theory LFex
imports "Nominal2_Atoms" "Nominal2_Eqvt" "Nominal2_Supp" "Abs" "Perm" "Fv"
begin

atom_decl name
atom_decl ident

datatype rkind =
    Type
  | KPi "rty" "name" "rkind"
and rty =
    TConst "ident"
  | TApp "rty" "rtrm"
  | TPi "rty" "name" "rty"
and rtrm =
    Const "ident"
  | Var "name"
  | App "rtrm" "rtrm"
  | Lam "rty" "name" "rtrm"


setup {* snd o define_raw_perms ["rkind", "rty", "rtrm"] ["LFex.rkind", "LFex.rty", "LFex.rtrm"] *}

local_setup {*
  snd o define_fv_alpha "LFex.rkind"
  [[ [], [[], [(NONE, 1)], [(NONE, 1)]] ],
   [ [[]], [[], []], [[], [(NONE, 1)], [(NONE, 1)]] ],
   [ [[]], [[]], [[], []], [[], [(NONE, 1)], [(NONE, 1)]]]] *}
notation
    alpha_rkind  ("_ \<approx>ki _" [100, 100] 100)
and alpha_rty    ("_ \<approx>ty _" [100, 100] 100)
and alpha_rtrm   ("_ \<approx>tr _" [100, 100] 100)
thm fv_rkind_fv_rty_fv_rtrm.simps alpha_rkind_alpha_rty_alpha_rtrm.intros
local_setup {* (fn ctxt => snd (Local_Theory.note ((@{binding alpha_rkind_alpha_rty_alpha_rtrm_inj}, []), (build_alpha_inj @{thms alpha_rkind_alpha_rty_alpha_rtrm.intros} @{thms rkind.distinct rty.distinct rtrm.distinct rkind.inject rty.inject rtrm.inject} @{thms alpha_rkind.cases alpha_rty.cases alpha_rtrm.cases} ctxt)) ctxt)) *}
thm alpha_rkind_alpha_rty_alpha_rtrm_inj

lemma rfv_eqvt[eqvt]:
  "((pi\<bullet>fv_rkind t1) = fv_rkind (pi\<bullet>t1))"
  "((pi\<bullet>fv_rty t2) = fv_rty (pi\<bullet>t2))"
  "((pi\<bullet>fv_rtrm t3) = fv_rtrm (pi\<bullet>t3))"
apply(induct t1 and t2 and t3 rule: rkind_rty_rtrm.inducts)
apply(simp_all add: union_eqvt Diff_eqvt)
apply(simp_all add: permute_set_eq atom_eqvt)
done

lemma alpha_eqvt:
  "t1 \<approx>ki s1 \<Longrightarrow> (pi \<bullet> t1) \<approx>ki (pi \<bullet> s1)"
  "t2 \<approx>ty s2 \<Longrightarrow> (pi \<bullet> t2) \<approx>ty (pi \<bullet> s2)"
  "t3 \<approx>tr s3 \<Longrightarrow> (pi \<bullet> t3) \<approx>tr (pi \<bullet> s3)"
apply(induct rule: alpha_rkind_alpha_rty_alpha_rtrm.inducts)
apply (simp_all add: alpha_rkind_alpha_rty_alpha_rtrm.intros)
apply (simp_all add: alpha_rkind_alpha_rty_alpha_rtrm_inj)
apply (rule alpha_gen_atom_eqvt)
apply (simp add: rfv_eqvt)
apply assumption
apply (rule alpha_gen_atom_eqvt)
apply (simp add: rfv_eqvt)
apply assumption
apply (rule alpha_gen_atom_eqvt)
apply (simp add: rfv_eqvt)
apply assumption
done

local_setup {* (fn ctxt => snd (Local_Theory.note ((@{binding alpha_equivps}, []),
  (build_equivps [@{term alpha_rkind}, @{term alpha_rty}, @{term alpha_rtrm}]
     @{thm rkind_rty_rtrm.induct} @{thm alpha_rkind_alpha_rty_alpha_rtrm.induct} 
     @{thms rkind.inject rty.inject rtrm.inject} @{thms alpha_rkind_alpha_rty_alpha_rtrm_inj}
     @{thms rkind.distinct rty.distinct rtrm.distinct}
     @{thms alpha_rkind.cases alpha_rty.cases alpha_rtrm.cases}
     @{thms alpha_eqvt} ctxt)) ctxt)) *}
thm alpha_equivps

quotient_type kind = rkind / alpha_rkind
  by (rule alpha_equivps)

quotient_type
    ty = rty / alpha_rty and
    trm = rtrm / alpha_rtrm
  by (auto intro: alpha_equivps)

quotient_definition
   "TYP :: kind"
is
  "Type"

quotient_definition
   "KPI :: ty \<Rightarrow> name \<Rightarrow> kind \<Rightarrow> kind"
is
  "KPi"

quotient_definition
   "TCONST :: ident \<Rightarrow> ty"
is
  "TConst"

quotient_definition
   "TAPP :: ty \<Rightarrow> trm \<Rightarrow> ty"
is
  "TApp"

quotient_definition
   "TPI :: ty \<Rightarrow> name \<Rightarrow> ty \<Rightarrow> ty"
is
  "TPi"

(* FIXME: does not work with CONST *)
quotient_definition
   "CONS :: ident \<Rightarrow> trm"
is
  "Const"

quotient_definition
   "VAR :: name \<Rightarrow> trm"
is
  "Var"

quotient_definition
   "APP :: trm \<Rightarrow> trm \<Rightarrow> trm"
is
  "App"

quotient_definition
   "LAM :: ty \<Rightarrow> name \<Rightarrow> trm \<Rightarrow> trm"
is
  "Lam"

(* FIXME: print out a warning if the type contains a liftet type, like rkind \<Rightarrow> name set *)
quotient_definition
   "fv_kind :: kind \<Rightarrow> atom set"
is
  "fv_rkind"

quotient_definition
   "fv_ty :: ty \<Rightarrow> atom set"
is
  "fv_rty"

quotient_definition
   "fv_trm :: trm \<Rightarrow> atom set"
is
  "fv_rtrm"

lemma alpha_rfv:
  shows "(t \<approx>ki s \<longrightarrow> fv_rkind t = fv_rkind s) \<and>
     (t1 \<approx>ty s1 \<longrightarrow> fv_rty t1 = fv_rty s1) \<and>
     (t2 \<approx>tr s2 \<longrightarrow> fv_rtrm t2 = fv_rtrm s2)"
  apply(rule alpha_rkind_alpha_rty_alpha_rtrm.induct)
  apply(simp_all add: alpha_gen)
  done

lemma perm_rsp[quot_respect]:
  "(op = ===> alpha_rkind ===> alpha_rkind) permute permute"
  "(op = ===> alpha_rty ===> alpha_rty) permute permute"
  "(op = ===> alpha_rtrm ===> alpha_rtrm) permute permute"
  by (simp_all add:alpha_eqvt)

lemma tconst_rsp[quot_respect]: 
  "(op = ===> alpha_rty) TConst TConst"
  apply (auto intro: alpha_rkind_alpha_rty_alpha_rtrm.intros) done
lemma tapp_rsp[quot_respect]: 
  "(alpha_rty ===> alpha_rtrm ===> alpha_rty) TApp TApp" 
  apply (auto intro: alpha_rkind_alpha_rty_alpha_rtrm.intros) done
lemma var_rsp[quot_respect]: 
  "(op = ===> alpha_rtrm) Var Var"
  apply (auto intro: alpha_rkind_alpha_rty_alpha_rtrm.intros) done
lemma app_rsp[quot_respect]: 
  "(alpha_rtrm ===> alpha_rtrm ===> alpha_rtrm) App App"
  apply (auto intro: alpha_rkind_alpha_rty_alpha_rtrm.intros) done
lemma const_rsp[quot_respect]: 
  "(op = ===> alpha_rtrm) Const Const"
  apply (auto intro: alpha_rkind_alpha_rty_alpha_rtrm.intros) done

lemma kpi_rsp[quot_respect]: 
  "(alpha_rty ===> op = ===> alpha_rkind ===> alpha_rkind) KPi KPi"
  apply (auto intro: alpha_rkind_alpha_rty_alpha_rtrm.intros)
  apply (rule alpha_rkind_alpha_rty_alpha_rtrm.intros(2)) apply simp_all
  apply (rule_tac x="0" in exI)
  apply (simp add: fresh_star_def fresh_zero_perm alpha_rfv alpha_gen)
  done

lemma tpi_rsp[quot_respect]: 
  "(alpha_rty ===> op = ===> alpha_rty ===> alpha_rty) TPi TPi"
  apply (auto intro: alpha_rkind_alpha_rty_alpha_rtrm.intros)
  apply (rule alpha_rkind_alpha_rty_alpha_rtrm.intros(5)) apply simp_all
  apply (rule_tac x="0" in exI)
  apply (simp add: fresh_star_def fresh_zero_perm alpha_rfv alpha_gen)
  done
lemma lam_rsp[quot_respect]: 
  "(alpha_rty ===> op = ===> alpha_rtrm ===> alpha_rtrm) Lam Lam"
  apply (auto intro: alpha_rkind_alpha_rty_alpha_rtrm.intros)
  apply (rule alpha_rkind_alpha_rty_alpha_rtrm.intros(9)) apply simp_all
  apply (rule_tac x="0" in exI)
  apply (simp add: fresh_star_def fresh_zero_perm alpha_rfv alpha_gen)
  done

lemma fv_rty_rsp[quot_respect]: 
  "(alpha_rty ===> op =) fv_rty fv_rty"
  by (simp add: alpha_rfv)
lemma fv_rkind_rsp[quot_respect]:
  "(alpha_rkind ===> op =) fv_rkind fv_rkind"
  by (simp add: alpha_rfv)
lemma fv_rtrm_rsp[quot_respect]:
  "(alpha_rtrm ===> op =) fv_rtrm fv_rtrm"
  by (simp add: alpha_rfv)

thm rkind_rty_rtrm.induct
lemmas kind_ty_trm_induct = rkind_rty_rtrm.induct[quot_lifted]

thm rkind_rty_rtrm.inducts
lemmas kind_ty_trm_inducts = rkind_rty_rtrm.inducts[quot_lifted]

instantiation kind and ty and trm :: pt
begin

quotient_definition
  "permute_kind :: perm \<Rightarrow> kind \<Rightarrow> kind"
is
  "permute :: perm \<Rightarrow> rkind \<Rightarrow> rkind"

quotient_definition
  "permute_ty :: perm \<Rightarrow> ty \<Rightarrow> ty"
is
  "permute :: perm \<Rightarrow> rty \<Rightarrow> rty"

quotient_definition
  "permute_trm :: perm \<Rightarrow> trm \<Rightarrow> trm"
is
  "permute :: perm \<Rightarrow> rtrm \<Rightarrow> rtrm"

lemmas permute_ktt[simp] = permute_rkind_permute_rty_permute_rtrm.simps[quot_lifted]

lemma perm_zero_ok: "0 \<bullet> (x :: kind) = x \<and> 0 \<bullet> (y :: ty) = y \<and> 0 \<bullet> (z :: trm) = z"
apply (induct rule: kind_ty_trm_induct)
apply (simp_all)
done

lemma perm_add_ok:
  "((p + q) \<bullet> (x1 :: kind) = (p \<bullet> q \<bullet> x1))"
  "((p + q) \<bullet> (x2 :: ty) = p \<bullet> q \<bullet> x2)"
  "((p + q) \<bullet> (x3 :: trm) = p \<bullet> q \<bullet> x3)"
apply (induct x1 and x2 and x3 rule: kind_ty_trm_inducts)
apply (simp_all)
done

instance
apply default
apply (simp_all add: perm_zero_ok perm_add_ok)
done

end

lemmas ALPHA_kind_ALPHA_ty_ALPHA_trm_inducts = alpha_rkind_alpha_rty_alpha_rtrm.inducts[unfolded alpha_gen, quot_lifted, folded alpha_gen]

lemmas kind_ty_trm_INJECT = alpha_rkind_alpha_rty_alpha_rtrm_inj[unfolded alpha_gen, quot_lifted, folded alpha_gen]

lemmas fv_kind_ty_trm = fv_rkind_fv_rty_fv_rtrm.simps[quot_lifted]

lemmas fv_eqvt = rfv_eqvt[quot_lifted]

lemma supp_rkind_rty_rtrm_easy:
 "supp TYP = {}"
 "supp (TCONST i) = {atom i}"
 "supp (TAPP A M) = supp A \<union> supp M"
 "supp (CONS i) = {atom i}"
 "supp (VAR x) = {atom x}"
 "supp (APP M N) = supp M \<union> supp N"
apply (simp_all add: supp_def permute_ktt)
apply (simp_all only: kind_ty_trm_INJECT)
apply (simp_all only: supp_at_base[simplified supp_def])
apply (simp_all add: Collect_imp_eq Collect_neg_eq)
done

lemma supp_bind:
  "(supp (atom na, (ty, ki))) supports (KPI ty na ki)"
  "(supp (atom na, (ty, ty2))) supports (TPI ty na ty2)"
  "(supp (atom na, (ty, rtrm))) supports (LAM ty na rtrm)"
apply(simp_all add: supports_def)
apply(fold fresh_def)
apply(simp_all add: fresh_Pair swap_fresh_fresh)
apply(clarify)
apply(subst swap_at_base_simps(3))
apply(simp_all add: fresh_atom)
apply(clarify)
apply(subst swap_at_base_simps(3))
apply(simp_all add: fresh_atom)
apply(clarify)
apply(subst swap_at_base_simps(3))
apply(simp_all add: fresh_atom)
done

lemma kind_ty_trm_fs:
  "finite (supp (x\<Colon>kind))"
  "finite (supp (y\<Colon>ty))"
  "finite (supp (z\<Colon>trm))"
apply(induct x and y and z rule: kind_ty_trm_inducts)
apply(simp_all add: supp_rkind_rty_rtrm_easy)
apply(rule supports_finite)
apply(rule supp_bind(1))
apply(simp add: supp_Pair supp_atom)
apply(rule supports_finite)
apply(rule supp_bind(2))
apply(simp add: supp_Pair supp_atom)
apply(rule supports_finite)
apply(rule supp_bind(3))
apply(simp add: supp_Pair supp_atom)
done

instance kind and ty and trm :: fs
apply(default)
apply(simp_all only: kind_ty_trm_fs)
done

lemma supp_fv:
 "supp t1 = fv_kind t1"
 "supp t2 = fv_ty t2"
 "supp t3 = fv_trm t3"
apply(induct t1 and t2 and t3 rule: kind_ty_trm_inducts)
apply (simp_all add: supp_rkind_rty_rtrm_easy)
apply (simp_all add: fv_kind_ty_trm)
apply(subgoal_tac "supp (KPI rty name rkind) = supp rty \<union> supp (Abs {atom name} rkind)")
apply(simp add: supp_Abs Set.Un_commute)
apply(simp (no_asm) add: supp_def)
apply(simp add: kind_ty_trm_INJECT)
apply(simp add: Abs_eq_iff)
apply(simp add: alpha_gen)
apply(simp add: Collect_imp_eq Collect_neg_eq Set.Un_commute insert_eqvt empty_eqvt atom_eqvt)
apply(simp add: supp_eqvt[symmetric] fv_eqvt[symmetric])
apply(subgoal_tac "supp (TPI rty1 name rty2) = supp rty1 \<union> supp (Abs {atom name} rty2)")
apply(simp add: supp_Abs Set.Un_commute)
apply(simp (no_asm) add: supp_def)
apply(simp add: kind_ty_trm_INJECT)
apply(simp add: Abs_eq_iff)
apply(simp add: alpha_gen)
apply(simp add: supp_eqvt[symmetric] fv_eqvt[symmetric] insert_eqvt empty_eqvt atom_eqvt)
apply(simp add: Collect_imp_eq Collect_neg_eq Set.Un_commute)
apply(subgoal_tac "supp (LAM rty name rtrm) = supp rty \<union> supp (Abs {atom name} rtrm)")
apply(simp add: supp_Abs Set.Un_commute)
apply(simp (no_asm) add: supp_def)
apply(simp add: kind_ty_trm_INJECT)
apply(simp add: Abs_eq_iff)
apply(simp add: alpha_gen)
apply(simp add: supp_eqvt[symmetric] fv_eqvt[symmetric] insert_eqvt empty_eqvt atom_eqvt)
apply(simp add: Collect_imp_eq Collect_neg_eq Set.Un_commute)
done

(* Not needed anymore *)
lemma supp_kpi_pre: "supp (KPI A x K) = (supp (Abs {atom x} K)) \<union> supp A"
apply (simp add: permute_set_eq supp_def Abs_eq_iff kind_ty_trm_INJECT)
apply (simp add: alpha_gen supp_fv)
apply (simp add: Collect_imp_eq Collect_neg_eq add: atom_eqvt Set.Un_commute)
done

lemma supp_rkind_rty_rtrm:
 "supp TYP = {}"
 "supp (KPI A x K) = supp A \<union> (supp K - {atom x})"
 "supp (TCONST i) = {atom i}"
 "supp (TAPP A M) = supp A \<union> supp M"
 "supp (TPI A x B) = supp A \<union> (supp B - {atom x})"
 "supp (CONS i) = {atom i}"
 "supp (VAR x) = {atom x}"
 "supp (APP M N) = supp M \<union> supp N"
 "supp (LAM A x M) = supp A \<union> (supp M - {atom x})"
apply (simp_all only: supp_rkind_rty_rtrm_easy)
apply (simp_all only: supp_fv fv_kind_ty_trm)
done

end