(* 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: sig include INTR_ELIM INTR_ELIM_AUX end) : INDRULE =
let
val sign = sign_of Inductive.thy;
val (Const(_,recT),rec_params) = strip_comb (hd Inductive.rec_tms);
val elem_type = Ind_Syntax.dest_setT (body_type recT);
val big_rec_name = space_implode "_" Intr_elim.rec_names;
val big_rec_tm = list_comb(Const(big_rec_name,recT), rec_params);
val _ = writeln " Proving the induction rule...";
(*** 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 :: Ind_Syntax.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,
Ind_Syntax.Int_const T $ rec_tm $
(Ind_Syntax.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)
(Logic.strip_imp_prems intr,[])
val (t,X) = Ind_Syntax.rule_concl intr
val (Some pred) = gen_assoc (op aconv) (ind_alist, X)
val concl = Ind_Syntax.mk_Trueprop (pred $ t)
in list_all_free (quantfrees, Logic.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 --> Ind_Syntax.boolT);
val ind_prems = map (induct_prem (map (rpair pred) Inductive.rec_tms))
Inductive.intr_tms;
(*Debugging code...
val _ = writeln "ind_prems = ";
val _ = seq (writeln o Sign.string_of_term sign) ind_prems;
*)
(*We use a MINIMAL simpset because others (such as HOL_ss) contain too many
simplifications. If the premises get simplified, then the proofs will
fail. This arose with a premise of the form {(F n,G n)|n . True}, which
expanded to something containing ...&True. *)
val min_ss = HOL_basic_ss;
val quant_induct =
prove_goalw_cterm part_rec_defs
(cterm_of sign
(Logic.list_implies (ind_prems,
Ind_Syntax.mk_Trueprop (Ind_Syntax.mk_all_imp
(big_rec_tm,pred)))))
(fn prems =>
[rtac (impI RS allI) 1,
DETERM (etac Intr_elim.raw_induct 1),
full_simp_tac (min_ss addsimps [Part_Collect]) 1,
REPEAT (FIRSTGOAL (eresolve_tac [IntE, CollectE, exE, conjE, disjE]
ORELSE' hyp_subst_tac)),
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 elem_type, however nested*)
(*The components of the element type, several if it is a product*)
val elem_factors = Prod_Syntax.factors elem_type;
val elem_frees = mk_frees "za" elem_factors;
val elem_tuple = Prod_Syntax.mk_tuple elem_type elem_frees;
(*Given a recursive set, return the "split" predicate
and a conclusion for the simultaneous induction rule*)
fun mk_predpair rec_tm =
let val rec_name = (#1 o dest_Const o head_of) rec_tm
val pfree = Free(pred_name ^ "_" ^ rec_name,
elem_factors ---> Ind_Syntax.boolT)
val qconcl =
foldr Ind_Syntax.mk_all
(elem_frees,
Ind_Syntax.imp $ (Ind_Syntax.mk_mem (elem_tuple, rec_tm))
$ (list_comb (pfree, elem_frees)))
in (Prod_Syntax.ap_split elem_type Ind_Syntax.boolT pfree,
qconcl)
end;
val (preds,qconcls) = split_list (map mk_predpair Inductive.rec_tms);
(*Used to form simultaneous induction lemma*)
fun mk_rec_imp (rec_tm,pred) =
Ind_Syntax.imp $ (Ind_Syntax.mk_mem (Bound 0, rec_tm)) $ (pred $ Bound 0);
(*To instantiate the main induction rule*)
val induct_concl =
Ind_Syntax.mk_Trueprop
(Ind_Syntax.mk_all_imp
(big_rec_tm,
Abs("z", elem_type,
fold_bal (app Ind_Syntax.conj)
(ListPair.map mk_rec_imp (Inductive.rec_tms,preds)))))
and mutual_induct_concl =
Ind_Syntax.mk_Trueprop (fold_bal (app Ind_Syntax.conj) qconcls);
val lemma_tac = FIRST' [eresolve_tac [asm_rl, conjE, PartE, mp],
resolve_tac [allI, impI, conjI, Part_eqI, refl],
dresolve_tac [spec, mp, splitD]];
val lemma = (*makes the link between the two induction rules*)
prove_goalw_cterm part_rec_defs
(cterm_of sign (Logic.mk_implies (induct_concl,
mutual_induct_concl)))
(fn prems =>
[cut_facts_tac prems 1,
REPEAT (rewrite_goals_tac [split RS eq_reflection] THEN
lemma_tac 1)])
handle e => print_sign_exn sign e;
(*Mutual induction follows by freeness of Inl/Inr.*)
(*Simplification largely reduces the mutual induction rule to the
standard rule*)
val mut_ss = min_ss addsimps [Inl_not_Inr, Inr_not_Inl, Inl_eq, Inr_eq, split];
val all_defs = [split RS eq_reflection] @ Inductive.con_defs @ part_rec_defs;
(*Removes Collects caused by M-operators in the intro rules*)
val cmonos = [subset_refl RS Int_Collect_mono] RL Inductive.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
(
(*Simplify the assumptions and goal by unfolding Part and
using freeness of the Sum constructors; proves all but one
conjunct by contradiction*)
rewrite_goals_tac all_defs THEN
simp_tac (mut_ss addsimps [Part_def]) 1 THEN
IF_UNSOLVED (*simp_tac may have finished it off!*)
((*simplify assumptions*)
full_simp_tac mut_ss 1 THEN
(*unpackage and use "prem" in the corresponding place*)
REPEAT (rtac impI 1) THEN
rtac (rewrite_rule all_defs prem) 1 THEN
(*prem must not be REPEATed below: could loop!*)
DEPTH_SOLVE (FIRSTGOAL (ares_tac [impI] ORELSE'
eresolve_tac (conjE::mp::cmonos))))
) i)
THEN mutual_ind_tac prems (i-1);
val _ = writeln " Proving the mutual induction rule...";
val mutual_induct_split =
prove_goalw_cterm []
(cterm_of sign
(Logic.list_implies (map (induct_prem (Inductive.rec_tms ~~ preds))
Inductive.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;
(** Uncurrying the predicate in the ordinary induction rule **)
(*The name "x.1" comes from the "RS spec" !*)
val xvar = cterm_of sign (Var(("x",1), elem_type));
(*strip quantifier and instantiate the variable to a tuple*)
val induct0 = quant_induct RS spec RSN (2,rev_mp) |>
freezeT |> (*Because elem_type contains TFrees not TVars*)
instantiate ([], [(xvar, cterm_of sign elem_tuple)]);
in
struct
val induct = standard
(Prod_Syntax.split_rule_var
(Var((pred_name,2), elem_type --> Ind_Syntax.boolT),
induct0));
(*Just "True" unless there's true mutual recursion. This saves storage.*)
val mutual_induct =
if length Intr_elim.rec_names > 1
then Prod_Syntax.remove_split mutual_induct_split
else TrueI;
end
end;