theory InductiveInternals
imports Main
begin

section {* Parsing *}

ML {*
fun parse p s =
  Scan.finite OuterLex.stopper (Scan.error (OuterParse.!!! p))
    (filter OuterLex.is_proper (OuterSyntax.scan Position.none s));
*}

ML {*
local structure P = OuterParse and K = OuterKeyword in

val ind_parser =
  P.fixes -- P.for_fixes --
  Scan.optional (P.$$$ "where" |--
    P.!!! (P.enum1 "|" (SpecParse.opt_thm_name ":" -- P.prop))) [];

end;
*}

ML {*
parse ind_parser
"even and odd \
\where\
\  even0: \"even 0\"\
\| evenS: \"odd n \<Longrightarrow> even (Suc n)\"\
\| oddS: \"even n \<Longrightarrow> odd (Suc n)\""
*}

ML {*
val (((preds_syn, params_syn), intro_srcs), []) =
parse ind_parser
"accpart for r :: \"'a \<Rightarrow> 'a \<Rightarrow> bool\"\
\where\
\  accpartI: \"(\<And>y. r y x \<Longrightarrow> accpart r y) \<Longrightarrow> accpart r x\""
*}


section {* Reading the Specification *}

ML {*
val lthy = TheoryTarget.context "-" @{theory}
*}

ML {*
val ((vars, intrs), _) = Specification.read_spec
  (preds_syn @ params_syn) intro_srcs lthy;
val (preds_syn, params_syn') = chop (length preds_syn) vars
val params = map fst params_syn';
*}


section {* Making the definition *}

ML {*
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_syn;
val Tss = map (binder_types o fastype_of) preds;

val intrs' = map
  (ObjectLogic.atomize_term (ProofContext.theory_of lthy) o snd) intrs;
*}

ML {*
fun mk_all x P = HOLogic.all_const (fastype_of x) $ lambda x P;
*}

ML {*
val (defs, lthy1) = fold_map (fn ((((R, _), syn), pred), Ts) =>
  let val zs = map Free (Variable.variant_frees lthy intrs'
    (map (pair "z") Ts))
  in
    LocalTheory.define Thm.internalK
      ((R, syn), (Attrib.empty_binding, fold_rev lambda (params' @ zs)
        (fold_rev mk_all preds (fold_rev (curry HOLogic.mk_imp)
           intrs' (list_comb (pred, zs)))))) #>> snd #>> snd
   end) (preds_syn ~~ preds ~~ Tss) lthy;
*}

ML {*
val (_, lthy2) = Variable.add_fixes
  (map (Binding.name_of o fst) params) lthy1;
*}


section {* Proving the induction rules *}

ML {*
val (Pnames, lthy3) =
  Variable.variant_fixes (replicate (length preds) "P") lthy2;
val Ps = map (fn (s, Ts) => Free (s, Ts ---> HOLogic.boolT))
  (Pnames ~~ Tss);
val cPs = map (cterm_of (ProofContext.theory_of lthy3)) Ps;
val intrs'' = map (subst_free (preds ~~ Ps) o snd) intrs;
*}

ML {*
fun inst_spec ct = Drule.instantiate'
  [SOME (ctyp_of_term ct)] [NONE, SOME ct] spec;
*}

ML {*
fun prove_indrule ((R, P), Ts) =
  let
    val (znames, lthy4) =
      Variable.variant_fixes (replicate (length Ts) "z") lthy3;
    val zs = map Free (znames ~~ Ts)
  in
    Goal.prove lthy4 []
      [HOLogic.mk_Trueprop (list_comb (R, zs))]
      (Logic.list_implies (intrs'',
         HOLogic.mk_Trueprop (list_comb (P, zs))))
      (fn {prems, ...} => EVERY
         ([ObjectLogic.full_atomize_tac 1,
           cut_facts_tac prems 1,
           rewrite_goals_tac defs] @
          map (fn ct => dtac (inst_spec ct) 1) cPs @
          [assume_tac 1])) |>
    singleton (ProofContext.export lthy4 lthy1)
  end;
*}

ML {*
val indrules = map prove_indrule (preds ~~ Ps ~~ Tss);
*}


section {* Proving the introduction rules *}

ML {*
val all_elims = fold (fn ct => fn th => th RS inst_spec ct);
val imp_elims = fold (fn th => fn th' => [th', th] MRS mp);
*}

ML {*
fun show_thms ctxt s ths = warning (s ^ ":\n" ^
  Pretty.string_of (Pretty.chunks (map (Display.pretty_thm ctxt) ths)));

fun show_cterms ctxt s cts = warning (s ^ ":\n" ^
  Pretty.string_of (Pretty.list "[" "]"
    (map (Syntax.pretty_term ctxt o term_of) cts)));
*}

ML {*
fun prove_intr (i, (_, r)) =
  Goal.prove lthy2 [] [] r
    (fn {prems, context = ctxt} => EVERY
       [ObjectLogic.rulify_tac 1,
        rewrite_goals_tac defs,
        REPEAT (resolve_tac [allI, impI] 1),
        SUBPROOF (fn {params, prems, context = ctxt', ...} =>
          let
            val (prems1, prems2) =
              chop (length prems - length intrs) prems;
            val _ = show_thms ctxt' "prems1" prems1;
            val _ = show_thms ctxt' "prems2" prems2;
            val (params1, params2) =
              chop (length params - length preds) (map snd params)
            val _ = show_cterms ctxt' "params1" params1;
            val _ = show_cterms ctxt' "params2" params2;
          in
            rtac (ObjectLogic.rulify
              (all_elims params1 (nth prems2 i))) 1 THEN
            EVERY (map (fn prem =>
              SUBPROOF (fn {prems = prems', concl, context = ctxt'', ...} =>
                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'
                  val _ = show_thms ctxt'' "prem" [prem];
                  val _ = show_thms ctxt'' "prem'" [prem'];
                  val _ = show_thms ctxt'' "prem''" [prem''];
                in rtac prem'' 1 end) ctxt' 1) prems1)
          end) ctxt 1]) |>
  singleton (ProofContext.export lthy2 lthy1);
*}

ML {*
val intr_ths = map_index prove_intr intrs;
*}

end