(*  Title:      HOL/Tools/Datatype/datatype_realizer.ML

    Author:     Stefan Berghofer, TU Muenchen

Program extraction from proofs involving datatypes:

realizers for induction and case analysis.

*)

signature DATATYPE_REALIZER =

sig

  val add_dt_realizers: Datatype.config -> string list -> theory -> theory

  val setup: theory -> theory

end;

structure Datatype_Realizer : DATATYPE_REALIZER =

struct

open Datatype_Aux;

fun subsets i j =

  if i <= j then

    let val is = subsets (i+1) j

    in map (fn ks => i::ks) is @ is end

  else [[]];

fun forall_intr_prf (t, prf) =

  let val (a, T) = (case t of Var ((a, _), T) => (a, T) | Free p => p)

  in Abst (a, SOME T, Proofterm.prf_abstract_over t prf) end;

fun prf_of thm =

  Reconstruct.reconstruct_proof (Thm.theory_of_thm thm) (Thm.prop_of thm) (Thm.proof_of thm);

fun is_unit t = snd (strip_type (fastype_of t)) = HOLogic.unitT;

fun tname_of (Type (s, _)) = s

  | tname_of _ = "";

fun make_ind sorts ({descr, rec_names, rec_rewrites, induct, ...} : info) is thy =

  let

    val recTs = get_rec_types descr sorts;

    val pnames = if length descr = 1 then ["P"]

      else map (fn i => "P" ^ string_of_int i) (1 upto length descr);

    val rec_result_Ts = map (fn ((i, _), P) =>

      if i mem is then TFree ("'" ^ P, HOLogic.typeS) else HOLogic.unitT)

        (descr ~~ pnames);

    fun make_pred i T U r x =

      if i mem is then

        Free (List.nth (pnames, i), T --> U --> HOLogic.boolT) $ r $ x

      else Free (List.nth (pnames, i), U --> HOLogic.boolT) $ x;

    fun mk_all i s T t =

      if i mem is then list_all_free ([(s, T)], t) else t;

    val (prems, rec_fns) = split_list (flat (fst (fold_map

      (fn ((i, (_, _, constrs)), T) => fold_map (fn (cname, cargs) => fn j =>

        let

          val Ts = map (typ_of_dtyp descr sorts) cargs;

          val tnames = Name.variant_list pnames (Datatype_Prop.make_tnames Ts);

          val recs = filter (is_rec_type o fst o fst) (cargs ~~ tnames ~~ Ts);

          val frees = tnames ~~ Ts;

          fun mk_prems vs [] = 

                let

                  val rT = nth (rec_result_Ts) i;

                  val vs' = filter_out is_unit vs;

                  val f = mk_Free "f" (map fastype_of vs' ---> rT) j;

                  val f' = Envir.eta_contract (list_abs_free

                    (map dest_Free vs, if i mem is then list_comb (f, vs')

                      else HOLogic.unit));

                in (HOLogic.mk_Trueprop (make_pred i rT T (list_comb (f, vs'))

                  (list_comb (Const (cname, Ts ---> T), map Free frees))), f')

                end

            | mk_prems vs (((dt, s), T) :: ds) = 

                let

                  val k = body_index dt;

                  val (Us, U) = strip_type T;

                  val i = length Us;

                  val rT = nth (rec_result_Ts) k;

                  val r = Free ("r" ^ s, Us ---> rT);

                  val (p, f) = mk_prems (vs @ [r]) ds

                in (mk_all k ("r" ^ s) (Us ---> rT) (Logic.mk_implies

                  (list_all (map (pair "x") Us, HOLogic.mk_Trueprop

                    (make_pred k rT U (app_bnds r i)

                      (app_bnds (Free (s, T)) i))), p)), f)

                end

        in (apfst (curry list_all_free frees) (mk_prems (map Free frees) recs), j + 1) end)

          constrs) (descr ~~ recTs) 1)));

    fun mk_proj j [] t = t

      | mk_proj j (i :: is) t = if null is then t else

          if (j: int) = i then HOLogic.mk_fst t

          else mk_proj j is (HOLogic.mk_snd t);

    val tnames = Datatype_Prop.make_tnames recTs;

    val fTs = map fastype_of rec_fns;

    val ps = map (fn ((((i, _), T), U), s) => Abs ("x", T, make_pred i U T

      (list_comb (Const (s, fTs ---> T --> U), rec_fns) $ Bound 0) (Bound 0)))

        (descr ~~ recTs ~~ rec_result_Ts ~~ rec_names);

    val r = if null is then Extraction.nullt else

      foldr1 HOLogic.mk_prod (map_filter (fn (((((i, _), T), U), s), tname) =>

        if i mem is then SOME

          (list_comb (Const (s, fTs ---> T --> U), rec_fns) $ Free (tname, T))

        else NONE) (descr ~~ recTs ~~ rec_result_Ts ~~ rec_names ~~ tnames));

    val concl = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop @{const_name "op &"})

      (map (fn ((((i, _), T), U), tname) =>

        make_pred i U T (mk_proj i is r) (Free (tname, T)))

          (descr ~~ recTs ~~ rec_result_Ts ~~ tnames)));

    val cert = cterm_of thy;

    val inst = map (pairself cert) (map head_of (HOLogic.dest_conj

      (HOLogic.dest_Trueprop (concl_of induct))) ~~ ps);

    val thm = OldGoals.simple_prove_goal_cterm (cert (Logic.list_implies (prems, concl)))

      (fn prems =>

         [rewrite_goals_tac (map mk_meta_eq [fst_conv, snd_conv]),

          rtac (cterm_instantiate inst induct) 1,

          ALLGOALS Object_Logic.atomize_prems_tac,

          rewrite_goals_tac (@{thm o_def} :: map mk_meta_eq rec_rewrites),

          REPEAT ((resolve_tac prems THEN_ALL_NEW (fn i =>

            REPEAT (etac allE i) THEN atac i)) 1)]);

    val ind_name = Thm.get_name induct;

    val vs = map (fn i => List.nth (pnames, i)) is;

    val (thm', thy') = thy

      |> Sign.root_path

      |> PureThy.store_thm

        (Binding.qualified_name (space_implode "_" (ind_name :: vs @ ["correctness"])), thm)

      ||> Sign.restore_naming thy;

    val ivs = rev (Term.add_vars (Logic.varify (Datatype_Prop.make_ind [descr] sorts)) []);

    val rvs = rev (Thm.fold_terms Term.add_vars thm' []);

    val ivs1 = map Var (filter_out (fn (_, T) =>  (* FIXME set (!??) *)

      tname_of (body_type T) mem [@{type_abbrev set}, @{type_name bool}]) ivs);

    val ivs2 = map (fn (ixn, _) => Var (ixn, the (AList.lookup (op =) rvs ixn))) ivs;

    val prf =

      List.foldr forall_intr_prf

        (fold_rev (fn (f, p) => fn prf =>

          (case head_of (strip_abs_body f) of

             Free (s, T) =>

               let val T' = Logic.varifyT T

               in Abst (s, SOME T', Proofterm.prf_abstract_over

                 (Var ((s, 0), T')) (AbsP ("H", SOME p, prf)))

               end

          | _ => AbsP ("H", SOME p, prf)))

            (rec_fns ~~ prems_of thm)

            (Proofterm.proof_combP (prf_of thm', map PBound (length prems - 1 downto 0)))) ivs2;

    val r' =

      if null is then r

      else Logic.varify (fold_rev lambda

        (map Logic.unvarify ivs1 @ filter_out is_unit

            (map (head_of o strip_abs_body) rec_fns)) r);

  in Extraction.add_realizers_i [(ind_name, (vs, r', prf))] thy' end;

fun make_casedists sorts ({index, descr, case_name, case_rewrites, exhaust, ...} : info) thy =

  let

    val cert = cterm_of thy;

    val rT = TFree ("'P", HOLogic.typeS);

    val rT' = TVar (("'P", 0), HOLogic.typeS);

    fun make_casedist_prem T (cname, cargs) =

      let

        val Ts = map (typ_of_dtyp descr sorts) cargs;

        val frees = Name.variant_list ["P", "y"] (Datatype_Prop.make_tnames Ts) ~~ Ts;

        val free_ts = map Free frees;

        val r = Free ("r" ^ Long_Name.base_name cname, Ts ---> rT)

      in (r, list_all_free (frees, Logic.mk_implies (HOLogic.mk_Trueprop

        (HOLogic.mk_eq (Free ("y", T), list_comb (Const (cname, Ts ---> T), free_ts))),

          HOLogic.mk_Trueprop (Free ("P", rT --> HOLogic.boolT) $

            list_comb (r, free_ts)))))

      end;

    val SOME (_, _, constrs) = AList.lookup (op =) descr index;

    val T = List.nth (get_rec_types descr sorts, index);

    val (rs, prems) = split_list (map (make_casedist_prem T) constrs);

    val r = Const (case_name, map fastype_of rs ---> T --> rT);

    val y = Var (("y", 0), Logic.varifyT T);

    val y' = Free ("y", T);

    val thm = OldGoals.prove_goalw_cterm [] (cert (Logic.list_implies (prems,

      HOLogic.mk_Trueprop (Free ("P", rT --> HOLogic.boolT) $

        list_comb (r, rs @ [y'])))))

      (fn prems =>

         [rtac (cterm_instantiate [(cert y, cert y')] exhaust) 1,

          ALLGOALS (EVERY'

            [asm_simp_tac (HOL_basic_ss addsimps case_rewrites),

             resolve_tac prems, asm_simp_tac HOL_basic_ss])]);

    val exh_name = Thm.get_name exhaust;

    val (thm', thy') = thy

      |> Sign.root_path

      |> PureThy.store_thm (Binding.qualified_name (exh_name ^ "_P_correctness"), thm)

      ||> Sign.restore_naming thy;

    val P = Var (("P", 0), rT' --> HOLogic.boolT);

    val prf = forall_intr_prf (y, forall_intr_prf (P,

      fold_rev (fn (p, r) => fn prf =>

          forall_intr_prf (Logic.varify r, AbsP ("H", SOME (Logic.varify p), prf)))

        (prems ~~ rs)

        (Proofterm.proof_combP (prf_of thm', map PBound (length prems - 1 downto 0)))));

    val r' = Logic.varify (Abs ("y", T,

      list_abs (map dest_Free rs, list_comb (r,

        map Bound ((length rs - 1 downto 0) @ [length rs])))));

  in Extraction.add_realizers_i

    [(exh_name, (["P"], r', prf)),

     (exh_name, ([], Extraction.nullt, prf_of exhaust))] thy'

  end;

fun add_dt_realizers config names thy =

  if ! Proofterm.proofs < 2 then thy

  else let

    val _ = message config "Adding realizers for induction and case analysis ..."

    val infos = map (Datatype.the_info thy) names;

    val info :: _ = infos;

  in

    thy

    |> fold_rev (make_ind (#sorts info) info) (subsets 0 (length (#descr info) - 1))

    |> fold_rev (make_casedists (#sorts info)) infos

  end;

val setup = Datatype.interpretation add_dt_realizers;

end;