theory LamEx

imports Nominal "../Quotient" "../Quotient_List"

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"

is

  "rVar"

quotient_definition

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

is

  "rApp"

quotient_definition

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

is

  "rLam"

quotient_definition

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

is

  "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"

is

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

end

lemma perm_rsp[quot_respect]:

  "(op = ===> alpha ===> alpha) op \<bullet> (op \<bullet> :: (name \<times> name) list \<Rightarrow> rlam \<Rightarrow> rlam)"

  apply auto

  apply(erule alpha_eqvt)

  done

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::"name 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)"

  by (lifting perm_rlam.simps[where 'a="name"])

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}"

  by(lifting rfv_var rfv_app 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>\<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 rlam_distinct:

  shows "\<not>(rVar nam \<approx> rApp rlam1' rlam2')"

  and   "\<not>(rApp rlam1' rlam2' \<approx> rVar nam)"

  and   "\<not>(rVar nam \<approx> rLam nam' rlam')"

  and   "\<not>(rLam nam' rlam' \<approx> rVar nam)"

  and   "\<not>(rApp rlam1 rlam2 \<approx> rLam nam' rlam')"

  and   "\<not>(rLam nam' rlam' \<approx> rApp rlam1 rlam2)"

  apply auto

  apply(erule alpha.cases)

  apply simp_all

  apply(erule alpha.cases)

  apply simp_all

  apply(erule alpha.cases)

  apply simp_all

  apply(erule alpha.cases)

  apply simp_all

  apply(erule alpha.cases)

  apply simp_all

  apply(erule alpha.cases)

  apply simp_all

  done

lemma lam_distinct[simp]:

  shows "Var nam \<noteq> App lam1' lam2'"

  and   "App lam1' lam2' \<noteq> Var nam"

  and   "Var nam \<noteq> Lam nam' lam'"

  and   "Lam nam' lam' \<noteq> Var nam"

  and   "App lam1 lam2 \<noteq> Lam nam' lam'"

  and   "Lam nam' lam' \<noteq> App lam1 lam2"

  by(lifting rlam_distinct(1) rlam_distinct(2) rlam_distinct(3) rlam_distinct(4) rlam_distinct(5) rlam_distinct(6))

lemma var_supp1:

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

  by (simp add: supp_def)

lemma var_supp:

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

  using var_supp1 by (simp add: supp_atm)

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)