src/ZF/add_ind_def.ML

ex_impE was incorrectly listed as Safe

(*  Title:      ZF/add_ind_def.ML
    ID:         $Id$
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1994  University of Cambridge

Fixedpoint definition module -- for Inductive/Coinductive Definitions

Features:
* least or greatest fixedpoints
* user-specified product and sum constructions
* mutually recursive definitions
* definitions involving arbitrary monotone operators
* automatically proves introduction and elimination rules

The recursive sets must *already* be declared as constants in parent theory!

  Introduction rules have the form
  [| ti:M(Sj), ..., P(x), ... |] ==> t: Sk |]
  where M is some monotone operator (usually the identity)
  P(x) is any (non-conjunctive) side condition on the free variables
  ti, t are any terms
  Sj, Sk are two of the sets being defined in mutual recursion

Sums are used only for mutual recursion;
Products are used only to derive "streamlined" induction rules for relations
*)

signature FP =          (** Description of a fixed point operator **)
  sig
  val oper      : term                  (*fixed point operator*)
  val bnd_mono  : term                  (*monotonicity predicate*)
  val bnd_monoI : thm                   (*intro rule for bnd_mono*)
  val subs      : thm                   (*subset theorem for fp*)
  val Tarski    : thm                   (*Tarski's fixed point theorem*)
  val induct    : thm                   (*induction/coinduction rule*)
  end;

signature SU =                  (** Description of a disjoint sum **)
  sig
  val sum       : term                  (*disjoint sum operator*)
  val inl       : term                  (*left injection*)
  val inr       : term                  (*right injection*)
  val elim      : term                  (*case operator*)
  val case_inl  : thm                   (*inl equality rule for case*)
  val case_inr  : thm                   (*inr equality rule for case*)
  val inl_iff   : thm                   (*injectivity of inl, using <->*)
  val inr_iff   : thm                   (*injectivity of inr, using <->*)
  val distinct  : thm                   (*distinctness of inl, inr using <->*)
  val distinct' : thm                   (*distinctness of inr, inl using <->*)
  val free_SEs  : thm list              (*elim rules for SU, and pair_iff!*)
  end;

signature ADD_INDUCTIVE_DEF =
  sig 
  val add_fp_def_i : term list * term * term list -> theory -> theory
  val add_constructs_def :
        string list * ((string*typ*mixfix) * 
                       string * term list * term list) list list ->
        theory -> theory
  end;



(*Declares functions to add fixedpoint/constructor defs to a theory*)
functor Add_inductive_def_Fun 
    (structure Fp: FP and Pr : PR and CP: CARTPROD and Su : SU)
    : ADD_INDUCTIVE_DEF =
struct
open Logic Ind_Syntax;

(*internal version*)
fun add_fp_def_i (rec_tms, dom_sum, intr_tms) thy = 
  let
    val sign = sign_of thy;

    (*recT and rec_params should agree for all mutually recursive components*)
    val rec_hds = map head_of rec_tms;

    val _ = assert_all is_Const rec_hds
            (fn t => "Recursive set not previously declared as constant: " ^ 
                     Sign.string_of_term sign t);

    (*Now we know they are all Consts, so get their names, type and params*)
    val rec_names = map (#1 o dest_Const) rec_hds
    and (Const(_,recT),rec_params) = strip_comb (hd rec_tms);

    val _ = assert_all Syntax.is_identifier rec_names
       (fn a => "Name of recursive set not an identifier: " ^ a);

    local (*Checking the introduction rules*)
      val intr_sets = map (#2 o rule_concl_msg sign) intr_tms;
      fun intr_ok set =
          case head_of set of Const(a,recT) => a mem rec_names | _ => false;
    in
      val _ =  assert_all intr_ok intr_sets
         (fn t => "Conclusion of rule does not name a recursive set: " ^ 
                  Sign.string_of_term sign t);
    end;

    val _ = assert_all is_Free rec_params
        (fn t => "Param in recursion term not a free variable: " ^
                 Sign.string_of_term sign t);

    (*** Construct the lfp definition ***)
    val mk_variant = variant (foldr add_term_names (intr_tms,[]));

    val z' = mk_variant"z" and X' = mk_variant"X" and w' = mk_variant"w";

    fun dest_tprop (Const("Trueprop",_) $ P) = P
      | dest_tprop Q = error ("Ill-formed premise of introduction rule: " ^ 
                              Sign.string_of_term sign Q);

    (*Makes a disjunct from an introduction rule*)
    fun lfp_part intr = (*quantify over rule's free vars except parameters*)
      let val prems = map dest_tprop (strip_imp_prems intr)
          val _ = seq (fn rec_hd => seq (chk_prem rec_hd) prems) rec_hds
          val exfrees = term_frees intr \\ rec_params
          val zeq = eq_const $ (Free(z',iT)) $ (#1 (rule_concl intr))
      in foldr mk_exists (exfrees, fold_bal (app conj) (zeq::prems)) end;

    (*The Part(A,h) terms -- compose injections to make h*)
    fun mk_Part (Bound 0) = Free(X',iT) (*no mutual rec, no Part needed*)
      | mk_Part h         = Part_const $ Free(X',iT) $ Abs(w',iT,h);

    (*Access to balanced disjoint sums via injections*)
    val parts = 
        map mk_Part (accesses_bal (ap Su.inl, ap Su.inr, Bound 0) 
                                  (length rec_tms));

    (*replace each set by the corresponding Part(A,h)*)
    val part_intrs = map (subst_free (rec_tms ~~ parts) o lfp_part) intr_tms;

    val lfp_abs = absfree(X', iT, 
                     mk_Collect(z', dom_sum, fold_bal (app disj) part_intrs));

    val lfp_rhs = Fp.oper $ dom_sum $ lfp_abs

    val _ = seq (fn rec_hd => deny (rec_hd occs lfp_rhs) 
                               "Illegal occurrence of recursion operator")
             rec_hds;

    (*** Make the new theory ***)

    (*A key definition:
      If no mutual recursion then it equals the one recursive set.
      If mutual recursion then it differs from all the recursive sets. *)
    val big_rec_name = space_implode "_" rec_names;

    (*Big_rec... is the union of the mutually recursive sets*)
    val big_rec_tm = list_comb(Const(big_rec_name,recT), rec_params);

    (*The individual sets must already be declared*)
    val axpairs = map mk_defpair 
          ((big_rec_tm, lfp_rhs) ::
           (case parts of 
               [_] => []                        (*no mutual recursion*)
             | _ => rec_tms ~~          (*define the sets as Parts*)
                    map (subst_atomic [(Free(X',iT),big_rec_tm)]) parts));

    val _ = seq (writeln o Sign.string_of_term sign o #2) axpairs
  
  in  thy |> add_defs_i axpairs  end


(*Expects the recursive sets to have been defined already.
  con_ty_lists specifies the constructors in the form (name,prems,mixfix) *)
fun add_constructs_def (rec_names, con_ty_lists) thy = 
  let
    val _ = writeln"  Defining the constructor functions...";
    val case_name = "f";                (*name for case variables*)

    (** Define the constructors **)

    (*The empty tuple is 0*)
    fun mk_tuple [] = Const("0",iT)
      | mk_tuple args = foldr1 (app Pr.pair) args;

    fun mk_inject n k u = access_bal(ap Su.inl, ap Su.inr, u) n k;

    val npart = length rec_names;       (*total # of mutually recursive parts*)

    (*Make constructor definition; kpart is # of this mutually recursive part*)
    fun mk_con_defs (kpart, con_ty_list) = 
      let val ncon = length con_ty_list    (*number of constructors*)
          fun mk_def (((id,T,syn), name, args, prems), kcon) =
                (*kcon is index of constructor*)
              mk_defpair (list_comb (Const(name,T), args),
                          mk_inject npart kpart
                          (mk_inject ncon kcon (mk_tuple args)))
      in  ListPair.map mk_def (con_ty_list, 1 upto ncon)  end;

    (** Define the case operator **)

    (*Combine split terms using case; yields the case operator for one part*)
    fun call_case case_list = 
      let fun call_f (free,[]) = Abs("null", iT, free)
            | call_f (free,args) =
                  CP.ap_split (foldr1 CP.mk_prod (map (#2 o dest_Free) args))
                              Ind_Syntax.iT 
                              free 
      in  fold_bal (app Su.elim) (map call_f case_list)  end;

    (** Generating function variables for the case definition
        Non-identifiers (e.g. infixes) get a name of the form f_op_nnn. **)

    (*Treatment of a single constructor*)
    fun add_case (((id,T,syn), name, args, prems), (opno,cases)) =
        if Syntax.is_identifier id
        then (opno,   
              (Free(case_name ^ "_" ^ id, T), args) :: cases)
        else (opno+1, 
              (Free(case_name ^ "_op_" ^ string_of_int opno, T), args) :: 
              cases)

    (*Treatment of a list of constructors, for one part*)
    fun add_case_list (con_ty_list, (opno,case_lists)) =
        let val (opno',case_list) = foldr add_case (con_ty_list, (opno,[]))
        in (opno', case_list :: case_lists) end;

    (*Treatment of all parts*)
    val (_, case_lists) = foldr add_case_list (con_ty_lists, (1,[]));

    val big_case_typ = flat (map (map (#2 o #1)) con_ty_lists) ---> (iT-->iT);

    val big_rec_name = space_implode "_" rec_names;

    val big_case_name = big_rec_name ^ "_case";

    (*The list of all the function variables*)
    val big_case_args = flat (map (map #1) case_lists);

    val big_case_tm = 
        list_comb (Const(big_case_name, big_case_typ), big_case_args); 

    val big_case_def = mk_defpair  
        (big_case_tm, fold_bal (app Su.elim) (map call_case case_lists)); 

    (** Build the new theory **)

    val const_decs =
        (big_case_name, big_case_typ, NoSyn) :: map #1 (flat con_ty_lists);

    val axpairs =
        big_case_def :: flat (ListPair.map mk_con_defs
			      (1 upto npart, con_ty_lists))

    in  thy |> add_consts_i const_decs |> add_defs_i axpairs  end;
end;