author  paulson 
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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 (i1); 
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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 Moperators 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 (i1); 
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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 
0  221 
end; 