theory LFex
imports Nominal QuotMain
begin

atom_decl name id

nominal_datatype kind = 
    Type
  | KPi "ty" "name" "kind"
and ty =  
    TConst "id"
  | TApp "ty" "trm"
  | TPi "ty" "name" "ty"
and trm = 
    Const "id"
  | Var "name"
  | App "trm" "trm"
  | Lam "ty" "name" "trm" 

function
    fv_kind :: "kind \<Rightarrow> name set"
and fv_ty   :: "ty \<Rightarrow> name set"
and fv_trm  :: "trm \<Rightarrow> name set"
where
  "fv_kind (Type) = {}"
| "fv_kind (KPi A x K) = (fv_ty A) \<union> ((fv_kind K) - {x})"
| "fv_ty (TConst i) = {}"
| "fv_ty (TApp A M) = (fv_ty A) \<union> (fv_trm M)"
| "fv_ty (TPi A x B) = (fv_ty A) \<union> ((fv_ty B) - {x})"
| "fv_trm (Const i) = {}"
| "fv_trm (Var x) = {x}"
| "fv_trm (App M N) = (fv_trm M) \<union> (fv_trm N)"
| "fv_trm (Lam A x M) = (fv_ty A) \<union> ((fv_trm M) - {x})"
sorry

termination fv_kind sorry

inductive
    akind :: "kind \<Rightarrow> kind \<Rightarrow> bool" ("_ \<approx>ki _" [100, 100] 100)
and aty   :: "ty \<Rightarrow> ty \<Rightarrow> bool"     ("_ \<approx>ty _" [100, 100] 100)
and atrm  :: "trm \<Rightarrow> trm \<Rightarrow> bool"   ("_ \<approx>tr _" [100, 100] 100)
where
  a1:  "(Type) \<approx>ki (Type)"
| a21: "\<lbrakk>A \<approx>ty A'; K \<approx>ki K'\<rbrakk> \<Longrightarrow> (KPi A x K) \<approx>ki (KPi A' x K')"
| a22: "\<lbrakk>A \<approx>ty A'; K \<approx>ki ([(x,x')]\<bullet>K'); x \<notin> (fv_ty A'); x \<notin> ((fv_kind K') - {x'})\<rbrakk> 
        \<Longrightarrow> (KPi A x K) \<approx>ki (KPi A' x' K')"
| a3:  "i = j \<Longrightarrow> (TConst i) \<approx>ty (TConst j)"
| a4:  "\<lbrakk>A \<approx>ty A'; M \<approx>tr M'\<rbrakk> \<Longrightarrow> (TApp A M) \<approx>ty (TApp A' M')"
| a51: "\<lbrakk>A \<approx>ty A'; B \<approx>ty B'\<rbrakk> \<Longrightarrow> (TPi A x B) \<approx>ty (TPi A' x B')"
| a52: "\<lbrakk>A \<approx>ty A'; B \<approx>ty ([(x,x')]\<bullet>B'); x \<notin> (fv_ty B'); x \<notin> ((fv_ty B') - {x'})\<rbrakk> 
        \<Longrightarrow> (TPi A x B) \<approx>ty (TPi A' x' B')"
| a6:  "i = j \<Longrightarrow> (Const i) \<approx>trm (Const j)"
| a7:  "x = y \<Longrightarrow> (Var x) \<approx>trm (Var y)"
| a8:  "\<lbrakk>M \<approx>trm M'; N \<approx>tr N'\<rbrakk> \<Longrightarrow> (App M N) \<approx>tr (App M' N')"
| a91: "\<lbrakk>A \<approx>ty A'; M \<approx>tr M'\<rbrakk> \<Longrightarrow> (Lam A x M) \<approx>tr (Lam A' x M')"
| a92: "\<lbrakk>A \<approx>ty A'; M \<approx>tr ([(x,x')]\<bullet>M'); x \<notin> (fv_ty B'); x \<notin> ((fv_trm M') - {x'})\<rbrakk> 
        \<Longrightarrow> (Lam A x M) \<approx>tr (Lam A' x' M')"

lemma al_refl:
  fixes K::"kind" 
  and   A::"ty"
  and   M::"trm"
  shows "K \<approx>ki K"
  and   "A \<approx>ty A"
  and   "M \<approx>tr M"
  apply(induct K and A and M rule: kind_ty_trm.inducts)
  apply(auto intro: akind_aty_atrm.intros)
  done

lemma alpha_EQUIVs:
  shows "EQUIV akind"
  and   "EQUIV aty"
  and   "EQUIV atrm"
sorry

quotient KIND = kind / akind
  by (rule alpha_EQUIVs)

quotient TY = ty / aty
   and   TRM = trm / atrm
  by (auto intro: alpha_EQUIVs)

print_quotients

quotient_def 
  TYP :: "KIND"
where
  "TYP \<equiv> Type"

quotient_def 
  KPI :: "TY \<Rightarrow> name \<Rightarrow> KIND \<Rightarrow> KIND"
where
  "KPI \<equiv> KPi"

quotient_def 
  TCONST :: "id \<Rightarrow> TY"
where
  "TCONST \<equiv> TConst"

quotient_def 
  TAPP :: "TY \<Rightarrow> TRM \<Rightarrow> TY"
where
  "TAPP \<equiv> TApp"

quotient_def 
  TPI :: "TY \<Rightarrow> name \<Rightarrow> TY \<Rightarrow> TY"
where
  "TPI \<equiv> TPi"

(* FIXME: does not work with CONST *)
quotient_def 
  CONS :: "id \<Rightarrow> TRM"
where
  "CONS \<equiv> Const"

quotient_def 
  VAR :: "name \<Rightarrow> TRM"
where
  "VAR \<equiv> Var"

quotient_def 
  APP :: "TRM \<Rightarrow> TRM \<Rightarrow> TRM"
where
  "APP \<equiv> App"

quotient_def 
  LAM :: "TY \<Rightarrow> name \<Rightarrow> TRM \<Rightarrow> TRM"
where
  "LAM \<equiv> Lam"

thm TYP_def
thm KPI_def
thm TCONST_def
thm TAPP_def
thm TPI_def
thm VAR_def
thm CONS_def
thm APP_def
thm LAM_def

(* FIXME: print out a warning if the type contains a liftet type, like kind \<Rightarrow> name set *)
quotient_def 
  FV_kind :: "KIND \<Rightarrow> name set"
where
  "FV_kind \<equiv> fv_kind"

quotient_def 
  FV_ty :: "TY \<Rightarrow> name set"
where
  "FV_ty \<equiv> fv_ty"

quotient_def 
  FV_trm :: "TRM \<Rightarrow> name set"
where
  "FV_trm \<equiv> fv_trm"

thm FV_kind_def
thm FV_ty_def
thm FV_trm_def

(* FIXME: does not work yet *)
overloading
    perm_kind \<equiv> "perm :: 'x prm \<Rightarrow> KIND \<Rightarrow> KIND"   (unchecked)
    perm_ty   \<equiv> "perm :: 'x prm \<Rightarrow> TY \<Rightarrow> TY"       (unchecked)
    perm_trm  \<equiv> "perm :: 'x prm \<Rightarrow> TRM \<Rightarrow> TRM"     (unchecked) 
begin

quotient_def 
  perm_kind :: "'x prm \<Rightarrow> KIND \<Rightarrow> KIND"
where
  "perm_kind \<equiv> (perm::'x prm \<Rightarrow> kind \<Rightarrow> kind)"

quotient_def 
  perm_ty :: "'x prm \<Rightarrow> TY \<Rightarrow> TY"
where
  "perm_ty \<equiv> (perm::'x prm \<Rightarrow> ty \<Rightarrow> ty)"

quotient_def 
  perm_trm :: "'x prm \<Rightarrow> TRM \<Rightarrow> TRM"
where
  "perm_trm \<equiv> (perm::'x prm \<Rightarrow> trm \<Rightarrow> trm)"










ML {* val defs =
  @{thms TYP_def KPI_def TCONST_def TAPP_def TPI_def VAR_def CONS_def APP_def LAM_def
    FV_kind_def FV_ty_def FV_trm_def perm_kind_def perm_ty_def perm_trm_def}
*}
ML {* val consts = lookup_quot_consts defs *}

thm akind_aty_atrm.induct

ML {*
fun regularize_monos_tac lthy eqvs =
  let 
    val subs1 = map (fn x => @{thm equiv_res_forall} OF [x]) eqvs
    val subs2 = map (fn x => @{thm equiv_res_exists} OF [x]) eqvs
  in
    REPEAT_ALL_NEW (FIRST' [
      (rtac @{thm impI} THEN' atac),
      (rtac @{thm my_equiv_res_forallR}),
      (rtac @{thm my_equiv_res_forallL}),
      (rtac @{thm Set.imp_mono}),
      (resolve_tac (Inductive.get_monos lthy)),
      (EqSubst.eqsubst_tac lthy [0] (subs1 @ subs2))
    ])
  end
*}

ML {*
  val subs1 = map (fn x => @{thm eq_reflection} OF [@{thm equiv_res_forall} OF [x]]) @{thms alpha_EQUIVs}
*}

ML {*
fun regularize_tac ctxt rel_eqvs rel_refls =
  let 
    val subs1 = map (fn x => @{thm equiv_res_forall} OF [x]) rel_eqvs
    val subs2 = map (fn x => @{thm equiv_res_exists} OF [x]) rel_eqvs
  in
  (ObjectLogic.full_atomize_tac) THEN'
  REPEAT_ALL_NEW (FIRST' [
    FIRST' (map rtac rel_refls),
    atac,
    rtac @{thm universal_twice},
    rtac @{thm impI} THEN' atac,
    rtac @{thm implication_twice},
    EqSubst.eqsubst_tac ctxt [0] (subs1 @ subs2),
    (* For a = b \<longrightarrow> a \<approx> b *)
    (rtac @{thm RIGHT_RES_FORALL_REGULAR})
  ])
  end
*}
thm RIGHT_RES_FORALL_REGULAR
thm my_equiv_res_forallR

(*
lemma "\<And>i j xb\<Colon>trm \<Rightarrow> trm \<Rightarrow> bool. Respects (atrm ===> atrm ===> op =) xb \<Longrightarrow> (\<forall>m\<Colon>trm \<Rightarrow> trm\<in>Respects (atrm ===> atrm). xb (Const i) (m (Const j))) \<longrightarrow> (\<forall>m\<Colon>trm \<Rightarrow> trm. xb (Const i) (m (Const j)))"
apply (simp add: Ball_def IN_RESPECTS Respects_def)
apply (metis COMBK_def al_refl(3))
*)

lemma move_quant: "((\<forall>y. \<forall>x\<in>P. A x y) \<longrightarrow> (\<forall>y. \<forall>x. B x y)) \<Longrightarrow> ((\<forall>x\<in>P. \<forall>y. A x y) \<longrightarrow> (\<forall>x. \<forall>y. B x y))"
by auto

lemma [mono]: "P \<longrightarrow> Q \<Longrightarrow> \<not>Q \<longrightarrow> \<not>P"
apply(auto)
done

lemma test: 
  fixes P Q::"'a \<Rightarrow> bool"  
  and x::"'a"
  assumes a: "REFL R2"
  and     b: "\<And>f. Q (f x) \<longrightarrow> P (f x)" 
  shows   "(Ball (Respects (R1 ===> R2)) (\<lambda>f. Q (f x)) \<longrightarrow> All (\<lambda>f. P (f x)))"
apply(rule impI)
apply(rule allI)
apply(drule_tac x="\<lambda>y. f x" in bspec)
apply(simp add: Respects_def IN_RESPECTS)
apply(rule impI)
using a
apply(simp add: REFL_def)
using b
apply(simp)
done

lemma "\<lbrakk>P1 TYP TYP; \<And>A A' K K' x. \<lbrakk>(A::TY) = A'; P2 A A'; (K::KIND) = K'; P1 K K'\<rbrakk> \<Longrightarrow> P1 (KPI A x K) (KPI A' x K');
 \<And>A A' K x x' K'.
    \<lbrakk>(A ::TY) = A'; P2 A A'; (K :: KIND) = ([(x, x')] \<bullet> K'); P1 K ([(x, x')] \<bullet> K'); x \<notin> FV_ty A'; x \<notin> FV_kind K' - {x'}\<rbrakk>
    \<Longrightarrow> P1 (KPI A x K) (KPI A' x' K');
 \<And>i j. i = j \<Longrightarrow> P2 (TCONST i) (TCONST j);
 \<And>A A' M M'. \<lbrakk>(A ::TY) = A'; P2 A A'; (M :: TRM) = M'; P3 M M'\<rbrakk> \<Longrightarrow> P2 (TAPP A M) (TAPP A' M');
 \<And>A A' B B' x. \<lbrakk>(A ::TY) = A'; P2 A A'; (B ::TY) = B'; P2 B B'\<rbrakk> \<Longrightarrow> P2 (TPI A x B) (TPI A' x B');
 \<And>A A' B x x' B'.
    \<lbrakk>(A ::TY) = A'; P2 A A'; (B ::TY) = ([(x, x')] \<bullet> B'); P2 B ([(x, x')] \<bullet> B'); x \<notin> FV_ty B'; x \<notin> FV_ty B' - {x'}\<rbrakk>
    \<Longrightarrow> P2 (TPI A x B) (TPI A' x' B');
 \<And>i j m. i = j \<Longrightarrow> P3 (CONS i) (m (CONS j)); \<And>x y m. x = y \<Longrightarrow> P3 (VAR x) (m (VAR y));
 \<And>M m M' N N'. \<lbrakk>(M :: TRM) = m M'; P3 M (m M'); (N :: TRM) = N'; P3 N N'\<rbrakk> \<Longrightarrow> P3 (APP M N) (APP M' N');
 \<And>A A' M M' x. \<lbrakk>(A ::TY) = A'; P2 A A'; (M :: TRM) = M'; P3 M M'\<rbrakk> \<Longrightarrow> P3 (LAM A x M) (LAM A' x M');
 \<And>A A' M x x' M' B'.
    \<lbrakk>(A ::TY) = A'; P2 A A'; (M :: TRM) = ([(x, x')] \<bullet> M'); P3 M ([(x, x')] \<bullet> M'); x \<notin> FV_ty B'; x \<notin> FV_trm M' - {x'}\<rbrakk>
    \<Longrightarrow> P3 (LAM A x M) (LAM A' x' M')\<rbrakk>
\<Longrightarrow> ((x1 :: KIND) = x2 \<longrightarrow> P1 x1 x2) \<and>
   ((x3 ::TY) = x4 \<longrightarrow> P2 x3 x4) \<and> ((x5 :: TRM) = x6 \<longrightarrow> P3 x5 x6)"
apply(tactic {* procedure_tac @{context} @{thm akind_aty_atrm.induct} 1 *})
apply(tactic {* (simp_tac ((Simplifier.context @{context} empty_ss) addsimps (subs1))) 1 *})
apply(atomize (full))
apply(rule RIGHT_RES_FORALL_REGULAR)
apply(rule RIGHT_RES_FORALL_REGULAR)
apply(rule RIGHT_RES_FORALL_REGULAR)
apply(tactic {* resolve_tac (Inductive.get_monos @{context}) 1 *})+
apply(rule test)
defer
apply(tactic {* resolve_tac (Inductive.get_monos @{context}) 1 *})+
apply(rule test)
defer
apply(tactic {* resolve_tac (Inductive.get_monos @{context}) 1 *})+
apply(rule move_quant)
apply(tactic {* resolve_tac (Inductive.get_monos @{context}) 1 *})+
apply(rule move_quant)
apply(tactic {* resolve_tac (Inductive.get_monos @{context}) 1 *})+
apply(rule move_quant)
apply(tactic {* resolve_tac (Inductive.get_monos @{context}) 1 *})+
apply(rule test)
defer
apply(tactic {* resolve_tac (Inductive.get_monos @{context}) 1 *})+


thm test[OF mp]


prefer 2
apply(tactic {* resolve_tac (Inductive.get_monos @{context}) 1 *})+
apply(thin_tac "Respects (akind ===> akind ===> op =) x") 
apply(thin_tac "Respects (aty ===> aty ===> op =) xa")
apply(thin_tac "Respects (atrm ===> atrm ===> op =) xb")
apply (simp add: Ball_def IN_RESPECTS Respects_def)
apply (metis COMBK_def al_refl(3))

apply(rule LEFT_RES_FORALL_REGULAR)
apply(rule conjI)
prefer 2
apply(tactic {* resolve_tac (Inductive.get_monos @{context}) 1 *})+
using al_refl
apply(simp add: Respects_def)


apply(tactic {* resolve_tac (Inductive.get_monos @{context}) 1 *})
apply(tactic {* resolve_tac (Inductive.get_monos @{context}) 1 *})
apply(tactic {* resolve_tac (Inductive.get_monos @{context}) 1 *})
apply(tactic {* resolve_tac (Inductive.get_monos @{context}) 1 *})
apply(tactic {* resolve_tac (Inductive.get_monos @{context}) 1 *})
apply(tactic {* resolve_tac (Inductive.get_monos @{context}) 1 *})
apply(tactic {* resolve_tac (Inductive.get_monos @{context}) 1 *})
apply(tactic {* resolve_tac (Inductive.get_monos @{context}) 1 *})
apply (simp add: Ball_def IN_RESPECTS Respects_def)
apply (metis COMBK_def al_refl(3))

apply(rule impI) apply(assumption)
apply(rule Set.imp_mono)
apply(rule impI) apply(assumption)
apply(rule Set.imp_mono)
apply(rule impI) apply(assumption)
apply(rule Set.imp_mono)
apply(rule impI) apply(assumption)
apply(rule Set.imp_mono)
apply(rule impI) apply(assumption)
apply(rule Set.imp_mono)
apply(rule impI) apply(assumption)
apply(rule Set.imp_mono)
apply(rule impI) apply(assumption)
apply(rule Set.imp_mono)
apply(tactic {* resolve_tac (Inductive.get_monos @{context}) 1 *})
apply(tactic {* resolve_tac (Inductive.get_monos @{context}) 1 *})
apply (simp add: Ball_def IN_RESPECTS Respects_def)
apply (metis COMBK_def al_refl(3))
apply(rule Set.imp_mono)
apply(tactic {* resolve_tac (Inductive.get_monos @{context}) 1 *})
apply(tactic {* resolve_tac (Inductive.get_monos @{context}) 1 *})
apply (simp add: Ball_def IN_RESPECTS Respects_def)
apply (metis COMBK_def al_refl(3))
apply(rule Set.imp_mono)
apply(tactic {* resolve_tac (Inductive.get_monos @{context}) 1 *})
apply(rule move_quant)
apply(tactic {* resolve_tac (Inductive.get_monos @{context}) 1 *})
apply(rule move_quant)
apply(tactic {* resolve_tac (Inductive.get_monos @{context}) 1 *})
apply(rule move_quant)
apply(tactic {* resolve_tac (Inductive.get_monos @{context}) 1 *})
apply (simp add: Ball_def IN_RESPECTS Respects_def)
apply (metis COMBK_def al_refl(3))
apply(rule impI) apply(assumption)

ML {*
val rty_qty_rel =
  [(@{typ kind}, (@{typ KIND}, @{term akind})),
   (@{typ ty}, (@{typ TY}, @{term aty})),
   (@{typ trm}, (@{typ TRM}, @{term atrm}))]
*}

print_quotients

ML {* val rty = [@{typ }] *}
ML {* val defs_sym = flat (map (add_lower_defs @{context}) defs) *}
ML {* val t_a = atomize_thm @{thm akind_aty_atrm.induct} *}
prove {* build_regularize_goal t_a rty rel @{context}

end