theory LFex
imports "Nominal2_Atoms" "Nominal2_Eqvt" "Nominal2_Supp" "Abs" "Perm" "Fv" "Rsp"
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"] *}
print_theorems

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

local_setup  {* define_quotient_type
  [(([], @{binding kind}, NoSyn), (@{typ rkind}, @{term alpha_rkind})),
   (([], @{binding ty},   NoSyn), (@{typ rty},   @{term alpha_rty}  )),
   (([], @{binding trm},  NoSyn), (@{typ rtrm},  @{term alpha_rtrm} ))]
  (ALLGOALS (resolve_tac @{thms alpha_equivps}))
*}

local_setup {*
(fn ctxt => ctxt
 |> snd o (Quotient_Def.quotient_lift_const ("TYP", @{term Type}))
 |> snd o (Quotient_Def.quotient_lift_const ("KPI", @{term KPi}))
 |> snd o (Quotient_Def.quotient_lift_const ("TCONST", @{term TConst}))
 |> snd o (Quotient_Def.quotient_lift_const ("TAPP", @{term TApp}))
 |> snd o (Quotient_Def.quotient_lift_const ("TPI", @{term TPi}))
 |> snd o (Quotient_Def.quotient_lift_const ("CONS", @{term Const}))
 |> snd o (Quotient_Def.quotient_lift_const ("VAR", @{term Var}))
 |> snd o (Quotient_Def.quotient_lift_const ("APP", @{term App}))
 |> snd o (Quotient_Def.quotient_lift_const ("LAM", @{term Lam}))
 |> snd o (Quotient_Def.quotient_lift_const ("fv_kind", @{term fv_rkind}))
 |> snd o (Quotient_Def.quotient_lift_const ("fv_ty", @{term fv_rty}))
 |> snd o (Quotient_Def.quotient_lift_const ("fv_trm", @{term fv_rtrm}))) *}
print_theorems

local_setup {* prove_const_rsp @{binding rfv_rsp} [@{term fv_rkind}, @{term fv_rty}, @{term fv_rtrm}]
  (fn _ => fvbv_rsp_tac @{thm alpha_rkind_alpha_rty_alpha_rtrm.induct} @{thms fv_rkind_fv_rty_fv_rtrm.simps} 1) *}
local_setup {* prove_const_rsp Binding.empty [@{term "permute :: perm \<Rightarrow> rkind \<Rightarrow> rkind"}]
  (fn _ => asm_simp_tac (HOL_ss addsimps @{thms alpha_eqvt}) 1) *}
local_setup {* prove_const_rsp Binding.empty [@{term "permute :: perm \<Rightarrow> rty \<Rightarrow> rty"}]
  (fn _ => asm_simp_tac (HOL_ss addsimps @{thms alpha_eqvt}) 1) *}
local_setup {* prove_const_rsp Binding.empty [@{term "permute :: perm \<Rightarrow> rtrm \<Rightarrow> rtrm"}]
  (fn _ => asm_simp_tac (HOL_ss addsimps @{thms alpha_eqvt}) 1) *}
ML {* fun const_rsp_tac _ = constr_rsp_tac @{thms alpha_rkind_alpha_rty_alpha_rtrm_inj}
  @{thms rfv_rsp} @{thms alpha_equivps} 1 *}
local_setup {* prove_const_rsp Binding.empty [@{term TConst}] const_rsp_tac *}
local_setup {* prove_const_rsp Binding.empty [@{term TApp}] const_rsp_tac *}
local_setup {* prove_const_rsp Binding.empty [@{term Var}] const_rsp_tac *}
local_setup {* prove_const_rsp Binding.empty [@{term App}] const_rsp_tac *}
local_setup {* prove_const_rsp Binding.empty [@{term Const}] const_rsp_tac *}
local_setup {* prove_const_rsp Binding.empty [@{term KPi}] const_rsp_tac *}
local_setup {* prove_const_rsp Binding.empty [@{term TPi}] const_rsp_tac *}
local_setup {* prove_const_rsp Binding.empty [@{term Lam}] const_rsp_tac *}

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]

setup {* define_lifted_perms ["LFex.kind", "LFex.ty", "LFex.trm"] 
  [("permute_kind", @{term "permute :: perm \<Rightarrow> rkind \<Rightarrow> rkind"}),
   ("permute_ty", @{term "permute :: perm \<Rightarrow> rty \<Rightarrow> rty"}),
   ("permute_trm", @{term "permute :: perm \<Rightarrow> rtrm \<Rightarrow> rtrm"})]
  @{thms permute_rkind_permute_rty_permute_rtrm_zero permute_rkind_permute_rty_permute_rtrm_append} *}

(*
Lifts, but slow and not needed?.
lemmas alpha_kind_alpha_ty_alpha_trm_induct = alpha_rkind_alpha_rty_alpha_rtrm.induct[unfolded alpha_gen, quot_lifted, folded alpha_gen]
*)

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

lemmas kind_ty_trm_inj = 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 supports:
  "{} supports TYP"
  "(supp (atom i)) supports (TCONST i)"
  "(supp A \<union> supp M) supports (TAPP A M)"
  "(supp (atom i)) supports (CONS i)"
  "(supp (atom x)) supports (VAR x)"
  "(supp M \<union> supp N) supports (APP M N)"
  "(supp ty \<union> supp (atom na) \<union> supp ki) supports (KPI ty na ki)"
  "(supp ty \<union> supp (atom na) \<union> supp ty2) supports (TPI ty na ty2)"
  "(supp ty \<union> supp (atom na) \<union> supp trm) supports (LAM ty na trm)"
apply(simp_all add: supports_def fresh_def[symmetric] swap_fresh_fresh)
apply(rule_tac [!] allI)+
apply(rule_tac [!] impI)
apply(tactic {* ALLGOALS (REPEAT o etac conjE) *})
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(tactic {* ALLGOALS (rtac @{thm supports_finite} THEN' resolve_tac @{thms supports}) *})
apply(simp_all add: supp_atom)
done

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

lemma supp_eqs:
  "supp TYP = {}"
  "supp rkind = fv_kind rkind \<Longrightarrow> supp (KPI rty name rkind) = supp rty \<union> supp (Abs {atom name} rkind)"
  "supp (TCONST i) = {atom i}"
  "supp (TAPP A M) = supp A \<union> supp M"
  "supp rty2 = fv_ty rty2 \<Longrightarrow> supp (TPI rty1 name rty2) = supp rty1 \<union> supp (Abs {atom name} rty2)"
  "supp (CONS i) = {atom i}"
  "supp (VAR x) = {atom x}"
  "supp (APP M N) = supp M \<union> supp N"
  "supp rtrm = fv_trm rtrm \<Longrightarrow> supp (LAM rty name rtrm) = supp rty \<union> supp (Abs {atom name} rtrm)"
  apply(simp_all (no_asm) add: supp_def)
  apply(simp_all only: kind_ty_trm_inj Abs_eq_iff alpha_gen)
  apply(simp_all only: insert_eqvt empty_eqvt atom_eqvt supp_eqvt[symmetric] fv_eqvt[symmetric])
  apply(simp_all add: Collect_imp_eq Collect_neg_eq[symmetric] Set.Un_commute)
  apply(simp_all add: supp_at_base[simplified supp_def])
  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 (no_asm) only: supp_eqs fv_kind_ty_trm)
  apply(simp_all)
  apply(subst supp_eqs)
  apply(simp_all add: supp_Abs)
  apply(subst supp_eqs)
  apply(simp_all add: supp_Abs)
  apply(subst supp_eqs)
  apply(simp_all add: supp_Abs)
  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})"
  by (simp_all only: supp_fv fv_kind_ty_trm)

end