theory ExLamimports "../Parser"beginatom_decl namenominal_datatype lm = Vr "name"| Ap "lm" "lm"| Lm x::"name" l::"lm" bind x in llemmas supp_fn' = lm.fv[simplified lm.supp]lemma fixes c::"'a::fs" assumes a1: "\<And>name c. P c (Vr name)" and a2: "\<And>lm1 lm2 c. \<lbrakk>\<And>d. P d lm1; \<And>d. P d lm2\<rbrakk> \<Longrightarrow> P c (Ap lm1 lm2)" and a3: "\<And>name lm c. \<lbrakk>atom name \<sharp> c; \<And>d. P d lm\<rbrakk> \<Longrightarrow> P c (Lm name lm)" shows "P c lm"proof - have "\<And>p. P c (p \<bullet> lm)" apply(induct lm arbitrary: c rule: lm.induct) apply(simp only: lm.perm) apply(rule a1) apply(simp only: lm.perm) apply(rule a2) apply(blast)[1] apply(assumption) apply(subgoal_tac "\<exists>new::name. (atom new) \<sharp> (c, Lm (p \<bullet> name) (p \<bullet> lm))") defer apply(simp add: fresh_def) apply(rule_tac X="supp (c, Lm (p \<bullet> name) (p \<bullet> lm))" in obtain_at_base) apply(simp add: supp_Pair finite_supp) apply(blast) apply(erule exE) apply(rule_tac t="p \<bullet> Lm name lm" and s="(((p \<bullet> name) \<leftrightarrow> new) + p) \<bullet> Lm name lm" in subst) apply(simp del: lm.perm) apply(subst lm.perm) apply(subst (2) lm.perm) apply(rule flip_fresh_fresh) apply(simp add: fresh_def) apply(simp only: supp_fn') apply(simp) apply(simp add: fresh_Pair) apply(simp) apply(rule a3) apply(simp add: fresh_Pair) apply(drule_tac x="((p \<bullet> name) \<leftrightarrow> new) + p" in meta_spec) apply(simp) done then have "P c (0 \<bullet> lm)" by blast then show "P c lm" by simpqedlemma fixes c::"'a::fs" assumes a1: "\<And>name c. P c (Vr name)" and a2: "\<And>lm1 lm2 c. \<lbrakk>\<And>d. P d lm1; \<And>d. P d lm2\<rbrakk> \<Longrightarrow> P c (Ap lm1 lm2)" and a3: "\<And>name lm c. \<lbrakk>atom name \<sharp> c; \<And>d. P d lm\<rbrakk> \<Longrightarrow> P c (Lm name lm)" shows "P c lm"proof - have "\<And>p. P c (p \<bullet> lm)" apply(induct lm arbitrary: c rule: lm.induct) apply(simp only: lm.perm) apply(rule a1) apply(simp only: lm.perm) apply(rule a2) apply(blast)[1] apply(assumption) thm at_set_avoiding apply(subgoal_tac "\<exists>q. (q \<bullet> {p \<bullet> atom name}) \<sharp>* c \<and> supp (p \<bullet> Lm name lm) \<sharp>* q") apply(erule exE) apply(rule_tac t="p \<bullet> Lm name lm" and s="q \<bullet> p \<bullet> Lm name lm" in subst) defer apply(simp add: lm.perm) apply(rule a3) apply(simp add: eqvts fresh_star_def) apply(drule_tac x="q + p" in meta_spec) apply(simp) sorry then have "P c (0 \<bullet> lm)" by blast then show "P c lm" by simpqedend