src/HOL/indrule.ML
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(*  Title: 	HOL/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 and
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	 Intr_elim: INTR_ELIM) : INDRULE  =
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struct
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open Logic Ind_Syntax Inductive Intr_elim;
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val sign = sign_of thy;
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val (Const(_,recT),rec_params) = strip_comb (hd rec_tms);
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val elem_type = dest_setT (body_type recT);
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val domTs = summands(elem_type);
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val big_rec_name = space_implode "_" 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 rules...";
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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 express induction rules: adds induction hypotheses.
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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 :: mk_Trueprop (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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		 (case binder_types (fastype_of pred) of
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		      [T] => (rec_tm, 
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			      Int_const T $ rec_tm $ (Collect_const T $ pred))
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		    | _ => error 
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		      "Bug: add_induct_prem called with non-unary predicate")
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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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			 (strip_imp_prems intr,[])
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      val (t,X) = rule_concl intr
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      val (Some pred) = gen_assoc (op aconv) (ind_alist, X)
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      val concl = mk_Trueprop (pred $ t)
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  in list_all_free (quantfrees, list_implies (iprems,concl)) end
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  handle Bind => error"Recursion term not found in conclusion";
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(*Avoids backtracking by delivering the correct premise to each goal*)
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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 [Part_eqI, prem, refl] i) THEN
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	ind_tac prems (i-1);
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val pred = Free(pred_name, elem_type --> boolT);
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val ind_prems = map (induct_prem (map (rpair pred) rec_tms)) 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 (list_implies (ind_prems, 
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				    mk_Trueprop (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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	etac raw_induct 1,
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	REPEAT (FIRSTGOAL (eresolve_tac [IntE, CollectE, exE, conjE, disjE] 
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			   ORELSE' hyp_subst_tac)),
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	REPEAT (FIRSTGOAL (eresolve_tac [PartE, CollectE])),
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	ind_tac (rev prems) (length prems)])
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    handle e => print_sign_exn sign e;
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(*** Prove the simultaneous induction rule ***)
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(*Make distinct predicates for each inductive set.
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  Splits cartesian products in domT, IF nested to the right! *)
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(*Given a recursive set and its domain, return the "split" predicate
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  and a conclusion for the simultaneous induction rule*)
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fun mk_predpair (rec_tm,domT) = 
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  let val rec_name = (#1 o dest_Const o head_of) rec_tm
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      val T = factors domT ---> boolT
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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 mk_all (frees, 
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		      imp $ (mk_mem (foldr1 mk_Pair frees, rec_tm))
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			  $ (list_comb (pfree,frees)))
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  in  (ap_split boolT 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 (rec_tms~~domTs));
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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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    imp $ (mk_mem (Bound 0, rec_tm)) $  (pred $ Bound 0);
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(*To instantiate the main induction rule*)
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val induct_concl = 
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 mk_Trueprop(mk_all_imp(big_rec_tm,
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		     Abs("z", elem_type, 
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			 fold_bal (app conj) 
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			          (map mk_rec_imp (rec_tms~~preds)))))
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and mutual_induct_concl = mk_Trueprop(fold_bal (app 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 (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, refl] 1
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	     ORELSE dresolve_tac [spec, mp, splitD] 1)])
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    handle e => print_sign_exn sign e;
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(*Mutual induction follows by freeness of Inl/Inr.*)
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(*Removes Collects caused by M-operators in the intro rules*)
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val cmonos = [subset_refl RS Int_Collect_mono] RL 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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	  ((*unpackage and use "prem" in the corresponding place*)
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	   REPEAT (FIRSTGOAL
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		   (etac conjE ORELSE' eq_mp_tac ORELSE' 
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		    ares_tac [impI, conjI]))
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	   (*prem is not allowed in the REPEAT, lest it loop!*)
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	   THEN TRYALL (rtac prem)
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	   THEN REPEAT
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		  (FIRSTGOAL (ares_tac [impI] ORELSE' 
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			      eresolve_tac (mp::cmonos)))
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	   (*prove remaining goals by contradiction*)
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	   THEN rewrite_goals_tac (con_defs@part_rec_defs)
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	   THEN DEPTH_SOLVE (eresolve_tac (PartE :: sumprod_free_SEs) 1))
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	  i)
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	THEN mutual_ind_tac prems (i-1);
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val mutual_induct_split = 
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    prove_goalw_cterm []
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	  (cterm_of sign
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	   (list_implies (map (induct_prem (rec_tms~~preds)) 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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    handle e => print_sign_exn sign e;
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(*Attempts to remove all occurrences of split*)
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val split_tac =
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    REPEAT (SOMEGOAL (FIRST' [rtac splitI, 
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			      dtac splitD,
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			      etac splitE,
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			      bound_hyp_subst_tac]))
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    THEN prune_params_tac;
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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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val mutual_induct = rule_by_tactic split_tac mutual_induct_split;
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end;