theory Ex4
imports "../NewParser"
begin

declare [[STEPS = 4]]
(* alpha does not work for this type *)

atom_decl name

nominal_datatype trm =
  Var "name"
| App "trm" "trm"
| Lam x::"name" t::"trm"        bind_set x in t
| Let p::"pat" "trm" t::"trm"   bind_set "f p" in t
| Foo1 p::"pat" q::"pat" t::"trm" bind_set "f p" "f q" in t
| Foo2 x::"name" p::"pat" t::"trm" bind_set x "f p" in t
and pat =
  PN
| PS "name"
| PD "pat" "pat"
binder
  f::"pat \<Rightarrow> atom set"
where
  "f PN = {}"
| "f (PS x) = {atom x}"
| "f (PD p1 p2) = (f p1) \<union> (f p2)"

thm permute_trm_raw_permute_pat_raw.simps
thm fv_trm_raw.simps fv_pat_raw.simps fv_f_raw.simps

(* thm alpha_trm_raw_alpha_pat_raw_alpha_f_raw.intros[no_vars]*)

inductive 
 alpha_trm_raw and alpha_pat_raw and alpha_f_raw
where
(* alpha_trm_raw *)
  "name = namea \<Longrightarrow> alpha_trm_raw (Var_raw name) (Var_raw namea)"
| "\<lbrakk>alpha_trm_raw trm_raw1 trm_raw1a; alpha_trm_raw trm_raw2 trm_raw2a\<rbrakk>
   \<Longrightarrow> alpha_trm_raw (App_raw trm_raw1 trm_raw2) (App_raw trm_raw1a trm_raw2a)"
| "\<exists>p. ({atom name}, trm_raw) \<approx>gen alpha_trm_raw fv_trm_raw p ({atom namea}, trm_rawa) \<Longrightarrow>
   alpha_trm_raw (Lam_raw name trm_raw) (Lam_raw namea trm_rawa)"
| "\<lbrakk>\<exists>p. (f_raw pat_raw, trm_raw2) \<approx>gen alpha_trm_raw fv_trm_raw p (f_raw pat_rawa, trm_raw2a);
   alpha_f_raw pat_raw pat_rawa; alpha_trm_raw trm_raw1 trm_raw1a\<rbrakk>
  \<Longrightarrow> alpha_trm_raw (Let_raw pat_raw trm_raw1 trm_raw2) (Let_raw pat_rawa trm_raw1a trm_raw2a)"
| "\<lbrakk>\<exists>p. (f_raw pat_raw1 \<union> f_raw pat_raw2, trm_raw) \<approx>gen alpha_trm_raw fv_trm_raw p (f_raw pat_raw1a \<union> f_raw pat_raw2a, trm_rawa);
   alpha_f_raw pat_raw1 pat_raw1a; alpha_f_raw pat_raw2 pat_raw2a\<rbrakk>
   \<Longrightarrow> alpha_trm_raw (Foo1_raw pat_raw1 pat_raw2 trm_raw) (Foo1_raw pat_raw1a pat_raw2a trm_rawa)"
| "\<lbrakk>\<exists>p. ({atom name} \<union> f_raw pat_raw, trm_raw) \<approx>gen alpha_trm_raw fv_trm_raw p ({atom namea} \<union> f_raw pat_rawa, trm_rawa);
   alpha_f_raw pat_raw pat_rawa\<rbrakk>
   \<Longrightarrow> alpha_trm_raw (Foo2_raw name pat_raw trm_raw) (Foo2_raw namea pat_rawa trm_rawa)"

| "alpha_pat_raw PN_raw PN_raw"
| "name = namea \<Longrightarrow> alpha_pat_raw (PS_raw name) (PS_raw namea)"
| "\<lbrakk>alpha_pat_raw pat_raw1 pat_raw1a; alpha_pat_raw pat_raw2 pat_raw2a\<rbrakk>
   \<Longrightarrow> alpha_pat_raw (PD_raw pat_raw1 pat_raw2) (PD_raw pat_raw1a pat_raw2a)"

| "alpha_f_raw PN_raw PN_raw"
| "alpha_f_raw (PS_raw name) (PS_raw namea)"
| "\<lbrakk>alpha_f_raw pat_raw1 pat_raw1a; alpha_f_raw pat_raw2 pat_raw2a\<rbrakk>
  \<Longrightarrow> alpha_f_raw (PD_raw pat_raw1 pat_raw2) (PD_raw pat_raw1a pat_raw2a)"

lemmas all = alpha_trm_raw_alpha_pat_raw_alpha_f_raw.intros

lemma
  shows "alpha_trm_raw (Foo2_raw x (PS_raw x) (Var_raw x))
                       (Foo2_raw y (PS_raw y) (Var_raw y))"
apply(rule all)
apply(rule_tac x="(atom x \<rightleftharpoons> atom y)" in exI)
apply(simp add: alphas)
apply(simp add: supp_at_base fresh_star_def)
apply(rule conjI)
apply(rule all)
apply(simp)
apply(perm_simp)
apply(simp)
apply(rule all)
done

 

end