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 

*)

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