(*notation ( output) "prop" ("#_" [1000] 1000) *)
notation ( output) "Trueprop" ("#_" [1000] 1000)

function(sequential)
    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) = True"
| a2: "(KPi A x K) \<approx>ki (KPi A' x' K') = (A \<approx>ty A' \<and> (\<exists>pi. (rfv_kind K - {atom x} = rfv_kind K' - {atom x'} \<and> (rfv_kind K - {atom x})\<sharp>* pi \<and> (pi \<bullet> K) \<approx>ki K' \<and> (pi \<bullet> x) = x')))"
| "_ \<approx>ki _ = False"
| a3: "(TConst i) \<approx>ty (TConst j) = (i = j)"
| a4: "(TApp A M) \<approx>ty (TApp A' M') = (A \<approx>ty A' \<and> M \<approx>tr M')"
| a5: "(TPi A x B) \<approx>ty (TPi A' x' B') = ((A \<approx>ty A') \<and> (\<exists>pi. rfv_ty B - {atom x} = rfv_ty B' - {atom x'} \<and> (rfv_ty B - {atom x})\<sharp>* pi \<and> (pi \<bullet> B) \<approx>ty B' \<and> (pi \<bullet> x) = x'))"
| "_ \<approx>ty _ = False"
| a6: "(Const i) \<approx>tr (Const j) = (i = j)"
| a7: "(Var x) \<approx>tr (Var y) = (x = y)"
| a8: "(App M N) \<approx>tr (App M' N') = (M \<approx>tr M' \<and> N \<approx>tr N')"
| a9: "(Lam A x M) \<approx>tr (Lam A' x' M') = (A \<approx>ty A' \<and> (\<exists>pi. rfv_trm M - {atom x} = rfv_trm M' - {atom x'} \<and> (rfv_trm M - {atom x})\<sharp>* pi \<and> (pi \<bullet> M) \<approx>tr M' \<and> (pi \<bullet> x) = x'))"
| "_ \<approx>tr _ = False"
apply (pat_completeness)
apply simp_all
done
termination
by (size_change)



lemma regularize_to_injection:
  shows "(QUOT_TRUE l \<Longrightarrow> y) \<Longrightarrow> (l = r) \<longrightarrow> y"
  by(auto simp add: QUOT_TRUE_def)

syntax
  "Bex1_rel" :: "id \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" ("(3\<exists>!!_\<in>_./ _)" [0, 0, 10] 10)
translations
  "\<exists>!!x\<in>A. P"  == "Bex1_rel A (%x. P)"


(* Atomize infrastructure *)
(* FIXME/TODO: is this really needed? *)
(*
lemma atomize_eqv:
  shows "(Trueprop A \<equiv> Trueprop B) \<equiv> (A \<equiv> B)"
proof
  assume "A \<equiv> B"
  then show "Trueprop A \<equiv> Trueprop B" by unfold
next
  assume *: "Trueprop A \<equiv> Trueprop B"
  have "A = B"
  proof (cases A)
    case True
    have "A" by fact
    then show "A = B" using * by simp
  next
    case False
    have "\<not>A" by fact
    then show "A = B" using * by auto
  qed
  then show "A \<equiv> B" by (rule eq_reflection)
qed
*)


ML {*
  fun dest_cbinop t =
    let
      val (t2, rhs) = Thm.dest_comb t;
      val (bop, lhs) = Thm.dest_comb t2;
    in
      (bop, (lhs, rhs))
    end
*}

ML {*
  fun dest_ceq t =
    let
      val (bop, pair) = dest_cbinop t;
      val (bop_s, _) = Term.dest_Const (Thm.term_of bop);
    in
      if bop_s = "op =" then pair else (raise CTERM ("Not an equality", [t]))
    end
*}

ML {*
  fun split_binop_conv t =
    let
      val (lhs, rhs) = dest_ceq t;
      val (bop, _) = dest_cbinop lhs;
      val [clT, cr2] = bop |> Thm.ctyp_of_term |> Thm.dest_ctyp;
      val [cmT, crT] = Thm.dest_ctyp cr2;
    in
      Drule.instantiate' [SOME clT, SOME cmT, SOME crT] [NONE, NONE, NONE, NONE, SOME bop] @{thm arg_cong2}
    end
*}


ML {*
  fun split_arg_conv t =
    let
      val (lhs, rhs) = dest_ceq t;
      val (lop, larg) = Thm.dest_comb lhs;
      val [caT, crT] = lop |> Thm.ctyp_of_term |> Thm.dest_ctyp;
    in
      Drule.instantiate' [SOME caT, SOME crT] [NONE, NONE, SOME lop] @{thm arg_cong}
    end
*}

ML {*
  fun split_binop_tac n thm =
    let
      val concl = Thm.cprem_of thm n;
      val (_, cconcl) = Thm.dest_comb concl;
      val rewr = split_binop_conv cconcl;
    in
      rtac rewr n thm
    end
      handle CTERM _ => Seq.empty
*}


ML {*
  fun split_arg_tac n thm =
    let
      val concl = Thm.cprem_of thm n;
      val (_, cconcl) = Thm.dest_comb concl;
      val rewr = split_arg_conv cconcl;
    in
      rtac rewr n thm
    end
      handle CTERM _ => Seq.empty
*}


lemma trueprop_cong:
  shows "(a \<equiv> b) \<Longrightarrow> (Trueprop a \<equiv> Trueprop b)"
  by auto

lemma list_induct_hol4:
  fixes P :: "'a list \<Rightarrow> bool"
  assumes a: "((P []) \<and> (\<forall>t. (P t) \<longrightarrow> (\<forall>h. (P (h # t)))))"
  shows "\<forall>l. (P l)"
  using a
  apply (rule_tac allI)
  apply (induct_tac "l")
  apply (simp)
  apply (metis)
  done

ML {*
val no_vars = Thm.rule_attribute (fn context => fn th =>
  let
    val ctxt = Variable.set_body false (Context.proof_of context);
    val ((_, [th']), _) = Variable.import true [th] ctxt;
  in th' end);
*}

(*lemma equality_twice:
  "a = c \<Longrightarrow> b = d \<Longrightarrow> (a = b \<longrightarrow> c = d)"
by auto*)


(*interpretation code *)
(*val bindd = ((Binding.make ("", Position.none)), ([]: Attrib.src list))
  val ((_, [eqn1pre]), lthy5) = Variable.import true [ABS_def] lthy4;
  val eqn1i = Thm.prop_of (symmetric eqn1pre)
  val ((_, [eqn2pre]), lthy6) = Variable.import true [REP_def] lthy5;
  val eqn2i = Thm.prop_of (symmetric eqn2pre)

  val exp_morphism = ProofContext.export_morphism lthy6 (ProofContext.init (ProofContext.theory_of lthy6));
  val exp_term = Morphism.term exp_morphism;
  val exp = Morphism.thm exp_morphism;

  val mthd = Method.SIMPLE_METHOD ((rtac quot_thm 1) THEN
    ALLGOALS (simp_tac (HOL_basic_ss addsimps [(symmetric (exp ABS_def)), (symmetric (exp REP_def))])))
  val mthdt = Method.Basic (fn _ => mthd)
  val bymt = Proof.global_terminal_proof (mthdt, NONE)
  val exp_i = [(@{const_name QUOT_TYPE}, ((("QUOT_TYPE_I_" ^ (Binding.name_of qty_name)), true),
    Expression.Named [("R", rel), ("Abs", abs), ("Rep", rep) ]))]*)

(*||> Local_Theory.theory (fn thy =>
      let
        val global_eqns = map exp_term [eqn2i, eqn1i];
        (* Not sure if the following context should not be used *)
        val (global_eqns2, lthy7) = Variable.import_terms true global_eqns lthy6;
        val global_eqns3 = map (fn t => (bindd, t)) global_eqns2;
      in ProofContext.theory_of (bymt (Expression.interpretation (exp_i, []) global_eqns3 thy)) end)*)