(* Title: HOL/Tools/Function/size.ML
Author: Stefan Berghofer, Florian Haftmann & Alexander Krauss, TU Muenchen
Size functions for datatypes.
*)
signature SIZE =
sig
val size_thms: theory -> string -> thm list
val setup: theory -> theory
end;
structure Size: SIZE =
struct
open Datatype_Aux;
structure SizeData = Theory_Data
(
type T = (string * thm list) Symtab.table;
val empty = Symtab.empty;
val extend = I
fun merge data = Symtab.merge (K true) data;
);
val lookup_size = SizeData.get #> Symtab.lookup;
fun plus (t1, t2) = Const (@{const_name Groups.plus},
HOLogic.natT --> HOLogic.natT --> HOLogic.natT) $ t1 $ t2;
fun size_of_type f g h (T as Type (s, Ts)) =
(case f s of
SOME t => SOME t
| NONE => (case g s of
SOME size_name =>
SOME (list_comb (Const (size_name,
map (fn U => U --> HOLogic.natT) Ts @ [T] ---> HOLogic.natT),
map (size_of_type' f g h) Ts))
| NONE => NONE))
| size_of_type f g h (TFree (s, _)) = h s
and size_of_type' f g h T = (case size_of_type f g h T of
NONE => Abs ("x", T, HOLogic.zero)
| SOME t => t);
fun is_poly thy (DtType (name, dts)) =
(case Datatype.get_info thy name of
NONE => false
| SOME _ => exists (is_poly thy) dts)
| is_poly _ _ = true;
fun constrs_of thy name =
let
val {descr, index, ...} = Datatype.the_info thy name
val SOME (_, _, constrs) = AList.lookup op = descr index
in constrs end;
val app = curry (list_comb o swap);
fun prove_size_thms (info : info) new_type_names thy =
let
val {descr, alt_names, sorts, rec_names, rec_rewrites, induct, ...} = info;
val l = length new_type_names;
val alt_names' = (case alt_names of
NONE => replicate l NONE | SOME names => map SOME names);
val descr' = List.take (descr, l);
val (rec_names1, rec_names2) = chop l rec_names;
val recTs = get_rec_types descr sorts;
val (recTs1, recTs2) = chop l recTs;
val (_, (_, paramdts, _)) :: _ = descr;
val paramTs = map (typ_of_dtyp descr sorts) paramdts;
val ((param_size_fs, param_size_fTs), f_names) = paramTs |>
map (fn T as TFree (s, _) =>
let
val name = "f" ^ implode (tl (explode s));
val U = T --> HOLogic.natT
in
(((s, Free (name, U)), U), name)
end) |> split_list |>> split_list;
val param_size = AList.lookup op = param_size_fs;
val extra_rewrites = descr |> map (#1 o snd) |> distinct op = |>
map_filter (Option.map snd o lookup_size thy) |> flat;
val extra_size = Option.map fst o lookup_size thy;
val (((size_names, size_fns), def_names), def_names') =
recTs1 ~~ alt_names' |>
map (fn (T as Type (s, _), optname) =>
let
val s' = the_default (Long_Name.base_name s) optname ^ "_size";
val s'' = Sign.full_bname thy s'
in
(s'',
(list_comb (Const (s'', param_size_fTs @ [T] ---> HOLogic.natT),
map snd param_size_fs),
(s' ^ "_def", s' ^ "_overloaded_def")))
end) |> split_list ||>> split_list ||>> split_list;
val overloaded_size_fns = map HOLogic.size_const recTs1;
(* instantiation for primrec combinator *)
fun size_of_constr b size_ofp ((_, cargs), (_, cargs')) =
let
val Ts = map (typ_of_dtyp descr sorts) cargs;
val k = length (filter is_rec_type cargs);
val (ts, _, _) = fold_rev (fn ((dt, dt'), T) => fn (us, i, j) =>
if is_rec_type dt then (Bound i :: us, i + 1, j + 1)
else
(if b andalso is_poly thy dt' then
case size_of_type (K NONE) extra_size size_ofp T of
NONE => us | SOME sz => sz $ Bound j :: us
else us, i, j + 1))
(cargs ~~ cargs' ~~ Ts) ([], 0, k);
val t =
if null ts andalso (not b orelse not (exists (is_poly thy) cargs'))
then HOLogic.zero
else foldl1 plus (ts @ [HOLogic.Suc_zero])
in
fold_rev (fn T => fn t' => Abs ("x", T, t')) (Ts @ replicate k HOLogic.natT) t
end;
val fs = maps (fn (_, (name, _, constrs)) =>
map (size_of_constr true param_size) (constrs ~~ constrs_of thy name)) descr;
val fs' = maps (fn (n, (name, _, constrs)) =>
map (size_of_constr (l <= n) (K NONE)) (constrs ~~ constrs_of thy name)) descr;
val fTs = map fastype_of fs;
val (rec_combs1, rec_combs2) = chop l (map (fn (T, rec_name) =>
Const (rec_name, fTs @ [T] ---> HOLogic.natT))
(recTs ~~ rec_names));
fun define_overloaded (def_name, eq) lthy =
let
val (Free (c, _), rhs) = (Logic.dest_equals o Syntax.check_term lthy) eq;
val (thm, lthy') = lthy
|> Local_Theory.define ((Binding.name c, NoSyn), ((Binding.name def_name, []), rhs))
|-> (fn (t, (_, thm)) => Spec_Rules.add Spec_Rules.Equational ([t], [thm]) #> pair thm);
val ctxt_thy = ProofContext.init_global (ProofContext.theory_of lthy');
val thm' = singleton (ProofContext.export lthy' ctxt_thy) thm;
in (thm', lthy') end;
val ((size_def_thms, size_def_thms'), thy') =
thy
|> Sign.add_consts_i (map (fn (s, T) =>
(Binding.name (Long_Name.base_name s), param_size_fTs @ [T] ---> HOLogic.natT, NoSyn))
(size_names ~~ recTs1))
|> PureThy.add_defs false
(map (Thm.no_attributes o apsnd (Logic.mk_equals o apsnd (app fs)))
(map Binding.name def_names ~~ (size_fns ~~ rec_combs1)))
||> Class.instantiation
(map (#1 o snd) descr', map dest_TFree paramTs, [HOLogic.class_size])
||>> fold_map define_overloaded
(def_names' ~~ map Logic.mk_equals (overloaded_size_fns ~~ map (app fs') rec_combs1))
||> Class.prove_instantiation_instance (K (Class.intro_classes_tac []))
||> Local_Theory.exit_global;
val ctxt = ProofContext.init_global thy';
val simpset1 = HOL_basic_ss addsimps @{thm Nat.add_0} :: @{thm Nat.add_0_right} ::
size_def_thms @ size_def_thms' @ rec_rewrites @ extra_rewrites;
val xs = map (fn i => "x" ^ string_of_int i) (1 upto length recTs2);
fun mk_unfolded_size_eq tab size_ofp fs (p as (x, T), r) =
HOLogic.mk_eq (app fs r $ Free p,
the (size_of_type tab extra_size size_ofp T) $ Free p);
fun prove_unfolded_size_eqs size_ofp fs =
if null recTs2 then []
else split_conj_thm (Skip_Proof.prove ctxt xs []
(HOLogic.mk_Trueprop (mk_conj (replicate l HOLogic.true_const @
map (mk_unfolded_size_eq (AList.lookup op =
(new_type_names ~~ map (app fs) rec_combs1)) size_ofp fs)
(xs ~~ recTs2 ~~ rec_combs2))))
(fn _ => (indtac induct xs THEN_ALL_NEW asm_simp_tac simpset1) 1));
val unfolded_size_eqs1 = prove_unfolded_size_eqs param_size fs;
val unfolded_size_eqs2 = prove_unfolded_size_eqs (K NONE) fs';
(* characteristic equations for size functions *)
fun gen_mk_size_eq p size_of size_ofp size_const T (cname, cargs) =
let
val Ts = map (typ_of_dtyp descr sorts) cargs;
val tnames = Name.variant_list f_names (Datatype_Prop.make_tnames Ts);
val ts = map_filter (fn (sT as (s, T), dt) =>
Option.map (fn sz => sz $ Free sT)
(if p dt then size_of_type size_of extra_size size_ofp T
else NONE)) (tnames ~~ Ts ~~ cargs)
in
HOLogic.mk_Trueprop (HOLogic.mk_eq
(size_const $ list_comb (Const (cname, Ts ---> T),
map2 (curry Free) tnames Ts),
if null ts then HOLogic.zero
else foldl1 plus (ts @ [HOLogic.Suc_zero])))
end;
val simpset2 = HOL_basic_ss addsimps
rec_rewrites @ size_def_thms @ unfolded_size_eqs1;
val simpset3 = HOL_basic_ss addsimps
rec_rewrites @ size_def_thms' @ unfolded_size_eqs2;
fun prove_size_eqs p size_fns size_ofp simpset =
maps (fn (((_, (_, _, constrs)), size_const), T) =>
map (fn constr => Drule.export_without_context (Skip_Proof.prove ctxt [] []
(gen_mk_size_eq p (AList.lookup op = (new_type_names ~~ size_fns))
size_ofp size_const T constr)
(fn _ => simp_tac simpset 1))) constrs)
(descr' ~~ size_fns ~~ recTs1);
val size_eqns = prove_size_eqs (is_poly thy') size_fns param_size simpset2 @
prove_size_eqs is_rec_type overloaded_size_fns (K NONE) simpset3;
val ([size_thms], thy'') = PureThy.add_thmss
[((Binding.name "size", size_eqns),
[Simplifier.simp_add, Nitpick_Simps.add,
Thm.declaration_attribute
(fn thm => Context.mapping (Code.add_default_eqn thm) I)])] thy'
in
SizeData.map (fold (Symtab.update_new o apsnd (rpair size_thms))
(new_type_names ~~ size_names)) thy''
end;
fun add_size_thms config (new_type_names as name :: _) thy =
let
val info as {descr, alt_names, ...} = Datatype.the_info thy name;
val prefix = Long_Name.map_base_name (K (space_implode "_"
(the_default (map Long_Name.base_name new_type_names) alt_names))) name;
val no_size = exists (fn (_, (_, _, constrs)) => exists (fn (_, cargs) => exists (fn dt =>
is_rec_type dt andalso not (null (fst (strip_dtyp dt)))) cargs) constrs) descr
in if no_size then thy
else
thy
|> Sign.root_path
|> Sign.add_path prefix
|> Theory.checkpoint
|> prove_size_thms info new_type_names
|> Sign.restore_naming thy
end;
val size_thms = snd oo (the oo lookup_size);
val setup = Datatype.interpretation add_size_thms;
end;