1(*  Title:      HOL/Nominal/nominal_primrec.ML
2    Author:     Norbert Voelker, FernUni Hagen
3    Author:     Stefan Berghofer, TU Muenchen
4
5Package for defining functions on nominal datatypes by primitive recursion.
6Taken from HOL/Tools/primrec.ML
7*)
8
9signature NOMINAL_PRIMREC =
10sig
11  val primrec: term list option -> term option ->
12    (binding * typ option * mixfix) list ->
13    (binding * typ option * mixfix) list ->
14    Specification.multi_specs -> local_theory -> Proof.state
15  val primrec_cmd: string list option -> string option ->
16    (binding * string option * mixfix) list ->
17    (binding * string option * mixfix) list ->
18    Specification.multi_specs_cmd -> local_theory -> Proof.state
19end;
20
21structure NominalPrimrec : NOMINAL_PRIMREC =
22struct
23
24exception RecError of string;
25
26fun primrec_err s = error ("Nominal primrec definition error:\n" ^ s);
27fun primrec_eq_err lthy s eq =
28  primrec_err (s ^ "\nin\n" ^ quote (Syntax.string_of_term lthy eq));
29
30
31(* preprocessing of equations *)
32
33fun unquantify t =
34  let
35    val (vs, Ts) = split_list (strip_qnt_vars \<^const_name>\<open>Pure.all\<close> t);
36    val body = strip_qnt_body \<^const_name>\<open>Pure.all\<close> t;
37    val (vs', _) = fold_map Name.variant vs (Name.make_context (fold_aterms
38      (fn Free (v, _) => insert (op =) v | _ => I) body []))
39  in (curry subst_bounds (map2 (curry Free) vs' Ts |> rev) body) end;
40
41fun process_eqn lthy is_fixed spec rec_fns =
42  let
43    val eq = unquantify spec;
44    val (lhs, rhs) =
45      HOLogic.dest_eq (HOLogic.dest_Trueprop (Logic.strip_imp_concl eq))
46      handle TERM _ => raise RecError "not a proper equation";
47
48    val (recfun, args) = strip_comb lhs;
49    val fname = case recfun of Free (v, _) => if is_fixed v then v
50          else raise RecError "illegal head of function equation"
51      | _ => raise RecError "illegal head of function equation";
52
53    val (ls', rest)  = chop_prefix is_Free args;
54    val (middle, rs') = chop_suffix is_Free rest;
55    val rpos = length ls';
56
57    val (constr, cargs') = if null middle then raise RecError "constructor missing"
58      else strip_comb (hd middle);
59    val (cname, T) = dest_Const constr
60      handle TERM _ => raise RecError "ill-formed constructor";
61    val (tname, _) = dest_Type (body_type T) handle TYPE _ =>
62      raise RecError "cannot determine datatype associated with function"
63
64    val (ls, cargs, rs) =
65      (map dest_Free ls', map dest_Free cargs', map dest_Free rs')
66      handle TERM _ => raise RecError "illegal argument in pattern";
67    val lfrees = ls @ rs @ cargs;
68
69    fun check_vars _ [] = ()
70      | check_vars s vars = raise RecError (s ^ commas_quote (map fst vars))
71  in
72    if length middle > 1 then
73      raise RecError "more than one non-variable in pattern"
74    else
75     (check_vars "repeated variable names in pattern: " (duplicates (op =) lfrees);
76      check_vars "extra variables on rhs: "
77        (map dest_Free (Misc_Legacy.term_frees rhs) |> subtract (op =) lfrees
78          |> filter_out (is_fixed o fst));
79      case AList.lookup (op =) rec_fns fname of
80        NONE =>
81          (fname, (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"
87          else
88            AList.update (op =)
89              (fname, (tname, rpos, (cname, (ls, cargs, rs, rhs, eq))::eqns))
90              rec_fns)
91  end
92  handle RecError s => primrec_eq_err lthy s spec;
93
94val param_err = "Parameters must be the same for all recursive functions";
95
96fun process_fun lthy descr eqns (i, fname) (fnames, fnss) =
97  let
98    val (_, (tname, _, constrs)) = nth descr i;
99
100    (* substitute "fname ls x rs" by "y" for (x, (_, y)) in subs *)
101
102    fun subst [] t fs = (t, fs)
103      | subst subs (Abs (a, T, t)) fs =
104          fs
105          |> subst subs t
106          |-> (fn t' => pair (Abs (a, T, t')))
107      | subst subs (t as (_ $ _)) fs =
108          let
109            val (f, ts) = strip_comb t;
110          in
111            if is_Free f
112              andalso member (fn ((v, _), (w, _)) => v = w) eqns (dest_Free f) then
113              let
114                val (fname', _) = dest_Free f;
115                val (_, rpos, eqns') = the (AList.lookup (op =) eqns fname');
116                val (ls, rs'') = chop rpos ts
117                val (x', rs) = case rs'' of
118                    x' :: rs => (x', rs)
119                  | [] => raise RecError ("not enough arguments in recursive application\n"
120                      ^ "of function " ^ quote fname' ^ " on rhs");
121                val rs' = (case eqns' of
122                    (_, (ls', _, rs', _, _)) :: _ =>
123                      let val (rs1, rs2) = chop (length rs') rs
124                      in
125                        if ls = map Free ls' andalso rs1 = map Free rs' then rs2
126                        else raise RecError param_err
127                      end
128                  | _ => raise RecError ("no equations for " ^ quote fname'));
129                val (x, xs) = strip_comb x'
130              in case AList.lookup (op =) subs x
131               of NONE =>
132                    fs
133                    |> fold_map (subst subs) ts
134                    |-> (fn ts' => pair (list_comb (f, ts')))
135                | SOME (i', y) =>
136                    fs
137                    |> fold_map (subst subs) (xs @ rs')
138                    ||> process_fun lthy descr eqns (i', fname')
139                    |-> (fn ts' => pair (list_comb (y, ts')))
140              end
141            else
142              fs
143              |> fold_map (subst subs) (f :: ts)
144              |-> (fn (f'::ts') => pair (list_comb (f', ts')))
145          end
146      | subst _ t fs = (t, fs);
147
148    (* translate rec equations into function arguments suitable for rec comb *)
149
150    fun trans eqns (cname, cargs) (fnames', fnss', fns) =
151      (case AList.lookup (op =) eqns cname of
152          NONE => (warning ("No equation for constructor " ^ quote cname ^
153            "\nin definition of function " ^ quote fname);
154              (fnames', fnss', (Const (\<^const_name>\<open>undefined\<close>, dummyT))::fns))
155        | SOME (ls, cargs', rs, rhs, eq) =>
156            let
157              val recs = filter (Old_Datatype_Aux.is_rec_type o snd) (cargs' ~~ cargs);
158              val rargs = map fst recs;
159              val subs = map (rpair dummyT o fst)
160                (rev (Term.rename_wrt_term rhs rargs));
161              val (rhs', (fnames'', fnss'')) = subst (map2 (fn (x, y) => fn z =>
162                (Free x, (Old_Datatype_Aux.body_index y, Free z))) recs subs) rhs (fnames', fnss')
163                  handle RecError s => primrec_eq_err lthy s eq
164            in (fnames'', fnss'', fold_rev absfree (cargs' @ subs) rhs' :: fns)
165            end)
166
167  in (case AList.lookup (op =) fnames i of
168      NONE =>
169        if exists (fn (_, v) => fname = v) fnames then
170          raise RecError ("inconsistent functions for datatype " ^ quote tname)
171        else
172          let
173            val SOME (_, _, eqns' as (_, (ls, _, rs, _, _)) :: _) =
174              AList.lookup (op =) eqns fname;
175            val (fnames', fnss', fns) = fold_rev (trans eqns') constrs
176              ((i, fname)::fnames, fnss, [])
177          in
178            (fnames', (i, (fname, ls, rs, fns))::fnss')
179          end
180    | SOME fname' =>
181        if fname = fname' then (fnames, fnss)
182        else raise RecError ("inconsistent functions for datatype " ^ quote tname))
183  end;
184
185
186(* prepare functions needed for definitions *)
187
188fun get_fns fns ((i : int, (tname, _, constrs)), rec_name) (fs, defs) =
189  case AList.lookup (op =) fns i of
190     NONE =>
191       let
192         val dummy_fns = map (fn (_, cargs) => Const (\<^const_name>\<open>undefined\<close>,
193           replicate (length cargs + length (filter Old_Datatype_Aux.is_rec_type cargs))
194             dummyT ---> HOLogic.unitT)) constrs;
195         val _ = warning ("No function definition for datatype " ^ quote tname)
196       in
197         (dummy_fns @ fs, defs)
198       end
199   | SOME (fname, ls, rs, fs') => (fs' @ fs, (fname, ls, rs, rec_name, tname) :: defs);
200
201
202(* make definition *)
203
204fun make_def ctxt fixes fs (fname, ls, rs, rec_name, tname) =
205  let
206    val used = map fst (fold Term.add_frees fs []);
207    val x = (singleton (Name.variant_list used) "x", dummyT);
208    val frees = ls @ x :: rs;
209    val raw_rhs = fold_rev absfree frees
210      (list_comb (Const (rec_name, dummyT), fs @ [Free x]))
211    val def_name = Thm.def_name (Long_Name.base_name fname);
212    val rhs = singleton (Syntax.check_terms ctxt) raw_rhs;
213    val SOME var = get_first (fn ((b, _), mx) =>
214      if Binding.name_of b = fname then SOME (b, mx) else NONE) fixes;
215  in
216    ((var, ((Binding.name def_name, []), rhs)),
217     subst_bounds (rev (map Free frees), strip_abs_body rhs))
218  end;
219
220
221(* find datatypes which contain all datatypes in tnames' *)
222
223fun find_dts (dt_info : NominalDatatype.nominal_datatype_info Symtab.table) _ [] = []
224  | find_dts dt_info tnames' (tname::tnames) =
225      (case Symtab.lookup dt_info tname of
226          NONE => primrec_err (quote tname ^ " is not a nominal datatype")
227        | SOME dt =>
228            if subset (op =) (tnames', map (#1 o snd) (#descr dt)) then
229              (tname, dt)::(find_dts dt_info tnames' tnames)
230            else find_dts dt_info tnames' tnames);
231
232fun common_prefix eq ([], _) = []
233  | common_prefix eq (_, []) = []
234  | common_prefix eq (x :: xs, y :: ys) =
235      if eq (x, y) then x :: common_prefix eq (xs, ys) else [];
236
237local
238
239fun gen_primrec prep_spec prep_term invs fctxt raw_fixes raw_params raw_spec lthy =
240  let
241    val (fixes', spec) = fst (prep_spec (raw_fixes @ raw_params) raw_spec lthy);
242    val fixes = List.take (fixes', length raw_fixes);
243    val (names_atts, spec') = split_list spec;
244    val eqns' = map unquantify spec'
245    val eqns = fold_rev (process_eqn lthy (fn v => Variable.is_fixed lthy v
246      orelse exists (fn ((w, _), _) => v = Binding.name_of w) fixes)) spec' [];
247    val dt_info = NominalDatatype.get_nominal_datatypes (Proof_Context.theory_of lthy);
248    val lsrs :: lsrss = maps (fn (_, (_, _, eqns)) =>
249      map (fn (_, (ls, _, rs, _, _)) => ls @ rs) eqns) eqns
250    val _ =
251      (if forall (curry (eq_set (op =)) lsrs) lsrss andalso forall
252         (fn (_, (_, _, (_, (ls, _, rs, _, _)) :: eqns)) =>
253               forall (fn (_, (ls', _, rs', _, _)) =>
254                 ls = ls' andalso rs = rs') eqns
255           | _ => true) eqns
256       then () else primrec_err param_err);
257    val tnames = distinct (op =) (map (#1 o snd) eqns);
258    val dts = find_dts dt_info tnames tnames;
259    val main_fns =
260      map (fn (tname, {index, ...}) =>
261        (index,
262          (fst o the o find_first (fn (_, x) => #1 x = tname)) eqns))
263      dts;
264    val {descr, rec_names, rec_rewrites, ...} =
265      if null dts then
266        primrec_err ("datatypes " ^ commas_quote tnames ^ "\nare not mutually recursive")
267      else snd (hd dts);
268    val descr = map (fn (i, (tname, args, constrs)) => (i, (tname, args,
269      map (fn (cname, cargs) => (cname, fold (fn (dTs, dT) => fn dTs' =>
270        dTs' @ dTs @ [dT]) cargs [])) constrs))) descr;
271    val (fnames, fnss) = fold_rev (process_fun lthy descr eqns) main_fns ([], []);
272    val (fs, defs) = fold_rev (get_fns fnss) (descr ~~ rec_names) ([], []);
273    val defs' = map (make_def lthy fixes fs) defs;
274    val names1 = map snd fnames;
275    val names2 = map fst eqns;
276    val _ = if eq_set (op =) (names1, names2) then ()
277      else primrec_err ("functions " ^ commas_quote names2 ^
278        "\nare not mutually recursive");
279    val (defs_thms, lthy') = lthy |>
280      fold_map (apfst (snd o snd) oo Local_Theory.define o fst) defs';
281    val qualify = Binding.qualify false
282      (space_implode "_" (map (Long_Name.base_name o #1) defs));
283    val names_atts' = map (apfst qualify) names_atts;
284
285    fun mk_idx eq =
286      let
287        val Free (name, _) = head_of (fst (HOLogic.dest_eq (HOLogic.dest_Trueprop
288          (Logic.strip_imp_concl eq))));
289        val SOME i = AList.lookup op = (map swap fnames) name;
290        val SOME (_, _, constrs) = AList.lookup op = descr i;
291        val SOME (_, _, eqns'') = AList.lookup op = eqns name;
292        val SOME (cname, (_, cargs, _, _, _)) = find_first
293          (fn (_, (_, _, _, _, eq')) => eq = eq') eqns''
294      in (i, find_index (fn (cname', _) => cname = cname') constrs, cargs) end;
295
296    val rec_rewritess =
297      unflat (map (fn (_, (_, _, constrs)) => constrs) descr) rec_rewrites;
298    val fvars = rec_rewrites |> hd |> Thm.concl_of |> HOLogic.dest_Trueprop |>
299      HOLogic.dest_eq |> fst |> strip_comb |> snd |> take_prefix is_Var;
300    val (pvars, ctxtvars) = List.partition
301      (equal HOLogic.boolT o body_type o snd)
302      (subtract (op =)
303        (Term.add_vars (Thm.concl_of (hd rec_rewrites)) [])
304        (fold_rev (Term.add_vars o Logic.strip_assums_concl)
305           (Thm.prems_of (hd rec_rewrites)) []));
306    val cfs = defs' |> hd |> snd |> strip_comb |> snd |>
307      curry (List.take o swap) (length fvars) |> map (Thm.cterm_of lthy');
308    val invs' = (case invs of
309        NONE => map (fn (i, _) =>
310          Abs ("x", fastype_of (snd (nth defs' i)), \<^term>\<open>True\<close>)) descr
311      | SOME invs' => map (prep_term lthy') invs');
312    val inst = (map (#1 o dest_Var) fvars ~~ cfs) @
313      (map #1 pvars ~~ map (Thm.cterm_of lthy') invs') @
314      (case ctxtvars of
315         [ctxtvar] => [(#1 ctxtvar,
316           Thm.cterm_of lthy' (the_default HOLogic.unit (Option.map (prep_term lthy') fctxt)))]
317       | _ => []);
318    val rec_rewrites' = map (fn eq =>
319      let
320        val (i, j, cargs) = mk_idx eq
321        val th = nth (nth rec_rewritess i) j;
322        val cargs' = th |> Thm.concl_of |> HOLogic.dest_Trueprop |>
323          HOLogic.dest_eq |> fst |> strip_comb |> snd |> List.last |>
324          strip_comb |> snd
325      in (cargs, Logic.strip_imp_prems eq,
326        infer_instantiate lthy' (inst @
327          (map (#1 o dest_Var) cargs' ~~ map (Thm.cterm_of lthy' o Free) cargs)) th)
328      end) eqns';
329
330    val prems = foldr1 (common_prefix op aconv) (map (Thm.prems_of o #3) rec_rewrites');
331    val cprems = map (Thm.cterm_of lthy') prems;
332    val asms = map Thm.assume cprems;
333    val premss = map (fn (cargs, eprems, eqn) =>
334      map (fn t => fold_rev (Logic.all o Free) cargs (Logic.list_implies (eprems, t)))
335        (List.drop (Thm.prems_of eqn, length prems))) rec_rewrites';
336    val cpremss = map (map (Thm.cterm_of lthy')) premss;
337    val asmss = map (map Thm.assume) cpremss;
338
339    fun mk_eqn ((cargs, eprems, eqn), asms') =
340      let
341        val ceprems = map (Thm.cterm_of lthy') eprems;
342        val asms'' = map Thm.assume ceprems;
343        val ccargs = map (Thm.cterm_of lthy' o Free) cargs;
344        val asms''' = map (fn th => implies_elim_list
345          (forall_elim_list ccargs th) asms'') asms'
346      in
347        implies_elim_list eqn (asms @ asms''') |>
348        implies_intr_list ceprems |>
349        forall_intr_list ccargs
350      end;
351
352    val rule_prems = cprems @ flat cpremss;
353    val rule = implies_intr_list rule_prems
354      (Conjunction.intr_balanced (map mk_eqn (rec_rewrites' ~~ asmss)));
355
356    val goals = map (fn ((cargs, _, _), eqn) =>
357      (fold_rev (Logic.all o Free) cargs eqn, [])) (rec_rewrites' ~~ eqns');
358
359  in
360    lthy' |>
361    Variable.add_fixes (map fst lsrs) |> snd |>
362    Proof.theorem NONE
363      (fn thss => fn goal_ctxt =>
364        let
365          val simps = Proof_Context.export goal_ctxt lthy' (flat thss);
366          val (simps', lthy'') =
367            fold_map Local_Theory.note (names_atts' ~~ map single simps) lthy';
368        in
369          lthy''
370          |> Local_Theory.note
371            ((qualify (Binding.name "simps"), @{attributes [simp, nitpick_simp]}), maps snd simps')
372          |> snd
373        end)
374      [goals] |>
375    Proof.refine_singleton (Method.Basic (fn ctxt => fn _ =>
376      CONTEXT_TACTIC
377       (rewrite_goals_tac ctxt defs_thms THEN
378        compose_tac ctxt (false, rule, length rule_prems) 1)))
379  end;
380
381in
382
383val primrec = gen_primrec Specification.check_multi_specs (K I);
384val primrec_cmd = gen_primrec Specification.read_multi_specs Syntax.read_term;
385
386end;
387
388
389(* outer syntax *)
390
391val freshness_context = Parse.reserved "freshness_context";
392val invariant = Parse.reserved "invariant";
393
394fun unless_flag scan = Scan.unless ((freshness_context || invariant) -- \<^keyword>\<open>:\<close>) scan;
395
396val parser1 = (freshness_context -- \<^keyword>\<open>:\<close>) |-- unless_flag Parse.term >> SOME;
397val parser2 = (invariant -- \<^keyword>\<open>:\<close>) |--
398    (Scan.repeat1 (unless_flag Parse.term) >> SOME) -- Scan.optional parser1 NONE ||
399  (parser1 >> pair NONE);
400val options =
401  Scan.optional (\<^keyword>\<open>(\<close> |-- Parse.!!! (parser2 --| \<^keyword>\<open>)\<close>)) (NONE, NONE);
402
403val _ =
404  Outer_Syntax.local_theory_to_proof \<^command_keyword>\<open>nominal_primrec\<close>
405    "define primitive recursive functions on nominal datatypes"
406    (options -- Parse.vars -- Parse.for_fixes -- Parse_Spec.where_multi_specs
407      >> (fn ((((invs, fctxt), vars), params), specs) =>
408        primrec_cmd invs fctxt vars params specs));
409
410end;
411