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

declare perm_rlam.simps[eqvt]

instance rlam::pt_name
apply(default)
apply(induct_tac [!] x rule: rlam.induct)
apply(simp_all add: pt_name2 pt_name3)
done

instance rlam::fs_name
apply(default)
apply(induct_tac [!] x rule: rlam.induct)
apply(simp add: supp_def)
apply(fold supp_def)
apply(simp add: supp_atm)
apply(simp add: supp_def Collect_imp_eq Collect_neg_eq)
apply(simp add: supp_def)
apply(simp add: supp_def Collect_imp_eq Collect_neg_eq[symmetric])
apply(fold supp_def)
apply(simp add: supp_atm)
done

declare set_diff_eqvt[eqvt]

lemma rfv_eqvt[eqvt]:
  fixes pi::"name prm"
  shows "(pi\<bullet>rfv t) = rfv (pi\<bullet>t)"
apply(induct t)
apply(simp_all)
apply(simp add: perm_set_eq)
apply(simp add: union_eqvt)
apply(simp add: set_diff_eqvt)
apply(simp add: perm_set_eq)
done

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: "\<exists>pi::name prm. (rfv t - {a} = rfv s - {b} \<and> (rfv t - {a})\<sharp>* pi \<and> (pi \<bullet> t) \<approx> s \<and> (pi \<bullet> a) = b) 
       \<Longrightarrow> rLam a t \<approx> rLam b s"

(* should be automatic with new version of eqvt-machinery *)
lemma alpha_eqvt:
  fixes pi::"name prm"
  shows "t \<approx> s \<Longrightarrow> (pi \<bullet> t) \<approx> (pi \<bullet> s)"
apply(induct rule: alpha.induct)
apply(simp add: a1)
apply(simp add: a2)
apply(simp)
apply(rule a3)
apply(erule conjE)
apply(erule exE)
apply(erule conjE)
apply(rule_tac x="pi \<bullet> pia" in exI)
apply(rule conjI)
apply(rule_tac pi1="rev pi" in perm_bij[THEN iffD1])
apply(perm_simp add: eqvts)
apply(rule conjI)
apply(rule_tac pi1="rev pi" in pt_fresh_star_bij(1)[OF pt_name_inst at_name_inst, THEN iffD1])
apply(perm_simp add: eqvts)
apply(rule conjI)
apply(subst perm_compose[symmetric])
apply(simp)
apply(subst perm_compose[symmetric])
apply(simp)
done

lemma alpha_refl:
  shows "t \<approx> t"
apply(induct t rule: rlam.induct)
apply(simp add: a1)
apply(simp add: a2)
apply(rule a3)
apply(rule_tac x="[]" in exI)
apply(simp_all add: fresh_star_def fresh_list_nil)
done

lemma alpha_sym:
  shows "t \<approx> s \<Longrightarrow> s \<approx> t"
apply(induct rule: alpha.induct)
apply(simp add: a1)
apply(simp add: a2)
apply(rule a3)
apply(erule exE)
apply(rule_tac x="rev pi" in exI)
apply(simp)
apply(simp add: fresh_star_def fresh_list_rev)
apply(rule conjI)
apply(erule conjE)+
apply(rotate_tac 3)
apply(drule_tac pi="rev pi" in alpha_eqvt)
apply(perm_simp)
apply(rule pt_bij2[OF pt_name_inst at_name_inst])
apply(simp)
done

lemma alpha_trans:
  shows "t1 \<approx> t2 \<Longrightarrow> t2 \<approx> t3 \<Longrightarrow> t1 \<approx> t3"
apply(induct arbitrary: t3 rule: alpha.induct)
apply(erule alpha.cases)
apply(simp_all)
apply(simp add: a1)
apply(rotate_tac 4)
apply(erule alpha.cases)
apply(simp_all)
apply(simp add: a2)
apply(rotate_tac 1)
apply(erule alpha.cases)
apply(simp_all)
apply(erule conjE)+
apply(erule exE)+
apply(erule conjE)+
apply(rule a3)
apply(rule_tac x="pia @ pi" in exI)
apply(simp add: fresh_star_def fresh_list_append)
apply(simp add: pt_name2)
apply(drule_tac x="rev pia \<bullet> sa" in spec)
apply(drule mp)
apply(rotate_tac 8)
apply(drule_tac pi="rev pia" in alpha_eqvt)
apply(perm_simp)
apply(rotate_tac 11)
apply(drule_tac pi="pia" in alpha_eqvt)
apply(perm_simp)
done

lemma alpha_equivp:
  shows "equivp alpha"
apply(rule equivpI)
unfolding reflp_def symp_def transp_def
apply(auto intro: alpha_refl alpha_sym alpha_trans)
done

lemma alpha_rfv:
  shows "t \<approx> s \<Longrightarrow> rfv t = rfv s"
apply(induct rule: alpha.induct)
apply(simp)
apply(simp)
apply(simp)
done

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

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(rule_tac x="[]" in exI)
  unfolding fresh_star_def
  apply(simp add: fresh_list_nil)
  apply(simp add: alpha_rfv)
  done

lemma rfv_rsp[quot_respect]: 
  "(alpha ===> op =) rfv rfv"
apply(simp add: alpha_rfv)
done

section {* lifted theorems *}

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)

lemma perm_lam [simp]:
  fixes pi::"'a prm"
  shows "pi \<bullet> Var a = Var (pi \<bullet> a)"
  and   "pi \<bullet> App t1 t2 = App (pi \<bullet> t1) (pi \<bullet> t2)"
  and   "pi \<bullet> Lam a t = Lam (pi \<bullet> a) (pi \<bullet> t)"
apply(lifting perm_rlam.simps)
done

instance lam::pt_name
apply(default)
apply(induct_tac [!] x rule: lam_induct)
apply(simp_all add: pt_name2 pt_name3)
done

lemma fv_lam [simp]: 
  shows "fv (Var a) = {a}"
  and   "fv (App t1 t2) = fv t1 \<union> fv t2"
  and   "fv (Lam a t) = fv t - {a}"
apply(lifting rfv_var rfv_app rfv_lam)
done


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>\<exists>pi::name prm. (fv t - {a} = fv s - {b} \<and> (fv t - {a})\<sharp>* pi \<and> (pi \<bullet> t) = s \<and> (pi \<bullet> a) = b)\<rbrakk> 
   \<Longrightarrow> Lam a t = Lam b s"
  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>t a s b. \<lbrakk>a1 = Lam a t; a2 = Lam b s; 
         \<exists>pi::name prm. fv t - {a} = fv s - {b} \<and> (fv t - {a}) \<sharp>* pi \<and> (pi \<bullet> t) = s \<and> pi \<bullet> a = b\<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>t a s b.
        \<lbrakk>\<exists>pi::name prm. fv t - {a} = fv s - {b} \<and>
         (fv t - {a}) \<sharp>* pi \<and> ((pi \<bullet> t) = s \<and> qxb (pi \<bullet> t) s) \<and> pi \<bullet> a = b\<rbrakk> \<Longrightarrow> qxb (Lam a t) (Lam b s)\<rbrakk>
    \<Longrightarrow> qxb qx qxa"
  by (lifting alpha.induct)

lemma lam_inject [simp]: 
  shows "(Var a = Var b) = (a = b)"
  and   "(App t1 t2 = App s1 s2) = (t1 = s1 \<and> t2 = s2)"
apply(lifting rlam.inject(1) rlam.inject(2))
apply(auto)
apply(drule alpha.cases)
apply(simp_all)
apply(simp add: alpha.a1)
apply(drule alpha.cases)
apply(simp_all)
apply(drule alpha.cases)
apply(simp_all)
apply(rule alpha.a2)
apply(simp_all)
done

lemma var_supp1:
  shows "(supp (Var a)) = ((supp a)::name set)"
apply(simp add: supp_def)
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: perm_lam supp_def lam_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)
apply(simp add: abs_perm)
sorry


instance lam::fs_name
apply(default)
apply(induct_tac x rule: lam_induct)
apply(simp add: var_supp)
apply(simp add: app_supp)
apply(simp add: lam_supp abs_supp)
done

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)
      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)
      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)
      apply(subst eq[symmetric])
      using p
      apply(simp only: perm_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 rlam_rec_eqvt:
  fixes pi::"name prm"
  and   f1::"name \<Rightarrow> ('a::pt_name)"
  shows "(pi\<bullet>rlam_rec f1 f2 f3 t) = rlam_rec (pi\<bullet>f1) (pi\<bullet>f2) (pi\<bullet>f3) (pi\<bullet>t)"
apply(induct t)
apply(simp_all)
apply(simp add: perm_fun_def)
apply(perm_simp)
apply(subst pt_fun_app_eq[OF pt_name_inst at_name_inst])
back
apply(subst pt_fun_app_eq[OF pt_name_inst at_name_inst])
apply(subst pt_fun_app_eq[OF pt_name_inst at_name_inst])
apply(subst pt_fun_app_eq[OF pt_name_inst at_name_inst])
apply(simp)
apply(subst pt_fun_app_eq[OF pt_name_inst at_name_inst])
back
apply(subst pt_fun_app_eq[OF pt_name_inst at_name_inst])
apply(subst pt_fun_app_eq[OF pt_name_inst at_name_inst])
apply(simp)
done
 

lemma rlam_rec_respects:
  assumes f1: "f_var \<in> Respects (op= ===> op=)"
  and     f2: "f_app \<in> Respects (alpha ===> alpha ===> op= ===> op= ===> op=)"
  and     f3: "f_lam \<in> Respects (op= ===> alpha ===> op= ===> op=)"
  shows "rlam_rec f_var f_app f_lam \<in> Respects (alpha ===> op =)"
apply(simp add: mem_def)
apply(simp add: Respects_def)
apply(rule allI)
apply(rule allI)
apply(rule impI)
apply(erule alpha.induct)
apply(simp)
apply(simp)
using f2
apply(simp add: mem_def)
apply(simp add: Respects_def)
using f3[simplified mem_def Respects_def]
apply(simp)
apply(case_tac "a=b")
apply(clarify)
apply(simp)
apply(subst pt_swap_bij'')
apply(rule pt_name_inst)
apply(rule at_name_inst)
apply(subst pt_swap_bij'')
apply(rule pt_name_inst)
apply(rule at_name_inst)
apply(simp)
apply(generate_fresh "name")
(* probably true *)
sorry

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

(* test test
lemma raw_hom_correct: 
  assumes f1: "f_var \<in> Respects (op= ===> op=)"
  and     f2: "f_app \<in> Respects (alpha ===> alpha ===> op= ===> op= ===> op=)"
  and     f3: "f_lam \<in> Respects ((op= ===> alpha) ===> (op= ===> op=) ===> op=)"
  shows "\<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))))"
unfolding Bex1_def
apply(rule ex1I)
sorry
*)


end