generate more theorems (e.g. for types with only one constructor)
(* Title: HOL/BNF/Tools/bnf_fp_rec_sugar.ML
Author: Lorenz Panny, TU Muenchen
Copyright 2013
Recursor and corecursor sugar.
*)
signature BNF_FP_REC_SUGAR =
sig
val add_primrec_cmd: (binding * string option * mixfix) list ->
(Attrib.binding * string) list -> local_theory -> local_theory;
val add_primcorec_cmd: bool ->
(binding * string option * mixfix) list * (Attrib.binding * string) list -> Proof.context ->
Proof.state
end;
structure BNF_FP_Rec_Sugar : BNF_FP_REC_SUGAR =
struct
open BNF_Util
open BNF_FP_Util
open BNF_FP_Rec_Sugar_Util
open BNF_FP_Rec_Sugar_Tactics
exception Primrec_Error of string * term list;
fun primrec_error str = raise Primrec_Error (str, []);
fun primrec_error_eqn str eqn = raise Primrec_Error (str, [eqn]);
fun primrec_error_eqns str eqns = raise Primrec_Error (str, eqns);
fun finds eq = fold_map (fn x => List.partition (curry eq x) #>> pair x);
val free_name = try (fn Free (v, _) => v);
val const_name = try (fn Const (v, _) => v);
val undef_const = Const (@{const_name undefined}, dummyT);
fun permute_args n t = list_comb (t, map Bound (0 :: (n downto 1)))
|> fold (K (Term.abs (Name.uu, dummyT))) (0 upto n);
val abs_tuple = HOLogic.tupled_lambda o HOLogic.mk_tuple;
fun drop_All t = subst_bounds (strip_qnt_vars @{const_name all} t |> map Free |> rev,
strip_qnt_body @{const_name all} t)
fun mk_not @{const True} = @{const False}
| mk_not @{const False} = @{const True}
| mk_not (@{const Not} $ t) = t
| mk_not (@{const Trueprop} $ t) = @{const Trueprop} $ mk_not t
| mk_not t = HOLogic.mk_not t
val mk_conjs = try (foldr1 HOLogic.mk_conj) #> the_default @{const True};
val mk_disjs = try (foldr1 HOLogic.mk_disj) #> the_default @{const False};
fun invert_prems [t] = map mk_not (HOLogic.disjuncts t)
| invert_prems ts = [mk_disjs (map mk_not ts)];
fun invert_prems_disj [t] = map mk_not (HOLogic.disjuncts t)
| invert_prems_disj ts = [mk_disjs (map (mk_conjs o map mk_not o HOLogic.disjuncts) ts)];
fun abstract vs =
let fun a n (t $ u) = a n t $ a n u
| a n (Abs (v, T, b)) = Abs (v, T, a (n + 1) b)
| a n t = let val idx = find_index (equal t) vs in
if idx < 0 then t else Bound (n + idx) end
in a 0 end;
val simp_attrs = @{attributes [simp]};
(* Primrec *)
type eqn_data = {
fun_name: string,
rec_type: typ,
ctr: term,
ctr_args: term list,
left_args: term list,
right_args: term list,
res_type: typ,
rhs_term: term,
user_eqn: term
};
fun dissect_eqn lthy fun_names eqn' =
let
val eqn = drop_All eqn' |> HOLogic.dest_Trueprop
handle TERM _ =>
primrec_error_eqn "malformed function equation (expected \"lhs = rhs\")" eqn';
val (lhs, rhs) = HOLogic.dest_eq eqn
handle TERM _ =>
primrec_error_eqn "malformed function equation (expected \"lhs = rhs\")" eqn';
val (fun_name, args) = strip_comb lhs
|>> (fn x => if is_Free x then fst (dest_Free x)
else primrec_error_eqn "malformed function equation (does not start with free)" eqn);
val (left_args, rest) = take_prefix is_Free args;
val (nonfrees, right_args) = take_suffix is_Free rest;
val _ = length nonfrees = 1 orelse if length nonfrees = 0 then
primrec_error_eqn "constructor pattern missing in left-hand side" eqn else
primrec_error_eqn "more than one non-variable argument in left-hand side" eqn;
val _ = member (op =) fun_names fun_name orelse
primrec_error_eqn "malformed function equation (does not start with function name)" eqn
val (ctr, ctr_args) = strip_comb (the_single nonfrees);
val _ = try (num_binder_types o fastype_of) ctr = SOME (length ctr_args) orelse
primrec_error_eqn "partially applied constructor in pattern" eqn;
val _ = let val d = duplicates (op =) (left_args @ ctr_args @ right_args) in null d orelse
primrec_error_eqn ("duplicate variable \"" ^ Syntax.string_of_term lthy (hd d) ^
"\" in left-hand side") eqn end;
val _ = forall is_Free ctr_args orelse
primrec_error_eqn "non-primitive pattern in left-hand side" eqn;
val _ =
let val b = fold_aterms (fn x as Free (v, _) =>
if (not (member (op =) (left_args @ ctr_args @ right_args) x) andalso
not (member (op =) fun_names v) andalso
not (Variable.is_fixed lthy v)) then cons x else I | _ => I) rhs []
in
null b orelse
primrec_error_eqn ("extra variable(s) in right-hand side: " ^
commas (map (Syntax.string_of_term lthy) b)) eqn
end;
in
{fun_name = fun_name,
rec_type = body_type (type_of ctr),
ctr = ctr,
ctr_args = ctr_args,
left_args = left_args,
right_args = right_args,
res_type = map fastype_of (left_args @ right_args) ---> fastype_of rhs,
rhs_term = rhs,
user_eqn = eqn'}
end;
fun rewrite_map_arg get_ctr_pos rec_type res_type =
let
val pT = HOLogic.mk_prodT (rec_type, res_type);
val maybe_suc = Option.map (fn x => x + 1);
fun subst d (t as Bound d') = t |> d = SOME d' ? curry (op $) (fst_const pT)
| subst d (Abs (v, T, b)) = Abs (v, if d = SOME ~1 then pT else T, subst (maybe_suc d) b)
| subst d t =
let
val (u, vs) = strip_comb t;
val ctr_pos = try (get_ctr_pos o the) (free_name u) |> the_default ~1;
in
if ctr_pos >= 0 then
if d = SOME ~1 andalso length vs = ctr_pos then
list_comb (permute_args ctr_pos (snd_const pT), vs)
else if length vs > ctr_pos andalso is_some d
andalso d = try (fn Bound n => n) (nth vs ctr_pos) then
list_comb (snd_const pT $ nth vs ctr_pos, map (subst d) (nth_drop ctr_pos vs))
else
primrec_error_eqn ("recursive call not directly applied to constructor argument") t
else if d = SOME ~1 andalso const_name u = SOME @{const_name comp} then
list_comb (map_types (K dummyT) u, map2 subst [NONE, d] vs)
else
list_comb (u, map (subst (d |> d = SOME ~1 ? K NONE)) vs)
end
in
subst (SOME ~1)
end;
fun subst_rec_calls lthy get_ctr_pos has_call ctr_args direct_calls indirect_calls t =
let
fun subst bound_Ts (Abs (v, T, b)) = Abs (v, T, subst (T :: bound_Ts) b)
| subst bound_Ts (t as g' $ y) =
let
val maybe_direct_y' = AList.lookup (op =) direct_calls y;
val maybe_indirect_y' = AList.lookup (op =) indirect_calls y;
val (g, g_args) = strip_comb g';
val ctr_pos = try (get_ctr_pos o the) (free_name g) |> the_default ~1;
val _ = ctr_pos < 0 orelse length g_args >= ctr_pos orelse
primrec_error_eqn "too few arguments in recursive call" t;
in
if not (member (op =) ctr_args y) then
pairself (subst bound_Ts) (g', y) |> (op $)
else if ctr_pos >= 0 then
list_comb (the maybe_direct_y', g_args)
else if is_some maybe_indirect_y' then
(if has_call g' then t else y)
|> massage_indirect_rec_call lthy has_call
(rewrite_map_arg get_ctr_pos) bound_Ts y (the maybe_indirect_y')
|> (if has_call g' then I else curry (op $) g')
else
t
end
| subst _ t = t
in
subst [] t
|> tap (fn u => has_call u andalso (* FIXME detect this case earlier *)
primrec_error_eqn "recursive call not directly applied to constructor argument" t)
end;
fun build_rec_arg lthy funs_data has_call ctr_spec maybe_eqn_data =
if is_none maybe_eqn_data then undef_const else
let
val eqn_data = the maybe_eqn_data;
val t = #rhs_term eqn_data;
val ctr_args = #ctr_args eqn_data;
val calls = #calls ctr_spec;
val n_args = fold (curry (op +) o (fn Direct_Rec _ => 2 | _ => 1)) calls 0;
val no_calls' = tag_list 0 calls
|> map_filter (try (apsnd (fn No_Rec n => n | Direct_Rec (n, _) => n)));
val direct_calls' = tag_list 0 calls
|> map_filter (try (apsnd (fn Direct_Rec (_, n) => n)));
val indirect_calls' = tag_list 0 calls
|> map_filter (try (apsnd (fn Indirect_Rec n => n)));
fun make_direct_type T = dummyT; (* FIXME? *)
val rec_res_type_list = map (fn (x :: _) => (#rec_type x, #res_type x)) funs_data;
fun make_indirect_type (Type (Tname, Ts)) = Type (Tname, Ts |> map (fn T =>
let val maybe_res_type = AList.lookup (op =) rec_res_type_list T in
if is_some maybe_res_type
then HOLogic.mk_prodT (T, the maybe_res_type)
else make_indirect_type T end))
| make_indirect_type T = T;
val args = replicate n_args ("", dummyT)
|> Term.rename_wrt_term t
|> map Free
|> fold (fn (ctr_arg_idx, arg_idx) =>
nth_map arg_idx (K (nth ctr_args ctr_arg_idx)))
no_calls'
|> fold (fn (ctr_arg_idx, arg_idx) =>
nth_map arg_idx (K (nth ctr_args ctr_arg_idx |> map_types make_direct_type)))
direct_calls'
|> fold (fn (ctr_arg_idx, arg_idx) =>
nth_map arg_idx (K (nth ctr_args ctr_arg_idx |> map_types make_indirect_type)))
indirect_calls';
val fun_name_ctr_pos_list =
map (fn (x :: _) => (#fun_name x, length (#left_args x))) funs_data;
val get_ctr_pos = try (the o AList.lookup (op =) fun_name_ctr_pos_list) #> the_default ~1;
val direct_calls = map (apfst (nth ctr_args) o apsnd (nth args)) direct_calls';
val indirect_calls = map (apfst (nth ctr_args) o apsnd (nth args)) indirect_calls';
val abstractions = args @ #left_args eqn_data @ #right_args eqn_data;
in
t
|> subst_rec_calls lthy get_ctr_pos has_call ctr_args direct_calls indirect_calls
|> fold_rev lambda abstractions
end;
fun build_defs lthy bs mxs funs_data rec_specs has_call =
let
val n_funs = length funs_data;
val ctr_spec_eqn_data_list' =
(take n_funs rec_specs |> map #ctr_specs) ~~ funs_data
|> maps (uncurry (finds (fn (x, y) => #ctr x = #ctr y))
##> (fn x => null x orelse
primrec_error_eqns "excess equations in definition" (map #rhs_term x)) #> fst);
val _ = ctr_spec_eqn_data_list' |> map (fn (_, x) => length x <= 1 orelse
primrec_error_eqns ("multiple equations for constructor") (map #user_eqn x));
val ctr_spec_eqn_data_list =
ctr_spec_eqn_data_list' @ (drop n_funs rec_specs |> maps #ctr_specs |> map (rpair []));
val recs = take n_funs rec_specs |> map #recx;
val rec_args = ctr_spec_eqn_data_list
|> sort ((op <) o pairself (#offset o fst) |> make_ord)
|> map (uncurry (build_rec_arg lthy funs_data has_call) o apsnd (try the_single));
val ctr_poss = map (fn x =>
if length (distinct ((op =) o pairself (length o #left_args)) x) <> 1 then
primrec_error ("inconstant constructor pattern position for function " ^
quote (#fun_name (hd x)))
else
hd x |> #left_args |> length) funs_data;
in
(recs, ctr_poss)
|-> map2 (fn recx => fn ctr_pos => list_comb (recx, rec_args) |> permute_args ctr_pos)
|> Syntax.check_terms lthy
|> map3 (fn b => fn mx => fn t => ((b, mx), ((Binding.map_name Thm.def_name b, []), t))) bs mxs
end;
fun find_rec_calls has_call eqn_data =
let
fun find (Abs (_, _, b)) ctr_arg = find b ctr_arg
| find (t as _ $ _) ctr_arg =
let
val (f', args') = strip_comb t;
val n = find_index (equal ctr_arg) args';
in
if n < 0 then
find f' ctr_arg @ maps (fn x => find x ctr_arg) args'
else
let val (f, args) = chop n args' |>> curry list_comb f' in
if has_call f then
f :: maps (fn x => find x ctr_arg) args
else
find f ctr_arg @ maps (fn x => find x ctr_arg) args
end
end
| find _ _ = [];
in
map (find (#rhs_term eqn_data)) (#ctr_args eqn_data)
|> (fn [] => NONE | callss => SOME (#ctr eqn_data, callss))
end;
fun add_primrec fixes specs lthy =
let
val (bs, mxs) = map_split (apfst fst) fixes;
val fun_names = map Binding.name_of bs;
val eqns_data = map (snd #> dissect_eqn lthy fun_names) specs;
val funs_data = eqns_data
|> partition_eq ((op =) o pairself #fun_name)
|> finds (fn (x, y) => x = #fun_name (hd y)) fun_names |> fst
|> map (fn (x, y) => the_single y handle List.Empty =>
primrec_error ("missing equations for function " ^ quote x));
val has_call = exists_subterm (map (fst #>> Binding.name_of #> Free) fixes |> member (op =));
val arg_Ts = map (#rec_type o hd) funs_data;
val res_Ts = map (#res_type o hd) funs_data;
val callssss = funs_data
|> map (partition_eq ((op =) o pairself #ctr))
|> map (maps (map_filter (find_rec_calls has_call)));
fun get_indices t = map (fst #>> Binding.name_of #> Free) fixes
|> map_index (fn (i, v) => if exists_subterm (equal v) t then SOME i else NONE)
|> map_filter I;
val ((nontriv, rec_specs, _, induct_thm, induct_thms), lthy') =
rec_specs_of bs arg_Ts res_Ts get_indices callssss lthy;
val actual_nn = length funs_data;
val _ = let val ctrs = (maps (map #ctr o #ctr_specs) rec_specs) in
map (fn {ctr, user_eqn, ...} => member (op =) ctrs ctr orelse
primrec_error_eqn ("argument " ^ quote (Syntax.string_of_term lthy' ctr) ^
" is not a constructor in left-hand side") user_eqn) eqns_data end;
val defs = build_defs lthy' bs mxs funs_data rec_specs has_call;
fun prove def_thms' {ctr_specs, nested_map_idents, nested_map_comps, ...} induct_thm fun_data
lthy =
let
val fun_name = #fun_name (hd fun_data);
val def_thms = map (snd o snd) def_thms';
val simp_thms = finds (fn (x, y) => #ctr x = #ctr y) fun_data ctr_specs
|> fst
|> map_filter (try (fn (x, [y]) =>
(#user_eqn x, length (#left_args x) + length (#right_args x), #rec_thm y)))
|> map (fn (user_eqn, num_extra_args, rec_thm) =>
mk_primrec_tac lthy num_extra_args nested_map_idents nested_map_comps def_thms rec_thm
|> K |> Goal.prove lthy [] [] user_eqn)
val notes =
[(inductN, if actual_nn > 1 then [induct_thm] else [], []),
(simpsN, simp_thms, simp_attrs)]
|> filter_out (null o #2)
|> map (fn (thmN, thms, attrs) =>
((Binding.qualify true fun_name (Binding.name thmN), attrs), [(thms, [])]));
in
lthy |> Local_Theory.notes notes
end;
val common_name = mk_common_name fun_names;
val common_notes =
[(inductN, if nontriv andalso actual_nn > 1 then [induct_thm] else [], [])]
|> filter_out (null o #2)
|> map (fn (thmN, thms, attrs) =>
((Binding.qualify true common_name (Binding.name thmN), attrs), [(thms, [])]));
in
lthy'
|> fold_map Local_Theory.define defs
|-> snd oo (fn def_thms' => fold_map3 (prove def_thms') (take actual_nn rec_specs)
(take actual_nn induct_thms) funs_data)
|> snd o Local_Theory.notes common_notes
end;
fun add_primrec_cmd raw_fixes raw_specs lthy =
let
val _ = let val d = duplicates (op =) (map (Binding.name_of o #1) raw_fixes) in null d orelse
primrec_error ("duplicate function name(s): " ^ commas d) end;
val (fixes, specs) = fst (Specification.read_spec raw_fixes raw_specs lthy);
in
add_primrec fixes specs lthy
handle ERROR str => primrec_error str
end
handle Primrec_Error (str, eqns) =>
if null eqns
then error ("primrec_new error:\n " ^ str)
else error ("primrec_new error:\n " ^ str ^ "\nin\n " ^
space_implode "\n " (map (quote o Syntax.string_of_term lthy) eqns))
(* Primcorec *)
type co_eqn_data_disc = {
fun_name: string,
fun_T: typ,
fun_args: term list,
ctr: term,
ctr_no: int, (*###*)
disc: term,
prems: term list,
user_eqn: term
};
type co_eqn_data_sel = {
fun_name: string,
fun_T: typ,
fun_args: term list,
ctr: term,
sel: term,
rhs_term: term,
user_eqn: term
};
datatype co_eqn_data =
Disc of co_eqn_data_disc |
Sel of co_eqn_data_sel;
fun co_dissect_eqn_disc sequential fun_names corec_specs prems' concl matchedsss =
let
fun find_subterm p = let (* FIXME \<exists>? *)
fun f (t as u $ v) = if p t then SOME t else merge_options (f u, f v)
| f t = if p t then SOME t else NONE
in f end;
val applied_fun = concl
|> find_subterm (member ((op =) o apsnd SOME) fun_names o try (fst o dest_Free o head_of))
|> the
handle Option.Option => primrec_error_eqn "malformed discriminator equation" concl;
val ((fun_name, fun_T), fun_args) = strip_comb applied_fun |>> dest_Free;
val {ctr_specs, ...} = the (AList.lookup (op =) (fun_names ~~ corec_specs) fun_name);
val discs = map #disc ctr_specs;
val ctrs = map #ctr ctr_specs;
val not_disc = head_of concl = @{term Not};
val _ = not_disc andalso length ctrs <> 2 andalso
primrec_error_eqn "\<not>ed discriminator for a type with \<noteq> 2 constructors" concl;
val disc = find_subterm (member (op =) discs o head_of) concl;
val eq_ctr0 = concl |> perhaps (try (HOLogic.dest_not)) |> try (HOLogic.dest_eq #> snd)
|> (fn SOME t => let val n = find_index (equal t) ctrs in
if n >= 0 then SOME n else NONE end | _ => NONE);
val _ = is_some disc orelse is_some eq_ctr0 orelse
primrec_error_eqn "no discriminator in equation" concl;
val ctr_no' =
if is_none disc then the eq_ctr0 else find_index (equal (head_of (the disc))) discs;
val ctr_no = if not_disc then 1 - ctr_no' else ctr_no';
val ctr = #ctr (nth ctr_specs ctr_no);
val catch_all = try (fst o dest_Free o the_single) prems' = SOME Name.uu_;
val matchedss = AList.lookup (op =) matchedsss fun_name |> the_default [];
val prems = map (abstract (List.rev fun_args)) prems';
val real_prems =
(if catch_all orelse sequential then maps invert_prems_disj matchedss else []) @
(if catch_all then [] else prems);
val matchedsss' = AList.delete (op =) fun_name matchedsss
|> cons (fun_name, if sequential then matchedss @ [prems] else matchedss @ [real_prems]);
val user_eqn =
(real_prems, betapply (#disc (nth ctr_specs ctr_no), applied_fun))
|>> map HOLogic.mk_Trueprop ||> HOLogic.mk_Trueprop
|> Logic.list_implies;
in
(Disc {
fun_name = fun_name,
fun_T = fun_T,
fun_args = fun_args,
ctr = ctr,
ctr_no = ctr_no,
disc = #disc (nth ctr_specs ctr_no),
prems = real_prems,
user_eqn = user_eqn
}, matchedsss')
end;
fun co_dissect_eqn_sel fun_names corec_specs eqn' eqn =
let
val (lhs, rhs) = HOLogic.dest_eq eqn
handle TERM _ =>
primrec_error_eqn "malformed function equation (expected \"lhs = rhs\")" eqn;
val sel = head_of lhs;
val ((fun_name, fun_T), fun_args) = dest_comb lhs |> snd |> strip_comb |> apfst dest_Free
handle TERM _ =>
primrec_error_eqn "malformed selector argument in left-hand side" eqn;
val corec_spec = the (AList.lookup (op =) (fun_names ~~ corec_specs) fun_name)
handle Option.Option => primrec_error_eqn "malformed selector argument in left-hand side" eqn;
val (ctr_spec, sel) = #ctr_specs corec_spec
|> the o get_index (try (the o find_first (equal sel) o #sels))
|>> nth (#ctr_specs corec_spec);
val user_eqn = drop_All eqn';
in
Sel {
fun_name = fun_name,
fun_T = fun_T,
fun_args = fun_args,
ctr = #ctr ctr_spec,
sel = sel,
rhs_term = rhs,
user_eqn = user_eqn
}
end;
fun co_dissect_eqn_ctr sequential fun_names corec_specs eqn' imp_prems imp_rhs matchedsss =
let
val (lhs, rhs) = HOLogic.dest_eq imp_rhs;
val fun_name = head_of lhs |> fst o dest_Free;
val {ctr_specs, ...} = the (AList.lookup (op =) (fun_names ~~ corec_specs) fun_name);
val (ctr, ctr_args) = strip_comb rhs;
val {disc, sels, ...} = the (find_first (equal ctr o #ctr) ctr_specs)
handle Option.Option => primrec_error_eqn "not a constructor" ctr;
val disc_imp_rhs = betapply (disc, lhs);
val (maybe_eqn_data_disc, matchedsss') = if length ctr_specs = 1
then (NONE, matchedsss)
else apfst SOME (co_dissect_eqn_disc
sequential fun_names corec_specs imp_prems disc_imp_rhs matchedsss);
val sel_imp_rhss = (sels ~~ ctr_args)
|> map (fn (sel, ctr_arg) => HOLogic.mk_eq (betapply (sel, lhs), ctr_arg));
val _ = tracing ("reduced\n " ^ Syntax.string_of_term @{context} imp_rhs ^ "\nto\n \<cdot> " ^
(is_some maybe_eqn_data_disc ? K (Syntax.string_of_term @{context} disc_imp_rhs ^ "\n \<cdot> ")) "" ^
space_implode "\n \<cdot> " (map (Syntax.string_of_term @{context}) sel_imp_rhss));
val eqns_data_sel =
map (co_dissect_eqn_sel fun_names corec_specs eqn') sel_imp_rhss;
in
(the_list maybe_eqn_data_disc @ eqns_data_sel, matchedsss')
end;
fun co_dissect_eqn sequential fun_names corec_specs eqn' matchedsss =
let
val eqn = drop_All eqn'
handle TERM _ => primrec_error_eqn "malformed function equation" eqn';
val (imp_prems, imp_rhs) = Logic.strip_horn eqn
|> apfst (map HOLogic.dest_Trueprop) o apsnd HOLogic.dest_Trueprop;
val head = imp_rhs
|> perhaps (try HOLogic.dest_not) |> perhaps (try (fst o HOLogic.dest_eq))
|> head_of;
val maybe_rhs = imp_rhs |> perhaps (try (HOLogic.dest_not)) |> try (snd o HOLogic.dest_eq);
val discs = maps #ctr_specs corec_specs |> map #disc;
val sels = maps #ctr_specs corec_specs |> maps #sels;
val ctrs = maps #ctr_specs corec_specs |> map #ctr;
in
if member (op =) discs head orelse
is_some maybe_rhs andalso
member (op =) (filter (null o binder_types o fastype_of) ctrs) (the maybe_rhs) then
co_dissect_eqn_disc sequential fun_names corec_specs imp_prems imp_rhs matchedsss
|>> single
else if member (op =) sels head then
([co_dissect_eqn_sel fun_names corec_specs eqn' imp_rhs], matchedsss)
else if is_Free head andalso member (op =) fun_names (fst (dest_Free head)) then
co_dissect_eqn_ctr sequential fun_names corec_specs eqn' imp_prems imp_rhs matchedsss
else
primrec_error_eqn "malformed function equation" eqn
end;
fun build_corec_arg_disc ctr_specs {fun_args, ctr_no, prems, ...} =
if is_none (#pred (nth ctr_specs ctr_no)) then I else
mk_conjs prems
|> curry subst_bounds (List.rev fun_args)
|> HOLogic.tupled_lambda (HOLogic.mk_tuple fun_args)
|> K |> nth_map (the (#pred (nth ctr_specs ctr_no)));
fun build_corec_arg_no_call sel_eqns sel =
find_first (equal sel o #sel) sel_eqns
|> try (fn SOME {fun_args, rhs_term, ...} => abs_tuple fun_args rhs_term)
|> the_default undef_const
|> K;
fun build_corec_arg_direct_call lthy has_call sel_eqns sel =
let
val maybe_sel_eqn = find_first (equal sel o #sel) sel_eqns;
fun rewrite is_end U _ t =
if U = @{typ bool} then @{term True} |> has_call t ? K @{term False} (* stop? *)
else if is_end = has_call t then undef_const
else if is_end then t (* end *)
else HOLogic.mk_tuple (snd (strip_comb t)); (* continue *)
fun massage rhs_term is_end t = massage_direct_corec_call
lthy has_call (rewrite is_end) [] (range_type (fastype_of t)) rhs_term;
in
if is_none maybe_sel_eqn then K I else
abs_tuple (#fun_args (the maybe_sel_eqn)) oo massage (#rhs_term (the maybe_sel_eqn))
end;
fun build_corec_arg_indirect_call lthy has_call sel_eqns sel =
let
val maybe_sel_eqn = find_first (equal sel o #sel) sel_eqns;
fun rewrite _ _ =
let
fun subst (Abs (v, T, b)) = Abs (v, T, subst b)
| subst (t as _ $ _) =
let val (u, vs) = strip_comb t in
if is_Free u andalso has_call u then
Const (@{const_name Inr}, dummyT) $
(try (foldr1 (fn (x, y) => Const (@{const_name Pair}, dummyT) $ x $ y)) vs
|> the_default (hd vs))
else if try (fst o dest_Const) u = SOME @{const_name prod_case} then
list_comb (u |> map_types (K dummyT), map subst vs)
else
list_comb (subst u, map subst vs)
end
| subst t = t;
in
subst
end;
fun massage rhs_term t = massage_indirect_corec_call
lthy has_call rewrite [] (range_type (fastype_of t)) rhs_term;
in
if is_none maybe_sel_eqn then I else
abs_tuple (#fun_args (the maybe_sel_eqn)) o massage (#rhs_term (the maybe_sel_eqn))
end;
fun build_corec_args_sel lthy has_call all_sel_eqns ctr_spec =
let val sel_eqns = filter (equal (#ctr ctr_spec) o #ctr) all_sel_eqns in
if null sel_eqns then I else
let
val sel_call_list = #sels ctr_spec ~~ #calls ctr_spec;
val no_calls' = map_filter (try (apsnd (fn No_Corec n => n))) sel_call_list;
val direct_calls' = map_filter (try (apsnd (fn Direct_Corec n => n))) sel_call_list;
val indirect_calls' = map_filter (try (apsnd (fn Indirect_Corec n => n))) sel_call_list;
in
I
#> fold (fn (sel, n) => nth_map n
(build_corec_arg_no_call sel_eqns sel)) no_calls'
#> fold (fn (sel, (q, g, h)) =>
let val f = build_corec_arg_direct_call lthy has_call sel_eqns sel in
nth_map h (f false) o nth_map g (f true) o nth_map q (f true) end) direct_calls'
#> fold (fn (sel, n) => nth_map n
(build_corec_arg_indirect_call lthy has_call sel_eqns sel)) indirect_calls'
end
end;
fun co_build_defs lthy bs mxs has_call arg_Tss corec_specs disc_eqnss sel_eqnss =
let
val corec_specs' = take (length bs) corec_specs;
val corecs = map #corec corec_specs';
val ctr_specss = map #ctr_specs corec_specs';
val corec_args = hd corecs
|> fst o split_last o binder_types o fastype_of
|> map (Const o pair @{const_name undefined})
|> fold2 (fold o build_corec_arg_disc) ctr_specss disc_eqnss
|> fold2 (fold o build_corec_args_sel lthy has_call) sel_eqnss ctr_specss;
fun currys Ts t = if length Ts <= 1 then t else
t $ foldr1 (fn (u, v) => HOLogic.pair_const dummyT dummyT $ u $ v)
(length Ts - 1 downto 0 |> map Bound)
|> fold_rev (Term.abs o pair Name.uu) Ts;
val _ = tracing ("corecursor arguments:\n \<cdot> " ^
space_implode "\n \<cdot> " (map (Syntax.string_of_term lthy) corec_args));
val exclss' =
disc_eqnss
|> map (map (fn {fun_args, ctr_no, prems, ...} => (fun_args, ctr_no, prems))
#> fst o (fn xs => fold_map (fn x => fn ys => ((x, ys), ys @ [x])) xs [])
#> maps (uncurry (map o pair)
#> map (fn ((fun_args, c, x), (_, c', y)) => ((c, c'), (x, mk_not (mk_conjs y)))
||> apfst (map HOLogic.mk_Trueprop) o apsnd HOLogic.mk_Trueprop
||> Logic.list_implies
||> curry Logic.list_all (map dest_Free fun_args))))
in
map (list_comb o rpair corec_args) corecs
|> map2 (fn Ts => fn t => if length Ts = 0 then t $ HOLogic.unit else t) arg_Tss
|> map2 currys arg_Tss
|> Syntax.check_terms lthy
|> map3 (fn b => fn mx => fn t => ((b, mx), ((Binding.map_name Thm.def_name b, []), t))) bs mxs
|> rpair exclss'
end;
fun mk_real_disc_eqns fun_binding arg_Ts {ctr_specs, ...} sel_eqns disc_eqns =
if length disc_eqns <> length ctr_specs - 1 then disc_eqns else
let
val n = 0 upto length ctr_specs
|> the o find_first (fn idx => not (exists (equal idx o #ctr_no) disc_eqns));
val fun_args = (try (#fun_args o hd) disc_eqns, try (#fun_args o hd) sel_eqns)
|> the_default (map (curry Free Name.uu) arg_Ts) o merge_options;
val extra_disc_eqn = {
fun_name = Binding.name_of fun_binding,
fun_T = arg_Ts ---> body_type (fastype_of (#ctr (hd ctr_specs))),
fun_args = fun_args,
ctr = #ctr (nth ctr_specs n),
ctr_no = n,
disc = #disc (nth ctr_specs n),
prems = maps (invert_prems o #prems) disc_eqns,
user_eqn = undef_const};
in
chop n disc_eqns ||> cons extra_disc_eqn |> (op @)
end;
fun add_primcorec sequential fixes specs lthy =
let
val (bs, mxs) = map_split (apfst fst) fixes;
val (arg_Ts, res_Ts) = map (strip_type o snd o fst #>> HOLogic.mk_tupleT) fixes |> split_list;
(* copied from primrec_new *)
fun get_indices t = map (fst #>> Binding.name_of #> Free) fixes
|> map_index (fn (i, v) => if exists_subterm (equal v) t then SOME i else NONE)
|> map_filter I;
val callssss = []; (* FIXME *)
val ((nontriv, corec_specs', _, coinduct_thm, strong_co_induct_thm, coinduct_thmss,
strong_coinduct_thmss), lthy') =
corec_specs_of bs arg_Ts res_Ts get_indices callssss lthy;
val fun_names = map Binding.name_of bs;
val corec_specs = take (length fun_names) corec_specs'; (*###*)
val (eqns_data, _) =
fold_map (co_dissect_eqn sequential fun_names corec_specs) (map snd specs) []
|>> flat;
val disc_eqnss' = map_filter (try (fn Disc x => x)) eqns_data
|> partition_eq ((op =) o pairself #fun_name)
|> fst o finds (fn (x, ({fun_name, ...} :: _)) => x = fun_name) fun_names
|> map (sort ((op <) o pairself #ctr_no |> make_ord) o flat o snd);
val _ = disc_eqnss' |> map (fn x =>
let val d = duplicates ((op =) o pairself #ctr_no) x in null d orelse
primrec_error_eqns "excess discriminator equations in definition"
(maps (fn t => filter (equal (#ctr_no t) o #ctr_no) x) d |> map #user_eqn) end);
val sel_eqnss = map_filter (try (fn Sel x => x)) eqns_data
|> partition_eq ((op =) o pairself #fun_name)
|> fst o finds (fn (x, ({fun_name, ...} :: _)) => x = fun_name) fun_names
|> map (flat o snd);
val has_call = exists_subterm (map (fst #>> Binding.name_of #> Free) fixes |> member (op =));
val arg_Tss = map (binder_types o snd o fst) fixes;
val disc_eqnss = map5 mk_real_disc_eqns bs arg_Tss corec_specs sel_eqnss disc_eqnss';
val (defs, exclss') =
co_build_defs lthy' bs mxs has_call arg_Tss corec_specs disc_eqnss sel_eqnss;
(* try to prove (automatically generated) tautologies by ourselves *)
val exclss'' = exclss'
|> map (map (apsnd
(`(try (fn t => Goal.prove lthy [] [] t (mk_primcorec_assumption_tac lthy |> K))))));
val taut_thmss = map (map (apsnd (the o fst)) o filter (is_some o fst o snd)) exclss'';
val (obligation_idxss, obligationss) = exclss''
|> map (map (apsnd (rpair [] o snd)) o filter (is_none o fst o snd))
|> split_list o map split_list;
fun prove thmss' def_thms' lthy =
let
val def_thms = map (snd o snd) def_thms';
val exclss' = map (op ~~) (obligation_idxss ~~ thmss');
fun mk_exclsss excls n =
(excls, map (fn k => replicate k [TrueI] @ replicate (n - k) []) (0 upto n - 1))
|-> fold (fn ((c, c'), thm) => nth_map c (nth_map c' (K [thm])));
val exclssss = (exclss' ~~ taut_thmss |> map (op @), fun_names ~~ corec_specs)
|-> map2 (fn excls => fn (_, {ctr_specs, ...}) => mk_exclsss excls (length ctr_specs));
fun prove_disc {ctr_specs, ...} exclsss
{fun_name, fun_T, fun_args, ctr_no, prems, user_eqn, ...} =
if Term.aconv_untyped (#disc (nth ctr_specs ctr_no), @{term "\<lambda>x. x = x"}) then [] else
let
val {disc_corec, ...} = nth ctr_specs ctr_no;
val k = 1 + ctr_no;
val m = length prems;
val t =
list_comb (Free (fun_name, fun_T), map Bound (length fun_args - 1 downto 0))
|> curry betapply (#disc (nth ctr_specs ctr_no)) (*###*)
|> HOLogic.mk_Trueprop
|> curry Logic.list_implies (map HOLogic.mk_Trueprop prems)
|> curry Logic.list_all (map dest_Free fun_args);
in
mk_primcorec_disc_tac lthy def_thms disc_corec k m exclsss
|> K |> Goal.prove lthy [] [] t
|> pair (#disc (nth ctr_specs ctr_no))
|> single
end;
fun prove_sel {ctr_specs, nested_maps, nested_map_idents, nested_map_comps, ...}
disc_eqns exclsss {fun_name, fun_T, fun_args, ctr, sel, rhs_term, ...} =
let
val (SOME ctr_spec) = find_first (equal ctr o #ctr) ctr_specs;
val ctr_no = find_index (equal ctr o #ctr) ctr_specs;
val prems = the_default (maps (invert_prems o #prems) disc_eqns)
(find_first (equal ctr_no o #ctr_no) disc_eqns |> Option.map #prems);
val sel_corec = find_index (equal sel) (#sels ctr_spec)
|> nth (#sel_corecs ctr_spec);
val k = 1 + ctr_no;
val m = length prems;
val t =
list_comb (Free (fun_name, fun_T), map Bound (length fun_args - 1 downto 0))
|> curry betapply sel
|> rpair (abstract (List.rev fun_args) rhs_term)
|> HOLogic.mk_Trueprop o HOLogic.mk_eq
|> curry Logic.list_implies (map HOLogic.mk_Trueprop prems)
|> curry Logic.list_all (map dest_Free fun_args);
in
mk_primcorec_ctr_or_sel_tac lthy def_thms sel_corec k m exclsss
nested_maps nested_map_idents nested_map_comps
|> K |> Goal.prove lthy [] [] t
|> pair sel
end;
fun prove_ctr (_, disc_thms) (_, sel_thms') disc_eqns sel_eqns
{ctr, disc, sels, collapse, ...} =
let val _ = tracing ("disc = " ^ @{make_string} disc); in
if not (exists (equal ctr o #ctr) disc_eqns)
andalso not (exists (equal ctr o #ctr) sel_eqns)
andalso (warning ("no eqns for ctr " ^ Syntax.string_of_term lthy ctr); true)
orelse (* don't try to prove theorems when some sel_eqns are missing *)
filter (equal ctr o #ctr) sel_eqns
|> fst o finds ((op =) o apsnd #sel) sels
|> exists (null o snd)
andalso (warning ("sel_eqn(s) missing for ctr " ^ Syntax.string_of_term lthy ctr); true)
then [] else
let
val _ = tracing ("ctr = " ^ Syntax.string_of_term lthy ctr);
val _ = tracing (the_default "NO disc_eqn" (Option.map (curry (op ^) "disc = " o Syntax.string_of_term lthy o #disc) (find_first (equal ctr o #ctr) disc_eqns)));
val (fun_name, fun_T, fun_args, prems) =
(find_first (equal ctr o #ctr) disc_eqns, find_first (equal ctr o #ctr) sel_eqns)
|>> Option.map (fn x => (#fun_name x, #fun_T x, #fun_args x, #prems x))
||> Option.map (fn x => (#fun_name x, #fun_T x, #fun_args x, []))
|> the o merge_options;
val m = length prems;
val t = sel_eqns
|> fst o finds ((op =) o apsnd #sel) sels
|> map (snd #> (fn [x] => (List.rev (#fun_args x), #rhs_term x)) #-> abstract)
|> curry list_comb ctr
|> curry HOLogic.mk_eq (list_comb (Free (fun_name, fun_T),
map Bound (length fun_args - 1 downto 0)))
|> HOLogic.mk_Trueprop
|> curry Logic.list_implies (map HOLogic.mk_Trueprop prems)
|> curry Logic.list_all (map dest_Free fun_args);
val maybe_disc_thm = AList.lookup (op =) disc_thms disc;
val sel_thms = map snd (filter (member (op =) sels o fst) sel_thms');
val _ = tracing ("t = " ^ Syntax.string_of_term lthy t);
val _ = tracing ("m = " ^ @{make_string} m);
val _ = tracing ("collapse = " ^ @{make_string} collapse);
val _ = tracing ("maybe_disc_thm = " ^ @{make_string} maybe_disc_thm);
val _ = tracing ("sel_thms = " ^ @{make_string} sel_thms);
in
mk_primcorec_ctr_of_dtr_tac lthy m collapse maybe_disc_thm sel_thms
|> K |> Goal.prove lthy [] [] t
|> single
(*handle ERROR x => (warning x; []))*)
end
end;
val (disc_notes, disc_thmss) =
fun_names ~~ map3 (maps oo prove_disc) corec_specs exclssss disc_eqnss
|> `(map (fn (fun_name, thms) =>
((Binding.qualify true fun_name (@{binding disc}), simp_attrs), [(map snd thms, [])])));
val (sel_notes, sel_thmss) =
fun_names ~~ map4 (map ooo prove_sel) corec_specs disc_eqnss exclssss sel_eqnss
|> `(map (fn (fun_name, thms) =>
((Binding.qualify true fun_name (@{binding sel}), simp_attrs), [(map snd thms, [])])));
val ctr_notes =
fun_names ~~ map5 (maps oooo prove_ctr) disc_thmss sel_thmss
disc_eqnss sel_eqnss (map #ctr_specs corec_specs)
|> map (fn (fun_name, thms) =>
((Binding.qualify true fun_name (@{binding ctr}), simp_attrs), [(thms, [])]));
in
lthy |> snd o Local_Theory.notes
(filter (not o null o fst o the_single o snd) (disc_notes @ sel_notes @ ctr_notes))
end;
in
lthy'
|> Proof.theorem NONE (curry (op #->) (fold_map Local_Theory.define defs) o prove) obligationss
|> Proof.refine (Method.primitive_text I)
|> Seq.hd
end
fun add_primcorec_cmd seq (raw_fixes, raw_specs) lthy =
let
val (fixes, specs) = fst (Specification.read_spec raw_fixes raw_specs lthy);
in
add_primcorec seq fixes specs lthy
handle ERROR str => primrec_error str
end
handle Primrec_Error (str, eqns) =>
if null eqns
then error ("primcorec error:\n " ^ str)
else error ("primcorec error:\n " ^ str ^ "\nin\n " ^
space_implode "\n " (map (quote o Syntax.string_of_term lthy) eqns))
end;