theory LamEx

imports Nominal "../QuotMain" "../QuotList"

begin

atom_decl name

datatype rlam =

  rVar "name"

| rApp "rlam" "rlam"

| rLam "name" "rlam"

fun

  rfv :: "rlam \<Rightarrow> name set"

where

  rfv_var: "rfv (rVar a) = {a}"

| rfv_app: "rfv (rApp t1 t2) = (rfv t1) \<union> (rfv t2)"

| rfv_lam: "rfv (rLam a t) = (rfv t) - {a}"

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

inductive

  alpha :: "rlam \<Rightarrow> rlam \<Rightarrow> bool" ("_ \<approx> _" [100, 100] 100)

where

  a1: "a = b \<Longrightarrow> (rVar a) \<approx> (rVar b)"

| a2: "\<lbrakk>t1 \<approx> t2; s1 \<approx> s2\<rbrakk> \<Longrightarrow> rApp t1 s1 \<approx> rApp t2 s2"

| 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)

lemma alpha_refl:

  fixes t::"rlam"

  shows "t \<approx> t"

  apply(induct t rule: rlam.induct)

  apply(simp add: a1)

  apply(simp add: a2)

  apply(rule a3)

  apply(simp add: helper)

  apply(simp)

  done

lemma alpha_equivp:

  shows "equivp alpha"

sorry

quotient_type lam = rlam / alpha

  by (rule alpha_equivp)

quotient_definition

  "Var :: name \<Rightarrow> lam"

as

  "rVar"

quotient_definition

   "App :: lam \<Rightarrow> lam \<Rightarrow> lam"

as

  "rApp"

quotient_definition

  "Lam :: name \<Rightarrow> lam \<Rightarrow> lam"

as

  "rLam"

quotient_definition

  "fv :: lam \<Rightarrow> name set"

as

  "rfv"

(* definition of overloaded permutation function *)

(* for the lifted type lam                       *)

overloading

  perm_lam \<equiv> "perm :: 'x prm \<Rightarrow> lam \<Rightarrow> lam"   (unchecked)

begin

quotient_definition

   "perm_lam :: 'x prm \<Rightarrow> lam \<Rightarrow> lam"

as

  "perm::'x prm \<Rightarrow> rlam \<Rightarrow> rlam"

end

(* lemmas that need to be lifted *)

lemma pi_var_eqvt1:

  fixes pi::"'x prm"

  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:

  fixes pi::"'x prm"

  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:

  fixes pi::"'x prm"

  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)

lemma real_alpha:

  assumes a: "t = [(a,b)]\<bullet>s" "a\<sharp>[b].s"

  shows "Lam a t = Lam b s"

using a

unfolding fresh_def supp_def

sorry

lemma perm_rsp[quot_respect]:

  "(op = ===> alpha ===> alpha) op \<bullet> op \<bullet>"

  apply(auto)

  (* this is propably true if some type conditions are imposed ;o) *)

  sorry

lemma fresh_rsp:

  "(op = ===> alpha ===> op =) fresh fresh"

  apply(auto)

  (* this is probably only true if some type conditions are imposed *)

  sorry

lemma rVar_rsp[quot_respect]:

  "(op = ===> alpha) rVar rVar"

  by (auto intro: a1)

lemma rApp_rsp[quot_respect]: "(alpha ===> alpha ===> alpha) rApp rApp"

  by (auto intro: a2)

lemma rLam_rsp[quot_respect]: "(op = ===> alpha ===> alpha) rLam rLam"

  apply(auto)

  apply(rule a3)

  apply(simp add: helper)

  apply(simp)

  done

lemma rfv_rsp[quot_respect]: 

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

  sorry

lemma rvar_inject: "rVar a \<approx> rVar b = (a = b)"

  apply (auto)

  apply (erule alpha.cases)

  apply (simp_all add: rlam.inject alpha_refl)

  done

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>

    \<Longrightarrow> P"

  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.

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

    \<Longrightarrow> qxb qx qxa"

  by (lifting alpha.induct)

lemma var_inject: 

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

  by (lifting rvar_inject)

lemma app_inject: 

  "(App t1 t2 = App s1 s2) = (t1 = s1 \<and> t2 = s2)"

sorry

lemma var_supp1:

  shows "(supp (Var a)) = ((supp a)::name set)"

apply(simp add: supp_def pi_var1 var_inject)

done

lemma var_supp:

  shows "(supp (Var a)) = {a::name}"

using var_supp1

apply(simp add: supp_atm)

done

lemma app_supp:

  shows "supp (App t1 t2) = (supp t1) \<union> ((supp t2)::name set)"

apply(simp only: supp_def pi_app app_inject)

apply(simp add: Collect_imp_eq Collect_neg_eq)

done

lemma lam_supp:

  shows "supp (Lam x t) = ((supp ([x].t))::name set)"

apply(simp add: supp_def pi_lam)

sorry

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)

instance lam::pt_name

apply(default)

apply(induct_tac x rule: lam_induct)

apply(simp add: pi_var1)

apply(simp add: pi_app)

apply(simp add: pi_lam)

apply(induct_tac x rule: lam_induct)

apply(simp add: pi_var1 pt_name2)

apply(simp add: pi_app)

apply(simp add: pi_lam pt_name2)

apply(induct_tac x rule: lam_induct)

apply(simp add: pi_var1 pt_name3)

apply(simp add: pi_app)

apply(simp add: pi_lam pt_name3)

done

instance lam::fs_name

apply(default)

apply(induct_tac x rule: lam_induct)

apply(simp add: var_supp)

apply(simp add: app_supp)

sorry

lemma fresh_lam:

  "(a \<sharp> Lam b t) \<longleftrightarrow> (a = b) \<or> (a \<noteq> b \<and> a \<sharp> t)"

apply(simp add: fresh_def)

apply(simp add: lam_supp abs_supp)

apply(auto)

done

lemma lam_induct_strong:

  fixes a::"'a::fs_name"

  assumes a1: "\<And>name b. P b (Var name)"

  and     a2: "\<And>lam1 lam2 b. \<lbrakk>\<And>c. P c lam1; \<And>c. P c lam2\<rbrakk> \<Longrightarrow> P b (App lam1 lam2)"

  and     a3: "\<And>name lam b. \<lbrakk>\<And>c. P c lam; name \<sharp> b\<rbrakk> \<Longrightarrow> P b (Lam name lam)"

  shows "P a lam"

proof -

  have "\<And>(pi::name prm) a. P a (pi \<bullet> lam)" 

  proof (induct lam rule: lam_induct)

    case (1 name pi)

    show "P a (pi \<bullet> Var name)"

      apply (simp only: pi_var1)

      apply (rule a1)

      done

  next

    case (2 lam1 lam2 pi)

    have b1: "\<And>(pi::name prm) a. P a (pi \<bullet> lam1)" by fact

    have b2: "\<And>(pi::name prm) a. P a (pi \<bullet> lam2)" by fact

    show "P a (pi \<bullet> App lam1 lam2)"

      apply (simp only: pi_app)

      apply (rule a2)

      apply (rule b1)

      apply (rule b2)

      done

  next

    case (3 name lam pi a)

    have b: "\<And>(pi::name prm) a. P a (pi \<bullet> lam)" by fact

    obtain c::name where fr: "c\<sharp>(a, pi\<bullet>name, pi\<bullet>lam)"

      apply(rule exists_fresh[of "(a, pi\<bullet>name, pi\<bullet>lam)"])

      apply(simp_all add: fs_name1)

      done

    from b fr have p: "P a (Lam c (([(c, pi\<bullet>name)]@pi)\<bullet>lam))" 

      apply -

      apply(rule a3)

      apply(blast)

      apply(simp)

      done

    have eq: "[(c, pi\<bullet>name)] \<bullet> Lam (pi \<bullet> name) (pi \<bullet> lam) = Lam (pi \<bullet> name) (pi \<bullet> lam)"

      apply(rule perm_fresh_fresh)

      using fr

      apply(simp add: fresh_lam)

      apply(simp add: fresh_lam)

      done

    show "P a (pi \<bullet> Lam name lam)" 

      apply (simp add: pi_lam)

      apply(subst eq[symmetric])

      using p

      apply(simp only: pi_lam pt_name2 swap_simps)

      done

  qed

  then have "P a (([]::name prm) \<bullet> lam)" by blast

  then show "P a lam" by simp 

qed

lemma var_fresh:

  fixes a::"name"

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

  apply(simp add: fresh_def)

  apply(simp add: var_supp1)

  done

(* lemma hom_reg: *)

lemma hom: 

  "\<exists>hom\<in>Respects (alpha ===> op =). 

    ((\<forall>x. hom (rVar x) = f_var x) \<and>

     (\<forall>l r. hom (rApp l r) = f_app l r (hom l) (hom r)) \<and>

     (\<forall>x a. hom (rLam a x) = f_lam (\<lambda>b. ([(a,b)]\<bullet> x)) (\<lambda>b. hom ([(a,b)] \<bullet> x))))"

apply(rule_tac x="rlam_rec f_var f_app XX" in bexI)

apply(rule conjI)

apply(simp)

apply(rule conjI)

apply(simp)

apply(simp)

sorry

lemma hom_reg:"

(\<exists>hom\<in>Respects (alpha ===> op =).

 (\<forall>x. hom (rVar x) = f_var x) \<and>

 (\<forall>l r. hom (rApp l r) = f_app l r (hom l) (hom r)) \<and>

 (\<forall>xa a. hom (rLam a xa) = f_lam (\<lambda>b. [(a, b)] \<bullet> xa) (\<lambda>b. hom ([(a, b)] \<bullet> xa)))) \<longrightarrow>

(\<exists>hom.

 (\<forall>x. hom (rVar x) = f_var x) \<and>

 (\<forall>l r. hom (rApp l r) = f_app l r (hom l) (hom r)) \<and>

 (\<forall>xa a. hom (rLam a xa) = f_lam (\<lambda>b. [(a, b)] \<bullet> xa) (\<lambda>b. hom ([(a, b)] \<bullet> xa))))"

by(regularize)

lemma hom_pre:"

(\<exists>hom.

 (\<forall>x. hom (rVar x) = f_var x) \<and>

 (\<forall>l r. hom (rApp l r) = f_app l r (hom l) (hom r)) \<and>

 (\<forall>xa a. hom (rLam a xa) = f_lam (\<lambda>b. [(a, b)] \<bullet> xa) (\<lambda>b. hom ([(a, b)] \<bullet> xa))))"

apply (rule impE[OF hom_reg])

apply (rule hom)

apply (assumption)

done

lemma hom':

"\<exists>hom. 

  ((\<forall>x. hom (Var x) = f_var x) \<and>

   (\<forall>l r. hom (App l r) = f_app l r (hom l) (hom r)) \<and>

   (\<forall>x a. hom (Lam a x) = f_lam (\<lambda>b. ([(a,b)] \<bullet> x)) (\<lambda>b. hom ([(a,b)] \<bullet> x))))"

apply (lifting hom)

done

thm bex_reg_eqv_range[OF identity_equivp, of "alpha"]

end