Deleted extra space in clos_mk.
(* Title: HOL/indrule.ML
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1994 University of Cambridge
Induction rule module -- for Inductive/Coinductive Definitions
Proves a strong induction rule and a mutual induction rule
*)
signature INDRULE =
sig
val induct : thm (*main induction rule*)
val mutual_induct : thm (*mutual induction rule*)
end;
functor Indrule_Fun
(structure Inductive: sig include INDUCTIVE_ARG INDUCTIVE_I end and
Intr_elim: INTR_ELIM) : INDRULE =
struct
open Logic Ind_Syntax Inductive Intr_elim;
val sign = sign_of thy;
val (Const(_,recT),rec_params) = strip_comb (hd rec_tms);
val elem_type = dest_setT (body_type recT);
val domTs = summands(elem_type);
val big_rec_name = space_implode "_" rec_names;
val big_rec_tm = list_comb(Const(big_rec_name,recT), rec_params);
val _ = writeln " Proving the induction rules...";
(*** Prove the main induction rule ***)
val pred_name = "P"; (*name for predicate variables*)
val big_rec_def::part_rec_defs = Intr_elim.defs;
(*Used to express induction rules: adds induction hypotheses.
ind_alist = [(rec_tm1,pred1),...] -- associates predicates with rec ops
prem is a premise of an intr rule*)
fun add_induct_prem ind_alist (prem as Const("Trueprop",_) $
(Const("op :",_)$t$X), iprems) =
(case gen_assoc (op aconv) (ind_alist, X) of
Some pred => prem :: mk_Trueprop (pred $ t) :: iprems
| None => (*possibly membership in M(rec_tm), for M monotone*)
let fun mk_sb (rec_tm,pred) =
(case binder_types (fastype_of pred) of
[T] => (rec_tm,
Int_const T $ rec_tm $ (Collect_const T $ pred))
| _ => error
"Bug: add_induct_prem called with non-unary predicate")
in subst_free (map mk_sb ind_alist) prem :: iprems end)
| add_induct_prem ind_alist (prem,iprems) = prem :: iprems;
(*Make a premise of the induction rule.*)
fun induct_prem ind_alist intr =
let val quantfrees = map dest_Free (term_frees intr \\ rec_params)
val iprems = foldr (add_induct_prem ind_alist)
(strip_imp_prems intr,[])
val (t,X) = rule_concl intr
val (Some pred) = gen_assoc (op aconv) (ind_alist, X)
val concl = mk_Trueprop (pred $ t)
in list_all_free (quantfrees, list_implies (iprems,concl)) end
handle Bind => error"Recursion term not found in conclusion";
(*Avoids backtracking by delivering the correct premise to each goal*)
fun ind_tac [] 0 = all_tac
| ind_tac(prem::prems) i =
DEPTH_SOLVE_1 (ares_tac [Part_eqI, prem, refl] i) THEN
ind_tac prems (i-1);
val pred = Free(pred_name, elem_type --> boolT);
val ind_prems = map (induct_prem (map (rpair pred) rec_tms)) intr_tms;
val quant_induct =
prove_goalw_cterm part_rec_defs
(cterm_of sign (list_implies (ind_prems,
mk_Trueprop (mk_all_imp(big_rec_tm,pred)))))
(fn prems =>
[rtac (impI RS allI) 1,
etac raw_induct 1,
REPEAT (FIRSTGOAL (eresolve_tac [IntE, CollectE, exE, conjE, disjE]
ORELSE' hyp_subst_tac)),
REPEAT (FIRSTGOAL (eresolve_tac [PartE, CollectE])),
ind_tac (rev prems) (length prems)])
handle e => print_sign_exn sign e;
(*** Prove the simultaneous induction rule ***)
(*Make distinct predicates for each inductive set.
Splits cartesian products in domT, IF nested to the right! *)
(*Given a recursive set and its domain, return the "split" predicate
and a conclusion for the simultaneous induction rule*)
fun mk_predpair (rec_tm,domT) =
let val rec_name = (#1 o dest_Const o head_of) rec_tm
val T = factors domT ---> boolT
val pfree = Free(pred_name ^ "_" ^ rec_name, T)
val frees = mk_frees "za" (binder_types T)
val qconcl =
foldr mk_all (frees,
imp $ (mk_mem (foldr1 mk_Pair frees, rec_tm))
$ (list_comb (pfree,frees)))
in (ap_split boolT pfree (binder_types T),
qconcl)
end;
val (preds,qconcls) = split_list (map mk_predpair (rec_tms~~domTs));
(*Used to form simultaneous induction lemma*)
fun mk_rec_imp (rec_tm,pred) =
imp $ (mk_mem (Bound 0, rec_tm)) $ (pred $ Bound 0);
(*To instantiate the main induction rule*)
val induct_concl =
mk_Trueprop(mk_all_imp(big_rec_tm,
Abs("z", elem_type,
fold_bal (app conj)
(map mk_rec_imp (rec_tms~~preds)))))
and mutual_induct_concl = mk_Trueprop(fold_bal (app conj) qconcls);
val lemma = (*makes the link between the two induction rules*)
prove_goalw_cterm part_rec_defs
(cterm_of sign (mk_implies (induct_concl,mutual_induct_concl)))
(fn prems =>
[cut_facts_tac prems 1,
REPEAT (eresolve_tac [asm_rl, conjE, PartE, mp] 1
ORELSE resolve_tac [allI, impI, conjI, Part_eqI, refl] 1
ORELSE dresolve_tac [spec, mp, splitD] 1)])
handle e => print_sign_exn sign e;
(*Mutual induction follows by freeness of Inl/Inr.*)
(*Removes Collects caused by M-operators in the intro rules*)
val cmonos = [subset_refl RS Int_Collect_mono] RL monos RLN (2,[rev_subsetD]);
(*Avoids backtracking by delivering the correct premise to each goal*)
fun mutual_ind_tac [] 0 = all_tac
| mutual_ind_tac(prem::prems) i =
DETERM
(SELECT_GOAL
((*unpackage and use "prem" in the corresponding place*)
REPEAT (FIRSTGOAL
(etac conjE ORELSE' eq_mp_tac ORELSE'
ares_tac [impI, conjI]))
(*prem is not allowed in the REPEAT, lest it loop!*)
THEN TRYALL (rtac prem)
THEN REPEAT
(FIRSTGOAL (ares_tac [impI] ORELSE'
eresolve_tac (mp::cmonos)))
(*prove remaining goals by contradiction*)
THEN rewrite_goals_tac (con_defs@part_rec_defs)
THEN DEPTH_SOLVE (eresolve_tac (PartE :: sumprod_free_SEs) 1))
i)
THEN mutual_ind_tac prems (i-1);
val mutual_induct_split =
prove_goalw_cterm []
(cterm_of sign
(list_implies (map (induct_prem (rec_tms~~preds)) intr_tms,
mutual_induct_concl)))
(fn prems =>
[rtac (quant_induct RS lemma) 1,
mutual_ind_tac (rev prems) (length prems)])
handle e => print_sign_exn sign e;
(*Attempts to remove all occurrences of split*)
val split_tac =
REPEAT (SOMEGOAL (FIRST' [rtac splitI,
dtac splitD,
etac splitE,
bound_hyp_subst_tac]))
THEN prune_params_tac;
(*strip quantifier*)
val induct = standard (quant_induct RS spec RSN (2,rev_mp));
val mutual_induct = rule_by_tactic split_tac mutual_induct_split;
end;