1 (* Title: HOL/Nominal/nominal_primrec.ML
3 Author: Stefan Berghofer, TU Muenchen and Norbert Voelker, FernUni Hagen
5 Package for defining functions on nominal datatypes by primitive recursion.
6 Taken from HOL/Tools/primrec_package.ML
9 signature NOMINAL_PRIMREC =
11 val quiet_mode: bool ref
12 val add_primrec: string -> string list option -> string option ->
13 ((bstring * string) * Attrib.src list) list -> theory -> Proof.state
14 val add_primrec_unchecked: string -> string list option -> string option ->
15 ((bstring * string) * Attrib.src list) list -> theory -> Proof.state
16 val add_primrec_i: string -> term list option -> term option ->
17 ((bstring * term) * attribute list) list -> theory -> Proof.state
18 val add_primrec_unchecked_i: string -> term list option -> term option ->
19 ((bstring * term) * attribute list) list -> theory -> Proof.state
22 structure NominalPrimrec : NOMINAL_PRIMREC =
27 exception RecError of string;
29 fun primrec_err s = error ("Nominal primrec definition error:\n" ^ s);
30 fun primrec_eq_err thy s eq =
31 primrec_err (s ^ "\nin\n" ^ quote (Sign.string_of_term thy eq));
36 val quiet_mode = ref false;
37 fun message s = if ! quiet_mode then () else writeln s;
40 (* preprocessing of equations *)
42 fun process_eqn thy eq rec_fns =
45 if null (term_vars eq) then
46 HOLogic.dest_eq (HOLogic.dest_Trueprop (Logic.strip_imp_concl eq))
47 handle TERM _ => raise RecError "not a proper equation"
48 else raise RecError "illegal schematic variable(s)";
50 val (recfun, args) = strip_comb lhs;
51 val fnameT = dest_Const recfun handle TERM _ =>
52 raise RecError "function is not declared as constant in theory";
54 val (ls', rest) = take_prefix is_Free args;
55 val (middle, rs') = take_suffix is_Free rest;
56 val rpos = length ls';
58 val (constr, cargs') = if null middle then raise RecError "constructor missing"
59 else strip_comb (hd middle);
60 val (cname, T) = dest_Const constr
61 handle TERM _ => raise RecError "ill-formed constructor";
62 val (tname, _) = dest_Type (body_type T) handle TYPE _ =>
63 raise RecError "cannot determine datatype associated with function"
66 (map dest_Free ls', map dest_Free cargs', map dest_Free rs')
67 handle TERM _ => raise RecError "illegal argument in pattern";
68 val lfrees = ls @ rs @ cargs;
70 fun check_vars _ [] = ()
71 | check_vars s vars = raise RecError (s ^ commas_quote (map fst vars))
73 if length middle > 1 then
74 raise RecError "more than one non-variable in pattern"
76 (check_vars "repeated variable names in pattern: " (duplicates (op =) lfrees);
77 check_vars "extra variables on rhs: "
78 (map dest_Free (term_frees rhs) \\ lfrees);
79 case AList.lookup (op =) rec_fns fnameT of
81 (fnameT, (tname, rpos, [(cname, (ls, cargs, rs, rhs, eq))]))::rec_fns
82 | SOME (_, rpos', eqns) =>
83 if AList.defined (op =) eqns cname then
84 raise RecError "constructor already occurred as pattern"
85 else if rpos <> rpos' then
86 raise RecError "position of recursive argument inconsistent"
88 AList.update (op =) (fnameT, (tname, rpos, (cname, (ls, cargs, rs, rhs, eq))::eqns))
91 handle RecError s => primrec_eq_err thy s eq;
93 val param_err = "Parameters must be the same for all recursive functions";
95 fun process_fun thy descr rec_eqns (i, fnameT as (fname, _)) (fnameTs, fnss) =
97 val (_, (tname, _, constrs)) = List.nth (descr, i);
99 (* substitute "fname ls x rs" by "y" for (x, (_, y)) in subs *)
101 fun subst [] t fs = (t, fs)
102 | subst subs (Abs (a, T, t)) fs =
105 |-> (fn t' => pair (Abs (a, T, t')))
106 | subst subs (t as (_ $ _)) fs =
108 val (f, ts) = strip_comb t;
110 if is_Const f andalso dest_Const f mem map fst rec_eqns then
112 val fnameT' as (fname', _) = dest_Const f;
113 val (_, rpos, eqns) = the (AList.lookup (op =) rec_eqns fnameT');
114 val ls = Library.take (rpos, ts);
115 val rest = Library.drop (rpos, ts);
116 val (x', rs) = (hd rest, tl rest)
117 handle Empty => raise RecError ("not enough arguments\
118 \ in recursive application\nof function " ^ quote fname' ^ " on rhs");
119 val rs' = (case eqns of
120 (_, (ls', _, rs', _, _)) :: _ =>
121 let val (rs1, rs2) = chop (length rs') rs
123 if ls = map Free ls' andalso rs1 = map Free rs' then rs2
124 else raise RecError param_err
126 | _ => raise RecError ("no equations for " ^ quote fname'));
127 val (x, xs) = strip_comb x'
128 in case AList.lookup (op =) subs x
131 |> fold_map (subst subs) ts
132 |-> (fn ts' => pair (list_comb (f, ts')))
135 |> fold_map (subst subs) (xs @ rs')
136 ||> process_fun thy descr rec_eqns (i', fnameT')
137 |-> (fn ts' => pair (list_comb (y, ts')))
141 |> fold_map (subst subs) (f :: ts)
142 |-> (fn (f'::ts') => pair (list_comb (f', ts')))
144 | subst _ t fs = (t, fs);
146 (* translate rec equations into function arguments suitable for rec comb *)
148 fun trans eqns (cname, cargs) (fnameTs', fnss', fns) =
149 (case AList.lookup (op =) eqns cname of
150 NONE => (warning ("No equation for constructor " ^ quote cname ^
151 "\nin definition of function " ^ quote fname);
152 (fnameTs', fnss', (Const ("arbitrary", dummyT))::fns))
153 | SOME (ls, cargs', rs, rhs, eq) =>
155 val recs = filter (is_rec_type o snd) (cargs' ~~ cargs);
156 val rargs = map fst recs;
157 val subs = map (rpair dummyT o fst)
158 (rev (rename_wrt_term rhs rargs));
159 val (rhs', (fnameTs'', fnss'')) =
160 (subst (map (fn ((x, y), z) =>
161 (Free x, (body_index y, Free z)))
162 (recs ~~ subs)) rhs (fnameTs', fnss'))
163 handle RecError s => primrec_eq_err thy s eq
164 in (fnameTs'', fnss'',
165 (list_abs_free (cargs' @ subs, rhs'))::fns)
168 in (case AList.lookup (op =) fnameTs i of
170 if exists (equal fnameT o snd) fnameTs then
171 raise RecError ("inconsistent functions for datatype " ^ quote tname)
174 val SOME (_, _, eqns as (_, (ls, _, rs, _, _)) :: _) =
175 AList.lookup (op =) rec_eqns fnameT;
176 val (fnameTs', fnss', fns) = fold_rev (trans eqns) constrs
177 ((i, fnameT)::fnameTs, fnss, [])
179 (fnameTs', (i, (fname, ls, rs, fns))::fnss')
182 if fnameT = fnameT' then (fnameTs, fnss)
183 else raise RecError ("inconsistent functions for datatype " ^ quote tname))
187 (* prepare functions needed for definitions *)
189 fun get_fns fns ((i : int, (tname, _, constrs)), rec_name) (fs, defs) =
190 case AList.lookup (op =) fns i of
193 val dummy_fns = map (fn (_, cargs) => Const ("arbitrary",
194 replicate ((length cargs) + (length (List.filter is_rec_type cargs)))
195 dummyT ---> HOLogic.unitT)) constrs;
196 val _ = warning ("No function definition for datatype " ^ quote tname)
198 (dummy_fns @ fs, defs)
200 | SOME (fname, ls, rs, fs') => (fs' @ fs, (fname, ls, rs, rec_name, tname) :: defs);
203 (* make definition *)
205 fun make_def thy fs (fname, ls, rs, rec_name, tname) =
207 val used = map fst (fold Term.add_frees fs []);
208 val x = (Name.variant used "x", dummyT);
209 val frees = ls @ x :: rs;
210 val rhs = list_abs_free (frees,
211 list_comb (Const (rec_name, dummyT), fs @ [Free x]))
212 val def_name = Sign.base_name fname ^ "_" ^ Sign.base_name tname ^ "_def";
213 val def_prop as _ $ _ $ t =
214 singleton (Syntax.check_terms (ProofContext.init thy))
215 (Logic.mk_equals (Const (fname, dummyT), rhs));
216 in ((def_name, def_prop), subst_bounds (rev (map Free frees), strip_abs_body t)) end;
219 (* find datatypes which contain all datatypes in tnames' *)
221 fun find_dts (dt_info : NominalPackage.nominal_datatype_info Symtab.table) _ [] = []
222 | find_dts dt_info tnames' (tname::tnames) =
223 (case Symtab.lookup dt_info tname of
224 NONE => primrec_err (quote tname ^ " is not a nominal datatype")
226 if tnames' subset (map (#1 o snd) (#descr dt)) then
227 (tname, dt)::(find_dts dt_info tnames' tnames)
228 else find_dts dt_info tnames' tnames);
230 fun common_prefix eq ([], _) = []
231 | common_prefix eq (_, []) = []
232 | common_prefix eq (x :: xs, y :: ys) =
233 if eq (x, y) then x :: common_prefix eq (xs, ys) else [];
237 fun gen_primrec_i note def alt_name invs fctxt eqns_atts thy =
239 val (eqns, atts) = split_list eqns_atts;
240 val dt_info = NominalPackage.get_nominal_datatypes thy;
241 val rec_eqns = fold_rev (process_eqn thy o snd) eqns [];
242 val lsrs :: lsrss = maps (fn (_, (_, _, eqns)) =>
243 map (fn (_, (ls, _, rs, _, _)) => ls @ rs) eqns) rec_eqns
245 (if forall (curry eq_set lsrs) lsrss andalso forall
246 (fn (_, (_, _, (_, (ls, _, rs, _, _)) :: eqns)) =>
247 forall (fn (_, (ls', _, rs', _, _)) =>
248 ls = ls' andalso rs = rs') eqns
249 | _ => true) rec_eqns
250 then () else primrec_err param_err);
251 val tnames = distinct (op =) (map (#1 o snd) rec_eqns);
252 val dts = find_dts dt_info tnames tnames;
254 map (fn (tname, {index, ...}) =>
256 (fst o the o find_first (fn f => (#1 o snd) f = tname)) rec_eqns))
258 val {descr, rec_names, rec_rewrites, ...} =
260 primrec_err ("datatypes " ^ commas_quote tnames ^ "\nare not mutually recursive")
262 val descr = map (fn (i, (tname, args, constrs)) => (i, (tname, args,
263 map (fn (cname, cargs) => (cname, fold (fn (dTs, dT) => fn dTs' =>
264 dTs' @ dTs @ [dT]) cargs [])) constrs))) descr;
265 val (fnameTs, fnss) =
266 fold_rev (process_fun thy descr rec_eqns) main_fns ([], []);
267 val (fs, defs) = fold_rev (get_fns fnss) (descr ~~ rec_names) ([], []);
268 val defs' = map (make_def thy fs) defs;
269 val nameTs1 = map snd fnameTs;
270 val nameTs2 = map fst rec_eqns;
271 val _ = if gen_eq_set (op =) (nameTs1, nameTs2) then ()
272 else primrec_err ("functions " ^ commas_quote (map fst nameTs2) ^
273 "\nare not mutually recursive");
275 if alt_name = "" then (space_implode "_" (map (Sign.base_name o #1) defs)) else alt_name;
276 val (defs_thms', thy') =
278 |> Sign.add_path primrec_name
279 |> fold_map def (map (fn ((name, t), _) => ((name, []), t)) defs');
280 val cert = cterm_of thy';
284 val Const c = head_of (fst (HOLogic.dest_eq (HOLogic.dest_Trueprop
285 (Logic.strip_imp_concl eq))));
286 val SOME i = AList.lookup op = (map swap fnameTs) c;
287 val SOME (_, _, constrs) = AList.lookup op = descr i;
288 val SOME (_, _, eqns) = AList.lookup op = rec_eqns c;
289 val SOME (cname, (_, cargs, _, _, _)) = find_first
290 (fn (_, (_, _, _, _, eq')) => eq = eq') eqns
291 in (i, find_index (fn (cname', _) => cname = cname') constrs, cargs) end;
294 unflat (map (fn (_, (_, _, constrs)) => constrs) descr) rec_rewrites;
295 val fvars = rec_rewrites |> hd |> concl_of |> HOLogic.dest_Trueprop |>
296 HOLogic.dest_eq |> fst |> strip_comb |> snd |> take_prefix is_Var |> fst;
297 val (pvars, ctxtvars) = List.partition
298 (equal HOLogic.boolT o body_type o snd)
299 (fold_rev Term.add_vars (map Logic.strip_assums_concl
300 (prems_of (hd rec_rewrites))) [] \\ map dest_Var fvars);
301 val cfs = defs' |> hd |> snd |> strip_comb |> snd |>
302 curry (List.take o swap) (length fvars) |> map cert;
303 val invs' = (case invs of
304 NONE => map (fn (i, _) =>
306 val SOME (_, T) = AList.lookup op = fnameTs i
307 val (Ts, U) = strip_type T
309 Abs ("x", List.drop (Ts, length lsrs + 1) ---> U, HOLogic.true_const)
311 | SOME invs' => invs');
312 val inst = (map cert fvars ~~ cfs) @
313 (map (cert o Var) pvars ~~ map cert invs') @
315 [ctxtvar] => [(cert (Var ctxtvar), cert (the_default HOLogic.unit fctxt))]
317 val rec_rewrites' = map (fn (_, eq) =>
319 val (i, j, cargs) = mk_idx eq
320 val th = nth (nth rec_rewritess i) j;
321 val cargs' = th |> concl_of |> HOLogic.dest_Trueprop |>
322 HOLogic.dest_eq |> fst |> strip_comb |> snd |> split_last |> snd |>
324 in (cargs, Logic.strip_imp_prems eq,
325 Drule.cterm_instantiate (inst @
326 (map (cterm_of thy') cargs' ~~ map (cterm_of thy' o Free) cargs)) th)
329 val prems = foldr1 (common_prefix op aconv) (map (prems_of o #3) rec_rewrites');
330 val cprems = map cert prems;
331 val asms = map Thm.assume cprems;
332 val premss = map (fn (cargs, eprems, eqn) =>
333 map (fn t => list_all_free (cargs, Logic.list_implies (eprems, t)))
334 (List.drop (prems_of eqn, length prems))) rec_rewrites';
335 val cpremss = map (map cert) premss;
336 val asmss = map (map Thm.assume) cpremss;
338 fun mk_eqn ((cargs, eprems, eqn), asms') =
340 val ceprems = map cert eprems;
341 val asms'' = map Thm.assume ceprems;
342 val ccargs = map (cert o Free) cargs;
343 val asms''' = map (fn th => implies_elim_list
344 (forall_elim_list ccargs th) asms'') asms'
346 implies_elim_list eqn (asms @ asms''') |>
347 implies_intr_list ceprems |>
348 forall_intr_list ccargs
351 val rule_prems = cprems @ flat cpremss;
352 val rule = implies_intr_list rule_prems
353 (Conjunction.intr_balanced (map mk_eqn (rec_rewrites' ~~ asmss)));
355 val goals = map (fn ((cargs, _, _), (_, eqn)) =>
356 (list_all_free (cargs, eqn), [])) (rec_rewrites' ~~ eqns);
362 (fn thss => ProofContext.theory (fn thy =>
364 val simps = map standard (flat thss);
366 fold_map note ((map fst eqns ~~ atts) ~~ map single simps) thy;
367 val simps'' = maps snd simps'
370 |> note (("simps", [Simplifier.simp_add]), simps'')
375 Proof.apply (Method.Basic (fn _ => Method.RAW_METHOD (fn _ =>
376 rewrite_goals_tac (map snd defs_thms') THEN
377 compose_tac (false, rule, length rule_prems) 1), Position.none)) |>
381 fun gen_primrec note def alt_name invs fctxt eqns thy =
383 val ((names, strings), srcss) = apfst split_list (split_list eqns);
384 val atts = map (map (Attrib.attribute thy)) srcss;
385 val eqn_ts = map (fn s => Syntax.read_prop_global thy s
386 handle ERROR msg => cat_error msg ("The error(s) above occurred for " ^ s)) strings;
387 val rec_ts = map (fn eq => head_of (fst (HOLogic.dest_eq
388 (HOLogic.dest_Trueprop (Logic.strip_imp_concl eq))))
389 handle TERM _ => primrec_eq_err thy "not a proper equation" eq) eqn_ts;
390 val (_, eqn_ts') = PrimrecPackage.unify_consts thy rec_ts eqn_ts
392 gen_primrec_i note def alt_name
393 (Option.map (map (Syntax.read_term_global thy)) invs)
394 (Option.map (Syntax.read_term_global thy) fctxt)
395 (names ~~ eqn_ts' ~~ atts) thy
398 fun thy_note ((name, atts), thms) =
399 PureThy.add_thmss [((name, thms), atts)] #-> (fn [thms] => pair (name, thms));
400 fun thy_def false ((name, atts), t) =
401 PureThy.add_defs_i false [((name, t), atts)] #-> (fn [thm] => pair (name, thm))
402 | thy_def true ((name, atts), t) =
403 PureThy.add_defs_unchecked_i false [((name, t), atts)] #-> (fn [thm] => pair (name, thm));
407 val add_primrec = gen_primrec thy_note (thy_def false);
408 val add_primrec_unchecked = gen_primrec thy_note (thy_def true);
409 val add_primrec_i = gen_primrec_i thy_note (thy_def false);
410 val add_primrec_unchecked_i = gen_primrec_i thy_note (thy_def true);
417 local structure P = OuterParse and K = OuterKeyword in
419 val _ = OuterSyntax.keywords ["invariant", "freshness_context"];
421 val parser1 = P.$$$ "freshness_context" |-- P.$$$ ":" |-- (P.term >> SOME);
423 P.$$$ "invariant" |-- P.$$$ ":" |--
424 (Scan.repeat1 P.term >> SOME) -- Scan.optional parser1 NONE ||
425 (parser1 >> pair NONE);
427 P.name -- Scan.optional parser2 (NONE, NONE) ||
428 (parser2 >> pair "");
430 (P.$$$ "unchecked" >> K true) -- Scan.optional parser3 ("", (NONE, NONE)) ||
431 (parser3 >> pair false);
433 Scan.optional (P.$$$ "(" |-- P.!!!
434 (parser4 --| P.$$$ ")")) (false, ("", (NONE, NONE)));
437 options -- Scan.repeat1 (SpecParse.opt_thm_name ":" -- P.prop);
440 OuterSyntax.command "nominal_primrec" "define primitive recursive functions on nominal datatypes" K.thy_goal
441 (primrec_decl >> (fn ((unchecked, (alt_name, (invs, fctxt))), eqns) =>
442 Toplevel.print o Toplevel.theory_to_proof
443 ((if unchecked then add_primrec_unchecked else add_primrec) alt_name invs fctxt
444 (map P.triple_swap eqns))));