theory Let
imports "../Nominal2" 
begin

atom_decl name

nominal_datatype trm =
  Var "name"
| App "trm" "trm"
| Lam x::"name" t::"trm"  bind  x in t
| Let as::"assn" t::"trm"   bind "bn as" in t
and assn =
  ANil
| ACons "name" "trm" "assn"
binder
  bn
where
  "bn ANil = []"
| "bn (ACons x t as) = (atom x) # (bn as)"

thm trm_assn.fv_defs
thm trm_assn.eq_iff 
thm trm_assn.bn_defs
thm trm_assn.perm_simps
thm trm_assn.induct
thm trm_assn.inducts
thm trm_assn.distinct
thm trm_assn.supp
thm trm_assn.fresh
thm trm_assn.exhaust
thm trm_assn.strong_exhaust


lemma lets_bla:
  "x \<noteq> z \<Longrightarrow> y \<noteq> z \<Longrightarrow> x \<noteq> y \<Longrightarrow>(Let (ACons x (Var y) ANil) (Var x)) \<noteq> (Let (ACons x (Var z) ANil) (Var x))"
  by (simp add: trm_assn.eq_iff)


lemma lets_ok:
  "(Let (ACons x (Var y) ANil) (Var x)) = (Let (ACons y (Var y) ANil) (Var y))"
  apply (simp add: trm_assn.eq_iff Abs_eq_iff )
  apply (rule_tac x="(x \<leftrightarrow> y)" in exI)
  apply (simp_all add: alphas atom_eqvt supp_at_base fresh_star_def trm_assn.bn_defs trm_assn.supp)
  done

lemma lets_ok3:
  "x \<noteq> y \<Longrightarrow>
   (Let (ACons x (App (Var y) (Var x)) (ACons y (Var y) ANil)) (App (Var x) (Var y))) \<noteq>
   (Let (ACons y (App (Var x) (Var y)) (ACons x (Var x) ANil)) (App (Var x) (Var y)))"
  apply (simp add: trm_assn.eq_iff)
  done


lemma lets_not_ok1:
  "x \<noteq> y \<Longrightarrow>
   (Let (ACons x (Var x) (ACons y (Var y) ANil)) (App (Var x) (Var y))) \<noteq>
   (Let (ACons y (Var x) (ACons x (Var y) ANil)) (App (Var x) (Var y)))"
  apply (simp add: alphas trm_assn.eq_iff trm_assn.supp fresh_star_def atom_eqvt Abs_eq_iff trm_assn.bn_defs)
  done

lemma lets_nok:
  "x \<noteq> y \<Longrightarrow> x \<noteq> z \<Longrightarrow> z \<noteq> y \<Longrightarrow>
   (Let (ACons x (App (Var z) (Var z)) (ACons y (Var z) ANil)) (App (Var x) (Var y))) \<noteq>
   (Let (ACons y (Var z) (ACons x (App (Var z) (Var z)) ANil)) (App (Var x) (Var y)))"
  apply (simp add: alphas trm_assn.eq_iff fresh_star_def trm_assn.bn_defs Abs_eq_iff trm_assn.supp trm_assn.distinct)
  done

lemma
  fixes a b c :: name
  assumes x: "a \<noteq> c" and y: "b \<noteq> c"
  shows "\<exists>p.([atom a], Var c) \<approx>lst (op =) supp p ([atom b], Var c)"
  apply (rule_tac x="(a \<leftrightarrow> b)" in exI)
  apply (simp add: alphas trm_assn.supp supp_at_base x y fresh_star_def atom_eqvt)
  by (metis Rep_name_inverse atom_name_def flip_fresh_fresh fresh_atom fresh_perm x y)

lemma alpha_bn_refl: "alpha_bn x x"
apply (induct x rule: trm_assn.inducts(2))
apply (rule TrueI)
apply (auto simp add: trm_assn.eq_iff)
done

lemma alpha_bn_inducts_raw:
  "\<lbrakk>alpha_bn_raw a b; P3 ANil_raw ANil_raw;
 \<And>trm_raw trm_rawa assn_raw assn_rawa name namea.
    \<lbrakk>alpha_trm_raw trm_raw trm_rawa; alpha_bn_raw assn_raw assn_rawa;
     P3 assn_raw assn_rawa\<rbrakk>
    \<Longrightarrow> P3 (ACons_raw name trm_raw assn_raw)
        (ACons_raw namea trm_rawa assn_rawa)\<rbrakk> \<Longrightarrow> P3 a b"
  by (erule alpha_trm_raw_alpha_assn_raw_alpha_bn_raw.inducts(3)[of _ _ "\<lambda>x y. True" _ "\<lambda>x y. True", simplified]) auto

lemmas alpha_bn_inducts = alpha_bn_inducts_raw[quot_lifted]

nominal_primrec
    subst  :: "name \<Rightarrow> trm \<Rightarrow> trm \<Rightarrow> trm"
and substa :: "name \<Rightarrow> trm \<Rightarrow> assn \<Rightarrow> assn"
where
  "subst s t (Var x) = (if (s = x) then t else (Var x))"
| "subst s t (App l r) = App (subst s t l) (subst s t r)"
| "atom v \<sharp> (s, t) \<Longrightarrow> subst s t (Lam v b) = Lam v (subst s t b)"
| "set (bn as) \<sharp>* (s, t) \<Longrightarrow> subst s t (Let as b) = Let (substa s t as) (subst s t b)"
| "substa s t ANil = ANil"
| "substa s t (ACons v t' as) = ACons v (subst v t t') as"

(*apply (subgoal_tac "\<forall>l. \<exists>!r. subst_substa_graph l r")
defer
apply rule
apply (simp only: Ex1_def)
apply (case_tac l)
apply (case_tac a)
apply (rule_tac y="c" and c="(aa,b)" in trm_assn.strong_exhaust(1))
apply simp_all[3]
apply rule
apply rule
apply (rule subst_substa_graph.intros)*)

defer
apply (case_tac x)
apply (case_tac a)
thm trm_assn.strong_exhaust(1)
apply (rule_tac y="c" and c="(aa,b)" in trm_assn.strong_exhaust(1))
apply (simp add: trm_assn.distinct trm_assn.eq_iff)
apply auto[1]
apply (simp add: trm_assn.distinct trm_assn.eq_iff)
apply auto[1]
apply (simp add: trm_assn.distinct trm_assn.eq_iff fresh_star_def)
apply (simp add: trm_assn.distinct trm_assn.eq_iff)
apply (drule_tac x="assn" in meta_spec)
apply (rotate_tac 3)
apply (drule_tac x="aa" in meta_spec)
apply (rotate_tac -1)
apply (drule_tac x="b" in meta_spec)
apply (rotate_tac -1)
apply (drule_tac x="trm" in meta_spec)
apply (auto simp add: alpha_bn_refl)[1]
apply (case_tac b)
apply (rule_tac ya="c" in trm_assn.strong_exhaust(2))
apply (simp add: trm_assn.distinct trm_assn.eq_iff)
apply auto[1]
apply blast
apply (simp add: trm_assn.distinct trm_assn.eq_iff)
apply auto[1]
apply blast
apply (simp_all only: sum.simps Pair_eq trm_assn.distinct trm_assn.eq_iff)
apply simp_all
apply (simp_all add: meta_eq_to_obj_eq[OF subst_def, symmetric, unfolded fun_eq_iff])
apply (simp_all add: meta_eq_to_obj_eq[OF substa_def, symmetric, unfolded fun_eq_iff])
apply clarify
prefer 2
apply clarify
apply (rule conjI)
prefer 2
apply (rename_tac a pp vv zzz a2 s t zz)
apply (erule alpha_bn_inducts)
apply (rule alpha_bn_refl)
apply clarify
apply (rename_tac t' a1 a2 n1 n2)
thm subst_substa_graph.intros[no_vars]
.
alpha_bn (substa s t (ACons n1 t' a1))
         (substa s t (ACons n2 t' a2))

alpha_bn (Acons s (subst a t t') a1)
         (Acons s (subst a t t') a2)

ACons v (subst v t t') as"

end