(* Title: ZF/constructor.ML
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
Constructor function module -- for Datatype Definitions
Defines constructors and a case-style eliminator (no primitive recursion)
Features:
* least or greatest fixedpoints
* user-specified product and sum constructions
* mutually recursive datatypes
* recursion over arbitrary monotone operators
* flexible: can derive any reasonable set of introduction rules
* automatically constructs a case analysis operator (but no recursion op)
* efficient treatment of large declarations (e.g. 60 constructors)
*)
(** STILL NEEDS: some treatment of recursion **)
signature CONSTRUCTOR =
sig
val thy : theory (*parent theory*)
val rec_specs : (string * string * (string list * string * mixfix)list) list
(*recursion ops, types, domains, constructors*)
val rec_styp : string (*common type of all recursion ops*)
val sintrs : string list (*desired introduction rules*)
val monos : thm list (*monotonicity of each M operator*)
val type_intrs : thm list (*type-checking intro rules*)
val type_elims : thm list (*type-checking elim rules*)
end;
signature CONSTRUCTOR_RESULT =
sig
val con_thy : theory (*theory defining the constructors*)
val con_defs : thm list (*definitions made in con_thy*)
val case_eqns : thm list (*equations for case operator*)
val free_iffs : thm list (*freeness rewrite rules*)
val free_SEs : thm list (*freeness destruct rules*)
val mk_free : string -> thm (*makes freeness theorems*)
end;
functor Constructor_Fun (structure Const: CONSTRUCTOR and
Pr : PR and Su : SU) : CONSTRUCTOR_RESULT =
struct
open Logic Const;
val dummy = writeln"Defining the constructor functions...";
val case_name = "f"; (*name for case variables*)
(** Extract basic information from arguments **)
val sign = sign_of thy;
val rdty = typ_of o read_ctyp sign;
val rec_names = map #1 rec_specs;
val dummy = assert_all Syntax.is_identifier rec_names
(fn a => "Name of recursive set not an identifier: " ^ a);
(*Expands multiple constant declarations*)
fun flatten_consts ((names, typ, mfix) :: cs) =
let fun mk_const name = (name, typ, mfix)
in (map mk_const names) @ (flatten_consts cs) end
| flatten_consts [] = [];
(*Constructors with types and arguments*)
fun mk_con_ty_list cons_pairs =
let fun mk_con_ty (id, st, syn) =
let val T = rdty st;
val args = mk_frees "xa" (binder_types T);
val name = const_name id syn;
(* because of mixfix annotations the internal name
can be different from 'id' *)
in (name, T, args) end;
fun pairtypes c = flatten_consts [c];
in map mk_con_ty (flat (map pairtypes cons_pairs)) end;
val con_ty_lists = map (mk_con_ty_list o #3) rec_specs;
(** Define the constructors **)
(*We identify 0 (the empty set) with the empty tuple*)
fun mk_tuple [] = Const("0",iT)
| mk_tuple args = foldr1 (app Pr.pair) args;
fun mk_inject n k u = access_bal(ap Su.inl, ap Su.inr, u) n k;
val npart = length rec_names; (*number of mutually recursive parts*)
(*Make constructor definition*)
fun mk_con_defs (kpart, con_ty_list) =
let val ncon = length con_ty_list (*number of constructors*)
fun mk_def ((a,T,args), kcon) = (*kcon = index of this constructor*)
mk_defpair sign
(list_comb (Const(a,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 Pr.split_const free (map (#2 o dest_Free) args)
in fold_bal (app Su.elim) (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 ((a,T,args), (opno,cases)) =
if Syntax.is_identifier a
then (opno,
(Free(case_name ^ "_" ^ a, 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) 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 sign
(big_case_tm, fold_bal (app Su.elim) (map call_case case_lists));
(** Build the new theory **)
val axpairs =
big_case_def :: flat (map mk_con_defs ((1 upto npart) ~~ con_ty_lists));
val const_decs = flatten_consts
(([big_case_name], flatten_typ sign big_case_typ, NoSyn) ::
(big_rec_name ins rec_names, rec_styp, NoSyn) ::
flat (map #3 rec_specs));
val con_thy = thy
|> add_consts const_decs
|> add_axioms_i axpairs
|> add_thyname (big_rec_name ^ "_Constructors");
(*1st element is the case definition; others are the constructors*)
val con_defs = map (get_axiom con_thy o #1) axpairs;
(** Prove the case theorem **)
(*Each equation has the form
rec_case(f_con1,...,f_conn)(coni(args)) = f_coni(args) *)
fun mk_case_equation ((a,T,args), case_free) =
mk_tprop
(eq_const $ (big_case_tm $ (list_comb (Const(a,T), args)))
$ (list_comb (case_free, args)));
val case_trans = hd con_defs RS def_trans;
(*proves a single case equation*)
fun case_tacsf con_def _ =
[rewtac con_def,
rtac case_trans 1,
REPEAT (resolve_tac [refl, Pr.split_eq RS trans,
Su.case_inl RS trans,
Su.case_inr RS trans] 1)];
fun prove_case_equation (arg,con_def) =
prove_term (sign_of con_thy) []
(mk_case_equation arg, case_tacsf con_def);
val free_iffs =
map standard (con_defs RL [def_swap_iff]) @
[Su.distinct, Su.distinct', Su.inl_iff, Su.inr_iff, Pr.pair_iff];
val free_SEs = map (gen_make_elim [conjE,FalseE]) (free_iffs RL [iffD1]);
val free_cs = ZF_cs addSEs free_SEs;
(*Typical theorems have the form ~con1=con2, con1=con2==>False,
con1(x)=con1(y) ==> x=y, con1(x)=con1(y) <-> x=y, etc. *)
fun mk_free s =
prove_goalw con_thy con_defs s
(fn prems => [cut_facts_tac prems 1, fast_tac free_cs 1]);
val case_eqns = map prove_case_equation
(flat con_ty_lists ~~ big_case_args ~~ tl con_defs);
end;