theory SingleLet
imports "../NewParser"
begin

atom_decl name

declare [[STEPS = 12]]


nominal_datatype trm =
  Var "name"
| App "trm" "trm"
| Lam x::"name" t::"trm"  bind_set x in t
| Let a::"assg" t::"trm"  bind_set "bn a" in t
| Foo x::"name" y::"name" t::"trm" t1::"trm" t2::"trm" bind_set x in y t t1 t2
| Bar x::"name" y::"name" t::"trm" bind y x in t x y
| Baz x::"name" t1::"trm" t2::"trm" bind x in t1, bind x in t2 
and assg =
  As "name" "name" "trm" "name"
binder
  bn::"assg \<Rightarrow> atom set"
where
  "bn (As x y t z) = {atom x}"


lemma
  shows "alpha_trm_raw x x"
  and "alpha_assg_raw y y"
  and "alpha_bn_raw y y"
apply(induct rule: trm_raw_assg_raw.inducts)
apply(rule alpha_trm_raw_alpha_assg_raw_alpha_bn_raw.intros)
apply(rule refl)
apply(rule alpha_trm_raw_alpha_assg_raw_alpha_bn_raw.intros)
apply(assumption)
apply(assumption)
apply(rule alpha_trm_raw_alpha_assg_raw_alpha_bn_raw.intros)
apply(rule_tac x="0" in exI)
apply(rule alpha_gen_refl)
apply(assumption)
apply(rule alpha_trm_raw_alpha_assg_raw_alpha_bn_raw.intros)
apply(rule_tac x="0" in exI)
apply(rule alpha_gen_refl)
apply(assumption)
apply(assumption)
apply(rule alpha_trm_raw_alpha_assg_raw_alpha_bn_raw.intros)
apply(rule_tac x="0" in exI)
apply(rule alpha_gen_refl)
apply(simp only: prod_alpha_def split_conv prod_rel.simps)
apply(simp)
apply(rule alpha_trm_raw_alpha_assg_raw_alpha_bn_raw.intros)
apply(rule_tac x="0" in exI)
apply(rule alpha_gen_refl)
apply(simp only: prod_alpha_def split_conv prod_rel.simps)
apply(simp)
apply(rule alpha_trm_raw_alpha_assg_raw_alpha_bn_raw.intros)
apply(rule refl)
apply(rule refl)
apply(assumption)
apply(rule refl)
apply(rule alpha_trm_raw_alpha_assg_raw_alpha_bn_raw.intros)
apply(rule refl)
apply(assumption)
apply(rule refl)
done



thm trm_assg.fv
thm trm_assg.supp
thm trm_assg.eq_iff
thm trm_assg.bn
thm trm_assg.perm
thm trm_assg.induct
thm trm_assg.inducts
thm trm_assg.distinct
ML {* Sign.of_sort @{theory} (@{typ trm}, @{sort fs}) *}

(* TEMPORARY
thm trm_assg.fv[simplified trm_assg.supp(1-2)]
*)

end