datatype rlam =

fun

overloading

  perm_rlam    \<equiv> "perm :: 'x prm \<Rightarrow> rlam \<Rightarrow> rlam"   (unchecked)

begin

fun

  perm_rlam

where

  "perm_rlam pi (rVar a) = rVar (pi \<bullet> a)"

| "perm_rlam pi (rApp t1 t2) = rApp (perm_rlam pi t1) (perm_rlam pi t2)"

| "perm_rlam pi (rLam a t) = rLam (pi \<bullet> a) (perm_rlam pi t)"

end

| a3: "\<lbrakk>t \<approx> ([(a,b)] \<bullet> s); a \<notin> rfv (rLam b t)\<rbrakk> \<Longrightarrow> rLam a t \<approx> rLam b s"

lemma helper:

  fixes t::"rlam"

  and   a::"name"

  shows "[(a, a)] \<bullet> t = t"

by (induct t)

   (auto simp add: calc_atm)

  apply(simp add: helper)

  by (rule alpha_equivp)

thm Var_def

thm App_def

thm Lam_def

(* lemmas that need to be lifted *)

lemma pi_var_eqvt1:

  shows "(pi \<bullet> rVar a) \<approx> rVar (pi \<bullet> a)"

  by (simp add: alpha_refl)

lemma pi_var_eqvt2:

  fixes pi::"'x prm"

  shows "(pi \<bullet> rVar a) = rVar (pi \<bullet> a)"

  by (simp)

lemma pi_app_eqvt1:

  shows "(pi \<bullet> rApp t1 t2) \<approx> rApp (pi \<bullet> t1) (pi \<bullet> t2)"

  by (simp add: alpha_refl)

lemma pi_app_eqvt2:

  fixes pi::"'x prm"

  shows "(pi \<bullet> rApp t1 t2) = rApp (pi \<bullet> t1) (pi \<bullet> t2)"

  by (simp)

lemma pi_lam_eqvt1:

  shows "(pi \<bullet> rLam a t) \<approx> rLam (pi \<bullet> a) (pi \<bullet> t)"

  by (simp add: alpha_refl)

lemma pi_lam_eqvt2:

  fixes pi::"'x prm"

  shows "(pi \<bullet> rLam a t) = rLam (pi \<bullet> a) (pi \<bullet> t)"

  by (simp add: alpha)

by (auto intro: a1)

by (auto intro: a2)

  apply(simp add: helper)

  apply(simp)

lemma rfv_rsp[quot_respect]: 

  "(alpha ===> op =) rfv rfv"

lemma pi_var1:

  fixes pi::"'x prm"

  shows "pi \<bullet> Var a = Var (pi \<bullet> a)"

  by (lifting pi_var_eqvt1)

lemma pi_var2:

  fixes pi::"'x prm"

  shows "pi \<bullet> Var a = Var (pi \<bullet> a)"

  by (lifting pi_var_eqvt2)

lemma pi_app: 

  fixes pi::"'x prm"

  shows "pi \<bullet> App t1 t2 = App (pi \<bullet> t1) (pi \<bullet> t2)"

  by (lifting pi_app_eqvt2)

lemma pi_lam: 

  fixes pi::"'x prm"

  shows "pi \<bullet> Lam a t = Lam (pi \<bullet> a) (pi \<bullet> t)"

  by (lifting pi_lam_eqvt2)

lemma fv_var: 

  shows "fv (Var a) = {a}"

  by  (lifting rfv_var)

lemma fv_app: 

  shows "fv (App t1 t2) = fv t1 \<union> fv t2"

  by (lifting rfv_app)

lemma fv_lam: 

  shows "fv (Lam a t) = fv t - {a}"

  by (lifting rfv_lam)

lemma a1: 

  "a = b \<Longrightarrow> Var a = Var b"

  by  (lifting a1)

lemma a2: 

  "\<lbrakk>x = xa; xb = xc\<rbrakk> \<Longrightarrow> App x xb = App xa xc"

  by  (lifting a2)

lemma a3: 

  "\<lbrakk>x = [(a, b)] \<bullet> xa; a \<notin> fv (Lam b x)\<rbrakk> \<Longrightarrow> Lam a x = Lam b xa"

  by  (lifting a3)

lemma alpha_cases: 

  "\<lbrakk>a1 = a2; \<And>a b. \<lbrakk>a1 = Var a; a2 = Var b; a = b\<rbrakk> \<Longrightarrow> P;

    \<And>x xa xb xc. \<lbrakk>a1 = App x xb; a2 = App xa xc; x = xa; xb = xc\<rbrakk> \<Longrightarrow> P;

    \<And>x a b xa. \<lbrakk>a1 = Lam a x; a2 = Lam b xa; x = [(a, b)] \<bullet> xa; a \<notin> fv (Lam b x)\<rbrakk> \<Longrightarrow> P\<rbrakk>

  by (lifting alpha.cases)

lemma alpha_induct: 

  "\<lbrakk>qx = qxa; \<And>a b. a = b \<Longrightarrow> qxb (Var a) (Var b);

    \<And>x xa xb xc. \<lbrakk>x = xa; qxb x xa; xb = xc; qxb xb xc\<rbrakk> \<Longrightarrow> qxb (App x xb) (App xa xc);

     \<And>x a b xa.

  by (lifting alpha.induct)

lemma var_inject: 

  "(Var a = Var b) = (a = b)"

  by (lifting rvar_inject)

lemma lam_induct:

  "\<lbrakk>\<And>name. P (Var name); 

    \<And>lam1 lam2. \<lbrakk>P lam1; P lam2\<rbrakk> \<Longrightarrow> P (App lam1 lam2);

    \<And>name lam. P lam \<Longrightarrow> P (Lam name lam)\<rbrakk> \<Longrightarrow> P lam"

  by (lifting rlam.induct)

  apply(simp add: pi_var2)

  shows "(a \<sharp> (Var b)) = (a \<sharp> b)"