author clasohm
Tue, 30 Jan 1996 13:42:57 +0100
changeset 1461 6bcb44e4d6e5
parent 1418 f5f97ee67cbb
child 1736 fe0b459273f2
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(*  Title:      ZF/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 =
  val induct        : thm                       (*main induction rule*)
  val mutual_induct : thm                       (*mutual induction rule*)

functor Indrule_Fun
    (structure Inductive: sig include INDUCTIVE_ARG INDUCTIVE_I end
     and Pr: PR and Su : SU and 
     Intr_elim: sig include INTR_ELIM INTR_ELIM_AUX end) : INDRULE  =

val sign = sign_of Inductive.thy;

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

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 make induction rules;
   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_tprop (pred $ t) :: iprems
        | None => (*possibly membership in M(rec_tm), for M monotone*)
            let fun mk_sb (rec_tm,pred) = 
                        (rec_tm, Ind_Syntax.Collect_const$rec_tm$pred)
            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_tprop (pred $ t)
  in list_all_free (quantfrees, Logic.list_implies (iprems,concl)) end
  handle Bind => error"Recursion term not found in conclusion";

(*Reduces backtracking by delivering the correct premise to each goal.
  Intro rules with extra Vars in premises still cause some backtracking *)
fun ind_tac [] 0 = all_tac
  | ind_tac(prem::prems) i = 
        DEPTH_SOLVE_1 (ares_tac [prem, refl] i) THEN
        ind_tac prems (i-1);

val pred = Free(pred_name, Ind_Syntax.iT --> Ind_Syntax.oT);

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

val quant_induct = 
    prove_goalw_cterm part_rec_defs 
      (cterm_of sign 
       (Logic.list_implies (ind_prems, 
                Ind_Syntax.mk_tprop (Ind_Syntax.mk_all_imp(big_rec_tm,pred)))))
      (fn prems =>
       [rtac (impI RS allI) 1,
        DETERM (etac Intr_elim.raw_induct 1),
        (*Push Part inside Collect*)
        asm_full_simp_tac (FOL_ss addsimps [Part_Collect]) 1,
        REPEAT (FIRSTGOAL (eresolve_tac [CollectE, exE, conjE, disjE] ORELSE'
        ind_tac (rev prems) (length prems) ]);

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

(*Make distinct predicates for each inductive set*)

(*Sigmas and Cartesian products may nest ONLY to the right!*)
fun mk_pred_typ (t $ A $ Abs(_,_,B)) = 
        if t = Pr.sigma  then  Ind_Syntax.iT --> mk_pred_typ B
                         else  Ind_Syntax.iT --> Ind_Syntax.oT
  | mk_pred_typ _           =  Ind_Syntax.iT --> Ind_Syntax.oT

(*For testing whether the inductive set is a relation*)
fun is_sigma (t$_$_) = (t = Pr.sigma)
  | is_sigma _       =  false;

(*Given a recursive set and its domain, return the "fsplit" predicate
  and a conclusion for the simultaneous induction rule.
  NOTE.  This will not work for mutually recursive predicates.  Previously
  a summand 'domt' was also an argument, but this required the domain of
  mutual recursion to invariably be a disjoint sum.*)
fun mk_predpair rec_tm = 
  let val rec_name = (#1 o dest_Const o head_of) rec_tm
      val T = mk_pred_typ Inductive.dom_sum
      val pfree = Free(pred_name ^ "_" ^ rec_name, T)
      val frees = mk_frees "za" (binder_types T)
      val qconcl = 
        foldr Ind_Syntax.mk_all (frees, 
                        Ind_Syntax.imp $ 
                          (Ind_Syntax.mem_const $ foldr1 (app Pr.pair) frees $
                          $ (list_comb (pfree,frees)))
  in  (Ind_Syntax.ap_split Pr.fsplit_const pfree (binder_types T), 

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.mem_const $ Bound 0 $ rec_tm) $ 
                     (pred $ Bound 0);

(*To instantiate the main induction rule*)
val induct_concl = 
             Abs("z", Ind_Syntax.iT, 
                 fold_bal (app Ind_Syntax.conj) 
                 (map mk_rec_imp (Inductive.rec_tms~~preds)))))
and mutual_induct_concl =
 Ind_Syntax.mk_tprop(fold_bal (app Ind_Syntax.conj) qconcls);

val lemma = (*makes the link between the two induction rules*)
    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 (eresolve_tac [asm_rl, conjE, PartE, mp] 1
             ORELSE resolve_tac [allI, impI, conjI, Part_eqI] 1
             ORELSE dresolve_tac [spec, mp, Pr.fsplitD] 1)]);

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

(*Simplification largely reduces the mutual induction rule to the 
  standard rule*)
val mut_ss = 
    FOL_ss addsimps   [Su.distinct, Su.distinct', Su.inl_iff, Su.inr_iff];

val all_defs = Inductive.con_defs @ part_rec_defs;

(*Removes Collects caused by M-operators in the intro rules.  It is very
  hard to simplify
    list({v: tf. (v : t --> P_t(v)) & (v : f --> P_f(v))}) 
  where t==Part(tf,Inl) and f==Part(tf,Inr) to  list({v: tf. P_t(v)}).
  Instead the following rules extract the relevant conjunct.
val cmonos = [subset_refl RS 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 = 
           (*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_iff]) 1  THEN
           IF_UNSOLVED (*simp_tac may have finished it off!*)
             ((*simplify assumptions, but don't accept new rewrite rules!*)
              asm_full_simp_tac (mut_ss setmksimps (fn _=>[])) 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 _ = writeln "  Proving the mutual induction rule...";

val mutual_induct_fsplit = 
    prove_goalw_cterm []
          (cterm_of sign
              (map (induct_prem (Inductive.rec_tms~~preds)) Inductive.intr_tms,
          (fn prems =>
           [rtac (quant_induct RS lemma) 1,
            mutual_ind_tac (rev prems) (length prems)]);

(*Attempts to remove all occurrences of fsplit*)
val fsplit_tac =
    REPEAT (SOMEGOAL (FIRST' [rtac Pr.fsplitI, 
                              dtac Pr.fsplitD,
                              etac Pr.fsplitE,  (*apparently never used!*)
    THEN prune_params_tac

  (*strip quantifier*)
  val induct = standard (quant_induct RS spec RSN (2,rev_mp));

  (*Just "True" unless significantly different from induct, with mutual
    recursion or because it involves tuples.  This saves storage.*)
  val mutual_induct = 
      if length Intr_elim.rec_names > 1 orelse
         is_sigma Inductive.dom_sum 
      then rule_by_tactic fsplit_tac mutual_induct_fsplit
      else TrueI;