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(5)
thm foo.fv_bn_eqvt
thm foo.size_eqvt
thm foo.supports
thm foo.fsupp
thm foo.supp
thm foo.fresh






(*
thm at_set_avoiding1[THEN exE]


lemma Let1_rename:
  fixes c::"'a::fs"
  shows "\<exists>name' trm'. {atom name'} \<sharp>* c \<and>  Lam name trm = Lam name' trm'"
apply(simp only: foo.eq_iff)
apply(rule at_set_avoiding1[where c="(c, atom name, trm)" and xs="set [atom name]", THEN exE])
apply(simp)
apply(simp add: finite_supp)
apply(perm_simp)
apply(rule Abs_rename_list[THEN exE])
apply(simp only: set_eqvt)
apply
sorry 
*)

thm foo.exhaust

ML {*
fun strip_prems_concl trm =
  (Logic.strip_imp_prems trm, Logic.strip_imp_concl trm)
*}


ML {*
  @{thms foo.exhaust}
  |> map prop_of
  |> map strip_prems_concl
  |> map fst
  |> map (map (Syntax.string_of_term @{context}))
  |> map cat_lines
  |> cat_lines
  |> writeln 
*}

lemma test6:
  fixes c::"'a::fs"
  assumes "\<exists>name. y = Var name \<Longrightarrow> P" 
  and     "\<exists>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     "\<exists>a1 trm1 a2 trm2.  ((set (bn a1)) \<union> (set (bn a2))) \<sharp>* c \<and> y = Let1 a1 trm1 a2 trm2 \<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(rule exI)+
apply(assumption)
apply(rule assms(2))
apply(rule exI)+
apply(assumption)
(* lam case *)
(*
apply(rule Let1_rename)
apply(rule assms(3))
apply(assumption)
apply(simp)
*)

apply(subgoal_tac "\<exists>p. (p \<bullet> {atom name}) \<sharp>* (c, [atom name], trm)")
apply(erule exE)
apply(insert Abs_rename_list)[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)
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(rule assms(4))
apply(simp only: foo.eq_iff)
apply(insert Abs_rename_list)[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_list)[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(simp only: union_eqvt fresh_star_Pair set.simps fresh_star_Un, simp)
apply(drule meta_mp)
apply(simp only: union_eqvt fresh_star_Pair set.simps fresh_star_Un, simp)
apply(erule exE)+
apply(rule exI)+
apply(perm_simp add: foo.permute_bn)
apply(rule conjI)
apply(simp add: fresh_star_Pair fresh_star_Un)
apply(erule conjE)+
apply(rule conjI)
apply(assumption)
apply(rotate_tac 4)
apply(assumption)
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)
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