src/ZF/indrule.ML
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(*  Title:      ZF/indrule.ML
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    ID:         $Id$
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1994  University of Cambridge
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Induction rule module -- for Inductive/Coinductive Definitions
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Proves a strong induction rule and a mutual induction rule
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*)
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signature INDRULE =
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  sig
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  val induct        : thm                       (*main induction rule*)
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  val mutual_induct : thm                       (*mutual induction rule*)
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  end;
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functor Indrule_Fun
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    (structure Inductive: sig include INDUCTIVE_ARG INDUCTIVE_I end
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     and Pr: PR and Su : SU and 
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     Intr_elim: sig include INTR_ELIM INTR_ELIM_AUX end) : INDRULE  =
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let
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val sign = sign_of Inductive.thy;
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val (Const(_,recT),rec_params) = strip_comb (hd Inductive.rec_tms);
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val big_rec_name = space_implode "_" Intr_elim.rec_names;
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val big_rec_tm = list_comb(Const(big_rec_name,recT), rec_params);
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val _ = writeln "  Proving the induction rule...";
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(*** Prove the main induction rule ***)
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val pred_name = "P";            (*name for predicate variables*)
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val big_rec_def::part_rec_defs = Intr_elim.defs;
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(*Used to make induction rules;
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   ind_alist = [(rec_tm1,pred1),...]  -- associates predicates with rec ops
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   prem is a premise of an intr rule*)
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fun add_induct_prem ind_alist (prem as Const("Trueprop",_) $ 
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                 (Const("op :",_)$t$X), iprems) =
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     (case gen_assoc (op aconv) (ind_alist, X) of
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          Some pred => prem :: Ind_Syntax.mk_tprop (pred $ t) :: iprems
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        | None => (*possibly membership in M(rec_tm), for M monotone*)
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            let fun mk_sb (rec_tm,pred) = 
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                        (rec_tm, Ind_Syntax.Collect_const$rec_tm$pred)
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            in  subst_free (map mk_sb ind_alist) prem :: iprems  end)
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  | add_induct_prem ind_alist (prem,iprems) = prem :: iprems;
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(*Make a premise of the induction rule.*)
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fun induct_prem ind_alist intr =
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  let val quantfrees = map dest_Free (term_frees intr \\ rec_params)
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      val iprems = foldr (add_induct_prem ind_alist)
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                         (Logic.strip_imp_prems intr,[])
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      val (t,X) = Ind_Syntax.rule_concl intr
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      val (Some pred) = gen_assoc (op aconv) (ind_alist, X)
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      val concl = Ind_Syntax.mk_tprop (pred $ t)
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  in list_all_free (quantfrees, Logic.list_implies (iprems,concl)) end
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  handle Bind => error"Recursion term not found in conclusion";
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(*Reduces backtracking by delivering the correct premise to each goal.
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  Intro rules with extra Vars in premises still cause some backtracking *)
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fun ind_tac [] 0 = all_tac
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  | ind_tac(prem::prems) i = 
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        DEPTH_SOLVE_1 (ares_tac [prem, refl] i) THEN
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        ind_tac prems (i-1);
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val pred = Free(pred_name, Ind_Syntax.iT --> Ind_Syntax.oT);
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val ind_prems = map (induct_prem (map (rpair pred) Inductive.rec_tms)) 
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                    Inductive.intr_tms;
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val quant_induct = 
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    prove_goalw_cterm part_rec_defs 
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      (cterm_of sign 
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       (Logic.list_implies (ind_prems, 
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                Ind_Syntax.mk_tprop (Ind_Syntax.mk_all_imp(big_rec_tm,pred)))))
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      (fn prems =>
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       [rtac (impI RS allI) 1,
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        DETERM (etac Intr_elim.raw_induct 1),
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        (*Push Part inside Collect*)
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        asm_full_simp_tac (FOL_ss addsimps [Part_Collect]) 1,
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        REPEAT (FIRSTGOAL (eresolve_tac [CollectE, exE, conjE, disjE] ORELSE'
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                           hyp_subst_tac)),
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        ind_tac (rev prems) (length prems) ]);
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(*** Prove the simultaneous induction rule ***)
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(*Make distinct predicates for each inductive set*)
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(*Sigmas and Cartesian products may nest ONLY to the right!*)
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fun mk_pred_typ (t $ A $ Abs(_,_,B)) = 
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        if t = Pr.sigma  then  Ind_Syntax.iT --> mk_pred_typ B
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                         else  Ind_Syntax.iT --> Ind_Syntax.oT
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  | mk_pred_typ _           =  Ind_Syntax.iT --> Ind_Syntax.oT
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(*For testing whether the inductive set is a relation*)
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fun is_sigma (t$_$_) = (t = Pr.sigma)
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  | is_sigma _       =  false;
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(*Given a recursive set and its domain, return the "fsplit" predicate
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  and a conclusion for the simultaneous induction rule.
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  NOTE.  This will not work for mutually recursive predicates.  Previously
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  a summand 'domt' was also an argument, but this required the domain of
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  mutual recursion to invariably be a disjoint sum.*)
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fun mk_predpair rec_tm = 
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  let val rec_name = (#1 o dest_Const o head_of) rec_tm
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      val T = mk_pred_typ Inductive.dom_sum
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      val pfree = Free(pred_name ^ "_" ^ rec_name, T)
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      val frees = mk_frees "za" (binder_types T)
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      val qconcl = 
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        foldr Ind_Syntax.mk_all (frees, 
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                        Ind_Syntax.imp $ 
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                          (Ind_Syntax.mem_const $ foldr1 (app Pr.pair) frees $
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                           rec_tm)
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                          $ (list_comb (pfree,frees)))
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  in  (Ind_Syntax.ap_split Pr.fsplit_const pfree (binder_types T), 
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      qconcl)  
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  end;
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val (preds,qconcls) = split_list (map mk_predpair Inductive.rec_tms);
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(*Used to form simultaneous induction lemma*)
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fun mk_rec_imp (rec_tm,pred) = 
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    Ind_Syntax.imp $ (Ind_Syntax.mem_const $ Bound 0 $ rec_tm) $ 
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                     (pred $ Bound 0);
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(*To instantiate the main induction rule*)
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val induct_concl = 
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 Ind_Syntax.mk_tprop(Ind_Syntax.mk_all_imp(big_rec_tm,
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             Abs("z", Ind_Syntax.iT, 
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                 fold_bal (app Ind_Syntax.conj) 
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                 (map mk_rec_imp (Inductive.rec_tms~~preds)))))
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and mutual_induct_concl =
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 Ind_Syntax.mk_tprop(fold_bal (app Ind_Syntax.conj) qconcls);
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val lemma = (*makes the link between the two induction rules*)
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    prove_goalw_cterm part_rec_defs 
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          (cterm_of sign (Logic.mk_implies (induct_concl,mutual_induct_concl)))
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          (fn prems =>
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           [cut_facts_tac prems 1, 
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            REPEAT (eresolve_tac [asm_rl, conjE, PartE, mp] 1
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             ORELSE resolve_tac [allI, impI, conjI, Part_eqI] 1
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             ORELSE dresolve_tac [spec, mp, Pr.fsplitD] 1)]);
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(*Mutual induction follows by freeness of Inl/Inr.*)
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(*Simplification largely reduces the mutual induction rule to the 
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  standard rule*)
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val mut_ss = 
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    FOL_ss addsimps   [Su.distinct, Su.distinct', Su.inl_iff, Su.inr_iff];
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val all_defs = Inductive.con_defs @ part_rec_defs;
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(*Removes Collects caused by M-operators in the intro rules.  It is very
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  hard to simplify
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    list({v: tf. (v : t --> P_t(v)) & (v : f --> P_f(v))}) 
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  where t==Part(tf,Inl) and f==Part(tf,Inr) to  list({v: tf. P_t(v)}).
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  Instead the following rules extract the relevant conjunct.
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*)
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val cmonos = [subset_refl RS Collect_mono] RL Inductive.monos RLN (2,[rev_subsetD]);
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(*Avoids backtracking by delivering the correct premise to each goal*)
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fun mutual_ind_tac [] 0 = all_tac
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  | mutual_ind_tac(prem::prems) i = 
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      DETERM
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       (SELECT_GOAL 
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          (
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           (*Simplify the assumptions and goal by unfolding Part and
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             using freeness of the Sum constructors; proves all but one
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             conjunct by contradiction*)
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           rewrite_goals_tac all_defs  THEN
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           simp_tac (mut_ss addsimps [Part_iff]) 1  THEN
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           IF_UNSOLVED (*simp_tac may have finished it off!*)
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             ((*simplify assumptions, but don't accept new rewrite rules!*)
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              asm_full_simp_tac (mut_ss setmksimps (fn _=>[])) 1  THEN
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              (*unpackage and use "prem" in the corresponding place*)
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              REPEAT (rtac impI 1)  THEN
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              rtac (rewrite_rule all_defs prem) 1  THEN
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              (*prem must not be REPEATed below: could loop!*)
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              DEPTH_SOLVE (FIRSTGOAL (ares_tac [impI] ORELSE' 
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                                      eresolve_tac (conjE::mp::cmonos))))
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          ) i)
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       THEN mutual_ind_tac prems (i-1);
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val _ = writeln "  Proving the mutual induction rule...";
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val mutual_induct_fsplit = 
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    prove_goalw_cterm []
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          (cterm_of sign
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           (Logic.list_implies 
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              (map (induct_prem (Inductive.rec_tms~~preds)) Inductive.intr_tms,
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               mutual_induct_concl)))
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          (fn prems =>
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           [rtac (quant_induct RS lemma) 1,
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            mutual_ind_tac (rev prems) (length prems)]);
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(*Attempts to remove all occurrences of fsplit*)
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val fsplit_tac =
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    REPEAT (SOMEGOAL (FIRST' [rtac Pr.fsplitI, 
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                              dtac Pr.fsplitD,
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                              etac Pr.fsplitE,  (*apparently never used!*)
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                              bound_hyp_subst_tac]))
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    THEN prune_params_tac
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in
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  struct
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  (*strip quantifier*)
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  val induct = standard (quant_induct RS spec RSN (2,rev_mp));
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  (*Just "True" unless significantly different from induct, with mutual
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    recursion or because it involves tuples.  This saves storage.*)
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  val mutual_induct = 
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      if length Intr_elim.rec_names > 1 orelse
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         is_sigma Inductive.dom_sum 
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      then rule_by_tactic fsplit_tac mutual_induct_fsplit
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      else TrueI;
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  end
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end;