theory Foo2
imports "../Nominal2" 
begin

(* 
  Contrived example that has more than one
  binding clause
*)

atom_decl name

nominal_datatype foo: trm =
  Var "name"
| App "trm" "trm"
| Lam x::"name" t::"trm"  bind x in t
| Let1 a1::"assg" t1::"trm" a2::"assg" t2::"trm" bind "bn a1" in t1, bind "bn a2" in t2
| Let2 x::"name" y::"name" t1::"trm" t2::"trm" bind x y in t1, bind y in t2
and assg =
  As_Nil
| As "name" x::"name" t::"trm" "assg" 
binder
  bn::"assg \<Rightarrow> atom list"
where
  "bn (As x y t a) = [atom x] @ bn a"
| "bn (As_Nil) = []"





thm foo.bn_defs
thm foo.permute_bn
thm foo.perm_bn_alpha
thm foo.perm_bn_simps
thm foo.bn_finite


thm foo.distinct
thm foo.induct
thm foo.inducts
thm foo.exhaust
thm foo.fv_defs
thm foo.bn_defs
thm foo.perm_simps
thm foo.eq_iff
thm foo.fv_bn_eqvt
thm foo.size_eqvt
thm foo.supports
thm foo.fsupp
thm foo.supp
thm foo.fresh

lemma at_set_avoiding5:
  assumes "finite xs"
  and     "finite (supp c)"
  shows "\<exists>p. (p \<bullet> xs) \<sharp>* c \<and> supp p = xs \<union> p \<bullet> xs"
using assms
apply(erule_tac c="c" in at_set_avoiding)
apply(auto)
done


lemma Abs_rename_lst3:
  fixes x::"'a::fs"
  assumes a: "(p \<bullet> (set bs)) \<sharp>* (bs, x)" 
  shows "\<exists>q. [bs]lst. x = [p \<bullet> bs]lst. (q \<bullet> x) \<and> p \<bullet> bs = q \<bullet> bs"
proof -
  from a have "p \<bullet> (set bs) \<inter> (set bs) = {}" using at_fresh_star_inter 
    by (simp add: fresh_star_Pair fresh_star_set)
  with list_renaming_perm 
  obtain q where *: "q \<bullet> bs = p \<bullet> bs" and **: "supp q \<subseteq> set bs \<union> (p \<bullet> set bs)" by metis 
  have "[bs]lst. x =  q \<bullet> ([bs]lst. x)"
    apply(rule perm_supp_eq[symmetric])
    using a **
    unfolding fresh_star_Pair
    unfolding Abs_fresh_star_iff
    unfolding fresh_star_def
    by auto
  also have "\<dots> = [q \<bullet> bs]lst. (q \<bullet> x)" by simp
  also have "\<dots> = [p \<bullet> bs]lst. (q \<bullet> x)" using * by simp
  finally have "[bs]lst. x = [p \<bullet> bs]lst. (q \<bullet> x)" .
  then show "\<exists>q. [bs]lst. x = [p \<bullet> bs]lst. (q \<bullet> x) \<and> p \<bullet> bs = q \<bullet> bs" 
    using * ** by metis
qed


lemma
  fixes c::"'a::fs"
  assumes a:  "\<And>assg1 trm1 assg2 trm2. \<lbrakk>((set (bn assg1)) \<union> (set (bn assg2))) \<sharp>* c; y = Let1 assg1 trm1 assg2 trm2\<rbrakk> \<Longrightarrow> P"
  shows "y = Let1 assg1 trm1 assg2 trm2 \<Longrightarrow> P"
apply -
apply(subgoal_tac "\<exists>p. (p \<bullet> ((set (bn assg1)) \<union> (set (bn assg2)))) \<sharp>* (c, bn assg1, bn assg2, trm1, trm2) \<and>
  supp p = (set (bn assg1)) \<union> (set (bn assg2)) \<union> (p \<bullet> ((set (bn assg1)) \<union> (set (bn assg2))))")
defer
apply(rule at_set_avoiding5)
apply(simp add: foo.bn_finite)
apply(simp add: supp_Pair finite_supp)
apply(erule exE)
apply(perm_simp add: foo.permute_bn)
apply(simp add: fresh_star_Pair)
apply(erule conjE)+
thm Abs_rename_lst
apply(insert Abs_rename_lst)[1]
apply(drule_tac x="p" in meta_spec)
apply(drule_tac x="bn assg1" in meta_spec)
apply(drule_tac x="trm1" in meta_spec)
apply(insert Abs_rename_lst)[1]
apply(drule_tac x="p" in meta_spec)
apply(drule_tac x="bn assg2" in meta_spec)
apply(drule_tac x="trm2" in meta_spec)
apply(drule meta_mp)
apply(perm_simp add: foo.permute_bn)
apply(simp add: fresh_star_Pair fresh_star_Un)
apply(drule meta_mp)
apply(perm_simp add: foo.permute_bn)
apply(simp add: fresh_star_Pair fresh_star_Un)
apply(erule exE)+
apply(rule a)
apply(assumption)
apply(simp only: foo.eq_iff)
apply(perm_simp add: foo.permute_bn)
apply(rule conjI)
apply(rule refl)
apply(rule conjI)
apply(rule foo.perm_bn_alpha)
apply(rule conjI)
apply(perm_simp add: foo.permute_bn)
apply(rule refl)
apply(rule foo.perm_bn_alpha)
done

lemma
  fixes c::"'a::fs"
  assumes a:  "\<And>assg1 trm1 assg2 trm2. \<lbrakk>((set (bn assg1)) \<union> (set (bn assg2))) \<sharp>* c; y = Let1 assg1 trm1 assg2 trm2\<rbrakk> \<Longrightarrow> P"
  shows "y = Let1 assg1 trm1 assg2 trm2 \<Longrightarrow> P"
apply -
apply(subgoal_tac "\<exists>p. (p \<bullet> ((set (bn assg1)) \<union> (set (bn assg2)))) \<sharp>* (c, bn assg1, bn assg2, trm1, trm2) \<and>
  supp p = (set (bn assg1)) \<union> (set (bn assg2)) \<union> (p \<bullet> ((set (bn assg1)) \<union> (set (bn assg2))))")
defer
apply(rule at_set_avoiding5)
apply(simp add: foo.bn_finite)
apply(simp add: supp_Pair finite_supp)
apply(erule exE)
apply(perm_simp add: foo.permute_bn)
apply(simp add: fresh_star_Pair)
apply(erule conjE)+
apply(insert Abs_rename_lst3)[1]
apply(drule_tac x="p" in meta_spec)
apply(drule_tac x="bn assg1" in meta_spec)
apply(drule_tac x="trm1" in meta_spec)
apply(insert Abs_rename_lst3)[1]
apply(drule_tac x="p" in meta_spec)
apply(drule_tac x="bn assg2" in meta_spec)
apply(drule_tac x="trm2" in meta_spec)
apply(drule meta_mp)
apply(perm_simp add: foo.permute_bn)
apply(simp add: fresh_star_Pair fresh_star_Un)
apply(drule meta_mp)
apply(perm_simp add: foo.permute_bn)
apply(simp add: fresh_star_Pair fresh_star_Un)
apply(erule exE)+
apply(erule conjE)+
apply(simp add: foo.permute_bn)
apply(rule a)
apply(assumption)
apply(clarify)
apply(simp (no_asm) only: foo.eq_iff)
apply(rule conjI)
apply(assumption)
apply(rule conjI)
apply(rule foo.perm_bn_alpha)
apply(rule conjI)
apply(assumption)
apply(rule foo.perm_bn_alpha)
done


lemma
  fixes c::"'a::fs"
  assumes a:  "\<And>assg1 trm1 assg2 trm2. \<lbrakk>((set (bn assg1)) \<union> (set (bn assg2))) \<sharp>* c; y = Let1 assg1 trm1 assg2 trm2\<rbrakk> \<Longrightarrow> P"
  shows "y = Let1 assg1 trm1 assg2 trm2 \<Longrightarrow> P"
apply -
apply(subgoal_tac "\<exists>p. (p \<bullet> ((set (bn assg1)) \<union> (set (bn assg2)))) \<sharp>* (c, bn assg1, bn assg2, trm1, trm2) \<and>
  supp p = (set (bn assg1)) \<union> (set (bn assg2)) \<union> (p \<bullet> ((set (bn assg1)) \<union> (set (bn assg2))))")
defer
apply(rule at_set_avoiding5)
apply(simp add: foo.bn_finite)
apply(simp add: supp_Pair finite_supp)
apply(erule exE)
apply(simp add: fresh_star_Pair)
apply(erule conjE)+
apply(simp (no_asm_use) only: foo.permute_bn set_eqvt union_eqvt)
apply(rule a)
apply(assumption)
apply(clarify)
apply(simp (no_asm) only: foo.eq_iff)
apply(rule conjI)
apply(rule trans)
apply(rule_tac p="p" in supp_perm_eq[symmetric])
apply(rule fresh_star_supp_conv)
apply(simp (no_asm_use) add: fresh_star_def)
apply(simp (no_asm_use) add: Abs_fresh_iff)[1]
apply(rule ballI)
apply(auto simp add: fresh_Pair)[1]
apply(simp (no_asm_use) only: permute_Abs)
apply(simp (no_asm_use) only: multi_recs.fv_bn_eqvt)
apply(simp)
apply(rule at_set_avoiding5)
apply(simp add: multi_recs.bn_finite)
apply(simp add: supp_Pair finite_supp)
apply(rule finite_sets_supp)
apply(simp add: multi_recs.bn_finite)
done



lemma test6:
  fixes c::"'a::fs"
  assumes "\<And>name. y = Var name \<Longrightarrow> P" 
  and     "\<And>trm1 trm2. y = App trm1 trm2 \<Longrightarrow> P"
  and     "\<And>name trm. \<lbrakk>{atom name} \<sharp>* c; y = Lam name trm\<rbrakk> \<Longrightarrow> P" 
  and     "\<And>a1 trm1 a2 trm2.  \<lbrakk>((set (bn a1)) \<union> (set (bn a2))) \<sharp>* c; y = Let1 a1 trm1 a2 trm2\<rbrakk> \<Longrightarrow> P"
  and     "\<exists>x1 x2 trm1 trm2. {atom x1, atom x2} \<sharp>* c \<and> y = Let2 x1 x2 trm1 trm2 \<Longrightarrow> P"
  shows "P"
apply(rule_tac y="y" in foo.exhaust(1))
apply(rule assms(1))
apply(assumption)
apply(rule assms(2))
apply(assumption)
(* lam case *)
apply(subgoal_tac "\<exists>p. (p \<bullet> {atom name}) \<sharp>* (c, name, trm)")
apply(erule exE)
apply(insert Abs_rename_lst)[1]
apply(drule_tac x="p" in meta_spec)
apply(drule_tac x="[atom name]" in meta_spec)
apply(drule_tac x="trm" in meta_spec)
apply(simp only: fresh_star_Pair set.simps)
apply(drule meta_mp)
apply(simp add: fresh_star_def fresh_Nil fresh_Cons fresh_atom_at_base)
apply(erule exE)
apply(rule assms(3))
apply(perm_simp)
apply(erule conjE)+
apply(assumption)
apply(clarify)
apply(simp (no_asm) add: foo.eq_iff)
apply(perm_simp)
apply(assumption)
apply(rule at_set_avoiding1)
apply(simp)
apply(simp add: finite_supp)
(* let1 case *)
apply(subgoal_tac "\<exists>p. (p \<bullet> ((set (bn assg1)) \<union> (set (bn assg2)))) \<sharp>* (c, bn assg1, bn assg2, trm1, trm2)")
apply(erule exE)
apply(perm_simp add: foo.permute_bn)
apply(simp add: fresh_star_Pair)
apply(erule conjE)+
apply(insert Abs_rename_lst)[1]
apply(drule_tac x="p" in meta_spec)
apply(drule_tac x="bn assg1" in meta_spec)
apply(drule_tac x="trm1" in meta_spec)
apply(insert Abs_rename_lst)[1]
apply(drule_tac x="p" in meta_spec)
apply(drule_tac x="bn assg2" in meta_spec)
apply(drule_tac x="trm2" in meta_spec)
apply(drule meta_mp)
apply(perm_simp add: foo.permute_bn)
apply(simp add: fresh_star_Pair fresh_star_Un)
apply(drule meta_mp)
apply(perm_simp add: foo.permute_bn)
apply(simp add: fresh_star_Pair fresh_star_Un)
apply(erule exE)+
apply(rule assms(4))
apply(assumption)
apply(simp add: foo.eq_iff refl)
apply(rule conjI)
apply(perm_simp add: foo.permute_bn)
apply(rule refl)
apply(rule conjI)
apply(rule foo.perm_bn_alpha)
apply(rule conjI)
apply(perm_simp add: foo.permute_bn)
apply(rule refl)
apply(rule foo.perm_bn_alpha)
apply(rule at_set_avoiding1)
apply(simp)
apply(simp add: finite_supp)
(* let2 case *)
apply(subgoal_tac "\<exists>p. (p \<bullet> ({atom name1} \<union> {atom name2})) \<sharp>* (c, atom name1, atom name2, trm1, trm2)")
apply(erule exE)
apply(rule assms(5))
apply(simp only:)
apply(simp only: foo.eq_iff)
apply(insert Abs_rename_list)[1]
apply(drule_tac x="p" in meta_spec)
apply(drule_tac x="[atom name1] @ [atom name2]" in meta_spec)
apply(drule_tac x="trm1" in meta_spec)
apply(insert Abs_rename_list)[1]
apply(drule_tac x="p" in meta_spec)
apply(drule_tac x="[atom name2]" in meta_spec)
apply(drule_tac x="trm2" in meta_spec)
apply(drule meta_mp)
apply(simp only: union_eqvt fresh_star_Pair fresh_star_list fresh_star_Un, simp)
apply(auto)[1]
apply(perm_simp)
apply(auto simp add: fresh_star_def)[1]
apply(perm_simp)
apply(auto simp add: fresh_star_def)[1]
apply(perm_simp)
apply(auto simp add: fresh_star_def)[1]
apply(drule meta_mp)
apply(perm_simp)
apply(auto simp add: fresh_star_def fresh_Pair fresh_Nil fresh_Cons)[1]
apply(erule exE)+
apply(rule exI)+
apply(perm_simp add: foo.permute_bn)
apply(rule conjI)
prefer 2
apply(rule conjI)
apply(assumption)
apply(assumption)
apply(simp add: fresh_star_Pair)
apply(simp add: fresh_star_def)
apply(rule at_set_avoiding1)
apply(simp)
apply(simp add: finite_supp)
done

lemma test5:
  fixes c::"'a::fs"
  assumes "\<And>name. y = Var name \<Longrightarrow> P" 
  and     "\<And>trm1 trm2. y = App trm1 trm2 \<Longrightarrow> P"
  and     "\<And>name trm. \<lbrakk>{atom name} \<sharp>* c; y = Lam name trm\<rbrakk> \<Longrightarrow> P" 
  and     "\<And>a1 trm1 a2 trm2. \<lbrakk>((set (bn a1)) \<union> (set (bn a2))) \<sharp>* c; y = Let1 a1 trm1 a2 trm2\<rbrakk> \<Longrightarrow> P"
  and     "\<And>x1 x2 trm1 trm2. \<lbrakk>{atom x1, atom x2} \<sharp>* c; y = Let2 x1 x2 trm1 trm2\<rbrakk> \<Longrightarrow> P"
  shows "P"
apply(rule_tac y="y" in test6)
apply(erule exE)+
apply(rule assms(1))
apply(assumption)
apply(erule exE)+
apply(rule assms(2))
apply(assumption)
apply(rule assms(3))
apply(auto)[2]
apply(erule exE)+
apply(rule assms(4))
apply(auto)[2]
apply(erule exE)+
apply(rule assms(5))
apply(auto)[2]
done


lemma strong_induct:
  fixes c :: "'a :: fs"
  and assg :: assg and trm :: trm
  assumes a0: "\<And>name c. P1 c (Var name)"
  and a1: "\<And>trm1 trm2 c. \<lbrakk>\<And>d. P1 d trm1; \<And>d. P1 d trm2\<rbrakk> \<Longrightarrow> P1 c (App trm1 trm2)"
  and a2: "\<And>name trm c. (\<And>d. P1 d trm) \<Longrightarrow> P1 c (Lam name trm)"
  and a3: "\<And>a1 t1 a2 t2 c. 
    \<lbrakk>((set (bn a1)) \<union> (set (bn a2))) \<sharp>* c; \<And>d. P2 c a1; \<And>d. P1 d t1; \<And>d. P2 d a2; \<And>d. P1 d t2\<rbrakk> 
    \<Longrightarrow> P1 c (Let1 a1 t1 a2 t2)"
  and a4: "\<And>n1 n2 t1 t2 c. 
    \<lbrakk>{atom n1, atom n2} \<sharp>* c; \<And>d. P1 d t1; \<And>d. P1 d t2\<rbrakk> \<Longrightarrow> P1 c (Let2 n1 n2 t1 t2)"
  and a5: "\<And>c. P2 c As_Nil"
  and a6: "\<And>name1 name2 trm assg c. \<lbrakk>\<And>d. P1 d trm; \<And>d. P2 d assg\<rbrakk> \<Longrightarrow> P2 c (As name1 name2 trm assg)"
  shows "P1 c trm" "P2 c assg"
  using assms
  apply(induction_schema)
  apply(rule_tac y="trm" and c="c" in test5)
  apply(simp_all)[5]
  apply(rule_tac y="assg" in foo.exhaust(2))
  apply(simp_all)[2]
  apply(relation "measure (sum_case (size o snd) (size o snd))")
  apply(simp_all add: foo.size)
  done

end