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.strong_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 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} \<union> {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 foo.strong_exhaust(1))

  apply(simp_all)[5]

  apply(rule_tac ya="assg" in foo.strong_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