src/HOL/indrule.ML

Tools/8bit: ./mk;

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

Induction rule module -- for Inductive/Coinductive Definitions

Proves a strong induction rule and a mutual induction rule
*)

signature INDRULE =
  sig
  val induct        : thm                       (*main induction rule*)
  val mutual_induct : thm                       (*mutual induction rule*)
  end;


functor Indrule_Fun
    (structure Inductive: sig include INDUCTIVE_ARG INDUCTIVE_I end and
         Intr_elim: sig include INTR_ELIM INTR_ELIM_AUX end) : INDRULE  =
let

val sign = sign_of Inductive.thy;

val (Const(_,recT),rec_params) = strip_comb (hd Inductive.rec_tms);

val elem_type = Ind_Syntax.dest_setT (body_type recT);
val big_rec_name = space_implode "_" Intr_elim.rec_names;
val big_rec_tm = list_comb(Const(big_rec_name,recT), rec_params);

val _ = writeln "  Proving the induction rule...";

(*** Prove the main induction rule ***)

val pred_name = "P";            (*name for predicate variables*)

val big_rec_def::part_rec_defs = Intr_elim.defs;

(*Used to express induction rules: adds induction hypotheses.
   ind_alist = [(rec_tm1,pred1),...]  -- associates predicates with rec ops
   prem is a premise of an intr rule*)
fun add_induct_prem ind_alist (prem as Const("Trueprop",_) $ 
                 (Const("op :",_)$t$X), iprems) =
     (case gen_assoc (op aconv) (ind_alist, X) of
          Some pred => prem :: Ind_Syntax.mk_Trueprop (pred $ t) :: iprems
        | None => (*possibly membership in M(rec_tm), for M monotone*)
            let fun mk_sb (rec_tm,pred) = 
                 (case binder_types (fastype_of pred) of
                      [T] => (rec_tm, 
                              Ind_Syntax.Int_const T $ rec_tm $ 
                                (Ind_Syntax.Collect_const T $ pred))
                    | _ => error 
                      "Bug: add_induct_prem called with non-unary predicate")
            in  subst_free (map mk_sb ind_alist) prem :: iprems  end)
  | add_induct_prem ind_alist (prem,iprems) = prem :: iprems;

(*Make a premise of the induction rule.*)
fun induct_prem ind_alist intr =
  let val quantfrees = map dest_Free (term_frees intr \\ rec_params)
      val iprems = foldr (add_induct_prem ind_alist)
                         (Logic.strip_imp_prems intr,[])
      val (t,X) = Ind_Syntax.rule_concl intr
      val (Some pred) = gen_assoc (op aconv) (ind_alist, X)
      val concl = Ind_Syntax.mk_Trueprop (pred $ t)
  in list_all_free (quantfrees, Logic.list_implies (iprems,concl)) end
  handle Bind => error"Recursion term not found in conclusion";

(*Avoids backtracking by delivering the correct premise to each goal*)
fun ind_tac [] 0 = all_tac
  | ind_tac(prem::prems) i = 
        DEPTH_SOLVE_1 (ares_tac [Part_eqI, prem, refl] i) THEN
        ind_tac prems (i-1);

val pred = Free(pred_name, elem_type --> Ind_Syntax.boolT);

val ind_prems = map (induct_prem (map (rpair pred) Inductive.rec_tms)) 
                    Inductive.intr_tms;

(*Debugging code...
val _ = writeln "ind_prems = ";
val _ = seq (writeln o Sign.string_of_term sign) ind_prems;
*)

(*We use a MINIMAL simpset because others (such as HOL_ss) contain too many
  simplifications.  If the premises get simplified, then the proofs will 
  fail.  This arose with a premise of the form {(F n,G n)|n . True}, which
  expanded to something containing ...&True. *)
val min_ss = HOL_basic_ss;

val quant_induct = 
    prove_goalw_cterm part_rec_defs 
      (cterm_of sign 
       (Logic.list_implies (ind_prems, 
                            Ind_Syntax.mk_Trueprop (Ind_Syntax.mk_all_imp 
                                                    (big_rec_tm,pred)))))
      (fn prems =>
       [rtac (impI RS allI) 1,
        DETERM (etac Intr_elim.raw_induct 1),
        full_simp_tac (min_ss addsimps [Part_Collect]) 1,
        REPEAT (FIRSTGOAL (eresolve_tac [IntE, CollectE, exE, conjE, disjE] 
                           ORELSE' hyp_subst_tac)),
        ind_tac (rev prems) (length prems)])
    handle e => print_sign_exn sign e;

(*** Prove the simultaneous induction rule ***)

(*Make distinct predicates for each inductive set.
  Splits cartesian products in elem_type, however nested*)

(*The components of the element type, several if it is a product*)
val elem_factors = Prod_Syntax.factors elem_type;
val elem_frees = mk_frees "za" elem_factors;
val elem_tuple = Prod_Syntax.mk_tuple elem_type elem_frees;

(*Given a recursive set, return the "split" predicate
  and a conclusion for the simultaneous induction rule*)
fun mk_predpair rec_tm = 
  let val rec_name = (#1 o dest_Const o head_of) rec_tm
      val pfree = Free(pred_name ^ "_" ^ rec_name, 
                       elem_factors ---> Ind_Syntax.boolT)
      val qconcl = 
        foldr Ind_Syntax.mk_all 
          (elem_frees, 
           Ind_Syntax.imp $ (Ind_Syntax.mk_mem (elem_tuple, rec_tm))
                $ (list_comb (pfree, elem_frees)))
  in  (Prod_Syntax.ap_split elem_type Ind_Syntax.boolT pfree, 
       qconcl)  
  end;

val (preds,qconcls) = split_list (map mk_predpair Inductive.rec_tms);

(*Used to form simultaneous induction lemma*)
fun mk_rec_imp (rec_tm,pred) = 
    Ind_Syntax.imp $ (Ind_Syntax.mk_mem (Bound 0, rec_tm)) $  (pred $ Bound 0);

(*To instantiate the main induction rule*)
val induct_concl = 
    Ind_Syntax.mk_Trueprop
      (Ind_Syntax.mk_all_imp
       (big_rec_tm,
        Abs("z", elem_type, 
            fold_bal (app Ind_Syntax.conj) 
            (ListPair.map mk_rec_imp (Inductive.rec_tms,preds)))))
and mutual_induct_concl = 
    Ind_Syntax.mk_Trueprop (fold_bal (app Ind_Syntax.conj) qconcls);

val lemma_tac = FIRST' [eresolve_tac [asm_rl, conjE, PartE, mp],
                        resolve_tac [allI, impI, conjI, Part_eqI, refl],
                        dresolve_tac [spec, mp, splitD]];

val need_mutual = length Intr_elim.rec_names > 1;

val lemma = (*makes the link between the two induction rules*)
  if need_mutual then
     (writeln "  Proving the mutual induction rule...";
      prove_goalw_cterm part_rec_defs 
	  (cterm_of sign (Logic.mk_implies (induct_concl,
					    mutual_induct_concl)))
	  (fn prems =>
	   [cut_facts_tac prems 1,
	    REPEAT (rewrite_goals_tac [split RS eq_reflection] THEN
		    lemma_tac 1)])
      handle e => print_sign_exn sign e)
  else TrueI;

(*Mutual induction follows by freeness of Inl/Inr.*)

(*Simplification largely reduces the mutual induction rule to the 
  standard rule*)
val mut_ss = min_ss addsimps [Inl_not_Inr, Inr_not_Inl, Inl_eq, Inr_eq, split];

val all_defs = [split RS eq_reflection] @ Inductive.con_defs @ part_rec_defs;

(*Removes Collects caused by M-operators in the intro rules*)
val cmonos = [subset_refl RS Int_Collect_mono] RL Inductive.monos RLN
             (2,[rev_subsetD]);

(*Avoids backtracking by delivering the correct premise to each goal*)
fun mutual_ind_tac [] 0 = all_tac
  | mutual_ind_tac(prem::prems) i = 
      DETERM
       (SELECT_GOAL 
          (
           (*Simplify the assumptions and goal by unfolding Part and
             using freeness of the Sum constructors; proves all but one
             conjunct by contradiction*)
           rewrite_goals_tac all_defs  THEN
           simp_tac (mut_ss addsimps [Part_def]) 1  THEN
           IF_UNSOLVED (*simp_tac may have finished it off!*)
             ((*simplify assumptions*)
              full_simp_tac mut_ss 1  THEN
              (*unpackage and use "prem" in the corresponding place*)
              REPEAT (rtac impI 1)  THEN
              rtac (rewrite_rule all_defs prem) 1  THEN
              (*prem must not be REPEATed below: could loop!*)
              DEPTH_SOLVE (FIRSTGOAL (ares_tac [impI] ORELSE' 
                                      eresolve_tac (conjE::mp::cmonos))))
          ) i)
       THEN mutual_ind_tac prems (i-1);

val mutual_induct_split = 
  if need_mutual then
    prove_goalw_cterm []
          (cterm_of sign
           (Logic.list_implies (map (induct_prem (Inductive.rec_tms ~~ preds)) 
                              Inductive.intr_tms,
                          mutual_induct_concl)))
          (fn prems =>
           [rtac (quant_induct RS lemma) 1,
            mutual_ind_tac (rev prems) (length prems)])
    handle e => print_sign_exn sign e
  else TrueI;

(** Uncurrying the predicate in the ordinary induction rule **)

val xvar = cterm_of sign (Var(("x",0), elem_type));

(*strip quantifier and instantiate the variable to a tuple*)
val induct0 = quant_induct RS spec RSN (2,rev_mp) |>
              zero_var_indexes |>
              freezeT |>     (*Because elem_type contains TFrees not TVars*)
              instantiate ([], [(xvar, cterm_of sign elem_tuple)]);

val Const ("Trueprop", _) $ (pred_var $ _) = concl_of induct0


in
  struct
  val induct = standard (Prod_Syntax.split_rule_var (pred_var, induct0));

  (*Just "True" unless there's true mutual recursion.  This saves storage.*)
  val mutual_induct = Prod_Syntax.remove_split mutual_induct_split
  end
end;