ZF/indrule/mk_pred_typ: corrected pattern to include Abs, allowing it to
handle any number of right-nested Sigmas.
(* Title: ZF/indrule.ML
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1993 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 Ind: INDUCTIVE and
Pr: PR and Intr_elim: INTR_ELIM) : INDRULE =
struct
open Logic Ind Intr_elim;
val dummy = writeln "Proving the induction rules...";
(*** Prove the main induction rule ***)
val pred_name = "P"; (*name for predicate variables*)
val prove = prove_term (sign_of Intr_elim.thy);
val big_rec_def::part_rec_defs = Intr_elim.defs;
(*Used to make induction rules;
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_tprop (pred $ t) :: iprems
| None => (*possibly membership in M(rec_tm), for M monotone*)
let fun mk_sb (rec_tm,pred) = (rec_tm, Collect_const$rec_tm$pred)
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_tprop (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 = REPEAT (ares_tac [Part_eqI,prem] i) THEN
ind_tac prems (i-1);
val pred = Free(pred_name, iT-->oT);
val ind_prems = map (induct_prem (map (rpair pred) rec_tms)) intr_tms;
val quant_induct =
prove part_rec_defs
(list_implies (ind_prems, mk_tprop (mk_all_imp(big_rec_tm,pred))),
fn prems =>
[rtac (impI RS allI) 1,
etac raw_induct 1,
REPEAT (FIRSTGOAL (eresolve_tac [CollectE,exE,conjE,disjE,ssubst])),
REPEAT (FIRSTGOAL (eresolve_tac [PartE,CollectE])),
ind_tac (rev prems) (length prems) ]);
(*** Prove the simultaneous induction rule ***)
(*Make distinct predicates for each inductive set*)
(*Sigmas and Cartesian products may nest ONLY to the right!*)
fun mk_pred_typ (t $ A $ Abs(_,_,B)) =
if t = Pr.sigma then iT --> mk_pred_typ B
else iT --> oT
| mk_pred_typ _ = iT --> oT
(*Given a recursive set and its domain, return the "fsplit" 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 = mk_pred_typ domt
val pfree = Free(pred_name ^ "_" ^ rec_name, T)
val frees = mk_frees "za" (binder_types T)
val qconcl =
foldr mk_all (frees,
imp $ (mem_const $ foldr1 (app Pr.pair) frees $ rec_tm)
$ (list_comb (pfree,frees)))
in (ap_split Pr.fsplit_const 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 $ (mem_const $ Bound 0 $ rec_tm) $ (pred $ Bound 0);
(*To instantiate the main induction rule*)
val induct_concl =
mk_tprop(mk_all_imp(big_rec_tm,
Abs("z", iT,
fold_bal (app conj)
(map mk_rec_imp (rec_tms~~preds)))))
and mutual_induct_concl = mk_tprop(fold_bal (app conj) qconcls);
val lemma = (*makes the link between the two induction rules*)
prove part_rec_defs
(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] 1
ORELSE dresolve_tac [spec, mp, Pr.fsplitD] 1)]);
(*Mutual induction follows by freeness of Inl/Inr.*)
(*Removes Collects caused by M-operators in the intro rules*)
val cmonos = [subset_refl RS 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 =
SELECT_GOAL
((*unpackage and use "prem" in the corresponding place*)
REPEAT (FIRSTGOAL
(eresolve_tac ([conjE,mp]@cmonos) ORELSE'
ares_tac [prem,impI,conjI]))
(*prove remaining goals by contradiction*)
THEN rewrite_goals_tac (con_defs@part_rec_defs)
THEN REPEAT (eresolve_tac (PartE :: sumprod_free_SEs) 1))
i THEN mutual_ind_tac prems (i-1);
val mutual_induct_fsplit =
prove []
(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)]);
(*Attempts to remove all occurrences of fsplit*)
val fsplit_tac =
REPEAT (SOMEGOAL (FIRST' [rtac Pr.fsplitI,
dtac Pr.fsplitD,
etac Pr.fsplitE,
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 fsplit_tac mutual_induct_fsplit;
end;