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"

end