theory Foo1

imports "../Nominal2" 

begin

text {* 

  Contrived example that has more than one

  binding function

*}

atom_decl name

nominal_datatype foo: trm =

  Var "name"

| App "trm" "trm"

| Lam x::"name" t::"trm"  bind x in t

| Let1 a::"assg" t::"trm"  bind "bn1 a" in t

| Let2 a::"assg" t::"trm"  bind "bn2 a" in t

| Let3 a::"assg" t::"trm"  bind "bn3 a" in t

| Let4 a::"assg'" t::"trm"  bind (set) "bn4 a" in t

and assg =

  As "name" "name" "trm"

and assg' =

  BNil

| BAs "name" "assg'"

binder

  bn1::"assg \<Rightarrow> atom list" and

  bn2::"assg \<Rightarrow> atom list" and

  bn3::"assg \<Rightarrow> atom list" and

  bn4::"assg' \<Rightarrow> atom set"

where

  "bn1 (As x y t) = [atom x]"

| "bn2 (As x y t) = [atom y]"

| "bn3 (As x y t) = [atom x, atom y]"

| "bn4 (BNil) = {}"

| "bn4 (BAs a as) = {atom a} \<union> bn4 as"

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

thm foo.bn_finite

lemma 

  fixes t::trm

  and   as::assg

  and   as'::assg'

  and   c::"'a::fs"

  assumes a1: "\<And>x c. P1 c (Var x)"

  and     a2: "\<And>t1 t2 c. \<lbrakk>\<And>d. P1 d t1; \<And>d. P1 d t2\<rbrakk> \<Longrightarrow> P1 c (App t1 t2)"

  and     a3: "\<And>x t c. \<lbrakk>{atom x} \<sharp>* c; \<And>d. P1 d t\<rbrakk> \<Longrightarrow> P1 c (Lam x t)"

  and     a4: "\<And>as t c. \<lbrakk>set (bn1 as) \<sharp>* c; \<And>d. P2 d as; \<And>d. P1 d t\<rbrakk> \<Longrightarrow> P1 c (Let1 as t)"

  and     a5: "\<And>as t c. \<lbrakk>set (bn2 as) \<sharp>* c; \<And>d. P2 d as; \<And>d. P1 d t\<rbrakk> \<Longrightarrow> P1 c (Let2 as t)"

  and     a6: "\<And>as t c. \<lbrakk>set (bn3 as) \<sharp>* c; \<And>d. P2 d as; \<And>d. P1 d t\<rbrakk> \<Longrightarrow> P1 c (Let3 as t)"

  and     a7: "\<And>as' t c. \<lbrakk>(bn4 as') \<sharp>* c; \<And>d. P3 d as'; \<And>d. P1 d t\<rbrakk> \<Longrightarrow> P1 c (Let4 as' t)" 

  and     a8: "\<And>x y t c. \<And>d. P1 d t \<Longrightarrow> P2 c (As x y t)"

  and     a9: "\<And>c. P3 c (BNil)"

  and     a10: "\<And>c a as. \<And>d. P3 d as \<Longrightarrow> P3 c (BAs a as)"

  shows "P1 c t" "P2 c as" "P3 c as'"

apply(rule_tac y="t" and c="c" in foo.strong_exhaust(1))

apply(simp_all)[7]

apply(rule_tac ya="as"in foo.strong_exhaust(2))

apply(simp_all)[1]

apply(rule_tac yb="as'" in foo.strong_exhaust(3))

apply(simp_all)[2]

apply(relation "measure (sum_case (size o snd) (sum_case (size o snd) (size o snd)))")

apply(simp_all add: foo.size)

done

end