_fresh)
wenzelm@36278	396	done
wenzelm@36278	397	also have "\<dots> = Cut <a1>.(Cut <c>.P (y). Ax y a1) (u).(N{y:=<c>.P})"
wenzelm@42764	398	using LAnd1 asm by simp
wenzelm@54152	399	also have "\<dots> \<longrightarrow>\<^sub>a* Cut <a1>.P[c\<turnstile>c>a1] (u).(N{y:=<c>.P})"
wenzelm@36278	400	proof (cases "fic P c")
wenzelm@36278	401	case True
wenzelm@36278	402	assume "fic P c"
wenzelm@42764	403	then show ?thesis using LAnd1
wenzelm@36278	404	apply -
wenzelm@36278	405	apply(rule a_starI)
wenzelm@36278	406	apply(rule better_CutL_intro)
wenzelm@36278	407	apply(rule al_redu)
wenzelm@36278	408	apply(rule better_LAxR_intro)
wenzelm@36278	409	apply(simp)
wenzelm@36278	410	done
wenzelm@36278	411	next
wenzelm@36278	412	case False
wenzelm@36278	413	assume "\<not>fic P c"
wenzelm@36278	414	then show ?thesis
wenzelm@36278	415	apply -
wenzelm@36278	416	apply(rule a_star_CutL)
wenzelm@36278	417	apply(rule a_star_trans)
wenzelm@36278	418	apply(rule a_starI)
wenzelm@36278	419	apply(rule ac_redu)
wenzelm@36278	420	apply(rule better_left)
wenzelm@36278	421	apply(simp)
wenzelm@36278	422	apply(simp add: subst_with_ax2)
wenzelm@36278	423	done
wenzelm@36278	424	qed
wenzelm@42764	425	also have "\<dots> = (Cut <a1>.M1 (u).N){y:=<c>.P}" using LAnd1 asm
wenzelm@36278	426	apply -
wenzelm@36278	427	apply(auto simp add: subst_fresh abs_fresh)
wenzelm@36278	428	apply(simp add: trm.inject)
wenzelm@36278	429	apply(simp add: alpha fresh_atm)
wenzelm@36278	430	apply(rule sym)
wenzelm@36278	431	apply(rule crename_swap)
wenzelm@36278	432	apply(simp)
wenzelm@36278	433	done
wenzelm@36278	434	finally
wenzelm@54152	435	have "(Cut <b>.AndR <a1>.M1 <a2>.M2 b (z).AndL1 (u).N z){y:=<c>.P} \<longrightarrow>\<^sub>a* (Cut <a1>.M1 (u).N){y:=<c>.P}"
wenzelm@36278	436	by simp
wenzelm@36278	437	}
wenzelm@36278	438	ultimately show ?thesis by blast
wenzelm@36278	439	qed
wenzelm@36278	440	next
wenzelm@36278	441	case (LAnd2 b a1 M1 a2 M2 N z u)
wenzelm@36278	442	then show ?case
wenzelm@36278	443	proof -
wenzelm@36278	444	{ assume asm: "M2\<noteq>Ax y a2"
wenzelm@36278	445	have "(Cut <b>.AndR <a1>.M1 <a2>.M2 b (z).AndL2 (u).N z){y:=<c>.P} =
wenzelm@36278	446	Cut <b>.AndR <a1>.(M1{y:=<c>.P}) <a2>.(M2{y:=<c>.P}) b (z).AndL2 (u).(N{y:=<c>.P}) z"
wenzelm@42764	447	using LAnd2 by (simp add: subst_fresh abs_fresh fresh_atm)
wenzelm@54152	448	also have "\<dots> \<longrightarrow>\<^sub>a* Cut <a2>.(M2{y:=<c>.P}) (u).(N{y:=<c>.P})"
wenzelm@42764	449	using LAnd2
wenzelm@36278	450	apply -
wenzelm@36278	451	apply(rule a_starI)
wenzelm@36278	452	apply(rule al_redu)
wenzelm@36278	453	apply(auto intro: l_redu.intros simp add: subst_fresh abs_fresh)
wenzelm@36278	454	done
wenzelm@42764	455	also have "\<dots> = (Cut <a2>.M2 (u).N){y:=<c>.P}" using LAnd2 asm
wenzelm@36278	456	by (simp add: subst_fresh abs_fresh fresh_atm)
wenzelm@36278	457	finally
wenzelm@54152	458	have "(Cut <b>.AndR <a1>.M1 <a2>.M2 b (z).AndL2 (u).N z){y:=<c>.P} \<longrightarrow>\<^sub>a* (Cut <a2>.M2 (u).N){y:=<c>.P}"
wenzelm@36278	459	by simp
wenzelm@36278	460	}
wenzelm@36278	461	moreover
wenzelm@36278	462	{ assume asm: "M2=Ax y a2"
wenzelm@36278	463	have "(Cut <b>.AndR <a1>.M1 <a2>.M2 b (z).AndL2 (u).N z){y:=<c>.P} =
wenzelm@36278	464	Cut <b>.AndR <a1>.(M1{y:=<c>.P}) <a2>.(M2{y:=<c>.P}) b (z).AndL2 (u).(N{y:=<c>.P}) z"
wenzelm@42764	465	using LAnd2 by (simp add: subst_fresh abs_fresh fresh_atm)
wenzelm@54152	466	also have "\<dots> \<longrightarrow>\<^sub>a* Cut <a2>.(M2{y:=<c>.P}) (u).(N{y:=<c>.P})"
wenzelm@42764	467	using LAnd2
wenzelm@36278	468	apply -
wenzelm@36278	469	apply(rule a_starI)
wenzelm@36278	470	apply(rule al_redu)
wenzelm@36278	471	apply(auto intro: l_redu.intros simp add: subst_fresh abs_fresh)
wenzelm@36278	472	done
wenzelm@36278	473	also have "\<dots> = Cut <a2>.(Cut <c>.P (y). Ax y a2) (u).(N{y:=<c>.P})"
wenzelm@42764	474	using LAnd2 asm by simp
wenzelm@54152	475	also have "\<dots> \<longrightarrow>\<^sub>a* Cut <a2>.P[c\<turnstile>c>a2] (u).(N{y:=<c>.P})"
wenzelm@36278	476	proof (cases "fic P c")
wenzelm@36278	477	case True
wenzelm@36278	478	assume "fic P c"
wenzelm@42764	479	then show ?thesis using LAnd2 asm
wenzelm@36278	480	apply -
wenzelm@36278	481	apply(rule a_starI)
wenzelm@36278	482	apply(rule better_CutL_intro)
wenzelm@36278	483	apply(rule al_redu)
wenzelm@36278	484	apply(rule better_LAxR_intro)
wenzelm@36278	485	apply(simp)
wenzelm@36278	486	done
wenzelm@36278	487	next
wenzelm@36278	488	case False
wenzelm@36278	489	assume "\<not>fic P c"
wenzelm@36278	490	then show ?thesis
wenzelm@36278	491	apply -
wenzelm@36278	492	apply(rule a_star_CutL)
wenzelm@36278	493	apply(rule a_star_trans)
wenzelm@36278	494	apply(rule a_starI)
wenzelm@36278	495	apply(rule ac_redu)
wenzelm@36278	496	apply(rule better_left)
wenzelm@36278	497	apply(simp)
wenzelm@36278	498	apply(simp add: subst_with_ax2)
wenzelm@36278	499	done
wenzelm@36278	500	qed
wenzelm@42764	501	also have "\<dots> = (Cut <a2>.M2 (u).N){y:=<c>.P}" using LAnd2 asm
wenzelm@36278	502	apply -
wenzelm@36278	503	apply(auto simp add: subst_fresh abs_fresh)
wenzelm@36278	504	apply(simp add: trm.inject)
wenzelm@36278	505	apply(simp add: alpha fresh_atm)
wenzelm@36278	506	apply(rule sym)
wenzelm@36278	507	apply(rule crename_swap)
wenzelm@36278	508	apply(simp)
wenzelm@36278	509	done
wenzelm@36278	510	finally
wenzelm@54152	511	have "(Cut <b>.AndR <a1>.M1 <a2>.M2 b (z).AndL2 (u).N z){y:=<c>.P} \<longrightarrow>\<^sub>a* (Cut <a2>.M2 (u).N){y:=<c>.P}"
wenzelm@36278	512	by simp
wenzelm@36278	513	}
wenzelm@36278	514	ultimately show ?thesis by blast
wenzelm@36278	515	qed
wenzelm@36278	516	next
wenzelm@36278	517	case (LOr1 b a M N1 N2 z x1 x2 y c P)
wenzelm@36278	518	then show ?case
wenzelm@36278	519	proof -
wenzelm@36278	520	{ assume asm: "M\<noteq>Ax y a"
wenzelm@36278	521	have "(Cut <b>.OrR1 <a>.M b (z).OrL (x1).N1 (x2).N2 z){y:=<c>.P} =
wenzelm@36278	522	Cut <b>.OrR1 <a>.(M{y:=<c>.P}) b (z).OrL (x1).(N1{y:=<c>.P}) (x2).(N2{y:=<c>.P}) z"
wenzelm@42764	523	using LOr1 by (simp add: subst_fresh abs_fresh fresh_atm)
wenzelm@54152	524	also have "\<dots> \<longrightarrow>\<^sub>a* Cut <a>.(M{y:=<c>.P}) (x1).(N1{y:=<c>.P})"
wenzelm@42764	525	using LOr1
wenzelm@36278	526	apply -
wenzelm@36278	527	apply(rule a_starI)
wenzelm@36278	528	apply(rule al_redu)
wenzelm@36278	529	apply(auto intro: l_redu.intros simp add: subst_fresh abs_fresh)
wenzelm@36278	530	done
wenzelm@42764	531	also have "\<dots> = (Cut <a>.M (x1).N1){y:=<c>.P}" using LOr1 asm
wenzelm@36278	532	by (simp add: subst_fresh abs_fresh fresh_atm)
wenzelm@36278	533	finally
wenzelm@54152	534	have "(Cut <b>.OrR1 <a>.M b (z).OrL (x1).N1 (x2).N2 z){y:=<c>.P} \<longrightarrow>\<^sub>a* (Cut <a>.M (x1).N1){y:=<c>.P}"
wenzelm@36278	535	by simp
wenzelm@36278	536	}
wenzelm@36278	537	moreover
wenzelm@36278	538	{ assume asm: "M=Ax y a"
wenzelm@36278	539	have "(Cut <b>.OrR1 <a>.M b (z).OrL (x1).N1 (x2).N2 z){y:=<c>.P} =
wenzelm@36278	540	Cut <b>.OrR1 <a>.(M{y:=<c>.P}) b (z).OrL (x1).(N1{y:=<c>.P}) (x2).(N2{y:=<c>.P}) z"
wenzelm@42764	541	using LOr1 by (simp add: subst_fresh abs_fresh fresh_atm)
wenzelm@54152	542	also have "\<dots> \<longrightarrow>\<^sub>a* Cut <a>.(M{y:=<c>.P}) (x1).(N1{y:=<c>.P})"
wenzelm@42764	543	using LOr1
wenzelm@36278	544	apply -
wenzelm@36278	545	apply(rule a_starI)
wenzelm@36278	546	apply(rule al_redu)
wenzelm@36278	547	apply(auto intro: l_redu.intros simp add: subst_fresh abs_fresh)
wenzelm@36278	548	done
wenzelm@36278	549	also have "\<dots> = Cut <a>.(Cut <c>.P (y). Ax y a) (x1).(N1{y:=<c>.P})"
wenzelm@42764	550	using LOr1 asm by simp
wenzelm@54152	551	also have "\<dots> \<longrightarrow>\<^sub>a* Cut <a>.P[c\<turnstile>c>a] (x1).(N1{y:=<c>.P})"
wenzelm@36278	552	proof (cases "fic P c")
wenzelm@36278	553	case True
wenzelm@36278	554	assume "fic P c"
wenzelm@42764	555	then show ?thesis using LOr1
wenzelm@36278	556	apply -
wenzelm@36278	557	apply(rule a_starI)
wenzelm@36278	558	apply(rule better_CutL_intro)
wenzelm@36278	559	apply(rule al_redu)
wenzelm@36278	560	apply(rule better_LAxR_intro)
wenzelm@36278	561	apply(simp)
wenzelm@36278	562	done
wenzelm@36278	563	next
wenzelm@36278	564	case False
wenzelm@36278	565	assume "\<not>fic P c"
wenzelm@36278	566	then show ?thesis
wenzelm@36278	567	apply -
wenzelm@36278	568	apply(rule a_star_CutL)
wenzelm@36278	569	apply(rule a_star_trans)
wenzelm@36278	570	apply(rule a_starI)
wenzelm@36278	571	apply(rule ac_redu)
wenzelm@36278	572	apply(rule better_left)
wenzelm@36278	573	apply(simp)
wenzelm@36278	574	apply(simp add: subst_with_ax2)
wenzelm@36278	575	done
wenzelm@36278	576	qed
wenzelm@42764	577	also have "\<dots> = (Cut <a>.M (x1).N1){y:=<c>.P}" using LOr1 asm
wenzelm@36278	578	apply -
wenzelm@36278	579	apply(auto simp add: subst_fresh abs_fresh)
wenzelm@36278	580	apply(simp add: trm.inject)
wenzelm@36278	581	apply(simp add: alpha fresh_atm)
wenzelm@36278	582	apply(rule sym)
wenzelm@36278	583	apply(rule crename_swap)
wenzelm@36278	584	apply(simp)
wenzelm@36278	585	done
wenzelm@36278	586	finally
wenzelm@54152	587	have "(Cut <b>.OrR1 <a>.M b (z).OrL (x1).N1 (x2).N2 z){y:=<c>.P} \<longrightarrow>\<^sub>a* (Cut <a>.M (x1).N1){y:=<c>.P}"
wenzelm@36278	588	by simp
wenzelm@36278	589	}
wenzelm@36278	590	ultimately show ?thesis by blast
wenzelm@36278	591	qed
wenzelm@36278	592	next
wenzelm@36278	593	case (LOr2 b a M N1 N2 z x1 x2 y c P)
wenzelm@36278	594	then show ?case
wenzelm@36278	595	proof -
wenzelm@36278	596	{ assume asm: "M\<noteq>Ax y a"
wenzelm@36278	597	have "(Cut <b>.OrR2 <a>.M b (z).OrL (x1).N1 (x2).N2 z){y:=<c>.P} =
wenzelm@36278	598	Cut <b>.OrR2 <a>.(M{y:=<c>.P}) b (z).OrL (x1).(N1{y:=<c>.P}) (x2).(N2{y:=<c>.P}) z"
wenzelm@42764	599	using LOr2 by (simp add: subst_fresh abs_fresh fresh_atm)
wenzelm@54152	600	also have "\<dots> \<longrightarrow>\<^sub>a* Cut <a>.(M{y:=<c>.P}) (x2).(N2{y:=<c>.P})"
wenzelm@42764	601	using LOr2
wenzelm@36278	602	apply -
wenzelm@36278	603	apply(rule a_starI)
wenzelm@36278	604	apply(rule al_redu)
wenzelm@36278	605	apply(auto intro: l_redu.intros simp add: subst_fresh abs_fresh)
wenzelm@36278	606	done
wenzelm@42764	607	also have "\<dots> = (Cut <a>.M (x2).N2){y:=<c>.P}" using LOr2 asm
wenzelm@36278	608	by (simp add: subst_fresh abs_fresh fresh_atm)
wenzelm@36278	609	finally
wenzelm@54152	610	have "(Cut <b>.OrR2 <a>.M b (z).OrL (x1).N1 (x2).N2 z){y:=<c>.P} \<longrightarrow>\<^sub>a* (Cut <a>.M (x2).N2){y:=<c>.P}"
wenzelm@36278	611	by simp
wenzelm@36278	612	}
wenzelm@36278	613	moreover
wenzelm@36278	614	{ assume asm: "M=Ax y a"
wenzelm@36278	615	have "(Cut <b>.OrR2 <a>.M b (z).OrL (x1).N1 (x2).N2 z){y:=<c>.P} =
wenzelm@36278	616	Cut <b>.OrR2 <a>.(M{y:=<c>.P}) b (z).OrL (x1).(N1{y:=<c>.P}) (x2).(N2{y:=<c>.P}) z"
wenzelm@42764	617	using LOr2 by (simp add: subst_fresh abs_fresh fresh_atm)
wenzelm@54152	618	also have "\<dots> \<longrightarrow>\<^sub>a* Cut <a>.(M{y:=<c>.P}) (x2).(N2{y:=<c>.P})"
wenzelm@42764	619	using LOr2
wenzelm@36278	620	apply -
wenzelm@36278	621	apply(rule a_starI)
wenzelm@36278	622	apply(rule al_redu)
wenzelm@36278	623	apply(auto intro: l_redu.intros simp add: subst_fresh abs_fresh)
wenzelm@36278	624	done
wenzelm@36278	625	also have "\<dots> = Cut <a>.(Cut <c>.P (y). Ax y a) (x2).(N2{y:=<c>.P})"
wenzelm@42764	626	using LOr2 asm by simp
wenzelm@54152	627	also have "\<dots> \<longrightarrow>\<^sub>a* Cut <a>.P[c\<turnstile>c>a] (x2).(N2{y:=<c>.P})"
wenzelm@36278	628	proof (cases "fic P c")
wenzelm@36278	629	case True
wenzelm@36278	630	assume "fic P c"
wenzelm@42764	631	then show ?thesis using LOr2
wenzelm@36278	632	apply -
wenzelm@36278	633	apply(rule a_starI)
wenzelm@36278	634	apply(rule better_CutL_intro)
wenzelm@36278	635	apply(rule al_redu)
wenzelm@36278	636	apply(rule better_LAxR_intro)
wenzelm@36278	637	apply(simp)
wenzelm@36278	638	done
wenzelm@36278	639	next
wenzelm@36278	640	case False
wenzelm@36278	641	assume "\<not>fic P c"
wenzelm@36278	642	then show ?thesis
wenzelm@36278	643	apply -
wenzelm@36278	644	apply(rule a_star_CutL)
wenzelm@36278	645	apply(rule a_star_trans)
wenzelm@36278	646	apply(rule a_starI)
wenzelm@36278	647	apply(rule ac_redu)
wenzelm@36278	648	apply(rule better_left)
wenzelm@36278	649	apply(simp)
wenzelm@36278	650	apply(simp add: subst_with_ax2)
wenzelm@36278	651	done
wenzelm@36278	652	qed
wenzelm@42764	653	also have "\<dots> = (Cut <a>.M (x2).N2){y:=<c>.P}" using LOr2 asm
wenzelm@36278	654	apply -
wenzelm@36278	655	apply(auto simp add: subst_fresh abs_fresh)
wenzelm@36278	656	apply(simp add: trm.inject)
wenzelm@36278	657	apply(simp add: alpha fresh_atm)
wenzelm@36278	658	apply(rule sym)
wenzelm@36278	659	apply(rule crename_swap)
wenzelm@36278	660	apply(simp)
wenzelm@36278	661	done
wenzelm@36278	662	finally
wenzelm@54152	663	have "(Cut <b>.OrR2 <a>.M b (z).OrL (x1).N1 (x2).N2 z){y:=<c>.P} \<longrightarrow>\<^sub>a* (Cut <a>.M (x2).N2){y:=<c>.P}"
wenzelm@36278	664	by simp
wenzelm@36278	665	}
wenzelm@36278	666	ultimately show ?thesis by blast
wenzelm@36278	667	qed
wenzelm@36278	668	next
wenzelm@36278	669	case (LImp z N u Q x M b a d y c P)
wenzelm@36278	670	then show ?case
wenzelm@36278	671	proof -
wenzelm@36278	672	{ assume asm: "N\<noteq>Ax y d"
wenzelm@36278	673	have "(Cut <b>.ImpR (x).<a>.M b (z).ImpL <d>.N (u).Q z){y:=<c>.P} =
wenzelm@36278	674	Cut <b>.ImpR (x).<a>.(M{y:=<c>.P}) b (z).ImpL <d>.(N{y:=<c>.P}) (u).(Q{y:=<c>.P}) z"
wenzelm@42764	675	using LImp by (simp add: fresh_prod abs_fresh fresh_atm)
wenzelm@54152	676	also have "\<dots> \<longrightarrow>\<^sub>a* Cut <a>.(Cut <d>.(N{y:=<c>.P}) (x).(M{y:=<c>.P})) (u).(Q{y:=<c>.P})"
wenzelm@42764	677	using LImp
wenzelm@36278	678	apply -
wenzelm@36278	679	apply(rule a_starI)
wenzelm@36278	680	apply(rule al_redu)
wenzelm@36278	681	apply(auto intro: l_redu.intros simp add: subst_fresh abs_fresh)
wenzelm@36278	682	done
wenzelm@42764	683	also have "\<dots> = (Cut <a>.(Cut <d>.N (x).M) (u).Q){y:=<c>.P}" using LImp asm
wenzelm@36278	684	by (simp add: subst_fresh abs_fresh fresh_atm)
wenzelm@36278	685	finally
wenzelm@54152	686	have "(Cut <b>.ImpR (x).<a>.M b (z).ImpL <d>.N (u).Q z){y:=<c>.P} \<longrightarrow>\<^sub>a*
wenzelm@36278	687	(Cut <a>.(Cut <d>.N (x).M) (u).Q){y:=<c>.P}"
wenzelm@36278	688	by simp
wenzelm@36278	689	}
wenzelm@36278	690	moreover
wenzelm@36278	691	{ assume asm: "N=Ax y d"
wenzelm@36278	692	have "(Cut <b>.ImpR (x).<a>.M b (z).ImpL <d>.N (u).Q z){y:=<c>.P} =
wenzelm@36278	693	Cut <b>.ImpR (x).<a>.(M{y:=<c>.P}) b (z).ImpL <d>.(N{y:=<c>.P}) (u).(Q{y:=<c>.P}) z"
wenzelm@42764	694	using LImp by (simp add: subst_fresh abs_fresh fresh_atm fresh_prod)
wenzelm@54152	695	also have "\<dots> \<longrightarrow>\<^sub>a* Cut <a>.(Cut <d>.(N{y:=<c>.P}) (x).(M{y:=<c>.P})) (u).(Q{y:=<c>.P})"
wenzelm@42764	696	using LImp
wenzelm@36278	697	apply -
wenzelm@36278	698	apply(rule a_starI)
wenzelm@36278	699	apply(rule al_redu)
wenzelm@36278	700	apply(auto intro: l_redu.intros simp add: subst_fresh abs_fresh)
wenzelm@36278	701	done
wenzelm@36278	702	also have "\<dots> = Cut <a>.(Cut <d>.(Cut <c>.P (y).Ax y d) (x).(M{y:=<c>.P})) (u).(Q{y:=<c>.P})"
wenzelm@42764	703	using LImp asm by simp
wenzelm@54152	704	also have "\<dots> \<longrightarrow>\<^sub>a* Cut <a>.(Cut <d>.(P[c\<turnstile>c>d]) (x).(M{y:=<c>.P})) (u).(Q{y:=<c>.P})"
wenzelm@36278	705	proof (cases "fic P c")
wenzelm@36278	706	case True
wenzelm@36278	707	assume "fic P c"
wenzelm@42764	708	then show ?thesis using LImp
wenzelm@36278	709	apply -
wenzelm@36278	710	apply(rule a_starI)
wenzelm@36278	711	apply(rule better_CutL_intro)
wenzelm@36278	712	apply(rule a_Cut_l)
wenzelm@36278	713	apply(simp add: subst_fresh abs_fresh)
wenzelm@36278	714	apply(simp add: abs_fresh fresh_atm)
wenzelm@36278	715	apply(rule al_redu)
wenzelm@36278	716	apply(rule better_LAxR_intro)
wenzelm@36278	717	apply(simp)
wenzelm@36278	718	done
wenzelm@36278	719	next
wenzelm@36278	720	case False
wenzelm@36278	721	assume "\<not>fic P c"
wenzelm@42764	722	then show ?thesis using LImp
wenzelm@36278	723	apply -
wenzelm@36278	724	apply(rule a_star_CutL)
wenzelm@36278	725	apply(rule a_star_CutL)
wenzelm@36278	726	apply(rule a_star_trans)
wenzelm@36278	727	apply(rule a_starI)
wenzelm@36278	728	apply(rule ac_redu)
wenzelm@36278	729	apply(rule better_left)
wenzelm@36278	730	apply(simp)
wenzelm@36278	731	apply(simp add: subst_with_ax2)
wenzelm@36278	732	done
wenzelm@36278	733	qed
wenzelm@42764	734	also have "\<dots> = (Cut <a>.(Cut <d>.N (x).M) (u).Q){y:=<c>.P}" using LImp asm
wenzelm@36278	735	apply -
wenzelm@36278	736	apply(auto simp add: subst_fresh abs_fresh)
wenzelm@36278	737	apply(simp add: trm.inject)
wenzelm@36278	738	apply(simp add: alpha fresh_atm)
wenzelm@36278	739	apply(simp add: trm.inject)
wenzelm@36278	740	apply(simp add: alpha)
wenzelm@36278	741	apply(rule sym)
wenzelm@36278	742	apply(rule crename_swap)
wenzelm@36278	743	apply(simp)
wenzelm@36278	744	done
wenzelm@36278	745	finally
wenzelm@54152	746	have "(Cut <b>.ImpR (x).<a>.M b (z).ImpL <d>.N (u).Q z){y:=<c>.P} \<longrightarrow>\<^sub>a*
wenzelm@36278	747	(Cut <a>.(Cut <d>.N (x).M) (u).Q){y:=<c>.P}"
wenzelm@36278	748	by simp
wenzelm@36278	749	}
wenzelm@36278	750	ultimately show ?thesis by blast
wenzelm@36278	751	qed
wenzelm@36278	752	qed
wenzelm@36278	753
wenzelm@36278	754	lemma l_redu_subst2:
wenzelm@54152	755	assumes a: "M \<longrightarrow>\<^sub>l M'"
wenzelm@54152	756	shows "M{c:=(y).P} \<longrightarrow>\<^sub>a* M'{c:=(y).P}"
wenzelm@36278	757	using a
wenzelm@36278	758	proof(nominal_induct M M' avoiding: y c P rule: l_redu.strong_induct)
wenzelm@36278	759	case LAxR
wenzelm@36278	760	then show ?case
wenzelm@36278	761	apply -
wenzelm@36278	762	apply(rule aux3)
wenzelm@36278	763	apply(rule better_Cut_substc)
wenzelm@36278	764	apply(simp add: abs_fresh)
wenzelm@36278	765	apply(simp add: abs_fresh)
wenzelm@36278	766	apply(simp add: trm.inject fresh_atm)
wenzelm@36278	767	apply(auto)
wenzelm@36278	768	apply(rule aux4)
wenzelm@36278	769	apply(rule sym)
wenzelm@36278	770	apply(rule fic_substc_crename)
wenzelm@36278	771	apply(simp_all)
wenzelm@36278	772	apply(rule a_starI)
wenzelm@36278	773	apply(rule al_redu)
wenzelm@36278	774	apply(rule aux2)
wenzelm@36278	775	apply(rule l_redu.intros)
wenzelm@36278	776	apply(simp add: subst_fresh)
wenzelm@36278	777	apply(simp add: fresh_atm)
wenzelm@36278	778	apply(rule fic_subst1)
wenzelm@36278	779	apply(simp_all)
wenzelm@36278	780	apply(rule subst_comm')
wenzelm@36278	781	apply(simp_all)
wenzelm@36278	782	done
wenzelm@36278	783	next
wenzelm@36278	784	case LAxL
wenzelm@36278	785	then show ?case
wenzelm@36278	786	apply -
wenzelm@36278	787	apply(rule aux3)
wenzelm@36278	788	apply(rule better_Cut_substc)
wenzelm@36278	789	apply(simp)
wenzelm@36278	790	apply(simp add: abs_fresh)
wenzelm@36278	791	apply(simp add: fresh_atm)
wenzelm@36278	792	apply(auto)
wenzelm@36278	793	apply(rule aux4)
wenzelm@36278	794	apply(simp add: trm.inject alpha calc_atm fresh_atm)
wenzelm@36278	795	apply(rule a_star_trans)
wenzelm@36278	796	apply(rule a_starI)
wenzelm@36278	797	apply(rule al_redu)
wenzelm@36278	798	apply(rule l_redu.intros)
wenzelm@36278	799	apply(simp add: subst_fresh)
wenzelm@36278	800	apply(simp add: fresh_atm)
wenzelm@36278	801	apply(rule fin_subst2)
wenzelm@36278	802	apply(simp_all)
wenzelm@36278	803	apply(rule aux4)
wenzelm@36278	804	apply(rule subst_comm')
wenzelm@36278	805	apply(simp_all)
wenzelm@36278	806	done
wenzelm@36278	807	next
wenzelm@36278	808	case (LNot v M N u a b)
wenzelm@36278	809	then show ?case
wenzelm@36278	810	proof -
wenzelm@36278	811	{ assume asm: "M\<noteq>Ax u c"
wenzelm@36278	812	have "(Cut <a>.NotR (u).M a (v).NotL <b>.N v){c:=(y).P} =
wenzelm@42764	813	(Cut <a>.NotR (u).(M{c:=(y).P}) a (v).NotL <b>.(N{c:=(y).P}) v)" using LNot
wenzelm@36278	814	by (simp add: subst_fresh abs_fresh fresh_atm)
wenzelm@54152	815	also have "\<dots> \<longrightarrow>\<^sub>l (Cut <b>.(N{c:=(y).P}) (u).(M{c:=(y).P}))" using LNot
wenzelm@36278	816	by (auto intro: l_redu.intros simp add: subst_fresh)
wenzelm@42764	817	also have "\<dots> = (Cut <b>.N (u).M){c:=(y).P}" using LNot asm
wenzelm@36278	818	by (simp add: subst_fresh abs_fresh fresh_atm)
wenzelm@36278	819	finally have ?thesis by auto
wenzelm@36278	820	}
wenzelm@36278	821	moreover
wenzelm@36278	822	{ assume asm: "M=Ax u c"
wenzelm@36278	823	have "(Cut <a>.NotR (u).M a (v).NotL <b>.N v){c:=(y).P} =
wenzelm@42764	824	(Cut <a>.NotR (u).(M{c:=(y).P}) a (v).NotL <b>.(N{c:=(y).P}) v)" using LNot
wenzelm@36278	825	by (simp add: subst_fresh abs_fresh fresh_atm)
wenzelm@54152	826	also have "\<dots> \<longrightarrow>\<^sub>a* (Cut <b>.(N{c:=(y).P}) (u).(M{c:=(y).P}))" using LNot
wenzelm@36278	827	apply -
wenzelm@36278	828	apply(rule a_starI)
wenzelm@36278	829	apply(rule al_redu)
wenzelm@36278	830	apply(auto intro: l_redu.intros simp add: subst_fresh abs_fresh)
wenzelm@36278	831	done
wenzelm@42764	832	also have "\<dots> = (Cut <b>.(N{c:=(y).P}) (u).(Cut <c>.(Ax u c) (y).P))" using LNot asm
wenzelm@36278	833	by simp
wenzelm@54152	834	also have "\<dots> \<longrightarrow>\<^sub>a* (Cut <b>.(N{c:=(y).P}) (u).(P[y\<turnstile>n>u]))"
wenzelm@36278	835	proof (cases "fin P y")
wenzelm@36278	836	case True
wenzelm@36278	837	assume "fin P y"
wenzelm@42764	838	then show ?thesis using LNot
wenzelm@36278	839	apply -
wenzelm@36278	840	apply(rule a_starI)
wenzelm@36278	841	apply(rule better_CutR_intro)
wenzelm@36278	842	apply(rule al_redu)
wenzelm@36278	843	apply(rule better_LAxL_intro)
wenzelm@36278	844	apply(simp)
wenzelm@36278	845	done
wenzelm@36278	846	next
wenzelm@36278	847	case False
wenzelm@36278	848	assume "\<not>fin P y"
wenzelm@36278	849	then show ?thesis
wenzelm@36278	850	apply -
wenzelm@36278	851	apply(rule a_star_CutR)
wenzelm@36278	852	apply(rule a_star_trans)
wenzelm@36278	853	apply(rule a_starI)
wenzelm@36278	854	apply(rule ac_redu)
wenzelm@36278	855	apply(rule better_right)
wenzelm@36278	856	apply(simp)
wenzelm@36278	857	apply(simp add: subst_with_ax1)
wenzelm@36278	858	done
wenzelm@36278	859	qed
wenzelm@42764	860	also have "\<dots> = (Cut <b>.N (u).M){c:=(y).P}" using LNot asm
wenzelm@36278	861	apply -
wenzelm@36278	862	apply(auto simp add: subst_fresh abs_fresh)
wenzelm@36278	863	apply(simp add: trm.inject)
wenzelm@36278	864	apply(simp add: alpha fresh_atm)
wenzelm@36278	865	apply(rule sym)
wenzelm@36278	866	apply(rule nrename_swap)
wenzelm@36278	867	apply(simp)
wenzelm@36278	868	done
wenzelm@54152	869	finally have "(Cut <a>.NotR (u).M a (v).NotL <b>.N v){c:=(y).P} \<longrightarrow>\<^sub>a* (Cut <b>.N (u).M){c:=(y).P}"
wenzelm@36278	870	by simp
wenzelm@36278	871	}
wenzelm@36278	872	ultimately show ?thesis by blast
wenzelm@36278	873	qed
wenzelm@36278	874	next
wenzelm@36278	875	case (LAnd1 b a1 M1 a2 M2 N z u)
wenzelm@36278	876	then show ?case
wenzelm@36278	877	proof -
wenzelm@36278	878	{ assume asm: "N\<noteq>Ax u c"
wenzelm@36278	879	have "(Cut <b>.AndR <a1>.M1 <a2>.M2 b (z).AndL1 (u).N z){c:=(y).P} =
wenzelm@36278	880	Cut <b>.AndR <a1>.(M1{c:=(y).P}) <a2>.(M2{c:=(y).P}) b (z).AndL1 (u).(N{c:=(y).P}) z"
wenzelm@42764	881	using LAnd1 by (simp add: subst_fresh abs_fresh fresh_atm)
wenzelm@54152	882	also have "\<dots> \<longrightarrow>\<^sub>a* Cut <a1>.(M1{c:=(y).P}) (u).(N{c:=(y).P})"
wenzelm@42764	883	using LAnd1
wenzelm@36278	884	apply -
wenzelm@36278	885	apply(rule a_starI)
wenzelm@36278	886	apply(rule al_redu)
wenzelm@36278	887	apply(auto intro: l_redu.intros simp add: subst_fresh abs_fresh)
wenzelm@36278	888	done
wenzelm@42764	889	also have "\<dots> = (Cut <a1>.M1 (u).N){c:=(y).P}" using LAnd1 asm
wenzelm@36278	890	by (simp add: subst_fresh abs_fresh fresh_atm)
wenzelm@36278	891	finally
wenzelm@54152	892	have "(Cut <b>.AndR <a1>.M1 <a2>.M2 b (z).AndL1 (u).N z){c:=(y).P} \<longrightarrow>\<^sub>a* (Cut <a1>.M1 (u).N){c:=(y).P}"
wenzelm@36278	893	by simp
wenzelm@36278	894	}
wenzelm@36278	895	moreover
wenzelm@36278	896	{ assume asm: "N=Ax u c"
wenzelm@36278	897	have "(Cut <b>.AndR <a1>.M1 <a2>.M2 b (z).AndL1 (u).N z){c:=(y).P} =
wenzelm@36278	898	Cut <b>.AndR <a1>.(M1{c:=(y).P}) <a2>.(M2{c:=(y).P}) b (z).AndL1 (u).(N{c:=(y).P}) z"
wenzelm@42764	899	using LAnd1 by (simp add: subst_fresh abs_fresh fresh_atm)
wenzelm@54152	900	also have "\<dots> \<longrightarrow>\<^sub>a* Cut <a1>.(M1{c:=(y).P}) (u).(N{c:=(y).P})"
wenzelm@42764	901	using LAnd1
wenzelm@36278	902	apply -
wenzelm@36278	903	apply(rule a_starI)
wenzelm@36278	904	apply(rule al_redu)
wenzelm@36278	905	apply(auto intro: l_redu.intros simp add: subst_fresh abs_fresh)
wenzelm@36278	906	done
wenzelm@36278	907	also have "\<dots> = Cut <a1>.(M1{c:=(y).P}) (u).(Cut <c>.(Ax u c) (y).P)"
wenzelm@42764	908	using LAnd1 asm by simp
wenzelm@54152	909	also have "\<dots> \<longrightarrow>\<^sub>a* Cut <a1>.(M1{c:=(y).P}) (u).(P[y\<turnstile>n>u])"
wenzelm@36278	910	proof (cases "fin P y")
wenzelm@36278	911	case True
wenzelm@36278	912	assume "fin P y"
wenzelm@42764	913	then show ?thesis using LAnd1
wenzelm@36278	914	apply -
wenzelm@36278	915	apply(rule a_starI)
wenzelm@36278	916	apply(rule better_CutR_intro)
wenzelm@36278	917	apply(rule al_redu)
wenzelm@36278	918	apply(rule better_LAxL_intro)
wenzelm@36278	919	apply(simp)
wenzelm@36278	920	done
wenzelm@36278	921	next
wenzelm@36278	922	case False
wenzelm@36278	923	assume "\<not>fin P y"
wenzelm@36278	924	then show ?thesis
wenzelm@36278	925	apply -
wenzelm@36278	926	apply(rule a_star_CutR)
wenzelm@36278	927	apply(rule a_star_trans)
wenzelm@36278	928	apply(rule a_starI)
wenzelm@36278	929	apply(rule ac_redu)
wenzelm@36278	930	apply(rule better_right)
wenzelm@36278	931	apply(simp)
wenzelm@36278	932	apply(simp add: subst_with_ax1)
wenzelm@36278	933	done
wenzelm@36278	934	qed
wenzelm@42764	935	also have "\<dots> = (Cut <a1>.M1 (u).N){c:=(y).P}" using LAnd1 asm
wenzelm@36278	936	apply -
wenzelm@36278	937	apply(auto simp add: subst_fresh abs_fresh)
wenzelm@36278	938	apply(simp add: trm.inject)
wenzelm@36278	939	apply(simp add: alpha fresh_atm)
wenzelm@36278	940	apply(rule sym)
wenzelm@36278	941	apply(rule nrename_swap)
wenzelm@36278	942	apply(simp)
wenzelm@36278	943	done
wenzelm@36278	944	finally
wenzelm@54152	945	have "(Cut <b>.AndR <a1>.M1 <a2>.M2 b (z).AndL1 (u).N z){c:=(y).P} \<longrightarrow>\<^sub>a* (Cut <a1>.M1 (u).N){c:=(y).P}"
wenzelm@36278	946	by simp
wenzelm@36278	947	}
wenzelm@36278	948	ultimately show ?thesis by blast
wenzelm@36278	949	qed
wenzelm@36278	950	next
wenzelm@36278	951	case (LAnd2 b a1 M1 a2 M2 N z u)
wenzelm@36278	952	then show ?case
wenzelm@36278	953	proof -
wenzelm@36278	954	{ assume asm: "N\<noteq>Ax u c"
wenzelm@36278	955	have "(Cut <b>.AndR <a1>.M1 <a2>.M2 b (z).AndL2 (u).N z){c:=(y).P} =
wenzelm@36278	956	Cut <b>.AndR <a1>.(M1{c:=(y).P}) <a2>.(M2{c:=(y).P}) b (z).AndL2 (u).(N{c:=(y).P}) z"
wenzelm@42764	957	using LAnd2 by (simp add: subst_fresh abs_fresh fresh_atm)
wenzelm@54152	958	also have "\<dots> \<longrightarrow>\<^sub>a* Cut <a2>.(M2{c:=(y).P}) (u).(N{c:=(y).P})"
wenzelm@42764	959	using LAnd2
wenzelm@36278	960	apply -
wenzelm@36278	961	apply(rule a_starI)
wenzelm@36278	962	apply(rule al_redu)
wenzelm@36278	963	apply(auto intro: l_redu.intros simp add: subst_fresh abs_fresh)
wenzelm@36278	964	done
wenzelm@42764	965	also have "\<dots> = (Cut <a2>.M2 (u).N){c:=(y).P}" using LAnd2 asm
wenzelm@36278	966	by (simp add: subst_fresh abs_fresh fresh_atm)
wenzelm@36278	967	finally
wenzelm@54152	968	have "(Cut <b>.AndR <a1>.M1 <a2>.M2 b (z).AndL2 (u).N z){c:=(y).P} \<longrightarrow>\<^sub>a* (Cut <a2>.M2 (u).N){c:=(y).P}"
wenzelm@36278	969	by simp
wenzelm@36278	970	}
wenzelm@36278	971	moreover
wenzelm@36278	972	{ assume asm: "N=Ax u c"
wenzelm@36278	973	have "(Cut <b>.AndR <a1>.M1 <a2>.M2 b (z).AndL2 (u).N z){c:=(y).P} =
wenzelm@36278	974	Cut <b>.AndR <a1>.(M1{c:=(y).P}) <a2>.(M2{c:=(y).P}) b (z).AndL2 (u).(N{c:=(y).P}) z"
wenzelm@42764	975	using LAnd2 by (simp add: subst_fresh abs_fresh fresh_atm)
wenzelm@54152	976	also have "\<dots> \<longrightarrow>\<^sub>a* Cut <a2>.(M2{c:=(y).P}) (u).(N{c:=(y).P})"
wenzelm@42764	977	using LAnd2
wenzelm@36278	978	apply -
wenzelm@36278	979	apply(rule a_starI)
wenzelm@36278	980	apply(rule al_redu)
wenzelm@36278	981	apply(auto intro: l_redu.intros simp add: subst_fresh abs_fresh)
wenzelm@36278	982	done
wenzelm@36278	983	also have "\<dots> = Cut <a2>.(M2{c:=(y).P}) (u).(Cut <c>.(Ax u c) (y).P)"
wenzelm@42764	984	using LAnd2 asm by simp
wenzelm@54152	985	also have "\<dots> \<longrightarrow>\<^sub>a* Cut <a2>.(M2{c:=(y).P}) (u).(P[y\<turnstile>n>u])"
wenzelm@36278	986	proof (cases "fin P y")
wenzelm@36278	987	case True
wenzelm@36278	988	assume "fin P y"
wenzelm@42764	989	then show ?thesis using LAnd2
wenzelm@36278	990	apply -
wenzelm@36278	991	apply(rule a_starI)
wenzelm@36278	992	apply(rule better_CutR_intro)
wenzelm@36278	993	apply(rule al_redu)
wenzelm@36278	994	apply(rule better_LAxL_intro)
wenzelm@36278	995	apply(simp)
wenzelm@36278	996	done
wenzelm@36278	997	next
wenzelm@36278	998	case False
wenzelm@36278	999	assume "\<not>fin P y"
wenzelm@36278	1000	then show ?thesis
wenzelm@36278	1001	apply -
wenzelm@36278	1002	apply(rule a_star_CutR)
wenzelm@36278	1003	apply(rule a_star_trans)
wenzelm@36278	1004	apply(rule a_starI)
wenzelm@36278	1005	apply(rule ac_redu)
wenzelm@36278	1006	apply(rule better_right)
wenzelm@36278	1007	apply(simp)
wenzelm@36278	1008	apply(simp add: subst_with_ax1)
wenzelm@36278	1009	done
wenzelm@36278	1010	qed
wenzelm@42764	1011	also have "\<dots> = (Cut <a2>.M2 (u).N){c:=(y).P}" using LAnd2 asm
wenzelm@36278	1012	apply -
wenzelm@36278	1013	apply(auto simp add: subst_fresh abs_fresh)
wenzelm@36278	1014	apply(simp add: trm.inject)
wenzelm@36278	1015	apply(simp add: alpha fresh_atm)
wenzelm@36278	1016	apply(rule sym)
wenzelm@36278	1017	apply(rule nrename_swap)
wenzelm@36278	1018	apply(simp)
wenzelm@36278	1019	done
wenzelm@36278	1020	finally
wenzelm@54152	1021	have "(Cut <b>.AndR <a1>.M1 <a2>.M2 b (z).AndL2 (u).N z){c:=(y).P} \<longrightarrow>\<^sub>a* (Cut <a2>.M2 (u).N){c:=(y).P}"
wenzelm@36278	1022	by simp
wenzelm@36278	1023	}
wenzelm@36278	1024	ultimately show ?thesis by blast
wenzelm@36278	1025	qed
wenzelm@36278	1026	next
wenzelm@36278	1027	case (LOr1 b a M N1 N2 z x1 x2 y c P)
wenzelm@36278	1028	then show ?case
wenzelm@36278	1029	proof -
wenzelm@36278	1030	{ assume asm: "N1\<noteq>Ax x1 c"
wenzelm@36278	1031	have "(Cut <b>.OrR1 <a>.M b (z).OrL (x1).N1 (x2).N2 z){c:=(y).P} =
wenzelm@36278	1032	Cut <b>.OrR1 <a>.(M{c:=(y).P}) b (z).OrL (x1).(N1{c:=(y).P}) (x2).(N2{c:=(y).P}) z"
wenzelm@42764	1033	using LOr1 by (simp add: subst_fresh abs_fresh fresh_atm)
wenzelm@54152	1034	also have "\<dots> \<longrightarrow>\<^sub>a* Cut <a>.(M{c:=(y).P}) (x1).(N1{c:=(y).P})"
wenzelm@42764	1035	using LOr1
wenzelm@36278	1036	apply -
wenzelm@36278	1037	apply(rule a_starI)
wenzelm@36278	1038	apply(rule al_redu)
wenzelm@36278	1039	apply(auto intro: l_redu.intros simp add: subst_fresh abs_fresh)
wenzelm@36278	1040	done
wenzelm@42764	1041	also have "\<dots> = (Cut <a>.M (x1).N1){c:=(y).P}" using LOr1 asm
wenzelm@36278	1042	by (simp add: subst_fresh abs_fresh fresh_atm)
wenzelm@36278	1043	finally
wenzelm@54152	1044	have "(Cut <b>.OrR1 <a>.M b (z).OrL (x1).N1 (x2).N2 z){c:=(y).P} \<longrightarrow>\<^sub>a* (Cut <a>.M (x1).N1){c:=(y).P}"
wenzelm@36278	1045	by simp
wenzelm@36278	1046	}
wenzelm@36278	1047	moreover
wenzelm@36278	1048	{ assume asm: "N1=Ax x1 c"
wenzelm@36278	1049	have "(Cut <b>.OrR1 <a>.M b (z).OrL (x1).N1 (x2).N2 z){c:=(y).P} =
wenzelm@36278	1050	Cut <b>.OrR1 <a>.(M{c:=(y).P}) b (z).OrL (x1).(N1{c:=(y).P}) (x2).(N2{c:=(y).P}) z"
wenzelm@42764	1051	using LOr1 by (simp add: subst_fresh abs_fresh fresh_atm)
wenzelm@54152	1052	also have "\<dots> \<longrightarrow>\<^sub>a* Cut <a>.(M{c:=(y).P}) (x1).(N1{c:=(y).P})"
wenzelm@42764	1053	using LOr1
wenzelm@36278	1054	apply -
wenzelm@36278	1055	apply(rule a_starI)
wenzelm@36278	1056	apply(rule al_redu)
wenzelm@36278	1057	apply(auto intro: l_redu.intros simp add: subst_fresh abs_fresh)
wenzelm@36278	1058	done
wenzelm@36278	1059	also have "\<dots> = Cut <a>.(M{c:=(y).P}) (x1).(Cut <c>.(Ax x1 c) (y).P)"
wenzelm@42764	1060	using LOr1 asm by simp
wenzelm@54152	1061	also have "\<dots> \<longrightarrow>\<^sub>a* Cut <a>.(M{c:=(y).P}) (x1).(P[y\<turnstile>n>x1])"
wenzelm@36278	1062	proof (cases "fin P y")
wenzelm@36278	1063	case True
wenzelm@36278	1064	assume "fin P y"
wenzelm@42764	1065	then show ?thesis using LOr1
wenzelm@36278	1066	apply -
wenzelm@36278	1067	apply(rule a_starI)
wenzelm@36278	1068	apply(rule better_CutR_intro)
wenzelm@36278	1069	apply(rule al_redu)
wenzelm@36278	1070	apply(rule better_LAxL_intro)
wenzelm@36278	1071	apply(simp)
wenzelm@36278	1072	done
wenzelm@36278	1073	next
wenzelm@36278	1074	case False
wenzelm@36278	1075	assume "\<not>fin P y"
wenzelm@36278	1076	then show ?thesis
wenzelm@36278	1077	apply -
wenzelm@36278	1078	apply(rule a_star_CutR)
wenzelm@36278	1079	apply(rule a_star_trans)
wenzelm@36278	1080	apply(rule a_starI)
wenzelm@36278	1081	apply(rule ac_redu)
wenzelm@36278	1082	apply(rule better_right)
wenzelm@36278	1083	apply(simp)
wenzelm@36278	1084	apply(simp add: subst_with_ax1)
wenzelm@36278	1085	done
wenzelm@36278	1086	qed
wenzelm@42764	1087	also have "\<dots> = (Cut <a>.M (x1).N1){c:=(y).P}" using LOr1 asm
wenzelm@36278	1088	apply -
wenzelm@36278	1089	apply(auto simp add: subst_fresh abs_fresh)
wenzelm@36278	1090	apply(simp add: trm.inject)
wenzelm@36278	1091	apply(simp add: alpha fresh_atm)
wenzelm@36278	1092	apply(rule sym)
wenzelm@36278	1093	apply(rule nrename_swap)
wenzelm@36278	1094	apply(simp)
wenzelm@36278	1095	done
wenzelm@36278	1096	finally
wenzelm@54152	1097	have "(Cut <b>.OrR1 <a>.M b (z).OrL (x1).N1 (x2).N2 z){c:=(y).P} \<longrightarrow>\<^sub>a* (Cut <a>.M (x1).N1){c:=(y).P}"
wenzelm@36278	1098	by simp
wenzelm@36278	1099	}
wenzelm@36278	1100	ultimately show ?thesis by blast
wenzelm@36278	1101	qed
wenzelm@36278	1102	next
wenzelm@36278	1103	case (LOr2 b a M N1 N2 z x1 x2 y c P)
wenzelm@36278	1104	then show ?case
wenzelm@36278	1105	proof -
wenzelm@36278	1106	{ assume asm: "N2\<noteq>Ax x2 c"
wenzelm@36278	1107	have "(Cut <b>.OrR2 <a>.M b (z).OrL (x1).N1 (x2).N2 z){c:=(y).P} =
wenzelm@36278	1108	Cut <b>.OrR2 <a>.(M{c:=(y).P}) b (z).OrL (x1).(N1{c:=(y).P}) (x2).(N2{c:=(y).P}) z"
wenzelm@42764	1109	using LOr2 by (simp add: subst_fresh abs_fresh fresh_atm)
wenzelm@54152	1110	also have "\<dots> \<longrightarrow>\<^sub>a* Cut <a>.(M{c:=(y).P}) (x2).(N2{c:=(y).P})"
wenzelm@42764	1111	using LOr2
wenzelm@36278	1112	apply -
wenzelm@36278	1113	apply(rule a_starI)
wenzelm@36278	1114	apply(rule al_redu)
wenzelm@36278	1115	apply(auto intro: l_redu.intros simp add: subst_fresh abs_fresh)
wenzelm@36278	1116	done
wenzelm@42764	1117	also have "\<dots> = (Cut <a>.M (x2).N2){c:=(y).P}" using LOr2 asm
wenzelm@36278	1118	by (simp add: subst_fresh abs_fresh fresh_atm)
wenzelm@36278	1119	finally
wenzelm@54152	1120	have "(Cut <b>.OrR2 <a>.M b (z).OrL (x1).N1 (x2).N2 z){c:=(y).P} \<longrightarrow>\<^sub>a* (Cut <a>.M (x2).N2){c:=(y).P}"
wenzelm@36278	1121	by simp
wenzelm@36278	1122	}
wenzelm@36278	1123	moreover
wenzelm@36278	1124	{ assume asm: "N2=Ax x2 c"
wenzelm@36278	1125	have "(Cut <b>.OrR2 <a>.M b (z).OrL (x1).N1 (x2).N2 z){c:=(y).P} =
wenzelm@36278	1126	Cut <b>.OrR2 <a>.(M{c:=(y).P}) b (z).OrL (x1).(N1{c:=(y).P}) (x2).(N2{c:=(y).P}) z"
wenzelm@42764	1127	using LOr2 by (simp add: subst_fresh abs_fresh fresh_atm)
wenzelm@54152	1128	also have "\<dots> \<longrightarrow>\<^sub>a* Cut <a>.(M{c:=(y).P}) (x2).(N2{c:=(y).P})"
wenzelm@42764	1129	using LOr2
wenzelm@36278	1130	apply -
wenzelm@36278	1131	apply(rule a_starI)
wenzelm@36278	1132	apply(rule al_redu)
wenzelm@36278	1133	apply(auto intro: l_redu.intros simp add: subst_fresh abs_fresh)
wenzelm@36278	1134	done
wenzelm@36278	1135	also have "\<dots> = Cut <a>.(M{c:=(y).P}) (x2).(Cut <c>.(Ax x2 c) (y).P)"
wenzelm@42764	1136	using LOr2 asm by simp
wenzelm@54152	1137	also have "\<dots> \<longrightarrow>\<^sub>a* Cut <a>.(M{c:=(y).P}) (x2).(P[y\<turnstile>n>x2])"
wenzelm@36278	1138	proof (cases "fin P y")
wenzelm@36278	1139	case True
wenzelm@36278	1140	assume "fin P y"
wenzelm@42764	1141	then show ?thesis using LOr2
wenzelm@36278	1142	apply -
wenzelm@36278	1143	apply(rule a_starI)
wenzelm@36278	1144	apply(rule better_CutR_intro)
wenzelm@36278	1145	apply(rule al_redu)
wenzelm@36278	1146	apply(rule better_LAxL_intro)
wenzelm@36278	1147	apply(simp)
wenzelm@36278	1148	done
wenzelm@36278	1149	next
wenzelm@36278	1150	case False
wenzelm@36278	1151	assume "\<not>fin P y"
wenzelm@36278	1152	then show ?thesis
wenzelm@36278	1153	apply -
wenzelm@36278	1154	apply(rule a_star_CutR)
wenzelm@36278	1155	apply(rule a_star_trans)
wenzelm@36278	1156	apply(rule a_starI)
wenzelm@36278	1157	apply(rule ac_redu)
wenzelm@36278	1158	apply(rule better_right)
wenzelm@36278	1159	apply(simp)
wenzelm@36278	1160	apply(simp add: subst_with_ax1)
wenzelm@36278	1161	done
wenzelm@36278	1162	qed
wenzelm@42764	1163	also have "\<dots> = (Cut <a>.M (x2).N2){c:=(y).P}" using LOr2 asm
wenzelm@36278	1164	apply -
wenzelm@36278	1165	apply(auto simp add: subst_fresh abs_fresh)
wenzelm@36278	1166	apply(simp add: trm.inject)
wenzelm@36278	1167	apply(simp add: alpha fresh_atm)
wenzelm@36278	1168	apply(rule sym)
wenzelm@36278	1169	apply(rule nrename_swap)
wenzelm@36278	1170	apply(simp)
wenzelm@36278	1171	done
wenzelm@36278	1172	finally
wenzelm@54152	1173	have "(Cut <b>.OrR2 <a>.M b (z).OrL (x1).N1 (x2).N2 z){c:=(y).P} \<longrightarrow>\<^sub>a* (Cut <a>.M (x2).N2){c:=(y).P}"
wenzelm@36278	1174	by simp
wenzelm@36278	1175	}
wenzelm@36278	1176	ultimately show ?thesis by blast
wenzelm@36278	1177	qed
wenzelm@36278	1178	next
wenzelm@36278	1179	case (LImp z N u Q x M b a d y c P)
wenzelm@36278	1180	then show ?case
wenzelm@36278	1181	proof -
wenzelm@36278	1182	{ assume asm: "M\<noteq>Ax x c \<and> Q\<noteq>Ax u c"
wenzelm@36278	1183	have "(Cut <b>.ImpR (x).<a>.M b (z).ImpL <d>.N (u).Q z){c:=(y).P} =
wenzelm@36278	1184	Cut <b>.ImpR (x).<a>.(M{c:=(y).P}) b (z).ImpL <d>.(N{c:=(y).P}) (u).(Q{c:=(y).P}) z"
wenzelm@42764	1185	using LImp by (simp add: fresh_prod abs_fresh fresh_atm)
wenzelm@54152	1186	also have "\<dots> \<longrightarrow>\<^sub>a* Cut <a>.(Cut <d>.(N{c:=(y).P}) (x).(M{c:=(y).P})) (u).(Q{c:=(y).P})"
wenzelm@42764	1187	using LImp
wenzelm@36278	1188	apply -
wenzelm@36278	1189	apply(rule a_starI)
wenzelm@36278	1190	apply(rule al_redu)
wenzelm@36278	1191	apply(auto intro: l_redu.intros simp add: subst_fresh abs_fresh)
wenzelm@36278	1192	done
wenzelm@42764	1193	also have "\<dots> = (Cut <a>.(Cut <d>.N (x).M) (u).Q){c:=(y).P}" using LImp asm
wenzelm@36278	1194	by (simp add: subst_fresh abs_fresh fresh_atm)
wenzelm@36278	1195	finally
wenzelm@54152	1196	have "(Cut <b>.ImpR (x).<a>.M b (z).ImpL <d>.N (u).Q z){c:=(y).P} \<longrightarrow>\<^sub>a*
wenzelm@36278	1197	(Cut <a>.(Cut <d>.N (x).M) (u).Q){c:=(y).P}"
wenzelm@36278	1198	by simp
wenzelm@36278	1199	}
wenzelm@36278	1200	moreover
wenzelm@36278	1201	{ assume asm: "M=Ax x c \<and> Q\<noteq>Ax u c"
wenzelm@36278	1202	have "(Cut <b>.ImpR (x).<a>.M b (z).ImpL <d>.N (u).Q z){c:=(y).P} =
wenzelm@36278	1203	Cut <b>.ImpR (x).<a>.(M{c:=(y).P}) b (z).ImpL <d>.(N{c:=(y).P}) (u).(Q{c:=(y).P}) z"
wenzelm@42764	1204	using LImp by (simp add: subst_fresh abs_fresh fresh_atm fresh_prod)
wenzelm@54152	1205	also have "\<dots> \<longrightarrow>\<^sub>a* Cut <a>.(Cut <d>.(N{c:=(y).P}) (x).(M{c:=(y).P})) (u).(Q{c:=(y).P})"
wenzelm@42764	1206	using LImp
wenzelm@36278	1207	apply -
wenzelm@36278	1208	apply(rule a_starI)
wenzelm@36278	1209	apply(rule al_redu)
wenzelm@36278	1210	apply(auto intro: l_redu.intros simp add: subst_fresh abs_fresh)
wenzelm@36278	1211	done
wenzelm@36278	1212	also have "\<dots> = Cut <a>.(Cut <d>.(N{c:=(y).P}) (x).(Cut <c>.Ax x c (y).P)) (u).(Q{c:=(y).P})"
wenzelm@42764	1213	using LImp asm by simp
wenzelm@54152	1214	also have "\<dots> \<longrightarrow>\<^sub>a* Cut <a>.(Cut <d>.(N{c:=(y).P}) (x).(P[y\<turnstile>n>x])) (u).(Q{c:=(y).P})"
wenzelm@36278	1215	proof (cases "fin P y")
wenzelm@36278	1216	case True
wenzelm@36278	1217	assume "fin P y"
wenzelm@42764	1218	then show ?thesis using LImp
wenzelm@36278	1219	apply -
wenzelm@36278	1220	apply(rule a_star_CutL)
wenzelm@36278	1221	apply(rule a_star_CutR)
wenzelm@36278	1222	apply(rule a_star_trans)
wenzelm@36278	1223	apply(rule a_starI)
wenzelm@36278	1224	apply(rule al_redu)
wenzelm@36278	1225	apply(rule better_LAxL_intro)
wenzelm@36278	1226	apply(simp)
wenzelm@36278	1227	apply(simp)
wenzelm@36278	1228	done
wenzelm@36278	1229	next
wenzelm@36278	1230	case False
wenzelm@36278	1231	assume "\<not>fin P y"
wenzelm@42764	1232	then show ?thesis using LImp
wenzelm@36278	1233	apply -
wenzelm@36278	1234	apply(rule a_star_CutL)
wenzelm@36278	1235	apply(rule a_star_CutR)
wenzelm@36278	1236	apply(rule a_star_trans)
wenzelm@36278	1237	apply(rule a_starI)
wenzelm@36278	1238	apply(rule ac_redu)
wenzelm@36278	1239	apply(rule better_right)
wenzelm@36278	1240	apply(simp)
wenzelm@36278	1241	apply(simp add: subst_with_ax1)
wenzelm@36278	1242	done
wenzelm@36278	1243	qed
wenzelm@42764	1244	also have "\<dots> = (Cut <a>.(Cut <d>.N (x).M) (u).Q){c:=(y).P}" using LImp asm
wenzelm@36278	1245	apply -
wenzelm@36278	1246	apply(auto simp add: subst_fresh abs_fresh)
wenzelm@36278	1247	apply(simp add: trm.inject)
wenzelm@36278	1248	apply(simp add: alpha fresh_atm)
wenzelm@36278	1249	apply(simp add: trm.inject)
wenzelm@36278	1250	apply(simp add: alpha)
wenzelm@36278	1251	apply(simp add: nrename_swap)
wenzelm@36278	1252	done
wenzelm@36278	1253	finally
wenzelm@54152	1254	have "(Cut <b>.ImpR (x).<a>.M b (z).ImpL <d>.N (u).Q z){c:=(y).P} \<longrightarrow>\<^sub>a*
wenzelm@36278	1255	(Cut <a>.(Cut <d>.N (x).M) (u).Q){c:=(y).P}"
wenzelm@36278	1256	by simp
wenzelm@36278	1257	}
wenzelm@36278	1258	moreover
wenzelm@36278	1259	{ assume asm: "M\<noteq>Ax x c \<and> Q=Ax u c"
wenzelm@36278	1260	have "(Cut <b>.ImpR (x).<a>.M b (z).ImpL <d>.N (u).Q z){c:=(y).P} =
wenzelm@36278	1261	Cut <b>.ImpR (x).<a>.(M{c:=(y).P}) b (z).ImpL <d>.(N{c:=(y).P}) (u).(Q{c:=(y).P}) z"
wenzelm@42764	1262	using LImp by (simp add: subst_fresh abs_fresh fresh_atm fresh_prod)
wenzelm@54152	1263	also have "\<dots> \<longrightarrow>\<^sub>a* Cut <a>.(Cut <d>.(N{c:=(y).P}) (x).(M{c:=(y).P})) (u).(Q{c:=(y).P})"
wenzelm@42764	1264	using LImp
wenzelm@36278	1265	apply -
wenzelm@36278	1266	apply(rule a_starI)
wenzelm@36278	1267	apply(rule al_redu)
wenzelm@36278	1268	apply(auto intro: l_redu.intros simp add: subst_fresh abs_fresh)
wenzelm@36278	1269	done
wenzelm@36278	1270	also have "\<dots> = Cut <a>.(Cut <d>.(N{c:=(y).P}) (x).(M{c:=(y).P})) (u).(Cut <c>.Ax u c (y).P)"
wenzelm@42764	1271	using LImp asm by simp
wenzelm@54152	1272	also have "\<dots> \<longrightarrow>\<^sub>a* Cut <a>.(Cut <d>.(N{c:=(y).P}) (x).(M{c:=(y).P})) (u).(P[y\<turnstile>n>u])"
wenzelm@36278	1273	proof (cases "fin P y")
wenzelm@36278	1274	case True
wenzelm@36278	1275	assume "fin P y"
wenzelm@42764	1276	then show ?thesis using LImp
wenzelm@36278	1277	apply -
wenzelm@36278	1278	apply(rule a_star_CutR)
wenzelm@36278	1279	apply(rule a_starI)
wenzelm@36278	1280	apply(rule al_redu)
wenzelm@36278	1281	apply(rule better_LAxL_intro)
wenzelm@36278	1282	apply(simp)
wenzelm@36278	1283	done
wenzelm@36278	1284	next
wenzelm@36278	1285	case False
wenzelm@36278	1286	assume "\<not>fin P y"
wenzelm@42764	1287	then show ?thesis using LImp
wenzelm@36278	1288	apply -
wenzelm@36278	1289	apply(rule a_star_CutR)
wenzelm@36278	1290	apply(rule a_star_trans)
wenzelm@36278	1291	apply(rule a_starI)
wenzelm@36278	1292	apply(rule ac_redu)
wenzelm@36278	1293	apply(rule better_right)
wenzelm@36278	1294	apply(simp)
wenzelm@36278	1295	apply(simp add: subst_with_ax1)
wenzelm@36278	1296	done
wenzelm@36278	1297	qed
wenzelm@42764	1298	also have "\<dots> = (Cut <a>.(Cut <d>.N (x).M) (u).Q){c:=(y).P}" using LImp asm
wenzelm@36278	1299	apply -
wenzelm@36278	1300	apply(auto simp add: subst_fresh abs_fresh)
wenzelm@36278	1301	apply(simp add: trm.inject)
wenzelm@36278	1302	apply(simp add: alpha fresh_atm)
wenzelm@36278	1303	apply(simp add: nrename_swap)
wenzelm@36278	1304	done
wenzelm@36278	1305	finally
wenzelm@54152	1306	have "(Cut <b>.ImpR (x).<a>.M b (z).ImpL <d>.N (u).Q z){c:=(y).P} \<longrightarrow>\<^sub>a*
wenzelm@36278	1307	(Cut <a>.(Cut <d>.N (x).M) (u).Q){c:=(y).P}"
wenzelm@36278	1308	by simp
wenzelm@36278	1309	}
wenzelm@36278	1310	moreover
wenzelm@36278	1311	{ assume asm: "M=Ax x c \<and> Q=Ax u c"
wenzelm@36278	1312	have "(Cut <b>.ImpR (x).<a>.M b (z).ImpL <d>.N (u).Q z){c:=(y).P} =
wenzelm@36278	1313	Cut <b>.ImpR (x).<a>.(M{c:=(y).P}) b (z).ImpL <d>.(N{c:=(y).P}) (u).(Q{c:=(y).P}) z"
wenzelm@42764	1314	using LImp by (simp add: subst_fresh abs_fresh fresh_atm fresh_prod)
wenzelm@54152	1315	also have "\<dots> \<longrightarrow>\<^sub>a* Cut <a>.(Cut <d>.(N{c:=(y).P}) (x).(M{c:=(y).P})) (u).(Q{c:=(y).P})"
wenzelm@42764	1316	using LImp
wenzelm@36278	1317	apply -
wenzelm@36278	1318	apply(rule a_starI)
wenzelm@36278	1319	apply(rule al_redu)
wenzelm@36278	1320	apply(auto intro: l_redu.intros simp add: subst_fresh abs_fresh)
wenzelm@36278	1321	done
wenzelm@36278	1322	also have "\<dots> = Cut <a>.(Cut <d>.(N{c:=(y).P}) (x).(Cut <c>.Ax x c (y).P)) (u).(Cut <c>.Ax u c (y).P)"
wenzelm@42764	1323	using LImp asm by simp
wenzelm@54152	1324	also have "\<dots> \<longrightarrow>\<^sub>a* Cut <a>.(Cut <d>.(N{c:=(y).P}) (x).(Cut <c>.Ax x c (y).P)) (u).(P[y\<turnstile>n>u])"
wenzelm@36278	1325	proof (cases "fin P y")
wenzelm@36278	1326	case True
wenzelm@36278	1327	assume "fin P y"
wenzelm@42764	1328	then show ?thesis using LImp
wenzelm@36278	1329	apply -
wenzelm@36278	1330	apply(rule a_star_CutR)
wenzelm@36278	1331	apply(rule a_starI)
wenzelm@36278	1332	apply(rule al_redu)
wenzelm@36278	1333	apply(rule better_LAxL_intro)
wenzelm@36278	1334	apply(simp)
wenzelm@36278	1335	done
wenzelm@36278	1336	next
wenzelm@36278	1337	case False
wenzelm@36278	1338	assume "\<not>fin P y"
wenzelm@42764	1339	then show ?thesis using LImp
wenzelm@36278	1340	apply -
wenzelm@36278	1341	apply(rule a_star_CutR)
wenzelm@36278	1342	apply(rule a_star_trans)
wenzelm@36278	1343	apply(rule a_starI)
wenzelm@36278	1344	apply(rule ac_redu)
wenzelm@36278	1345	apply(rule better_right)
wenzelm@36278	1346	apply(simp)
wenzelm@36278	1347	apply(simp add: subst_with_ax1)
wenzelm@36278	1348	done
wenzelm@36278	1349	qed
wenzelm@54152	1350	also have "\<dots> \<longrightarrow>\<^sub>a* Cut <a>.(Cut <d>.(N{c:=(y).P}) (x).(P[y\<turnstile>n>x])) (u).(P[y\<turnstile>n>u])"
wenzelm@36278	1351	proof (cases "fin P y")
wenzelm@36278	1352	case True
wenzelm@36278	1353	assume "fin P y"
wenzelm@42764	1354	then show ?thesis using LImp
wenzelm@36278	1355	apply -
wenzelm@36278	1356	apply(rule a_star_CutL)
wenzelm@36278	1357	apply(rule a_star_CutR)
wenzelm@36278	1358	apply(rule a_starI)
wenzelm@36278	1359	apply(rule al_redu)
wenzelm@36278	1360	apply(rule better_LAxL_intro)
wenzelm@36278	1361	apply(simp)
wenzelm@36278	1362	done
wenzelm@36278	1363	next
wenzelm@36278	1364	case False
wenzelm@36278	1365	assume "\<not>fin P y"
wenzelm@42764	1366	then show ?thesis using LImp
wenzelm@36278	1367	apply -
wenzelm@36278	1368	apply(rule a_star_CutL)
wenzelm@36278	1369	apply(rule a_star_CutR)
wenzelm@36278	1370	apply(rule a_star_trans)
wenzelm@36278	1371	apply(rule a_starI)
wenzelm@36278	1372	apply(rule ac_redu)
wenzelm@36278	1373	apply(rule better_right)
wenzelm@36278	1374	apply(simp)
wenzelm@36278	1375	apply(simp add: subst_with_ax1)
wenzelm@36278	1376	done
wenzelm@36278	1377	qed
wenzelm@42764	1378	also have "\<dots> = (Cut <a>.(Cut <d>.N (x).M) (u).Q){c:=(y).P}" using LImp asm
wenzelm@36278	1379	apply -
wenzelm@36278	1380	apply(auto simp add: subst_fresh abs_fresh)
wenzelm@36278	1381	apply(simp add: trm.inject)
wenzelm@36278	1382	apply(rule conjI)
wenzelm@36278	1383	apply(simp add: alpha fresh_atm trm.inject)
wenzelm@36278	1384	apply(simp add: nrename_swap)
wenzelm@36278	1385	apply(simp add: alpha fresh_atm trm.inject)
wenzelm@36278	1386	apply(simp add: nrename_swap)
wenzelm@36278	1387	done
wenzelm@36278	1388	finally
wenzelm@54152	1389	have "(Cut <b>.ImpR (x).<a>.M b (z).ImpL <d>.N (u).Q z){c:=(y).P} \<longrightarrow>\<^sub>a*
wenzelm@36278	1390	(Cut <a>.(Cut <d>.N (x).M) (u).Q){c:=(y).P}"
wenzelm@36278	1391	by simp
wenzelm@36278	1392	}
wenzelm@36278	1393	ultimately show ?thesis by blast
wenzelm@36278	1394	qed
wenzelm@36278	1395	qed
wenzelm@36278	1396
wenzelm@36278	1397	lemma a_redu_subst1:
wenzelm@54152	1398	assumes a: "M \<longrightarrow>\<^sub>a M'"
wenzelm@54152	1399	shows "M{y:=<c>.P} \<longrightarrow>\<^sub>a* M'{y:=<c>.P}"
wenzelm@36278	1400	using a
wenzelm@36278	1401	proof(nominal_induct avoiding: y c P rule: a_redu.strong_induct)
wenzelm@36278	1402	case al_redu
wenzelm@36278	1403	then show ?case by (simp only: l_redu_subst1)
wenzelm@36278	1404	next
wenzelm@36278	1405	case ac_redu
wenzelm@36278	1406	then show ?case
wenzelm@36278	1407	apply -
wenzelm@36278	1408	apply(rule a_starI)
wenzelm@36278	1409	apply(rule a_redu.ac_redu)
wenzelm@36278	1410	apply(simp only: c_redu_subst1')
wenzelm@36278	1411	done
wenzelm@36278	1412	next
wenzelm@36278	1413	case (a_Cut_l a N x M M' y c P)
wenzelm@36278	1414	then show ?case
wenzelm@36278	1415	apply(simp add: subst_fresh fresh_a_redu)
wenzelm@36278	1416	apply(rule conjI)
wenzelm@36278	1417	apply(rule impI)+
wenzelm@36278	1418	apply(simp)
wenzelm@36278	1419	apply(drule ax_do_not_a_reduce)
wenzelm@36278	1420	apply(simp)
wenzelm@36278	1421	apply(rule impI)
wenzelm@36278	1422	apply(rule conjI)
wenzelm@36278	1423	apply(rule impI)
wenzelm@36278	1424	apply(simp)
wenzelm@36278	1425	apply(drule_tac x="y" in meta_spec)
wenzelm@36278	1426	apply(drule_tac x="c" in meta_spec)
wenzelm@36278	1427	apply(drule_tac x="P" in meta_spec)
wenzelm@36278	1428	apply(simp)
wenzelm@36278	1429	apply(rule a_star_trans)
wenzelm@36278	1430	apply(rule a_star_CutL)
wenzelm@36278	1431	apply(assumption)
wenzelm@36278	1432	apply(rule a_star_trans)
wenzelm@36278	1433	apply(rule_tac M'="P[c\<turnstile>c>a]" in a_star_CutL)
wenzelm@36278	1434	apply(case_tac "fic P c")
wenzelm@36278	1435	apply(rule a_starI)
wenzelm@36278	1436	apply(rule al_redu)
wenzelm@36278	1437	apply(rule better_LAxR_intro)
wenzelm@36278	1438	apply(simp)
wenzelm@36278	1439	apply(rule a_star_trans)
wenzelm@36278	1440	apply(rule a_starI)
wenzelm@36278	1441	apply(rule ac_redu)
wenzelm@36278	1442	apply(rule better_left)
wenzelm@36278	1443	apply(simp)
wenzelm@36278	1444	apply(rule subst_with_ax2)
wenzelm@36278	1445	apply(rule aux4)
wenzelm@36278	1446	apply(simp add: trm.inject)
wenzelm@36278	1447	apply(simp add: alpha fresh_atm)
wenzelm@36278	1448	apply(simp add: crename_swap)
wenzelm@36278	1449	apply(rule impI)
wenzelm@36278	1450	apply(rule a_star_CutL)
wenzelm@36278	1451	apply(auto)
wenzelm@36278	1452	done
wenzelm@36278	1453	next
wenzelm@36278	1454	case (a_Cut_r a N x M M' y c P)
wenzelm@36278	1455	then show ?case
wenzelm@36278	1456	apply(auto simp add: subst_fresh fresh_a_redu)
wenzelm@36278	1457	apply(rule a_star_CutR)
wenzelm@36278	1458	apply(auto)[1]
wenzelm@36278	1459	apply(rule a_star_CutR)
wenzelm@36278	1460	apply(auto)[1]
wenzelm@36278	1461	done
wenzelm@36278	1462	next
wenzelm@36278	1463	case a_NotL
wenzelm@36278	1464	then show ?case
wenzelm@36278	1465	apply(auto)
wenzelm@36278	1466	apply(generate_fresh "name")
wenzelm@36278	1467	apply(fresh_fun_simp)
wenzelm@36278	1468	apply(fresh_fun_simp)
wenzelm@36278	1469	apply(simp add: subst_fresh)
wenzelm@36278	1470	apply(rule a_star_CutR)
wenzelm@36278	1471	apply(rule a_star_NotL)
wenzelm@36278	1472	apply(auto)[1]
wenzelm@36278	1473	apply(rule a_star_NotL)
wenzelm@36278	1474	apply(auto)[1]
wenzelm@36278	1475	done
wenzelm@36278	1476	next
wenzelm@36278	1477	case a_NotR
wenzelm@36278	1478	then show ?case
wenzelm@36278	1479	apply(auto)
wenzelm@36278	1480	apply(rule a_star_NotR)
wenzelm@36278	1481	apply(auto)[1]
wenzelm@36278	1482	done
wenzelm@36278	1483	next
wenzelm@36278	1484	case a_AndR_l
wenzelm@36278	1485	then show ?case
wenzelm@36278	1486	apply(auto simp add: subst_fresh fresh_a_redu)
wenzelm@36278	1487	apply(rule a_star_AndR)
wenzelm@36278	1488	apply(auto)
wenzelm@36278	1489	done
wenzelm@36278	1490	next
wenzelm@36278	1491	case a_AndR_r
wenzelm@36278	1492	then show ?case
wenzelm@36278	1493	apply(auto simp add: subst_fresh fresh_a_redu)
wenzelm@36278	1494	apply(rule a_star_AndR)
wenzelm@36278	1495	apply(auto)
wenzelm@36278	1496	done
wenzelm@36278	1497	next
wenzelm@36278	1498	case a_AndL1
wenzelm@36278	1499	then show ?case
wenzelm@36278	1500	apply(auto)
wenzelm@36278	1501	apply(generate_fresh "name")
wenzelm@36278	1502	apply(fresh_fun_simp)
wenzelm@36278	1503	apply(fresh_fun_simp)
wenzelm@36278	1504	apply(simp add: subst_fresh)
wenzelm@36278	1505	apply(rule a_star_CutR)
wenzelm@36278	1506	apply(rule a_star_AndL1)
wenzelm@36278	1507	apply(auto)[1]
wenzelm@36278	1508	apply(rule a_star_AndL1)
wenzelm@36278	1509	apply(auto)[1]
wenzelm@36278	1510	done
wenzelm@36278	1511	next
wenzelm@36278	1512	case a_AndL2
wenzelm@36278	1513	then show ?case
wenzelm@36278	1514	apply(auto)
wenzelm@36278	1515	apply(generate_fresh "name")
wenzelm@36278	1516	apply(fresh_fun_simp)
wenzelm@36278	1517	apply(fresh_fun_simp)
wenzelm@36278	1518	apply(simp add: subst_fresh)
wenzelm@36278	1519	apply(rule a_star_CutR)
wenzelm@36278	1520	apply(rule a_star_AndL2)
wenzelm@36278	1521	apply(auto)[1]
wenzelm@36278	1522	apply(rule a_star_AndL2)
wenzelm@36278	1523	apply(auto)[1]
wenzelm@36278	1524	done
wenzelm@36278	1525	next
wenzelm@36278	1526	case a_OrR1
wenzelm@36278	1527	then show ?case
wenzelm@36278	1528	apply(auto simp add: subst_fresh fresh_a_redu)
wenzelm@36278	1529	apply(rule a_star_OrR1)
wenzelm@36278	1530	apply(auto)
wenzelm@36278	1531	done
wenzelm@36278	1532	next
wenzelm@36278	1533	case a_OrR2
wenzelm@36278	1534	then show ?case
wenzelm@36278	1535	apply(auto simp add: subst_fresh fresh_a_redu)
wenzelm@36278	1536	apply(rule a_star_OrR2)
wenzelm@36278	1537	apply(auto)
wenzelm@36278	1538	done
wenzelm@36278	1539	next
wenzelm@36278	1540	case a_OrL_l
wenzelm@36278	1541	then show ?case
wenzelm@36278	1542	apply(auto simp add: subst_fresh fresh_a_redu)
wenzelm@36278	1543	apply(generate_fresh "name")
wenzelm@36278	1544	apply(fresh_fun_simp)
wenzelm@36278	1545	apply(fresh_fun_simp)
wenzelm@36278	1546	apply(simp add: subst_fresh)
wenzelm@36278	1547	apply(rule a_star_CutR)
wenzelm@36278	1548	apply(rule a_star_OrL)
wenzelm@36278	1549	apply(auto)
wenzelm@36278	1550	apply(rule a_star_OrL)
wenzelm@36278	1551	apply(auto)
wenzelm@36278	1552	done
wenzelm@36278	1553	next
wenzelm@36278	1554	case a_OrL_r
wenzelm@36278	1555	then show ?case
wenzelm@36278	1556	apply(auto simp add: subst_fresh fresh_a_redu)
wenzelm@36278	1557	apply(generate_fresh "name")
wenzelm@36278	1558	apply(fresh_fun_simp)
wenzelm@36278	1559	apply(fresh_fun_simp)
wenzelm@36278	1560	apply(simp add: subst_fresh)
wenzelm@36278	1561	apply(rule a_star_CutR)
wenzelm@36278	1562	apply(rule a_star_OrL)
wenzelm@36278	1563	apply(auto)
wenzelm@36278	1564	apply(rule a_star_OrL)
wenzelm@36278	1565	apply(auto)
wenzelm@36278	1566	done
wenzelm@36278	1567	next
wenzelm@36278	1568	case a_ImpR
wenzelm@36278	1569	then show ?case
wenzelm@36278	1570	apply(auto simp add: subst_fresh fresh_a_redu)
wenzelm@36278	1571	apply(rule a_star_ImpR)
wenzelm@36278	1572	apply(auto)
wenzelm@36278	1573	done
wenzelm@36278	1574	next
wenzelm@36278	1575	case a_ImpL_r
wenzelm@36278	1576	then show ?case
wenzelm@36278	1577	apply(auto simp add: subst_fresh fresh_a_redu)
wenzelm@36278	1578	apply(generate_fresh "name")
wenzelm@36278	1579	apply(fresh_fun_simp)
wenzelm@36278	1580	apply(fresh_fun_simp)
wenzelm@36278	1581	apply(simp add: subst_fresh)
wenzelm@36278	1582	apply(rule a_star_CutR)
wenzelm@36278	1583	apply(rule a_star_ImpL)
wenzelm@36278	1584	apply(auto)
wenzelm@36278	1585	apply(rule a_star_ImpL)
wenzelm@36278	1586	apply(auto)
wenzelm@36278	1587	done
wenzelm@36278	1588	next
wenzelm@36278	1589	case a_ImpL_l
wenzelm@36278	1590	then show ?case
wenzelm@36278	1591	apply(auto simp add: subst_fresh fresh_a_redu)
wenzelm@36278	1592	apply(generate_fresh "name")
wenzelm@36278	1593	apply(fresh_fun_simp)
wenzelm@36278	1594	apply(fresh_fun_simp)
wenzelm@36278	1595	apply(simp add: subst_fresh)
wenzelm@36278	1596	apply(rule a_star_CutR)
wenzelm@36278	1597	apply(rule a_star_ImpL)
wenzelm@36278	1598	apply(auto)
wenzelm@36278	1599	apply(rule a_star_ImpL)
wenzelm@36278	1600	apply(auto)
wenzelm@36278	1601	done
wenzelm@36278	1602	qed
wenzelm@36278	1603
wenzelm@36278	1604	lemma a_redu_subst2:
wenzelm@54152	1605	assumes a: "M \<longrightarrow>\<^sub>a M'"
wenzelm@54152	1606	shows "M{c:=(y).P} \<longrightarrow>\<^sub>a* M'{c:=(y).P}"
wenzelm@36278	1607	using a
wenzelm@36278	1608	proof(nominal_induct avoiding: y c P rule: a_redu.strong_induct)
wenzelm@36278	1609	case al_redu
wenzelm@36278	1610	then show ?case by (simp only: l_redu_subst2)
wenzelm@36278	1611	next
wenzelm@36278	1612	case ac_redu
wenzelm@36278	1613	then show ?case
wenzelm@36278	1614	apply -
wenzelm@36278	1615	apply(rule a_starI)
wenzelm@36278	1616	apply(rule a_redu.ac_redu)
wenzelm@36278	1617	apply(simp only: c_redu_subst2')
wenzelm@36278	1618	done
wenzelm@36278	1619	next
wenzelm@36278	1620	case (a_Cut_r a N x M M' y c P)
wenzelm@36278	1621	then show ?case
wenzelm@36278	1622	apply(simp add: subst_fresh fresh_a_redu)
wenzelm@36278	1623	apply(rule conjI)
wenzelm@36278	1624	apply(rule impI)+
wenzelm@36278	1625	apply(simp)
wenzelm@36278	1626	apply(drule ax_do_not_a_reduce)
wenzelm@36278	1627	apply(simp)
wenzelm@36278	1628	apply(rule impI)
wenzelm@36278	1629	apply(rule conjI)
wenzelm@36278	1630	apply(rule impI)
wenzelm@36278	1631	apply(simp)
wenzelm@36278	1632	apply(drule_tac x="c" in meta_spec)
wenzelm@36278	1633	apply(drule_tac x="y" in meta_spec)
wenzelm@36278	1634	apply(drule_tac x="P" in meta_spec)
wenzelm@36278	1635	apply(simp)
wenzelm@36278	1636	apply(rule a_star_trans)
wenzelm@36278	1637	apply(rule a_star_CutR)
wenzelm@36278	1638	apply(assumption)
wenzelm@36278	1639	apply(rule a_star_trans)
wenzelm@36278	1640	apply(rule_tac N'="P[y\<turnstile>n>x]" in a_star_CutR)
wenzelm@36278	1641	apply(case_tac "fin P y")
wenzelm@36278	1642	apply(rule a_starI)
wenzelm@36278	1643	apply(rule al_redu)
wenzelm@36278	1644	apply(rule better_LAxL_intro)
wenzelm@36278	1645	apply(simp)
wenzelm@36278	1646	apply(rule a_star_trans)
wenzelm@36278	1647	apply(rule a_starI)
wenzelm@36278	1648	apply(rule ac_redu)
wenzelm@36278	1649	apply(rule better_right)
wenzelm@36278	1650	apply(simp)
wenzelm@36278	1651	apply(rule subst_with_ax1)
wenzelm@36278	1652	apply(rule aux4)
wenzelm@36278	1653	apply(simp add: trm.inject)
wenzelm@36278	1654	apply(simp add: alpha fresh_atm)
wenzelm@36278	1655	apply(simp add: nrename_swap)
wenzelm@36278	1656	apply(rule impI)
wenzelm@36278	1657	apply(rule a_star_CutR)
wenzelm@36278	1658	apply(auto)
wenzelm@36278	1659	done
wenzelm@36278	1660	next
wenzelm@36278	1661	case (a_Cut_l a N x M M' y c P)
wenzelm@36278	1662	then show ?case
wenzelm@36278	1663	apply(auto simp add: subst_fresh fresh_a_redu)
wenzelm@36278	1664	apply(rule a_star_CutL)
wenzelm@36278	1665	apply(auto)[1]
wenzelm@36278	1666	apply(rule a_star_CutL)
wenzelm@36278	1667	apply(auto)[1]
wenzelm@36278	1668	done
wenzelm@36278	1669	next
wenzelm@36278	1670	case a_NotR
wenzelm@36278	1671	then show ?case
wenzelm@36278	1672	apply(auto)
wenzelm@36278	1673	apply(generate_fresh "coname")
wenzelm@36278	1674	apply(fresh_fun_simp)
wenzelm@36278	1675	apply(fresh_fun_simp)
wenzelm@36278	1676	apply(simp add: subst_fresh)
wenzelm@36278	1677	apply(rule a_star_CutL)
wenzelm@36278	1678	apply(rule a_star_NotR)
wenzelm@36278	1679	apply(auto)[1]
wenzelm@36278	1680	apply(rule a_star_NotR)
wenzelm@36278	1681	apply(auto)[1]
wenzelm@36278	1682	done
wenzelm@36278	1683	next
wenzelm@36278	1684	case a_NotL
wenzelm@36278	1685	then show ?case
wenzelm@36278	1686	apply(auto)
wenzelm@36278	1687	apply(rule a_star_NotL)
wenzelm@36278	1688	apply(auto)[1]
wenzelm@36278	1689	done
wenzelm@36278	1690	next
wenzelm@36278	1691	case a_AndR_l
wenzelm@36278	1692	then show ?case
wenzelm@36278	1693	apply(auto simp add: subst_fresh fresh_a_redu)
wenzelm@36278	1694	apply(generate_fresh "coname")
wenzelm@36278	1695	apply(fresh_fun_simp)
wenzelm@36278	1696	apply(fresh_fun_simp)
wenzelm@36278	1697	apply(simp add: subst_fresh)
wenzelm@36278	1698	apply(rule a_star_CutL)
wenzelm@36278	1699	apply(rule a_star_AndR)
wenzelm@36278	1700	apply(auto)
wenzelm@36278	1701	apply(rule a_star_AndR)
wenzelm@36278	1702	apply(auto)
wenzelm@36278	1703	done
wenzelm@36278	1704	next
wenzelm@36278	1705	case a_AndR_r
wenzelm@36278	1706	then show ?case
wenzelm@36278	1707	apply(auto simp add: subst_fresh fresh_a_redu)
wenzelm@36278	1708	apply(generate_fresh "coname")
wenzelm@36278	1709	apply(fresh_fun_simp)
wenzelm@36278	1710	apply(fresh_fun_simp)
wenzelm@36278	1711	apply(simp add: subst_fresh)
wenzelm@36278	1712	apply(rule a_star_CutL)
wenzelm@36278	1713	apply(rule a_star_AndR)
wenzelm@36278	1714	apply(auto)
wenzelm@36278	1715	apply(rule a_star_AndR)
wenzelm@36278	1716	apply(auto)
wenzelm@36278	1717	done
wenzelm@36278	1718	next
wenzelm@36278	1719	case a_AndL1
wenzelm@36278	1720	then show ?case
wenzelm@36278	1721	apply(auto simp add: subst_fresh fresh_a_redu)
wenzelm@36278	1722	apply(rule a_star_AndL1)
wenzelm@36278	1723	apply(auto)
wenzelm@36278	1724	done
wenzelm@36278	1725	next
wenzelm@36278	1726	case a_AndL2
wenzelm@36278	1727	then show ?case
wenzelm@36278	1728	apply(auto simp add: subst_fresh fresh_a_redu)
wenzelm@36278	1729	apply(rule a_star_AndL2)
wenzelm@36278	1730	apply(auto)
wenzelm@36278	1731	done
wenzelm@36278	1732	next
wenzelm@36278	1733	case a_OrR1
wenzelm@36278	1734	then show ?case
wenzelm@36278	1735	apply(auto)
wenzelm@36278	1736	apply(generate_fresh "coname")
wenzelm@36278	1737	apply(fresh_fun_simp)
wenzelm@36278	1738	apply(fresh_fun_simp)
wenzelm@36278	1739	apply(simp add: subst_fresh)
wenzelm@36278	1740	apply(rule a_star_CutL)
wenzelm@36278	1741	apply(rule a_star_OrR1)
wenzelm@36278	1742	apply(auto)[1]
wenzelm@36278	1743	apply(rule a_star_OrR1)
wenzelm@36278	1744	apply(auto)[1]
wenzelm@36278	1745	done
wenzelm@36278	1746	next
wenzelm@36278	1747	case a_OrR2
wenzelm@36278	1748	then show ?case
wenzelm@36278	1749	apply(auto)
wenzelm@36278	1750	apply(generate_fresh "coname")
wenzelm@36278	1751	apply(fresh_fun_simp)
wenzelm@36278	1752	apply(fresh_fun_simp)
wenzelm@36278	1753	apply(simp add: subst_fresh)
wenzelm@36278	1754	apply(rule a_star_CutL)
wenzelm@36278	1755	apply(rule a_star_OrR2)
wenzelm@36278	1756	apply(auto)[1]
wenzelm@36278	1757	apply(rule a_star_OrR2)
wenzelm@36278	1758	apply(auto)[1]
wenzelm@36278	1759	done
wenzelm@36278	1760	next
wenzelm@36278	1761	case a_OrL_l
wenzelm@36278	1762	then show ?case
wenzelm@36278	1763	apply(auto simp add: subst_fresh fresh_a_redu)
wenzelm@36278	1764	apply(rule a_star_OrL)
wenzelm@36278	1765	apply(auto)
wenzelm@36278	1766	done
wenzelm@36278	1767	next
wenzelm@36278	1768	case a_OrL_r
wenzelm@36278	1769	then show ?case
wenzelm@36278	1770	apply(auto simp add: subst_fresh fresh_a_redu)
wenzelm@36278	1771	apply(rule a_star_OrL)
wenzelm@36278	1772	apply(auto)
wenzelm@36278	1773	done
wenzelm@36278	1774	next
wenzelm@36278	1775	case a_ImpR
wenzelm@36278	1776	then show ?case
wenzelm@36278	1777	apply(auto simp add: subst_fresh fresh_a_redu)
wenzelm@36278	1778	apply(generate_fresh "coname")
wenzelm@36278	1779	apply(fresh_fun_simp)
wenzelm@36278	1780	apply(fresh_fun_simp)
wenzelm@36278	1781	apply(simp add: subst_fresh)
wenzelm@36278	1782	apply(rule a_star_CutL)
wenzelm@36278	1783	apply(rule a_star_ImpR)
wenzelm@36278	1784	apply(auto)
wenzelm@36278	1785	apply(rule a_star_ImpR)
wenzelm@36278	1786	apply(auto)
wenzelm@36278	1787	done
wenzelm@36278	1788	next
wenzelm@36278	1789	case a_ImpL_l
wenzelm@36278	1790	then show ?case
wenzelm@36278	1791	apply(auto simp add: subst_fresh fresh_a_redu)
wenzelm@36278	1792	apply(rule a_star_ImpL)
wenzelm@36278	1793	apply(auto)
wenzelm@36278	1794	done
wenzelm@36278	1795	next
wenzelm@36278	1796	case a_ImpL_r
wenzelm@36278	1797	then show ?case
wenzelm@36278	1798	apply(auto simp add: subst_fresh fresh_a_redu)
wenzelm@36278	1799	apply(rule a_star_ImpL)
wenzelm@36278	1800	apply(auto)
wenzelm@36278	1801	done
wenzelm@36278	1802	qed
wenzelm@36278	1803
wenzelm@36278	1804	lemma a_star_subst1:
wenzelm@54152	1805	assumes a: "M \<longrightarrow>\<^sub>a* M'"
wenzelm@54152	1806	shows "M{y:=<c>.P} \<longrightarrow>\<^sub>a* M'{y:=<c>.P}"
wenzelm@36278	1807	using a
wenzelm@36278	1808	apply(induct)
wenzelm@36278	1809	apply(blast)
wenzelm@36278	1810	apply(drule_tac y="y" and c="c" and P="P" in a_redu_subst1)
wenzelm@36278	1811	apply(auto)
wenzelm@36278	1812	done
wenzelm@36278	1813
wenzelm@36278	1814	lemma a_star_subst2:
wenzelm@54152	1815	assumes a: "M \<longrightarrow>\<^sub>a* M'"
wenzelm@54152	1816	shows "M{c:=(y).P} \<longrightarrow>\<^sub>a* M'{c:=(y).P}"
wenzelm@36278	1817	using a
wenzelm@36278	1818	apply(induct)
wenzelm@36278	1819	apply(blast)
wenzelm@36278	1820	apply(drule_tac y="y" and c="c" and P="P" in a_redu_subst2)
wenzelm@36278	1821	apply(auto)
wenzelm@36278	1822	done
wenzelm@36278	1823
wenzelm@36278	1824	text {* Candidates and SN *}
wenzelm@36278	1825
wenzelm@36278	1826	text {* SNa *}
wenzelm@36278	1827
wenzelm@36278	1828	inductive
wenzelm@36278	1829	SNa :: "trm \<Rightarrow> bool"
wenzelm@36278	1830	where
wenzelm@54152	1831	SNaI: "(\<And>M'. M \<longrightarrow>\<^sub>a M' \<Longrightarrow> SNa M') \<Longrightarrow> SNa M"
wenzelm@36278	1832
wenzelm@36278	1833	lemma SNa_induct[consumes 1]:
wenzelm@36278	1834	assumes major: "SNa M"
wenzelm@54152	1835	assumes hyp: "\<And>M'. SNa M' \<Longrightarrow> (\<forall>M''. M'\<longrightarrow>\<^sub>a M'' \<longrightarrow> P M'' \<Longrightarrow> P M')"
wenzelm@36278	1836	shows "P M"
wenzelm@36278	1837	apply (rule major[THEN SNa.induct])
wenzelm@36278	1838	apply (rule hyp)
wenzelm@36278	1839	apply (rule SNaI)
wenzelm@36278	1840	apply (blast)+
wenzelm@36278	1841	done
wenzelm@36278	1842
wenzelm@36278	1843
wenzelm@36278	1844	lemma double_SNa_aux:
wenzelm@36278	1845	assumes a_SNa: "SNa a"
wenzelm@36278	1846	and b_SNa: "SNa b"
wenzelm@36278	1847	and hyp: "\<And>x z.
wenzelm@54152	1848	(\<And>y. x\<longrightarrow>\<^sub>a y \<Longrightarrow> SNa y) \<Longrightarrow>
wenzelm@54152	1849	(\<And>y. x\<longrightarrow>\<^sub>a y \<Longrightarrow> P y z) \<Longrightarrow>
wenzelm@54152	1850	(\<And>u. z\<longrightarrow>\<^sub>a u \<Longrightarrow> SNa u) \<Longrightarrow>
wenzelm@54152	1851	(\<And>u. z\<longrightarrow>\<^sub>a u \<Longrightarrow> P x u) \<Longrightarrow> P x z"
wenzelm@36278	1852	shows "P a b"
wenzelm@36278	1853	proof -
wenzelm@36278	1854	from a_SNa
wenzelm@36278	1855	have r: "\<And>b. SNa b \<Longrightarrow> P a b"
wenzelm@36278	1856	proof (induct a rule: SNa.induct)
wenzelm@36278	1857	case (SNaI x)
wenzelm@36278	1858	note SNa' = this
wenzelm@36278	1859	have "SNa b" by fact
wenzelm@36278	1860	thus ?case
wenzelm@36278	1861	proof (induct b rule: SNa.induct)
wenzelm@36278	1862	case (SNaI y)
wenzelm@36278	1863	show ?case
wenzelm@36278	1864	apply (rule hyp)
wenzelm@36278	1865	apply (erule SNa')
wenzelm@36278	1866	apply (erule SNa')
wenzelm@36278	1867	apply (rule SNa.SNaI)
wenzelm@36278	1868	apply (erule SNaI)+
wenzelm@36278	1869	done
wenzelm@36278	1870	qed
wenzelm@36278	1871	qed
wenzelm@36278	1872	from b_SNa show ?thesis by (rule r)
wenzelm@36278	1873	qed
wenzelm@36278	1874
wenzelm@36278	1875	lemma double_SNa:
wenzelm@54152	1876	"\<lbrakk>SNa a; SNa b; \<forall>x z. ((\<forall>y. x\<longrightarrow>\<^sub>ay \<longrightarrow> P y z) \<and> (\<forall>u. z\<longrightarrow>\<^sub>a u \<longrightarrow> P x u)) \<longrightarrow> P x z\<rbrakk> \<Longrightarrow> P a b"
wenzelm@36278	1877	apply(rule_tac double_SNa_aux)
wenzelm@36278	1878	apply(assumption)+
wenzelm@36278	1879	apply(blast)
wenzelm@36278	1880	done
wenzelm@36278	1881
wenzelm@36278	1882	lemma a_preserves_SNa:
wenzelm@54152	1883	assumes a: "SNa M" "M\<longrightarrow>\<^sub>a M'"
wenzelm@36278	1884	shows "SNa M'"
wenzelm@36278	1885	using a
wenzelm@36278	1886	by (erule_tac SNa.cases) (simp)
wenzelm@36278	1887
wenzelm@36278	1888	lemma a_star_preserves_SNa:
wenzelm@54152	1889	assumes a: "SNa M" and b: "M\<longrightarrow>\<^sub>a* M'"
wenzelm@36278	1890	shows "SNa M'"
wenzelm@36278	1891	using b a
wenzelm@36278	1892	by (induct) (auto simp add: a_preserves_SNa)
wenzelm@36278	1893
wenzelm@36278	1894	lemma Ax_in_SNa:
wenzelm@36278	1895	shows "SNa (Ax x a)"
wenzelm@36278	1896	apply(rule SNaI)
wenzelm@36278	1897	apply(erule a_redu.cases, auto)
wenzelm@36278	1898	apply(erule l_redu.cases, auto)
wenzelm@36278	1899	apply(erule c_redu.cases, auto)
wenzelm@36278	1900	done
wenzelm@36278	1901
wenzelm@36278	1902	lemma NotL_in_SNa:
wenzelm@36278	1903	assumes a: "SNa M"
wenzelm@36278	1904	shows "SNa (NotL <a>.M x)"
wenzelm@36278	1905	using a
wenzelm@36278	1906	apply(induct)
wenzelm@36278	1907	apply(rule SNaI)
wenzelm@36278	1908	apply(erule a_redu.cases, auto)
wenzelm@36278	1909	apply(erule l_redu.cases, auto)
wenzelm@36278	1910	apply(erule c_redu.cases, auto)
wenzelm@36278	1911	apply(auto simp add: trm.inject alpha)
wenzelm@36278	1912	apply(rotate_tac 1)
wenzelm@36278	1913	apply(drule_tac x="[(a,aa)]\<bullet>M'a" in meta_spec)
wenzelm@36278	1914	apply(simp add: a_redu.eqvt)
wenzelm@36278	1915	apply(subgoal_tac "NotL <a>.([(a,aa)]\<bullet>M'a) x = NotL <aa>.M'a x")
wenzelm@36278	1916	apply(simp)
wenzelm@36278	1917	apply(simp add: trm.inject alpha fresh_a_redu)
wenzelm@36278	1918	done
wenzelm@36278	1919
wenzelm@36278	1920	lemma NotR_in_SNa:
wenzelm@36278	1921	assumes a: "SNa M"
wenzelm@36278	1922	shows "SNa (NotR (x).M a)"
wenzelm@36278	1923	using a
wenzelm@36278	1924	apply(induct)
wenzelm@36278	1925	apply(rule SNaI)
wenzelm@36278	1926	apply(erule a_redu.cases, auto)
wenzelm@36278	1927	apply(erule l_redu.cases, auto)
wenzelm@36278	1928	apply(erule c_redu.cases, auto)
wenzelm@36278	1929	apply(auto simp add: trm.inject alpha)
wenzelm@36278	1930	apply(rotate_tac 1)
wenzelm@36278	1931	apply(drule_tac x="[(x,xa)]\<bullet>M'a" in meta_spec)
wenzelm@36278	1932	apply(simp add: a_redu.eqvt)
wenzelm@36278	1933	apply(rule_tac s="NotR (x).([(x,xa)]\<bullet>M'a) a" in subst)
wenzelm@36278	1934	apply(simp add: trm.inject alpha fresh_a_redu)
wenzelm@36278	1935	apply(simp)
wenzelm@36278	1936	done
wenzelm@36278	1937
wenzelm@36278	1938	lemma AndL1_in_SNa:
wenzelm@36278	1939	assumes a: "SNa M"
wenzelm@36278	1940	shows "SNa (AndL1 (x).M y)"
wenzelm@36278	1941	using a
wenzelm@36278	1942	apply(induct)
wenzelm@36278	1943	apply(rule SNaI)
wenzelm@36278	1944	apply(erule a_redu.cases, auto)
wenzelm@36278	1945	apply(erule l_redu.cases, auto)
wenzelm@36278	1946	apply(erule c_redu.cases, auto)
wenzelm@36278	1947	apply(auto simp add: trm.inject alpha)
wenzelm@36278	1948	apply(rotate_tac 1)
wenzelm@36278	1949	apply(drule_tac x="[(x,xa)]\<bullet>M'a" in meta_spec)
wenzelm@36278	1950	apply(simp add: a_redu.eqvt)
wenzelm@36278	1951	apply(rule_tac s="AndL1 x.([(x,xa)]\<bullet>M'a) y" in subst)
wenzelm@36278	1952	apply(simp add: trm.inject alpha fresh_a_redu)
wenzelm@36278	1953	apply(simp)
wenzelm@36278	1954	done
wenzelm@36278	1955
wenzelm@36278	1956	lemma AndL2_in_SNa:
wenzelm@36278	1957	assumes a: "SNa M"
wenzelm@36278	1958	shows "SNa (AndL2 (x).M y)"
wenzelm@36278	1959	using a
wenzelm@36278	1960	apply(induct)
wenzelm@36278	1961	apply(rule SNaI)
wenzelm@36278	1962	apply(erule a_redu.cases, auto)
wenzelm@36278	1963	apply(erule l_redu.cases, auto)
wenzelm@36278	1964	apply(erule c_redu.cases, auto)
wenzelm@36278	1965	apply(auto simp add: trm.inject alpha)
wenzelm@36278	1966	apply(rotate_tac 1)
wenzelm@36278	1967	apply(drule_tac x="[(x,xa)]\<bullet>M'a" in meta_spec)
wenzelm@36278	1968	apply(simp add: a_redu.eqvt)
wenzelm@36278	1969	apply(rule_tac s="AndL2 x.([(x,xa)]\<bullet>M'a) y" in subst)
wenzelm@36278	1970	apply(simp add: trm.inject alpha fresh_a_redu)
wenzelm@36278	1971	apply(simp)
wenzelm@36278	1972	done
wenzelm@36278	1973
wenzelm@36278	1974	lemma OrR1_in_SNa:
wenzelm@36278	1975	assumes a: "SNa M"
wenzelm@36278	1976	shows "SNa (OrR1 <a>.M b)"
wenzelm@36278	1977	using a
wenzelm@36278	1978	apply(induct)
wenzelm@36278	1979	apply(rule SNaI)
wenzelm@36278	1980	apply(erule a_redu.cases, auto)
wenzelm@36278	1981	apply(erule l_redu.cases, auto)
wenzelm@36278	1982	apply(erule c_redu.cases, auto)
wenzelm@36278	1983	apply(auto simp add: trm.inject alpha)
wenzelm@36278	1984	apply(rotate_tac 1)
wenzelm@36278	1985	apply(drule_tac x="[(a,aa)]\<bullet>M'a" in meta_spec)
wenzelm@36278	1986	apply(simp add: a_redu.eqvt)
wenzelm@36278	1987	apply(rule_tac s="OrR1 <a>.([(a,aa)]\<bullet>M'a) b" in subst)
wenzelm@36278	1988	apply(simp add: trm.inject alpha fresh_a_redu)
wenzelm@36278	1989	apply(simp)
wenzelm@36278	1990	done
wenzelm@36278	1991
wenzelm@36278	1992	lemma OrR2_in_SNa:
wenzelm@36278	1993	assumes a: "SNa M"
wenzelm@36278	1994	shows "SNa (OrR2 <a>.M b)"
wenzelm@36278	1995	using a
wenzelm@36278	1996	apply(induct)
wenzelm@36278	1997	apply(rule SNaI)
wenzelm@36278	1998	apply(erule a_redu.cases, auto)
wenzelm@36278	1999	apply(erule l_redu.cases, auto)
wenzelm@36278	2000	apply(erule c_redu.cases, auto)
wenzelm@36278	2001	apply(auto simp add: trm.inject alpha)
wenzelm@36278	2002	apply(rotate_tac 1)
wenzelm@36278	2003	apply(drule_tac x="[(a,aa)]\<bullet>M'a" in meta_spec)
wenzelm@36278	2004	apply(simp add: a_redu.eqvt)
wenzelm@36278	2005	apply(rule_tac s="OrR2 <a>.([(a,aa)]\<bullet>M'a) b" in subst)
wenzelm@36278	2006	apply(simp add: trm.inject alpha fresh_a_redu)
wenzelm@36278	2007	apply(simp)
wenzelm@36278	2008	done
wenzelm@36278	2009
wenzelm@36278	2010	lemma ImpR_in_SNa:
wenzelm@36278	2011	assumes a: "SNa M"
wenzelm@36278	2012	shows "SNa (ImpR (x).<a>.M b)"
wenzelm@36278	2013	using a
wenzelm@36278	2014	apply(induct)
wenzelm@36278	2015	apply(rule SNaI)
wenzelm@36278	2016	apply(erule a_redu.cases, auto)
wenzelm@36278	2017	apply(erule l_redu.cases, auto)
wenzelm@36278	2018	apply(erule c_redu.cases, auto)
wenzelm@36278	2019	apply(auto simp add: trm.inject alpha abs_fresh abs_perm calc_atm)
wenzelm@36278	2020	apply(rotate_tac 1)
wenzelm@36278	2021	apply(drule_tac x="[(a,aa)]\<bullet>M'a" in meta_spec)
wenzelm@36278	2022	apply(simp add: a_redu.eqvt)
wenzelm@36278	2023	apply(rule_tac s="ImpR (x).<a>.([(a,aa)]\<bullet>M'a) b" in subst)
wenzelm@36278	2024	apply(simp add: trm.inject alpha fresh_a_redu)
wenzelm@36278	2025	apply(simp)
wenzelm@36278	2026	apply(rotate_tac 1)
wenzelm@36278	2027	apply(drule_tac x="[(x,xa)]\<bullet>M'a" in meta_spec)
wenzelm@36278	2028	apply(simp add: a_redu.eqvt)
wenzelm@36278	2029	apply(rule_tac s="ImpR (x).<a>.([(x,xa)]\<bullet>M'a) b" in subst)
wenzelm@36278	2030	apply(simp add: trm.inject alpha fresh_a_redu abs_fresh abs_perm calc_atm)
wenzelm@36278	2031	apply(simp)
wenzelm@36278	2032	apply(rotate_tac 1)
wenzelm@36278	2033	apply(drule_tac x="[(a,aa)]\<bullet>[(x,xa)]\<bullet>M'a" in meta_spec)
wenzelm@36278	2034	apply(simp add: a_redu.eqvt)
wenzelm@36278	2035	apply(rule_tac s="ImpR (x).<a>.([(a,aa)]\<bullet>[(x,xa)]\<bullet>M'a) b" in subst)
wenzelm@36278	2036	apply(simp add: trm.inject alpha fresh_a_redu abs_fresh abs_perm calc_atm)
wenzelm@36278	2037	apply(simp add: fresh_left calc_atm fresh_a_redu)
wenzelm@36278	2038	apply(simp)
wenzelm@36278	2039	done
wenzelm@36278	2040
wenzelm@36278	2041	lemma AndR_in_SNa:
wenzelm@36278	2042	assumes a: "SNa M" "SNa N"
wenzelm@36278	2043	shows "SNa (AndR <a>.M <b>.N c)"
wenzelm@36278	2044	apply(rule_tac a="M" and b="N" in double_SNa)
wenzelm@36278	2045	apply(rule a)+
wenzelm@36278	2046	apply(auto)
wenzelm@36278	2047	apply(rule SNaI)
wenzelm@36278	2048	apply(drule a_redu_AndR_elim)
wenzelm@36278	2049	apply(auto)
wenzelm@36278	2050	done
wenzelm@36278	2051
wenzelm@36278	2052	lemma OrL_in_SNa:
wenzelm@36278	2053	assumes a: "SNa M" "SNa N"
wenzelm@36278	2054	shows "SNa (OrL (x).M (y).N z)"
wenzelm@36278	2055	apply(rule_tac a="M" and b="N" in double_SNa)
wenzelm@36278	2056	apply(rule a)+
wenzelm@36278	2057	apply(auto)
wenzelm@36278	2058	apply(rule SNaI)
wenzelm@36278	2059	apply(drule a_redu_OrL_elim)
wenzelm@36278	2060	apply(auto)
wenzelm@36278	2061	done
wenzelm@36278	2062
wenzelm@36278	2063	lemma ImpL_in_SNa:
wenzelm@36278	2064	assumes a: "SNa M" "SNa N"
wenzelm@36278	2065	shows "SNa (ImpL <a>.M (y).N z)"
wenzelm@36278	2066	apply(rule_tac a="M" and b="N" in double_SNa)
wenzelm@36278	2067	apply(rule a)+
wenzelm@36278	2068	apply(auto)
wenzelm@36278	2069	apply(rule SNaI)
wenzelm@36278	2070	apply(drule a_redu_ImpL_elim)
wenzelm@36278	2071	apply(auto)
wenzelm@36278	2072	done
wenzelm@36278	2073
wenzelm@36278	2074	lemma SNa_eqvt:
wenzelm@36278	2075	fixes pi1::"name prm"
wenzelm@36278	2076	and pi2::"coname prm"
wenzelm@36278	2077	shows "SNa M \<Longrightarrow> SNa (pi1\<bullet>M)"
wenzelm@36278	2078	and "SNa M \<Longrightarrow> SNa (pi2\<bullet>M)"
wenzelm@36278	2079	apply -
wenzelm@36278	2080	apply(induct rule: SNa.induct)
wenzelm@36278	2081	apply(rule SNaI)
wenzelm@36278	2082	apply(drule_tac pi="(rev pi1)" in a_redu.eqvt(1))
wenzelm@36278	2083	apply(rotate_tac 1)
wenzelm@36278	2084	apply(drule_tac x="(rev pi1)\<bullet>M'" in meta_spec)
wenzelm@36278	2085	apply(perm_simp)
wenzelm@36278	2086	apply(induct rule: SNa.induct)
wenzelm@36278	2087	apply(rule SNaI)
wenzelm@36278	2088	apply(drule_tac pi="(rev pi2)" in a_redu.eqvt(2))
wenzelm@36278	2089	apply(rotate_tac 1)
wenzelm@36278	2090	apply(drule_tac x="(rev pi2)\<bullet>M'" in meta_spec)
wenzelm@36278	2091	apply(perm_simp)
wenzelm@36278	2092	done
wenzelm@36278	2093
wenzelm@36278	2094	text {* set operators *}
wenzelm@36278	2095
wenzelm@36278	2096	definition AXIOMSn :: "ty \<Rightarrow> ntrm set" where
wenzelm@36278	2097	"AXIOMSn B \<equiv> { (x):(Ax y b) \| x y b. True }"
wenzelm@36278	2098
wenzelm@36278	2099	definition AXIOMSc::"ty \<Rightarrow> ctrm set" where
wenzelm@36278	2100	"AXIOMSc B \<equiv> { <a>:(Ax y b) \| a y b. True }"
wenzelm@36278	2101
wenzelm@36278	2102	definition BINDINGn::"ty \<Rightarrow> ctrm set \<Rightarrow> ntrm set" where
wenzelm@36278	2103	"BINDINGn B X \<equiv> { (x):M \| x M. \<forall>a P. <a>:P\<in>X \<longrightarrow> SNa (M{x:=<a>.P})}"
wenzelm@36278	2104
wenzelm@36278	2105	definition BINDINGc::"ty \<Rightarrow> ntrm set \<Rightarrow> ctrm set" where
wenzelm@36278	2106	"BINDINGc B X \<equiv> { <a>:M \| a M. \<forall>x P. (x):P\<in>X \<longrightarrow> SNa (M{a:=(x).P})}"
wenzelm@36278	2107
wenzelm@36278	2108	lemma BINDINGn_decreasing:
wenzelm@36278	2109	shows "X\<subseteq>Y \<Longrightarrow> BINDINGn B Y \<subseteq> BINDINGn B X"
wenzelm@36278	2110	by (simp add: BINDINGn_def) (blast)
wenzelm@36278	2111
wenzelm@36278	2112	lemma BINDINGc_decreasing:
wenzelm@36278	2113	shows "X\<subseteq>Y \<Longrightarrow> BINDINGc B Y \<subseteq> BINDINGc B X"
wenzelm@36278	2114	by (simp add: BINDINGc_def) (blast)
wenzelm@36278	2115
wenzelm@36278	2116	nominal_primrec
wenzelm@36278	2117	NOTRIGHT :: "ty \<Rightarrow> ntrm set \<Rightarrow> ctrm set"
wenzelm@36278	2118	where
wenzelm@36278	2119	"NOTRIGHT (NOT B) X = { <a>:NotR (x).M a \| a x M. fic (NotR (x).M a) a \<and> (x):M \<in> X }"
wenzelm@36278	2120	apply(rule TrueI)+
wenzelm@36278	2121	done
wenzelm@36278	2122
wenzelm@36278	2123	lemma NOTRIGHT_eqvt_name:
wenzelm@36278	2124	fixes pi::"name prm"
wenzelm@36278	2125	shows "(pi\<bullet>(NOTRIGHT (NOT B) X)) = NOTRIGHT (NOT B) (pi\<bullet>X)"
berghofe@47052	2126	apply(auto simp add: perm_set_def)
wenzelm@36278	2127	apply(rule_tac x="pi\<bullet>a" in exI)
wenzelm@36278	2128	apply(rule_tac x="pi\<bullet>xb" in exI)
wenzelm@36278	2129	apply(rule_tac x="pi\<bullet>M" in exI)
wenzelm@36278	2130	apply(simp)
wenzelm@36278	2131	apply(rule conjI)
wenzelm@36278	2132	apply(drule_tac pi="pi" in fic.eqvt(1))
wenzelm@36278	2133	apply(simp)
wenzelm@36278	2134	apply(rule_tac x="(xb):M" in exI)
wenzelm@36278	2135	apply(simp)
wenzelm@36278	2136	apply(rule_tac x="(rev pi)\<bullet>(<a>:NotR (xa).M a)" in exI)
wenzelm@36278	2137	apply(perm_simp)
wenzelm@36278	2138	apply(rule_tac x="(rev pi)\<bullet>a" in exI)
wenzelm@36278	2139	apply(rule_tac x="(rev pi)\<bullet>xa" in exI)
wenzelm@36278	2140	apply(rule_tac x="(rev pi)\<bullet>M" in exI)
wenzelm@36278	2141	apply(simp add: swap_simps)
wenzelm@36278	2142	apply(drule_tac pi="rev pi" in fic.eqvt(1))
wenzelm@36278	2143	apply(simp)
wenzelm@36278	2144	apply(drule sym)
wenzelm@36278	2145	apply(drule pt_bij1[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	2146	apply(simp add: swap_simps)
wenzelm@36278	2147	done
wenzelm@36278	2148
wenzelm@36278	2149	lemma NOTRIGHT_eqvt_coname:
wenzelm@36278	2150	fixes pi::"coname prm"
wenzelm@36278	2151	shows "(pi\<bullet>(NOTRIGHT (NOT B) X)) = NOTRIGHT (NOT B) (pi\<bullet>X)"
berghofe@47052	2152	apply(auto simp add: perm_set_def)
wenzelm@36278	2153	apply(rule_tac x="pi\<bullet>a" in exI)
wenzelm@36278	2154	apply(rule_tac x="pi\<bullet>xb" in exI)
wenzelm@36278	2155	apply(rule_tac x="pi\<bullet>M" in exI)
wenzelm@36278	2156	apply(simp)
wenzelm@36278	2157	apply(rule conjI)
wenzelm@36278	2158	apply(drule_tac pi="pi" in fic.eqvt(2))
wenzelm@36278	2159	apply(simp)
wenzelm@36278	2160	apply(rule_tac x="(xb):M" in exI)
wenzelm@36278	2161	apply(simp)
wenzelm@36278	2162	apply(rule_tac x="<((rev pi)\<bullet>a)>:NotR ((rev pi)\<bullet>xa).((rev pi)\<bullet>M) ((rev pi)\<bullet>a)" in exI)
wenzelm@36278	2163	apply(perm_simp)
wenzelm@36278	2164	apply(rule_tac x="(rev pi)\<bullet>a" in exI)
wenzelm@36278	2165	apply(rule_tac x="(rev pi)\<bullet>xa" in exI)
wenzelm@36278	2166	apply(rule_tac x="(rev pi)\<bullet>M" in exI)
wenzelm@36278	2167	apply(simp add: swap_simps)
wenzelm@36278	2168	apply(drule_tac pi="rev pi" in fic.eqvt(2))
wenzelm@36278	2169	apply(simp)
wenzelm@36278	2170	apply(drule sym)
wenzelm@36278	2171	apply(drule pt_bij1[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	2172	apply(simp add: swap_simps)
wenzelm@36278	2173	done
wenzelm@36278	2174
wenzelm@36278	2175	nominal_primrec
wenzelm@36278	2176	NOTLEFT :: "ty \<Rightarrow> ctrm set \<Rightarrow> ntrm set"
wenzelm@36278	2177	where
wenzelm@36278	2178	"NOTLEFT (NOT B) X = { (x):NotL <a>.M x \| a x M. fin (NotL <a>.M x) x \<and> <a>:M \<in> X }"
wenzelm@36278	2179	apply(rule TrueI)+
wenzelm@36278	2180	done
wenzelm@36278	2181
wenzelm@36278	2182	lemma NOTLEFT_eqvt_name:
wenzelm@36278	2183	fixes pi::"name prm"
wenzelm@36278	2184	shows "(pi\<bullet>(NOTLEFT (NOT B) X)) = NOTLEFT (NOT B) (pi\<bullet>X)"
berghofe@47052	2185	apply(auto simp add: perm_set_def)
wenzelm@36278	2186	apply(rule_tac x="pi\<bullet>a" in exI)
wenzelm@36278	2187	apply(rule_tac x="pi\<bullet>xb" in exI)
wenzelm@36278	2188	apply(rule_tac x="pi\<bullet>M" in exI)
wenzelm@36278	2189	apply(simp)
wenzelm@36278	2190	apply(rule conjI)
wenzelm@36278	2191	apply(drule_tac pi="pi" in fin.eqvt(1))
wenzelm@36278	2192	apply(simp)
wenzelm@36278	2193	apply(rule_tac x="<a>:M" in exI)
wenzelm@36278	2194	apply(simp)
wenzelm@36278	2195	apply(rule_tac x="(((rev pi)\<bullet>xa)):NotL <((rev pi)\<bullet>a)>.((rev pi)\<bullet>M) ((rev pi)\<bullet>xa)" in exI)
wenzelm@36278	2196	apply(perm_simp)
wenzelm@36278	2197	apply(rule_tac x="(rev pi)\<bullet>a" in exI)
wenzelm@36278	2198	apply(rule_tac x="(rev pi)\<bullet>xa" in exI)
wenzelm@36278	2199	apply(rule_tac x="(rev pi)\<bullet>M" in exI)
wenzelm@36278	2200	apply(simp add: swap_simps)
wenzelm@36278	2201	apply(drule_tac pi="rev pi" in fin.eqvt(1))
wenzelm@36278	2202	apply(simp)
wenzelm@36278	2203	apply(drule sym)
wenzelm@36278	2204	apply(drule pt_bij1[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	2205	apply(simp add: swap_simps)
wenzelm@36278	2206	done
wenzelm@36278	2207
wenzelm@36278	2208	lemma NOTLEFT_eqvt_coname:
wenzelm@36278	2209	fixes pi::"coname prm"
wenzelm@36278	2210	shows "(pi\<bullet>(NOTLEFT (NOT B) X)) = NOTLEFT (NOT B) (pi\<bullet>X)"
berghofe@47052	2211	apply(auto simp add: perm_set_def)
wenzelm@36278	2212	apply(rule_tac x="pi\<bullet>a" in exI)
wenzelm@36278	2213	apply(rule_tac x="pi\<bullet>xb" in exI)
wenzelm@36278	2214	apply(rule_tac x="pi\<bullet>M" in exI)
wenzelm@36278	2215	apply(simp)
wenzelm@36278	2216	apply(rule conjI)
wenzelm@36278	2217	apply(drule_tac pi="pi" in fin.eqvt(2))
wenzelm@36278	2218	apply(simp)
wenzelm@36278	2219	apply(rule_tac x="<a>:M" in exI)
wenzelm@36278	2220	apply(simp)
wenzelm@36278	2221	apply(rule_tac x="(((rev pi)\<bullet>xa)):NotL <((rev pi)\<bullet>a)>.((rev pi)\<bullet>M) ((rev pi)\<bullet>xa)" in exI)
wenzelm@36278	2222	apply(perm_simp)
wenzelm@36278	2223	apply(rule_tac x="(rev pi)\<bullet>a" in exI)
wenzelm@36278	2224	apply(rule_tac x="(rev pi)\<bullet>xa" in exI)
wenzelm@36278	2225	apply(rule_tac x="(rev pi)\<bullet>M" in exI)
wenzelm@36278	2226	apply(simp add: swap_simps)
wenzelm@36278	2227	apply(drule_tac pi="rev pi" in fin.eqvt(2))
wenzelm@36278	2228	apply(simp)
wenzelm@36278	2229	apply(drule sym)
wenzelm@36278	2230	apply(drule pt_bij1[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	2231	apply(simp add: swap_simps)
wenzelm@36278	2232	done
wenzelm@36278	2233
wenzelm@36278	2234	nominal_primrec
wenzelm@36278	2235	ANDRIGHT :: "ty \<Rightarrow> ctrm set \<Rightarrow> ctrm set \<Rightarrow> ctrm set"
wenzelm@36278	2236	where
wenzelm@36278	2237	"ANDRIGHT (B AND C) X Y =
wenzelm@36278	2238	{ <c>:AndR <a>.M <b>.N c \| c a b M N. fic (AndR <a>.M <b>.N c) c \<and> <a>:M \<in> X \<and> <b>:N \<in> Y }"
wenzelm@36278	2239	apply(rule TrueI)+
wenzelm@36278	2240	done
wenzelm@36278	2241
wenzelm@36278	2242	lemma ANDRIGHT_eqvt_name:
wenzelm@36278	2243	fixes pi::"name prm"
wenzelm@36278	2244	shows "(pi\<bullet>(ANDRIGHT (A AND B) X Y)) = ANDRIGHT (A AND B) (pi\<bullet>X) (pi\<bullet>Y)"
berghofe@47052	2245	apply(auto simp add: perm_set_def)
wenzelm@36278	2246	apply(rule_tac x="pi\<bullet>c" in exI)
wenzelm@36278	2247	apply(rule_tac x="pi\<bullet>a" in exI)
wenzelm@36278	2248	apply(rule_tac x="pi\<bullet>b" in exI)
wenzelm@36278	2249	apply(rule_tac x="pi\<bullet>M" in exI)
wenzelm@36278	2250	apply(rule_tac x="pi\<bullet>N" in exI)
wenzelm@36278	2251	apply(simp)
wenzelm@36278	2252	apply(rule conjI)
wenzelm@36278	2253	apply(drule_tac pi="pi" in fic.eqvt(1))
wenzelm@36278	2254	apply(simp)
wenzelm@36278	2255	apply(rule conjI)
wenzelm@36278	2256	apply(rule_tac x="<a>:M" in exI)
wenzelm@36278	2257	apply(simp)
wenzelm@36278	2258	apply(rule_tac x="<b>:N" in exI)
wenzelm@36278	2259	apply(simp)
wenzelm@36278	2260	apply(rule_tac x="(rev pi)\<bullet>(<c>:AndR <a>.M <b>.N c)" in exI)
wenzelm@36278	2261	apply(perm_simp)
wenzelm@36278	2262	apply(rule_tac x="(rev pi)\<bullet>c" in exI)
wenzelm@36278	2263	apply(rule_tac x="(rev pi)\<bullet>a" in exI)
wenzelm@36278	2264	apply(rule_tac x="(rev pi)\<bullet>b" in exI)
wenzelm@36278	2265	apply(rule_tac x="(rev pi)\<bullet>M" in exI)
wenzelm@36278	2266	apply(rule_tac x="(rev pi)\<bullet>N" in exI)
wenzelm@36278	2267	apply(simp add: swap_simps)
wenzelm@36278	2268	apply(drule_tac pi="rev pi" in fic.eqvt(1))
wenzelm@36278	2269	apply(simp)
wenzelm@36278	2270	apply(drule sym)
wenzelm@36278	2271	apply(drule sym)
wenzelm@36278	2272	apply(drule pt_bij1[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	2273	apply(drule pt_bij1[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	2274	apply(simp add: swap_simps)
wenzelm@36278	2275	done
wenzelm@36278	2276
wenzelm@36278	2277	lemma ANDRIGHT_eqvt_coname:
wenzelm@36278	2278	fixes pi::"coname prm"
wenzelm@36278	2279	shows "(pi\<bullet>(ANDRIGHT (A AND B) X Y)) = ANDRIGHT (A AND B) (pi\<bullet>X) (pi\<bullet>Y)"
berghofe@47052	2280	apply(auto simp add: perm_set_def)
wenzelm@36278	2281	apply(rule_tac x="pi\<bullet>c" in exI)
wenzelm@36278	2282	apply(rule_tac x="pi\<bullet>a" in exI)
wenzelm@36278	2283	apply(rule_tac x="pi\<bullet>b" in exI)
wenzelm@36278	2284	apply(rule_tac x="pi\<bullet>M" in exI)
wenzelm@36278	2285	apply(rule_tac x="pi\<bullet>N" in exI)
wenzelm@36278	2286	apply(simp)
wenzelm@36278	2287	apply(rule conjI)
wenzelm@36278	2288	apply(drule_tac pi="pi" in fic.eqvt(2))
wenzelm@36278	2289	apply(simp)
wenzelm@36278	2290	apply(rule conjI)
wenzelm@36278	2291	apply(rule_tac x="<a>:M" in exI)
wenzelm@36278	2292	apply(simp)
wenzelm@36278	2293	apply(rule_tac x="<b>:N" in exI)
wenzelm@36278	2294	apply(simp)
wenzelm@36278	2295	apply(rule_tac x="(rev pi)\<bullet>(<c>:AndR <a>.M <b>.N c)" in exI)
wenzelm@36278	2296	apply(perm_simp)
wenzelm@36278	2297	apply(rule_tac x="(rev pi)\<bullet>c" in exI)
wenzelm@36278	2298	apply(rule_tac x="(rev pi)\<bullet>a" in exI)
wenzelm@36278	2299	apply(rule_tac x="(rev pi)\<bullet>b" in exI)
wenzelm@36278	2300	apply(rule_tac x="(rev pi)\<bullet>M" in exI)
wenzelm@36278	2301	apply(rule_tac x="(rev pi)\<bullet>N" in exI)
wenzelm@36278	2302	apply(simp)
wenzelm@36278	2303	apply(drule_tac pi="rev pi" in fic.eqvt(2))
wenzelm@36278	2304	apply(simp)
wenzelm@36278	2305	apply(drule sym)
wenzelm@36278	2306	apply(drule sym)
wenzelm@36278	2307	apply(drule pt_bij1[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	2308	apply(drule pt_bij1[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	2309	apply(simp)
wenzelm@36278	2310	done
wenzelm@36278	2311
wenzelm@36278	2312	nominal_primrec
wenzelm@36278	2313	ANDLEFT1 :: "ty \<Rightarrow> ntrm set \<Rightarrow> ntrm set"
wenzelm@36278	2314	where
wenzelm@36278	2315	"ANDLEFT1 (B AND C) X = { (y):AndL1 (x).M y \| x y M. fin (AndL1 (x).M y) y \<and> (x):M \<in> X }"
wenzelm@36278	2316	apply(rule TrueI)+
wenzelm@36278	2317	done
wenzelm@36278	2318
wenzelm@36278	2319	lemma ANDLEFT1_eqvt_name:
wenzelm@36278	2320	fixes pi::"name prm"
wenzelm@36278	2321	shows "(pi\<bullet>(ANDLEFT1 (A AND B) X)) = ANDLEFT1 (A AND B) (pi\<bullet>X)"
berghofe@47052	2322	apply(auto simp add: perm_set_def)
wenzelm@36278	2323	apply(rule_tac x="pi\<bullet>xb" in exI)
wenzelm@36278	2324	apply(rule_tac x="pi\<bullet>y" in exI)
wenzelm@36278	2325	apply(rule_tac x="pi\<bullet>M" in exI)
wenzelm@36278	2326	apply(simp)
wenzelm@36278	2327	apply(rule conjI)
wenzelm@36278	2328	apply(drule_tac pi="pi" in fin.eqvt(1))
wenzelm@36278	2329	apply(simp)
wenzelm@36278	2330	apply(rule_tac x="(xb):M" in exI)
wenzelm@36278	2331	apply(simp)
wenzelm@36278	2332	apply(rule_tac x="(rev pi)\<bullet>((y):AndL1 (xa).M y)" in exI)
wenzelm@36278	2333	apply(perm_simp)
wenzelm@36278	2334	apply(rule_tac x="(rev pi)\<bullet>xa" in exI)
wenzelm@36278	2335	apply(rule_tac x="(rev pi)\<bullet>y" in exI)
wenzelm@36278	2336	apply(rule_tac x="(rev pi)\<bullet>M" in exI)
wenzelm@36278	2337	apply(simp)
wenzelm@36278	2338	apply(drule_tac pi="rev pi" in fin.eqvt(1))
wenzelm@36278	2339	apply(simp)
wenzelm@36278	2340	apply(drule sym)
wenzelm@36278	2341	apply(drule pt_bij1[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	2342	apply(simp)
wenzelm@36278	2343	done
wenzelm@36278	2344
wenzelm@36278	2345	lemma ANDLEFT1_eqvt_coname:
wenzelm@36278	2346	fixes pi::"coname prm"
wenzelm@36278	2347	shows "(pi\<bullet>(ANDLEFT1 (A AND B) X)) = ANDLEFT1 (A AND B) (pi\<bullet>X)"
berghofe@47052	2348	apply(auto simp add: perm_set_def)
wenzelm@36278	2349	apply(rule_tac x="pi\<bullet>xb" in exI)
wenzelm@36278	2350	apply(rule_tac x="pi\<bullet>y" in exI)
wenzelm@36278	2351	apply(rule_tac x="pi\<bullet>M" in exI)
wenzelm@36278	2352	apply(simp)
wenzelm@36278	2353	apply(rule conjI)
wenzelm@36278	2354	apply(drule_tac pi="pi" in fin.eqvt(2))
wenzelm@36278	2355	apply(simp)
wenzelm@36278	2356	apply(rule_tac x="(xb):M" in exI)
wenzelm@36278	2357	apply(simp)
wenzelm@36278	2358	apply(rule_tac x="(rev pi)\<bullet>((y):AndL1 (xa).M y)" in exI)
wenzelm@36278	2359	apply(perm_simp)
wenzelm@36278	2360	apply(rule_tac x="(rev pi)\<bullet>xa" in exI)
wenzelm@36278	2361	apply(rule_tac x="(rev pi)\<bullet>y" in exI)
wenzelm@36278	2362	apply(rule_tac x="(rev pi)\<bullet>M" in exI)
wenzelm@36278	2363	apply(simp add: swap_simps)
wenzelm@36278	2364	apply(drule_tac pi="rev pi" in fin.eqvt(2))
wenzelm@36278	2365	apply(simp)
wenzelm@36278	2366	apply(drule sym)
wenzelm@36278	2367	apply(drule pt_bij1[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	2368	apply(simp add: swap_simps)
wenzelm@36278	2369	done
wenzelm@36278	2370
wenzelm@36278	2371	nominal_primrec
wenzelm@36278	2372	ANDLEFT2 :: "ty \<Rightarrow> ntrm set \<Rightarrow> ntrm set"
wenzelm@36278	2373	where
wenzelm@36278	2374	"ANDLEFT2 (B AND C) X = { (y):AndL2 (x).M y \| x y M. fin (AndL2 (x).M y) y \<and> (x):M \<in> X }"
wenzelm@36278	2375	apply(rule TrueI)+
wenzelm@36278	2376	done
wenzelm@36278	2377
wenzelm@36278	2378	lemma ANDLEFT2_eqvt_name:
wenzelm@36278	2379	fixes pi::"name prm"
wenzelm@36278	2380	shows "(pi\<bullet>(ANDLEFT2 (A AND B) X)) = ANDLEFT2 (A AND B) (pi\<bullet>X)"
berghofe@47052	2381	apply(auto simp add: perm_set_def)
wenzelm@36278	2382	apply(rule_tac x="pi\<bullet>xb" in exI)
wenzelm@36278	2383	apply(rule_tac x="pi\<bullet>y" in exI)
wenzelm@36278	2384	apply(rule_tac x="pi\<bullet>M" in exI)
wenzelm@36278	2385	apply(simp)
wenzelm@36278	2386	apply(rule conjI)
wenzelm@36278	2387	apply(drule_tac pi="pi" in fin.eqvt(1))
wenzelm@36278	2388	apply(simp)
wenzelm@36278	2389	apply(rule_tac x="(xb):M" in exI)
wenzelm@36278	2390	apply(simp)
wenzelm@36278	2391	apply(rule_tac x="(rev pi)\<bullet>((y):AndL2 (xa).M y)" in exI)
wenzelm@36278	2392	apply(perm_simp)
wenzelm@36278	2393	apply(rule_tac x="(rev pi)\<bullet>xa" in exI)
wenzelm@36278	2394	apply(rule_tac x="(rev pi)\<bullet>y" in exI)
wenzelm@36278	2395	apply(rule_tac x="(rev pi)\<bullet>M" in exI)
wenzelm@36278	2396	apply(simp)
wenzelm@36278	2397	apply(drule_tac pi="rev pi" in fin.eqvt(1))
wenzelm@36278	2398	apply(simp)
wenzelm@36278	2399	apply(drule sym)
wenzelm@36278	2400	apply(drule pt_bij1[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	2401	apply(simp)
wenzelm@36278	2402	done
wenzelm@36278	2403
wenzelm@36278	2404	lemma ANDLEFT2_eqvt_coname:
wenzelm@36278	2405	fixes pi::"coname prm"
wenzelm@36278	2406	shows "(pi\<bullet>(ANDLEFT2 (A AND B) X)) = ANDLEFT2 (A AND B) (pi\<bullet>X)"
berghofe@47052	2407	apply(auto simp add: perm_set_def)
wenzelm@36278	2408	apply(rule_tac x="pi\<bullet>xb" in exI)
wenzelm@36278	2409	apply(rule_tac x="pi\<bullet>y" in exI)
wenzelm@36278	2410	apply(rule_tac x="pi\<bullet>M" in exI)
wenzelm@36278	2411	apply(simp)
wenzelm@36278	2412	apply(rule conjI)
wenzelm@36278	2413	apply(drule_tac pi="pi" in fin.eqvt(2))
wenzelm@36278	2414	apply(simp)
wenzelm@36278	2415	apply(rule_tac x="(xb):M" in exI)
wenzelm@36278	2416	apply(simp)
wenzelm@36278	2417	apply(rule_tac x="(rev pi)\<bullet>((y):AndL2 (xa).M y)" in exI)
wenzelm@36278	2418	apply(perm_simp)
wenzelm@36278	2419	apply(rule_tac x="(rev pi)\<bullet>xa" in exI)
wenzelm@36278	2420	apply(rule_tac x="(rev pi)\<bullet>y" in exI)
wenzelm@36278	2421	apply(rule_tac x="(rev pi)\<bullet>M" in exI)
wenzelm@36278	2422	apply(simp add: swap_simps)
wenzelm@36278	2423	apply(drule_tac pi="rev pi" in fin.eqvt(2))
wenzelm@36278	2424	apply(simp)
wenzelm@36278	2425	apply(drule sym)
wenzelm@36278	2426	apply(drule pt_bij1[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	2427	apply(simp add: swap_simps)
wenzelm@36278	2428	done
wenzelm@36278	2429
wenzelm@36278	2430	nominal_primrec
wenzelm@36278	2431	ORLEFT :: "ty \<Rightarrow> ntrm set \<Rightarrow> ntrm set \<Rightarrow> ntrm set"
wenzelm@36278	2432	where
wenzelm@36278	2433	"ORLEFT (B OR C) X Y =
wenzelm@36278	2434	{ (z):OrL (x).M (y).N z \| x y z M N. fin (OrL (x).M (y).N z) z \<and> (x):M \<in> X \<and> (y):N \<in> Y }"
wenzelm@36278	2435	apply(rule TrueI)+
wenzelm@36278	2436	done
wenzelm@36278	2437
wenzelm@36278	2438	lemma ORLEFT_eqvt_name:
wenzelm@36278	2439	fixes pi::"name prm"
wenzelm@36278	2440	shows "(pi\<bullet>(ORLEFT (A OR B) X Y)) = ORLEFT (A OR B) (pi\<bullet>X) (pi\<bullet>Y)"
berghofe@47052	2441	apply(auto simp add: perm_set_def)
wenzelm@36278	2442	apply(rule_tac x="pi\<bullet>xb" in exI)
wenzelm@36278	2443	apply(rule_tac x="pi\<bullet>y" in exI)
wenzelm@36278	2444	apply(rule_tac x="pi\<bullet>z" in exI)
wenzelm@36278	2445	apply(rule_tac x="pi\<bullet>M" in exI)
wenzelm@36278	2446	apply(rule_tac x="pi\<bullet>N" in exI)
wenzelm@36278	2447	apply(simp)
wenzelm@36278	2448	apply(rule conjI)
wenzelm@36278	2449	apply(drule_tac pi="pi" in fin.eqvt(1))
wenzelm@36278	2450	apply(simp)
wenzelm@36278	2451	apply(rule conjI)
wenzelm@36278	2452	apply(rule_tac x="(xb):M" in exI)
wenzelm@36278	2453	apply(simp)
wenzelm@36278	2454	apply(rule_tac x="(y):N" in exI)
wenzelm@36278	2455	apply(simp)
wenzelm@36278	2456	apply(rule_tac x="(rev pi)\<bullet>((z):OrL (xa).M (y).N z)" in exI)
wenzelm@36278	2457	apply(perm_simp)
wenzelm@36278	2458	apply(rule_tac x="(rev pi)\<bullet>xa" in exI)
wenzelm@36278	2459	apply(rule_tac x="(rev pi)\<bullet>y" in exI)
wenzelm@36278	2460	apply(rule_tac x="(rev pi)\<bullet>z" in exI)
wenzelm@36278	2461	apply(rule_tac x="(rev pi)\<bullet>M" in exI)
wenzelm@36278	2462	apply(rule_tac x="(rev pi)\<bullet>N" in exI)
wenzelm@36278	2463	apply(simp)
wenzelm@36278	2464	apply(drule_tac pi="rev pi" in fin.eqvt(1))
wenzelm@36278	2465	apply(simp)
wenzelm@36278	2466	apply(drule sym)
wenzelm@36278	2467	apply(drule sym)
wenzelm@36278	2468	apply(drule pt_bij1[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	2469	apply(drule pt_bij1[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	2470	apply(simp)
wenzelm@36278	2471	done
wenzelm@36278	2472
wenzelm@36278	2473	lemma ORLEFT_eqvt_coname:
wenzelm@36278	2474	fixes pi::"coname prm"
wenzelm@36278	2475	shows "(pi\<bullet>(ORLEFT (A OR B) X Y)) = ORLEFT (A OR B) (pi\<bullet>X) (pi\<bullet>Y)"
berghofe@47052	2476	apply(auto simp add: perm_set_def)
wenzelm@36278	2477	apply(rule_tac x="pi\<bullet>xb" in exI)
wenzelm@36278	2478	apply(rule_tac x="pi\<bullet>y" in exI)
wenzelm@36278	2479	apply(rule_tac x="pi\<bullet>z" in exI)
wenzelm@36278	2480	apply(rule_tac x="pi\<bullet>M" in exI)
wenzelm@36278	2481	apply(rule_tac x="pi\<bullet>N" in exI)
wenzelm@36278	2482	apply(simp)
wenzelm@36278	2483	apply(rule conjI)
wenzelm@36278	2484	apply(drule_tac pi="pi" in fin.eqvt(2))
wenzelm@36278	2485	apply(simp)
wenzelm@36278	2486	apply(rule conjI)
wenzelm@36278	2487	apply(rule_tac x="(xb):M" in exI)
wenzelm@36278	2488	apply(simp)
wenzelm@36278	2489	apply(rule_tac x="(y):N" in exI)
wenzelm@36278	2490	apply(simp)
wenzelm@36278	2491	apply(rule_tac x="(rev pi)\<bullet>((z):OrL (xa).M (y).N z)" in exI)
wenzelm@36278	2492	apply(perm_simp)
wenzelm@36278	2493	apply(rule_tac x="(rev pi)\<bullet>xa" in exI)
wenzelm@36278	2494	apply(rule_tac x="(rev pi)\<bullet>y" in exI)
wenzelm@36278	2495	apply(rule_tac x="(rev pi)\<bullet>z" in exI)
wenzelm@36278	2496	apply(rule_tac x="(rev pi)\<bullet>M" in exI)
wenzelm@36278	2497	apply(rule_tac x="(rev pi)\<bullet>N" in exI)
wenzelm@36278	2498	apply(simp add: swap_simps)
wenzelm@36278	2499	apply(drule_tac pi="rev pi" in fin.eqvt(2))
wenzelm@36278	2500	apply(simp)
wenzelm@36278	2501	apply(drule sym)
wenzelm@36278	2502	apply(drule sym)
wenzelm@36278	2503	apply(drule pt_bij1[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	2504	apply(drule pt_bij1[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	2505	apply(simp add: swap_simps)
wenzelm@36278	2506	done
wenzelm@36278	2507
wenzelm@36278	2508	nominal_primrec
wenzelm@36278	2509	ORRIGHT1 :: "ty \<Rightarrow> ctrm set \<Rightarrow> ctrm set"
wenzelm@36278	2510	where
wenzelm@36278	2511	"ORRIGHT1 (B OR C) X = { <b>:OrR1 <a>.M b \| a b M. fic (OrR1 <a>.M b) b \<and> <a>:M \<in> X }"
wenzelm@36278	2512	apply(rule TrueI)+
wenzelm@36278	2513	done
wenzelm@36278	2514
wenzelm@36278	2515	lemma ORRIGHT1_eqvt_name:
wenzelm@36278	2516	fixes pi::"name prm"
wenzelm@36278	2517	shows "(pi\<bullet>(ORRIGHT1 (A OR B) X)) = ORRIGHT1 (A OR B) (pi\<bullet>X)"
berghofe@47052	2518	apply(auto simp add: perm_set_def)
wenzelm@36278	2519	apply(rule_tac x="pi\<bullet>a" in exI)
wenzelm@36278	2520	apply(rule_tac x="pi\<bullet>b" in exI)
wenzelm@36278	2521	apply(rule_tac x="pi\<bullet>M" in exI)
wenzelm@36278	2522	apply(simp)
wenzelm@36278	2523	apply(rule conjI)
wenzelm@36278	2524	apply(drule_tac pi="pi" in fic.eqvt(1))
wenzelm@36278	2525	apply(simp)
wenzelm@36278	2526	apply(rule_tac x="<a>:M" in exI)
wenzelm@36278	2527	apply(perm_simp)
wenzelm@36278	2528	apply(rule_tac x="(rev pi)\<bullet>(<b>:OrR1 <a>.M b)" in exI)
wenzelm@36278	2529	apply(perm_simp)
wenzelm@36278	2530	apply(rule_tac x="(rev pi)\<bullet>a" in exI)
wenzelm@36278	2531	apply(rule_tac x="(rev pi)\<bullet>b" in exI)
wenzelm@36278	2532	apply(rule_tac x="(rev pi)\<bullet>M" in exI)
wenzelm@36278	2533	apply(simp add: swap_simps)
wenzelm@36278	2534	apply(drule_tac pi="rev pi" in fic.eqvt(1))
wenzelm@36278	2535	apply(simp)
wenzelm@36278	2536	apply(drule sym)
wenzelm@36278	2537	apply(drule pt_bij1[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	2538	apply(simp add: swap_simps)
wenzelm@36278	2539	done
wenzelm@36278	2540
wenzelm@36278	2541	lemma ORRIGHT1_eqvt_coname:
wenzelm@36278	2542	fixes pi::"coname prm"
wenzelm@36278	2543	shows "(pi\<bullet>(ORRIGHT1 (A OR B) X)) = ORRIGHT1 (A OR B) (pi\<bullet>X)"
berghofe@47052	2544	apply(auto simp add: perm_set_def)
wenzelm@36278	2545	apply(rule_tac x="pi\<bullet>a" in exI)
wenzelm@36278	2546	apply(rule_tac x="pi\<bullet>b" in exI)
wenzelm@36278	2547	apply(rule_tac x="pi\<bullet>M" in exI)
wenzelm@36278	2548	apply(simp)
wenzelm@36278	2549	apply(rule conjI)
wenzelm@36278	2550	apply(drule_tac pi="pi" in fic.eqvt(2))
wenzelm@36278	2551	apply(simp)
wenzelm@36278	2552	apply(rule_tac x="<a>:M" in exI)
wenzelm@36278	2553	apply(perm_simp)
wenzelm@36278	2554	apply(rule_tac x="(rev pi)\<bullet>(<b>:OrR1 <a>.M b)" in exI)
wenzelm@36278	2555	apply(perm_simp)
wenzelm@36278	2556	apply(rule_tac x="(rev pi)\<bullet>a" in exI)
wenzelm@36278	2557	apply(rule_tac x="(rev pi)\<bullet>b" in exI)
wenzelm@36278	2558	apply(rule_tac x="(rev pi)\<bullet>M" in exI)
wenzelm@36278	2559	apply(simp)
wenzelm@36278	2560	apply(drule_tac pi="rev pi" in fic.eqvt(2))
wenzelm@36278	2561	apply(simp)
wenzelm@36278	2562	apply(drule sym)
wenzelm@36278	2563	apply(drule pt_bij1[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	2564	apply(simp)
wenzelm@36278	2565	done
wenzelm@36278	2566
wenzelm@36278	2567	nominal_primrec
wenzelm@36278	2568	ORRIGHT2 :: "ty \<Rightarrow> ctrm set \<Rightarrow> ctrm set"
wenzelm@36278	2569	where
wenzelm@36278	2570	"ORRIGHT2 (B OR C) X = { <b>:OrR2 <a>.M b \| a b M. fic (OrR2 <a>.M b) b \<and> <a>:M \<in> X }"
wenzelm@36278	2571	apply(rule TrueI)+
wenzelm@36278	2572	done
wenzelm@36278	2573
wenzelm@36278	2574	lemma ORRIGHT2_eqvt_name:
wenzelm@36278	2575	fixes pi::"name prm"
wenzelm@36278	2576	shows "(pi\<bullet>(ORRIGHT2 (A OR B) X)) = ORRIGHT2 (A OR B) (pi\<bullet>X)"
berghofe@47052	2577	apply(auto simp add: perm_set_def)
wenzelm@36278	2578	apply(rule_tac x="pi\<bullet>a" in exI)
wenzelm@36278	2579	apply(rule_tac x="pi\<bullet>b" in exI)
wenzelm@36278	2580	apply(rule_tac x="pi\<bullet>M" in exI)
wenzelm@36278	2581	apply(simp)
wenzelm@36278	2582	apply(rule conjI)
wenzelm@36278	2583	apply(drule_tac pi="pi" in fic.eqvt(1))
wenzelm@36278	2584	apply(simp)
wenzelm@36278	2585	apply(rule_tac x="<a>:M" in exI)
wenzelm@36278	2586	apply(perm_simp)
wenzelm@36278	2587	apply(rule_tac x="(rev pi)\<bullet>(<b>:OrR2 <a>.M b)" in exI)
wenzelm@36278	2588	apply(perm_simp)
wenzelm@36278	2589	apply(rule_tac x="(rev pi)\<bullet>a" in exI)
wenzelm@36278	2590	apply(rule_tac x="(rev pi)\<bullet>b" in exI)
wenzelm@36278	2591	apply(rule_tac x="(rev pi)\<bullet>M" in exI)
wenzelm@36278	2592	apply(simp add: swap_simps)
wenzelm@36278	2593	apply(drule_tac pi="rev pi" in fic.eqvt(1))
wenzelm@36278	2594	apply(simp)
wenzelm@36278	2595	apply(drule sym)
wenzelm@36278	2596	apply(drule pt_bij1[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	2597	apply(simp add: swap_simps)
wenzelm@36278	2598	done
wenzelm@36278	2599
wenzelm@36278	2600	lemma ORRIGHT2_eqvt_coname:
wenzelm@36278	2601	fixes pi::"coname prm"
wenzelm@36278	2602	shows "(pi\<bullet>(ORRIGHT2 (A OR B) X)) = ORRIGHT2 (A OR B) (pi\<bullet>X)"
berghofe@47052	2603	apply(auto simp add: perm_set_def)
wenzelm@36278	2604	apply(rule_tac x="pi\<bullet>a" in exI)
wenzelm@36278	2605	apply(rule_tac x="pi\<bullet>b" in exI)
wenzelm@36278	2606	apply(rule_tac x="pi\<bullet>M" in exI)
wenzelm@36278	2607	apply(simp)
wenzelm@36278	2608	apply(rule conjI)
wenzelm@36278	2609	apply(drule_tac pi="pi" in fic.eqvt(2))
wenzelm@36278	2610	apply(simp)
wenzelm@36278	2611	apply(rule_tac x="<a>:M" in exI)
wenzelm@36278	2612	apply(perm_simp)
wenzelm@36278	2613	apply(rule_tac x="(rev pi)\<bullet>(<b>:OrR2 <a>.M b)" in exI)
wenzelm@36278	2614	apply(perm_simp)
wenzelm@36278	2615	apply(rule_tac x="(rev pi)\<bullet>a" in exI)
wenzelm@36278	2616	apply(rule_tac x="(rev pi)\<bullet>b" in exI)
wenzelm@36278	2617	apply(rule_tac x="(rev pi)\<bullet>M" in exI)
wenzelm@36278	2618	apply(simp)
wenzelm@36278	2619	apply(drule_tac pi="rev pi" in fic.eqvt(2))
wenzelm@36278	2620	apply(simp)
wenzelm@36278	2621	apply(drule sym)
wenzelm@36278	2622	apply(drule pt_bij1[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	2623	apply(simp)
wenzelm@36278	2624	done
wenzelm@36278	2625
wenzelm@36278	2626	nominal_primrec
wenzelm@36278	2627	IMPRIGHT :: "ty \<Rightarrow> ntrm set \<Rightarrow> ctrm set \<Rightarrow> ntrm set \<Rightarrow> ctrm set \<Rightarrow> ctrm set"
wenzelm@36278	2628	where
wenzelm@36278	2629	"IMPRIGHT (B IMP C) X Y Z U=
wenzelm@36278	2630	{ <b>:ImpR (x).<a>.M b \| x a b M. fic (ImpR (x).<a>.M b) b
wenzelm@36278	2631	\<and> (\<forall>z P. x\<sharp>(z,P) \<and> (z):P \<in> Z \<longrightarrow> (x):(M{a:=(z).P}) \<in> X)
wenzelm@36278	2632	\<and> (\<forall>c Q. a\<sharp>(c,Q) \<and> <c>:Q \<in> U \<longrightarrow> <a>:(M{x:=<c>.Q}) \<in> Y)}"
wenzelm@36278	2633	apply(rule TrueI)+
wenzelm@36278	2634	done
wenzelm@36278	2635
wenzelm@36278	2636	lemma IMPRIGHT_eqvt_name:
wenzelm@36278	2637	fixes pi::"name prm"
wenzelm@36278	2638	shows "(pi\<bullet>(IMPRIGHT (A IMP B) X Y Z U)) = IMPRIGHT (A IMP B) (pi\<bullet>X) (pi\<bullet>Y) (pi\<bullet>Z) (pi\<bullet>U)"
berghofe@47052	2639	apply(auto simp add: perm_set_def)
wenzelm@36278	2640	apply(rule_tac x="pi\<bullet>xb" in exI)
wenzelm@36278	2641	apply(rule_tac x="pi\<bullet>a" in exI)
wenzelm@36278	2642	apply(rule_tac x="pi\<bullet>b" in exI)
wenzelm@36278	2643	apply(rule_tac x="pi\<bullet>M" in exI)
wenzelm@36278	2644	apply(simp)
wenzelm@36278	2645	apply(rule conjI)
wenzelm@36278	2646	apply(drule_tac pi="pi" in fic.eqvt(1))
wenzelm@36278	2647	apply(simp)
wenzelm@36278	2648	apply(rule conjI)
wenzelm@36278	2649	apply(auto)[1]
wenzelm@36278	2650	apply(rule_tac x="(xb):(M{a:=((rev pi)\<bullet>z).((rev pi)\<bullet>P)})" in exI)
wenzelm@36278	2651	apply(perm_simp add: csubst_eqvt)
wenzelm@36278	2652	apply(drule sym)
wenzelm@36278	2653	apply(drule pt_bij1[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	2654	apply(simp)
wenzelm@36278	2655	apply(simp add: fresh_right)
wenzelm@36278	2656	apply(auto)[1]
wenzelm@36278	2657	apply(rule_tac x="<a>:(M{xb:=<((rev pi)\<bullet>c)>.((rev pi)\<bullet>Q)})" in exI)
wenzelm@36278	2658	apply(perm_simp add: nsubst_eqvt)
wenzelm@36278	2659	apply(drule sym)
wenzelm@36278	2660	apply(drule pt_bij1[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	2661	apply(simp add: swap_simps fresh_left)
wenzelm@36278	2662	apply(rule_tac x="(rev pi)\<bullet>(<b>:ImpR xa.<a>.M b)" in exI)
wenzelm@36278	2663	apply(perm_simp)
wenzelm@36278	2664	apply(rule_tac x="(rev pi)\<bullet>xa" in exI)
wenzelm@36278	2665	apply(rule_tac x="(rev pi)\<bullet>a" in exI)
wenzelm@36278	2666	apply(rule_tac x="(rev pi)\<bullet>b" in exI)
wenzelm@36278	2667	apply(rule_tac x="(rev pi)\<bullet>M" in exI)
wenzelm@36278	2668	apply(simp add: swap_simps)
wenzelm@36278	2669	apply(drule_tac pi="rev pi" in fic.eqvt(1))
wenzelm@36278	2670	apply(simp add: swap_simps)
wenzelm@36278	2671	apply(rule conjI)
wenzelm@36278	2672	apply(auto)[1]
wenzelm@36278	2673	apply(drule_tac x="pi\<bullet>z" in spec)
wenzelm@36278	2674	apply(drule_tac x="pi\<bullet>P" in spec)
wenzelm@36278	2675	apply(drule mp)
wenzelm@36278	2676	apply(simp add: fresh_right)
wenzelm@36278	2677	apply(rule_tac x="(z):P" in exI)
wenzelm@36278	2678	apply(simp)
wenzelm@36278	2679	apply(auto)[1]
wenzelm@36278	2680	apply(drule sym)
wenzelm@36278	2681	apply(drule pt_bij1[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	2682	apply(perm_simp add: csubst_eqvt fresh_right)
wenzelm@36278	2683	apply(auto)[1]
wenzelm@36278	2684	apply(drule_tac x="pi\<bullet>c" in spec)
wenzelm@36278	2685	apply(drule_tac x="pi\<bullet>Q" in spec)
wenzelm@36278	2686	apply(drule mp)
wenzelm@36278	2687	apply(simp add: swap_simps fresh_left)
wenzelm@36278	2688	apply(rule_tac x="<c>:Q" in exI)
wenzelm@36278	2689	apply(simp add: swap_simps)
wenzelm@36278	2690	apply(auto)[1]
wenzelm@36278	2691	apply(drule sym)
wenzelm@36278	2692	apply(drule pt_bij1[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	2693	apply(perm_simp add: nsubst_eqvt)
wenzelm@36278	2694	done
wenzelm@36278	2695
wenzelm@36278	2696	lemma IMPRIGHT_eqvt_coname:
wenzelm@36278	2697	fixes pi::"coname prm"
wenzelm@36278	2698	shows "(pi\<bullet>(IMPRIGHT (A IMP B) X Y Z U)) = IMPRIGHT (A IMP B) (pi\<bullet>X) (pi\<bullet>Y) (pi\<bullet>Z) (pi\<bullet>U)"
berghofe@47052	2699	apply(auto simp add: perm_set_def)
wenzelm@36278	2700	apply(rule_tac x="pi\<bullet>xb" in exI)
wenzelm@36278	2701	apply(rule_tac x="pi\<bullet>a" in exI)
wenzelm@36278	2702	apply(rule_tac x="pi\<bullet>b" in exI)
wenzelm@36278	2703	apply(rule_tac x="pi\<bullet>M" in exI)
wenzelm@36278	2704	apply(simp)
wenzelm@36278	2705	apply(rule conjI)
wenzelm@36278	2706	apply(drule_tac pi="pi" in fic.eqvt(2))
wenzelm@36278	2707	apply(simp)
wenzelm@36278	2708	apply(rule conjI)
wenzelm@36278	2709	apply(auto)[1]
wenzelm@36278	2710	apply(rule_tac x="(xb):(M{a:=((rev pi)\<bullet>z).((rev pi)\<bullet>P)})" in exI)
wenzelm@36278	2711	apply(perm_simp add: csubst_eqvt)
wenzelm@36278	2712	apply(drule sym)
wenzelm@36278	2713	apply(drule pt_bij1[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	2714	apply(simp add: swap_simps fresh_left)
wenzelm@36278	2715	apply(auto)[1]
wenzelm@36278	2716	apply(rule_tac x="<a>:(M{xb:=<((rev pi)\<bullet>c)>.((rev pi)\<bullet>Q)})" in exI)
wenzelm@36278	2717	apply(perm_simp add: nsubst_eqvt)
wenzelm@36278	2718	apply(drule sym)
wenzelm@36278	2719	apply(drule pt_bij1[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	2720	apply(simp add: fresh_right)
wenzelm@36278	2721	apply(rule_tac x="(rev pi)\<bullet>(<b>:ImpR xa.<a>.M b)" in exI)
wenzelm@36278	2722	apply(perm_simp)
wenzelm@36278	2723	apply(rule_tac x="(rev pi)\<bullet>xa" in exI)
wenzelm@36278	2724	apply(rule_tac x="(rev pi)\<bullet>a" in exI)
wenzelm@36278	2725	apply(rule_tac x="(rev pi)\<bullet>b" in exI)
wenzelm@36278	2726	apply(rule_tac x="(rev pi)\<bullet>M" in exI)
wenzelm@36278	2727	apply(simp add: swap_simps)
wenzelm@36278	2728	apply(drule_tac pi="rev pi" in fic.eqvt(2))
wenzelm@36278	2729	apply(simp add: swap_simps)
wenzelm@36278	2730	apply(rule conjI)
wenzelm@36278	2731	apply(auto)[1]
wenzelm@36278	2732	apply(drule_tac x="pi\<bullet>z" in spec)
wenzelm@36278	2733	apply(drule_tac x="pi\<bullet>P" in spec)
wenzelm@36278	2734	apply(simp add: swap_simps fresh_left)
wenzelm@36278	2735	apply(drule mp)
wenzelm@36278	2736	apply(rule_tac x="(z):P" in exI)
wenzelm@36278	2737	apply(simp add: swap_simps)
wenzelm@36278	2738	apply(auto)[1]
wenzelm@36278	2739	apply(drule sym)
wenzelm@36278	2740	apply(drule pt_bij1[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	2741	apply(perm_simp add: csubst_eqvt fresh_right)
wenzelm@36278	2742	apply(auto)[1]
wenzelm@36278	2743	apply(drule_tac x="pi\<bullet>c" in spec)
wenzelm@36278	2744	apply(drule_tac x="pi\<bullet>Q" in spec)
wenzelm@36278	2745	apply(simp add: fresh_right)
wenzelm@36278	2746	apply(drule mp)
wenzelm@36278	2747	apply(rule_tac x="<c>:Q" in exI)
wenzelm@36278	2748	apply(simp)
wenzelm@36278	2749	apply(auto)[1]
wenzelm@36278	2750	apply(drule sym)
wenzelm@36278	2751	apply(drule pt_bij1[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	2752	apply(perm_simp add: nsubst_eqvt fresh_right)
wenzelm@36278	2753	done
wenzelm@36278	2754
wenzelm@36278	2755	nominal_primrec
wenzelm@36278	2756	IMPLEFT :: "ty \<Rightarrow> ctrm set \<Rightarrow> ntrm set \<Rightarrow> ntrm set"
wenzelm@36278	2757	where
wenzelm@36278	2758	"IMPLEFT (B IMP C) X Y =
wenzelm@36278	2759	{ (y):ImpL <a>.M (x).N y \| x a y M N. fin (ImpL <a>.M (x).N y) y \<and> <a>:M \<in> X \<and> (x):N \<in> Y }"
wenzelm@36278	2760	apply(rule TrueI)+
wenzelm@36278	2761	done
wenzelm@36278	2762
wenzelm@36278	2763	lemma IMPLEFT_eqvt_name:
wenzelm@36278	2764	fixes pi::"name prm"
wenzelm@36278	2765	shows "(pi\<bullet>(IMPLEFT (A IMP B) X Y)) = IMPLEFT (A IMP B) (pi\<bullet>X) (pi\<bullet>Y)"
berghofe@47052	2766	apply(auto simp add: perm_set_def)
wenzelm@36278	2767	apply(rule_tac x="pi\<bullet>xb" in exI)
wenzelm@36278	2768	apply(rule_tac x="pi\<bullet>a" in exI)
wenzelm@36278	2769	apply(rule_tac x="pi\<bullet>y" in exI)
wenzelm@36278	2770	apply(rule_tac x="pi\<bullet>M" in exI)
wenzelm@36278	2771	apply(rule_tac x="pi\<bullet>N" in exI)
wenzelm@36278	2772	apply(simp)
wenzelm@36278	2773	apply(rule conjI)
wenzelm@36278	2774	apply(drule_tac pi="pi" in fin.eqvt(1))
wenzelm@36278	2775	apply(simp)
wenzelm@36278	2776	apply(rule conjI)
wenzelm@36278	2777	apply(rule_tac x="<a>:M" in exI)
wenzelm@36278	2778	apply(simp)
wenzelm@36278	2779	apply(rule_tac x="(xb):N" in exI)
wenzelm@36278	2780	apply(simp)
wenzelm@36278	2781	apply(rule_tac x="(rev pi)\<bullet>((y):ImpL <a>.M (xa).N y)" in exI)
wenzelm@36278	2782	apply(perm_simp)
wenzelm@36278	2783	apply(rule_tac x="(rev pi)\<bullet>xa" in exI)
wenzelm@36278	2784	apply(rule_tac x="(rev pi)\<bullet>a" in exI)
wenzelm@36278	2785	apply(rule_tac x="(rev pi)\<bullet>y" in exI)
wenzelm@36278	2786	apply(rule_tac x="(rev pi)\<bullet>M" in exI)
wenzelm@36278	2787	apply(rule_tac x="(rev pi)\<bullet>N" in exI)
wenzelm@36278	2788	apply(simp add: swap_simps)
wenzelm@36278	2789	apply(drule_tac pi="rev pi" in fin.eqvt(1))
wenzelm@36278	2790	apply(simp)
wenzelm@36278	2791	apply(drule sym)
wenzelm@36278	2792	apply(drule sym)
wenzelm@36278	2793	apply(drule pt_bij1[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	2794	apply(drule pt_bij1[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	2795	apply(simp add: swap_simps)
wenzelm@36278	2796	done
wenzelm@36278	2797
wenzelm@36278	2798	lemma IMPLEFT_eqvt_coname:
wenzelm@36278	2799	fixes pi::"coname prm"
wenzelm@36278	2800	shows "(pi\<bullet>(IMPLEFT (A IMP B) X Y)) = IMPLEFT (A IMP B) (pi\<bullet>X) (pi\<bullet>Y)"
berghofe@47052	2801	apply(auto simp add: perm_set_def)
wenzelm@36278	2802	apply(rule_tac x="pi\<bullet>xb" in exI)
wenzelm@36278	2803	apply(rule_tac x="pi\<bullet>a" in exI)
wenzelm@36278	2804	apply(rule_tac x="pi\<bullet>y" in exI)
wenzelm@36278	2805	apply(rule_tac x="pi\<bullet>M" in exI)
wenzelm@36278	2806	apply(rule_tac x="pi\<bullet>N" in exI)
wenzelm@36278	2807	apply(simp)
wenzelm@36278	2808	apply(rule conjI)
wenzelm@36278	2809	apply(drule_tac pi="pi" in fin.eqvt(2))
wenzelm@36278	2810	apply(simp)
wenzelm@36278	2811	apply(rule conjI)
wenzelm@36278	2812	apply(rule_tac x="<a>:M" in exI)
wenzelm@36278	2813	apply(simp)
wenzelm@36278	2814	apply(rule_tac x="(xb):N" in exI)
wenzelm@36278	2815	apply(simp)
wenzelm@36278	2816	apply(rule_tac x="(rev pi)\<bullet>((y):ImpL <a>.M (xa).N y)" in exI)
wenzelm@36278	2817	apply(perm_simp)
wenzelm@36278	2818	apply(rule_tac x="(rev pi)\<bullet>xa" in exI)
wenzelm@36278	2819	apply(rule_tac x="(rev pi)\<bullet>a" in exI)
wenzelm@36278	2820	apply(rule_tac x="(rev pi)\<bullet>y" in exI)
wenzelm@36278	2821	apply(rule_tac x="(rev pi)\<bullet>M" in exI)
wenzelm@36278	2822	apply(rule_tac x="(rev pi)\<bullet>N" in exI)
wenzelm@36278	2823	apply(simp add: swap_simps)
wenzelm@36278	2824	apply(drule_tac pi="rev pi" in fin.eqvt(2))
wenzelm@36278	2825	apply(simp)
wenzelm@36278	2826	apply(drule sym)
wenzelm@36278	2827	apply(drule sym)
wenzelm@36278	2828	apply(drule pt_bij1[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	2829	apply(drule pt_bij1[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	2830	apply(simp add: swap_simps)
wenzelm@36278	2831	done
wenzelm@36278	2832
wenzelm@36278	2833	lemma sum_cases:
wenzelm@36278	2834	shows "(\<exists>y. x=Inl y) \<or> (\<exists>y. x=Inr y)"
wenzelm@36278	2835	apply(rule_tac s="x" in sumE)
wenzelm@36278	2836	apply(auto)
wenzelm@36278	2837	done
wenzelm@36278	2838
wenzelm@36278	2839	function
wenzelm@36278	2840	NEGc::"ty \<Rightarrow> ntrm set \<Rightarrow> ctrm set"
wenzelm@36278	2841	and
wenzelm@36278	2842	NEGn::"ty \<Rightarrow> ctrm set \<Rightarrow> ntrm set"
wenzelm@36278	2843	where
wenzelm@36278	2844	"NEGc (PR A) X = AXIOMSc (PR A) \<union> BINDINGc (PR A) X"
wenzelm@36278	2845	\| "NEGc (NOT C) X = AXIOMSc (NOT C) \<union> BINDINGc (NOT C) X
wenzelm@36278	2846	\<union> NOTRIGHT (NOT C) (lfp (NEGn C \<circ> NEGc C))"
wenzelm@36278	2847	\| "NEGc (C AND D) X = AXIOMSc (C AND D) \<union> BINDINGc (C AND D) X
wenzelm@36278	2848	\<union> ANDRIGHT (C AND D) (NEGc C (lfp (NEGn C \<circ> NEGc C))) (NEGc D (lfp (NEGn D \<circ> NEGc D)))"
wenzelm@36278	2849	\| "NEGc (C OR D) X = AXIOMSc (C OR D) \<union> BINDINGc (C OR D) X
wenzelm@36278	2850	\<union> ORRIGHT1 (C OR D) (NEGc C (lfp (NEGn C \<circ> NEGc C)))
wenzelm@36278	2851	\<union> ORRIGHT2 (C OR D) (NEGc D (lfp (NEGn D \<circ> NEGc D)))"
wenzelm@36278	2852	\| "NEGc (C IMP D) X = AXIOMSc (C IMP D) \<union> BINDINGc (C IMP D) X
wenzelm@36278	2853	\<union> IMPRIGHT (C IMP D) (lfp (NEGn C \<circ> NEGc C)) (NEGc D (lfp (NEGn D \<circ> NEGc D)))
wenzelm@36278	2854	(lfp (NEGn D \<circ> NEGc D)) (NEGc C (lfp (NEGn C \<circ> NEGc C)))"
wenzelm@36278	2855	\| "NEGn (PR A) X = AXIOMSn (PR A) \<union> BINDINGn (PR A) X"
wenzelm@36278	2856	\| "NEGn (NOT C) X = AXIOMSn (NOT C) \<union> BINDINGn (NOT C) X
wenzelm@36278	2857	\<union> NOTLEFT (NOT C) (NEGc C (lfp (NEGn C \<circ> NEGc C)))"
wenzelm@36278	2858	\| "NEGn (C AND D) X = AXIOMSn (C AND D) \<union> BINDINGn (C AND D) X
wenzelm@36278	2859	\<union> ANDLEFT1 (C AND D) (lfp (NEGn C \<circ> NEGc C))
wenzelm@36278	2860	\<union> ANDLEFT2 (C AND D) (lfp (NEGn D \<circ> NEGc D))"
wenzelm@36278	2861	\| "NEGn (C OR D) X = AXIOMSn (C OR D) \<union> BINDINGn (C OR D) X
wenzelm@36278	2862	\<union> ORLEFT (C OR D) (lfp (NEGn C \<circ> NEGc C)) (lfp (NEGn D \<circ> NEGc D))"
wenzelm@36278	2863	\| "NEGn (C IMP D) X = AXIOMSn (C IMP D) \<union> BINDINGn (C IMP D) X
wenzelm@36278	2864	\<union> IMPLEFT (C IMP D) (NEGc C (lfp (NEGn C \<circ> NEGc C))) (lfp (NEGn D \<circ> NEGc D))"
wenzelm@36278	2865	using ty_cases sum_cases
wenzelm@36278	2866	apply(auto simp add: ty.inject)
wenzelm@36278	2867	apply(drule_tac x="x" in meta_spec)
wenzelm@36278	2868	apply(auto simp add: ty.inject)
wenzelm@36278	2869	apply(rotate_tac 10)
wenzelm@36278	2870	apply(drule_tac x="a" in meta_spec)
wenzelm@36278	2871	apply(auto simp add: ty.inject)
wenzelm@36278	2872	apply(rotate_tac 10)
wenzelm@36278	2873	apply(drule_tac x="a" in meta_spec)
wenzelm@36278	2874	apply(auto simp add: ty.inject)
wenzelm@36278	2875	done
wenzelm@36278	2876
wenzelm@36278	2877	termination
blanchet@56756	2878	apply(relation "measure (case_sum (size\<circ>fst) (size\<circ>fst))")
wenzelm@36278	2879	apply(simp_all)
wenzelm@36278	2880	done
wenzelm@36278	2881
wenzelm@36278	2882	text {* Candidates *}
wenzelm@36278	2883
wenzelm@36278	2884	lemma test1:
wenzelm@36278	2885	shows "x\<in>(X\<union>Y) = (x\<in>X \<or> x\<in>Y)"
wenzelm@36278	2886	by blast
wenzelm@36278	2887
wenzelm@36278	2888	lemma test2:
wenzelm@36278	2889	shows "x\<in>(X\<inter>Y) = (x\<in>X \<and> x\<in>Y)"
wenzelm@36278	2890	by blast
wenzelm@36278	2891
wenzelm@36278	2892	lemma big_inter_eqvt:
wenzelm@36278	2893	fixes pi1::"name prm"
wenzelm@36278	2894	and X::"('a::pt_name) set set"
wenzelm@36278	2895	and pi2::"coname prm"
wenzelm@36278	2896	and Y::"('b::pt_coname) set set"
wenzelm@36278	2897	shows "(pi1\<bullet>(\<Inter> X)) = \<Inter> (pi1\<bullet>X)"
wenzelm@36278	2898	and "(pi2\<bullet>(\<Inter> Y)) = \<Inter> (pi2\<bullet>Y)"
berghofe@47052	2899	apply(auto simp add: perm_set_def)
wenzelm@36278	2900	apply(rule_tac x="(rev pi1)\<bullet>x" in exI)
wenzelm@36278	2901	apply(perm_simp)
wenzelm@36278	2902	apply(rule ballI)
wenzelm@36278	2903	apply(drule_tac x="pi1\<bullet>xa" in spec)
wenzelm@36278	2904	apply(auto)
wenzelm@36278	2905	apply(drule_tac x="xa" in spec)
wenzelm@36278	2906	apply(auto)[1]
wenzelm@36278	2907	apply(rule_tac x="(rev pi1)\<bullet>xb" in exI)
wenzelm@36278	2908	apply(perm_simp)
wenzelm@36278	2909	apply(simp add: pt_set_bij1[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	2910	apply(simp add: pt_set_bij[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	2911	apply(simp add: pt_set_bij1[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	2912	apply(rule_tac x="(rev pi2)\<bullet>x" in exI)
wenzelm@36278	2913	apply(perm_simp)
wenzelm@36278	2914	apply(rule ballI)
wenzelm@36278	2915	apply(drule_tac x="pi2\<bullet>xa" in spec)
wenzelm@36278	2916	apply(auto)
wenzelm@36278	2917	apply(drule_tac x="xa" in spec)
wenzelm@36278	2918	apply(auto)[1]
wenzelm@36278	2919	apply(rule_tac x="(rev pi2)\<bullet>xb" in exI)
wenzelm@36278	2920	apply(perm_simp)
wenzelm@36278	2921	apply(simp add: pt_set_bij1[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	2922	apply(simp add: pt_set_bij[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	2923	apply(simp add: pt_set_bij1[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	2924	done
wenzelm@36278	2925
wenzelm@36278	2926	lemma lfp_eqvt:
wenzelm@36278	2927	fixes pi1::"name prm"
wenzelm@36278	2928	and f::"'a set \<Rightarrow> ('a::pt_name) set"
wenzelm@36278	2929	and pi2::"coname prm"
wenzelm@36278	2930	and g::"'b set \<Rightarrow> ('b::pt_coname) set"
wenzelm@36278	2931	shows "pi1\<bullet>(lfp f) = lfp (pi1\<bullet>f)"
wenzelm@36278	2932	and "pi2\<bullet>(lfp g) = lfp (pi2\<bullet>g)"
wenzelm@36278	2933	apply(simp add: lfp_def)
wenzelm@36278	2934	apply(simp add: big_inter_eqvt)
wenzelm@36278	2935	apply(simp add: pt_Collect_eqvt[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	2936	apply(subgoal_tac "{u. (pi1\<bullet>f) u \<subseteq> u} = {u. ((rev pi1)\<bullet>((pi1\<bullet>f) u)) \<subseteq> ((rev pi1)\<bullet>u)}")
wenzelm@36278	2937	apply(perm_simp)
wenzelm@36278	2938	apply(rule Collect_cong)
wenzelm@36278	2939	apply(rule iffI)
wenzelm@36278	2940	apply(rule subseteq_eqvt(1)[THEN iffD1])
wenzelm@36278	2941	apply(simp add: perm_bool)
wenzelm@36278	2942	apply(drule subseteq_eqvt(1)[THEN iffD2])
wenzelm@36278	2943	apply(simp add: perm_bool)
wenzelm@36278	2944	apply(simp add: lfp_def)
wenzelm@36278	2945	apply(simp add: big_inter_eqvt)
wenzelm@36278	2946	apply(simp add: pt_Collect_eqvt[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	2947	apply(subgoal_tac "{u. (pi2\<bullet>g) u \<subseteq> u} = {u. ((rev pi2)\<bullet>((pi2\<bullet>g) u)) \<subseteq> ((rev pi2)\<bullet>u)}")
wenzelm@36278	2948	apply(perm_simp)
wenzelm@36278	2949	apply(rule Collect_cong)
wenzelm@36278	2950	apply(rule iffI)
wenzelm@36278	2951	apply(rule subseteq_eqvt(2)[THEN iffD1])
wenzelm@36278	2952	apply(simp add: perm_bool)
wenzelm@36278	2953	apply(drule subseteq_eqvt(2)[THEN iffD2])
wenzelm@36278	2954	apply(simp add: perm_bool)
wenzelm@36278	2955	done
wenzelm@36278	2956
wenzelm@36278	2957	abbreviation
wenzelm@36278	2958	CANDn::"ty \<Rightarrow> ntrm set" ("\<parallel>'(_')\<parallel>" [60] 60)
wenzelm@36278	2959	where
wenzelm@36278	2960	"\<parallel>(B)\<parallel> \<equiv> lfp (NEGn B \<circ> NEGc B)"
wenzelm@36278	2961
wenzelm@36278	2962	abbreviation
wenzelm@36278	2963	CANDc::"ty \<Rightarrow> ctrm set" ("\<parallel><_>\<parallel>" [60] 60)
wenzelm@36278	2964	where
wenzelm@36278	2965	"\<parallel><B>\<parallel> \<equiv> NEGc B (\<parallel>(B)\<parallel>)"
wenzelm@36278	2966
wenzelm@36278	2967	lemma NEGn_decreasing:
wenzelm@36278	2968	shows "X\<subseteq>Y \<Longrightarrow> NEGn B Y \<subseteq> NEGn B X"
wenzelm@36278	2969	by (nominal_induct B rule: ty.strong_induct)
wenzelm@36278	2970	(auto dest: BINDINGn_decreasing)
wenzelm@36278	2971
wenzelm@36278	2972	lemma NEGc_decreasing:
wenzelm@36278	2973	shows "X\<subseteq>Y \<Longrightarrow> NEGc B Y \<subseteq> NEGc B X"
wenzelm@36278	2974	by (nominal_induct B rule: ty.strong_induct)
wenzelm@36278	2975	(auto dest: BINDINGc_decreasing)
wenzelm@36278	2976
wenzelm@36278	2977	lemma mono_NEGn_NEGc:
wenzelm@36278	2978	shows "mono (NEGn B \<circ> NEGc B)"
wenzelm@36278	2979	and "mono (NEGc B \<circ> NEGn B)"
wenzelm@36278	2980	proof -
wenzelm@36278	2981	have "\<forall>X Y. X\<subseteq>Y \<longrightarrow> NEGn B (NEGc B X) \<subseteq> NEGn B (NEGc B Y)"
wenzelm@36278	2982	proof (intro strip)
wenzelm@36278	2983	fix X::"ntrm set" and Y::"ntrm set"
wenzelm@36278	2984	assume "X\<subseteq>Y"
wenzelm@36278	2985	then have "NEGc B Y \<subseteq> NEGc B X" by (simp add: NEGc_decreasing)
wenzelm@36278	2986	then show "NEGn B (NEGc B X) \<subseteq> NEGn B (NEGc B Y)" by (simp add: NEGn_decreasing)
wenzelm@36278	2987	qed
wenzelm@36278	2988	then show "mono (NEGn B \<circ> NEGc B)" by (simp add: mono_def)
wenzelm@36278	2989	next
wenzelm@36278	2990	have "\<forall>X Y. X\<subseteq>Y \<longrightarrow> NEGc B (NEGn B X) \<subseteq> NEGc B (NEGn B Y)"
wenzelm@36278	2991	proof (intro strip)
wenzelm@36278	2992	fix X::"ctrm set" and Y::"ctrm set"
wenzelm@36278	2993	assume "X\<subseteq>Y"
wenzelm@36278	2994	then have "NEGn B Y \<subseteq> NEGn B X" by (simp add: NEGn_decreasing)
wenzelm@36278	2995	then show "NEGc B (NEGn B X) \<subseteq> NEGc B (NEGn B Y)" by (simp add: NEGc_decreasing)
wenzelm@36278	2996	qed
wenzelm@36278	2997	then show "mono (NEGc B \<circ> NEGn B)" by (simp add: mono_def)
wenzelm@36278	2998	qed
wenzelm@36278	2999
wenzelm@36278	3000	lemma NEG_simp:
wenzelm@36278	3001	shows "\<parallel><B>\<parallel> = NEGc B (\<parallel>(B)\<parallel>)"
wenzelm@36278	3002	and "\<parallel>(B)\<parallel> = NEGn B (\<parallel><B>\<parallel>)"
wenzelm@36278	3003	proof -
wenzelm@36278	3004	show "\<parallel><B>\<parallel> = NEGc B (\<parallel>(B)\<parallel>)" by simp
wenzelm@36278	3005	next
wenzelm@36278	3006	have "\<parallel>(B)\<parallel> \<equiv> lfp (NEGn B \<circ> NEGc B)" by simp
wenzelm@36278	3007	then have "\<parallel>(B)\<parallel> = (NEGn B \<circ> NEGc B) (\<parallel>(B)\<parallel>)" using mono_NEGn_NEGc def_lfp_unfold by blast
wenzelm@36278	3008	then show "\<parallel>(B)\<parallel> = NEGn B (\<parallel><B>\<parallel>)" by simp
wenzelm@36278	3009	qed
wenzelm@36278	3010
wenzelm@36278	3011	lemma NEG_elim:
wenzelm@36278	3012	shows "M \<in> \<parallel><B>\<parallel> \<Longrightarrow> M \<in> NEGc B (\<parallel>(B)\<parallel>)"
wenzelm@36278	3013	and "N \<in> \<parallel>(B)\<parallel> \<Longrightarrow> N \<in> NEGn B (\<parallel><B>\<parallel>)"
wenzelm@36278	3014	using NEG_simp by (blast)+
wenzelm@36278	3015
wenzelm@36278	3016	lemma NEG_intro:
wenzelm@36278	3017	shows "M \<in> NEGc B (\<parallel>(B)\<parallel>) \<Longrightarrow> M \<in> \<parallel><B>\<parallel>"
wenzelm@36278	3018	and "N \<in> NEGn B (\<parallel><B>\<parallel>) \<Longrightarrow> N \<in> \<parallel>(B)\<parallel>"
wenzelm@36278	3019	using NEG_simp by (blast)+
wenzelm@36278	3020
wenzelm@36278	3021	lemma NEGc_simps:
wenzelm@36278	3022	shows "NEGc (PR A) (\<parallel>(PR A)\<parallel>) = AXIOMSc (PR A) \<union> BINDINGc (PR A) (\<parallel>(PR A)\<parallel>)"
wenzelm@36278	3023	and "NEGc (NOT C) (\<parallel>(NOT C)\<parallel>) = AXIOMSc (NOT C) \<union> BINDINGc (NOT C) (\<parallel>(NOT C)\<parallel>)
wenzelm@36278	3024	\<union> (NOTRIGHT (NOT C) (\<parallel>(C)\<parallel>))"
wenzelm@36278	3025	and "NEGc (C AND D) (\<parallel>(C AND D)\<parallel>) = AXIOMSc (C AND D) \<union> BINDINGc (C AND D) (\<parallel>(C AND D)\<parallel>)
wenzelm@36278	3026	\<union> (ANDRIGHT (C AND D) (\<parallel><C>\<parallel>) (\<parallel><D>\<parallel>))"
wenzelm@36278	3027	and "NEGc (C OR D) (\<parallel>(C OR D)\<parallel>) = AXIOMSc (C OR D) \<union> BINDINGc (C OR D) (\<parallel>(C OR D)\<parallel>)
wenzelm@36278	3028	\<union> (ORRIGHT1 (C OR D) (\<parallel><C>\<parallel>)) \<union> (ORRIGHT2 (C OR D) (\<parallel><D>\<parallel>))"
wenzelm@36278	3029	and "NEGc (C IMP D) (\<parallel>(C IMP D)\<parallel>) = AXIOMSc (C IMP D) \<union> BINDINGc (C IMP D) (\<parallel>(C IMP D)\<parallel>)
wenzelm@36278	3030	\<union> (IMPRIGHT (C IMP D) (\<parallel>(C)\<parallel>) (\<parallel><D>\<parallel>) (\<parallel>(D)\<parallel>) (\<parallel><C>\<parallel>))"
wenzelm@36278	3031	by (simp_all only: NEGc.simps)
wenzelm@36278	3032
wenzelm@36278	3033	lemma AXIOMS_in_CANDs:
wenzelm@36278	3034	shows "AXIOMSn B \<subseteq> (\<parallel>(B)\<parallel>)"
wenzelm@36278	3035	and "AXIOMSc B \<subseteq> (\<parallel><B>\<parallel>)"
wenzelm@36278	3036	proof -
wenzelm@36278	3037	have "AXIOMSn B \<subseteq> NEGn B (\<parallel><B>\<parallel>)"
wenzelm@36278	3038	by (nominal_induct B rule: ty.strong_induct) (auto)
wenzelm@36278	3039	then show "AXIOMSn B \<subseteq> \<parallel>(B)\<parallel>" using NEG_simp by blast
wenzelm@36278	3040	next
wenzelm@36278	3041	have "AXIOMSc B \<subseteq> NEGc B (\<parallel>(B)\<parallel>)"
wenzelm@36278	3042	by (nominal_induct B rule: ty.strong_induct) (auto)
wenzelm@36278	3043	then show "AXIOMSc B \<subseteq> \<parallel><B>\<parallel>" using NEG_simp by blast
wenzelm@36278	3044	qed
wenzelm@36278	3045
wenzelm@36278	3046	lemma Ax_in_CANDs:
wenzelm@36278	3047	shows "(y):Ax x a \<in> \<parallel>(B)\<parallel>"
wenzelm@36278	3048	and "<b>:Ax x a \<in> \<parallel><B>\<parallel>"
wenzelm@36278	3049	proof -
wenzelm@36278	3050	have "(y):Ax x a \<in> AXIOMSn B" by (auto simp add: AXIOMSn_def)
wenzelm@36278	3051	also have "AXIOMSn B \<subseteq> \<parallel>(B)\<parallel>" by (rule AXIOMS_in_CANDs)
wenzelm@36278	3052	finally show "(y):Ax x a \<in> \<parallel>(B)\<parallel>" by simp
wenzelm@36278	3053	next
wenzelm@36278	3054	have "<b>:Ax x a \<in> AXIOMSc B" by (auto simp add: AXIOMSc_def)
wenzelm@36278	3055	also have "AXIOMSc B \<subseteq> \<parallel><B>\<parallel>" by (rule AXIOMS_in_CANDs)
wenzelm@36278	3056	finally show "<b>:Ax x a \<in> \<parallel><B>\<parallel>" by simp
wenzelm@36278	3057	qed
wenzelm@36278	3058
wenzelm@36278	3059	lemma AXIOMS_eqvt_aux_name:
wenzelm@36278	3060	fixes pi::"name prm"
wenzelm@36278	3061	shows "M \<in> AXIOMSn B \<Longrightarrow> (pi\<bullet>M) \<in> AXIOMSn B"
wenzelm@36278	3062	and "N \<in> AXIOMSc B \<Longrightarrow> (pi\<bullet>N) \<in> AXIOMSc B"
wenzelm@36278	3063	apply(auto simp add: AXIOMSn_def AXIOMSc_def)
wenzelm@36278	3064	apply(rule_tac x="pi\<bullet>x" in exI)
wenzelm@36278	3065	apply(rule_tac x="pi\<bullet>y" in exI)
wenzelm@36278	3066	apply(rule_tac x="pi\<bullet>b" in exI)
wenzelm@36278	3067	apply(simp)
wenzelm@36278	3068	apply(rule_tac x="pi\<bullet>a" in exI)
wenzelm@36278	3069	apply(rule_tac x="pi\<bullet>y" in exI)
wenzelm@36278	3070	apply(rule_tac x="pi\<bullet>b" in exI)
wenzelm@36278	3071	apply(simp)
wenzelm@36278	3072	done
wenzelm@36278	3073
wenzelm@36278	3074	lemma AXIOMS_eqvt_aux_coname:
wenzelm@36278	3075	fixes pi::"coname prm"
wenzelm@36278	3076	shows "M \<in> AXIOMSn B \<Longrightarrow> (pi\<bullet>M) \<in> AXIOMSn B"
wenzelm@36278	3077	and "N \<in> AXIOMSc B \<Longrightarrow> (pi\<bullet>N) \<in> AXIOMSc B"
wenzelm@36278	3078	apply(auto simp add: AXIOMSn_def AXIOMSc_def)
wenzelm@36278	3079	apply(rule_tac x="pi\<bullet>x" in exI)
wenzelm@36278	3080	apply(rule_tac x="pi\<bullet>y" in exI)
wenzelm@36278	3081	apply(rule_tac x="pi\<bullet>b" in exI)
wenzelm@36278	3082	apply(simp)
wenzelm@36278	3083	apply(rule_tac x="pi\<bullet>a" in exI)
wenzelm@36278	3084	apply(rule_tac x="pi\<bullet>y" in exI)
wenzelm@36278	3085	apply(rule_tac x="pi\<bullet>b" in exI)
wenzelm@36278	3086	apply(simp)
wenzelm@36278	3087	done
wenzelm@36278	3088
wenzelm@36278	3089	lemma AXIOMS_eqvt_name:
wenzelm@36278	3090	fixes pi::"name prm"
wenzelm@36278	3091	shows "(pi\<bullet>AXIOMSn B) = AXIOMSn B"
wenzelm@36278	3092	and "(pi\<bullet>AXIOMSc B) = AXIOMSc B"
wenzelm@36278	3093	apply(auto)
wenzelm@36278	3094	apply(simp add: pt_set_bij1a[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	3095	apply(drule_tac pi="pi" in AXIOMS_eqvt_aux_name(1))
wenzelm@36278	3096	apply(perm_simp)
wenzelm@36278	3097	apply(drule_tac pi="rev pi" in AXIOMS_eqvt_aux_name(1))
wenzelm@36278	3098	apply(simp add: pt_set_bij1[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	3099	apply(simp add: pt_set_bij1a[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	3100	apply(drule_tac pi="pi" in AXIOMS_eqvt_aux_name(2))
wenzelm@36278	3101	apply(perm_simp)
wenzelm@36278	3102	apply(drule_tac pi="rev pi" in AXIOMS_eqvt_aux_name(2))
wenzelm@36278	3103	apply(simp add: pt_set_bij1[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	3104	done
wenzelm@36278	3105
wenzelm@36278	3106	lemma AXIOMS_eqvt_coname:
wenzelm@36278	3107	fixes pi::"coname prm"
wenzelm@36278	3108	shows "(pi\<bullet>AXIOMSn B) = AXIOMSn B"
wenzelm@36278	3109	and "(pi\<bullet>AXIOMSc B) = AXIOMSc B"
wenzelm@36278	3110	apply(auto)
wenzelm@36278	3111	apply(simp add: pt_set_bij1a[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	3112	apply(drule_tac pi="pi" in AXIOMS_eqvt_aux_coname(1))
wenzelm@36278	3113	apply(perm_simp)
wenzelm@36278	3114	apply(drule_tac pi="rev pi" in AXIOMS_eqvt_aux_coname(1))
wenzelm@36278	3115	apply(simp add: pt_set_bij1[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	3116	apply(simp add: pt_set_bij1a[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	3117	apply(drule_tac pi="pi" in AXIOMS_eqvt_aux_coname(2))
wenzelm@36278	3118	apply(perm_simp)
wenzelm@36278	3119	apply(drule_tac pi="rev pi" in AXIOMS_eqvt_aux_coname(2))
wenzelm@36278	3120	apply(simp add: pt_set_bij1[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	3121	done
wenzelm@36278	3122
wenzelm@36278	3123	lemma BINDING_eqvt_name:
wenzelm@36278	3124	fixes pi::"name prm"
wenzelm@36278	3125	shows "(pi\<bullet>(BINDINGn B X)) = BINDINGn B (pi\<bullet>X)"
wenzelm@36278	3126	and "(pi\<bullet>(BINDINGc B Y)) = BINDINGc B (pi\<bullet>Y)"
berghofe@47052	3127	apply(auto simp add: BINDINGn_def BINDINGc_def perm_set_def)
wenzelm@36278	3128	apply(rule_tac x="pi\<bullet>xb" in exI)
wenzelm@36278	3129	apply(rule_tac x="pi\<bullet>M" in exI)
wenzelm@36278	3130	apply(simp)
wenzelm@36278	3131	apply(auto)[1]
wenzelm@36278	3132	apply(drule_tac x="(rev pi)\<bullet>a" in spec)
wenzelm@36278	3133	apply(drule_tac x="(rev pi)\<bullet>P" in spec)
wenzelm@36278	3134	apply(drule mp)
wenzelm@36278	3135	apply(drule sym)
wenzelm@36278	3136	apply(drule pt_bij1[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	3137	apply(simp)
wenzelm@36278	3138	apply(drule_tac ?pi1.0="pi" in SNa_eqvt(1))
wenzelm@36278	3139	apply(perm_simp add: nsubst_eqvt)
wenzelm@36278	3140	apply(rule_tac x="(rev pi\<bullet>xa):(rev pi\<bullet>M)" in exI)
wenzelm@36278	3141	apply(perm_simp)
wenzelm@36278	3142	apply(rule_tac x="rev pi\<bullet>xa" in exI)
wenzelm@36278	3143	apply(rule_tac x="rev pi\<bullet>M" in exI)
wenzelm@36278	3144	apply(simp)
wenzelm@36278	3145	apply(auto)[1]
wenzelm@36278	3146	apply(drule_tac x="pi\<bullet>a" in spec)
wenzelm@36278	3147	apply(drule_tac x="pi\<bullet>P" in spec)
wenzelm@36278	3148	apply(drule mp)
wenzelm@36278	3149	apply(force)
wenzelm@36278	3150	apply(drule_tac ?pi1.0="rev pi" in SNa_eqvt(1))
wenzelm@36278	3151	apply(perm_simp add: nsubst_eqvt)
wenzelm@36278	3152	apply(rule_tac x="pi\<bullet>a" in exI)
wenzelm@36278	3153	apply(rule_tac x="pi\<bullet>M" in exI)
wenzelm@36278	3154	apply(simp)
wenzelm@36278	3155	apply(auto)[1]
wenzelm@36278	3156	apply(drule_tac x="(rev pi)\<bullet>x" in spec)
wenzelm@36278	3157	apply(drule_tac x="(rev pi)\<bullet>P" in spec)
wenzelm@36278	3158	apply(drule mp)
wenzelm@36278	3159	apply(drule sym)
wenzelm@36278	3160	apply(drule pt_bij1[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	3161	apply(simp)
wenzelm@36278	3162	apply(drule_tac ?pi1.0="pi" in SNa_eqvt(1))
wenzelm@36278	3163	apply(perm_simp add: csubst_eqvt)
wenzelm@36278	3164	apply(rule_tac x="<(rev pi\<bullet>a)>:(rev pi\<bullet>M)" in exI)
wenzelm@36278	3165	apply(perm_simp)
wenzelm@36278	3166	apply(rule_tac x="rev pi\<bullet>a" in exI)
wenzelm@36278	3167	apply(rule_tac x="rev pi\<bullet>M" in exI)
wenzelm@36278	3168	apply(simp add: swap_simps)
wenzelm@36278	3169	apply(auto)[1]
wenzelm@36278	3170	apply(drule_tac x="pi\<bullet>x" in spec)
wenzelm@36278	3171	apply(drule_tac x="pi\<bullet>P" in spec)
wenzelm@36278	3172	apply(drule mp)
wenzelm@36278	3173	apply(force)
wenzelm@36278	3174	apply(drule_tac ?pi1.0="rev pi" in SNa_eqvt(1))
wenzelm@36278	3175	apply(perm_simp add: csubst_eqvt)
wenzelm@36278	3176	done
wenzelm@36278	3177
wenzelm@36278	3178	lemma BINDING_eqvt_coname:
wenzelm@36278	3179	fixes pi::"coname prm"
wenzelm@36278	3180	shows "(pi\<bullet>(BINDINGn B X)) = BINDINGn B (pi\<bullet>X)"
wenzelm@36278	3181	and "(pi\<bullet>(BINDINGc B Y)) = BINDINGc B (pi\<bullet>Y)"
berghofe@47052	3182	apply(auto simp add: BINDINGn_def BINDINGc_def perm_set_def)
wenzelm@36278	3183	apply(rule_tac x="pi\<bullet>xb" in exI)
wenzelm@36278	3184	apply(rule_tac x="pi\<bullet>M" in exI)
wenzelm@36278	3185	apply(simp)
wenzelm@36278	3186	apply(auto)[1]
wenzelm@36278	3187	apply(drule_tac x="(rev pi)\<bullet>a" in spec)
wenzelm@36278	3188	apply(drule_tac x="(rev pi)\<bullet>P" in spec)
wenzelm@36278	3189	apply(drule mp)
wenzelm@36278	3190	apply(drule sym)
wenzelm@36278	3191	apply(drule pt_bij1[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	3192	apply(simp)
wenzelm@36278	3193	apply(drule_tac ?pi2.0="pi" in SNa_eqvt(2))
wenzelm@36278	3194	apply(perm_simp add: nsubst_eqvt)
wenzelm@36278	3195	apply(rule_tac x="(rev pi\<bullet>xa):(rev pi\<bullet>M)" in exI)
wenzelm@36278	3196	apply(perm_simp)
wenzelm@36278	3197	apply(rule_tac x="rev pi\<bullet>xa" in exI)
wenzelm@36278	3198	apply(rule_tac x="rev pi\<bullet>M" in exI)
wenzelm@36278	3199	apply(simp add: swap_simps)
wenzelm@36278	3200	apply(auto)[1]
wenzelm@36278	3201	apply(drule_tac x="pi\<bullet>a" in spec)
wenzelm@36278	3202	apply(drule_tac x="pi\<bullet>P" in spec)
wenzelm@36278	3203	apply(drule mp)
wenzelm@36278	3204	apply(force)
wenzelm@36278	3205	apply(drule_tac ?pi2.0="rev pi" in SNa_eqvt(2))
wenzelm@36278	3206	apply(perm_simp add: nsubst_eqvt)
wenzelm@36278	3207	apply(rule_tac x="pi\<bullet>a" in exI)
wenzelm@36278	3208	apply(rule_tac x="pi\<bullet>M" in exI)
wenzelm@36278	3209	apply(simp)
wenzelm@36278	3210	apply(auto)[1]
wenzelm@36278	3211	apply(drule_tac x="(rev pi)\<bullet>x" in spec)
wenzelm@36278	3212	apply(drule_tac x="(rev pi)\<bullet>P" in spec)
wenzelm@36278	3213	apply(drule mp)
wenzelm@36278	3214	apply(drule sym)
wenzelm@36278	3215	apply(drule pt_bij1[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	3216	apply(simp)
wenzelm@36278	3217	apply(drule_tac ?pi2.0="pi" in SNa_eqvt(2))
wenzelm@36278	3218	apply(perm_simp add: csubst_eqvt)
wenzelm@36278	3219	apply(rule_tac x="<(rev pi\<bullet>a)>:(rev pi\<bullet>M)" in exI)
wenzelm@36278	3220	apply(perm_simp)
wenzelm@36278	3221	apply(rule_tac x="rev pi\<bullet>a" in exI)
wenzelm@36278	3222	apply(rule_tac x="rev pi\<bullet>M" in exI)
wenzelm@36278	3223	apply(simp)
wenzelm@36278	3224	apply(auto)[1]
wenzelm@36278	3225	apply(drule_tac x="pi\<bullet>x" in spec)
wenzelm@36278	3226	apply(drule_tac x="pi\<bullet>P" in spec)
wenzelm@36278	3227	apply(drule mp)
wenzelm@36278	3228	apply(force)
wenzelm@36278	3229	apply(drule_tac ?pi2.0="rev pi" in SNa_eqvt(2))
wenzelm@36278	3230	apply(perm_simp add: csubst_eqvt)
wenzelm@36278	3231	done
wenzelm@36278	3232
wenzelm@36278	3233	lemma CAND_eqvt_name:
wenzelm@36278	3234	fixes pi::"name prm"
wenzelm@36278	3235	shows "(pi\<bullet>(\<parallel>(B)\<parallel>)) = (\<parallel>(B)\<parallel>)"
wenzelm@36278	3236	and "(pi\<bullet>(\<parallel><B>\<parallel>)) = (\<parallel><B>\<parallel>)"
wenzelm@36278	3237	proof (nominal_induct B rule: ty.strong_induct)
wenzelm@36278	3238	case (PR X)
wenzelm@36278	3239	{ case 1 show ?case
wenzelm@36278	3240	apply -
wenzelm@36278	3241	apply(simp add: lfp_eqvt)
wenzelm@46895	3242	apply(simp add: perm_fun_def)
wenzelm@36278	3243	apply(simp add: union_eqvt AXIOMS_eqvt_name BINDING_eqvt_name)
wenzelm@36278	3244	apply(perm_simp)
wenzelm@36278	3245	done
wenzelm@36278	3246	next
wenzelm@36278	3247	case 2 show ?case
wenzelm@36278	3248	apply -
wenzelm@36278	3249	apply(simp only: NEGc_simps)
wenzelm@36278	3250	apply(simp add: union_eqvt AXIOMS_eqvt_name BINDING_eqvt_name)
wenzelm@36278	3251	apply(simp add: lfp_eqvt)
wenzelm@36278	3252	apply(simp add: comp_def)
wenzelm@46895	3253	apply(simp add: perm_fun_def)
wenzelm@36278	3254	apply(simp add: union_eqvt AXIOMS_eqvt_name BINDING_eqvt_name)
wenzelm@36278	3255	apply(perm_simp)
wenzelm@36278	3256	done
wenzelm@36278	3257	}
wenzelm@36278	3258	next
wenzelm@36278	3259	case (NOT B)
wenzelm@36278	3260	have ih1: "pi\<bullet>(\<parallel>(B)\<parallel>) = (\<parallel>(B)\<parallel>)" by fact
wenzelm@36278	3261	have ih2: "pi\<bullet>(\<parallel><B>\<parallel>) = (\<parallel><B>\<parallel>)" by fact
wenzelm@36278	3262	have g: "pi\<bullet>(\<parallel>(NOT B)\<parallel>) = (\<parallel>(NOT B)\<parallel>)"
wenzelm@36278	3263	apply -
wenzelm@36278	3264	apply(simp only: lfp_eqvt)
wenzelm@36278	3265	apply(simp only: comp_def)
wenzelm@46895	3266	apply(simp only: perm_fun_def)
wenzelm@36278	3267	apply(simp only: NEGc.simps NEGn.simps)
wenzelm@36278	3268	apply(simp only: union_eqvt AXIOMS_eqvt_name BINDING_eqvt_name NOTRIGHT_eqvt_name NOTLEFT_eqvt_name)
wenzelm@36278	3269	apply(perm_simp add: ih1 ih2)
wenzelm@36278	3270	done
wenzelm@36278	3271	{ case 1 show ?case by (rule g)
wenzelm@36278	3272	next
wenzelm@36278	3273	case 2 show ?case
wenzelm@36278	3274	by (simp only: NEGc_simps union_eqvt AXIOMS_eqvt_name BINDING_eqvt_name NOTRIGHT_eqvt_name ih1 ih2 g)
wenzelm@36278	3275	}
wenzelm@36278	3276	next
wenzelm@36278	3277	case (AND A B)
wenzelm@36278	3278	have ih1: "pi\<bullet>(\<parallel>(A)\<parallel>) = (\<parallel>(A)\<parallel>)" by fact
wenzelm@36278	3279	have ih2: "pi\<bullet>(\<parallel><A>\<parallel>) = (\<parallel><A>\<parallel>)" by fact
wenzelm@36278	3280	have ih3: "pi\<bullet>(\<parallel>(B)\<parallel>) = (\<parallel>(B)\<parallel>)" by fact
wenzelm@36278	3281	have ih4: "pi\<bullet>(\<parallel><B>\<parallel>) = (\<parallel><B>\<parallel>)" by fact
wenzelm@36278	3282	have g: "pi\<bullet>(\<parallel>(A AND B)\<parallel>) = (\<parallel>(A AND B)\<parallel>)"
wenzelm@36278	3283	apply -
wenzelm@36278	3284	apply(simp only: lfp_eqvt)
wenzelm@36278	3285	apply(simp only: comp_def)
wenzelm@46895	3286	apply(simp only: perm_fun_def)
wenzelm@36278	3287	apply(simp only: NEGc.simps NEGn.simps)
wenzelm@36278	3288	apply(simp only: union_eqvt AXIOMS_eqvt_name BINDING_eqvt_name ANDRIGHT_eqvt_name
wenzelm@36278	3289	ANDLEFT2_eqvt_name ANDLEFT1_eqvt_name)
wenzelm@36278	3290	apply(perm_simp add: ih1 ih2 ih3 ih4)
wenzelm@36278	3291	done
wenzelm@36278	3292	{ case 1 show ?case by (rule g)
wenzelm@36278	3293	next
wenzelm@36278	3294	case 2 show ?case
wenzelm@36278	3295	by (simp only: NEGc_simps union_eqvt AXIOMS_eqvt_name BINDING_eqvt_name
wenzelm@36278	3296	ANDRIGHT_eqvt_name ANDLEFT1_eqvt_name ANDLEFT2_eqvt_name ih1 ih2 ih3 ih4 g)
wenzelm@36278	3297	}
wenzelm@36278	3298	next
wenzelm@36278	3299	case (OR A B)
wenzelm@36278	3300	have ih1: "pi\<bullet>(\<parallel>(A)\<parallel>) = (\<parallel>(A)\<parallel>)" by fact
wenzelm@36278	3301	have ih2: "pi\<bullet>(\<parallel><A>\<parallel>) = (\<parallel><A>\<parallel>)" by fact
wenzelm@36278	3302	have ih3: "pi\<bullet>(\<parallel>(B)\<parallel>) = (\<parallel>(B)\<parallel>)" by fact
wenzelm@36278	3303	have ih4: "pi\<bullet>(\<parallel><B>\<parallel>) = (\<parallel><B>\<parallel>)" by fact
wenzelm@36278	3304	have g: "pi\<bullet>(\<parallel>(A OR B)\<parallel>) = (\<parallel>(A OR B)\<parallel>)"
wenzelm@36278	3305	apply -
wenzelm@36278	3306	apply(simp only: lfp_eqvt)
wenzelm@36278	3307	apply(simp only: comp_def)
wenzelm@46895	3308	apply(simp only: perm_fun_def)
wenzelm@36278	3309	apply(simp only: NEGc.simps NEGn.simps)
wenzelm@36278	3310	apply(simp only: union_eqvt AXIOMS_eqvt_name BINDING_eqvt_name ORRIGHT1_eqvt_name
wenzelm@36278	3311	ORRIGHT2_eqvt_name ORLEFT_eqvt_name)
wenzelm@36278	3312	apply(perm_simp add: ih1 ih2 ih3 ih4)
wenzelm@36278	3313	done
wenzelm@36278	3314	{ case 1 show ?case by (rule g)
wenzelm@36278	3315	next
wenzelm@36278	3316	case 2 show ?case
wenzelm@36278	3317	by (simp only: NEGc_simps union_eqvt AXIOMS_eqvt_name BINDING_eqvt_name
wenzelm@36278	3318	ORRIGHT1_eqvt_name ORRIGHT2_eqvt_name ORLEFT_eqvt_name ih1 ih2 ih3 ih4 g)
wenzelm@36278	3319	}
wenzelm@36278	3320	next
wenzelm@36278	3321	case (IMP A B)
wenzelm@36278	3322	have ih1: "pi\<bullet>(\<parallel>(A)\<parallel>) = (\<parallel>(A)\<parallel>)" by fact
wenzelm@36278	3323	have ih2: "pi\<bullet>(\<parallel><A>\<parallel>) = (\<parallel><A>\<parallel>)" by fact
wenzelm@36278	3324	have ih3: "pi\<bullet>(\<parallel>(B)\<parallel>) = (\<parallel>(B)\<parallel>)" by fact
wenzelm@36278	3325	have ih4: "pi\<bullet>(\<parallel><B>\<parallel>) = (\<parallel><B>\<parallel>)" by fact
wenzelm@36278	3326	have g: "pi\<bullet>(\<parallel>(A IMP B)\<parallel>) = (\<parallel>(A IMP B)\<parallel>)"
wenzelm@36278	3327	apply -
wenzelm@36278	3328	apply(simp only: lfp_eqvt)
wenzelm@36278	3329	apply(simp only: comp_def)
wenzelm@46895	3330	apply(simp only: perm_fun_def)
wenzelm@36278	3331	apply(simp only: NEGc.simps NEGn.simps)
wenzelm@36278	3332	apply(simp only: union_eqvt AXIOMS_eqvt_name BINDING_eqvt_name IMPRIGHT_eqvt_name IMPLEFT_eqvt_name)
wenzelm@36278	3333	apply(perm_simp add: ih1 ih2 ih3 ih4)
wenzelm@36278	3334	done
wenzelm@36278	3335	{ case 1 show ?case by (rule g)
wenzelm@36278	3336	next
wenzelm@36278	3337	case 2 show ?case
wenzelm@36278	3338	by (simp only: NEGc_simps union_eqvt AXIOMS_eqvt_name BINDING_eqvt_name
wenzelm@36278	3339	IMPRIGHT_eqvt_name IMPLEFT_eqvt_name ih1 ih2 ih3 ih4 g)
wenzelm@36278	3340	}
wenzelm@36278	3341	qed
wenzelm@36278	3342
wenzelm@36278	3343	lemma CAND_eqvt_coname:
wenzelm@36278	3344	fixes pi::"coname prm"
wenzelm@36278	3345	shows "(pi\<bullet>(\<parallel>(B)\<parallel>)) = (\<parallel>(B)\<parallel>)"
wenzelm@36278	3346	and "(pi\<bullet>(\<parallel><B>\<parallel>)) = (\<parallel><B>\<parallel>)"
wenzelm@36278	3347	proof (nominal_induct B rule: ty.strong_induct)
wenzelm@36278	3348	case (PR X)
wenzelm@36278	3349	{ case 1 show ?case
wenzelm@36278	3350	apply -
wenzelm@36278	3351	apply(simp add: lfp_eqvt)
wenzelm@46895	3352	apply(simp add: perm_fun_def)
wenzelm@36278	3353	apply(simp add: union_eqvt AXIOMS_eqvt_coname BINDING_eqvt_coname)
wenzelm@36278	3354	apply(perm_simp)
wenzelm@36278	3355	done
wenzelm@36278	3356	next
wenzelm@36278	3357	case 2 show ?case
wenzelm@36278	3358	apply -
wenzelm@36278	3359	apply(simp only: NEGc_simps)
wenzelm@36278	3360	apply(simp add: union_eqvt AXIOMS_eqvt_coname BINDING_eqvt_coname)
wenzelm@36278	3361	apply(simp add: lfp_eqvt)
wenzelm@36278	3362	apply(simp add: comp_def)
wenzelm@46895	3363	apply(simp add: perm_fun_def)
wenzelm@36278	3364	apply(simp add: union_eqvt AXIOMS_eqvt_coname BINDING_eqvt_coname)
wenzelm@36278	3365	apply(perm_simp)
wenzelm@36278	3366	done
wenzelm@36278	3367	}
wenzelm@36278	3368	next
wenzelm@36278	3369	case (NOT B)
wenzelm@36278	3370	have ih1: "pi\<bullet>(\<parallel>(B)\<parallel>) = (\<parallel>(B)\<parallel>)" by fact
wenzelm@36278	3371	have ih2: "pi\<bullet>(\<parallel><B>\<parallel>) = (\<parallel><B>\<parallel>)" by fact
wenzelm@36278	3372	have g: "pi\<bullet>(\<parallel>(NOT B)\<parallel>) = (\<parallel>(NOT B)\<parallel>)"
wenzelm@36278	3373	apply -
wenzelm@36278	3374	apply(simp only: lfp_eqvt)
wenzelm@36278	3375	apply(simp only: comp_def)
wenzelm@46895	3376	apply(simp only: perm_fun_def)
wenzelm@36278	3377	apply(simp only: NEGc.simps NEGn.simps)
wenzelm@36278	3378	apply(simp only: union_eqvt AXIOMS_eqvt_coname BINDING_eqvt_coname
wenzelm@36278	3379	NOTRIGHT_eqvt_coname NOTLEFT_eqvt_coname)
wenzelm@36278	3380	apply(perm_simp add: ih1 ih2)
wenzelm@36278	3381	done
wenzelm@36278	3382	{ case 1 show ?case by (rule g)
wenzelm@36278	3383	next
wenzelm@36278	3384	case 2 show ?case
wenzelm@36278	3385	by (simp only: NEGc_simps union_eqvt AXIOMS_eqvt_coname BINDING_eqvt_coname
wenzelm@36278	3386	NOTRIGHT_eqvt_coname ih1 ih2 g)
wenzelm@36278	3387	}
wenzelm@36278	3388	next
wenzelm@36278	3389	case (AND A B)
wenzelm@36278	3390	have ih1: "pi\<bullet>(\<parallel>(A)\<parallel>) = (\<parallel>(A)\<parallel>)" by fact
wenzelm@36278	3391	have ih2: "pi\<bullet>(\<parallel><A>\<parallel>) = (\<parallel><A>\<parallel>)" by fact
wenzelm@36278	3392	have ih3: "pi\<bullet>(\<parallel>(B)\<parallel>) = (\<parallel>(B)\<parallel>)" by fact
wenzelm@36278	3393	have ih4: "pi\<bullet>(\<parallel><B>\<parallel>) = (\<parallel><B>\<parallel>)" by fact
wenzelm@36278	3394	have g: "pi\<bullet>(\<parallel>(A AND B)\<parallel>) = (\<parallel>(A AND B)\<parallel>)"
wenzelm@36278	3395	apply -
wenzelm@36278	3396	apply(simp only: lfp_eqvt)
wenzelm@36278	3397	apply(simp only: comp_def)
wenzelm@46895	3398	apply(simp only: perm_fun_def)
wenzelm@36278	3399	apply(simp only: NEGc.simps NEGn.simps)
wenzelm@36278	3400	apply(simp only: union_eqvt AXIOMS_eqvt_coname BINDING_eqvt_coname ANDRIGHT_eqvt_coname
wenzelm@36278	3401	ANDLEFT2_eqvt_coname ANDLEFT1_eqvt_coname)
wenzelm@36278	3402	apply(perm_simp add: ih1 ih2 ih3 ih4)
wenzelm@36278	3403	done
wenzelm@36278	3404	{ case 1 show ?case by (rule g)
wenzelm@36278	3405	next
wenzelm@36278	3406	case 2 show ?case
wenzelm@36278	3407	by (simp only: NEGc_simps union_eqvt AXIOMS_eqvt_coname BINDING_eqvt_coname
wenzelm@36278	3408	ANDRIGHT_eqvt_coname ANDLEFT1_eqvt_coname ANDLEFT2_eqvt_coname ih1 ih2 ih3 ih4 g)
wenzelm@36278	3409	}
wenzelm@36278	3410	next
wenzelm@36278	3411	case (OR A B)
wenzelm@36278	3412	have ih1: "pi\<bullet>(\<parallel>(A)\<parallel>) = (\<parallel>(A)\<parallel>)" by fact
wenzelm@36278	3413	have ih2: "pi\<bullet>(\<parallel><A>\<parallel>) = (\<parallel><A>\<parallel>)" by fact
wenzelm@36278	3414	have ih3: "pi\<bullet>(\<parallel>(B)\<parallel>) = (\<parallel>(B)\<parallel>)" by fact
wenzelm@36278	3415	have ih4: "pi\<bullet>(\<parallel><B>\<parallel>) = (\<parallel><B>\<parallel>)" by fact
wenzelm@36278	3416	have g: "pi\<bullet>(\<parallel>(A OR B)\<parallel>) = (\<parallel>(A OR B)\<parallel>)"
wenzelm@36278	3417	apply -
wenzelm@36278	3418	apply(simp only: lfp_eqvt)
wenzelm@36278	3419	apply(simp only: comp_def)
wenzelm@46895	3420	apply(simp only: perm_fun_def)
wenzelm@36278	3421	apply(simp only: NEGc.simps NEGn.simps)
wenzelm@36278	3422	apply(simp only: union_eqvt AXIOMS_eqvt_coname BINDING_eqvt_coname ORRIGHT1_eqvt_coname
wenzelm@36278	3423	ORRIGHT2_eqvt_coname ORLEFT_eqvt_coname)
wenzelm@36278	3424	apply(perm_simp add: ih1 ih2 ih3 ih4)
wenzelm@36278	3425	done
wenzelm@36278	3426	{ case 1 show ?case by (rule g)
wenzelm@36278	3427	next
wenzelm@36278	3428	case 2 show ?case
wenzelm@36278	3429	by (simp only: NEGc_simps union_eqvt AXIOMS_eqvt_coname BINDING_eqvt_coname
wenzelm@36278	3430	ORRIGHT1_eqvt_coname ORRIGHT2_eqvt_coname ORLEFT_eqvt_coname ih1 ih2 ih3 ih4 g)
wenzelm@36278	3431	}
wenzelm@36278	3432	next
wenzelm@36278	3433	case (IMP A B)
wenzelm@36278	3434	have ih1: "pi\<bullet>(\<parallel>(A)\<parallel>) = (\<parallel>(A)\<parallel>)" by fact
wenzelm@36278	3435	have ih2: "pi\<bullet>(\<parallel><A>\<parallel>) = (\<parallel><A>\<parallel>)" by fact
wenzelm@36278	3436	have ih3: "pi\<bullet>(\<parallel>(B)\<parallel>) = (\<parallel>(B)\<parallel>)" by fact
wenzelm@36278	3437	have ih4: "pi\<bullet>(\<parallel><B>\<parallel>) = (\<parallel><B>\<parallel>)" by fact
wenzelm@36278	3438	have g: "pi\<bullet>(\<parallel>(A IMP B)\<parallel>) = (\<parallel>(A IMP B)\<parallel>)"
wenzelm@36278	3439	apply -
wenzelm@36278	3440	apply(simp only: lfp_eqvt)
wenzelm@36278	3441	apply(simp only: comp_def)
wenzelm@46895	3442	apply(simp only: perm_fun_def)
wenzelm@36278	3443	apply(simp only: NEGc.simps NEGn.simps)
wenzelm@36278	3444	apply(simp only: union_eqvt AXIOMS_eqvt_coname BINDING_eqvt_coname IMPRIGHT_eqvt_coname
wenzelm@36278	3445	IMPLEFT_eqvt_coname)
wenzelm@36278	3446	apply(perm_simp add: ih1 ih2 ih3 ih4)
wenzelm@36278	3447	done
wenzelm@36278	3448	{ case 1 show ?case by (rule g)
wenzelm@36278	3449	next
wenzelm@36278	3450	case 2 show ?case
wenzelm@36278	3451	by (simp only: NEGc_simps union_eqvt AXIOMS_eqvt_coname BINDING_eqvt_coname
wenzelm@36278	3452	IMPRIGHT_eqvt_coname IMPLEFT_eqvt_coname ih1 ih2 ih3 ih4 g)
wenzelm@36278	3453	}
wenzelm@36278	3454	qed
wenzelm@36278	3455
wenzelm@36278	3456	text {* Elimination rules for the set-operators *}
wenzelm@36278	3457
wenzelm@36278	3458	lemma BINDINGc_elim:
wenzelm@36278	3459	assumes a: "<a>:M \<in> BINDINGc B (\<parallel>(B)\<parallel>)"
wenzelm@36278	3460	shows "\<forall>x P. ((x):P)\<in>(\<parallel>(B)\<parallel>) \<longrightarrow> SNa (M{a:=(x).P})"
wenzelm@36278	3461	using a
wenzelm@36278	3462	apply(auto simp add: BINDINGc_def)
wenzelm@36278	3463	apply(auto simp add: ctrm.inject alpha)
wenzelm@36278	3464	apply(drule_tac x="[(a,aa)]\<bullet>x" in spec)
wenzelm@36278	3465	apply(drule_tac x="[(a,aa)]\<bullet>P" in spec)
wenzelm@36278	3466	apply(drule mp)
wenzelm@36278	3467	apply(drule_tac pi="[(a,aa)]" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	3468	apply(simp add: CAND_eqvt_coname)
wenzelm@36278	3469	apply(drule_tac ?pi2.0="[(a,aa)]" in SNa_eqvt(2))
wenzelm@36278	3470	apply(perm_simp add: csubst_eqvt)
wenzelm@36278	3471	done
wenzelm@36278	3472
wenzelm@36278	3473	lemma BINDINGn_elim:
wenzelm@36278	3474	assumes a: "(x):M \<in> BINDINGn B (\<parallel><B>\<parallel>)"
wenzelm@36278	3475	shows "\<forall>c P. (<c>:P)\<in>(\<parallel><B>\<parallel>) \<longrightarrow> SNa (M{x:=<c>.P})"
wenzelm@36278	3476	using a
wenzelm@36278	3477	apply(auto simp add: BINDINGn_def)
wenzelm@36278	3478	apply(auto simp add: ntrm.inject alpha)
wenzelm@36278	3479	apply(drule_tac x="[(x,xa)]\<bullet>c" in spec)
wenzelm@36278	3480	apply(drule_tac x="[(x,xa)]\<bullet>P" in spec)
wenzelm@36278	3481	apply(drule mp)
wenzelm@36278	3482	apply(drule_tac pi="[(x,xa)]" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	3483	apply(simp add: CAND_eqvt_name)
wenzelm@36278	3484	apply(drule_tac ?pi1.0="[(x,xa)]" in SNa_eqvt(1))
wenzelm@36278	3485	apply(perm_simp add: nsubst_eqvt)
wenzelm@36278	3486	done
wenzelm@36278	3487
wenzelm@36278	3488	lemma NOTRIGHT_elim:
wenzelm@36278	3489	assumes a: "<a>:M \<in> NOTRIGHT (NOT B) (\<parallel>(B)\<parallel>)"
wenzelm@36278	3490	obtains x' M' where "M = NotR (x').M' a" and "fic (NotR (x').M' a) a" and "(x'):M' \<in> (\<parallel>(B)\<parallel>)"
wenzelm@36278	3491	using a
wenzelm@36278	3492	apply(auto simp add: ctrm.inject alpha abs_fresh calc_atm)
wenzelm@36278	3493	apply(drule_tac x="x" in meta_spec)
thomas@58834	3494	apply(drule_tac x="[(a,aa)]\<bullet>Ma" in meta_spec)
wenzelm@36278	3495	apply(simp)
wenzelm@36278	3496	apply(drule meta_mp)
wenzelm@36278	3497	apply(drule_tac pi="[(a,aa)]" in fic.eqvt(2))
wenzelm@36278	3498	apply(simp add: calc_atm)
wenzelm@36278	3499	apply(drule meta_mp)
wenzelm@36278	3500	apply(drule_tac pi="[(a,aa)]" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	3501	apply(simp add: calc_atm CAND_eqvt_coname)
wenzelm@36278	3502	apply(simp)
wenzelm@36278	3503	done
wenzelm@36278	3504
wenzelm@36278	3505	lemma NOTLEFT_elim:
wenzelm@36278	3506	assumes a: "(x):M \<in> NOTLEFT (NOT B) (\<parallel><B>\<parallel>)"
wenzelm@36278	3507	obtains a' M' where "M = NotL <a'>.M' x" and "fin (NotL <a'>.M' x) x" and "<a'>:M' \<in> (\<parallel><B>\<parallel>)"
wenzelm@36278	3508	using a
wenzelm@36278	3509	apply(auto simp add: ntrm.inject alpha abs_fresh calc_atm)
wenzelm@36278	3510	apply(drule_tac x="a" in meta_spec)
thomas@58834	3511	apply(drule_tac x="[(x,xa)]\<bullet>Ma" in meta_spec)
wenzelm@36278	3512	apply(simp)
wenzelm@36278	3513	apply(drule meta_mp)
wenzelm@36278	3514	apply(drule_tac pi="[(x,xa)]" in fin.eqvt(1))
wenzelm@36278	3515	apply(simp add: calc_atm)
wenzelm@36278	3516	apply(drule meta_mp)
wenzelm@36278	3517	apply(drule_tac pi="[(x,xa)]" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	3518	apply(simp add: calc_atm CAND_eqvt_name)
wenzelm@36278	3519	apply(simp)
wenzelm@36278	3520	done
wenzelm@36278	3521
wenzelm@36278	3522	lemma ANDRIGHT_elim:
wenzelm@36278	3523	assumes a: "<a>:M \<in> ANDRIGHT (B AND C) (\<parallel><B>\<parallel>) (\<parallel><C>\<parallel>)"
wenzelm@36278	3524	obtains d' M' e' N' where "M = AndR <d'>.M' <e'>.N' a" and "fic (AndR <d'>.M' <e'>.N' a) a"
wenzelm@36278	3525	and "<d'>:M' \<in> (\<parallel><B>\<parallel>)" and "<e'>:N' \<in> (\<parallel><C>\<parallel>)"
wenzelm@36278	3526	using a
wenzelm@36278	3527	apply(auto simp add: ctrm.inject alpha abs_fresh calc_atm fresh_atm)
wenzelm@36278	3528	apply(drule_tac x="c" in meta_spec)
thomas@58834	3529	apply(drule_tac x="[(a,c)]\<bullet>Ma" in meta_spec)
wenzelm@36278	3530	apply(drule_tac x="c" in meta_spec)
wenzelm@36278	3531	apply(drule_tac x="[(a,c)]\<bullet>N" in meta_spec)
wenzelm@36278	3532	apply(simp)
wenzelm@36278	3533	apply(drule meta_mp)
wenzelm@36278	3534	apply(drule_tac pi="[(a,c)]" in fic.eqvt(2))
wenzelm@36278	3535	apply(simp add: calc_atm)
wenzelm@36278	3536	apply(drule meta_mp)
wenzelm@36278	3537	apply(drule_tac pi="[(a,c)]" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	3538	apply(simp add: calc_atm CAND_eqvt_coname)
wenzelm@36278	3539	apply(drule meta_mp)
wenzelm@36278	3540	apply(drule_tac pi="[(a,c)]" and x="<a>:N" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	3541	apply(simp add: calc_atm CAND_eqvt_coname)
wenzelm@36278	3542	apply(simp)
wenzelm@36278	3543	apply(case_tac "a=b")
wenzelm@36278	3544	apply(simp)
wenzelm@36278	3545	apply(drule_tac x="c" in meta_spec)
thomas@58834	3546	apply(drule_tac x="[(b,c)]\<bullet>Ma" in meta_spec)
wenzelm@36278	3547	apply(drule_tac x="c" in meta_spec)
wenzelm@36278	3548	apply(drule_tac x="[(b,c)]\<bullet>N" in meta_spec)
wenzelm@36278	3549	apply(simp)
wenzelm@36278	3550	apply(drule meta_mp)
wenzelm@36278	3551	apply(drule_tac pi="[(b,c)]" in fic.eqvt(2))
wenzelm@36278	3552	apply(simp add: calc_atm)
wenzelm@36278	3553	apply(drule meta_mp)
wenzelm@36278	3554	apply(drule_tac pi="[(b,c)]" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	3555	apply(simp add: calc_atm CAND_eqvt_coname)
wenzelm@36278	3556	apply(drule meta_mp)
wenzelm@36278	3557	apply(drule_tac pi="[(b,c)]" and x="<b>:N" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	3558	apply(simp add: calc_atm CAND_eqvt_coname)
wenzelm@36278	3559	apply(simp)
wenzelm@36278	3560	apply(simp)
wenzelm@36278	3561	apply(case_tac "c=b")
wenzelm@36278	3562	apply(simp)
wenzelm@36278	3563	apply(drule_tac x="b" in meta_spec)
thomas@58834	3564	apply(drule_tac x="[(a,b)]\<bullet>Ma" in meta_spec)
wenzelm@36278	3565	apply(drule_tac x="a" in meta_spec)
wenzelm@36278	3566	apply(drule_tac x="[(a,b)]\<bullet>N" in meta_spec)
wenzelm@36278	3567	apply(simp)
wenzelm@36278	3568	apply(drule meta_mp)
wenzelm@36278	3569	apply(drule_tac pi="[(a,b)]" in fic.eqvt(2))
wenzelm@36278	3570	apply(simp add: calc_atm)
wenzelm@36278	3571	apply(drule meta_mp)
wenzelm@36278	3572	apply(drule_tac pi="[(a,b)]" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	3573	apply(simp add: calc_atm CAND_eqvt_coname)
wenzelm@36278	3574	apply(drule meta_mp)
wenzelm@36278	3575	apply(drule_tac pi="[(a,b)]" and x="<b>:N" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	3576	apply(simp add: calc_atm CAND_eqvt_coname)
wenzelm@36278	3577	apply(simp)
wenzelm@36278	3578	apply(simp)
wenzelm@36278	3579	apply(drule_tac x="c" in meta_spec)
thomas@58834	3580	apply(drule_tac x="[(a,c)]\<bullet>Ma" in meta_spec)
wenzelm@36278	3581	apply(drule_tac x="b" in meta_spec)
wenzelm@36278	3582	apply(drule_tac x="[(a,c)]\<bullet>N" in meta_spec)
wenzelm@36278	3583	apply(simp)
wenzelm@36278	3584	apply(drule meta_mp)
wenzelm@36278	3585	apply(drule_tac pi="[(a,c)]" in fic.eqvt(2))
wenzelm@36278	3586	apply(simp add: calc_atm)
wenzelm@36278	3587	apply(drule meta_mp)
wenzelm@36278	3588	apply(drule_tac pi="[(a,c)]" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	3589	apply(simp add: calc_atm CAND_eqvt_coname)
wenzelm@36278	3590	apply(drule meta_mp)
wenzelm@36278	3591	apply(drule_tac pi="[(a,c)]" and x="<b>:N" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	3592	apply(simp add: calc_atm CAND_eqvt_coname)
wenzelm@36278	3593	apply(simp)
wenzelm@36278	3594	apply(case_tac "a=aa")
wenzelm@36278	3595	apply(simp)
wenzelm@36278	3596	apply(drule_tac x="c" in meta_spec)
thomas@58834	3597	apply(drule_tac x="[(aa,c)]\<bullet>Ma" in meta_spec)
wenzelm@36278	3598	apply(drule_tac x="c" in meta_spec)
wenzelm@36278	3599	apply(drule_tac x="[(aa,c)]\<bullet>N" in meta_spec)
wenzelm@36278	3600	apply(simp)
wenzelm@36278	3601	apply(drule meta_mp)
wenzelm@36278	3602	apply(drule_tac pi="[(aa,c)]" in fic.eqvt(2))
wenzelm@36278	3603	apply(simp add: calc_atm)
wenzelm@36278	3604	apply(drule meta_mp)
wenzelm@36278	3605	apply(drule_tac pi="[(aa,c)]" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	3606	apply(simp add: calc_atm CAND_eqvt_coname)
wenzelm@36278	3607	apply(drule meta_mp)
wenzelm@36278	3608	apply(drule_tac pi="[(aa,c)]" and x="<aa>:N" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	3609	apply(simp add: calc_atm CAND_eqvt_coname)
wenzelm@36278	3610	apply(simp)
wenzelm@36278	3611	apply(simp)
wenzelm@36278	3612	apply(case_tac "c=aa")
wenzelm@36278	3613	apply(simp)
wenzelm@36278	3614	apply(drule_tac x="a" in meta_spec)
thomas@58834	3615	apply(drule_tac x="[(a,aa)]\<bullet>Ma" in meta_spec)
wenzelm@36278	3616	apply(drule_tac x="aa" in meta_spec)
wenzelm@36278	3617	apply(drule_tac x="[(a,aa)]\<bullet>N" in meta_spec)
wenzelm@36278	3618	apply(simp)
wenzelm@36278	3619	apply(drule meta_mp)
wenzelm@36278	3620	apply(drule_tac pi="[(a,aa)]" in fic.eqvt(2))
wenzelm@36278	3621	apply(simp add: calc_atm)
wenzelm@36278	3622	apply(drule meta_mp)
wenzelm@36278	3623	apply(drule_tac pi="[(a,aa)]" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	3624	apply(simp add: calc_atm CAND_eqvt_coname)
wenzelm@36278	3625	apply(drule meta_mp)
wenzelm@36278	3626	apply(drule_tac pi="[(a,aa)]" and x="<a>:N" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	3627	apply(simp add: calc_atm CAND_eqvt_coname)
wenzelm@36278	3628	apply(simp)
wenzelm@36278	3629	apply(simp)
wenzelm@36278	3630	apply(drule_tac x="aa" in meta_spec)
thomas@58834	3631	apply(drule_tac x="[(a,c)]\<bullet>Ma" in meta_spec)
wenzelm@36278	3632	apply(drule_tac x="c" in meta_spec)
wenzelm@36278	3633	apply(drule_tac x="[(a,c)]\<bullet>N" in meta_spec)
wenzelm@36278	3634	apply(simp)
wenzelm@36278	3635	apply(drule meta_mp)
wenzelm@36278	3636	apply(drule_tac pi="[(a,c)]" in fic.eqvt(2))
wenzelm@36278	3637	apply(simp add: calc_atm)
wenzelm@36278	3638	apply(drule meta_mp)
wenzelm@36278	3639	apply(drule_tac pi="[(a,c)]" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	3640	apply(simp add: calc_atm CAND_eqvt_coname)
wenzelm@36278	3641	apply(drule meta_mp)
wenzelm@36278	3642	apply(drule_tac pi="[(a,c)]" and x="<a>:N" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	3643	apply(simp add: calc_atm CAND_eqvt_coname)
wenzelm@36278	3644	apply(simp)
wenzelm@36278	3645	apply(case_tac "a=aa")
wenzelm@36278	3646	apply(simp)
wenzelm@36278	3647	apply(case_tac "aa=b")
wenzelm@36278	3648	apply(simp)
wenzelm@36278	3649	apply(drule_tac x="c" in meta_spec)
thomas@58834	3650	apply(drule_tac x="[(b,c)]\<bullet>Ma" in meta_spec)
wenzelm@36278	3651	apply(drule_tac x="c" in meta_spec)
wenzelm@36278	3652	apply(drule_tac x="[(b,c)]\<bullet>N" in meta_spec)
wenzelm@36278	3653	apply(simp)
wenzelm@36278	3654	apply(drule meta_mp)
wenzelm@36278	3655	apply(drule_tac pi="[(b,c)]" in fic.eqvt(2))
wenzelm@36278	3656	apply(simp add: calc_atm)
wenzelm@36278	3657	apply(drule meta_mp)
wenzelm@36278	3658	apply(drule_tac pi="[(b,c)]" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	3659	apply(simp add: calc_atm CAND_eqvt_coname)
wenzelm@36278	3660	apply(drule meta_mp)
wenzelm@36278	3661	apply(drule_tac pi="[(b,c)]" and x="<b>:N" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	3662	apply(simp add: calc_atm CAND_eqvt_coname)
wenzelm@36278	3663	apply(simp)
wenzelm@36278	3664	apply(simp)
wenzelm@36278	3665	apply(case_tac "c=b")
wenzelm@36278	3666	apply(simp)
wenzelm@36278	3667	apply(drule_tac x="b" in meta_spec)
thomas@58834	3668	apply(drule_tac x="[(aa,b)]\<bullet>Ma" in meta_spec)
wenzelm@36278	3669	apply(drule_tac x="aa" in meta_spec)
wenzelm@36278	3670	apply(drule_tac x="[(aa,b)]\<bullet>N" in meta_spec)
wenzelm@36278	3671	apply(simp)
wenzelm@36278	3672	apply(drule meta_mp)
wenzelm@36278	3673	apply(drule_tac pi="[(aa,b)]" in fic.eqvt(2))
wenzelm@36278	3674	apply(simp add: calc_atm)
wenzelm@36278	3675	apply(drule meta_mp)
wenzelm@36278	3676	apply(drule_tac pi="[(aa,b)]" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	3677	apply(simp add: calc_atm CAND_eqvt_coname)
wenzelm@36278	3678	apply(drule meta_mp)
wenzelm@36278	3679	apply(drule_tac pi="[(aa,b)]" and x="<b>:N" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	3680	apply(simp add: calc_atm CAND_eqvt_coname)
wenzelm@36278	3681	apply(simp)
wenzelm@36278	3682	apply(simp)
wenzelm@36278	3683	apply(drule_tac x="c" in meta_spec)
thomas@58834	3684	apply(drule_tac x="[(aa,c)]\<bullet>Ma" in meta_spec)
wenzelm@36278	3685	apply(drule_tac x="b" in meta_spec)
wenzelm@36278	3686	apply(drule_tac x="[(aa,c)]\<bullet>N" in meta_spec)
wenzelm@36278	3687	apply(simp)
wenzelm@36278	3688	apply(drule meta_mp)
wenzelm@36278	3689	apply(drule_tac pi="[(aa,c)]" in fic.eqvt(2))
wenzelm@36278	3690	apply(simp add: calc_atm)
wenzelm@36278	3691	apply(drule meta_mp)
wenzelm@36278	3692	apply(drule_tac pi="[(aa,c)]" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	3693	apply(simp add: calc_atm CAND_eqvt_coname)
wenzelm@36278	3694	apply(drule meta_mp)
wenzelm@36278	3695	apply(drule_tac pi="[(aa,c)]" and x="<b>:N" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	3696	apply(simp add: calc_atm CAND_eqvt_coname)
wenzelm@36278	3697	apply(simp)
wenzelm@36278	3698	apply(simp)
wenzelm@36278	3699	apply(case_tac "c=aa")
wenzelm@36278	3700	apply(simp)
wenzelm@36278	3701	apply(case_tac "a=b")
wenzelm@36278	3702	apply(simp)
wenzelm@36278	3703	apply(drule_tac x="b" in meta_spec)
thomas@58834	3704	apply(drule_tac x="[(b,aa)]\<bullet>Ma" in meta_spec)
wenzelm@36278	3705	apply(drule_tac x="aa" in meta_spec)
wenzelm@36278	3706	apply(drule_tac x="[(b,aa)]\<bullet>N" in meta_spec)
wenzelm@36278	3707	apply(simp)
wenzelm@36278	3708	apply(drule meta_mp)
wenzelm@36278	3709	apply(drule_tac pi="[(b,aa)]" in fic.eqvt(2))
wenzelm@36278	3710	apply(simp add: calc_atm)
wenzelm@36278	3711	apply(drule meta_mp)
wenzelm@36278	3712	apply(drule_tac pi="[(b,aa)]" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	3713	apply(simp add: calc_atm CAND_eqvt_coname)
wenzelm@36278	3714	apply(drule meta_mp)
wenzelm@36278	3715	apply(drule_tac pi="[(b,aa)]" and x="<b>:N" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	3716	apply(simp add: calc_atm CAND_eqvt_coname)
wenzelm@36278	3717	apply(simp)
wenzelm@36278	3718	apply(simp)
wenzelm@36278	3719	apply(case_tac "aa=b")
wenzelm@36278	3720	apply(simp)
wenzelm@36278	3721	apply(drule_tac x="a" in meta_spec)
thomas@58834	3722	apply(drule_tac x="[(a,b)]\<bullet>Ma" in meta_spec)
wenzelm@36278	3723	apply(drule_tac x="a" in meta_spec)
wenzelm@36278	3724	apply(drule_tac x="[(a,b)]\<bullet>N" in meta_spec)
wenzelm@36278	3725	apply(simp)
wenzelm@36278	3726	apply(drule meta_mp)
wenzelm@36278	3727	apply(drule_tac pi="[(a,b)]" in fic.eqvt(2))
wenzelm@36278	3728	apply(simp add: calc_atm)
wenzelm@36278	3729	apply(drule meta_mp)
wenzelm@36278	3730	apply(drule_tac pi="[(a,b)]" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	3731	apply(simp add: calc_atm CAND_eqvt_coname)
wenzelm@36278	3732	apply(drule meta_mp)
wenzelm@36278	3733	apply(drule_tac pi="[(a,b)]" and x="<b>:N" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	3734	apply(simp add: calc_atm CAND_eqvt_coname)
wenzelm@36278	3735	apply(simp)
wenzelm@36278	3736	apply(simp)
wenzelm@36278	3737	apply(drule_tac x="a" in meta_spec)
thomas@58834	3738	apply(drule_tac x="[(a,aa)]\<bullet>Ma" in meta_spec)
wenzelm@36278	3739	apply(drule_tac x="b" in meta_spec)
wenzelm@36278	3740	apply(drule_tac x="[(a,aa)]\<bullet>N" in meta_spec)
wenzelm@36278	3741	apply(simp)
wenzelm@36278	3742	apply(drule meta_mp)
wenzelm@36278	3743	apply(drule_tac pi="[(a,aa)]" in fic.eqvt(2))
wenzelm@36278	3744	apply(simp add: calc_atm)
wenzelm@36278	3745	apply(drule meta_mp)
wenzelm@36278	3746	apply(drule_tac pi="[(a,aa)]" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	3747	apply(simp add: calc_atm CAND_eqvt_coname)
wenzelm@36278	3748	apply(drule meta_mp)
wenzelm@36278	3749	apply(drule_tac pi="[(a,aa)]" and x="<b>:N" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	3750	apply(simp add: calc_atm CAND_eqvt_coname)
wenzelm@36278	3751	apply(simp)
wenzelm@36278	3752	apply(simp)
wenzelm@36278	3753	apply(case_tac "a=b")
wenzelm@36278	3754	apply(simp)
wenzelm@36278	3755	apply(drule_tac x="aa" in meta_spec)
thomas@58834	3756	apply(drule_tac x="[(b,c)]\<bullet>Ma" in meta_spec)
wenzelm@36278	3757	apply(drule_tac x="c" in meta_spec)
wenzelm@36278	3758	apply(drule_tac x="[(b,c)]\<bullet>N" in meta_spec)
wenzelm@36278	3759	apply(simp)
wenzelm@36278	3760	apply(drule meta_mp)
wenzelm@36278	3761	apply(drule_tac pi="[(b,c)]" in fic.eqvt(2))
wenzelm@36278	3762	apply(simp add: calc_atm)
wenzelm@36278	3763	apply(drule meta_mp)
wenzelm@36278	3764	apply(drule_tac pi="[(b,c)]" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	3765	apply(simp add: calc_atm CAND_eqvt_coname)
wenzelm@36278	3766	apply(drule meta_mp)
wenzelm@36278	3767	apply(drule_tac pi="[(b,c)]" and x="<b>:N" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	3768	apply(simp add: calc_atm CAND_eqvt_coname)
wenzelm@36278	3769	apply(simp)
wenzelm@36278	3770	apply(simp)
wenzelm@36278	3771	apply(case_tac "c=b")
wenzelm@36278	3772	apply(simp)
wenzelm@36278	3773	apply(drule_tac x="aa" in meta_spec)
thomas@58834	3774	apply(drule_tac x="[(a,b)]\<bullet>Ma" in meta_spec)
wenzelm@36278	3775	apply(drule_tac x="a" in meta_spec)
wenzelm@36278	3776	apply(drule_tac x="[(a,b)]\<bullet>N" in meta_spec)
wenzelm@36278	3777	apply(simp)
wenzelm@36278	3778	apply(drule meta_mp)
wenzelm@36278	3779	apply(drule_tac pi="[(a,b)]" in fic.eqvt(2))
wenzelm@36278	3780	apply(simp add: calc_atm)
wenzelm@36278	3781	apply(drule meta_mp)
wenzelm@36278	3782	apply(drule_tac pi="[(a,b)]" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	3783	apply(simp add: calc_atm CAND_eqvt_coname)
wenzelm@36278	3784	apply(drule meta_mp)
wenzelm@36278	3785	apply(drule_tac pi="[(a,b)]" and x="<b>:N" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	3786	apply(simp add: calc_atm CAND_eqvt_coname)
wenzelm@36278	3787	apply(simp)
wenzelm@36278	3788	apply(simp)
wenzelm@36278	3789	apply(drule_tac x="aa" in meta_spec)
thomas@58834	3790	apply(drule_tac x="[(a,c)]\<bullet>Ma" in meta_spec)
wenzelm@36278	3791	apply(drule_tac x="b" in meta_spec)
wenzelm@36278	3792	apply(drule_tac x="[(a,c)]\<bullet>N" in meta_spec)
wenzelm@36278	3793	apply(simp)
wenzelm@36278	3794	apply(drule meta_mp)
wenzelm@36278	3795	apply(drule_tac pi="[(a,c)]" in fic.eqvt(2))
wenzelm@36278	3796	apply(simp add: calc_atm)
wenzelm@36278	3797	apply(drule meta_mp)
wenzelm@36278	3798	apply(drule_tac pi="[(a,c)]" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	3799	apply(simp add: calc_atm CAND_eqvt_coname)
wenzelm@36278	3800	apply(drule meta_mp)
wenzelm@36278	3801	apply(drule_tac pi="[(a,c)]" and x="<b>:N" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	3802	apply(simp add: calc_atm CAND_eqvt_coname)
wenzelm@36278	3803	apply(simp)
wenzelm@36278	3804	done
wenzelm@36278	3805
wenzelm@36278	3806	lemma ANDLEFT1_elim:
wenzelm@36278	3807	assumes a: "(x):M \<in> ANDLEFT1 (B AND C) (\<parallel>(B)\<parallel>)"
wenzelm@36278	3808	obtains x' M' where "M = AndL1 (x').M' x" and "fin (AndL1 (x').M' x) x" and "(x'):M' \<in> (\<parallel>(B)\<parallel>)"
thomas@58834	3809	using a [[ hypsubst_thin = true ]]
wenzelm@36278	3810	apply(auto simp add: ntrm.inject alpha abs_fresh calc_atm)
wenzelm@36278	3811	apply(drule_tac x="y" in meta_spec)
wenzelm@36278	3812	apply(drule_tac x="[(x,y)]\<bullet>M" in meta_spec)
wenzelm@36278	3813	apply(simp)
wenzelm@36278	3814	apply(drule meta_mp)
wenzelm@36278	3815	apply(drule_tac pi="[(x,y)]" in fin.eqvt(1))
wenzelm@36278	3816	apply(simp add: calc_atm)
wenzelm@36278	3817	apply(drule meta_mp)
wenzelm@36278	3818	apply(drule_tac pi="[(x,y)]" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	3819	apply(simp add: calc_atm CAND_eqvt_name)
wenzelm@36278	3820	apply(simp)
wenzelm@36278	3821	apply(case_tac "x=xa")
wenzelm@36278	3822	apply(simp)
wenzelm@36278	3823	apply(drule_tac x="y" in meta_spec)
wenzelm@36278	3824	apply(drule_tac x="[(xa,y)]\<bullet>M" in meta_spec)
wenzelm@36278	3825	apply(simp)
wenzelm@36278	3826	apply(drule meta_mp)
wenzelm@36278	3827	apply(drule_tac pi="[(xa,y)]" in fin.eqvt(1))
wenzelm@36278	3828	apply(simp add: calc_atm)
wenzelm@36278	3829	apply(drule meta_mp)
wenzelm@36278	3830	apply(drule_tac pi="[(xa,y)]" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	3831	apply(simp add: calc_atm CAND_eqvt_name)
wenzelm@36278	3832	apply(simp)
wenzelm@36278	3833	apply(simp)
wenzelm@36278	3834	apply(case_tac "y=xa")
wenzelm@36278	3835	apply(simp)
wenzelm@36278	3836	apply(drule_tac x="x" in meta_spec)
wenzelm@36278	3837	apply(drule_tac x="[(x,xa)]\<bullet>M" in meta_spec)
wenzelm@36278	3838	apply(simp)
wenzelm@36278	3839	apply(drule meta_mp)
wenzelm@36278	3840	apply(drule_tac pi="[(x,xa)]" in fin.eqvt(1))
wenzelm@36278	3841	apply(simp add: calc_atm)
wenzelm@36278	3842	apply(drule meta_mp)
wenzelm@36278	3843	apply(drule_tac pi="[(x,xa)]" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	3844	apply(simp add: calc_atm CAND_eqvt_name)
wenzelm@36278	3845	apply(simp)
wenzelm@36278	3846	apply(simp)
wenzelm@36278	3847	apply(drule_tac x="xa" in meta_spec)
wenzelm@36278	3848	apply(drule_tac x="[(x,y)]\<bullet>M" in meta_spec)
wenzelm@36278	3849	apply(simp)
wenzelm@36278	3850	apply(drule meta_mp)
wenzelm@36278	3851	apply(drule_tac pi="[(x,y)]" in fin.eqvt(1))
wenzelm@36278	3852	apply(simp add: calc_atm)
wenzelm@36278	3853	apply(drule meta_mp)
wenzelm@36278	3854	apply(drule_tac pi="[(x,y)]" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	3855	apply(simp add: calc_atm CAND_eqvt_name)
wenzelm@36278	3856	apply(simp)
wenzelm@36278	3857	done
wenzelm@36278	3858
wenzelm@36278	3859	lemma ANDLEFT2_elim:
wenzelm@36278	3860	assumes a: "(x):M \<in> ANDLEFT2 (B AND C) (\<parallel>(C)\<parallel>)"
wenzelm@36278	3861	obtains x' M' where "M = AndL2 (x').M' x" and "fin (AndL2 (x').M' x) x" and "(x'):M' \<in> (\<parallel>(C)\<parallel>)"
thomas@58834	3862	using a [[ hypsubst_thin = true ]]
wenzelm@36278	3863	apply(auto simp add: ntrm.inject alpha abs_fresh calc_atm)
wenzelm@36278	3864	apply(drule_tac x="y" in meta_spec)
wenzelm@36278	3865	apply(drule_tac x="[(x,y)]\<bullet>M" in meta_spec)
wenzelm@36278	3866	apply(simp)
wenzelm@36278	3867	apply(drule meta_mp)
wenzelm@36278	3868	apply(drule_tac pi="[(x,y)]" in fin.eqvt(1))
wenzelm@36278	3869	apply(simp add: calc_atm)
wenzelm@36278	3870	apply(drule meta_mp)
wenzelm@36278	3871	apply(drule_tac pi="[(x,y)]" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	3872	apply(simp add: calc_atm CAND_eqvt_name)
wenzelm@36278	3873	apply(simp)
wenzelm@36278	3874	apply(case_tac "x=xa")
wenzelm@36278	3875	apply(simp)
wenzelm@36278	3876	apply(drule_tac x="y" in meta_spec)
wenzelm@36278	3877	apply(drule_tac x="[(xa,y)]\<bullet>M" in meta_spec)
wenzelm@36278	3878	apply(simp)
wenzelm@36278	3879	apply(drule meta_mp)
wenzelm@36278	3880	apply(drule_tac pi="[(xa,y)]" in fin.eqvt(1))
wenzelm@36278	3881	apply(simp add: calc_atm)
wenzelm@36278	3882	apply(drule meta_mp)
wenzelm@36278	3883	apply(drule_tac pi="[(xa,y)]" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	3884	apply(simp add: calc_atm CAND_eqvt_name)
wenzelm@36278	3885	apply(simp)
wenzelm@36278	3886	apply(simp)
wenzelm@36278	3887	apply(case_tac "y=xa")
wenzelm@36278	3888	apply(simp)
wenzelm@36278	3889	apply(drule_tac x="x" in meta_spec)
wenzelm@36278	3890	apply(drule_tac x="[(x,xa)]\<bullet>M" in meta_spec)
wenzelm@36278	3891	apply(simp)
wenzelm@36278	3892	apply(drule meta_mp)
wenzelm@36278	3893	apply(drule_tac pi="[(x,xa)]" in fin.eqvt(1))
wenzelm@36278	3894	apply(simp add: calc_atm)
wenzelm@36278	3895	apply(drule meta_mp)
wenzelm@36278	3896	apply(drule_tac pi="[(x,xa)]" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	3897	apply(simp add: calc_atm CAND_eqvt_name)
wenzelm@36278	3898	apply(simp)
wenzelm@36278	3899	apply(simp)
wenzelm@36278	3900	apply(drule_tac x="xa" in meta_spec)
wenzelm@36278	3901	apply(drule_tac x="[(x,y)]\<bullet>M" in meta_spec)
wenzelm@36278	3902	apply(simp)
wenzelm@36278	3903	apply(drule meta_mp)
wenzelm@36278	3904	apply(drule_tac pi="[(x,y)]" in fin.eqvt(1))
wenzelm@36278	3905	apply(simp add: calc_atm)
wenzelm@36278	3906	apply(drule meta_mp)
wenzelm@36278	3907	apply(drule_tac pi="[(x,y)]" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	3908	apply(simp add: calc_atm CAND_eqvt_name)
wenzelm@36278	3909	apply(simp)
wenzelm@36278	3910	done
wenzelm@36278	3911
wenzelm@36278	3912	lemma ORRIGHT1_elim:
wenzelm@36278	3913	assumes a: "<a>:M \<in> ORRIGHT1 (B OR C) (\<parallel><B>\<parallel>)"
wenzelm@36278	3914	obtains a' M' where "M = OrR1 <a'>.M' a" and "fic (OrR1 <a'>.M' a) a" and "<a'>:M' \<in> (\<parallel><B>\<parallel>)"
wenzelm@36278	3915	using a
wenzelm@36278	3916	apply(auto simp add: ctrm.inject alpha abs_fresh calc_atm)
wenzelm@36278	3917	apply(drule_tac x="b" in meta_spec)
thomas@58834	3918	apply(drule_tac x="[(a,b)]\<bullet>Ma" in meta_spec)
wenzelm@36278	3919	apply(simp)
wenzelm@36278	3920	apply(drule meta_mp)
wenzelm@36278	3921	apply(drule_tac pi="[(a,b)]" in fic.eqvt(2))
wenzelm@36278	3922	apply(simp add: calc_atm)
wenzelm@36278	3923	apply(drule meta_mp)
wenzelm@36278	3924	apply(drule_tac pi="[(a,b)]" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	3925	apply(simp add: calc_atm CAND_eqvt_coname)
wenzelm@36278	3926	apply(simp)
wenzelm@36278	3927	apply(case_tac "a=aa")
wenzelm@36278	3928	apply(simp)
wenzelm@36278	3929	apply(drule_tac x="b" in meta_spec)
thomas@58834	3930	apply(drule_tac x="[(aa,b)]\<bullet>Ma" in meta_spec)
wenzelm@36278	3931	apply(simp)
wenzelm@36278	3932	apply(drule meta_mp)
wenzelm@36278	3933	apply(drule_tac pi="[(aa,b)]" in fic.eqvt(2))
wenzelm@36278	3934	apply(simp add: calc_atm)
wenzelm@36278	3935	apply(drule meta_mp)
wenzelm@36278	3936	apply(drule_tac pi="[(aa,b)]" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	3937	apply(simp add: calc_atm CAND_eqvt_coname)
wenzelm@36278	3938	apply(simp)
wenzelm@36278	3939	apply(simp)
wenzelm@36278	3940	apply(case_tac "b=aa")
wenzelm@36278	3941	apply(simp)
wenzelm@36278	3942	apply(drule_tac x="a" in meta_spec)
thomas@58834	3943	apply(drule_tac x="[(a,aa)]\<bullet>Ma" in meta_spec)
wenzelm@36278	3944	apply(simp)
wenzelm@36278	3945	apply(drule meta_mp)
wenzelm@36278	3946	apply(drule_tac pi="[(a,aa)]" in fic.eqvt(2))
wenzelm@36278	3947	apply(simp add: calc_atm)
wenzelm@36278	3948	apply(drule meta_mp)
wenzelm@36278	3949	apply(drule_tac pi="[(a,aa)]" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	3950	apply(simp add: calc_atm CAND_eqvt_coname)
wenzelm@36278	3951	apply(simp)
wenzelm@36278	3952	apply(simp)
wenzelm@36278	3953	apply(drule_tac x="aa" in meta_spec)
thomas@58834	3954	apply(drule_tac x="[(a,b)]\<bullet>Ma" in meta_spec)
wenzelm@36278	3955	apply(simp)
wenzelm@36278	3956	apply(drule meta_mp)
wenzelm@36278	3957	apply(drule_tac pi="[(a,b)]" in fic.eqvt(2))
wenzelm@36278	3958	apply(simp add: calc_atm)
wenzelm@36278	3959	apply(drule meta_mp)
wenzelm@36278	3960	apply(drule_tac pi="[(a,b)]" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	3961	apply(simp add: calc_atm CAND_eqvt_coname)
wenzelm@36278	3962	apply(simp)
wenzelm@36278	3963	done
wenzelm@36278	3964
wenzelm@36278	3965	lemma ORRIGHT2_elim:
wenzelm@36278	3966	assumes a: "<a>:M \<in> ORRIGHT2 (B OR C) (\<parallel><C>\<parallel>)"
wenzelm@36278	3967	obtains a' M' where "M = OrR2 <a'>.M' a" and "fic (OrR2 <a'>.M' a) a" and "<a'>:M' \<in> (\<parallel><C>\<parallel>)"
wenzelm@36278	3968	using a
wenzelm@36278	3969	apply(auto simp add: ctrm.inject alpha abs_fresh calc_atm)
wenzelm@36278	3970	apply(drule_tac x="b" in meta_spec)
thomas@58834	3971	apply(drule_tac x="[(a,b)]\<bullet>Ma" in meta_spec)
wenzelm@36278	3972	apply(simp)
wenzelm@36278	3973	apply(drule meta_mp)
wenzelm@36278	3974	apply(drule_tac pi="[(a,b)]" in fic.eqvt(2))
wenzelm@36278	3975	apply(simp add: calc_atm)
wenzelm@36278	3976	apply(drule meta_mp)
wenzelm@36278	3977	apply(drule_tac pi="[(a,b)]" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	3978	apply(simp add: calc_atm CAND_eqvt_coname)
wenzelm@36278	3979	apply(simp)
wenzelm@36278	3980	apply(case_tac "a=aa")
wenzelm@36278	3981	apply(simp)
wenzelm@36278	3982	apply(drule_tac x="b" in meta_spec)
thomas@58834	3983	apply(drule_tac x="[(aa,b)]\<bullet>Ma" in meta_spec)
wenzelm@36278	3984	apply(simp)
wenzelm@36278	3985	apply(drule meta_mp)
wenzelm@36278	3986	apply(drule_tac pi="[(aa,b)]" in fic.eqvt(2))
wenzelm@36278	3987	apply(simp add: calc_atm)
wenzelm@36278	3988	apply(drule meta_mp)
wenzelm@36278	3989	apply(drule_tac pi="[(aa,b)]" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	3990	apply(simp add: calc_atm CAND_eqvt_coname)
wenzelm@36278	3991	apply(simp)
wenzelm@36278	3992	apply(simp)
wenzelm@36278	3993	apply(case_tac "b=aa")
wenzelm@36278	3994	apply(simp)
wenzelm@36278	3995	apply(drule_tac x="a" in meta_spec)
thomas@58834	3996	apply(drule_tac x="[(a,aa)]\<bullet>Ma" in meta_spec)
wenzelm@36278	3997	apply(simp)
wenzelm@36278	3998	apply(drule meta_mp)
wenzelm@36278	3999	apply(drule_tac pi="[(a,aa)]" in fic.eqvt(2))
wenzelm@36278	4000	apply(simp add: calc_atm)
wenzelm@36278	4001	apply(drule meta_mp)
wenzelm@36278	4002	apply(drule_tac pi="[(a,aa)]" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4003	apply(simp add: calc_atm CAND_eqvt_coname)
wenzelm@36278	4004	apply(simp)
wenzelm@36278	4005	apply(simp)
wenzelm@36278	4006	apply(drule_tac x="aa" in meta_spec)
thomas@58834	4007	apply(drule_tac x="[(a,b)]\<bullet>Ma" in meta_spec)
wenzelm@36278	4008	apply(simp)
wenzelm@36278	4009	apply(drule meta_mp)
wenzelm@36278	4010	apply(drule_tac pi="[(a,b)]" in fic.eqvt(2))
wenzelm@36278	4011	apply(simp add: calc_atm)
wenzelm@36278	4012	apply(drule meta_mp)
wenzelm@36278	4013	apply(drule_tac pi="[(a,b)]" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4014	apply(simp add: calc_atm CAND_eqvt_coname)
wenzelm@36278	4015	apply(simp)
wenzelm@36278	4016	done
wenzelm@36278	4017
wenzelm@36278	4018	lemma ORLEFT_elim:
wenzelm@36278	4019	assumes a: "(x):M \<in> ORLEFT (B OR C) (\<parallel>(B)\<parallel>) (\<parallel>(C)\<parallel>)"
wenzelm@36278	4020	obtains y' M' z' N' where "M = OrL (y').M' (z').N' x" and "fin (OrL (y').M' (z').N' x) x"
wenzelm@36278	4021	and "(y'):M' \<in> (\<parallel>(B)\<parallel>)" and "(z'):N' \<in> (\<parallel>(C)\<parallel>)"
wenzelm@36278	4022	using a
wenzelm@36278	4023	apply(auto simp add: ntrm.inject alpha abs_fresh calc_atm fresh_atm)
wenzelm@36278	4024	apply(drule_tac x="z" in meta_spec)
thomas@58834	4025	apply(drule_tac x="[(x,z)]\<bullet>Ma" in meta_spec)
wenzelm@36278	4026	apply(drule_tac x="z" in meta_spec)
wenzelm@36278	4027	apply(drule_tac x="[(x,z)]\<bullet>N" in meta_spec)
wenzelm@36278	4028	apply(simp)
wenzelm@36278	4029	apply(drule meta_mp)
wenzelm@36278	4030	apply(drule_tac pi="[(x,z)]" in fin.eqvt(1))
wenzelm@36278	4031	apply(simp add: calc_atm)
wenzelm@36278	4032	apply(drule meta_mp)
wenzelm@36278	4033	apply(drule_tac pi="[(x,z)]" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4034	apply(simp add: calc_atm CAND_eqvt_name)
wenzelm@36278	4035	apply(drule meta_mp)
wenzelm@36278	4036	apply(drule_tac pi="[(x,z)]" and x="(x):N" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4037	apply(simp add: calc_atm CAND_eqvt_name)
wenzelm@36278	4038	apply(simp)
wenzelm@36278	4039	apply(case_tac "x=y")
wenzelm@36278	4040	apply(simp)
wenzelm@36278	4041	apply(drule_tac x="z" in meta_spec)
thomas@58834	4042	apply(drule_tac x="[(y,z)]\<bullet>Ma" in meta_spec)
wenzelm@36278	4043	apply(drule_tac x="z" in meta_spec)
wenzelm@36278	4044	apply(drule_tac x="[(y,z)]\<bullet>N" in meta_spec)
wenzelm@36278	4045	apply(simp)
wenzelm@36278	4046	apply(drule meta_mp)
wenzelm@36278	4047	apply(drule_tac pi="[(y,z)]" in fin.eqvt(1))
wenzelm@36278	4048	apply(simp add: calc_atm)
wenzelm@36278	4049	apply(drule meta_mp)
wenzelm@36278	4050	apply(drule_tac pi="[(y,z)]" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4051	apply(simp add: calc_atm CAND_eqvt_name)
wenzelm@36278	4052	apply(drule meta_mp)
wenzelm@36278	4053	apply(drule_tac pi="[(y,z)]" and x="(y):N" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4054	apply(simp add: calc_atm CAND_eqvt_name)
wenzelm@36278	4055	apply(simp)
wenzelm@36278	4056	apply(simp)
wenzelm@36278	4057	apply(case_tac "z=y")
wenzelm@36278	4058	apply(simp)
wenzelm@36278	4059	apply(drule_tac x="y" in meta_spec)
thomas@58834	4060	apply(drule_tac x="[(x,y)]\<bullet>Ma" in meta_spec)
wenzelm@36278	4061	apply(drule_tac x="x" in meta_spec)
wenzelm@36278	4062	apply(drule_tac x="[(x,y)]\<bullet>N" in meta_spec)
wenzelm@36278	4063	apply(simp)
wenzelm@36278	4064	apply(drule meta_mp)
wenzelm@36278	4065	apply(drule_tac pi="[(x,y)]" in fin.eqvt(1))
wenzelm@36278	4066	apply(simp add: calc_atm)
wenzelm@36278	4067	apply(drule meta_mp)
wenzelm@36278	4068	apply(drule_tac pi="[(x,y)]" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4069	apply(simp add: calc_atm CAND_eqvt_name)
wenzelm@36278	4070	apply(drule meta_mp)
wenzelm@36278	4071	apply(drule_tac pi="[(x,y)]" and x="(y):N" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4072	apply(simp add: calc_atm CAND_eqvt_name)
wenzelm@36278	4073	apply(simp)
wenzelm@36278	4074	apply(simp)
wenzelm@36278	4075	apply(drule_tac x="z" in meta_spec)
thomas@58834	4076	apply(drule_tac x="[(x,z)]\<bullet>Ma" in meta_spec)
wenzelm@36278	4077	apply(drule_tac x="y" in meta_spec)
wenzelm@36278	4078	apply(drule_tac x="[(x,z)]\<bullet>N" in meta_spec)
wenzelm@36278	4079	apply(simp)
wenzelm@36278	4080	apply(drule meta_mp)
wenzelm@36278	4081	apply(drule_tac pi="[(x,z)]" in fin.eqvt(1))
wenzelm@36278	4082	apply(simp add: calc_atm)
wenzelm@36278	4083	apply(drule meta_mp)
wenzelm@36278	4084	apply(drule_tac pi="[(x,z)]" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4085	apply(simp add: calc_atm CAND_eqvt_name)
wenzelm@36278	4086	apply(drule meta_mp)
wenzelm@36278	4087	apply(drule_tac pi="[(x,z)]" and x="(y):N" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4088	apply(simp add: calc_atm CAND_eqvt_name)
wenzelm@36278	4089	apply(simp)
wenzelm@36278	4090	apply(case_tac "x=xa")
wenzelm@36278	4091	apply(simp)
wenzelm@36278	4092	apply(drule_tac x="z" in meta_spec)
thomas@58834	4093	apply(drule_tac x="[(xa,z)]\<bullet>Ma" in meta_spec)
wenzelm@36278	4094	apply(drule_tac x="z" in meta_spec)
wenzelm@36278	4095	apply(drule_tac x="[(xa,z)]\<bullet>N" in meta_spec)
wenzelm@36278	4096	apply(simp)
wenzelm@36278	4097	apply(drule meta_mp)
wenzelm@36278	4098	apply(drule_tac pi="[(xa,z)]" in fin.eqvt(1))
wenzelm@36278	4099	apply(simp add: calc_atm)
wenzelm@36278	4100	apply(drule meta_mp)
wenzelm@36278	4101	apply(drule_tac pi="[(xa,z)]" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4102	apply(simp add: calc_atm CAND_eqvt_name)
wenzelm@36278	4103	apply(drule meta_mp)
wenzelm@36278	4104	apply(drule_tac pi="[(xa,z)]" and x="(xa):N" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4105	apply(simp add: calc_atm CAND_eqvt_name)
wenzelm@36278	4106	apply(simp)
wenzelm@36278	4107	apply(simp)
wenzelm@36278	4108	apply(case_tac "z=xa")
wenzelm@36278	4109	apply(simp)
wenzelm@36278	4110	apply(drule_tac x="x" in meta_spec)
thomas@58834	4111	apply(drule_tac x="[(x,xa)]\<bullet>Ma" in meta_spec)
wenzelm@36278	4112	apply(drule_tac x="xa" in meta_spec)
wenzelm@36278	4113	apply(drule_tac x="[(x,xa)]\<bullet>N" in meta_spec)
wenzelm@36278	4114	apply(simp)
wenzelm@36278	4115	apply(drule meta_mp)
wenzelm@36278	4116	apply(drule_tac pi="[(x,xa)]" in fin.eqvt(1))
wenzelm@36278	4117	apply(simp add: calc_atm)
wenzelm@36278	4118	apply(drule meta_mp)
wenzelm@36278	4119	apply(drule_tac pi="[(x,xa)]" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4120	apply(simp add: calc_atm CAND_eqvt_name)
wenzelm@36278	4121	apply(drule meta_mp)
wenzelm@36278	4122	apply(drule_tac pi="[(x,xa)]" and x="(x):N" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4123	apply(simp add: calc_atm CAND_eqvt_name)
wenzelm@36278	4124	apply(simp)
wenzelm@36278	4125	apply(simp)
wenzelm@36278	4126	apply(drule_tac x="xa" in meta_spec)
thomas@58834	4127	apply(drule_tac x="[(x,z)]\<bullet>Ma" in meta_spec)
wenzelm@36278	4128	apply(drule_tac x="z" in meta_spec)
wenzelm@36278	4129	apply(drule_tac x="[(x,z)]\<bullet>N" in meta_spec)
wenzelm@36278	4130	apply(simp)
wenzelm@36278	4131	apply(drule meta_mp)
wenzelm@36278	4132	apply(drule_tac pi="[(x,z)]" in fin.eqvt(1))
wenzelm@36278	4133	apply(simp add: calc_atm)
wenzelm@36278	4134	apply(drule meta_mp)
wenzelm@36278	4135	apply(drule_tac pi="[(x,z)]" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4136	apply(simp add: calc_atm CAND_eqvt_name)
wenzelm@36278	4137	apply(drule meta_mp)
wenzelm@36278	4138	apply(drule_tac pi="[(x,z)]" and x="(x):N" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4139	apply(simp add: calc_atm CAND_eqvt_name)
wenzelm@36278	4140	apply(simp)
wenzelm@36278	4141	apply(case_tac "x=xa")
wenzelm@36278	4142	apply(simp)
wenzelm@36278	4143	apply(case_tac "xa=y")
wenzelm@36278	4144	apply(simp)
wenzelm@36278	4145	apply(drule_tac x="z" in meta_spec)
thomas@58834	4146	apply(drule_tac x="[(y,z)]\<bullet>Ma" in meta_spec)
wenzelm@36278	4147	apply(drule_tac x="z" in meta_spec)
wenzelm@36278	4148	apply(drule_tac x="[(y,z)]\<bullet>N" in meta_spec)
wenzelm@36278	4149	apply(simp)
wenzelm@36278	4150	apply(drule meta_mp)
wenzelm@36278	4151	apply(drule_tac pi="[(y,z)]" in fin.eqvt(1))
wenzelm@36278	4152	apply(simp add: calc_atm)
wenzelm@36278	4153	apply(drule meta_mp)
wenzelm@36278	4154	apply(drule_tac pi="[(y,z)]" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4155	apply(simp add: calc_atm CAND_eqvt_name)
wenzelm@36278	4156	apply(drule meta_mp)
wenzelm@36278	4157	apply(drule_tac pi="[(y,z)]" and x="(y):N" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4158	apply(simp add: calc_atm CAND_eqvt_name)
wenzelm@36278	4159	apply(simp)
wenzelm@36278	4160	apply(simp)
wenzelm@36278	4161	apply(case_tac "z=y")
wenzelm@36278	4162	apply(simp)
wenzelm@36278	4163	apply(drule_tac x="y" in meta_spec)
thomas@58834	4164	apply(drule_tac x="[(xa,y)]\<bullet>Ma" in meta_spec)
wenzelm@36278	4165	apply(drule_tac x="xa" in meta_spec)
wenzelm@36278	4166	apply(drule_tac x="[(xa,y)]\<bullet>N" in meta_spec)
wenzelm@36278	4167	apply(simp)
wenzelm@36278	4168	apply(drule meta_mp)
wenzelm@36278	4169	apply(drule_tac pi="[(xa,y)]" in fin.eqvt(1))
wenzelm@36278	4170	apply(simp add: calc_atm)
wenzelm@36278	4171	apply(drule meta_mp)
wenzelm@36278	4172	apply(drule_tac pi="[(xa,y)]" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4173	apply(simp add: calc_atm CAND_eqvt_name)
wenzelm@36278	4174	apply(drule meta_mp)
wenzelm@36278	4175	apply(drule_tac pi="[(xa,y)]" and x="(y):N" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4176	apply(simp add: calc_atm CAND_eqvt_name)
wenzelm@36278	4177	apply(simp)
wenzelm@36278	4178	apply(simp)
wenzelm@36278	4179	apply(drule_tac x="z" in meta_spec)
thomas@58834	4180	apply(drule_tac x="[(xa,z)]\<bullet>Ma" in meta_spec)
wenzelm@36278	4181	apply(drule_tac x="y" in meta_spec)
wenzelm@36278	4182	apply(drule_tac x="[(xa,z)]\<bullet>N" in meta_spec)
wenzelm@36278	4183	apply(simp)
wenzelm@36278	4184	apply(drule meta_mp)
wenzelm@36278	4185	apply(drule_tac pi="[(xa,z)]" in fin.eqvt(1))
wenzelm@36278	4186	apply(simp add: calc_atm)
wenzelm@36278	4187	apply(drule meta_mp)
wenzelm@36278	4188	apply(drule_tac pi="[(xa,z)]" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4189	apply(simp add: calc_atm CAND_eqvt_name)
wenzelm@36278	4190	apply(drule meta_mp)
wenzelm@36278	4191	apply(drule_tac pi="[(xa,z)]" and x="(y):N" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4192	apply(simp add: calc_atm CAND_eqvt_name)
wenzelm@36278	4193	apply(simp)
wenzelm@36278	4194	apply(simp)
wenzelm@36278	4195	apply(case_tac "z=xa")
wenzelm@36278	4196	apply(simp)
wenzelm@36278	4197	apply(case_tac "x=y")
wenzelm@36278	4198	apply(simp)
wenzelm@36278	4199	apply(drule_tac x="y" in meta_spec)
thomas@58834	4200	apply(drule_tac x="[(y,xa)]\<bullet>Ma" in meta_spec)
wenzelm@36278	4201	apply(drule_tac x="xa" in meta_spec)
wenzelm@36278	4202	apply(drule_tac x="[(y,xa)]\<bullet>N" in meta_spec)
wenzelm@36278	4203	apply(simp)
wenzelm@36278	4204	apply(drule meta_mp)
wenzelm@36278	4205	apply(drule_tac pi="[(y,xa)]" in fin.eqvt(1))
wenzelm@36278	4206	apply(simp add: calc_atm)
wenzelm@36278	4207	apply(drule meta_mp)
wenzelm@36278	4208	apply(drule_tac pi="[(y,xa)]" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4209	apply(simp add: calc_atm CAND_eqvt_name)
wenzelm@36278	4210	apply(drule meta_mp)
wenzelm@36278	4211	apply(drule_tac pi="[(y,xa)]" and x="(y):N" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4212	apply(simp add: calc_atm CAND_eqvt_name)
wenzelm@36278	4213	apply(simp)
wenzelm@36278	4214	apply(simp)
wenzelm@36278	4215	apply(case_tac "xa=y")
wenzelm@36278	4216	apply(simp)
wenzelm@36278	4217	apply(drule_tac x="x" in meta_spec)
thomas@58834	4218	apply(drule_tac x="[(x,y)]\<bullet>Ma" in meta_spec)
wenzelm@36278	4219	apply(drule_tac x="x" in meta_spec)
wenzelm@36278	4220	apply(drule_tac x="[(x,y)]\<bullet>N" in meta_spec)
wenzelm@36278	4221	apply(simp)
wenzelm@36278	4222	apply(drule meta_mp)
wenzelm@36278	4223	apply(drule_tac pi="[(x,y)]" in fin.eqvt(1))
wenzelm@36278	4224	apply(simp add: calc_atm)
wenzelm@36278	4225	apply(drule meta_mp)
wenzelm@36278	4226	apply(drule_tac pi="[(x,y)]" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4227	apply(simp add: calc_atm CAND_eqvt_name)
wenzelm@36278	4228	apply(drule meta_mp)
wenzelm@36278	4229	apply(drule_tac pi="[(x,y)]" and x="(y):N" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4230	apply(simp add: calc_atm CAND_eqvt_name)
wenzelm@36278	4231	apply(simp)
wenzelm@36278	4232	apply(simp)
wenzelm@36278	4233	apply(drule_tac x="x" in meta_spec)
thomas@58834	4234	apply(drule_tac x="[(x,xa)]\<bullet>Ma" in meta_spec)
wenzelm@36278	4235	apply(drule_tac x="y" in meta_spec)
wenzelm@36278	4236	apply(drule_tac x="[(x,xa)]\<bullet>N" in meta_spec)
wenzelm@36278	4237	apply(simp)
wenzelm@36278	4238	apply(drule meta_mp)
wenzelm@36278	4239	apply(drule_tac pi="[(x,xa)]" in fin.eqvt(1))
wenzelm@36278	4240	apply(simp add: calc_atm)
wenzelm@36278	4241	apply(drule meta_mp)
wenzelm@36278	4242	apply(drule_tac pi="[(x,xa)]" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4243	apply(simp add: calc_atm CAND_eqvt_name)
wenzelm@36278	4244	apply(drule meta_mp)
wenzelm@36278	4245	apply(drule_tac pi="[(x,xa)]" and x="(y):N" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4246	apply(simp add: calc_atm CAND_eqvt_name)
wenzelm@36278	4247	apply(simp)
wenzelm@36278	4248	apply(simp)
wenzelm@36278	4249	apply(case_tac "x=y")
wenzelm@36278	4250	apply(simp)
wenzelm@36278	4251	apply(drule_tac x="xa" in meta_spec)
thomas@58834	4252	apply(drule_tac x="[(y,z)]\<bullet>Ma" in meta_spec)
wenzelm@36278	4253	apply(drule_tac x="z" in meta_spec)
wenzelm@36278	4254	apply(drule_tac x="[(y,z)]\<bullet>N" in meta_spec)
wenzelm@36278	4255	apply(simp)
wenzelm@36278	4256	apply(drule meta_mp)
wenzelm@36278	4257	apply(drule_tac pi="[(y,z)]" in fin.eqvt(1))
wenzelm@36278	4258	apply(simp add: calc_atm)
wenzelm@36278	4259	apply(drule meta_mp)
wenzelm@36278	4260	apply(drule_tac pi="[(y,z)]" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4261	apply(simp add: calc_atm CAND_eqvt_name)
wenzelm@36278	4262	apply(drule meta_mp)
wenzelm@36278	4263	apply(drule_tac pi="[(y,z)]" and x="(y):N" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4264	apply(simp add: calc_atm CAND_eqvt_name)
wenzelm@36278	4265	apply(simp)
wenzelm@36278	4266	apply(simp)
wenzelm@36278	4267	apply(case_tac "z=y")
wenzelm@36278	4268	apply(simp)
wenzelm@36278	4269	apply(drule_tac x="xa" in meta_spec)
thomas@58834	4270	apply(drule_tac x="[(x,y)]\<bullet>Ma" in meta_spec)
wenzelm@36278	4271	apply(drule_tac x="x" in meta_spec)
wenzelm@36278	4272	apply(drule_tac x="[(x,y)]\<bullet>N" in meta_spec)
wenzelm@36278	4273	apply(simp)
wenzelm@36278	4274	apply(drule meta_mp)
wenzelm@36278	4275	apply(drule_tac pi="[(x,y)]" in fin.eqvt(1))
wenzelm@36278	4276	apply(simp add: calc_atm)
wenzelm@36278	4277	apply(drule meta_mp)
wenzelm@36278	4278	apply(drule_tac pi="[(x,y)]" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4279	apply(simp add: calc_atm CAND_eqvt_name)
wenzelm@36278	4280	apply(drule meta_mp)
wenzelm@36278	4281	apply(drule_tac pi="[(x,y)]" and x="(y):N" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4282	apply(simp add: calc_atm CAND_eqvt_name)
wenzelm@36278	4283	apply(simp)
wenzelm@36278	4284	apply(simp)
wenzelm@36278	4285	apply(drule_tac x="xa" in meta_spec)
thomas@58834	4286	apply(drule_tac x="[(x,z)]\<bullet>Ma" in meta_spec)
wenzelm@36278	4287	apply(drule_tac x="y" in meta_spec)
wenzelm@36278	4288	apply(drule_tac x="[(x,z)]\<bullet>N" in meta_spec)
wenzelm@36278	4289	apply(simp)
wenzelm@36278	4290	apply(drule meta_mp)
wenzelm@36278	4291	apply(drule_tac pi="[(x,z)]" in fin.eqvt(1))
wenzelm@36278	4292	apply(simp add: calc_atm)
wenzelm@36278	4293	apply(drule meta_mp)
wenzelm@36278	4294	apply(drule_tac pi="[(x,z)]" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4295	apply(simp add: calc_atm CAND_eqvt_name)
wenzelm@36278	4296	apply(drule meta_mp)
wenzelm@36278	4297	apply(drule_tac pi="[(x,z)]" and x="(y):N" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4298	apply(simp add: calc_atm CAND_eqvt_name)
wenzelm@36278	4299	apply(simp)
wenzelm@36278	4300	done
wenzelm@36278	4301
wenzelm@36278	4302	lemma IMPRIGHT_elim:
wenzelm@36278	4303	assumes a: "<a>:M \<in> IMPRIGHT (B IMP C) (\<parallel>(B)\<parallel>) (\<parallel><C>\<parallel>) (\<parallel>(C)\<parallel>) (\<parallel><B>\<parallel>)"
wenzelm@36278	4304	obtains x' a' M' where "M = ImpR (x').<a'>.M' a" and "fic (ImpR (x').<a'>.M' a) a"
wenzelm@36278	4305	and "\<forall>z P. x'\<sharp>(z,P) \<and> (z):P \<in> \<parallel>(C)\<parallel> \<longrightarrow> (x'):(M'{a':=(z).P}) \<in> \<parallel>(B)\<parallel>"
wenzelm@36278	4306	and "\<forall>c Q. a'\<sharp>(c,Q) \<and> <c>:Q \<in> \<parallel><B>\<parallel> \<longrightarrow> <a'>:(M'{x':=<c>.Q}) \<in> \<parallel><C>\<parallel>"
wenzelm@36278	4307	using a
wenzelm@36278	4308	apply(auto simp add: ctrm.inject alpha abs_fresh calc_atm)
wenzelm@36278	4309	apply(drule_tac x="x" in meta_spec)
wenzelm@36278	4310	apply(drule_tac x="b" in meta_spec)
thomas@58834	4311	apply(drule_tac x="[(a,b)]\<bullet>Ma" in meta_spec)
wenzelm@36278	4312	apply(simp)
wenzelm@36278	4313	apply(drule_tac pi="[(a,b)]" in fic.eqvt(2))
wenzelm@36278	4314	apply(simp add: calc_atm)
wenzelm@36278	4315	apply(drule meta_mp)
wenzelm@36278	4316	apply(auto)[1]
wenzelm@36278	4317	apply(drule_tac pi="[(a,b)]" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4318	apply(simp add: calc_atm CAND_eqvt_coname)
wenzelm@36278	4319	apply(drule_tac x="z" in spec)
wenzelm@36278	4320	apply(drule_tac x="[(a,b)]\<bullet>P" in spec)
wenzelm@36278	4321	apply(simp add: fresh_prod fresh_left calc_atm)
thomas@58834	4322	apply(drule_tac pi="[(a,b)]" and x="(x):Ma{a:=(z).([(a,b)]\<bullet>P)}"
wenzelm@36278	4323	in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4324	apply(perm_simp add: calc_atm csubst_eqvt CAND_eqvt_coname)
wenzelm@36278	4325	apply(drule meta_mp)
wenzelm@36278	4326	apply(auto)[1]
wenzelm@36278	4327	apply(drule_tac pi="[(a,b)]" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4328	apply(simp add: CAND_eqvt_coname)
wenzelm@36278	4329	apply(rotate_tac 2)
wenzelm@36278	4330	apply(drule_tac x="[(a,b)]\<bullet>c" in spec)
wenzelm@36278	4331	apply(drule_tac x="[(a,b)]\<bullet>Q" in spec)
wenzelm@36278	4332	apply(simp add: fresh_prod fresh_left)
wenzelm@36278	4333	apply(drule mp)
wenzelm@36278	4334	apply(simp add: calc_atm)
thomas@58834	4335	apply(drule_tac pi="[(a,b)]" and x="<a>:Ma{x:=<([(a,b)]\<bullet>c)>.([(a,b)]\<bullet>Q)}"
wenzelm@36278	4336	in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4337	apply(perm_simp add: nsubst_eqvt CAND_eqvt_coname)
wenzelm@36278	4338	apply(simp add: calc_atm)
wenzelm@36278	4339	apply(case_tac "a=aa")
wenzelm@36278	4340	apply(simp)
wenzelm@36278	4341	apply(drule_tac x="x" in meta_spec)
wenzelm@36278	4342	apply(drule_tac x="b" in meta_spec)
thomas@58834	4343	apply(drule_tac x="[(aa,b)]\<bullet>Ma" in meta_spec)
wenzelm@36278	4344	apply(simp)
wenzelm@36278	4345	apply(drule meta_mp)
wenzelm@36278	4346	apply(drule_tac pi="[(aa,b)]" in fic.eqvt(2))
wenzelm@36278	4347	apply(simp add: calc_atm)
wenzelm@36278	4348	apply(drule meta_mp)
wenzelm@36278	4349	apply(auto)[1]
wenzelm@36278	4350	apply(drule_tac pi="[(a,b)]" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4351	apply(simp add: calc_atm CAND_eqvt_coname)
wenzelm@36278	4352	apply(drule_tac x="z" in spec)
wenzelm@36278	4353	apply(drule_tac x="[(a,b)]\<bullet>P" in spec)
wenzelm@36278	4354	apply(simp add: fresh_prod fresh_left calc_atm)
thomas@58834	4355	apply(drule_tac pi="[(a,b)]" and x="(x):Ma{a:=(z).([(a,b)]\<bullet>P)}"
wenzelm@36278	4356	in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4357	apply(perm_simp add: calc_atm csubst_eqvt CAND_eqvt_coname)
wenzelm@36278	4358	apply(drule meta_mp)
wenzelm@36278	4359	apply(auto)[1]
wenzelm@36278	4360	apply(drule_tac pi="[(a,b)]" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4361	apply(simp add: CAND_eqvt_coname)
wenzelm@36278	4362	apply(drule_tac x="[(a,b)]\<bullet>c" in spec)
wenzelm@36278	4363	apply(drule_tac x="[(a,b)]\<bullet>Q" in spec)
wenzelm@36278	4364	apply(simp)
wenzelm@36278	4365	apply(simp add: fresh_prod fresh_left)
wenzelm@36278	4366	apply(drule mp)
wenzelm@36278	4367	apply(simp add: calc_atm)
thomas@58834	4368	apply(drule_tac pi="[(a,b)]" and x="<a>:Ma{x:=<([(a,b)]\<bullet>c)>.([(a,b)]\<bullet>Q)}"
wenzelm@36278	4369	in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4370	apply(perm_simp add: nsubst_eqvt CAND_eqvt_coname)
wenzelm@36278	4371	apply(simp add: calc_atm)
wenzelm@36278	4372	apply(simp)
wenzelm@36278	4373	apply(case_tac "b=aa")
wenzelm@36278	4374	apply(simp)
wenzelm@36278	4375	apply(drule_tac x="x" in meta_spec)
wenzelm@36278	4376	apply(drule_tac x="a" in meta_spec)
thomas@58834	4377	apply(drule_tac x="[(a,aa)]\<bullet>Ma" in meta_spec)
wenzelm@36278	4378	apply(simp)
wenzelm@36278	4379	apply(drule meta_mp)
wenzelm@36278	4380	apply(drule_tac pi="[(a,aa)]" in fic.eqvt(2))
wenzelm@36278	4381	apply(simp add: calc_atm)
wenzelm@36278	4382	apply(drule meta_mp)
wenzelm@36278	4383	apply(auto)[1]
wenzelm@36278	4384	apply(drule_tac pi="[(a,aa)]" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4385	apply(simp add: calc_atm CAND_eqvt_coname)
wenzelm@36278	4386	apply(drule_tac x="z" in spec)
wenzelm@36278	4387	apply(drule_tac x="[(a,aa)]\<bullet>P" in spec)
wenzelm@36278	4388	apply(simp add: fresh_prod fresh_left calc_atm)
thomas@58834	4389	apply(drule_tac pi="[(a,aa)]" and x="(x):Ma{aa:=(z).([(a,aa)]\<bullet>P)}"
wenzelm@36278	4390	in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4391	apply(perm_simp add: calc_atm csubst_eqvt CAND_eqvt_coname)
wenzelm@36278	4392	apply(drule meta_mp)
wenzelm@36278	4393	apply(auto)[1]
wenzelm@36278	4394	apply(drule_tac pi="[(a,aa)]" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4395	apply(simp add: CAND_eqvt_coname)
wenzelm@36278	4396	apply(drule_tac x="[(a,aa)]\<bullet>c" in spec)
wenzelm@36278	4397	apply(drule_tac x="[(a,aa)]\<bullet>Q" in spec)
wenzelm@36278	4398	apply(simp)
wenzelm@36278	4399	apply(simp add: fresh_prod fresh_left)
wenzelm@36278	4400	apply(drule mp)
wenzelm@36278	4401	apply(simp add: calc_atm)
thomas@58834	4402	apply(drule_tac pi="[(a,aa)]" and x="<aa>:Ma{x:=<([(a,aa)]\<bullet>c)>.([(a,aa)]\<bullet>Q)}"
wenzelm@36278	4403	in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4404	apply(perm_simp add: nsubst_eqvt CAND_eqvt_coname)
wenzelm@36278	4405	apply(simp add: calc_atm)
wenzelm@36278	4406	apply(simp)
wenzelm@36278	4407	apply(drule_tac x="x" in meta_spec)
wenzelm@36278	4408	apply(drule_tac x="aa" in meta_spec)
thomas@58834	4409	apply(drule_tac x="[(a,b)]\<bullet>Ma" in meta_spec)
wenzelm@36278	4410	apply(simp)
wenzelm@36278	4411	apply(drule meta_mp)
wenzelm@36278	4412	apply(drule_tac pi="[(a,b)]" in fic.eqvt(2))
wenzelm@36278	4413	apply(simp add: calc_atm)
wenzelm@36278	4414	apply(drule meta_mp)
wenzelm@36278	4415	apply(auto)[1]
wenzelm@36278	4416	apply(drule_tac pi="[(a,b)]" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4417	apply(simp add: calc_atm CAND_eqvt_coname)
wenzelm@36278	4418	apply(drule_tac x="z" in spec)
wenzelm@36278	4419	apply(drule_tac x="[(a,b)]\<bullet>P" in spec)
wenzelm@36278	4420	apply(simp add: fresh_prod fresh_left calc_atm)
thomas@58834	4421	apply(drule_tac pi="[(a,b)]" and x="(x):Ma{aa:=(z).([(a,b)]\<bullet>P)}"
wenzelm@36278	4422	in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4423	apply(perm_simp add: calc_atm csubst_eqvt CAND_eqvt_coname)
wenzelm@36278	4424	apply(drule meta_mp)
wenzelm@36278	4425	apply(auto)[1]
wenzelm@36278	4426	apply(drule_tac pi="[(a,b)]" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4427	apply(simp add: CAND_eqvt_coname)
wenzelm@36278	4428	apply(drule_tac x="[(a,b)]\<bullet>c" in spec)
wenzelm@36278	4429	apply(drule_tac x="[(a,b)]\<bullet>Q" in spec)
wenzelm@36278	4430	apply(simp add: fresh_prod fresh_left)
wenzelm@36278	4431	apply(drule mp)
wenzelm@36278	4432	apply(simp add: calc_atm)
thomas@58834	4433	apply(drule_tac pi="[(a,b)]" and x="<aa>:Ma{x:=<([(a,b)]\<bullet>c)>.([(a,b)]\<bullet>Q)}"
wenzelm@36278	4434	in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4435	apply(perm_simp add: nsubst_eqvt CAND_eqvt_coname)
wenzelm@36278	4436	apply(simp add: calc_atm)
wenzelm@36278	4437	done
wenzelm@36278	4438
wenzelm@36278	4439	lemma IMPLEFT_elim:
wenzelm@36278	4440	assumes a: "(x):M \<in> IMPLEFT (B IMP C) (\<parallel><B>\<parallel>) (\<parallel>(C)\<parallel>)"
wenzelm@36278	4441	obtains x' a' M' N' where "M = ImpL <a'>.M' (x').N' x" and "fin (ImpL <a'>.M' (x').N' x) x"
wenzelm@36278	4442	and "<a'>:M' \<in> \<parallel><B>\<parallel>" and "(x'):N' \<in> \<parallel>(C)\<parallel>"
wenzelm@36278	4443	using a
wenzelm@36278	4444	apply(auto simp add: ntrm.inject alpha abs_fresh calc_atm)
wenzelm@36278	4445	apply(drule_tac x="a" in meta_spec)
thomas@58834	4446	apply(drule_tac x="[(x,y)]\<bullet>Ma" in meta_spec)
wenzelm@36278	4447	apply(drule_tac x="y" in meta_spec)
wenzelm@36278	4448	apply(drule_tac x="[(x,y)]\<bullet>N" in meta_spec)
wenzelm@36278	4449	apply(simp)
wenzelm@36278	4450	apply(drule meta_mp)
wenzelm@36278	4451	apply(drule_tac pi="[(x,y)]" in fin.eqvt(1))
wenzelm@36278	4452	apply(simp add: calc_atm)
wenzelm@36278	4453	apply(drule meta_mp)
wenzelm@36278	4454	apply(drule_tac pi="[(x,y)]" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4455	apply(simp add: calc_atm CAND_eqvt_name)
wenzelm@36278	4456	apply(drule meta_mp)
wenzelm@36278	4457	apply(drule_tac pi="[(x,y)]" and x="(x):N" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4458	apply(perm_simp add: calc_atm CAND_eqvt_name)
wenzelm@36278	4459	apply(simp)
wenzelm@36278	4460	apply(case_tac "x=xa")
wenzelm@36278	4461	apply(simp)
wenzelm@36278	4462	apply(drule_tac x="a" in meta_spec)
thomas@58834	4463	apply(drule_tac x="[(xa,y)]\<bullet>Ma" in meta_spec)
wenzelm@36278	4464	apply(drule_tac x="y" in meta_spec)
wenzelm@36278	4465	apply(drule_tac x="[(xa,y)]\<bullet>N" in meta_spec)
wenzelm@36278	4466	apply(simp)
wenzelm@36278	4467	apply(drule meta_mp)
wenzelm@36278	4468	apply(drule_tac pi="[(xa,y)]" in fin.eqvt(1))
wenzelm@36278	4469	apply(simp add: calc_atm)
wenzelm@36278	4470	apply(drule meta_mp)
wenzelm@36278	4471	apply(drule_tac pi="[(xa,y)]" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4472	apply(simp add: calc_atm CAND_eqvt_name)
wenzelm@36278	4473	apply(drule meta_mp)
wenzelm@36278	4474	apply(drule_tac pi="[(xa,y)]" and x="(xa):N" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4475	apply(simp add: calc_atm CAND_eqvt_name)
wenzelm@36278	4476	apply(simp)
wenzelm@36278	4477	apply(simp)
wenzelm@36278	4478	apply(case_tac "y=xa")
wenzelm@36278	4479	apply(simp)
wenzelm@36278	4480	apply(drule_tac x="a" in meta_spec)
thomas@58834	4481	apply(drule_tac x="[(x,xa)]\<bullet>Ma" in meta_spec)
wenzelm@36278	4482	apply(drule_tac x="x" in meta_spec)
wenzelm@36278	4483	apply(drule_tac x="[(x,xa)]\<bullet>N" in meta_spec)
wenzelm@36278	4484	apply(simp)
wenzelm@36278	4485	apply(drule meta_mp)
wenzelm@36278	4486	apply(drule_tac pi="[(x,xa)]" in fin.eqvt(1))
wenzelm@36278	4487	apply(simp add: calc_atm)
wenzelm@36278	4488	apply(drule meta_mp)
thomas@58834	4489	apply(drule_tac pi="[(x,xa)]" and x="<a>:Ma" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4490	apply(simp add: calc_atm CAND_eqvt_name)
wenzelm@36278	4491	apply(drule meta_mp)
wenzelm@36278	4492	apply(drule_tac pi="[(x,xa)]" and x="(xa):N" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4493	apply(simp add: calc_atm CAND_eqvt_name)
wenzelm@36278	4494	apply(simp)
wenzelm@36278	4495	apply(simp)
wenzelm@36278	4496	apply(drule_tac x="a" in meta_spec)
thomas@58834	4497	apply(drule_tac x="[(x,y)]\<bullet>Ma" in meta_spec)
wenzelm@36278	4498	apply(drule_tac x="xa" in meta_spec)
wenzelm@36278	4499	apply(drule_tac x="[(x,y)]\<bullet>N" in meta_spec)
wenzelm@36278	4500	apply(simp)
wenzelm@36278	4501	apply(drule meta_mp)
wenzelm@36278	4502	apply(drule_tac pi="[(x,y)]" in fin.eqvt(1))
wenzelm@36278	4503	apply(simp add: calc_atm)
wenzelm@36278	4504	apply(drule meta_mp)
wenzelm@36278	4505	apply(drule_tac pi="[(x,y)]" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4506	apply(simp add: calc_atm CAND_eqvt_name)
wenzelm@36278	4507	apply(drule meta_mp)
wenzelm@36278	4508	apply(drule_tac pi="[(x,y)]" and x="(xa):N" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4509	apply(simp add: calc_atm CAND_eqvt_name)
wenzelm@36278	4510	apply(simp)
wenzelm@36278	4511	done
wenzelm@36278	4512
wenzelm@36278	4513	lemma CANDs_alpha:
wenzelm@36278	4514	shows "<a>:M \<in> (\<parallel><B>\<parallel>) \<Longrightarrow> [a].M = [b].N \<Longrightarrow> <b>:N \<in> (\<parallel><B>\<parallel>)"
wenzelm@36278	4515	and "(x):M \<in> (\<parallel>(B)\<parallel>) \<Longrightarrow> [x].M = [y].N \<Longrightarrow> (y):N \<in> (\<parallel>(B)\<parallel>)"
wenzelm@36278	4516	apply(auto simp add: alpha)
wenzelm@36278	4517	apply(drule_tac pi="[(a,b)]" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4518	apply(perm_simp add: CAND_eqvt_coname calc_atm)
wenzelm@36278	4519	apply(drule_tac pi="[(x,y)]" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4520	apply(perm_simp add: CAND_eqvt_name calc_atm)
wenzelm@36278	4521	done
wenzelm@36278	4522
wenzelm@36278	4523	lemma CAND_NotR_elim:
wenzelm@36278	4524	assumes a: "<a>:NotR (x).M a \<in> (\<parallel><B>\<parallel>)" "<a>:NotR (x).M a \<notin> BINDINGc B (\<parallel>(B)\<parallel>)"
wenzelm@36278	4525	shows "\<exists>B'. B = NOT B' \<and> (x):M \<in> (\<parallel>(B')\<parallel>)"
wenzelm@36278	4526	using a
wenzelm@36278	4527	apply(nominal_induct B rule: ty.strong_induct)
wenzelm@36278	4528	apply(simp_all add: ty.inject AXIOMSc_def ctrm.inject alpha)
wenzelm@36278	4529	apply(auto intro: CANDs_alpha simp add: trm.inject calc_atm abs_fresh fresh_atm)
wenzelm@36278	4530	apply(drule_tac pi="[(a,aa)]" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4531	apply(auto simp add: CAND_eqvt_coname calc_atm intro: CANDs_alpha)
wenzelm@36278	4532	done
wenzelm@36278	4533
wenzelm@36278	4534	lemma CAND_NotL_elim_aux:
wenzelm@36278	4535	assumes a: "(x):NotL <a>.M x \<in> NEGn B (\<parallel><B>\<parallel>)" "(x):NotL <a>.M x \<notin> BINDINGn B (\<parallel><B>\<parallel>)"
wenzelm@36278	4536	shows "\<exists>B'. B = NOT B' \<and> <a>:M \<in> (\<parallel><B'>\<parallel>)"
wenzelm@36278	4537	using a
wenzelm@36278	4538	apply(nominal_induct B rule: ty.strong_induct)
wenzelm@36278	4539	apply(simp_all add: ty.inject AXIOMSn_def ntrm.inject alpha)
wenzelm@36278	4540	apply(auto intro: CANDs_alpha simp add: trm.inject calc_atm abs_fresh fresh_atm)
wenzelm@36278	4541	apply(drule_tac pi="[(x,xa)]" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4542	apply(auto simp add: CAND_eqvt_name calc_atm intro: CANDs_alpha)
wenzelm@36278	4543	done
wenzelm@36278	4544
wenzelm@36278	4545	lemmas CAND_NotL_elim = CAND_NotL_elim_aux[OF NEG_elim(2)]
wenzelm@36278	4546
wenzelm@36278	4547	lemma CAND_AndR_elim:
wenzelm@36278	4548	assumes a: "<a>:AndR <b>.M <c>.N a \<in> (\<parallel><B>\<parallel>)" "<a>:AndR <b>.M <c>.N a \<notin> BINDINGc B (\<parallel>(B)\<parallel>)"
wenzelm@36278	4549	shows "\<exists>B1 B2. B = B1 AND B2 \<and> <b>:M \<in> (\<parallel><B1>\<parallel>) \<and> <c>:N \<in> (\<parallel><B2>\<parallel>)"
wenzelm@36278	4550	using a
wenzelm@36278	4551	apply(nominal_induct B rule: ty.strong_induct)
wenzelm@36278	4552	apply(simp_all add: ty.inject AXIOMSc_def ctrm.inject alpha)
wenzelm@36278	4553	apply(auto intro: CANDs_alpha simp add: trm.inject calc_atm abs_fresh fresh_atm)
wenzelm@36278	4554	apply(drule_tac pi="[(a,ca)]" and x="<a>:Ma" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4555	apply(simp add: CAND_eqvt_coname calc_atm)
wenzelm@36278	4556	apply(auto intro: CANDs_alpha)[1]
wenzelm@36278	4557	apply(drule_tac pi="[(a,ca)]" and x="<a>:Na" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4558	apply(simp add: CAND_eqvt_coname calc_atm)
wenzelm@36278	4559	apply(auto intro: CANDs_alpha)[1]
wenzelm@36278	4560	apply(drule_tac pi="[(a,ca)]" and x="<a>:Ma" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4561	apply(simp add: CAND_eqvt_coname calc_atm)
wenzelm@36278	4562	apply(auto intro: CANDs_alpha)[1]
wenzelm@36278	4563	apply(case_tac "a=ba")
wenzelm@36278	4564	apply(simp)
wenzelm@36278	4565	apply(drule_tac pi="[(ba,ca)]" and x="<ba>:Na" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4566	apply(simp add: CAND_eqvt_coname calc_atm)
wenzelm@36278	4567	apply(auto intro: CANDs_alpha)[1]
wenzelm@36278	4568	apply(simp)
wenzelm@36278	4569	apply(case_tac "ca=ba")
wenzelm@36278	4570	apply(simp)
wenzelm@36278	4571	apply(drule_tac pi="[(a,ba)]" and x="<ba>:Na" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4572	apply(simp add: CAND_eqvt_coname calc_atm)
wenzelm@36278	4573	apply(auto intro: CANDs_alpha)[1]
wenzelm@36278	4574	apply(simp)
wenzelm@36278	4575	apply(drule_tac pi="[(a,ca)]" and x="<ba>:Na" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4576	apply(simp add: CAND_eqvt_coname calc_atm)
wenzelm@36278	4577	apply(auto intro: CANDs_alpha)[1]
wenzelm@36278	4578	apply(case_tac "a=aa")
wenzelm@36278	4579	apply(simp)
wenzelm@36278	4580	apply(drule_tac pi="[(aa,ca)]" and x="<aa>:Ma" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4581	apply(simp add: CAND_eqvt_coname calc_atm)
wenzelm@36278	4582	apply(auto intro: CANDs_alpha)[1]
wenzelm@36278	4583	apply(simp)
wenzelm@36278	4584	apply(case_tac "ca=aa")
wenzelm@36278	4585	apply(simp)
wenzelm@36278	4586	apply(drule_tac pi="[(a,aa)]" and x="<aa>:Ma" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4587	apply(simp add: CAND_eqvt_coname calc_atm)
wenzelm@36278	4588	apply(auto intro: CANDs_alpha)[1]
wenzelm@36278	4589	apply(simp)
wenzelm@36278	4590	apply(drule_tac pi="[(a,ca)]" and x="<aa>:Ma" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4591	apply(simp add: CAND_eqvt_coname calc_atm)
wenzelm@36278	4592	apply(auto intro: CANDs_alpha)[1]
wenzelm@36278	4593	apply(drule_tac pi="[(a,ca)]" and x="<a>:Na" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4594	apply(simp add: CAND_eqvt_coname calc_atm)
wenzelm@36278	4595	apply(auto intro: CANDs_alpha)[1]
wenzelm@36278	4596	apply(case_tac "a=aa")
wenzelm@36278	4597	apply(simp)
wenzelm@36278	4598	apply(drule_tac pi="[(aa,ca)]" and x="<aa>:Ma" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4599	apply(simp add: CAND_eqvt_coname calc_atm)
wenzelm@36278	4600	apply(auto intro: CANDs_alpha)[1]
wenzelm@36278	4601	apply(simp)
wenzelm@36278	4602	apply(case_tac "ca=aa")
wenzelm@36278	4603	apply(simp)
wenzelm@36278	4604	apply(drule_tac pi="[(a,aa)]" and x="<aa>:Ma" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4605	apply(simp add: CAND_eqvt_coname calc_atm)
wenzelm@36278	4606	apply(auto intro: CANDs_alpha)[1]
wenzelm@36278	4607	apply(simp)
wenzelm@36278	4608	apply(drule_tac pi="[(a,ca)]" and x="<aa>:Ma" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4609	apply(simp add: CAND_eqvt_coname calc_atm)
wenzelm@36278	4610	apply(auto intro: CANDs_alpha)[1]
wenzelm@36278	4611	apply(case_tac "a=ba")
wenzelm@36278	4612	apply(simp)
wenzelm@36278	4613	apply(drule_tac pi="[(ba,ca)]" and x="<ba>:Na" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4614	apply(simp add: CAND_eqvt_coname calc_atm)
wenzelm@36278	4615	apply(auto intro: CANDs_alpha)[1]
wenzelm@36278	4616	apply(simp)
wenzelm@36278	4617	apply(case_tac "ca=ba")
wenzelm@36278	4618	apply(simp)
wenzelm@36278	4619	apply(drule_tac pi="[(a,ba)]" and x="<ba>:Na" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4620	apply(simp add: CAND_eqvt_coname calc_atm)
wenzelm@36278	4621	apply(auto intro: CANDs_alpha)[1]
wenzelm@36278	4622	apply(simp)
wenzelm@36278	4623	apply(drule_tac pi="[(a,ca)]" and x="<ba>:Na" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4624	apply(simp add: CAND_eqvt_coname calc_atm)
wenzelm@36278	4625	apply(auto intro: CANDs_alpha)[1]
wenzelm@36278	4626	done
wenzelm@36278	4627
wenzelm@36278	4628	lemma CAND_OrR1_elim:
wenzelm@36278	4629	assumes a: "<a>:OrR1 <b>.M a \<in> (\<parallel><B>\<parallel>)" "<a>:OrR1 <b>.M a \<notin> BINDINGc B (\<parallel>(B)\<parallel>)"
wenzelm@36278	4630	shows "\<exists>B1 B2. B = B1 OR B2 \<and> <b>:M \<in> (\<parallel><B1>\<parallel>)"
wenzelm@36278	4631	using a
wenzelm@36278	4632	apply(nominal_induct B rule: ty.strong_induct)
wenzelm@36278	4633	apply(simp_all add: ty.inject AXIOMSc_def ctrm.inject alpha)
wenzelm@36278	4634	apply(auto intro: CANDs_alpha simp add: trm.inject calc_atm abs_fresh fresh_atm)
wenzelm@36278	4635	apply(drule_tac pi="[(a,ba)]" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4636	apply(auto simp add: CAND_eqvt_coname calc_atm intro: CANDs_alpha)
wenzelm@36278	4637	apply(case_tac "a=aa")
wenzelm@36278	4638	apply(simp)
wenzelm@36278	4639	apply(drule_tac pi="[(aa,ba)]" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4640	apply(auto simp add: CAND_eqvt_coname calc_atm intro: CANDs_alpha)
wenzelm@36278	4641	apply(case_tac "ba=aa")
wenzelm@36278	4642	apply(simp)
wenzelm@36278	4643	apply(drule_tac pi="[(a,aa)]" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4644	apply(auto simp add: CAND_eqvt_coname calc_atm intro: CANDs_alpha)
wenzelm@36278	4645	apply(drule_tac pi="[(a,ba)]" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4646	apply(auto simp add: CAND_eqvt_coname calc_atm intro: CANDs_alpha)
wenzelm@36278	4647	done
wenzelm@36278	4648
wenzelm@36278	4649	lemma CAND_OrR2_elim:
wenzelm@36278	4650	assumes a: "<a>:OrR2 <b>.M a \<in> (\<parallel><B>\<parallel>)" "<a>:OrR2 <b>.M a \<notin> BINDINGc B (\<parallel>(B)\<parallel>)"
wenzelm@36278	4651	shows "\<exists>B1 B2. B = B1 OR B2 \<and> <b>:M \<in> (\<parallel><B2>\<parallel>)"
wenzelm@36278	4652	using a
wenzelm@36278	4653	apply(nominal_induct B rule: ty.strong_induct)
wenzelm@36278	4654	apply(simp_all add: ty.inject AXIOMSc_def ctrm.inject alpha)
wenzelm@36278	4655	apply(auto intro: CANDs_alpha simp add: trm.inject calc_atm abs_fresh fresh_atm)
wenzelm@36278	4656	apply(drule_tac pi="[(a,ba)]" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4657	apply(auto simp add: CAND_eqvt_coname calc_atm intro: CANDs_alpha)
wenzelm@36278	4658	apply(case_tac "a=aa")
wenzelm@36278	4659	apply(simp)
wenzelm@36278	4660	apply(drule_tac pi="[(aa,ba)]" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4661	apply(auto simp add: CAND_eqvt_coname calc_atm intro: CANDs_alpha)
wenzelm@36278	4662	apply(case_tac "ba=aa")
wenzelm@36278	4663	apply(simp)
wenzelm@36278	4664	apply(drule_tac pi="[(a,aa)]" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4665	apply(auto simp add: CAND_eqvt_coname calc_atm intro: CANDs_alpha)
wenzelm@36278	4666	apply(drule_tac pi="[(a,ba)]" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4667	apply(auto simp add: CAND_eqvt_coname calc_atm intro: CANDs_alpha)
wenzelm@36278	4668	done
wenzelm@36278	4669
wenzelm@36278	4670	lemma CAND_OrL_elim_aux:
wenzelm@36278	4671	assumes a: "(x):(OrL (y).M (z).N x) \<in> NEGn B (\<parallel><B>\<parallel>)" "(x):(OrL (y).M (z).N x) \<notin> BINDINGn B (\<parallel><B>\<parallel>)"
wenzelm@36278	4672	shows "\<exists>B1 B2. B = B1 OR B2 \<and> (y):M \<in> (\<parallel>(B1)\<parallel>) \<and> (z):N \<in> (\<parallel>(B2)\<parallel>)"
wenzelm@36278	4673	using a
wenzelm@36278	4674	apply(nominal_induct B rule: ty.strong_induct)
wenzelm@36278	4675	apply(simp_all add: ty.inject AXIOMSn_def ntrm.inject alpha)
wenzelm@36278	4676	apply(auto intro: CANDs_alpha simp add: trm.inject calc_atm abs_fresh fresh_atm)
wenzelm@36278	4677	apply(drule_tac pi="[(x,za)]" and x="(x):Ma" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4678	apply(simp add: CAND_eqvt_name calc_atm)
wenzelm@36278	4679	apply(auto intro: CANDs_alpha)[1]
wenzelm@36278	4680	apply(drule_tac pi="[(x,za)]" and x="(x):Nb" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4681	apply(simp add: CAND_eqvt_name calc_atm)
wenzelm@36278	4682	apply(auto intro: CANDs_alpha)[1]
wenzelm@36278	4683	apply(drule_tac pi="[(x,za)]" and x="(x):Ma" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4684	apply(simp add: CAND_eqvt_name calc_atm)
wenzelm@36278	4685	apply(auto intro: CANDs_alpha)[1]
wenzelm@36278	4686	apply(case_tac "x=ya")
wenzelm@36278	4687	apply(simp)
wenzelm@36278	4688	apply(drule_tac pi="[(ya,za)]" and x="(ya):Nb" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4689	apply(simp add: CAND_eqvt_name calc_atm)
wenzelm@36278	4690	apply(auto intro: CANDs_alpha)[1]
wenzelm@36278	4691	apply(simp)
wenzelm@36278	4692	apply(case_tac "za=ya")
wenzelm@36278	4693	apply(simp)
wenzelm@36278	4694	apply(drule_tac pi="[(x,ya)]" and x="(ya):Nb" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4695	apply(simp add: CAND_eqvt_name calc_atm)
wenzelm@36278	4696	apply(auto intro: CANDs_alpha)[1]
wenzelm@36278	4697	apply(simp)
wenzelm@36278	4698	apply(drule_tac pi="[(x,za)]" and x="(ya):Nb" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4699	apply(simp add: CAND_eqvt_name calc_atm)
wenzelm@36278	4700	apply(auto intro: CANDs_alpha)[1]
wenzelm@36278	4701	apply(case_tac "x=xa")
wenzelm@36278	4702	apply(simp)
wenzelm@36278	4703	apply(drule_tac pi="[(xa,za)]" and x="(xa):Ma" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4704	apply(simp add: CAND_eqvt_name calc_atm)
wenzelm@36278	4705	apply(auto intro: CANDs_alpha)[1]
wenzelm@36278	4706	apply(simp)
wenzelm@36278	4707	apply(case_tac "za=xa")
wenzelm@36278	4708	apply(simp)
wenzelm@36278	4709	apply(drule_tac pi="[(x,xa)]" and x="(xa):Ma" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4710	apply(simp add: CAND_eqvt_name calc_atm)
wenzelm@36278	4711	apply(auto intro: CANDs_alpha)[1]
wenzelm@36278	4712	apply(simp)
wenzelm@36278	4713	apply(drule_tac pi="[(x,za)]" and x="(xa):Ma" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4714	apply(simp add: CAND_eqvt_name calc_atm)
wenzelm@36278	4715	apply(auto intro: CANDs_alpha)[1]
wenzelm@36278	4716	apply(drule_tac pi="[(x,za)]" and x="(x):Nb" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4717	apply(simp add: CAND_eqvt_name calc_atm)
wenzelm@36278	4718	apply(auto intro: CANDs_alpha)[1]
wenzelm@36278	4719	apply(case_tac "x=xa")
wenzelm@36278	4720	apply(simp)
wenzelm@36278	4721	apply(drule_tac pi="[(xa,za)]" and x="(xa):Ma" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4722	apply(simp add: CAND_eqvt_name calc_atm)
wenzelm@36278	4723	apply(auto intro: CANDs_alpha)[1]
wenzelm@36278	4724	apply(simp)
wenzelm@36278	4725	apply(case_tac "za=xa")
wenzelm@36278	4726	apply(simp)
wenzelm@36278	4727	apply(drule_tac pi="[(x,xa)]" and x="(xa):Ma" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4728	apply(simp add: CAND_eqvt_name calc_atm)
wenzelm@36278	4729	apply(auto intro: CANDs_alpha)[1]
wenzelm@36278	4730	apply(simp)
wenzelm@36278	4731	apply(drule_tac pi="[(x,za)]" and x="(xa):Ma" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4732	apply(simp add: CAND_eqvt_name calc_atm)
wenzelm@36278	4733	apply(auto intro: CANDs_alpha)[1]
wenzelm@36278	4734	apply(case_tac "x=ya")
wenzelm@36278	4735	apply(simp)
wenzelm@36278	4736	apply(drule_tac pi="[(ya,za)]" and x="(ya):Nb" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4737	apply(simp add: CAND_eqvt_name calc_atm)
wenzelm@36278	4738	apply(auto intro: CANDs_alpha)[1]
wenzelm@36278	4739	apply(simp)
wenzelm@36278	4740	apply(case_tac "za=ya")
wenzelm@36278	4741	apply(simp)
wenzelm@36278	4742	apply(drule_tac pi="[(x,ya)]" and x="(ya):Nb" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4743	apply(simp add: CAND_eqvt_name calc_atm)
wenzelm@36278	4744	apply(auto intro: CANDs_alpha)[1]
wenzelm@36278	4745	apply(simp)
wenzelm@36278	4746	apply(drule_tac pi="[(x,za)]" and x="(ya):Nb" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4747	apply(simp add: CAND_eqvt_name calc_atm)
wenzelm@36278	4748	apply(auto intro: CANDs_alpha)[1]
wenzelm@36278	4749	done
wenzelm@36278	4750
wenzelm@36278	4751	lemmas CAND_OrL_elim = CAND_OrL_elim_aux[OF NEG_elim(2)]
wenzelm@36278	4752
wenzelm@36278	4753	lemma CAND_AndL1_elim_aux:
wenzelm@36278	4754	assumes a: "(x):(AndL1 (y).M x) \<in> NEGn B (\<parallel><B>\<parallel>)" "(x):(AndL1 (y).M x) \<notin> BINDINGn B (\<parallel><B>\<parallel>)"
wenzelm@36278	4755	shows "\<exists>B1 B2. B = B1 AND B2 \<and> (y):M \<in> (\<parallel>(B1)\<parallel>)"
wenzelm@36278	4756	using a
wenzelm@36278	4757	apply(nominal_induct B rule: ty.strong_induct)
wenzelm@36278	4758	apply(simp_all add: ty.inject AXIOMSn_def ntrm.inject alpha)
wenzelm@36278	4759	apply(auto intro: CANDs_alpha simp add: trm.inject calc_atm abs_fresh fresh_atm)
wenzelm@36278	4760	apply(drule_tac pi="[(x,ya)]" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4761	apply(auto simp add: CAND_eqvt_name calc_atm intro: CANDs_alpha)
wenzelm@36278	4762	apply(case_tac "x=xa")
wenzelm@36278	4763	apply(simp)
wenzelm@36278	4764	apply(drule_tac pi="[(xa,ya)]" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4765	apply(auto simp add: CAND_eqvt_name calc_atm intro: CANDs_alpha)
wenzelm@36278	4766	apply(case_tac "ya=xa")
wenzelm@36278	4767	apply(simp)
wenzelm@36278	4768	apply(drule_tac pi="[(x,xa)]" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4769	apply(auto simp add: CAND_eqvt_name calc_atm intro: CANDs_alpha)
wenzelm@36278	4770	apply(drule_tac pi="[(x,ya)]" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4771	apply(auto simp add: CAND_eqvt_name calc_atm intro: CANDs_alpha)
wenzelm@36278	4772	done
wenzelm@36278	4773
wenzelm@36278	4774	lemmas CAND_AndL1_elim = CAND_AndL1_elim_aux[OF NEG_elim(2)]
wenzelm@36278	4775
wenzelm@36278	4776	lemma CAND_AndL2_elim_aux:
wenzelm@36278	4777	assumes a: "(x):(AndL2 (y).M x) \<in> NEGn B (\<parallel><B>\<parallel>)" "(x):(AndL2 (y).M x) \<notin> BINDINGn B (\<parallel><B>\<parallel>)"
wenzelm@36278	4778	shows "\<exists>B1 B2. B = B1 AND B2 \<and> (y):M \<in> (\<parallel>(B2)\<parallel>)"
wenzelm@36278	4779	using a
wenzelm@36278	4780	apply(nominal_induct B rule: ty.strong_induct)
wenzelm@36278	4781	apply(simp_all add: ty.inject AXIOMSn_def ntrm.inject alpha)
wenzelm@36278	4782	apply(auto intro: CANDs_alpha simp add: trm.inject calc_atm abs_fresh fresh_atm)
wenzelm@36278	4783	apply(drule_tac pi="[(x,ya)]" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4784	apply(auto simp add: CAND_eqvt_name calc_atm intro: CANDs_alpha)
wenzelm@36278	4785	apply(case_tac "x=xa")
wenzelm@36278	4786	apply(simp)
wenzelm@36278	4787	apply(drule_tac pi="[(xa,ya)]" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4788	apply(auto simp add: CAND_eqvt_name calc_atm intro: CANDs_alpha)
wenzelm@36278	4789	apply(case_tac "ya=xa")
wenzelm@36278	4790	apply(simp)
wenzelm@36278	4791	apply(drule_tac pi="[(x,xa)]" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4792	apply(auto simp add: CAND_eqvt_name calc_atm intro: CANDs_alpha)
wenzelm@36278	4793	apply(drule_tac pi="[(x,ya)]" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4794	apply(auto simp add: CAND_eqvt_name calc_atm intro: CANDs_alpha)
wenzelm@36278	4795	done
wenzelm@36278	4796
wenzelm@36278	4797	lemmas CAND_AndL2_elim = CAND_AndL2_elim_aux[OF NEG_elim(2)]
wenzelm@36278	4798
wenzelm@36278	4799	lemma CAND_ImpL_elim_aux:
wenzelm@36278	4800	assumes a: "(x):(ImpL <a>.M (z).N x) \<in> NEGn B (\<parallel><B>\<parallel>)" "(x):(ImpL <a>.M (z).N x) \<notin> BINDINGn B (\<parallel><B>\<parallel>)"
wenzelm@36278	4801	shows "\<exists>B1 B2. B = B1 IMP B2 \<and> <a>:M \<in> (\<parallel><B1>\<parallel>) \<and> (z):N \<in> (\<parallel>(B2)\<parallel>)"
wenzelm@36278	4802	using a
wenzelm@36278	4803	apply(nominal_induct B rule: ty.strong_induct)
wenzelm@36278	4804	apply(simp_all add: ty.inject AXIOMSn_def ntrm.inject alpha)
wenzelm@36278	4805	apply(auto intro: CANDs_alpha simp add: trm.inject calc_atm abs_fresh fresh_atm)
wenzelm@36278	4806	apply(drule_tac pi="[(x,y)]" and x="<aa>:Ma" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4807	apply(simp add: CAND_eqvt_name calc_atm)
wenzelm@36278	4808	apply(auto intro: CANDs_alpha)[1]
wenzelm@36278	4809	apply(drule_tac pi="[(x,y)]" and x="(x):Nb" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4810	apply(simp add: CAND_eqvt_name calc_atm)
wenzelm@36278	4811	apply(auto intro: CANDs_alpha)[1]
wenzelm@36278	4812	apply(drule_tac pi="[(x,y)]" and x="<aa>:Ma" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4813	apply(simp add: CAND_eqvt_name calc_atm)
wenzelm@36278	4814	apply(auto intro: CANDs_alpha)[1]
wenzelm@36278	4815	apply(case_tac "x=xa")
wenzelm@36278	4816	apply(simp)
wenzelm@36278	4817	apply(drule_tac pi="[(xa,y)]" and x="(xa):Nb" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4818	apply(simp add: CAND_eqvt_name calc_atm)
wenzelm@36278	4819	apply(auto intro: CANDs_alpha)[1]
wenzelm@36278	4820	apply(simp)
wenzelm@36278	4821	apply(case_tac "y=xa")
wenzelm@36278	4822	apply(simp)
wenzelm@36278	4823	apply(drule_tac pi="[(x,xa)]" and x="(xa):Nb" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4824	apply(simp add: CAND_eqvt_name calc_atm)
wenzelm@36278	4825	apply(auto intro: CANDs_alpha)[1]
wenzelm@36278	4826	apply(simp)
wenzelm@36278	4827	apply(drule_tac pi="[(x,y)]" and x="(xa):Nb" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4828	apply(simp add: CAND_eqvt_name calc_atm)
wenzelm@36278	4829	apply(auto intro: CANDs_alpha)[1]
wenzelm@36278	4830	done
wenzelm@36278	4831
wenzelm@36278	4832	lemmas CAND_ImpL_elim = CAND_ImpL_elim_aux[OF NEG_elim(2)]
wenzelm@36278	4833
wenzelm@36278	4834	lemma CAND_ImpR_elim:
wenzelm@36278	4835	assumes a: "<a>:ImpR (x).<b>.M a \<in> (\<parallel><B>\<parallel>)" "<a>:ImpR (x).<b>.M a \<notin> BINDINGc B (\<parallel>(B)\<parallel>)"
wenzelm@36278	4836	shows "\<exists>B1 B2. B = B1 IMP B2 \<and>
wenzelm@36278	4837	(\<forall>z P. x\<sharp>(z,P) \<and> (z):P \<in> \<parallel>(B2)\<parallel> \<longrightarrow> (x):(M{b:=(z).P}) \<in> \<parallel>(B1)\<parallel>) \<and>
wenzelm@36278	4838	(\<forall>c Q. b\<sharp>(c,Q) \<and> <c>:Q \<in> \<parallel><B1>\<parallel> \<longrightarrow> <b>:(M{x:=<c>.Q}) \<in> \<parallel><B2>\<parallel>)"
wenzelm@36278	4839	using a
wenzelm@36278	4840	apply(nominal_induct B rule: ty.strong_induct)
wenzelm@36278	4841	apply(simp_all add: ty.inject AXIOMSc_def ctrm.inject alpha)
wenzelm@36278	4842	apply(auto intro: CANDs_alpha simp add: trm.inject calc_atm abs_fresh fresh_atm fresh_prod fresh_bij)
wenzelm@36278	4843	apply(generate_fresh "name")
wenzelm@36278	4844	apply(generate_fresh "coname")
wenzelm@36278	4845	apply(drule_tac a="ca" and z="c" in alpha_name_coname)
wenzelm@36278	4846	apply(simp)
wenzelm@36278	4847	apply(simp)
wenzelm@36278	4848	apply(simp)
wenzelm@36278	4849	apply(drule_tac x="[(xa,c)]\<bullet>[(aa,ca)]\<bullet>[(b,ca)]\<bullet>[(x,c)]\<bullet>z" in spec)
wenzelm@36278	4850	apply(drule_tac x="[(xa,c)]\<bullet>[(aa,ca)]\<bullet>[(b,ca)]\<bullet>[(x,c)]\<bullet>P" in spec)
wenzelm@36278	4851	apply(drule mp)
wenzelm@36278	4852	apply(rule conjI)
wenzelm@36278	4853	apply(auto simp add: calc_atm fresh_prod fresh_atm)[1]
wenzelm@36278	4854	apply(rule conjI)
wenzelm@36278	4855	apply(auto simp add: fresh_left calc_atm fresh_prod fresh_atm)[1]
wenzelm@36278	4856	apply(drule_tac pi="[(x,c)]" and X="\<parallel>(ty2)\<parallel>" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4857	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	4858	apply(drule_tac pi="[(b,ca)]" and X="\<parallel>(ty2)\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4859	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	4860	apply(drule_tac pi="[(aa,ca)]" and X="\<parallel>(ty2)\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4861	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	4862	apply(drule_tac pi="[(xa,c)]" and X="\<parallel>(ty2)\<parallel>" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4863	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	4864	apply(drule_tac pi="[(xa,c)]" and X="\<parallel>(ty1)\<parallel>" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4865	apply(simp add: CAND_eqvt_name CAND_eqvt_coname csubst_eqvt nsubst_eqvt)
wenzelm@36278	4866	apply(drule_tac pi="[(aa,ca)]" and X="\<parallel>(ty1)\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4867	apply(simp add: CAND_eqvt_name CAND_eqvt_coname csubst_eqvt nsubst_eqvt)
wenzelm@36278	4868	apply(drule_tac pi="[(b,ca)]" and X="\<parallel>(ty1)\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4869	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	4870	apply(drule_tac pi="[(x,c)]" and X="\<parallel>(ty1)\<parallel>" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4871	apply(perm_simp add: CAND_eqvt_name CAND_eqvt_coname csubst_eqvt nsubst_eqvt)
wenzelm@36278	4872	apply(generate_fresh "name")
wenzelm@36278	4873	apply(generate_fresh "coname")
wenzelm@36278	4874	apply(drule_tac a="cb" and z="ca" in alpha_name_coname)
wenzelm@36278	4875	apply(simp)
wenzelm@36278	4876	apply(simp)
wenzelm@36278	4877	apply(simp)
wenzelm@36278	4878	apply(drule_tac x="[(xa,ca)]\<bullet>[(aa,cb)]\<bullet>[(b,cb)]\<bullet>[(x,ca)]\<bullet>c" in spec)
wenzelm@36278	4879	apply(drule_tac x="[(xa,ca)]\<bullet>[(aa,cb)]\<bullet>[(b,cb)]\<bullet>[(x,ca)]\<bullet>Q" in spec)
wenzelm@36278	4880	apply(drule mp)
wenzelm@36278	4881	apply(rule conjI)
wenzelm@36278	4882	apply(auto simp add: calc_atm fresh_prod fresh_atm)[1]
wenzelm@36278	4883	apply(rule conjI)
wenzelm@36278	4884	apply(auto simp add: fresh_left calc_atm fresh_prod fresh_atm)[1]
wenzelm@36278	4885	apply(drule_tac pi="[(x,ca)]" and X="\<parallel><ty1>\<parallel>" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4886	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	4887	apply(drule_tac pi="[(b,cb)]" and X="\<parallel><ty1>\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4888	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	4889	apply(drule_tac pi="[(aa,cb)]" and X="\<parallel><ty1>\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4890	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	4891	apply(drule_tac pi="[(xa,ca)]" and X="\<parallel><ty1>\<parallel>" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4892	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	4893	apply(drule_tac pi="[(xa,ca)]" and X="\<parallel><ty2>\<parallel>" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4894	apply(simp add: CAND_eqvt_name CAND_eqvt_coname csubst_eqvt nsubst_eqvt)
wenzelm@36278	4895	apply(drule_tac pi="[(aa,cb)]" and X="\<parallel><ty2>\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4896	apply(simp add: CAND_eqvt_name CAND_eqvt_coname csubst_eqvt nsubst_eqvt)
wenzelm@36278	4897	apply(drule_tac pi="[(b,cb)]" and X="\<parallel><ty2>\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4898	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	4899	apply(drule_tac pi="[(x,ca)]" and X="\<parallel><ty2>\<parallel>" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4900	apply(perm_simp add: CAND_eqvt_name CAND_eqvt_coname csubst_eqvt nsubst_eqvt)
wenzelm@36278	4901	apply(generate_fresh "name")
wenzelm@36278	4902	apply(generate_fresh "coname")
wenzelm@36278	4903	apply(drule_tac a="ca" and z="c" in alpha_name_coname)
wenzelm@36278	4904	apply(simp add: fresh_left calc_atm fresh_prod fresh_atm)
wenzelm@36278	4905	apply(simp add: fresh_left calc_atm fresh_prod fresh_atm)
wenzelm@36278	4906	apply(auto)[1]
wenzelm@36278	4907	apply(simp)
wenzelm@36278	4908	apply(drule_tac x="[(a,ba)]\<bullet>[(xa,c)]\<bullet>[(ba,ca)]\<bullet>[(b,ca)]\<bullet>[(x,c)]\<bullet>z" in spec)
wenzelm@36278	4909	apply(drule_tac x="[(a,ba)]\<bullet>[(xa,c)]\<bullet>[(ba,ca)]\<bullet>[(b,ca)]\<bullet>[(x,c)]\<bullet>P" in spec)
wenzelm@36278	4910	apply(drule mp)
wenzelm@36278	4911	apply(rule conjI)
wenzelm@36278	4912	apply(auto simp add: calc_atm fresh_prod fresh_atm)[1]
wenzelm@36278	4913	apply(rule conjI)
wenzelm@36278	4914	apply(auto simp add: fresh_left calc_atm fresh_prod fresh_atm)[1]
wenzelm@36278	4915	apply(drule_tac pi="[(x,c)]" and X="\<parallel>(ty2)\<parallel>" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4916	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	4917	apply(drule_tac pi="[(b,ca)]" and X="\<parallel>(ty2)\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4918	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	4919	apply(drule_tac pi="[(ba,ca)]" and X="\<parallel>(ty2)\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4920	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	4921	apply(drule_tac pi="[(xa,c)]" and X="\<parallel>(ty2)\<parallel>" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4922	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	4923	apply(drule_tac pi="[(a,ba)]" and X="\<parallel>(ty2)\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4924	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	4925	apply(drule_tac pi="[(a,ba)]" and X="\<parallel>(ty1)\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4926	apply(simp add: CAND_eqvt_name CAND_eqvt_coname csubst_eqvt nsubst_eqvt)
wenzelm@36278	4927	apply(drule_tac pi="[(xa,c)]" and X="\<parallel>(ty1)\<parallel>" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4928	apply(simp add: CAND_eqvt_name CAND_eqvt_coname csubst_eqvt nsubst_eqvt)
wenzelm@36278	4929	apply(drule_tac pi="[(ba,ca)]" and X="\<parallel>(ty1)\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4930	apply(simp add: CAND_eqvt_name CAND_eqvt_coname csubst_eqvt nsubst_eqvt)
wenzelm@36278	4931	apply(drule_tac pi="[(b,ca)]" and X="\<parallel>(ty1)\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4932	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	4933	apply(drule_tac pi="[(x,c)]" and X="\<parallel>(ty1)\<parallel>" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4934	apply(perm_simp add: CAND_eqvt_name CAND_eqvt_coname csubst_eqvt nsubst_eqvt)
wenzelm@36278	4935	apply(generate_fresh "name")
wenzelm@36278	4936	apply(generate_fresh "coname")
wenzelm@36278	4937	apply(drule_tac a="cb" and z="ca" in alpha_name_coname)
wenzelm@36278	4938	apply(simp add: fresh_left calc_atm fresh_prod fresh_atm)
wenzelm@36278	4939	apply(simp add: fresh_left calc_atm fresh_prod fresh_atm)
wenzelm@36278	4940	apply(auto)[1]
wenzelm@36278	4941	apply(simp)
wenzelm@36278	4942	apply(drule_tac x="[(a,ba)]\<bullet>[(xa,ca)]\<bullet>[(ba,cb)]\<bullet>[(b,cb)]\<bullet>[(x,ca)]\<bullet>c" in spec)
wenzelm@36278	4943	apply(drule_tac x="[(a,ba)]\<bullet>[(xa,ca)]\<bullet>[(ba,cb)]\<bullet>[(b,cb)]\<bullet>[(x,ca)]\<bullet>Q" in spec)
wenzelm@36278	4944	apply(drule mp)
wenzelm@36278	4945	apply(rule conjI)
wenzelm@36278	4946	apply(auto simp add: calc_atm fresh_prod fresh_atm)[1]
wenzelm@36278	4947	apply(rule conjI)
wenzelm@36278	4948	apply(auto simp add: fresh_left calc_atm fresh_prod fresh_atm)[1]
wenzelm@36278	4949	apply(drule_tac pi="[(x,ca)]" and X="\<parallel><ty1>\<parallel>" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4950	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	4951	apply(drule_tac pi="[(b,cb)]" and X="\<parallel><ty1>\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4952	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	4953	apply(drule_tac pi="[(ba,cb)]" and X="\<parallel><ty1>\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4954	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	4955	apply(drule_tac pi="[(xa,ca)]" and X="\<parallel><ty1>\<parallel>" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4956	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	4957	apply(drule_tac pi="[(a,ba)]" and X="\<parallel><ty1>\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4958	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	4959	apply(drule_tac pi="[(a,ba)]" and X="\<parallel><ty2>\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4960	apply(simp add: CAND_eqvt_name CAND_eqvt_coname csubst_eqvt nsubst_eqvt)
wenzelm@36278	4961	apply(drule_tac pi="[(xa,ca)]" and X="\<parallel><ty2>\<parallel>" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4962	apply(simp add: CAND_eqvt_name CAND_eqvt_coname csubst_eqvt nsubst_eqvt)
wenzelm@36278	4963	apply(drule_tac pi="[(ba,cb)]" and X="\<parallel><ty2>\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4964	apply(simp add: CAND_eqvt_name CAND_eqvt_coname csubst_eqvt nsubst_eqvt)
wenzelm@36278	4965	apply(drule_tac pi="[(b,cb)]" and X="\<parallel><ty2>\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4966	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	4967	apply(drule_tac pi="[(x,ca)]" and X="\<parallel><ty2>\<parallel>" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4968	apply(perm_simp add: CAND_eqvt_name CAND_eqvt_coname csubst_eqvt nsubst_eqvt)
wenzelm@36278	4969	apply(case_tac "a=aa")
wenzelm@36278	4970	apply(simp)
wenzelm@36278	4971	apply(generate_fresh "name")
wenzelm@36278	4972	apply(generate_fresh "coname")
wenzelm@36278	4973	apply(drule_tac a="ca" and z="c" in alpha_name_coname)
wenzelm@36278	4974	apply(simp add: fresh_left calc_atm fresh_prod fresh_atm)
wenzelm@36278	4975	apply(simp add: fresh_left calc_atm fresh_prod fresh_atm)
wenzelm@36278	4976	apply(auto)[1]
wenzelm@36278	4977	apply(simp)
wenzelm@36278	4978	apply(drule_tac x="[(aa,ba)]\<bullet>[(xa,c)]\<bullet>[(ba,ca)]\<bullet>[(b,ca)]\<bullet>[(x,c)]\<bullet>z" in spec)
wenzelm@36278	4979	apply(drule_tac x="[(aa,ba)]\<bullet>[(xa,c)]\<bullet>[(ba,ca)]\<bullet>[(b,ca)]\<bullet>[(x,c)]\<bullet>P" in spec)
wenzelm@36278	4980	apply(drule mp)
wenzelm@36278	4981	apply(rule conjI)
wenzelm@36278	4982	apply(auto simp add: calc_atm fresh_prod fresh_atm)[1]
wenzelm@36278	4983	apply(rule conjI)
wenzelm@36278	4984	apply(auto simp add: fresh_left calc_atm fresh_prod fresh_atm)[1]
wenzelm@36278	4985	apply(drule_tac pi="[(x,c)]" and X="\<parallel>(ty2)\<parallel>" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4986	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	4987	apply(drule_tac pi="[(b,ca)]" and X="\<parallel>(ty2)\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4988	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	4989	apply(drule_tac pi="[(ba,ca)]" and X="\<parallel>(ty2)\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4990	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	4991	apply(drule_tac pi="[(xa,c)]" and X="\<parallel>(ty2)\<parallel>" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4992	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	4993	apply(drule_tac pi="[(aa,ba)]" and X="\<parallel>(ty2)\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4994	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	4995	apply(drule_tac pi="[(aa,ba)]" and X="\<parallel>(ty1)\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	4996	apply(simp add: CAND_eqvt_name CAND_eqvt_coname csubst_eqvt nsubst_eqvt)
wenzelm@36278	4997	apply(drule_tac pi="[(xa,c)]" and X="\<parallel>(ty1)\<parallel>" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	4998	apply(simp add: CAND_eqvt_name CAND_eqvt_coname csubst_eqvt nsubst_eqvt)
wenzelm@36278	4999	apply(drule_tac pi="[(ba,ca)]" and X="\<parallel>(ty1)\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	5000	apply(simp add: CAND_eqvt_name CAND_eqvt_coname csubst_eqvt nsubst_eqvt)
wenzelm@36278	5001	apply(drule_tac pi="[(b,ca)]" and X="\<parallel>(ty1)\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	5002	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	5003	apply(drule_tac pi="[(x,c)]" and X="\<parallel>(ty1)\<parallel>" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	5004	apply(perm_simp add: CAND_eqvt_name CAND_eqvt_coname csubst_eqvt nsubst_eqvt)
wenzelm@36278	5005	apply(simp)
wenzelm@36278	5006	apply(case_tac "ba=aa")
wenzelm@36278	5007	apply(simp)
wenzelm@36278	5008	apply(generate_fresh "name")
wenzelm@36278	5009	apply(generate_fresh "coname")
wenzelm@36278	5010	apply(drule_tac a="ca" and z="c" in alpha_name_coname)
wenzelm@36278	5011	apply(simp add: fresh_left calc_atm fresh_prod fresh_atm)
wenzelm@36278	5012	apply(simp add: fresh_left calc_atm fresh_prod fresh_atm)
wenzelm@36278	5013	apply(auto)[1]
wenzelm@36278	5014	apply(simp)
wenzelm@36278	5015	apply(drule_tac x="[(a,aa)]\<bullet>[(xa,c)]\<bullet>[(a,ca)]\<bullet>[(b,ca)]\<bullet>[(x,c)]\<bullet>z" in spec)
wenzelm@36278	5016	apply(drule_tac x="[(a,aa)]\<bullet>[(xa,c)]\<bullet>[(a,ca)]\<bullet>[(b,ca)]\<bullet>[(x,c)]\<bullet>P" in spec)
wenzelm@36278	5017	apply(drule mp)
wenzelm@36278	5018	apply(rule conjI)
wenzelm@36278	5019	apply(auto simp add: calc_atm fresh_prod fresh_atm)[1]
wenzelm@36278	5020	apply(rule conjI)
wenzelm@36278	5021	apply(auto simp add: fresh_left calc_atm fresh_prod fresh_atm)[1]
wenzelm@36278	5022	apply(drule_tac pi="[(x,c)]" and X="\<parallel>(ty2)\<parallel>" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	5023	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	5024	apply(drule_tac pi="[(b,ca)]" and X="\<parallel>(ty2)\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	5025	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	5026	apply(drule_tac pi="[(a,ca)]" and X="\<parallel>(ty2)\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	5027	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	5028	apply(drule_tac pi="[(xa,c)]" and X="\<parallel>(ty2)\<parallel>" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	5029	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	5030	apply(drule_tac pi="[(a,aa)]" and X="\<parallel>(ty2)\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	5031	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	5032	apply(drule_tac pi="[(a,aa)]" and X="\<parallel>(ty1)\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	5033	apply(simp add: CAND_eqvt_name CAND_eqvt_coname csubst_eqvt nsubst_eqvt)
wenzelm@36278	5034	apply(drule_tac pi="[(xa,c)]" and X="\<parallel>(ty1)\<parallel>" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	5035	apply(simp add: CAND_eqvt_name CAND_eqvt_coname csubst_eqvt nsubst_eqvt)
wenzelm@36278	5036	apply(drule_tac pi="[(a,ca)]" and X="\<parallel>(ty1)\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	5037	apply(simp add: CAND_eqvt_name CAND_eqvt_coname csubst_eqvt nsubst_eqvt)
wenzelm@36278	5038	apply(drule_tac pi="[(b,ca)]" and X="\<parallel>(ty1)\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	5039	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	5040	apply(drule_tac pi="[(x,c)]" and X="\<parallel>(ty1)\<parallel>" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	5041	apply(perm_simp add: CAND_eqvt_name CAND_eqvt_coname csubst_eqvt nsubst_eqvt)
wenzelm@36278	5042	apply(simp)
wenzelm@36278	5043	apply(generate_fresh "name")
wenzelm@36278	5044	apply(generate_fresh "coname")
wenzelm@36278	5045	apply(drule_tac a="ca" and z="c" in alpha_name_coname)
wenzelm@36278	5046	apply(simp add: fresh_left calc_atm fresh_prod fresh_atm)
wenzelm@36278	5047	apply(simp add: fresh_left calc_atm fresh_prod fresh_atm)
wenzelm@36278	5048	apply(auto)[1]
wenzelm@36278	5049	apply(simp)
wenzelm@36278	5050	apply(drule_tac x="[(a,ba)]\<bullet>[(xa,c)]\<bullet>[(aa,ca)]\<bullet>[(b,ca)]\<bullet>[(x,c)]\<bullet>z" in spec)
wenzelm@36278	5051	apply(drule_tac x="[(a,ba)]\<bullet>[(xa,c)]\<bullet>[(aa,ca)]\<bullet>[(b,ca)]\<bullet>[(x,c)]\<bullet>P" in spec)
wenzelm@36278	5052	apply(drule mp)
wenzelm@36278	5053	apply(rule conjI)
wenzelm@36278	5054	apply(auto simp add: calc_atm fresh_prod fresh_atm)[1]
wenzelm@36278	5055	apply(rule conjI)
wenzelm@36278	5056	apply(auto simp add: fresh_left calc_atm fresh_prod fresh_atm)[1]
wenzelm@36278	5057	apply(drule_tac pi="[(x,c)]" and X="\<parallel>(ty2)\<parallel>" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	5058	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	5059	apply(drule_tac pi="[(b,ca)]" and X="\<parallel>(ty2)\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	5060	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	5061	apply(drule_tac pi="[(aa,ca)]" and X="\<parallel>(ty2)\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	5062	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	5063	apply(drule_tac pi="[(xa,c)]" and X="\<parallel>(ty2)\<parallel>" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	5064	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	5065	apply(drule_tac pi="[(a,ba)]" and X="\<parallel>(ty2)\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	5066	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	5067	apply(drule_tac pi="[(a,ba)]" and X="\<parallel>(ty1)\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	5068	apply(simp add: CAND_eqvt_name CAND_eqvt_coname csubst_eqvt nsubst_eqvt)
wenzelm@36278	5069	apply(drule_tac pi="[(xa,c)]" and X="\<parallel>(ty1)\<parallel>" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	5070	apply(simp add: CAND_eqvt_name CAND_eqvt_coname csubst_eqvt nsubst_eqvt)
wenzelm@36278	5071	apply(drule_tac pi="[(aa,ca)]" and X="\<parallel>(ty1)\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	5072	apply(simp add: CAND_eqvt_name CAND_eqvt_coname csubst_eqvt nsubst_eqvt)
wenzelm@36278	5073	apply(drule_tac pi="[(b,ca)]" and X="\<parallel>(ty1)\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	5074	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	5075	apply(drule_tac pi="[(x,c)]" and X="\<parallel>(ty1)\<parallel>" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	5076	apply(perm_simp add: CAND_eqvt_name CAND_eqvt_coname csubst_eqvt nsubst_eqvt)
wenzelm@36278	5077	apply(case_tac "a=aa")
wenzelm@36278	5078	apply(simp)
wenzelm@36278	5079	apply(generate_fresh "name")
wenzelm@36278	5080	apply(generate_fresh "coname")
wenzelm@36278	5081	apply(drule_tac a="cb" and z="ca" in alpha_name_coname)
wenzelm@36278	5082	apply(simp add: fresh_left calc_atm fresh_prod fresh_atm)
wenzelm@36278	5083	apply(simp add: fresh_left calc_atm fresh_prod fresh_atm)
wenzelm@36278	5084	apply(auto)[1]
wenzelm@36278	5085	apply(simp)
wenzelm@36278	5086	apply(drule_tac x="[(aa,ba)]\<bullet>[(xa,ca)]\<bullet>[(ba,cb)]\<bullet>[(b,cb)]\<bullet>[(x,ca)]\<bullet>c" in spec)
wenzelm@36278	5087	apply(drule_tac x="[(aa,ba)]\<bullet>[(xa,ca)]\<bullet>[(ba,cb)]\<bullet>[(b,cb)]\<bullet>[(x,ca)]\<bullet>Q" in spec)
wenzelm@36278	5088	apply(drule mp)
wenzelm@36278	5089	apply(rule conjI)
wenzelm@36278	5090	apply(auto simp add: calc_atm fresh_prod fresh_atm)[1]
wenzelm@36278	5091	apply(rule conjI)
wenzelm@36278	5092	apply(auto simp add: fresh_left calc_atm fresh_prod fresh_atm)[1]
wenzelm@36278	5093	apply(drule_tac pi="[(x,ca)]" and X="\<parallel><ty1>\<parallel>" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	5094	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	5095	apply(drule_tac pi="[(b,cb)]" and X="\<parallel><ty1>\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	5096	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	5097	apply(drule_tac pi="[(ba,cb)]" and X="\<parallel><ty1>\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	5098	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	5099	apply(drule_tac pi="[(xa,ca)]" and X="\<parallel><ty1>\<parallel>" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	5100	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	5101	apply(drule_tac pi="[(aa,ba)]" and X="\<parallel><ty1>\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	5102	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	5103	apply(drule_tac pi="[(aa,ba)]" and X="\<parallel><ty2>\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	5104	apply(simp add: CAND_eqvt_name CAND_eqvt_coname csubst_eqvt nsubst_eqvt)
wenzelm@36278	5105	apply(drule_tac pi="[(xa,ca)]" and X="\<parallel><ty2>\<parallel>" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	5106	apply(simp add: CAND_eqvt_name CAND_eqvt_coname csubst_eqvt nsubst_eqvt)
wenzelm@36278	5107	apply(drule_tac pi="[(ba,cb)]" and X="\<parallel><ty2>\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	5108	apply(simp add: CAND_eqvt_name CAND_eqvt_coname csubst_eqvt nsubst_eqvt)
wenzelm@36278	5109	apply(drule_tac pi="[(b,cb)]" and X="\<parallel><ty2>\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	5110	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	5111	apply(drule_tac pi="[(x,ca)]" and X="\<parallel><ty2>\<parallel>" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	5112	apply(perm_simp add: CAND_eqvt_name CAND_eqvt_coname csubst_eqvt nsubst_eqvt)
wenzelm@36278	5113	apply(simp)
wenzelm@36278	5114	apply(case_tac "ba=aa")
wenzelm@36278	5115	apply(simp)
wenzelm@36278	5116	apply(generate_fresh "name")
wenzelm@36278	5117	apply(generate_fresh "coname")
wenzelm@36278	5118	apply(drule_tac a="cb" and z="ca" in alpha_name_coname)
wenzelm@36278	5119	apply(simp add: fresh_left calc_atm fresh_prod fresh_atm)
wenzelm@36278	5120	apply(simp add: fresh_left calc_atm fresh_prod fresh_atm)
wenzelm@36278	5121	apply(auto)[1]
wenzelm@36278	5122	apply(simp)
wenzelm@36278	5123	apply(drule_tac x="[(a,aa)]\<bullet>[(xa,ca)]\<bullet>[(a,cb)]\<bullet>[(b,cb)]\<bullet>[(x,ca)]\<bullet>c" in spec)
wenzelm@36278	5124	apply(drule_tac x="[(a,aa)]\<bullet>[(xa,ca)]\<bullet>[(a,cb)]\<bullet>[(b,cb)]\<bullet>[(x,ca)]\<bullet>Q" in spec)
wenzelm@36278	5125	apply(drule mp)
wenzelm@36278	5126	apply(rule conjI)
wenzelm@36278	5127	apply(auto simp add: calc_atm fresh_prod fresh_atm)[1]
wenzelm@36278	5128	apply(rule conjI)
wenzelm@36278	5129	apply(auto simp add: fresh_left calc_atm fresh_prod fresh_atm)[1]
wenzelm@36278	5130	apply(drule_tac pi="[(x,ca)]" and X="\<parallel><ty1>\<parallel>" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	5131	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	5132	apply(drule_tac pi="[(b,cb)]" and X="\<parallel><ty1>\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	5133	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	5134	apply(drule_tac pi="[(a,cb)]" and X="\<parallel><ty1>\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	5135	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	5136	apply(drule_tac pi="[(xa,ca)]" and X="\<parallel><ty1>\<parallel>" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	5137	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	5138	apply(drule_tac pi="[(a,aa)]" and X="\<parallel><ty1>\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	5139	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	5140	apply(drule_tac pi="[(a,aa)]" and X="\<parallel><ty2>\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	5141	apply(simp add: CAND_eqvt_name CAND_eqvt_coname csubst_eqvt nsubst_eqvt)
wenzelm@36278	5142	apply(drule_tac pi="[(xa,ca)]" and X="\<parallel><ty2>\<parallel>" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	5143	apply(simp add: CAND_eqvt_name CAND_eqvt_coname csubst_eqvt nsubst_eqvt)
wenzelm@36278	5144	apply(drule_tac pi="[(a,cb)]" and X="\<parallel><ty2>\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	5145	apply(simp add: CAND_eqvt_name CAND_eqvt_coname csubst_eqvt nsubst_eqvt)
wenzelm@36278	5146	apply(drule_tac pi="[(b,cb)]" and X="\<parallel><ty2>\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	5147	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	5148	apply(drule_tac pi="[(x,ca)]" and X="\<parallel><ty2>\<parallel>" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	5149	apply(perm_simp add: CAND_eqvt_name CAND_eqvt_coname csubst_eqvt nsubst_eqvt)
wenzelm@36278	5150	apply(simp)
wenzelm@36278	5151	apply(generate_fresh "name")
wenzelm@36278	5152	apply(generate_fresh "coname")
wenzelm@36278	5153	apply(drule_tac a="cb" and z="ca" in alpha_name_coname)
wenzelm@36278	5154	apply(simp add: fresh_left calc_atm fresh_prod fresh_atm)
wenzelm@36278	5155	apply(simp add: fresh_left calc_atm fresh_prod fresh_atm)
wenzelm@36278	5156	apply(auto)[1]
wenzelm@36278	5157	apply(simp)
wenzelm@36278	5158	apply(drule_tac x="[(a,ba)]\<bullet>[(xa,ca)]\<bullet>[(aa,cb)]\<bullet>[(b,cb)]\<bullet>[(x,ca)]\<bullet>c" in spec)
wenzelm@36278	5159	apply(drule_tac x="[(a,ba)]\<bullet>[(xa,ca)]\<bullet>[(aa,cb)]\<bullet>[(b,cb)]\<bullet>[(x,ca)]\<bullet>Q" in spec)
wenzelm@36278	5160	apply(drule mp)
wenzelm@36278	5161	apply(rule conjI)
wenzelm@36278	5162	apply(auto simp add: calc_atm fresh_prod fresh_atm)[1]
wenzelm@36278	5163	apply(rule conjI)
wenzelm@36278	5164	apply(auto simp add: fresh_left calc_atm fresh_prod fresh_atm)[1]
wenzelm@36278	5165	apply(drule_tac pi="[(x,ca)]" and X="\<parallel><ty1>\<parallel>" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	5166	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	5167	apply(drule_tac pi="[(b,cb)]" and X="\<parallel><ty1>\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	5168	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	5169	apply(drule_tac pi="[(aa,cb)]" and X="\<parallel><ty1>\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	5170	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	5171	apply(drule_tac pi="[(xa,ca)]" and X="\<parallel><ty1>\<parallel>" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	5172	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	5173	apply(drule_tac pi="[(a,ba)]" and X="\<parallel><ty1>\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	5174	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	5175	apply(drule_tac pi="[(a,ba)]" and X="\<parallel><ty2>\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	5176	apply(simp add: CAND_eqvt_name CAND_eqvt_coname csubst_eqvt nsubst_eqvt)
wenzelm@36278	5177	apply(drule_tac pi="[(xa,ca)]" and X="\<parallel><ty2>\<parallel>" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	5178	apply(simp add: CAND_eqvt_name CAND_eqvt_coname csubst_eqvt nsubst_eqvt)
wenzelm@36278	5179	apply(drule_tac pi="[(aa,cb)]" and X="\<parallel><ty2>\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	5180	apply(simp add: CAND_eqvt_name CAND_eqvt_coname csubst_eqvt nsubst_eqvt)
wenzelm@36278	5181	apply(drule_tac pi="[(b,cb)]" and X="\<parallel><ty2>\<parallel>" in pt_set_bij2[OF pt_coname_inst, OF at_coname_inst])
wenzelm@36278	5182	apply(simp add: CAND_eqvt_name CAND_eqvt_coname)
wenzelm@36278	5183	apply(drule_tac pi="[(x,ca)]" and X="\<parallel><ty2>\<parallel>" in pt_set_bij2[OF pt_name_inst, OF at_name_inst])
wenzelm@36278	5184	apply(perm_simp add: CAND_eqvt_name CAND_eqvt_coname csubst_eqvt nsubst_eqvt)
wenzelm@36278	5185	done
wenzelm@36278	5186
wenzelm@36278	5187	text {* Main lemma 1 *}
wenzelm@36278	5188
wenzelm@36278	5189	lemma AXIOMS_imply_SNa:
wenzelm@36278	5190	shows "<a>:M \<in> AXIOMSc B \<Longrightarrow> SNa M"
wenzelm@36278	5191	and "(x):M \<in> AXIOMSn B \<Longrightarrow> SNa M"
wenzelm@36278	5192	apply -
wenzelm@36278	5193	apply(auto simp add: AXIOMSn_def AXIOMSc_def ntrm.inject ctrm.inject alpha)
wenzelm@36278	5194	apply(rule Ax_in_SNa)+
wenzelm@36278	5195	done
wenzelm@36278	5196
wenzelm@36278	5197	lemma BINDING_imply_SNa:
wenzelm@36278	5198	shows "<a>:M \<in> BINDINGc B (\<parallel>(B)\<parallel>) \<Longrightarrow> SNa M"
wenzelm@36278	5199	and "(x):M \<in> BINDINGn B (\<parallel><B>\<parallel>) \<Longrightarrow> SNa M"
wenzelm@36278	5200	apply -
wenzelm@36278	5201	apply(auto simp add: BINDINGn_def BINDINGc_def ntrm.inject ctrm.inject alpha)
wenzelm@36278	5202	apply(drule_tac x="x" in spec)
wenzelm@36278	5203	apply(drule_tac x="Ax x a" in spec)
wenzelm@36278	5204	apply(drule mp)
wenzelm@36278	5205	apply(rule Ax_in_CANDs)
wenzelm@36278	5206	apply(drule a_star_preserves_SNa)
wenzelm@36278	5207	apply(rule subst_with_ax2)
wenzelm@36278	5208	apply(simp add: crename_id)
wenzelm@36278	5209	apply(drule_tac x="x" in spec)
wenzelm@36278	5210	apply(drule_tac x="Ax x aa" in spec)
wenzelm@36278	5211	apply(drule mp)
wenzelm@36278	5212	apply(rule Ax_in_CANDs)
wenzelm@36278	5213	apply(drule a_star_preserves_SNa)
wenzelm@36278	5214	apply(rule subst_with_ax2)
wenzelm@36278	5215	apply(simp add: crename_id SNa_eqvt)
wenzelm@36278	5216	apply(drule_tac x="a" in spec)
wenzelm@36278	5217	apply(drule_tac x="Ax x a" in spec)
wenzelm@36278	5218	apply(drule mp)
wenzelm@36278	5219	apply(rule Ax_in_CANDs)
wenzelm@36278	5220	apply(drule a_star_preserves_SNa)
wenzelm@36278	5221	apply(rule subst_with_ax1)
wenzelm@36278	5222	apply(simp add: nrename_id)
wenzelm@36278	5223	apply(drule_tac x="a" in spec)
wenzelm@36278	5224	apply(drule_tac x="Ax xa a" in spec)
wenzelm@36278	5225	apply(drule mp)
wenzelm@36278	5226	apply(rule Ax_in_CANDs)
wenzelm@36278	5227	apply(drule a_star_preserves_SNa)
wenzelm@36278	5228	apply(rule subst_with_ax1)
wenzelm@36278	5229	apply(simp add: nrename_id SNa_eqvt)
wenzelm@36278	5230	done
wenzelm@36278	5231
wenzelm@36278	5232	lemma CANDs_imply_SNa:
wenzelm@36278	5233	shows "<a>:M \<in> \<parallel><B>\<parallel> \<Longrightarrow> SNa M"
wenzelm@36278	5234	and "(x):M \<in> \<parallel>(B)\<parallel> \<Longrightarrow> SNa M"
wenzelm@36278	5235	proof(induct B arbitrary: a x M rule: ty.induct)
wenzelm@36278	5236	case (PR X)
wenzelm@36278	5237	{ case 1
wenzelm@36278	5238	have "<a>:M \<in> \<parallel><PR X>\<parallel>" by fact
wenzelm@36278	5239	then have "<a>:M \<in> NEGc (PR X) (\<parallel>(PR X)\<parallel>)" by simp
wenzelm@36278	5240	then have "<a>:M \<in> AXIOMSc (PR X) \<union> BINDINGc (PR X) (\<parallel>(PR X)\<parallel>)" by simp
wenzelm@36278	5241	moreover
wenzelm@36278	5242	{ assume "<a>:M \<in> AXIOMSc (PR X)"
wenzelm@36278	5243	then have "SNa M" by (simp add: AXIOMS_imply_SNa)
wenzelm@36278	5244	}
wenzelm@36278	5245	moreover
wenzelm@36278	5246	{ assume "<a>:M \<in> BINDINGc (PR X) (\<parallel>(PR X)\<parallel>)"
wenzelm@36278	5247	then have "SNa M" by (simp add: BINDING_imply_SNa)
wenzelm@36278	5248	}
wenzelm@36278	5249	ultimately show "SNa M" by blast
wenzelm@36278	5250	next
wenzelm@36278	5251	case 2
wenzelm@36278	5252	have "(x):M \<in> (\<parallel>(PR X)\<parallel>)" by fact
wenzelm@36278	5253	then have "(x):M \<in> NEGn (PR X) (\<parallel><PR X>\<parallel>)" using NEG_simp by blast
wenzelm@36278	5254	then have "(x):M \<in> AXIOMSn (PR X) \<union> BINDINGn (PR X) (\<parallel><PR X>\<parallel>)" by simp
wenzelm@36278	5255	moreover
wenzelm@36278	5256	{ assume "(x):M \<in> AXIOMSn (PR X)"
wenzelm@36278	5257	then have "SNa M" by (simp add: AXIOMS_imply_SNa)
wenzelm@36278	5258	}
wenzelm@36278	5259	moreover
wenzelm@36278	5260	{ assume "(x):M \<in> BINDINGn (PR X) (\<parallel><PR X>\<parallel>)"
wenzelm@36278	5261	then have "SNa M" by (simp only: BINDING_imply_SNa)
wenzelm@36278	5262	}
wenzelm@36278	5263	ultimately show "SNa M" by blast
wenzelm@36278	5264	}
wenzelm@36278	5265	next
wenzelm@36278	5266	case (NOT B)
wenzelm@36278	5267	have ih1: "\<And>a M. <a>:M \<in> \<parallel><B>\<parallel> \<Longrightarrow> SNa M" by fact
wenzelm@36278	5268	have ih2: "\<And>x M. (x):M \<in> \<parallel>(B)\<parallel> \<Longrightarrow> SNa M" by fact
wenzelm@36278	5269	{ case 1
wenzelm@36278	5270	have "<a>:M \<in> (\<parallel><NOT B>\<parallel>)" by fact
wenzelm@36278	5271	then have "<a>:M \<in> NEGc (NOT B) (\<parallel>(NOT B)\<parallel>)" by simp
wenzelm@36278	5272	then have "<a>:M \<in> AXIOMSc (NOT B) \<union> BINDINGc (NOT B) (\<parallel>(NOT B)\<parallel>) \<union> NOTRIGHT (NOT B) (\<parallel>(B)\<parallel>)" by simp
wenzelm@36278	5273	moreover
wenzelm@36278	5274	{ assume "<a>:M \<in> AXIOMSc (NOT B)"
wenzelm@36278	5275	then have "SNa M" by (simp add: AXIOMS_imply_SNa)
wenzelm@36278	5276	}
wenzelm@36278	5277	moreover
wenzelm@36278	5278	{ assume "<a>:M \<in> BINDINGc (NOT B) (\<parallel>(NOT B)\<parallel>)"
wenzelm@36278	5279	then have "SNa M" by (simp only: BINDING_imply_SNa)
wenzelm@36278	5280	}
wenzelm@36278	5281	moreover
wenzelm@36278	5282	{ assume "<a>:M \<in> NOTRIGHT (NOT B) (\<parallel>(B)\<parallel>)"
wenzelm@36278	5283	then obtain x' M' where eq: "M = NotR (x').M' a" and "(x'):M' \<in> (\<parallel>(B)\<parallel>)"
wenzelm@36278	5284	using NOTRIGHT_elim by blast
wenzelm@36278	5285	then have "SNa M'" using ih2 by blast
wenzelm@36278	5286	then have "SNa M" using eq by (simp add: NotR_in_SNa)
wenzelm@36278	5287	}
wenzelm@36278	5288	ultimately show "SNa M" by blast
wenzelm@36278	5289	next
wenzelm@36278	5290	case 2
wenzelm@36278	5291	have "(x):M \<in> (\<parallel>(NOT B)\<parallel>)" by fact
wenzelm@36278	5292	then have "(x):M \<in> NEGn (NOT B) (\<parallel><NOT B>\<parallel>)" using NEG_simp by blast
wenzelm@36278	5293	then have "(x):M \<in> AXIOMSn (NOT B) \<union> BINDINGn (NOT B) (\<parallel><NOT B>\<parallel>) \<union> NOTLEFT (NOT B) (\<parallel><B>\<parallel>)"
wenzelm@36278	5294	by (simp only: NEGn.simps)
wenzelm@36278	5295	moreover
wenzelm@36278	5296	{ assume "(x):M \<in> AXIOMSn (NOT B)"
wenzelm@36278	5297	then have "SNa M" by (simp add: AXIOMS_imply_SNa)
wenzelm@36278	5298	}
wenzelm@36278	5299	moreover
wenzelm@36278	5300	{ assume "(x):M \<in> BINDINGn (NOT B) (\<parallel><NOT B>\<parallel>)"
wenzelm@36278	5301	then have "SNa M" by (simp only: BINDING_imply_SNa)
wenzelm@36278	5302	}
wenzelm@36278	5303	moreover
wenzelm@36278	5304	{ assume "(x):M \<in> NOTLEFT (NOT B) (\<parallel><B>\<parallel>)"
wenzelm@36278	5305	then obtain a' M' where eq: "M = NotL <a'>.M' x" and "<a'>:M' \<in> (\<parallel><B>\<parallel>)"
wenzelm@36278	5306	using NOTLEFT_elim by blast
wenzelm@36278	5307	then have "SNa M'" using ih1 by blast
wenzelm@36278	5308	then have "SNa M" using eq by (simp add: NotL_in_SNa)
wenzelm@36278	5309	}
wenzelm@36278	5310	ultimately show "SNa M" by blast
wenzelm@36278	5311	}
wenzelm@36278	5312	next
wenzelm@36278	5313	case (AND A B)
wenzelm@36278	5314	have ih1: "\<And>a M. <a>:M \<in> \<parallel><A>\<parallel> \<Longrightarrow> SNa M" by fact
wenzelm@36278	5315	have ih2: "\<And>x M. (x):M \<in> \<parallel>(A)\<parallel> \<Longrightarrow> SNa M" by fact
wenzelm@36278	5316	have ih3: "\<And>a M. <a>:M \<in> \<parallel><B>\<parallel> \<Longrightarrow> SNa M" by fact
wenzelm@36278	5317	have ih4: "\<And>x M. (x):M \<in> \<parallel>(B)\<parallel> \<Longrightarrow> SNa M" by fact
wenzelm@36278	5318	{ case 1
wenzelm@36278	5319	have "<a>:M \<in> (\<parallel><A AND B>\<parallel>)" by fact
wenzelm@36278	5320	then have "<a>:M \<in> NEGc (A AND B) (\<parallel>(A AND B)\<parallel>)" by simp
wenzelm@36278	5321	then have "<a>:M \<in> AXIOMSc (A AND B) \<union> BINDINGc (A AND B) (\<parallel>(A AND B)\<parallel>)
wenzelm@36278	5322	\<union> ANDRIGHT (A AND B) (\<parallel><A>\<parallel>) (\<parallel><B>\<parallel>)" by simp
wenzelm@36278	5323	moreover
wenzelm@36278	5324	{ assume "<a>:M \<in> AXIOMSc (A AND B)"
wenzelm@36278	5325	then have "SNa M" by (simp add: AXIOMS_imply_SNa)
wenzelm@36278	5326	}
wenzelm@36278	5327	moreover
wenzelm@36278	5328	{ assume "<a>:M \<in> BINDINGc (A AND B) (\<parallel>(A AND B)\<parallel>)"
wenzelm@36278	5329	then have "SNa M" by (simp only: BINDING_imply_SNa)
wenzelm@36278	5330	}
wenzelm@36278	5331	moreover
wenzelm@36278	5332	{ assume "<a>:M \<in> ANDRIGHT (A AND B) (\<parallel><A>\<parallel>) (\<parallel><B>\<parallel>)"
wenzelm@36278	5333	then obtain a' M' b' N' where eq: "M = AndR <a'>.M' <b'>.N' a"
wenzelm@36278	5334	and "<a'>:M' \<in> (\<parallel><A>\<parallel>)" and "<b'>:N' \<in> (\<parallel><B>\<parallel>)"
wenzelm@36278	5335	by (erule_tac ANDRIGHT_elim, blast)
wenzelm@36278	5336	then have "SNa M'" and "SNa N'" using ih1 ih3 by blast+
wenzelm@36278	5337	then have "SNa M" using eq by (simp add: AndR_in_SNa)
wenzelm@36278	5338	}
wenzelm@36278	5339	ultimately show "SNa M" by blast
wenzelm@36278	5340	next
wenzelm@36278	5341	case 2
wenzelm@36278	5342	have "(x):M \<in> (\<parallel>(A AND B)\<parallel>)" by fact
wenzelm@36278	5343	then have "(x):M \<in> NEGn (A AND B) (\<parallel><A AND B>\<parallel>)" using NEG_simp by blast
wenzelm@36278	5344	then have "(x):M \<in> AXIOMSn (A AND B) \<union> BINDINGn (A AND B) (\<parallel><A AND B>\<parallel>)
wenzelm@36278	5345	\<union> ANDLEFT1 (A AND B) (\<parallel>(A)\<parallel>) \<union> ANDLEFT2 (A AND B) (\<parallel>(B)\<parallel>)"
wenzelm@36278	5346	by (simp only: NEGn.simps)
wenzelm@36278	5347	moreover
wenzelm@36278	5348	{ assume "(x):M \<in> AXIOMSn (A AND B)"
wenzelm@36278	5349	then have "SNa M" by (simp add: AXIOMS_imply_SNa)
wenzelm@36278	5350	}
wenzelm@36278	5351	moreover
wenzelm@36278	5352	{ assume "(x):M \<in> BINDINGn (A AND B) (\<parallel><A AND B>\<parallel>)"
wenzelm@36278	5353	then have "SNa M" by (simp only: BINDING_imply_SNa)
wenzelm@36278	5354	}
wenzelm@36278	5355	moreover
wenzelm@36278	5356	{ assume "(x):M \<in> ANDLEFT1 (A AND B) (\<parallel>(A)\<parallel>)"
wenzelm@36278	5357	then obtain x' M' where eq: "M = AndL1 (x').M' x" and "(x'):M' \<in> (\<parallel>(A)\<parallel>)"
wenzelm@36278	5358	using ANDLEFT1_elim by blast
wenzelm@36278	5359	then have "SNa M'" using ih2 by blast
wenzelm@36278	5360	then have "SNa M" using eq by (simp add: AndL1_in_SNa)
wenzelm@36278	5361	}
wenzelm@36278	5362	moreover
wenzelm@36278	5363	{ assume "(x):M \<in> ANDLEFT2 (A AND B) (\<parallel>(B)\<parallel>)"
wenzelm@36278	5364	then obtain x' M' where eq: "M = AndL2 (x').M' x" and "(x'):M' \<in> (\<parallel>(B)\<parallel>)"
wenzelm@36278	5365	using ANDLEFT2_elim by blast
wenzelm@36278	5366	then have "SNa M'" using ih4 by blast
wenzelm@36278	5367	then have "SNa M" using eq by (simp add: AndL2_in_SNa)
wenzelm@36278	5368	}
wenzelm@36278	5369	ultimately show "SNa M" by blast
wenzelm@36278	5370	}
wenzelm@36278	5371	next
wenzelm@36278	5372	case (OR A B)
wenzelm@36278	5373	have ih1: "\<And>a M. <a>:M \<in> \<parallel><A>\<parallel> \<Longrightarrow> SNa M" by fact
wenzelm@36278	5374	have ih2: "\<And>x M. (x):M \<in> \<parallel>(A)\<parallel> \<Longrightarrow> SNa M" by fact
wenzelm@36278	5375	have ih3: "\<And>a M. <a>:M \<in> \<parallel><B>\<parallel> \<Longrightarrow> SNa M" by fact
wenzelm@36278	5376	have ih4: "\<And>x M. (x):M \<in> \<parallel>(B)\<parallel> \<Longrightarrow> SNa M" by fact
wenzelm@36278	5377	{ case 1
wenzelm@36278	5378	have "<a>:M \<in> (\<parallel><A OR B>\<parallel>)" by fact
wenzelm@36278	5379	then have "<a>:M \<in> NEGc (A OR B) (\<parallel>(A OR B)\<parallel>)" by simp
wenzelm@36278	5380	then have "<a>:M \<in> AXIOMSc (A OR B) \<union> BINDINGc (A OR B) (\<parallel>(A OR B)\<parallel>)
wenzelm@36278	5381	\<union> ORRIGHT1 (A OR B) (\<parallel><A>\<parallel>) \<union> ORRIGHT2 (A OR B) (\<parallel><B>\<parallel>)" by simp
wenzelm@36278	5382	moreover
wenzelm@36278	5383	{ assume "<a>:M \<in> AXIOMSc (A OR B)"
wenzelm@36278	5384	then have "SNa M" by (simp add: AXIOMS_imply_SNa)
wenzelm@36278	5385	}
wenzelm@36278	5386	moreover
wenzelm@36278	5387	{ assume "<a>:M \<in> BINDINGc (A OR B) (\<parallel>(A OR B)\<parallel>)"
wenzelm@36278	5388	then have "SNa M" by (simp only: BINDING_imply_SNa)
wenzelm@36278	5389	}
wenzelm@36278	5390	moreover
wenzelm@36278	5391	{ assume "<a>:M \<in> ORRIGHT1 (A OR B) (\<parallel><A>\<parallel>)"
wenzelm@36278	5392	then obtain a' M' where eq: "M = OrR1 <a'>.M' a"
wenzelm@36278	5393	and "<a'>:M' \<in> (\<parallel><A>\<parallel>)"
wenzelm@36278	5394	by (erule_tac ORRIGHT1_elim, blast)
wenzelm@36278	5395	then have "SNa M'" using ih1 by blast
wenzelm@36278	5396	then have "SNa M" using eq by (simp add: OrR1_in_SNa)
wenzelm@36278	5397	}
wenzelm@36278	5398	moreover
wenzelm@36278	5399	{ assume "<a>:M \<in> ORRIGHT2 (A OR B) (\<parallel><B>\<parallel>)"
wenzelm@36278	5400	then obtain a' M' where eq: "M = OrR2 <a'>.M' a" and "<a'>:M' \<in> (\<parallel><B>\<parallel>)"
wenzelm@36278	5401	using ORRIGHT2_elim by blast
wenzelm@36278	5402	then have "SNa M'" using ih3 by blast
wenzelm@36278	5403	then have "SNa M" using eq by (simp add: OrR2_in_SNa)
wenzelm@36278	5404	}
wenzelm@36278	5405	ultimately show "SNa M" by blast
wenzelm@36278	5406	next
wenzelm@36278	5407	case 2
wenzelm@36278	5408	have "(x):M \<in> (\<parallel>(A OR B)\<parallel>)" by fact
wenzelm@36278	5409	then have "(x):M \<in> NEGn (A OR B) (\<parallel><A OR B>\<parallel>)" using NEG_simp by blast
wenzelm@36278	5410	then have "(x):M \<in> AXIOMSn (A OR B) \<union> BINDINGn (A OR B) (\<parallel><A OR B>\<parallel>)
wenzelm@36278	5411	\<union> ORLEFT (A OR B) (\<parallel>(A)\<parallel>) (\<parallel>(B)\<parallel>)"
wenzelm@36278	5412	by (simp only: NEGn.simps)
wenzelm@36278	5413	moreover
wenzelm@36278	5414	{ assume "(x):M \<in> AXIOMSn (A OR B)"
wenzelm@36278	5415	then have "SNa M" by (simp add: AXIOMS_imply_SNa)
wenzelm@36278	5416	}
wenzelm@36278	5417	moreover
wenzelm@36278	5418	{ assume "(x):M \<in> BINDINGn (A OR B) (\<parallel><A OR B>\<parallel>)"
wenzelm@36278	5419	then have "SNa M" by (simp only: BINDING_imply_SNa)
wenzelm@36278	5420	}
wenzelm@36278	5421	moreover
wenzelm@36278	5422	{ assume "(x):M \<in> ORLEFT (A OR B) (\<parallel>(A)\<parallel>) (\<parallel>(B)\<parallel>)"
wenzelm@36278	5423	then obtain x' M' y' N' where eq: "M = OrL (x').M' (y').N' x"
wenzelm@36278	5424	and "(x'):M' \<in> (\<parallel>(A)\<parallel>)" and "(y'):N' \<in> (\<parallel>(B)\<parallel>)"
wenzelm@36278	5425	by (erule_tac ORLEFT_elim, blast)
wenzelm@36278	5426	then have "SNa M'" and "SNa N'" using ih2 ih4 by blast+
wenzelm@36278	5427	then have "SNa M" using eq by (simp add: OrL_in_SNa)
wenzelm@36278	5428	}
wenzelm@36278	5429	ultimately show "SNa M" by blast
wenzelm@36278	5430	}
wenzelm@36278	5431	next
wenzelm@36278	5432	case (IMP A B)
wenzelm@36278	5433	have ih1: "\<And>a M. <a>:M \<in> \<parallel><A>\<parallel> \<Longrightarrow> SNa M" by fact
wenzelm@36278	5434	have ih2: "\<And>x M. (x):M \<in> \<parallel>(A)\<parallel> \<Longrightarrow> SNa M" by fact
wenzelm@36278	5435	have ih3: "\<And>a M. <a>:M \<in> \<parallel><B>\<parallel> \<Longrightarrow> SNa M" by fact
wenzelm@36278	5436	have ih4: "\<And>x M. (x):M \<in> \<parallel>(B)\<parallel> \<Longrightarrow> SNa M" by fact
wenzelm@36278	5437	{ case 1
wenzelm@36278	5438	have "<a>:M \<in> (\<parallel><A IMP B>\<parallel>)" by fact
wenzelm@36278	5439	then have "<a>:M \<in> NEGc (A IMP B) (\<parallel>(A IMP B)\<parallel>)" by simp
wenzelm@36278	5440	then have "<a>:M \<in> AXIOMSc (A IMP B) \<union> BINDINGc (A IMP B) (\<parallel>(A IMP B)\<parallel>)
wenzelm@36278	5441	\<union> IMPRIGHT (A IMP B) (\<parallel>(A)\<parallel>) (\<parallel><B>\<parallel>) (\<parallel>(B)\<parallel>) (\<parallel><A>\<parallel>)" by simp
wenzelm@36278	5442	moreover
wenzelm@36278	5443	{ assume "<a>:M \<in> AXIOMSc (A IMP B)"
wenzelm@36278	5444	then have "SNa M" by (simp add: AXIOMS_imply_SNa)
wenzelm@36278	5445	}
wenzelm@36278	5446	moreover
wenzelm@36278	5447	{ assume "<a>:M \<in> BINDINGc (A IMP B) (\<parallel>(A IMP B)\<parallel>)"
wenzelm@36278	5448	then have "SNa M" by (simp only: BINDING_imply_SNa)
wenzelm@36278	5449	}
wenzelm@36278	5450	moreover
wenzelm@36278	5451	{ assume "<a>:M \<in> IMPRIGHT (A IMP B) (\<parallel>(A)\<parallel>) (\<parallel><B>\<parallel>) (\<parallel>(B)\<parallel>) (\<parallel><A>\<parallel>)"
wenzelm@36278	5452	then obtain x' a' M' where eq: "M = ImpR (x').<a'>.M' a"
wenzelm@36278	5453	and imp: "\<forall>z P. x'\<sharp>(z,P) \<and> (z):P \<in> \<parallel>(B)\<parallel> \<longrightarrow> (x'):(M'{a':=(z).P}) \<in> \<parallel>(A)\<parallel>"
wenzelm@36278	5454	by (erule_tac IMPRIGHT_elim, blast)
wenzelm@36278	5455	obtain z::"name" where fs: "z\<sharp>x'" by (rule_tac exists_fresh, rule fin_supp, blast)
wenzelm@36278	5456	have "(z):Ax z a'\<in> \<parallel>(B)\<parallel>" by (simp add: Ax_in_CANDs)
wenzelm@36278	5457	with imp fs have "(x'):(M'{a':=(z).Ax z a'}) \<in> \<parallel>(A)\<parallel>" by (simp add: fresh_prod fresh_atm)
wenzelm@36278	5458	then have "SNa (M'{a':=(z).Ax z a'})" using ih2 by blast
wenzelm@36278	5459	moreover
wenzelm@54152	5460	have "M'{a':=(z).Ax z a'} \<longrightarrow>\<^sub>a* M'[a'\<turnstile>c>a']" by (simp add: subst_with_ax2)
wenzelm@36278	5461	ultimately have "SNa (M'[a'\<turnstile>c>a'])" by (simp add: a_star_preserves_SNa)
wenzelm@36278	5462	then have "SNa M'" by (simp add: crename_id)
wenzelm@36278	5463	then have "SNa M" using eq by (simp add: ImpR_in_SNa)
wenzelm@36278	5464	}
wenzelm@36278	5465	ultimately show "SNa M" by blast
wenzelm@36278	5466	next
wenzelm@36278	5467	case 2
wenzelm@36278	5468	have "(x):M \<in> (\<parallel>(A IMP B)\<parallel>)" by fact
wenzelm@36278	5469	then have "(x):M \<in> NEGn (A IMP B) (\<parallel><A IMP B>\<parallel>)" using NEG_simp by blast
wenzelm@36278	5470	then have "(x):M \<in> AXIOMSn (A IMP B) \<union> BINDINGn (A IMP B) (\<parallel><A IMP B>\<parallel>)
wenzelm@36278	5471	\<union> IMPLEFT (A IMP B) (\<parallel><A>\<parallel>) (\<parallel>(B)\<parallel>)"
wenzelm@36278	5472	by (simp only: NEGn.simps)
wenzelm@36278	5473	moreover
wenzelm@36278	5474	{ assume "(x):M \<in> AXIOMSn (A IMP B)"
wenzelm@36278	5475	then have "SNa M" by (simp add: AXIOMS_imply_SNa)
wenzelm@36278	5476	}
wenzelm@36278	5477	moreover
wenzelm@36278	5478	{ assume "(x):M \<in> BINDINGn (A IMP B) (\<parallel><A IMP B>\<parallel>)"
wenzelm@36278	5479	then have "SNa M" by (simp only: BINDING_imply_SNa)
wenzelm@36278	5480	}
wenzelm@36278	5481	moreover
wenzelm@36278	5482	{ assume "(x):M \<in> IMPLEFT (A IMP B) (\<parallel><A>\<parallel>) (\<parallel>(B)\<parallel>)"
wenzelm@36278	5483	then obtain a' M' y' N' where eq: "M = ImpL <a'>.M' (y').N' x"
wenzelm@36278	5484	and "<a'>:M' \<in> (\<parallel><A>\<parallel>)" and "(y'):N' \<in> (\<parallel>(B)\<parallel>)"
wenzelm@36278	5485	by (erule_tac IMPLEFT_elim, blast)
wenzelm@36278	5486	then have "SNa M'" and "SNa N'" using ih1 ih4 by blast+
wenzelm@36278	5487	then have "SNa M" using eq by (simp add: ImpL_in_SNa)
wenzelm@36278	5488	}
wenzelm@36278	5489	ultimately show "SNa M" by blast
wenzelm@36278	5490	}
wenzelm@36278	5491	qed
wenzelm@36278	5492
wenzelm@36278	5493	text {* Main lemma 2 *}
wenzelm@36278	5494
wenzelm@36278	5495	lemma AXIOMS_preserved:
wenzelm@54152	5496	shows "<a>:M \<in> AXIOMSc B \<Longrightarrow> M \<longrightarrow>\<^sub>a* M' \<Longrightarrow> <a>:M' \<in> AXIOMSc B"
wenzelm@54152	5497	and "(x):M \<in> AXIOMSn B \<Longrightarrow> M \<longrightarrow>\<^sub>a* M' \<Longrightarrow> (x):M' \<in> AXIOMSn B"
wenzelm@36278	5498	apply(simp_all add: AXIOMSc_def AXIOMSn_def)
wenzelm@36278	5499	apply(auto simp add: ntrm.inject ctrm.inject alpha)
wenzelm@36278	5500	apply(drule ax_do_not_a_star_reduce)
wenzelm@36278	5501	apply(auto)
wenzelm@36278	5502	apply(drule ax_do_not_a_star_reduce)
wenzelm@36278	5503	apply(auto)
wenzelm@36278	5504	apply(drule ax_do_not_a_star_reduce)
wenzelm@36278	5505	apply(auto)
wenzelm@36278	5506	apply(drule ax_do_not_a_star_reduce)
wenzelm@36278	5507	apply(auto)
wenzelm@36278	5508	done
wenzelm@36278	5509
wenzelm@36278	5510	lemma BINDING_preserved:
wenzelm@54152	5511	shows "<a>:M \<in> BINDINGc B (\<parallel>(B)\<parallel>) \<Longrightarrow> M \<longrightarrow>\<^sub>a* M' \<Longrightarrow> <a>:M' \<in> BINDINGc B (\<parallel>(B)\<parallel>)"
wenzelm@54152	5512	and "(x):M \<in> BINDINGn B (\<parallel><B>\<parallel>) \<Longrightarrow> M \<longrightarrow>\<^sub>a* M' \<Longrightarrow> (x):M' \<in> BINDINGn B (\<parallel><B>\<parallel>)"
wenzelm@36278	5513	proof -
wenzelm@54152	5514	assume red: "M \<longrightarrow>\<^sub>a* M'"
wenzelm@36278	5515	assume asm: "<a>:M \<in> BINDINGc B (\<parallel>(B)\<parallel>)"
wenzelm@36278	5516	{
wenzelm@36278	5517	fix x::"name" and P::"trm"
wenzelm@36278	5518	from asm have "((x):P) \<in> (\<parallel>(B)\<parallel>) \<Longrightarrow> SNa (M{a:=(x).P})" by (simp add: BINDINGc_elim)
wenzelm@36278	5519	moreover
wenzelm@54152	5520	have "M{a:=(x).P} \<longrightarrow>\<^sub>a* M'{a:=(x).P}" using red by (simp add: a_star_subst2)
wenzelm@36278	5521	ultimately
wenzelm@36278	5522	have "((x):P) \<in> (\<parallel>(B)\<parallel>) \<Longrightarrow> SNa (M'{a:=(x).P})" by (simp add: a_star_preserves_SNa)
wenzelm@36278	5523	}
wenzelm@36278	5524	then show "<a>:M' \<in> BINDINGc B (\<parallel>(B)\<parallel>)" by (auto simp add: BINDINGc_def)
wenzelm@36278	5525	next
wenzelm@54152	5526	assume red: "M \<longrightarrow>\<^sub>a* M'"
wenzelm@36278	5527	assume asm: "(x):M \<in> BINDINGn B (\<parallel><B>\<parallel>)"
wenzelm@36278	5528	{
wenzelm@36278	5529	fix c::"coname" and P::"trm"
wenzelm@36278	5530	from asm have "(<c>:P) \<in> (\<parallel><B>\<parallel>) \<Longrightarrow> SNa (M{x:=<c>.P})" by (simp add: BINDINGn_elim)
wenzelm@36278	5531	moreover
wenzelm@54152	5532	have "M{x:=<c>.P} \<longrightarrow>\<^sub>a* M'{x:=<c>.P}" using red by (simp add: a_star_subst1)
wenzelm@36278	5533	ultimately
wenzelm@36278	5534	have "(<c>:P) \<in> (\<parallel><B>\<parallel>) \<Longrightarrow> SNa (M'{x:=<c>.P})" by (simp add: a_star_preserves_SNa)
wenzelm@36278	5535	}
wenzelm@36278	5536	then show "(x):M' \<in> BINDINGn B (\<parallel><B>\<parallel>)" by (auto simp add: BINDINGn_def)
wenzelm@36278	5537	qed
wenzelm@36278	5538
wenzelm@36278	5539	lemma CANDs_preserved:
wenzelm@54152	5540	shows "<a>:M \<in> \<parallel><B>\<parallel> \<Longrightarrow> M \<longrightarrow>\<^sub>a* M' \<Longrightarrow> <a>:M' \<in> \<parallel><B>\<parallel>"
wenzelm@54152	5541	and "(x):M \<in> \<parallel>(B)\<parallel> \<Longrightarrow> M \<longrightarrow>\<^sub>a* M' \<Longrightarrow> (x):M' \<in> \<parallel>(B)\<parallel>"
wenzelm@36278	5542	proof(nominal_induct B arbitrary: a x M M' rule: ty.strong_induct)
wenzelm@36278	5543	case (PR X)
wenzelm@36278	5544	{ case 1
wenzelm@54152	5545	have asm: "M \<longrightarrow>\<^sub>a* M'" by fact
wenzelm@36278	5546	have "<a>:M \<in> \<parallel><PR X>\<parallel>" by fact
wenzelm@36278	5547	then have "<a>:M \<in> NEGc (PR X) (\<parallel>(PR X)\<parallel>)" by simp
wenzelm@36278	5548	then have "<a>:M \<in> AXIOMSc (PR X) \<union> BINDINGc (PR X) (\<parallel>(PR X)\<parallel>)" by simp
wenzelm@36278	5549	moreover
wenzelm@36278	5550	{ assume "<a>:M \<in> AXIOMSc (PR X)"
wenzelm@36278	5551	then have "<a>:M' \<in> AXIOMSc (PR X)" using asm by (simp only: AXIOMS_preserved)
wenzelm@36278	5552	}
wenzelm@36278	5553	moreover
wenzelm@36278	5554	{ assume "<a>:M \<in> BINDINGc (PR X) (\<parallel>(PR X)\<parallel>)"
wenzelm@36278	5555	then have "<a>:M' \<in> BINDINGc (PR X) (\<parallel>(PR X)\<parallel>)" using asm by (simp add: BINDING_preserved)
wenzelm@36278	5556	}
wenzelm@36278	5557	ultimately have "<a>:M' \<in> AXIOMSc (PR X) \<union> BINDINGc (PR X) (\<parallel>(PR X)\<parallel>)" by blast
wenzelm@36278	5558	then have "<a>:M' \<in> NEGc (PR X) (\<parallel>(PR X)\<parallel>)" by simp
wenzelm@36278	5559	then show "<a>:M' \<in> (\<parallel><PR X>\<parallel>)" using NEG_simp by blast
wenzelm@36278	5560	next
wenzelm@36278	5561	case 2
wenzelm@54152	5562	have asm: "M \<longrightarrow>\<^sub>a* M'" by fact
wenzelm@36278	5563	have "(x):M \<in> \<parallel>(PR X)\<parallel>" by fact
wenzelm@36278	5564	then have "(x):M \<in> NEGn (PR X) (\<parallel><PR X>\<parallel>)" using NEG_simp by blast
wenzelm@36278	5565	then have "(x):M \<in> AXIOMSn (PR X) \<union> BINDINGn (PR X) (\<parallel><PR X>\<parallel>)" by simp
wenzelm@36278	5566	moreover
wenzelm@36278	5567	{ assume "(x):M \<in> AXIOMSn (PR X)"
wenzelm@36278	5568	then have "(x):M' \<in> AXIOMSn (PR X)" using asm by (simp only: AXIOMS_preserved)
wenzelm@36278	5569	}
wenzelm@36278	5570	moreover
wenzelm@36278	5571	{ assume "(x):M \<in> BINDINGn (PR X) (\<parallel><PR X>\<parallel>)"
wenzelm@36278	5572	then have "(x):M' \<in> BINDINGn (PR X) (\<parallel><PR X>\<parallel>)" using asm by (simp only: BINDING_preserved)
wenzelm@36278	5573	}
wenzelm@36278	5574	ultimately have "(x):M' \<in> AXIOMSn (PR X) \<union> BINDINGn (PR X) (\<parallel><PR X>\<parallel>)" by blast
wenzelm@36278	5575	then have "(x):M' \<in> NEGn (PR X) (\<parallel><PR X>\<parallel>)" by simp
wenzelm@36278	5576	then show "(x):M' \<in> (\<parallel>(PR X)\<parallel>)" using NEG_simp by blast
wenzelm@36278	5577	}
wenzelm@36278	5578	next
wenzelm@36278	5579	case (IMP A B)
wenzelm@54152	5580	have ih1: "\<And>a M M'. \<lbrakk><a>:M \<in> \<parallel><A>\<parallel>; M \<longrightarrow>\<^sub>a* M'\<rbrakk> \<Longrightarrow> <a>:M' \<in> \<parallel><A>\<parallel>" by fact
wenzelm@54152	5581	have ih2: "\<And>x M M'. \<lbrakk>(x):M \<in> \<parallel>(A)\<parallel>; M \<longrightarrow>\<^sub>a* M'\<rbrakk> \<Longrightarrow> (x):M' \<in> \<parallel>(A)\<parallel>" by fact
wenzelm@54152	5582	have ih3: "\<And>a M M'. \<lbrakk><a>:M \<in> \<parallel><B>\<parallel>; M \<longrightarrow>\<^sub>a* M'\<rbrakk> \<Longrightarrow> <a>:M' \<in> \<parallel><B>\<parallel>" by fact
wenzelm@54152	5583	have ih4: "\<And>x M M'. \<lbrakk>(x):M \<in> \<parallel>(B)\<parallel>; M \<longrightarrow>\<^sub>a* M'\<rbrakk> \<Longrightarrow> (x):M' \<in> \<parallel>(B)\<parallel>" by fact
wenzelm@36278	5584	{ case 1
wenzelm@54152	5585	have asm: "M \<longrightarrow>\<^sub>a* M'" by fact
wenzelm@36278	5586	have "<a>:M \<in> \<parallel><A IMP B>\<parallel>" by fact
wenzelm@36278	5587	then have "<a>:M \<in> NEGc (A IMP B) (\<parallel>(A IMP B)\<parallel>)" by simp
wenzelm@36278	5588	then have "<a>:M \<in> AXIOMSc (A IMP B) \<union> BINDINGc (A IMP B) (\<parallel>(A IMP B)\<parallel>)
wenzelm@36278	5589	\<union> IMPRIGHT (A IMP B) (\<parallel>(A)\<parallel>) (\<parallel><B>\<parallel>) (\<parallel>(B)\<parallel>) (\<parallel><A>\<parallel>)" by simp
wenzelm@36278	5590	moreover
wenzelm@36278	5591	{ assume "<a>:M \<in> AXIOMSc (A IMP B)"
wenzelm@36278	5592	then have "<a>:M' \<in> AXIOMSc (A IMP B)" using asm by (simp only: AXIOMS_preserved)
wenzelm@36278	5593	}
wenzelm@36278	5594	moreover
wenzelm@36278	5595	{ assume "<a>:M \<in> BINDINGc (A IMP B) (\<parallel>(A IMP B)\<parallel>)"
wenzelm@36278	5596	then have "<a>:M' \<in> BINDINGc (A IMP B) (\<parallel>(A IMP B)\<parallel>)" using asm by (simp only: BINDING_preserved)
wenzelm@36278	5597	}
wenzelm@36278	5598	moreover
wenzelm@36278	5599	{ assume "<a>:M \<in> IMPRIGHT (A IMP B) (\<parallel>(A)\<parallel>) (\<parallel><B>\<parallel>) (\<parallel>(B)\<parallel>) (\<parallel><A>\<parallel>)"
wenzelm@36278	5600	then obtain x' a' N' where eq: "M = ImpR (x').<a'>.N' a" and fic: "fic (ImpR (x').<a'>.N' a) a"
wenzelm@36278	5601	and imp1: "\<forall>z P. x'\<sharp>(z,P) \<and> (z):P \<in> \<parallel>(B)\<parallel> \<longrightarrow> (x'):(N'{a':=(z).P}) \<in> \<parallel>(A)\<parallel>"
wenzelm@36278	5602	and imp2: "\<forall>c Q. a'\<sharp>(c,Q) \<and> <c>:Q \<in> \<parallel><A>\<parallel> \<longrightarrow> <a'>:(N'{x':=<c>.Q}) \<in> \<parallel><B>\<parallel>"
wenzelm@36278	5603	using IMPRIGHT_elim by blast
wenzelm@54152	5604	from eq asm obtain N'' where eq': "M' = ImpR (x').<a'>.N'' a" and red: "N' \<longrightarrow>\<^sub>a* N''"
wenzelm@36278	5605	using a_star_redu_ImpR_elim by (blast)
wenzelm@36278	5606	from imp1 have "\<forall>z P. x'\<sharp>(z,P) \<and> (z):P \<in> \<parallel>(B)\<parallel> \<longrightarrow> (x'):(N''{a':=(z).P}) \<in> \<parallel>(A)\<parallel>" using red ih2
wenzelm@36278	5607	apply(auto)
wenzelm@36278	5608	apply(drule_tac x="z" in spec)
wenzelm@36278	5609	apply(drule_tac x="P" in spec)
wenzelm@36278	5610	apply(simp)
wenzelm@36278	5611	apply(drule_tac a_star_subst2)
wenzelm@36278	5612	apply(blast)
wenzelm@36278	5613	done
wenzelm@36278	5614	moreover
wenzelm@36278	5615	from imp2 have "\<forall>c Q. a'\<sharp>(c,Q) \<and> <c>:Q \<in> \<parallel><A>\<parallel> \<longrightarrow> <a'>:(N''{x':=<c>.Q}) \<in> \<parallel><B>\<parallel>" using red ih3
wenzelm@36278	5616	apply(auto)
wenzelm@36278	5617	apply(drule_tac x="c" in spec)
wenzelm@36278	5618	apply(drule_tac x="Q" in spec)
wenzelm@36278	5619	apply(simp)
wenzelm@36278	5620	apply(drule_tac a_star_subst1)
wenzelm@36278	5621	apply(blast)
wenzelm@36278	5622	done
wenzelm@36278	5623	moreover
wenzelm@36278	5624	from fic have "fic M' a" using eq asm by (simp add: fic_a_star_reduce)
wenzelm@36278	5625	ultimately have "<a>:M' \<in> IMPRIGHT (A IMP B) (\<parallel>(A)\<parallel>) (\<parallel><B>\<parallel>) (\<parallel>(B)\<parallel>) (\<parallel><A>\<parallel>)" using eq' by auto
wenzelm@36278	5626	}
wenzelm@36278	5627	ultimately have "<a>:M' \<in> AXIOMSc (A IMP B) \<union> BINDINGc (A IMP B) (\<parallel>(A IMP B)\<parallel>)
wenzelm@36278	5628	\<union> IMPRIGHT (A IMP B) (\<parallel>(A)\<parallel>) (\<parallel><B>\<parallel>) (\<parallel>(B)\<parallel>) (\<parallel><A>\<parallel>)" by blast
wenzelm@36278	5629	then have "<a>:M' \<in> NEGc (A IMP B) (\<parallel>(A IMP B)\<parallel>)" by simp
wenzelm@36278	5630	then show "<a>:M' \<in> (\<parallel><A IMP B>\<parallel>)" using NEG_simp by blast
wenzelm@36278	5631	next
wenzelm@36278	5632	case 2
wenzelm@54152	5633	have asm: "M \<longrightarrow>\<^sub>a* M'" by fact
wenzelm@36278	5634	have "(x):M \<in> \<parallel>(A IMP B)\<parallel>" by fact
wenzelm@36278	5635	then have "(x):M \<in> NEGn (A IMP B) (\<parallel><A IMP B>\<parallel>)" using NEG_simp by blast
wenzelm@36278	5636	then have "(x):M \<in> AXIOMSn (A IMP B) \<union> BINDINGn (A IMP B) (\<parallel><A IMP B>\<parallel>)
wenzelm@36278	5637	\<union> IMPLEFT (A IMP B) (\<parallel><A>\<parallel>) (\<parallel>(B)\<parallel>)" by simp
wenzelm@36278	5638	moreover
wenzelm@36278	5639	{ assume "(x):M \<in> AXIOMSn (A IMP B)"
wenzelm@36278	5640	then have "(x):M' \<in> AXIOMSn (A IMP B)" using asm by (simp only: AXIOMS_preserved)
wenzelm@36278	5641	}
wenzelm@36278	5642	moreover
wenzelm@36278	5643	{ assume "(x):M \<in> BINDINGn (A IMP B) (\<parallel><A IMP B>\<parallel>)"
wenzelm@36278	5644	then have "(x):M' \<in> BINDINGn (A IMP B) (\<parallel><A IMP B>\<parallel>)" using asm by (simp only: BINDING_preserved)
wenzelm@36278	5645	}
wenzelm@36278	5646	moreover
wenzelm@36278	5647	{ assume "(x):M \<in> IMPLEFT (A IMP B) (\<parallel><A>\<parallel>) (\<parallel>(B)\<parallel>)"
wenzelm@36278	5648	then obtain a' T' y' N' where eq: "M = ImpL <a'>.T' (y').N' x"
wenzelm@36278	5649	and fin: "fin (ImpL <a'>.T' (y').N' x) x"
wenzelm@36278	5650	and imp1: "<a'>:T' \<in> \<parallel><A>\<parallel>" and imp2: "(y'):N' \<in> \<parallel>(B)\<parallel>"
wenzelm@36278	5651	by (erule_tac IMPLEFT_elim, blast)
wenzelm@36278	5652	from eq asm obtain T'' N'' where eq': "M' = ImpL <a'>.T'' (y').N'' x"
wenzelm@54152	5653	and red1: "T' \<longrightarrow>\<^sub>a* T''" and red2: "N' \<longrightarrow>\<^sub>a* N''"
wenzelm@36278	5654	using a_star_redu_ImpL_elim by blast
wenzelm@36278	5655	from fin have "fin M' x" using eq asm by (simp add: fin_a_star_reduce)
wenzelm@36278	5656	moreover
wenzelm@36278	5657	from imp1 red1 have "<a'>:T'' \<in> \<parallel><A>\<parallel>" using ih1 by simp
wenzelm@36278	5658	moreover
wenzelm@36278	5659	from imp2 red2 have "(y'):N'' \<in> \<parallel>(B)\<parallel>" using ih4 by simp
wenzelm@36278	5660	ultimately have "(x):M' \<in> IMPLEFT (A IMP B) (\<parallel><A>\<parallel>) (\<parallel>(B)\<parallel>)" using eq' by (simp, blast)
wenzelm@36278	5661	}
wenzelm@36278	5662	ultimately have "(x):M' \<in> AXIOMSn (A IMP B) \<union> BINDINGn (A IMP B) (\<parallel><A IMP B>\<parallel>)
wenzelm@36278	5663	\<union> IMPLEFT (A IMP B) (\<parallel><A>\<parallel>) (\<parallel>(B)\<parallel>)" by blast
wenzelm@36278	5664	then have "(x):M' \<in> NEGn (A IMP B) (\<parallel><A IMP B>\<parallel>)" by simp
wenzelm@36278	5665	then show "(x):M' \<in> (\<parallel>(A IMP B)\<parallel>)" using NEG_simp by blast
wenzelm@36278	5666	}
wenzelm@36278	5667	next
wenzelm@36278	5668	case (AND A B)
wenzelm@54152	5669	have ih1: "\<And>a M M'. \<lbrakk><a>:M \<in> \<parallel><A>\<parallel>; M \<longrightarrow>\<^sub>a* M'\<rbrakk> \<Longrightarrow> <a>:M' \<in> \<parallel><A>\<parallel>" by fact
wenzelm@54152	5670	have ih2: "\<And>x M M'. \<lbrakk>(x):M \<in> \<parallel>(A)\<parallel>; M \<longrightarrow>\<^sub>a* M'\<rbrakk> \<Longrightarrow> (x):M' \<in> \<parallel>(A)\<parallel>" by fact
wenzelm@54152	5671	have ih3: "\<And>a M M'. \<lbrakk><a>:M \<in> \<parallel><B>\<parallel>; M \<longrightarrow>\<^sub>a* M'\<rbrakk> \<Longrightarrow> <a>:M' \<in> \<parallel><B>\<parallel>" by fact
wenzelm@54152	5672	have ih4: "\<And>x M M'. \<lbrakk>(x):M \<in> \<parallel>(B)\<parallel>; M \<longrightarrow>\<^sub>a* M'\<rbrakk> \<Longrightarrow> (x):M' \<in> \<parallel>(B)\<parallel>" by fact
wenzelm@36278	5673	{ case 1
wenzelm@54152	5674	have asm: "M \<longrightarrow>\<^sub>a* M'" by fact
wenzelm@36278	5675	have "<a>:M \<in> \<parallel><A AND B>\<parallel>" by fact
wenzelm@36278	5676	then have "<a>:M \<in> NEGc (A AND B) (\<parallel>(A AND B)\<parallel>)" by simp
wenzelm@36278	5677	then have "<a>:M \<in> AXIOMSc (A AND B) \<union> BINDINGc (A AND B) (\<parallel>(A AND B)\<parallel>)
wenzelm@36278	5678	\<union> ANDRIGHT (A AND B) (\<parallel><A>\<parallel>) (\<parallel><B>\<parallel>)" by simp
wenzelm@36278	5679	moreover
wenzelm@36278	5680	{ assume "<a>:M \<in> AXIOMSc (A AND B)"
wenzelm@36278	5681	then have "<a>:M' \<in> AXIOMSc (A AND B)" using asm by (simp only: AXIOMS_preserved)
wenzelm@36278	5682	}
wenzelm@36278	5683	moreover
wenzelm@36278	5684	{ assume "<a>:M \<in> BINDINGc (A AND B) (\<parallel>(A AND B)\<parallel>)"
wenzelm@36278	5685	then have "<a>:M' \<in> BINDINGc (A AND B) (\<parallel>(A AND B)\<parallel>)" using asm by (simp only: BINDING_preserved)
wenzelm@36278	5686	}
wenzelm@36278	5687	moreover
wenzelm@36278	5688	{ assume "<a>:M \<in> ANDRIGHT (A AND B) (\<parallel><A>\<parallel>) (\<parallel><B>\<parallel>)"
wenzelm@36278	5689	then obtain a' T' b' N' where eq: "M = AndR <a'>.T' <b'>.N' a"
wenzelm@36278	5690	and fic: "fic (AndR <a'>.T' <b'>.N' a) a"
wenzelm@36278	5691	and imp1: "<a'>:T' \<in> \<parallel><A>\<parallel>" and imp2: "<b'>:N' \<in> \<parallel><B>\<parallel>"
wenzelm@36278	5692	using ANDRIGHT_elim by blast
wenzelm@36278	5693	from eq asm obtain T'' N'' where eq': "M' = AndR <a'>.T'' <b'>.N'' a"
wenzelm@54152	5694	and red1: "T' \<longrightarrow>\<^sub>a* T''" and red2: "N' \<longrightarrow>\<^sub>a* N''"
wenzelm@36278	5695	using a_star_redu_AndR_elim by blast
wenzelm@36278	5696	from fic have "fic M' a" using eq asm by (simp add: fic_a_star_reduce)
wenzelm@36278	5697	moreover
wenzelm@36278	5698	from imp1 red1 have "<a'>:T'' \<in> \<parallel><A>\<parallel>" using ih1 by simp
wenzelm@36278	5699	moreover
wenzelm@36278	5700	from imp2 red2 have "<b'>:N'' \<in> \<parallel><B>\<parallel>" using ih3 by simp
wenzelm@36278	5701	ultimately have "<a>:M' \<in> ANDRIGHT (A AND B) (\<parallel><A>\<parallel>) (\<parallel><B>\<parallel>)" using eq' by (simp, blast)
wenzelm@36278	5702	}
wenzelm@36278	5703	ultimately have "<a>:M' \<in> AXIOMSc (A AND B) \<union> BINDINGc (A AND B) (\<parallel>(A AND B)\<parallel>)
wenzelm@36278	5704	\<union> ANDRIGHT (A AND B) (\<parallel><A>\<parallel>) (\<parallel><B>\<parallel>)" by blast
wenzelm@36278	5705	then have "<a>:M' \<in> NEGc (A AND B) (\<parallel>(A AND B)\<parallel>)" by simp
wenzelm@36278	5706	then show "<a>:M' \<in> (\<parallel><A AND B>\<parallel>)" using NEG_simp by blast
wenzelm@36278	5707	next
wenzelm@36278	5708	case 2
wenzelm@54152	5709	have asm: "M \<longrightarrow>\<^sub>a* M'" by fact
wenzelm@36278	5710	have "(x):M \<in> \<parallel>(A AND B)\<parallel>" by fact
wenzelm@36278	5711	then have "(x):M \<in> NEGn (A AND B) (\<parallel><A AND B>\<parallel>)" using NEG_simp by blast
wenzelm@36278	5712	then have "(x):M \<in> AXIOMSn (A AND B) \<union> BINDINGn (A AND B) (\<parallel><A AND B>\<parallel>)
wenzelm@36278	5713	\<union> ANDLEFT1 (A AND B) (\<parallel>(A)\<parallel>) \<union> ANDLEFT2 (A AND B) (\<parallel>(B)\<parallel>)" by simp
wenzelm@36278	5714	moreover
wenzelm@36278	5715	{ assume "(x):M \<in> AXIOMSn (A AND B)"
wenzelm@36278	5716	then have "(x):M' \<in> AXIOMSn (A AND B)" using asm by (simp only: AXIOMS_preserved)
wenzelm@36278	5717	}
wenzelm@36278	5718	moreover
wenzelm@36278	5719	{ assume "(x):M \<in> BINDINGn (A AND B) (\<parallel><A AND B>\<parallel>)"
wenzelm@36278	5720	then have "(x):M' \<in> BINDINGn (A AND B) (\<parallel><A AND B>\<parallel>)" using asm by (simp only: BINDING_preserved)
wenzelm@36278	5721	}
wenzelm@36278	5722	moreover
wenzelm@36278	5723	{ assume "(x):M \<in> ANDLEFT1 (A AND B) (\<parallel>(A)\<parallel>)"
wenzelm@36278	5724	then obtain y' N' where eq: "M = AndL1 (y').N' x"
wenzelm@36278	5725	and fin: "fin (AndL1 (y').N' x) x" and imp: "(y'):N' \<in> \<parallel>(A)\<parallel>"
wenzelm@36278	5726	by (erule_tac ANDLEFT1_elim, blast)
wenzelm@54152	5727	from eq asm obtain N'' where eq': "M' = AndL1 (y').N'' x" and red1: "N' \<longrightarrow>\<^sub>a* N''"
wenzelm@36278	5728	using a_star_redu_AndL1_elim by blast
wenzelm@36278	5729	from fin have "fin M' x" using eq asm by (simp add: fin_a_star_reduce)
wenzelm@36278	5730	moreover
wenzelm@36278	5731	from imp red1 have "(y'):N'' \<in> \<parallel>(A)\<parallel>" using ih2 by simp
wenzelm@36278	5732	ultimately have "(x):M' \<in> ANDLEFT1 (A AND B) (\<parallel>(A)\<parallel>)" using eq' by (simp, blast)
wenzelm@36278	5733	}
wenzelm@36278	5734	moreover
wenzelm@36278	5735	{ assume "(x):M \<in> ANDLEFT2 (A AND B) (\<parallel>(B)\<parallel>)"
wenzelm@36278	5736	then obtain y' N' where eq: "M = AndL2 (y').N' x"
wenzelm@36278	5737	and fin: "fin (AndL2 (y').N' x) x" and imp: "(y'):N' \<in> \<parallel>(B)\<parallel>"
wenzelm@36278	5738	by (erule_tac ANDLEFT2_elim, blast)
wenzelm@54152	5739	from eq asm obtain N'' where eq': "M' = AndL2 (y').N'' x" and red1: "N' \<longrightarrow>\<^sub>a* N''"
wenzelm@36278	5740	using a_star_redu_AndL2_elim by blast
wenzelm@36278	5741	from fin have "fin M' x" using eq asm by (simp add: fin_a_star_reduce)
wenzelm@36278	5742	moreover
wenzelm@36278	5743	from imp red1 have "(y'):N'' \<in> \<parallel>(B)\<parallel>" using ih4 by simp
wenzelm@36278	5744	ultimately have "(x):M' \<in> ANDLEFT2 (A AND B) (\<parallel>(B)\<parallel>)" using eq' by (simp, blast)
wenzelm@36278	5745	}
wenzelm@36278	5746	ultimately have "(x):M' \<in> AXIOMSn (A AND B) \<union> BINDINGn (A AND B) (\<parallel><A AND B>\<parallel>)
wenzelm@36278	5747	\<union> ANDLEFT1 (A AND B) (\<parallel>(A)\<parallel>) \<union> ANDLEFT2 (A AND B) (\<parallel>(B)\<parallel>)" by blast
wenzelm@36278	5748	then have "(x):M' \<in> NEGn (A AND B) (\<parallel><A AND B>\<parallel>)" by simp
wenzelm@36278	5749	then show "(x):M' \<in> (\<parallel>(A AND B)\<parallel>)" using NEG_simp by blast
wenzelm@36278	5750	}
wenzelm@36278	5751	next
wenzelm@36278	5752	case (OR A B)
wenzelm@54152	5753	have ih1: "\<And>a M M'. \<lbrakk><a>:M \<in> \<parallel><A>\<parallel>; M \<longrightarrow>\<^sub>a* M'\<rbrakk> \<Longrightarrow> <a>:M' \<in> \<parallel><A>\<parallel>" by fact
wenzelm@54152	5754	have ih2: "\<And>x M M'. \<lbrakk>(x):M \<in> \<parallel>(A)\<parallel>; M \<longrightarrow>\<^sub>a* M'\<rbrakk> \<Longrightarrow> (x):M' \<in> \<parallel>(A)\<parallel>" by fact
wenzelm@54152	5755	have ih3: "\<And>a M M'. \<lbrakk><a>:M \<in> \<parallel><B>\<parallel>; M \<longrightarrow>\<^sub>a* M'\<rbrakk> \<Longrightarrow> <a>:M' \<in> \<parallel><B>\<parallel>" by fact
wenzelm@54152	5756	have ih4: "\<And>x M M'. \<lbrakk>(x):M \<in> \<parallel>(B)\<parallel>; M \<longrightarrow>\<^sub>a* M'\<rbrakk> \<Longrightarrow> (x):M' \<in> \<parallel>(B)\<parallel>" by fact
wenzelm@36278	5757	{ case 1
wenzelm@54152	5758	have asm: "M \<longrightarrow>\<^sub>a* M'" by fact
wenzelm@36278	5759	have "<a>:M \<in> \<parallel><A OR B>\<parallel>" by fact
wenzelm@36278	5760	then have "<a>:M \<in> NEGc (A OR B) (\<parallel>(A OR B)\<parallel>)" by simp
wenzelm@36278	5761	then have "<a>:M \<in> AXIOMSc (A OR B) \<union> BINDINGc (A OR B) (\<parallel>(A OR B)\<parallel>)
wenzelm@36278	5762	\<union> ORRIGHT1 (A OR B) (\<parallel><A>\<parallel>) \<union> ORRIGHT2 (A OR B) (\<parallel><B>\<parallel>)" by simp
wenzelm@36278	5763	moreover
wenzelm@36278	5764	{ assume "<a>:M \<in> AXIOMSc (A OR B)"
wenzelm@36278	5765	then have "<a>:M' \<in> AXIOMSc (A OR B)" using asm by (simp only: AXIOMS_preserved)
wenzelm@36278	5766	}
wenzelm@36278	5767	moreover
wenzelm@36278	5768	{ assume "<a>:M \<in> BINDINGc (A OR B) (\<parallel>(A OR B)\<parallel>)"
wenzelm@36278	5769	then have "<a>:M' \<in> BINDINGc (A OR B) (\<parallel>(A OR B)\<parallel>)" using asm by (simp only: BINDING_preserved)
wenzelm@36278	5770	}
wenzelm@36278	5771	moreover
wenzelm@36278	5772	{ assume "<a>:M \<in> ORRIGHT1 (A OR B) (\<parallel><A>\<parallel>)"
wenzelm@36278	5773	then obtain a' N' where eq: "M = OrR1 <a'>.N' a"
wenzelm@36278	5774	and fic: "fic (OrR1 <a'>.N' a) a" and imp1: "<a'>:N' \<in> \<parallel><A>\<parallel>"
wenzelm@36278	5775	using ORRIGHT1_elim by blast
wenzelm@54152	5776	from eq asm obtain N'' where eq': "M' = OrR1 <a'>.N'' a" and red1: "N' \<longrightarrow>\<^sub>a* N''"
wenzelm@36278	5777	using a_star_redu_OrR1_elim by blast
wenzelm@36278	5778	from fic have "fic M' a" using eq asm by (simp add: fic_a_star_reduce)
wenzelm@36278	5779	moreover
wenzelm@36278	5780	from imp1 red1 have "<a'>:N'' \<in> \<parallel><A>\<parallel>" using ih1 by simp
wenzelm@36278	5781	ultimately have "<a>:M' \<in> ORRIGHT1 (A OR B) (\<parallel><A>\<parallel>)" using eq' by (simp, blast)
wenzelm@36278	5782	}
wenzelm@36278	5783	moreover
wenzelm@36278	5784	{ assume "<a>:M \<in> ORRIGHT2 (A OR B) (\<parallel><B>\<parallel>)"
wenzelm@36278	5785	then obtain a' N' where eq: "M = OrR2 <a'>.N' a"
wenzelm@36278	5786	and fic: "fic (OrR2 <a'>.N' a) a" and imp1: "<a'>:N' \<in> \<parallel><B>\<parallel>"
wenzelm@36278	5787	using ORRIGHT2_elim by blast
wenzelm@54152	5788	from eq asm obtain N'' where eq': "M' = OrR2 <a'>.N'' a" and red1: "N' \<longrightarrow>\<^sub>a* N''"
wenzelm@36278	5789	using a_star_redu_OrR2_elim by blast
wenzelm@36278	5790	from fic have "fic M' a" using eq asm by (simp add: fic_a_star_reduce)
wenzelm@36278	5791	moreover
wenzelm@36278	5792	from imp1 red1 have "<a'>:N'' \<in> \<parallel><B>\<parallel>" using ih3 by simp
wenzelm@36278	5793	ultimately have "<a>:M' \<in> ORRIGHT2 (A OR B) (\<parallel><B>\<parallel>)" using eq' by (simp, blast)
wenzelm@36278	5794	}
wenzelm@36278	5795	ultimately have "<a>:M' \<in> AXIOMSc (A OR B) \<union> BINDINGc (A OR B) (\<parallel>(A OR B)\<parallel>)
wenzelm@36278	5796	\<union> ORRIGHT1 (A OR B) (\<parallel><A>\<parallel>) \<union> ORRIGHT2 (A OR B) (\<parallel><B>\<parallel>)" by blast
wenzelm@36278	5797	then have "<a>:M' \<in> NEGc (A OR B) (\<parallel>(A OR B)\<parallel>)" by simp
wenzelm@36278	5798	then show "<a>:M' \<in> (\<parallel><A OR B>\<parallel>)" using NEG_simp by blast
wenzelm@36278	5799	next
wenzelm@36278	5800	case 2
wenzelm@54152	5801	have asm: "M \<longrightarrow>\<^sub>a* M'" by fact
wenzelm@36278	5802	have "(x):M \<in> \<parallel>(A OR B)\<parallel>" by fact
wenzelm@36278	5803	then have "(x):M \<in> NEGn (A OR B) (\<parallel><A OR B>\<parallel>)" using NEG_simp by blast
wenzelm@36278	5804	then have "(x):M \<in> AXIOMSn (A OR B) \<union> BINDINGn (A OR B) (\<parallel><A OR B>\<parallel>)
wenzelm@36278	5805	\<union> ORLEFT (A OR B) (\<parallel>(A)\<parallel>) (\<parallel>(B)\<parallel>)" by simp
wenzelm@36278	5806	moreover
wenzelm@36278	5807	{ assume "(x):M \<in> AXIOMSn (A OR B)"
wenzelm@36278	5808	then have "(x):M' \<in> AXIOMSn (A OR B)" using asm by (simp only: AXIOMS_preserved)
wenzelm@36278	5809	}
wenzelm@36278	5810	moreover
wenzelm@36278	5811	{ assume "(x):M \<in> BINDINGn (A OR B) (\<parallel><A OR B>\<parallel>)"
wenzelm@36278	5812	then have "(x):M' \<in> BINDINGn (A OR B) (\<parallel><A OR B>\<parallel>)" using asm by (simp only: BINDING_preserved)
wenzelm@36278	5813	}
wenzelm@36278	5814	moreover
wenzelm@36278	5815	{ assume "(x):M \<in> ORLEFT (A OR B) (\<parallel>(A)\<parallel>) (\<parallel>(B)\<parallel>)"
wenzelm@36278	5816	then obtain y' T' z' N' where eq: "M = OrL (y').T' (z').N' x"
wenzelm@36278	5817	and fin: "fin (OrL (y').T' (z').N' x) x"
wenzelm@36278	5818	and imp1: "(y'):T' \<in> \<parallel>(A)\<parallel>" and imp2: "(z'):N' \<in> \<parallel>(B)\<parallel>"
wenzelm@36278	5819	by (erule_tac ORLEFT_elim, blast)
wenzelm@36278	5820	from eq asm obtain T'' N'' where eq': "M' = OrL (y').T'' (z').N'' x"
wenzelm@54152	5821	and red1: "T' \<longrightarrow>\<^sub>a* T''" and red2: "N' \<longrightarrow>\<^sub>a* N''"
wenzelm@36278	5822	using a_star_redu_OrL_elim by blast
wenzelm@36278	5823	from fin have "fin M' x" using eq asm by (simp add: fin_a_star_reduce)
wenzelm@36278	5824	moreover
wenzelm@36278	5825	from imp1 red1 have "(y'):T'' \<in> \<parallel>(A)\<parallel>" using ih2 by simp
wenzelm@36278	5826	moreover
wenzelm@36278	5827	from imp2 red2 have "(z'):N'' \<in> \<parallel>(B)\<parallel>" using ih4 by simp
wenzelm@36278	5828	ultimately have "(x):M' \<in> ORLEFT (A OR B) (\<parallel>(A)\<parallel>) (\<parallel>(B)\<parallel>)" using eq' by (simp, blast)
wenzelm@36278	5829	}
wenzelm@36278	5830	ultimately have "(x):M' \<in> AXIOMSn (A OR B) \<union> BINDINGn (A OR B) (\<parallel><A OR B>\<parallel>)
wenzelm@36278	5831	\<union> ORLEFT (A OR B) (\<parallel>(A)\<parallel>) (\<parallel>(B)\<parallel>)" by blast
wenzelm@36278	5832	then have "(x):M' \<in> NEGn (A OR B) (\<parallel><A OR B>\<parallel>)" by simp
wenzelm@36278	5833	then show "(x):M' \<in> (\<parallel>(A OR B)\<parallel>)" using NEG_simp by blast
wenzelm@36278	5834	}
wenzelm@36278	5835	next
wenzelm@36278	5836	case (NOT A)
wenzelm@54152	5837	have ih1: "\<And>a M M'. \<lbrakk><a>:M \<in> \<parallel><A>\<parallel>; M \<longrightarrow>\<^sub>a* M'\<rbrakk> \<Longrightarrow> <a>:M' \<in> \<parallel><A>\<parallel>" by fact
wenzelm@54152	5838	have ih2: "\<And>x M M'. \<lbrakk>(x):M \<in> \<parallel>(A)\<parallel>; M \<longrightarrow>\<^sub>a* M'\<rbrakk> \<Longrightarrow> (x):M' \<in> \<parallel>(A)\<parallel>" by fact
wenzelm@36278	5839	{ case 1
wenzelm@54152	5840	have asm: "M \<longrightarrow>\<^sub>a* M'" by fact
wenzelm@36278	5841	have "<a>:M \<in> \<parallel><NOT A>\<parallel>" by fact
wenzelm@36278	5842	then have "<a>:M \<in> NEGc (NOT A) (\<parallel>(NOT A)\<parallel>)" by simp
wenzelm@36278	5843	then have "<a>:M \<in> AXIOMSc (NOT A) \<union> BINDINGc (NOT A) (\<parallel>(NOT A)\<parallel>)
wenzelm@36278	5844	\<union> NOTRIGHT (NOT A) (\<parallel>(A)\<parallel>)" by simp
wenzelm@36278	5845	moreover
wenzelm@36278	5846	{ assume "<a>:M \<in> AXIOMSc (NOT A)"
wenzelm@36278	5847	then have "<a>:M' \<in> AXIOMSc (NOT A)" using asm by (simp only: AXIOMS_preserved)
wenzelm@36278	5848	}
wenzelm@36278	5849	moreover
wenzelm@36278	5850	{ assume "<a>:M \<in> BINDINGc (NOT A) (\<parallel>(NOT A)\<parallel>)"
wenzelm@36278	5851	then have "<a>:M' \<in> BINDINGc (NOT A) (\<parallel>(NOT A)\<parallel>)" using asm by (simp only: BINDING_preserved)
wenzelm@36278	5852	}
wenzelm@36278	5853	moreover
wenzelm@36278	5854	{ assume "<a>:M \<in> NOTRIGHT (NOT A) (\<parallel>(A)\<parallel>)"
wenzelm@36278	5855	then obtain y' N' where eq: "M = NotR (y').N' a"
wenzelm@36278	5856	and fic: "fic (NotR (y').N' a) a" and imp: "(y'):N' \<in> \<parallel>(A)\<parallel>"
wenzelm@36278	5857	using NOTRIGHT_elim by blast
wenzelm@54152	5858	from eq asm obtain N'' where eq': "M' = NotR (y').N'' a" and red: "N' \<longrightarrow>\<^sub>a* N''"
wenzelm@36278	5859	using a_star_redu_NotR_elim by blast
wenzelm@36278	5860	from fic have "fic M' a" using eq asm by (simp add: fic_a_star_reduce)
wenzelm@36278	5861	moreover
wenzelm@36278	5862	from imp red have "(y'):N'' \<in> \<parallel>(A)\<parallel>" using ih2 by simp
wenzelm@36278	5863	ultimately have "<a>:M' \<in> NOTRIGHT (NOT A) (\<parallel>(A)\<parallel>)" using eq' by (simp, blast)
wenzelm@36278	5864	}
wenzelm@36278	5865	ultimately have "<a>:M' \<in> AXIOMSc (NOT A) \<union> BINDINGc (NOT A) (\<parallel>(NOT A)\<parallel>)
wenzelm@36278	5866	\<union> NOTRIGHT (NOT A) (\<parallel>(A)\<parallel>)" by blast
wenzelm@36278	5867	then have "<a>:M' \<in> NEGc (NOT A) (\<parallel>(NOT A)\<parallel>)" by simp
wenzelm@36278	5868	then show "<a>:M' \<in> (\<parallel><NOT A>\<parallel>)" using NEG_simp by blast
wenzelm@36278	5869	next
wenzelm@36278	5870	case 2
wenzelm@54152	5871	have asm: "M \<longrightarrow>\<^sub>a* M'" by fact
wenzelm@36278	5872	have "(x):M \<in> \<parallel>(NOT A)\<parallel>" by fact
wenzelm@36278	5873	then have "(x):M \<in> NEGn (NOT A) (\<parallel><NOT A>\<parallel>)" using NEG_simp by blast
wenzelm@36278	5874	then have "(x):M \<in> AXIOMSn (NOT A) \<union> BINDINGn (NOT A) (\<parallel><NOT A>\<parallel>)
wenzelm@36278	5875	\<union> NOTLEFT (NOT A) (\<parallel><A>\<parallel>)" by simp
wenzelm@36278	5876	moreover
wenzelm@36278	5877	{ assume "(x):M \<in> AXIOMSn (NOT A)"
wenzelm@36278	5878	then have "(x):M' \<in> AXIOMSn (NOT A)" using asm by (simp only: AXIOMS_preserved)
wenzelm@36278	5879	}
wenzelm@36278	5880	moreover
wenzelm@36278	5881	{ assume "(x):M \<in> BINDINGn (NOT A) (\<parallel><NOT A>\<parallel>)"
wenzelm@36278	5882	then have "(x):M' \<in> BINDINGn (NOT A) (\<parallel><NOT A>\<parallel>)" using asm by (simp only: BINDING_preserved)
wenzelm@36278	5883	}
wenzelm@36278	5884	moreover
wenzelm@36278	5885	{ assume "(x):M \<in> NOTLEFT (NOT A) (\<parallel><A>\<parallel>)"
wenzelm@36278	5886	then obtain a' N' where eq: "M = NotL <a'>.N' x"
wenzelm@36278	5887	and fin: "fin (NotL <a'>.N' x) x" and imp: "<a'>:N' \<in> \<parallel><A>\<parallel>"
wenzelm@36278	5888	by (erule_tac NOTLEFT_elim, blast)
wenzelm@54152	5889	from eq asm obtain N'' where eq': "M' = NotL <a'>.N'' x" and red1: "N' \<longrightarrow>\<^sub>a* N''"
wenzelm@36278	5890	using a_star_redu_NotL_elim by blast
wenzelm@36278	5891	from fin have "fin M' x" using eq asm by (simp add: fin_a_star_reduce)
wenzelm@36278	5892	moreover
wenzelm@36278	5893	from imp red1 have "<a'>:N'' \<in> \<parallel><A>\<parallel>" using ih1 by simp
wenzelm@36278	5894	ultimately have "(x):M' \<in> NOTLEFT (NOT A) (\<parallel><A>\<parallel>)" using eq' by (simp, blast)
wenzelm@36278	5895	}
wenzelm@36278	5896	ultimately have "(x):M' \<in> AXIOMSn (NOT A) \<union> BINDINGn (NOT A) (\<parallel><NOT A>\<parallel>)
wenzelm@36278	5897	\<union> NOTLEFT (NOT A) (\<parallel><A>\<parallel>)" by blast
wenzelm@36278	5898	then have "(x):M' \<in> NEGn (NOT A) (\<parallel><NOT A>\<parallel>)" by simp
wenzelm@36278	5899	then show "(x):M' \<in> (\<parallel>(NOT A)\<parallel>)" using NEG_simp by blast
wenzelm@36278	5900	}
wenzelm@36278	5901	qed
wenzelm@36278	5902
wenzelm@36278	5903	lemma CANDs_preserved_single:
wenzelm@54152	5904	shows "<a>:M \<in> \<parallel><B>\<parallel> \<Longrightarrow> M \<longrightarrow>\<^sub>a M' \<Longrightarrow> <a>:M' \<in> \<parallel><B>\<parallel>"
wenzelm@54152	5905	and "(x):M \<in> \<parallel>(B)\<parallel> \<Longrightarrow> M \<longrightarrow>\<^sub>a M' \<Longrightarrow> (x):M' \<in> \<parallel>(B)\<parallel>"
wenzelm@36278	5906	by (auto simp add: a_starI CANDs_preserved)
wenzelm@36278	5907
wenzelm@36278	5908	lemma fic_CANDS:
wenzelm@36278	5909	assumes a: "\<not>fic M a"
wenzelm@36278	5910	and b: "<a>:M \<in> \<parallel><B>\<parallel>"
wenzelm@36278	5911	shows "<a>:M \<in> AXIOMSc B \<or> <a>:M \<in> BINDINGc B (\<parallel>(B)\<parallel>)"
wenzelm@36278	5912	using a b
wenzelm@36278	5913	apply(nominal_induct B rule: ty.strong_induct)
wenzelm@36278	5914	apply(simp)
wenzelm@36278	5915	apply(simp)
wenzelm@36278	5916	apply(erule disjE)
wenzelm@36278	5917	apply(simp)
wenzelm@36278	5918	apply(erule disjE)
wenzelm@36278	5919	apply(simp)
wenzelm@36278	5920	apply(auto simp add: ctrm.inject)[1]
wenzelm@36278	5921	apply(simp add: alpha)
wenzelm@36278	5922	apply(erule disjE)
wenzelm@36278	5923	apply(simp)
wenzelm@36278	5924	apply(auto simp add: calc_atm)[1]
wenzelm@36278	5925	apply(drule_tac pi="[(a,aa)]" in fic.eqvt(2))
wenzelm@36278	5926	apply(simp add: calc_atm)
wenzelm@36278	5927	apply(simp)
wenzelm@36278	5928	apply(erule disjE)
wenzelm@36278	5929	apply(simp)
wenzelm@36278	5930	apply(erule disjE)
wenzelm@36278	5931	apply(simp)
wenzelm@36278	5932	apply(auto simp add: ctrm.inject)[1]
wenzelm@36278	5933	apply(simp add: alpha)
wenzelm@36278	5934	apply(erule disjE)
wenzelm@36278	5935	apply(simp)
wenzelm@36278	5936	apply(erule conjE)+
wenzelm@36278	5937	apply(simp)
wenzelm@36278	5938	apply(drule_tac pi="[(a,c)]" in fic.eqvt(2))
wenzelm@36278	5939	apply(simp add: calc_atm)
wenzelm@36278	5940	apply(simp)
wenzelm@36278	5941	apply(erule disjE)
wenzelm@36278	5942	apply(simp)
wenzelm@36278	5943	apply(erule disjE)
wenzelm@36278	5944	apply(simp)
wenzelm@36278	5945	apply(auto simp add: ctrm.inject)[1]
wenzelm@36278	5946	apply(simp add: alpha)
wenzelm@36278	5947	apply(erule disjE)
wenzelm@36278	5948	apply(simp)
wenzelm@36278	5949	apply(erule conjE)+
wenzelm@36278	5950	apply(simp)
wenzelm@36278	5951	apply(drule_tac pi="[(a,b)]" in fic.eqvt(2))
wenzelm@36278	5952	apply(simp add: calc_atm)
wenzelm@36278	5953	apply(simp add: alpha)
wenzelm@36278	5954	apply(erule disjE)
wenzelm@36278	5955	apply(simp)
wenzelm@36278	5956	apply(erule conjE)+
wenzelm@36278	5957	apply(simp)
wenzelm@36278	5958	apply(drule_tac pi="[(a,b)]" in fic.eqvt(2))
wenzelm@36278	5959	apply(simp add: calc_atm)
wenzelm@36278	5960	apply(simp)
wenzelm@36278	5961	apply(erule disjE)
wenzelm@36278	5962	apply(simp)
wenzelm@36278	5963	apply(erule disjE)
wenzelm@36278	5964	apply(simp)
wenzelm@36278	5965	apply(auto simp add: ctrm.inject)[1]
wenzelm@36278	5966	apply(simp add: alpha)
wenzelm@36278	5967	apply(erule disjE)
wenzelm@36278	5968	apply(simp)
wenzelm@36278	5969	apply(erule conjE)+
wenzelm@36278	5970	apply(simp)
wenzelm@36278	5971	apply(drule_tac pi="[(a,b)]" in fic.eqvt(2))
wenzelm@36278	5972	apply(simp add: calc_atm)
wenzelm@36278	5973	done
wenzelm@36278	5974
wenzelm@36278	5975	lemma fin_CANDS_aux:
wenzelm@36278	5976	assumes a: "\<not>fin M x"
wenzelm@36278	5977	and b: "(x):M \<in> (NEGn B (\<parallel><B>\<parallel>))"
wenzelm@36278	5978	shows "(x):M \<in> AXIOMSn B \<or> (x):M \<in> BINDINGn B (\<parallel><B>\<parallel>)"
wenzelm@36278	5979	using a b
wenzelm@36278	5980	apply(nominal_induct B rule: ty.strong_induct)
wenzelm@36278	5981	apply(simp)
wenzelm@36278	5982	apply(simp)
wenzelm@36278	5983	apply(erule disjE)
wenzelm@36278	5984	apply(simp)
wenzelm@36278	5985	apply(erule disjE)
wenzelm@36278	5986	apply(simp)
wenzelm@36278	5987	apply(auto simp add: ntrm.inject)[1]
wenzelm@36278	5988	apply(simp add: alpha)
wenzelm@36278	5989	apply(erule disjE)
wenzelm@36278	5990	apply(simp)
wenzelm@36278	5991	apply(auto simp add: calc_atm)[1]
wenzelm@36278	5992	apply(drule_tac pi="[(x,xa)]" in fin.eqvt(1))
wenzelm@36278	5993	apply(simp add: calc_atm)
wenzelm@36278	5994	apply(simp)
wenzelm@36278	5995	apply(erule disjE)
wenzelm@36278	5996	apply(simp)
wenzelm@36278	5997	apply(erule disjE)
wenzelm@36278	5998	apply(simp)
wenzelm@36278	5999	apply(auto simp add: ntrm.inject)[1]
wenzelm@36278	6000	apply(simp add: alpha)
wenzelm@36278	6001	apply(erule disjE)
wenzelm@36278	6002	apply(simp)
wenzelm@36278	6003	apply(erule conjE)+
wenzelm@36278	6004	apply(simp)
wenzelm@36278	6005	apply(drule_tac pi="[(x,y)]" in fin.eqvt(1))
wenzelm@36278	6006	apply(simp add: calc_atm)
wenzelm@36278	6007	apply(simp add: alpha)
wenzelm@36278	6008	apply(erule disjE)
wenzelm@36278	6009	apply(simp)
wenzelm@36278	6010	apply(erule conjE)+
wenzelm@36278	6011	apply(simp)
wenzelm@36278	6012	apply(drule_tac pi="[(x,y)]" in fin.eqvt(1))
wenzelm@36278	6013	apply(simp add: calc_atm)
wenzelm@36278	6014	apply(simp)
wenzelm@36278	6015	apply(erule disjE)
wenzelm@36278	6016	apply(simp)
wenzelm@36278	6017	apply(erule disjE)
wenzelm@36278	6018	apply(simp)
wenzelm@36278	6019	apply(auto simp add: ntrm.inject)[1]
wenzelm@36278	6020	apply(simp add: alpha)
wenzelm@36278	6021	apply(erule disjE)
wenzelm@36278	6022	apply(simp)
wenzelm@36278	6023	apply(erule conjE)+
wenzelm@36278	6024	apply(simp)
wenzelm@36278	6025	apply(drule_tac pi="[(x,z)]" in fin.eqvt(1))
wenzelm@36278	6026	apply(simp add: calc_atm)
wenzelm@36278	6027	apply(simp)
wenzelm@36278	6028	apply(erule disjE)
wenzelm@36278	6029	apply(simp)
wenzelm@36278	6030	apply(erule disjE)
wenzelm@36278	6031	apply(simp)
wenzelm@36278	6032	apply(auto simp add: ntrm.inject)[1]
wenzelm@36278	6033	apply(simp add: alpha)
wenzelm@36278	6034	apply(erule disjE)
wenzelm@36278	6035	apply(simp)
wenzelm@36278	6036	apply(erule conjE)+
wenzelm@36278	6037	apply(simp)
wenzelm@36278	6038	apply(drule_tac pi="[(x,y)]" in fin.eqvt(1))
wenzelm@36278	6039	apply(simp add: calc_atm)
wenzelm@36278	6040	done
wenzelm@36278	6041
wenzelm@36278	6042	lemma fin_CANDS:
wenzelm@36278	6043	assumes a: "\<not>fin M x"
wenzelm@36278	6044	and b: "(x):M \<in> (\<parallel>(B)\<parallel>)"
wenzelm@36278	6045	shows "(x):M \<in> AXIOMSn B \<or> (x):M \<in> BINDINGn B (\<parallel><B>\<parallel>)"
wenzelm@36278	6046	apply(rule fin_CANDS_aux)
wenzelm@36278	6047	apply(rule a)
wenzelm@36278	6048	apply(rule NEG_elim)
wenzelm@36278	6049	apply(rule b)
wenzelm@36278	6050	done
wenzelm@36278	6051
wenzelm@36278	6052	lemma BINDING_implies_CAND:
wenzelm@36278	6053	shows "<c>:M \<in> BINDINGc B (\<parallel>(B)\<parallel>) \<Longrightarrow> <c>:M \<in> (\<parallel><B>\<parallel>)"
wenzelm@36278	6054	and "(x):N \<in> BINDINGn B (\<parallel><B>\<parallel>) \<Longrightarrow> (x):N \<in> (\<parallel>(B)\<parallel>)"
wenzelm@36278	6055	apply -
wenzelm@36278	6056	apply(nominal_induct B rule: ty.strong_induct)
wenzelm@36278	6057	apply(auto)
wenzelm@36278	6058	apply(rule NEG_intro)
wenzelm@36278	6059	apply(nominal_induct B rule: ty.strong_induct)
wenzelm@36278	6060	apply(auto)
wenzelm@36278	6061	done
wenzelm@36278	6062
wenzelm@36278	6063	end

author	Thomas Sewell <thomas.sewell@nicta.com.au>
	Wed, 11 Jun 2014 14:24:23 +1000
changeset 58834	74bf65a1910a
parent 57415	29e308b56d23
child 59324	ec559c6ab5ba
permissions	-rw-r--r--