(* @chunk SIMPLE_INDUCTIVE_PACKAGE *)

signature SIMPLE_INDUCTIVE_PACKAGE =

sig

  val add_inductive_i:

    ((Binding.binding * typ) * mixfix) list ->  (*{predicates}*)

    (Binding.binding * typ) list ->  (*{parameters}*)

    (Attrib.binding * term) list ->  (*{rules}*)

    local_theory -> local_theory

  val add_inductive:

    (Binding.binding * string option * mixfix) list ->  (*{predicates}*)

    (Binding.binding * string option * mixfix) list ->  (*{parameters}*)

    (Attrib.binding * string) list ->  (*{rules}*)

    local_theory -> local_theory

end;

(* @end *)

structure SimpleInductivePackage: SIMPLE_INDUCTIVE_PACKAGE =

struct

(* @chunk make_definitions *) 

fun make_defs ((binding, syn), trm) lthy =

let 

  val arg = ((binding, syn), ((Binding.suffix_name "_def" binding, []), trm))

  val ((_, (_ , thm)), lthy) = Local_Theory.define arg lthy

in 

  (thm, lthy) 

end

(* @end *)

(* @chunk definitions_aux *) 

fun defs_aux lthy orules preds params (pred, arg_types) =

let 

  fun mk_all x P = HOLogic.all_const (fastype_of x) $ lambda x P 

  val fresh_args = 

        arg_types 

        |> map (pair "z")

        |> Variable.variant_frees lthy orules 

        |> map Free

in

  list_comb (pred, fresh_args)

  |> fold_rev (curry HOLogic.mk_imp) orules

  |> fold_rev mk_all preds

  |> fold_rev lambda (params @ fresh_args) 

end

(* @end *)

(* @chunk definitions *) 

fun definitions rules params preds prednames syns arg_typess lthy =

let

  val thy = Proof_Context.theory_of lthy

  val orules = map (Object_Logic.atomize_term thy) rules

  val defs = 

    map (defs_aux lthy orules preds params) (preds ~~ arg_typess) 

in

  fold_map make_defs (prednames ~~ syns ~~ defs) lthy

end

(* @end *)

fun inst_spec ct = 

  Drule.instantiate' [SOME (ctyp_of_term ct)] [NONE, SOME ct] @{thm spec};

(* @chunk induction_tac *)

          cut_facts_tac prems,

          EVERY' (map (dtac o inst_spec) insts),

          assume_tac]

(* @end *)

(* @chunk induction_rules *)

fun inductions rules defs parnames preds Tss lthy1  =

let

  val (_, lthy2) = Variable.add_fixes parnames lthy1

  val Ps = replicate (length preds) "P"

  val (newprednames, lthy3) = Variable.variant_fixes Ps lthy2

  val thy = Proof_Context.theory_of lthy3			      

  val Tss' = map (fn Ts => Ts ---> HOLogic.boolT) Tss

  val newpreds = map Free (newprednames ~~ Tss')

  val cnewpreds = map (cterm_of thy) newpreds

  val rules' = map (subst_free (preds ~~ newpreds)) rules

  fun prove_induction ((pred, newpred), Ts)  =

  let

    val (newargnames, lthy4) = 

          Variable.variant_fixes (replicate (length Ts) "z") lthy3;

    val newargs = map Free (newargnames ~~ Ts)

    val prem = HOLogic.mk_Trueprop (list_comb (pred, newargs))

    val goal = Logic.list_implies 

         (rules', HOLogic.mk_Trueprop (list_comb (newpred, newargs)))

  in

    Goal.prove lthy4 [] [prem] goal

      |> singleton (Proof_Context.export lthy4 lthy1)

  end 

in

  map prove_induction (preds ~~ newpreds ~~ Tss)

end

(* @end *)

val all_elims = fold (fn ct => fn th => th RS inst_spec ct);

val imp_elims = fold (fn th => fn th' => [th', th] MRS @{thm mp});

(* @chunk subproof1 *) 

fun subproof2 prem params2 prems2 =  

 SUBPROOF (fn {prems, ...} =>

  let

    val prem' = prems MRS prem;

    val prem'' = 

       case prop_of prem' of

           _ $ (Const (@{const_name All}, _) $ _) =>

             prem' |> all_elims params2 

                   |> imp_elims prems2

         | _ => prem';

  in 

    rtac prem'' 1 

  end)

(* @end *)

(* @chunk subproof2 *) 

fun subproof1 rules preds i = 

 SUBPROOF (fn {params, prems, context = ctxt', ...} =>

  let

    val (prems1, prems2) = chop (length prems - length rules) prems;

    val (params1, params2) = chop (length params - length preds) (map snd params);

  in

    THEN

    EVERY1 (map (fn prem => subproof2 prem params2 prems2 ctxt') prems1)

  end)

(* @end *)

fun introductions_tac defs rules preds i ctxt =

          REPEAT o (resolve_tac [@{thm allI},@{thm impI}]),

          subproof1 rules preds i ctxt]

(* @chunk intro_rules *) 

fun introductions rules parnames preds defs lthy1 = 

let

  val (_, lthy2) = Variable.add_fixes parnames lthy1

  fun prove_intro (i, goal) =

    Goal.prove lthy2 [] [] goal

       (fn {context, ...} => introductions_tac defs rules preds i context)

       |> singleton (Proof_Context.export lthy2 lthy1)

in

  map_index prove_intro rules

end

(* @end *)

(* @chunk add_inductive_i *)

fun add_inductive_i preds params specs lthy =

let

  val params' = map (fn (p, T) => Free (Binding.name_of p, T)) params;

  val preds' = map (fn ((R, T), _) => list_comb (Free (Binding.name_of R, T), params')) preds;

  val Tss = map (binder_types o fastype_of) preds';   

  val (ass, rules) = split_list specs;  (* FIXME: ass not used? *)  

  val prednames = map (fst o fst) preds

  val syns = map snd preds

  val parnames = map (Binding.name_of o fst) params

  val (defs, lthy1) = definitions rules params' preds' prednames syns Tss lthy;

  val inducts = inductions rules defs parnames preds' Tss lthy1 	

  val intros = introductions rules parnames preds' defs lthy1

  val mut_name = space_implode "_" (map (Binding.name_of o fst o fst) preds);

  val case_names = map (Binding.name_of o fst o fst) specs

in

    lthy1 

    |> Local_Theory.notes (map (fn (((a, atts), _), th) =>

        ((Binding.qualify false mut_name a, atts), [([th], [])])) (specs ~~ intros)) 

    |-> (fn intross => Local_Theory.note 

         ((Binding.qualify false mut_name (Binding.name "intros"), []), maps snd intross)) 

    |>> snd 

    ||>> (Local_Theory.notes (map (fn (((R, _), _), th) =>

         ((Binding.qualify false (Binding.name_of R) (Binding.name "induct"),

          [Attrib.internal (K (Rule_Cases.case_names case_names)),

           Attrib.internal (K (Rule_Cases.consumes 1)),

           Attrib.internal (K (Induct.induct_pred ""))]), [([th], [])]))

          (preds ~~ inducts)) #>> maps snd) 

    |> snd

end

(* @end *)

(* @chunk read_specification *)

fun read_specification' vars specs lthy =

let 

  val specs' = map (fn (a, s) => (a, [s])) specs

  val ((varst, specst), _) = 

                   Specification.read_specification vars specs' lthy

  val specst' = map (apsnd the_single) specst

in   

  (varst, specst')

end 

(* @end *)

(* @chunk add_inductive *)

fun add_inductive preds params specs lthy =

let

  val (vars, specs') = read_specification' (preds @ params) specs lthy;

  val (preds', params') = chop (length preds) vars;

  val params'' = map fst params'

in

  add_inductive_i preds' params'' specs' lthy

end;

(* @end *)

(* @chunk parser *)

val spec_parser = 

   Parse.opt_target --

   Parse.fixes -- 

   Parse.for_fixes --

   Scan.optional 

     (Parse.$$$ "where" |--

        Parse.!!! 

          (Parse.enum1 "|" 

             (Parse_Spec.opt_thm_name ":" -- Parse.prop))) []

(* @end *)

(* @chunk syntax *)

val specification =

  spec_parser >>

    (fn (((loc, preds), params), specs) =>

      Toplevel.local_theory loc (add_inductive preds params specs))

val _ = Outer_Syntax.command @{command_spec "simple_inductive"} "define inductive predicates"

          specification

(* @end *)

end;