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>"

  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)

ML {* show_all_types := true *}

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)

ML_prf {*

  List.last (map (symmetric o #def) (Quotient_Info.qconsts_dest @{context}));

  List.last (map (Thm.varifyT o symmetric o #def) (Quotient_Info.qconsts_dest @{context}))

*}

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 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"

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

done

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)"