src/HOL/indrule.ML
author clasohm
Wed Mar 13 11:55:25 1996 +0100 (1996-03-13 ago)
changeset 1574 5a63ab90ee8a
parent 1465 5d7a7e439cec
child 1653 1a2ffa2fbf7d
permissions -rw-r--r--
modified primrec so it can be used in MiniML/Type.thy
     1 (*  Title:      HOL/indrule.ML
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Copyright   1994  University of Cambridge
     5 
     6 Induction rule module -- for Inductive/Coinductive Definitions
     7 
     8 Proves a strong induction rule and a mutual induction rule
     9 *)
    10 
    11 signature INDRULE =
    12   sig
    13   val induct        : thm                       (*main induction rule*)
    14   val mutual_induct : thm                       (*mutual induction rule*)
    15   end;
    16 
    17 
    18 functor Indrule_Fun
    19     (structure Inductive: sig include INDUCTIVE_ARG INDUCTIVE_I end and
    20          Intr_elim: sig include INTR_ELIM INTR_ELIM_AUX end) : INDRULE  =
    21 let
    22 
    23 val sign = sign_of Inductive.thy;
    24 
    25 val (Const(_,recT),rec_params) = strip_comb (hd Inductive.rec_tms);
    26 
    27 val elem_type = Ind_Syntax.dest_setT (body_type recT);
    28 val big_rec_name = space_implode "_" Intr_elim.rec_names;
    29 val big_rec_tm = list_comb(Const(big_rec_name,recT), rec_params);
    30 
    31 val _ = writeln "  Proving the induction rule...";
    32 
    33 (*** Prove the main induction rule ***)
    34 
    35 val pred_name = "P";            (*name for predicate variables*)
    36 
    37 val big_rec_def::part_rec_defs = Intr_elim.defs;
    38 
    39 (*Used to express induction rules: adds induction hypotheses.
    40    ind_alist = [(rec_tm1,pred1),...]  -- associates predicates with rec ops
    41    prem is a premise of an intr rule*)
    42 fun add_induct_prem ind_alist (prem as Const("Trueprop",_) $ 
    43                  (Const("op :",_)$t$X), iprems) =
    44      (case gen_assoc (op aconv) (ind_alist, X) of
    45           Some pred => prem :: Ind_Syntax.mk_Trueprop (pred $ t) :: iprems
    46         | None => (*possibly membership in M(rec_tm), for M monotone*)
    47             let fun mk_sb (rec_tm,pred) = 
    48                  (case binder_types (fastype_of pred) of
    49                       [T] => (rec_tm, 
    50                               Ind_Syntax.Int_const T $ rec_tm $ 
    51                                 (Ind_Syntax.Collect_const T $ pred))
    52                     | _ => error 
    53                       "Bug: add_induct_prem called with non-unary predicate")
    54             in  subst_free (map mk_sb ind_alist) prem :: iprems  end)
    55   | add_induct_prem ind_alist (prem,iprems) = prem :: iprems;
    56 
    57 (*Make a premise of the induction rule.*)
    58 fun induct_prem ind_alist intr =
    59   let val quantfrees = map dest_Free (term_frees intr \\ rec_params)
    60       val iprems = foldr (add_induct_prem ind_alist)
    61                          (Logic.strip_imp_prems intr,[])
    62       val (t,X) = Ind_Syntax.rule_concl intr
    63       val (Some pred) = gen_assoc (op aconv) (ind_alist, X)
    64       val concl = Ind_Syntax.mk_Trueprop (pred $ t)
    65   in list_all_free (quantfrees, Logic.list_implies (iprems,concl)) end
    66   handle Bind => error"Recursion term not found in conclusion";
    67 
    68 (*Avoids backtracking by delivering the correct premise to each goal*)
    69 fun ind_tac [] 0 = all_tac
    70   | ind_tac(prem::prems) i = 
    71         DEPTH_SOLVE_1 (ares_tac [Part_eqI, prem, refl] i) THEN
    72         ind_tac prems (i-1);
    73 
    74 val pred = Free(pred_name, elem_type --> Ind_Syntax.boolT);
    75 
    76 val ind_prems = map (induct_prem (map (rpair pred) Inductive.rec_tms)) 
    77                     Inductive.intr_tms;
    78 
    79 (*Debugging code...
    80 val _ = writeln "ind_prems = ";
    81 val _ = seq (writeln o Sign.string_of_term sign) ind_prems;
    82 *)
    83 
    84 val quant_induct = 
    85     prove_goalw_cterm part_rec_defs 
    86       (cterm_of sign 
    87        (Logic.list_implies (ind_prems, 
    88                             Ind_Syntax.mk_Trueprop (Ind_Syntax.mk_all_imp 
    89                                                     (big_rec_tm,pred)))))
    90       (fn prems =>
    91        [rtac (impI RS allI) 1,
    92         DETERM (etac Intr_elim.raw_induct 1),
    93         asm_full_simp_tac (!simpset addsimps [Part_Collect]) 1,
    94         REPEAT (FIRSTGOAL (eresolve_tac [IntE, CollectE, exE, conjE, disjE] 
    95                            ORELSE' hyp_subst_tac)),
    96         ind_tac (rev prems) (length prems)])
    97     handle e => print_sign_exn sign e;
    98 
    99 (*** Prove the simultaneous induction rule ***)
   100 
   101 (*Make distinct predicates for each inductive set.
   102   Splits cartesian products in elem_type, IF nested to the right! *)
   103 
   104 (*Given a recursive set, return the "split" predicate
   105   and a conclusion for the simultaneous induction rule*)
   106 fun mk_predpair rec_tm = 
   107   let val rec_name = (#1 o dest_Const o head_of) rec_tm
   108       val T = Ind_Syntax.factors elem_type ---> Ind_Syntax.boolT
   109       val pfree = Free(pred_name ^ "_" ^ rec_name, T)
   110       val frees = mk_frees "za" (binder_types T)
   111       val qconcl = 
   112         foldr Ind_Syntax.mk_all 
   113           (frees, 
   114            Ind_Syntax.imp $ (Ind_Syntax.mk_mem 
   115                              (foldr1 Ind_Syntax.mk_Pair frees, rec_tm))
   116                 $ (list_comb (pfree,frees)))
   117   in  (Ind_Syntax.ap_split Ind_Syntax.boolT pfree (binder_types T), 
   118       qconcl)  
   119   end;
   120 
   121 val (preds,qconcls) = split_list (map mk_predpair Inductive.rec_tms);
   122 
   123 (*Used to form simultaneous induction lemma*)
   124 fun mk_rec_imp (rec_tm,pred) = 
   125     Ind_Syntax.imp $ (Ind_Syntax.mk_mem (Bound 0, rec_tm)) $  (pred $ Bound 0);
   126 
   127 (*To instantiate the main induction rule*)
   128 val induct_concl = 
   129     Ind_Syntax.mk_Trueprop
   130       (Ind_Syntax.mk_all_imp
   131        (big_rec_tm,
   132         Abs("z", elem_type, 
   133             fold_bal (app Ind_Syntax.conj) 
   134             (map mk_rec_imp (Inductive.rec_tms~~preds)))))
   135 and mutual_induct_concl = 
   136     Ind_Syntax.mk_Trueprop (fold_bal (app Ind_Syntax.conj) qconcls);
   137 
   138 val lemma = (*makes the link between the two induction rules*)
   139     prove_goalw_cterm part_rec_defs 
   140           (cterm_of sign (Logic.mk_implies (induct_concl,
   141                                             mutual_induct_concl)))
   142           (fn prems =>
   143            [cut_facts_tac prems 1,
   144             REPEAT (eresolve_tac [asm_rl, conjE, PartE, mp] 1
   145              ORELSE resolve_tac [allI, impI, conjI, Part_eqI, refl] 1
   146              ORELSE dresolve_tac [spec, mp, splitD] 1)])
   147     handle e => print_sign_exn sign e;
   148 
   149 (*Mutual induction follows by freeness of Inl/Inr.*)
   150 
   151 (*Simplification largely reduces the mutual induction rule to the 
   152   standard rule*)
   153 val mut_ss = simpset_of "Fun"
   154              addsimps [Inl_Inr_eq, Inr_Inl_eq, Inl_eq, Inr_eq];
   155 
   156 val all_defs = Inductive.con_defs @ part_rec_defs;
   157 
   158 (*Removes Collects caused by M-operators in the intro rules*)
   159 val cmonos = [subset_refl RS Int_Collect_mono] RL Inductive.monos RLN
   160              (2,[rev_subsetD]);
   161 
   162 (*Avoids backtracking by delivering the correct premise to each goal*)
   163 fun mutual_ind_tac [] 0 = all_tac
   164   | mutual_ind_tac(prem::prems) i = 
   165       DETERM
   166        (SELECT_GOAL 
   167           (
   168            (*Simplify the assumptions and goal by unfolding Part and
   169              using freeness of the Sum constructors; proves all but one
   170              conjunct by contradiction*)
   171            rewrite_goals_tac all_defs  THEN
   172            simp_tac (mut_ss addsimps [Part_def]) 1  THEN
   173            IF_UNSOLVED (*simp_tac may have finished it off!*)
   174              ((*simplify assumptions, but don't accept new rewrite rules!*)
   175               asm_full_simp_tac (mut_ss setmksimps K[]) 1  THEN
   176               (*unpackage and use "prem" in the corresponding place*)
   177               REPEAT (rtac impI 1)  THEN
   178               rtac (rewrite_rule all_defs prem) 1  THEN
   179               (*prem must not be REPEATed below: could loop!*)
   180               DEPTH_SOLVE (FIRSTGOAL (ares_tac [impI] ORELSE' 
   181                                       eresolve_tac (conjE::mp::cmonos))))
   182           ) i)
   183        THEN mutual_ind_tac prems (i-1);
   184 
   185 val _ = writeln "  Proving the mutual induction rule...";
   186 
   187 val mutual_induct_split = 
   188     prove_goalw_cterm []
   189           (cterm_of sign
   190            (Logic.list_implies (map (induct_prem (Inductive.rec_tms ~~ preds)) 
   191                               Inductive.intr_tms,
   192                           mutual_induct_concl)))
   193           (fn prems =>
   194            [rtac (quant_induct RS lemma) 1,
   195             mutual_ind_tac (rev prems) (length prems)])
   196     handle e => print_sign_exn sign e;
   197 
   198 (*Attempts to remove all occurrences of split*)
   199 val split_tac =
   200     REPEAT (SOMEGOAL (FIRST' [rtac splitI, 
   201                               dtac splitD,
   202                               etac splitE,
   203                               bound_hyp_subst_tac]))
   204     THEN prune_params_tac;
   205 
   206 in
   207   struct
   208   (*strip quantifier*)
   209   val induct = standard (quant_induct RS spec RSN (2,rev_mp));
   210 
   211   val mutual_induct = 
   212       if length Intr_elim.rec_names > 1 orelse
   213          length (Ind_Syntax.factors elem_type) > 1
   214       then rule_by_tactic split_tac mutual_induct_split
   215       else TrueI;
   216   end
   217 end;