author | paulson |
Thu, 12 Sep 1996 10:40:05 +0200 | |
changeset 1985 | 84cf16192e03 |
parent 1867 | 37615e73f2d8 |
child 2031 | 03a843f0f447 |
permissions | -rw-r--r-- |
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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: 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 elem_type = Ind_Syntax.dest_setT (body_type recT); |
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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 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 :: Ind_Syntax.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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Ind_Syntax.Int_const T $ rec_tm $ |
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(Ind_Syntax.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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(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_Trueprop (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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(*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 --> Ind_Syntax.boolT); |
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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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(*Debugging code... |
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val _ = writeln "ind_prems = "; |
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val _ = seq (writeln o Sign.string_of_term sign) ind_prems; |
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*) |
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(*We use a MINIMAL simpset because others (such as HOL_ss) contain too many |
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simplifications. If the premises get simplified, then the proofs will |
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fail. This arose with a premise of the form {(F n,G n)|n . True}, which |
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expanded to something containing ...&True. *) |
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val min_ss = empty_ss |
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setmksimps (mksimps mksimps_pairs) |
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setsolver (fn prems => resolve_tac (TrueI::refl::prems) ORELSE' atac |
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ORELSE' etac FalseE); |
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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_Trueprop (Ind_Syntax.mk_all_imp |
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(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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full_simp_tac (min_ss addsimps [Part_Collect]) 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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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 elem_type, however nested*) |
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(*The components of the element type, several if it is a product*) |
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val elem_factors = Prod_Syntax.factors elem_type; |
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val elem_frees = mk_frees "za" elem_factors; |
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val elem_tuple = Prod_Syntax.mk_tuple elem_type elem_frees; |
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(*Given a recursive set, 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 = |
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let val rec_name = (#1 o dest_Const o head_of) rec_tm |
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val pfree = Free(pred_name ^ "_" ^ rec_name, |
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elem_factors ---> Ind_Syntax.boolT) |
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val qconcl = |
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foldr Ind_Syntax.mk_all |
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(elem_frees, |
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Ind_Syntax.imp $ (Ind_Syntax.mk_mem (elem_tuple, rec_tm)) |
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$ (list_comb (pfree, elem_frees))) |
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in (Prod_Syntax.ap_split elem_type Ind_Syntax.boolT pfree, |
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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.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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Ind_Syntax.mk_Trueprop |
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(Ind_Syntax.mk_all_imp |
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(big_rec_tm, |
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Abs("z", elem_type, |
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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_Trueprop (fold_bal (app Ind_Syntax.conj) qconcls); |
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val lemma_tac = FIRST' [eresolve_tac [asm_rl, conjE, PartE, mp], |
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resolve_tac [allI, impI, conjI, Part_eqI, refl], |
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dresolve_tac [spec, mp, splitD]]; |
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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, |
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mutual_induct_concl))) |
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(fn prems => |
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[cut_facts_tac prems 1, |
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REPEAT (rewrite_goals_tac [split RS eq_reflection] THEN |
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lemma_tac 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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(*Simplification largely reduces the mutual induction rule to the |
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standard rule*) |
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val mut_ss = min_ss addsimps [Inl_not_Inr, Inr_not_Inl, Inl_eq, Inr_eq, split]; |
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val all_defs = [split RS eq_reflection] @ Inductive.con_defs @ part_rec_defs; |
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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 Inductive.monos RLN |
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(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_def]) 1 THEN |
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IF_UNSOLVED (*simp_tac may have finished it off!*) |
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((*simplify assumptions*) |
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full_simp_tac mut_ss 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_split = |
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prove_goalw_cterm [] |
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(cterm_of sign |
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(Logic.list_implies (map (induct_prem (Inductive.rec_tms ~~ preds)) |
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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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handle e => print_sign_exn sign e; |
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(** Uncurrying the predicate in the ordinary induction rule **) |
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(*The name "x.1" comes from the "RS spec" !*) |
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val xvar = cterm_of sign (Var(("x",1), elem_type)); |
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(*strip quantifier and instantiate the variable to a tuple*) |
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val induct0 = quant_induct RS spec RSN (2,rev_mp) |> |
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freezeT |> (*Because elem_type contains TFrees not TVars*) |
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instantiate ([], [(xvar, cterm_of sign elem_tuple)]); |
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in |
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struct |
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val induct = standard |
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(Prod_Syntax.split_rule_var |
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(Var((pred_name,2), elem_type --> Ind_Syntax.boolT), |
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induct0)); |
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|
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(*Just "True" unless there's true mutual recursion. This saves storage.*) |
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val mutual_induct = |
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if length Intr_elim.rec_names > 1 |
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Moved split_rule et al from ind_syntax.ML to Prod.ML.
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then Prod_Syntax.remove_split mutual_induct_split |
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else TrueI; |
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end |
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end; |