src/HOL/indrule.ML
 author paulson Tue, 07 May 1996 18:15:51 +0200 changeset 1728 01beef6262aa parent 1653 1a2ffa2fbf7d child 1746 f0c6aabc6c02 permissions -rw-r--r--
Unfolding of arbitrarily nested tuples in induction rules
```
(*  Title:      HOL/indrule.ML
ID:         \$Id\$
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

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

(*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;
*)

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),
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 = Ind_Syntax.factors elem_type;
val elem_frees = mk_frees "za" elem_factors;
val elem_tuple = Ind_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  (Ind_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)
(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 = simpset_of "Fun"
addsimps [Inl_Inr_eq, Inr_Inl_eq, 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
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, but don't accept new rewrite rules!*)
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
(Ind_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 Ind_Syntax.remove_split mutual_induct_split
else TrueI;
end
end;
```