(* Title: HOL/add_ind_def.ML
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1994 University of Cambridge
Fixedpoint definition module -- for Inductive/Coinductive Definitions
Features:
* least or greatest fixedpoints
* user-specified product and sum constructions
* mutually recursive definitions
* definitions involving arbitrary monotone operators
* automatically proves introduction and elimination rules
The recursive sets must *already* be declared as constants in parent theory!
Introduction rules have the form
[| ti:M(Sj), ..., P(x), ... |] ==> t: Sk |]
where M is some monotone operator (usually the identity)
P(x) is any (non-conjunctive) side condition on the free variables
ti, t are any terms
Sj, Sk are two of the sets being defined in mutual recursion
Sums are used only for mutual recursion;
Products are used only to derive "streamlined" induction rules for relations
Nestings of disjoint sum types:
(a+(b+c)) for 3, ((a+b)+(c+d)) for 4, ((a+b)+(c+(d+e))) for 5,
((a+(b+c))+(d+(e+f))) for 6
*)
signature FP = (** Description of a fixed point operator **)
sig
val oper : string * typ * term -> term (*fixed point operator*)
val Tarski : thm (*Tarski's fixed point theorem*)
val induct : thm (*induction/coinduction rule*)
end;
signature ADD_INDUCTIVE_DEF =
sig
val add_fp_def_i : term list * term list -> theory -> theory
end;
(*Declares functions to add fixedpoint/constructor defs to a theory*)
functor Add_inductive_def_Fun (Fp: FP) : ADD_INDUCTIVE_DEF =
struct
open Ind_Syntax;
(*internal version*)
fun add_fp_def_i (rec_tms, intr_tms) thy =
let
val sign = sign_of thy;
(*rec_params should agree for all mutually recursive components*)
val rec_hds = map head_of rec_tms;
val _ = assert_all is_Const rec_hds
(fn t => "Recursive set not previously declared as constant: " ^
Sign.string_of_term sign t);
(*Now we know they are all Consts, so get their names, type and params*)
val rec_names = map (#1 o dest_Const) rec_hds
and (Const(_,recT),rec_params) = strip_comb (hd rec_tms);
val _ = assert_all Syntax.is_identifier rec_names
(fn a => "Name of recursive set not an identifier: " ^ a);
local (*Checking the introduction rules*)
val intr_sets = map (#2 o rule_concl_msg sign) intr_tms;
fun intr_ok set =
case head_of set of Const(a,_) => a mem rec_names | _ => false;
in
val _ = assert_all intr_ok intr_sets
(fn t => "Conclusion of rule does not name a recursive set: " ^
Sign.string_of_term sign t);
end;
val _ = assert_all is_Free rec_params
(fn t => "Param in recursion term not a free variable: " ^
Sign.string_of_term sign t);
(*** Construct the lfp definition ***)
val mk_variant = variant (foldr add_term_names (intr_tms,[]));
val z = mk_variant"z" and X = mk_variant"X" and w = mk_variant"w";
(*Mutual recursion ?? *)
val domTs = summands (dest_setT (body_type recT));
(*alternative defn: map (dest_setT o fastype_of) rec_tms *)
val dom_sumT = fold_bal mk_sum domTs;
val dom_set = mk_setT dom_sumT;
val freez = Free(z, dom_sumT)
and freeX = Free(X, dom_set);
(*type of w may be any of the domTs*)
fun dest_tprop (Const("Trueprop",_) $ P) = P
| dest_tprop Q = error ("Ill-formed premise of introduction rule: " ^
Sign.string_of_term sign Q);
(*Makes a disjunct from an introduction rule*)
fun lfp_part intr = (*quantify over rule's free vars except parameters*)
let val prems = map dest_tprop (Logic.strip_imp_prems intr)
val _ = seq (fn rec_hd => seq (chk_prem rec_hd) prems) rec_hds
val exfrees = term_frees intr \\ rec_params
val zeq = eq_const dom_sumT $ freez $ (#1 (rule_concl intr))
in foldr mk_exists (exfrees, fold_bal (app conj) (zeq::prems)) end;
(*The Part(A,h) terms -- compose injections to make h*)
fun mk_Part (Bound 0, _) = freeX (*no mutual rec, no Part needed*)
| mk_Part (h, domT) =
let val goodh = mend_sum_types (h, dom_sumT)
and Part_const =
Const("Part", [dom_set, domT-->dom_sumT]---> dom_set)
in Part_const $ freeX $ Abs(w,domT,goodh) end;
(*Access to balanced disjoint sums via injections*)
val parts = map mk_Part
(accesses_bal (ap Inl, ap Inr, Bound 0) (length domTs) ~~
domTs);
(*replace each set by the corresponding Part(A,h)*)
val part_intrs = map (subst_free (rec_tms ~~ parts) o lfp_part) intr_tms;
val lfp_rhs = Fp.oper(X, dom_sumT,
mk_Collect(z, dom_sumT,
fold_bal (app disj) part_intrs))
(*** Make the new theory ***)
(*A key definition:
If no mutual recursion then it equals the one recursive set.
If mutual recursion then it differs from all the recursive sets. *)
val big_rec_name = space_implode "_" rec_names;
(*Big_rec... is the union of the mutually recursive sets*)
val big_rec_tm = list_comb(Const(big_rec_name,recT), rec_params);
(*The individual sets must already be declared*)
val axpairs = map mk_defpair
((big_rec_tm, lfp_rhs) ::
(case parts of
[_] => [] (*no mutual recursion*)
| _ => rec_tms ~~ (*define the sets as Parts*)
map (subst_atomic [(freeX, big_rec_tm)]) parts));
val _ = seq (writeln o Sign.string_of_term sign o #2) axpairs
(*Detect occurrences of operator, even with other types!*)
val _ = (case rec_names inter (add_term_names (lfp_rhs,[])) of
[] => ()
| x::_ => error ("Illegal occurrence of recursion op: " ^ x ^
"\n\t*Consider adding type constraints*"))
in thy |> add_defs_i axpairs end
(****************************************************************OMITTED
(*Expects the recursive sets to have been defined already.
con_ty_lists specifies the constructors in the form (name,prems,mixfix) *)
fun add_constructs_def (rec_names, con_ty_lists) thy =
* let
* val _ = writeln" Defining the constructor functions...";
* val case_name = "f"; (*name for case variables*)
* (** Define the constructors **)
* (*The empty tuple is 0*)
* fun mk_tuple [] = Const("0",iT)
* | mk_tuple args = foldr1 mk_Pair args;
* fun mk_inject n k u = access_bal(ap Inl, ap Inr, u) n k;
* val npart = length rec_names; (*total # of mutually recursive parts*)
* (*Make constructor definition; kpart is # of this mutually recursive part*)
* fun mk_con_defs (kpart, con_ty_list) =
* let val ncon = length con_ty_list (*number of constructors*)
fun mk_def (((id,T,syn), name, args, prems), kcon) =
(*kcon is index of constructor*)
mk_defpair (list_comb (Const(name,T), args),
mk_inject npart kpart
(mk_inject ncon kcon (mk_tuple args)))
* in map mk_def (con_ty_list ~~ (1 upto ncon)) end;
* (** Define the case operator **)
* (*Combine split terms using case; yields the case operator for one part*)
* fun call_case case_list =
* let fun call_f (free,args) =
ap_split T free (map (#2 o dest_Free) args)
* in fold_bal (app sum_case) (map call_f case_list) end;
* (** Generating function variables for the case definition
Non-identifiers (e.g. infixes) get a name of the form f_op_nnn. **)
* (*Treatment of a single constructor*)
* fun add_case (((id,T,syn), name, args, prems), (opno,cases)) =
if Syntax.is_identifier id
then (opno,
(Free(case_name ^ "_" ^ id, T), args) :: cases)
else (opno+1,
(Free(case_name ^ "_op_" ^ string_of_int opno, T), args) ::
cases)
* (*Treatment of a list of constructors, for one part*)
* fun add_case_list (con_ty_list, (opno,case_lists)) =
let val (opno',case_list) = foldr add_case (con_ty_list, (opno,[]))
in (opno', case_list :: case_lists) end;
* (*Treatment of all parts*)
* val (_, case_lists) = foldr add_case_list (con_ty_lists, (1,[]));
* val big_case_typ = flat (map (map (#2 o #1)) con_ty_lists) ---> (iT-->iT);
* val big_rec_name = space_implode "_" rec_names;
* val big_case_name = big_rec_name ^ "_case";
* (*The list of all the function variables*)
* val big_case_args = flat (map (map #1) case_lists);
* val big_case_tm =
list_comb (Const(big_case_name, big_case_typ), big_case_args);
* val big_case_def = mk_defpair
(big_case_tm, fold_bal (app sum_case) (map call_case case_lists));
* (** Build the new theory **)
* val const_decs =
(big_case_name, big_case_typ, NoSyn) :: map #1 (flat con_ty_lists);
* val axpairs =
big_case_def :: flat (map mk_con_defs ((1 upto npart) ~~ con_ty_lists))
* in thy |> add_consts_i const_decs |> add_defs_i axpairs end;
****************************************************************)
end;