1 (* Title: HOL/SPARK/Tools/spark_vcs.ML
2 Author: Stefan Berghofer
3 Copyright: secunet Security Networks AG
5 Store for verification conditions generated by SPARK/Ada.
10 val set_vcs: Fdl_Parser.decls -> Fdl_Parser.rules ->
11 (string * string) * Fdl_Parser.vcs -> Path.T -> string -> theory -> theory
12 val add_proof_fun: (typ option -> 'a -> term) ->
13 string * ((string list * string) option * 'a) ->
15 val add_type: string * typ -> theory -> theory
16 val lookup_vc: theory -> string -> (Element.context_i list *
17 (string * thm list option * Element.context_i * Element.statement_i)) option
18 val get_vcs: theory ->
19 Element.context_i list * (binding * thm) list * (string *
20 (string * thm list option * Element.context_i * Element.statement_i)) list
21 val mark_proved: string -> thm list -> theory -> theory
22 val close: theory -> theory
23 val is_closed: theory -> bool
26 structure SPARK_VCs: SPARK_VCS =
34 fun err_unfinished () = error "An unfinished SPARK environment is still open."
36 val strip_number = pairself implode o take_suffix Fdl_Lexer.is_digit o raw_explode;
38 val name_ord = prod_ord string_ord (option_ord int_ord) o
39 pairself (strip_number ##> Int.fromString);
41 structure VCtab = Table(type key = string val ord = name_ord);
43 structure VCs = Theory_Data
46 {pfuns: ((string list * string) option * term) Symtab.table,
47 type_map: typ Symtab.table,
49 {ctxt: Element.context_i list,
50 defs: (binding * thm) list,
52 funs: (string list * string) tab,
53 ids: (term * string) Symtab.table * Name.context,
55 vcs: (string * thm list option *
56 (string * expr) list * (string * expr) list) VCtab.table,
58 prefix: string} option}
59 val empty : T = {pfuns = Symtab.empty, type_map = Symtab.empty, env = NONE}
61 fun merge ({pfuns = pfuns1, type_map = type_map1, env = NONE},
62 {pfuns = pfuns2, type_map = type_map2, env = NONE}) =
63 {pfuns = Symtab.merge (eq_pair (op =) (op aconv)) (pfuns1, pfuns2),
64 type_map = Symtab.merge (op =) (type_map1, type_map2),
66 | merge _ = err_unfinished ()
72 val to_lower = raw_explode #> map Symbol.to_ascii_lower #> implode;
74 val lcase_eq = (op =) o pairself (to_lower o Long_Name.base_name);
76 fun lookup_prfx "" tab s = Symtab.lookup tab s
77 | lookup_prfx prfx tab s = (case Symtab.lookup tab s of
78 NONE => Symtab.lookup tab (prfx ^ "__" ^ s)
83 fun strip ys [] = ("", implode ys)
84 | strip ys ("_" :: "_" :: xs) = (implode (rev xs), implode ys)
85 | strip ys (x :: xs) = strip (x :: ys) xs
86 in strip [] (rev (raw_explode s)) end;
89 let val T = fastype_of t
90 in Const (s, T --> T) $ t end;
94 val setT = fastype_of t;
95 val T = HOLogic.dest_setT setT;
96 val U = HOLogic.dest_setT (fastype_of u)
98 Const (@{const_name Sigma}, setT --> (T --> HOLogic.mk_setT U) -->
99 HOLogic.mk_setT (HOLogic.mk_prodT (T, U))) $ t $ Abs ("", T, u)
102 fun get_type thy prfx ty =
103 let val {type_map, ...} = VCs.get thy
104 in lookup_prfx prfx type_map ty end;
106 fun mk_type _ _ "integer" = HOLogic.intT
107 | mk_type _ _ "boolean" = HOLogic.boolT
108 | mk_type thy prfx ty =
109 (case get_type thy prfx ty of
111 Syntax.check_typ (Proof_Context.init_global thy)
112 (Type (Sign.full_name thy (Binding.name ty), []))
115 val booleanN = "boolean";
116 val integerN = "integer";
118 fun define_overloaded (def_name, eq) lthy =
120 val ((c, _), rhs) = eq |> Syntax.check_term lthy |>
121 Logic.dest_equals |>> dest_Free;
122 val ((_, (_, thm)), lthy') = Local_Theory.define
123 ((Binding.name c, NoSyn), ((Binding.name def_name, []), rhs)) lthy
124 val ctxt_thy = Proof_Context.init_global (Proof_Context.theory_of lthy');
125 val thm' = singleton (Proof_Context.export lthy' ctxt_thy) thm
126 in (thm', lthy') end;
128 fun strip_underscores s =
129 strip_underscores (unsuffix "_" s) handle Fail _ => s;
132 unsuffix "~" s ^ "_init" handle Fail _ => s;
134 val mangle_name = strip_underscores #> strip_tilde;
136 fun mk_variables thy prfx xs ty (tab, ctxt) =
138 val T = mk_type thy prfx ty;
139 val (ys, ctxt') = fold_map Name.variant (map mangle_name xs) ctxt;
140 val zs = map (Free o rpair T) ys;
141 in (zs, (fold (Symtab.update o apsnd (rpair ty)) (xs ~~ zs) tab, ctxt')) end;
143 fun get_record_info thy T = (case Record.dest_recTs T of
144 [(tyname, [@{typ unit}])] =>
145 Record.get_info thy (Long_Name.qualifier tyname)
148 fun find_field fname = find_first (curry lcase_eq fname o fst);
150 fun find_field' fname = get_first (fn (flds, fldty) =>
151 if member (op =) flds fname then SOME fldty else NONE);
153 fun assoc_ty_err thy T s msg =
154 error ("Type " ^ Syntax.string_of_typ_global thy T ^
155 " associated with SPARK type " ^ s ^ "\n" ^ msg);
158 (** generate properties of enumeration types **)
160 fun add_enum_type tyname tyname' thy =
162 val {case_name, ...} = the (Datatype.get_info thy tyname');
163 val cs = map Const (the (Datatype.get_constrs thy tyname'));
165 val T = Type (tyname', []);
166 val p = Const (@{const_name pos}, T --> HOLogic.intT);
167 val v = Const (@{const_name val}, HOLogic.intT --> T);
168 val card = Const (@{const_name card},
169 HOLogic.mk_setT T --> HOLogic.natT) $ HOLogic.mk_UNIV T;
171 fun mk_binrel_def s f = Logic.mk_equals
172 (Const (s, T --> T --> HOLogic.boolT),
173 Abs ("x", T, Abs ("y", T,
174 Const (s, HOLogic.intT --> HOLogic.intT --> HOLogic.boolT) $
175 (f $ Bound 1) $ (f $ Bound 0))));
177 val (((def1, def2), def3), lthy) = thy |>
179 Class.instantiation ([tyname'], [], @{sort spark_enum}) |>
181 define_overloaded ("pos_" ^ tyname ^ "_def", Logic.mk_equals
183 list_comb (Const (case_name, replicate k HOLogic.intT @
184 [T] ---> HOLogic.intT),
185 map (HOLogic.mk_number HOLogic.intT) (0 upto k - 1)))) ||>>
187 define_overloaded ("less_eq_" ^ tyname ^ "_def",
188 mk_binrel_def @{const_name less_eq} p) ||>>
189 define_overloaded ("less_" ^ tyname ^ "_def",
190 mk_binrel_def @{const_name less} p);
192 val UNIV_eq = Goal.prove lthy [] []
193 (HOLogic.mk_Trueprop (HOLogic.mk_eq
194 (HOLogic.mk_UNIV T, HOLogic.mk_set T cs)))
196 rtac @{thm subset_antisym} 1 THEN
197 rtac @{thm subsetI} 1 THEN
198 Datatype_Aux.exh_tac (K (#exhaust (Datatype.the_info
199 (Proof_Context.theory_of lthy) tyname'))) 1 THEN
200 ALLGOALS (asm_full_simp_tac (simpset_of lthy)));
202 val finite_UNIV = Goal.prove lthy [] []
203 (HOLogic.mk_Trueprop (Const (@{const_name finite},
204 HOLogic.mk_setT T --> HOLogic.boolT) $ HOLogic.mk_UNIV T))
205 (fn _ => simp_tac (simpset_of lthy addsimps [UNIV_eq]) 1);
207 val card_UNIV = Goal.prove lthy [] []
208 (HOLogic.mk_Trueprop (HOLogic.mk_eq
209 (card, HOLogic.mk_number HOLogic.natT k)))
210 (fn _ => simp_tac (simpset_of lthy addsimps [UNIV_eq]) 1);
212 val range_pos = Goal.prove lthy [] []
213 (HOLogic.mk_Trueprop (HOLogic.mk_eq
214 (Const (@{const_name image}, (T --> HOLogic.intT) -->
215 HOLogic.mk_setT T --> HOLogic.mk_setT HOLogic.intT) $
216 p $ HOLogic.mk_UNIV T,
217 Const (@{const_name atLeastLessThan}, HOLogic.intT -->
218 HOLogic.intT --> HOLogic.mk_setT HOLogic.intT) $
219 HOLogic.mk_number HOLogic.intT 0 $
220 (@{term int} $ card))))
222 simp_tac (simpset_of lthy addsimps [card_UNIV]) 1 THEN
223 simp_tac (simpset_of lthy addsimps [UNIV_eq, def1]) 1 THEN
224 rtac @{thm subset_antisym} 1 THEN
225 simp_tac (simpset_of lthy) 1 THEN
226 rtac @{thm subsetI} 1 THEN
227 asm_full_simp_tac (simpset_of lthy addsimps @{thms interval_expand}
228 delsimps @{thms atLeastLessThan_iff}) 1);
231 Class.prove_instantiation_instance (fn _ =>
232 Class.intro_classes_tac [] THEN
233 rtac finite_UNIV 1 THEN
234 rtac range_pos 1 THEN
235 simp_tac (HOL_basic_ss addsimps [def3]) 1 THEN
236 simp_tac (HOL_basic_ss addsimps [def2]) 1) lthy;
238 val (pos_eqs, val_eqs) = split_list (map_index (fn (i, c) =>
240 val n = HOLogic.mk_number HOLogic.intT i;
241 val th = Goal.prove lthy' [] []
242 (HOLogic.mk_Trueprop (HOLogic.mk_eq (p $ c, n)))
243 (fn _ => simp_tac (simpset_of lthy' addsimps [def1]) 1);
244 val th' = Goal.prove lthy' [] []
245 (HOLogic.mk_Trueprop (HOLogic.mk_eq (v $ n, c)))
247 rtac (@{thm inj_pos} RS @{thm injD}) 1 THEN
248 simp_tac (simpset_of lthy' addsimps
249 [@{thm pos_val}, range_pos, card_UNIV, th]) 1)
250 in (th, th') end) cs);
252 val first_el = Goal.prove lthy' [] []
253 (HOLogic.mk_Trueprop (HOLogic.mk_eq
254 (Const (@{const_name first_el}, T), hd cs)))
255 (fn _ => simp_tac (simpset_of lthy' addsimps
256 [@{thm first_el_def}, hd val_eqs]) 1);
258 val last_el = Goal.prove lthy' [] []
259 (HOLogic.mk_Trueprop (HOLogic.mk_eq
260 (Const (@{const_name last_el}, T), List.last cs)))
261 (fn _ => simp_tac (simpset_of lthy' addsimps
262 [@{thm last_el_def}, List.last val_eqs, card_UNIV]) 1);
266 ((Binding.name (tyname ^ "_card"), @{attributes [simp]}), [card_UNIV]) ||>>
268 ((Binding.name (tyname ^ "_pos"), @{attributes [simp]}), pos_eqs) ||>>
270 ((Binding.name (tyname ^ "_val"), @{attributes [simp]}), val_eqs) ||>>
272 ((Binding.name (tyname ^ "_first_el"), @{attributes [simp]}), [first_el]) ||>>
274 ((Binding.name (tyname ^ "_last_el"), @{attributes [simp]}), [last_el]) |> snd |>
275 Local_Theory.exit_global
279 fun check_no_assoc thy prfx s = case get_type thy prfx s of
281 | SOME _ => error ("Cannot associate a type with " ^ s ^
282 "\nsince it is no record or enumeration type");
284 fun check_enum [] [] = NONE
285 | check_enum els [] = SOME ("has no element(s) " ^ commas els)
286 | check_enum [] cs = SOME ("has extra element(s) " ^
287 commas (map (Long_Name.base_name o fst) cs))
288 | check_enum (el :: els) ((cname, _) :: cs) =
289 if lcase_eq (el, cname) then check_enum els cs
290 else SOME ("either has no element " ^ el ^
291 " or it is at the wrong position");
293 fun add_type_def prfx (s, Basic_Type ty) (ids, thy) =
294 (check_no_assoc thy prfx s;
296 Typedecl.abbrev_global (Binding.name s, [], NoSyn)
297 (mk_type thy prfx ty) thy |> snd))
299 | add_type_def prfx (s, Enum_Type els) ((tab, ctxt), thy) =
301 val (thy', tyname) = (case get_type thy prfx s of
304 val tyb = Binding.name s;
305 val tyname = Sign.full_name thy tyb
308 Datatype.add_datatype {strict = true, quiet = true}
309 [((tyb, [], NoSyn), map (fn s => (Binding.name s, [], NoSyn)) els)] |> snd |>
310 add_enum_type s tyname,
313 | SOME (T as Type (tyname, [])) =>
314 (case Datatype.get_constrs thy tyname of
315 NONE => assoc_ty_err thy T s "is not a datatype"
318 val (prfx', _) = strip_prfx s;
320 if prfx' = "" then els
321 else map (unprefix (prfx' ^ "__")) els
322 handle Fail _ => error ("Bad enumeration type " ^ s)
324 case check_enum els' cs of
325 NONE => (thy, tyname)
326 | SOME msg => assoc_ty_err thy T s msg
328 val cs = map Const (the (Datatype.get_constrs thy' tyname));
330 ((fold (Symtab.update_new o apsnd (rpair s)) (els ~~ cs) tab,
331 fold Name.declare els ctxt),
335 | add_type_def prfx (s, Array_Type (argtys, resty)) (ids, thy) =
336 (check_no_assoc thy prfx s;
338 Typedecl.abbrev_global (Binding.name s, [], NoSyn)
339 (foldr1 HOLogic.mk_prodT (map (mk_type thy prfx) argtys) -->
340 mk_type thy prfx resty) thy |> snd))
342 | add_type_def prfx (s, Record_Type fldtys) (ids, thy) =
344 let val fldTs = maps (fn (flds, ty) =>
345 map (rpair (mk_type thy prfx ty)) flds) fldtys
346 in case get_type thy prfx s of
348 Record.add_record ([], Binding.name s) NONE
349 (map (fn (fld, T) => (Binding.name fld, T, NoSyn)) fldTs) thy
351 (case get_record_info thy rT of
352 NONE => assoc_ty_err thy rT s "is not a record type"
353 | SOME {fields, ...} =>
354 (case subtract (lcase_eq o pairself fst) fldTs fields of
356 | flds => assoc_ty_err thy rT s ("has extra field(s) " ^
357 commas (map (Long_Name.base_name o fst) flds));
359 case AList.lookup lcase_eq fields fld of
360 NONE => assoc_ty_err thy rT s ("has no field " ^ fld)
361 | SOME U => T = U orelse assoc_ty_err thy rT s
363 fld ^ " whose type\n" ^
364 Syntax.string_of_typ_global thy U ^
365 "\ndoes not match declared type\n" ^
366 Syntax.string_of_typ_global thy T)) fldTs;
370 | add_type_def prfx (s, Pending_Type) (ids, thy) =
372 case get_type thy prfx s of
374 | NONE => Typedecl.typedecl_global
375 (Binding.name s, [], NoSyn) thy |> snd);
378 fun term_of_expr thy prfx types pfuns =
380 fun tm_of vs (Funct ("->", [e, e'])) =
381 (HOLogic.mk_imp (fst (tm_of vs e), fst (tm_of vs e')), booleanN)
383 | tm_of vs (Funct ("<->", [e, e'])) =
384 (HOLogic.mk_eq (fst (tm_of vs e), fst (tm_of vs e')), booleanN)
386 | tm_of vs (Funct ("or", [e, e'])) =
387 (HOLogic.mk_disj (fst (tm_of vs e), fst (tm_of vs e')), booleanN)
389 | tm_of vs (Funct ("and", [e, e'])) =
390 (HOLogic.mk_conj (fst (tm_of vs e), fst (tm_of vs e')), booleanN)
392 | tm_of vs (Funct ("not", [e])) =
393 (HOLogic.mk_not (fst (tm_of vs e)), booleanN)
395 | tm_of vs (Funct ("=", [e, e'])) =
396 (HOLogic.mk_eq (fst (tm_of vs e), fst (tm_of vs e')), booleanN)
398 | tm_of vs (Funct ("<>", [e, e'])) = (HOLogic.mk_not
399 (HOLogic.mk_eq (fst (tm_of vs e), fst (tm_of vs e'))), booleanN)
401 | tm_of vs (Funct ("<", [e, e'])) = (HOLogic.mk_binrel @{const_name less}
402 (fst (tm_of vs e), fst (tm_of vs e')), booleanN)
404 | tm_of vs (Funct (">", [e, e'])) = (HOLogic.mk_binrel @{const_name less}
405 (fst (tm_of vs e'), fst (tm_of vs e)), booleanN)
407 | tm_of vs (Funct ("<=", [e, e'])) = (HOLogic.mk_binrel @{const_name less_eq}
408 (fst (tm_of vs e), fst (tm_of vs e')), booleanN)
410 | tm_of vs (Funct (">=", [e, e'])) = (HOLogic.mk_binrel @{const_name less_eq}
411 (fst (tm_of vs e'), fst (tm_of vs e)), booleanN)
413 | tm_of vs (Funct ("+", [e, e'])) = (HOLogic.mk_binop @{const_name plus}
414 (fst (tm_of vs e), fst (tm_of vs e')), integerN)
416 | tm_of vs (Funct ("-", [e, e'])) = (HOLogic.mk_binop @{const_name minus}
417 (fst (tm_of vs e), fst (tm_of vs e')), integerN)
419 | tm_of vs (Funct ("*", [e, e'])) = (HOLogic.mk_binop @{const_name times}
420 (fst (tm_of vs e), fst (tm_of vs e')), integerN)
422 | tm_of vs (Funct ("/", [e, e'])) = (HOLogic.mk_binop @{const_name divide}
423 (fst (tm_of vs e), fst (tm_of vs e')), integerN)
425 | tm_of vs (Funct ("div", [e, e'])) = (HOLogic.mk_binop @{const_name sdiv}
426 (fst (tm_of vs e), fst (tm_of vs e')), integerN)
428 | tm_of vs (Funct ("mod", [e, e'])) = (HOLogic.mk_binop @{const_name mod}
429 (fst (tm_of vs e), fst (tm_of vs e')), integerN)
431 | tm_of vs (Funct ("-", [e])) =
432 (mk_unop @{const_name uminus} (fst (tm_of vs e)), integerN)
434 | tm_of vs (Funct ("**", [e, e'])) =
435 (Const (@{const_name power}, HOLogic.intT --> HOLogic.natT -->
436 HOLogic.intT) $ fst (tm_of vs e) $
437 (@{const nat} $ fst (tm_of vs e')), integerN)
439 | tm_of (tab, _) (Ident s) =
440 (case Symtab.lookup tab s of
442 | NONE => (case lookup_prfx prfx pfuns s of
443 SOME (SOME ([], resty), t) => (t, resty)
444 | _ => error ("Undeclared identifier " ^ s)))
446 | tm_of _ (Number i) = (HOLogic.mk_number HOLogic.intT i, integerN)
448 | tm_of vs (Quantifier (s, xs, ty, e)) =
450 val (ys, vs') = mk_variables thy prfx xs ty vs;
452 "for_all" => HOLogic.mk_all
453 | "for_some" => HOLogic.mk_exists)
455 (fold_rev (fn Free (x, T) => fn t => q (x, T, t))
456 ys (fst (tm_of vs' e)),
460 | tm_of vs (Funct (s, es)) =
462 (* record field selection *)
463 (case try (unprefix "fld_") s of
464 SOME fname => (case es of
467 val (t, rcdty) = tm_of vs e;
468 val rT = mk_type thy prfx rcdty
469 in case (get_record_info thy rT, lookup types rcdty) of
470 (SOME {fields, ...}, SOME (Record_Type fldtys)) =>
471 (case (find_field fname fields,
472 find_field' fname fldtys) of
473 (SOME (fname', fT), SOME fldty) =>
474 (Const (fname', rT --> fT) $ t, fldty)
475 | _ => error ("Record " ^ rcdty ^
476 " has no field named " ^ fname))
477 | _ => error (rcdty ^ " is not a record type")
479 | _ => error ("Function " ^ s ^ " expects one argument"))
482 (* record field update *)
483 (case try (unprefix "upf_") s of
484 SOME fname => (case es of
487 val (t, rcdty) = tm_of vs e;
488 val rT = mk_type thy prfx rcdty;
489 val (u, fldty) = tm_of vs e';
490 val fT = mk_type thy prfx fldty
491 in case get_record_info thy rT of
492 SOME {fields, ...} =>
493 (case find_field fname fields of
496 (Const (fname' ^ "_update",
497 (fT --> fT) --> rT --> rT) $
498 Abs ("x", fT, u) $ t,
500 else error ("Type\n" ^
501 Syntax.string_of_typ_global thy fT ^
502 "\ndoes not match type\n" ^
503 Syntax.string_of_typ_global thy fU ^
504 "\nof field " ^ fname)
505 | NONE => error ("Record " ^ rcdty ^
506 " has no field named " ^ fname))
507 | _ => error (rcdty ^ " is not a record type")
509 | _ => error ("Function " ^ s ^ " expects two arguments"))
512 (* enumeration type to integer *)
513 (case try (unsuffix "__pos") s of
514 SOME tyname => (case es of
515 [e] => (Const (@{const_name pos},
516 mk_type thy prfx tyname --> HOLogic.intT) $ fst (tm_of vs e),
518 | _ => error ("Function " ^ s ^ " expects one argument"))
521 (* integer to enumeration type *)
522 (case try (unsuffix "__val") s of
523 SOME tyname => (case es of
524 [e] => (Const (@{const_name val},
525 HOLogic.intT --> mk_type thy prfx tyname) $ fst (tm_of vs e),
527 | _ => error ("Function " ^ s ^ " expects one argument"))
530 (* successor / predecessor of enumeration type element *)
531 if s = "succ" orelse s = "pred" then (case es of
534 val (t, tyname) = tm_of vs e;
535 val T = mk_type thy prfx tyname
537 (if s = "succ" then @{const_name succ}
538 else @{const_name pred}, T --> T) $ t, tyname)
540 | _ => error ("Function " ^ s ^ " expects one argument"))
542 (* user-defined proof function *)
544 (case lookup_prfx prfx pfuns s of
545 SOME (SOME (_, resty), t) =>
546 (list_comb (t, map (fst o tm_of vs) es), resty)
547 | _ => error ("Undeclared proof function " ^ s))))))
549 | tm_of vs (Element (e, es)) =
550 let val (t, ty) = tm_of vs e
551 in case lookup types ty of
552 SOME (Array_Type (_, elty)) =>
553 (t $ foldr1 HOLogic.mk_prod (map (fst o tm_of vs) es), elty)
554 | _ => error (ty ^ " is not an array type")
557 | tm_of vs (Update (e, es, e')) =
558 let val (t, ty) = tm_of vs e
559 in case lookup types ty of
560 SOME (Array_Type (idxtys, elty)) =>
562 val T = foldr1 HOLogic.mk_prodT
563 (map (mk_type thy prfx) idxtys);
564 val U = mk_type thy prfx elty;
567 (Const (@{const_name fun_upd}, fT --> T --> U --> fT) $
568 t $ foldr1 HOLogic.mk_prod (map (fst o tm_of vs) es) $
572 | _ => error (ty ^ " is not an array type")
575 | tm_of vs (Record (s, flds)) =
577 val T = mk_type thy prfx s;
578 val {extension = (ext_name, _), fields, ...} =
579 (case get_record_info thy T of
580 NONE => error (s ^ " is not a record type")
581 | SOME info => info);
582 val flds' = map (apsnd (tm_of vs)) flds;
583 val fnames = map (Long_Name.base_name o fst) fields;
584 val fnames' = map fst flds;
585 val (fvals, ftys) = split_list (map (fn s' =>
586 case AList.lookup lcase_eq flds' s' of
587 SOME fval_ty => fval_ty
588 | NONE => error ("Field " ^ s' ^ " missing in record " ^ s))
590 val _ = (case subtract lcase_eq fnames fnames' of
592 | xs => error ("Extra field(s) " ^ commas xs ^
594 val _ = (case duplicates (op =) fnames' of
596 | xs => error ("Duplicate field(s) " ^ commas xs ^
601 map (mk_type thy prfx) ftys @ [HOLogic.unitT] ---> T),
602 fvals @ [HOLogic.unit]),
606 | tm_of vs (Array (s, default, assocs)) =
607 (case lookup types s of
608 SOME (Array_Type (idxtys, elty)) =>
610 val Ts = map (mk_type thy prfx) idxtys;
611 val T = foldr1 HOLogic.mk_prodT Ts;
612 val U = mk_type thy prfx elty;
613 fun mk_idx' T (e, NONE) = HOLogic.mk_set T [fst (tm_of vs e)]
614 | mk_idx' T (e, SOME e') = Const (@{const_name atLeastAtMost},
615 T --> T --> HOLogic.mk_setT T) $
616 fst (tm_of vs e) $ fst (tm_of vs e');
618 if length Ts <> length idx then
619 error ("Arity mismatch in construction of array " ^ s)
620 else foldr1 mk_times (map2 mk_idx' Ts idx);
621 fun mk_upd (idxs, e) t =
622 if length idxs = 1 andalso forall (is_none o snd) (hd idxs)
624 Const (@{const_name fun_upd}, (T --> U) -->
625 T --> U --> T --> U) $ t $
626 foldl1 HOLogic.mk_prod
627 (map (fst o tm_of vs o fst) (hd idxs)) $
630 Const (@{const_name fun_upds}, (T --> U) -->
631 HOLogic.mk_setT T --> U --> T --> U) $ t $
632 foldl1 (HOLogic.mk_binop @{const_name sup})
638 SOME e => Abs ("x", T, fst (tm_of vs e))
639 | NONE => Const (@{const_name undefined}, T --> U)),
642 | _ => error (s ^ " is not an array type"))
647 fun term_of_rule thy prfx types pfuns ids rule =
648 let val tm_of = fst o term_of_expr thy prfx types pfuns ids
650 Inference_Rule (es, e) => Logic.list_implies
651 (map (HOLogic.mk_Trueprop o tm_of) es, HOLogic.mk_Trueprop (tm_of e))
652 | Substitution_Rule (es, e, e') => Logic.list_implies
653 (map (HOLogic.mk_Trueprop o tm_of) es,
654 HOLogic.mk_Trueprop (HOLogic.mk_eq (tm_of e, tm_of e')))
658 val builtin = Symtab.make (map (rpair ())
659 ["->", "<->", "or", "and", "not", "=", "<>", "<", ">", "<=", ">=",
660 "+", "-", "*", "/", "div", "mod", "**"]);
662 fun complex_expr (Number _) = false
663 | complex_expr (Ident _) = false
664 | complex_expr (Funct (s, es)) =
665 not (Symtab.defined builtin s) orelse exists complex_expr es
666 | complex_expr (Quantifier (_, _, _, e)) = complex_expr e
667 | complex_expr _ = true;
669 fun complex_rule (Inference_Rule (es, e)) =
670 complex_expr e orelse exists complex_expr es
671 | complex_rule (Substitution_Rule (es, e, e')) =
672 complex_expr e orelse complex_expr e' orelse
673 exists complex_expr es;
676 Symtab.defined builtin orf
677 can (unprefix "fld_") orf can (unprefix "upf_") orf
678 can (unsuffix "__pos") orf can (unsuffix "__val") orf
679 equal "succ" orf equal "pred";
681 fun fold_opt f = the_default I o Option.map f;
682 fun fold_pair f g (x, y) = f x #> g y;
684 fun fold_expr f g (Funct (s, es)) = f s #> fold (fold_expr f g) es
685 | fold_expr f g (Ident s) = g s
686 | fold_expr f g (Number _) = I
687 | fold_expr f g (Quantifier (_, _, _, e)) = fold_expr f g e
688 | fold_expr f g (Element (e, es)) =
689 fold_expr f g e #> fold (fold_expr f g) es
690 | fold_expr f g (Update (e, es, e')) =
691 fold_expr f g e #> fold (fold_expr f g) es #> fold_expr f g e'
692 | fold_expr f g (Record (_, flds)) = fold (fold_expr f g o snd) flds
693 | fold_expr f g (Array (_, default, assocs)) =
694 fold_opt (fold_expr f g) default #>
696 (fold (fold (fold_pair
697 (fold_expr f g) (fold_opt (fold_expr f g)))))
698 (fold_expr f g)) assocs;
700 fun add_expr_pfuns funs = fold_expr
701 (fn s => if is_pfun s then I else insert (op =) s)
702 (fn s => if is_none (lookup funs s) then I else insert (op =) s);
704 val add_expr_idents = fold_expr (K I) (insert (op =));
706 fun pfun_type thy prfx (argtys, resty) =
707 map (mk_type thy prfx) argtys ---> mk_type thy prfx resty;
709 fun check_pfun_type thy prfx s t optty1 optty2 =
711 val T = fastype_of t;
713 let val U = pfun_type thy prfx ty
717 Syntax.string_of_typ_global thy T ^
719 Syntax.string_of_term_global thy t ^
720 " associated with proof function " ^ s ^
721 "\ndoes not match declared type\n" ^
722 Syntax.string_of_typ_global thy U)
724 in (Option.map check optty1; Option.map check optty2; ()) end;
726 fun upd_option x y = if is_some x then x else y;
728 fun check_pfuns_types thy prfx funs =
729 Symtab.map (fn s => fn (optty, t) =>
730 let val optty' = lookup funs
732 else unprefix (prfx ^ "__") s handle Fail _ => s)
734 (check_pfun_type thy prfx s t optty optty';
735 (NONE |> upd_option optty |> upd_option optty', t))
741 fun err_vcs names = error (Pretty.string_of
742 (Pretty.big_list "The following verification conditions have not been proved:"
743 (map Pretty.str names)))
745 fun set_env (env as {funs, prefix, ...}) thy = VCs.map (fn
746 {pfuns, type_map, env = NONE} =>
747 {pfuns = check_pfuns_types thy prefix funs pfuns,
750 | _ => err_unfinished ()) thy;
752 fun mk_pat s = (case Int.fromString s of
753 SOME i => [HOLogic.mk_Trueprop (Var (("C", i), HOLogic.boolT))]
754 | NONE => error ("Bad conclusion identifier: C" ^ s));
756 fun mk_vc thy prfx types pfuns ids (tr, proved, ps, cs) =
758 HOLogic.mk_Trueprop o fst o term_of_expr thy prfx types pfuns ids
761 Element.Assumes (map (fn (s', e) =>
762 ((Binding.name ("H" ^ s'), []), [(prop_of e, [])])) ps),
763 Element.Shows (map (fn (s', e) =>
764 (Attrib.empty_binding, [(prop_of e, mk_pat s')])) cs))
768 VCtab.fold (fn (_, (_, _, ps, cs)) => fold f ps #> fold f cs) vcs;
770 fun pfuns_of_vcs prfx funs pfuns vcs =
771 fold_vcs (add_expr_pfuns funs o snd) vcs [] |>
772 filter (is_none o lookup_prfx prfx pfuns);
774 fun declare_missing_pfuns thy prfx funs pfuns vcs (tab, ctxt) =
776 val (fs, (tys, Ts)) =
777 pfuns_of_vcs prfx funs pfuns vcs |>
778 map_filter (fn s => lookup funs s |>
779 Option.map (fn ty => (s, (SOME ty, pfun_type thy prfx ty)))) |>
780 split_list ||> split_list;
781 val (fs', ctxt') = fold_map Name.variant fs ctxt
783 (fold Symtab.update_new (fs ~~ (tys ~~ map Free (fs' ~~ Ts))) pfuns,
784 Element.Fixes (map2 (fn s => fn T =>
785 (Binding.name s, SOME T, NoSyn)) fs' Ts),
789 fun add_proof_fun prep (s, (optty, raw_t)) thy =
791 {env = SOME {proving = true, ...}, ...} => err_unfinished ()
792 | {pfuns, type_map, env} =>
794 val (optty', prfx) = (case env of
795 SOME {funs, prefix, ...} => (lookup funs s, prefix)
796 | NONE => (NONE, ""));
797 val optty'' = NONE |> upd_option optty |> upd_option optty';
798 val t = prep (Option.map (pfun_type thy prfx) optty'') raw_t;
799 val _ = (case fold_aterms (fn u =>
800 if is_Var u orelse is_Free u then insert (op =) u else I) t [] of
802 | ts => error ("Term\n" ^ Syntax.string_of_term_global thy t ^
803 "\nto be associated with proof function " ^ s ^
804 " contains free variable(s) " ^
805 commas (map (Syntax.string_of_term_global thy) ts)));
807 (check_pfun_type thy prfx s t optty optty';
808 if is_some optty'' orelse is_none env then
809 {pfuns = Symtab.update_new (s, (optty'', t)) pfuns,
812 handle Symtab.DUP _ => error ("Proof function " ^ s ^
813 " already associated with function")
814 else error ("Undeclared proof function " ^ s))
817 fun add_type (s, T as Type (tyname, Ts)) thy =
820 {env = SOME _, ...} => err_unfinished ()
821 | {pfuns, type_map, env} =>
823 type_map = Symtab.update_new (s, T) type_map,
825 handle Symtab.DUP _ => error ("SPARK type " ^ s ^
826 " already associated with type")) |>
828 case Datatype.get_constrs thy' tyname of
833 (fn (_, Type (_, [])) => ()
834 | (cname, _) => assoc_ty_err thy T s
835 ("has illegal constructor " ^
836 Long_Name.base_name cname)) cs;
837 add_enum_type s tyname thy')
838 else assoc_ty_err thy T s "is illegal")
839 | add_type (s, T) thy = assoc_ty_err thy T s "is illegal";
841 val is_closed = is_none o #env o VCs.get;
843 fun lookup_vc thy name =
845 {env = SOME {vcs, types, funs, ids, ctxt, prefix, ...}, pfuns, ...} =>
846 (case VCtab.lookup vcs name of
848 let val (pfuns', ctxt', ids') =
849 declare_missing_pfuns thy prefix funs pfuns vcs ids
850 in SOME (ctxt @ [ctxt'], mk_vc thy prefix types pfuns' ids' vc) end
854 fun get_vcs thy = (case VCs.get thy of
855 {env = SOME {vcs, types, funs, ids, ctxt, defs, prefix, ...}, pfuns, ...} =>
856 let val (pfuns', ctxt', ids') =
857 declare_missing_pfuns thy prefix funs pfuns vcs ids
859 (ctxt @ [ctxt'], defs,
861 map (apsnd (mk_vc thy prefix types pfuns' ids')))
863 | _ => ([], [], []));
865 fun mark_proved name thms = VCs.map (fn
867 env = SOME {ctxt, defs, types, funs, ids, vcs, path, prefix, ...}} =>
870 env = SOME {ctxt = ctxt, defs = defs,
871 types = types, funs = funs, ids = ids,
873 vcs = VCtab.map_entry name (fn (trace, _, ps, cs) =>
874 (trace, SOME thms, ps, cs)) vcs,
882 {pfuns, type_map, env = SOME {vcs, path, ...}} =>
883 (case VCtab.fold_rev (fn vc as (_, (_, p, _, _)) =>
884 (if is_some p then apfst else apsnd) (cons vc)) vcs ([], []) of
886 (Thm.join_proofs (maps (the o #2 o snd) proved);
887 File.write (Path.ext "prv" path)
888 (implode (map (fn (s, _) => snd (strip_number s) ^
889 " -- proved by " ^ Distribution.version ^ "\n") proved));
890 {pfuns = pfuns, type_map = type_map, env = NONE})
891 | (_, unproved) => err_vcs (map fst unproved))
892 | _ => error "No SPARK environment is currently open") |>
896 (** set up verification conditions **)
898 fun partition_opt f =
900 fun part ys zs [] = (rev ys, rev zs)
901 | part ys zs (x :: xs) = (case f x of
902 SOME y => part (y :: ys) zs xs
903 | NONE => part ys (x :: zs) xs)
906 fun dest_def (id, (Substitution_Rule ([], Ident s, rhs))) = SOME (id, (s, rhs))
909 fun mk_rulename (s, i) = Binding.name (s ^ string_of_int i);
911 fun add_const prfx (s, ty) ((tab, ctxt), thy) =
913 val T = mk_type thy prfx ty;
914 val b = Binding.name s;
915 val c = Const (Sign.full_name thy b, T)
918 ((Symtab.update (s, (c, ty)) tab, Name.declare s ctxt),
919 Sign.add_consts_i [(b, T, NoSyn)] thy))
922 fun add_def prfx types pfuns consts (id, (s, e)) (ids as (tab, ctxt), thy) =
923 (case lookup consts s of
926 val (t, ty') = term_of_expr thy prfx types pfuns ids e;
927 val T = mk_type thy prfx ty;
928 val T' = mk_type thy prfx ty';
929 val _ = T = T' orelse
930 error ("Declared type " ^ ty ^ " of " ^ s ^
931 "\ndoes not match type " ^ ty' ^ " in definition");
932 val id' = mk_rulename id;
933 val lthy = Named_Target.theory_init thy;
934 val ((t', (_, th)), lthy') = Specification.definition
935 (NONE, ((id', []), HOLogic.mk_Trueprop (HOLogic.mk_eq
936 (Free (s, T), t)))) lthy;
937 val phi = Proof_Context.export_morphism lthy' lthy
939 ((id', Morphism.thm phi th),
940 ((Symtab.update (s, (Morphism.term phi t', ty)) tab,
941 Name.declare s ctxt),
942 Local_Theory.exit_global lthy'))
944 | NONE => error ("Undeclared constant " ^ s));
946 fun add_var prfx (s, ty) (ids, thy) =
947 let val ([Free p], ids') = mk_variables thy prfx [s] ty ids
948 in (p, (ids', thy)) end;
950 fun add_init_vars prfx vcs (ids_thy as ((tab, _), _)) =
951 fold_map (add_var prfx)
953 (fn s => case try (unsuffix "~") s of
954 SOME s' => (case Symtab.lookup tab s' of
955 SOME (_, ty) => SOME (s, ty)
956 | NONE => error ("Undeclared identifier " ^ s'))
958 (fold_vcs (add_expr_idents o snd) vcs []))
961 fun is_trivial_vc ([], [(_, Ident "true")]) = true
962 | is_trivial_vc _ = false;
964 fun rulenames rules = commas
965 (map (fn ((s, i), _) => s ^ "(" ^ string_of_int i ^ ")") rules);
967 (* sort definitions according to their dependency *)
968 fun sort_defs _ _ _ _ [] sdefs = rev sdefs
969 | sort_defs prfx funs pfuns consts defs sdefs =
970 (case find_first (fn (_, (_, e)) =>
971 forall (is_some o lookup_prfx prfx pfuns)
972 (add_expr_pfuns funs e []) andalso
974 member (fn (s, (_, (s', _))) => s = s') sdefs id orelse
976 (add_expr_idents e [])) defs of
977 SOME d => sort_defs prfx funs pfuns consts
978 (remove (op =) d defs) (d :: sdefs)
979 | NONE => error ("Bad definitions: " ^ rulenames defs));
981 fun set_vcs ({types, vars, consts, funs} : decls)
982 (rules, _) ((_, ident), vcs) path prfx thy =
984 val {pfuns, ...} = VCs.get thy;
985 val (defs, rules') = partition_opt dest_def rules;
987 subtract (fn ((_, (s, _)), (s', _)) => s = s') defs (items consts);
988 (* ignore all complex rules in rls files *)
989 val (rules'', other_rules) =
990 List.partition (complex_rule o snd) rules';
991 val _ = if null rules'' then ()
992 else warning ("Ignoring rules: " ^ rulenames rules'');
994 val vcs' = VCtab.make (maps (fn (tr, vcs) =>
995 map (fn (s, (ps, cs)) => (s, (tr, NONE, ps, cs)))
996 (filter_out (is_trivial_vc o snd) vcs)) vcs);
998 val _ = (case filter_out (is_some o lookup funs)
999 (pfuns_of_vcs prfx funs pfuns vcs') of
1001 | fs => error ("Undeclared proof function(s) " ^ commas fs));
1003 val (((defs', vars''), ivars), (ids, thy')) =
1005 Symtab.update ("false", (@{term False}, booleanN)) |>
1006 Symtab.update ("true", (@{term True}, booleanN)),
1008 thy |> Sign.add_path (Long_Name.base_name ident)) |>
1009 fold (add_type_def prfx) (items types) |>
1010 fold (snd oo add_const prfx) consts' |> (fn ids_thy as ((tab, _), _) =>
1012 fold_map (add_def prfx types pfuns consts)
1013 (sort_defs prfx funs pfuns (Symtab.defined tab) defs []) ||>>
1014 fold_map (add_var prfx) (items vars) ||>>
1015 add_init_vars prfx vcs');
1018 [Element.Fixes (map (fn (s, T) =>
1019 (Binding.name s, SOME T, NoSyn)) (vars'' @ ivars)),
1020 Element.Assumes (map (fn (id, rl) =>
1021 ((mk_rulename id, []),
1022 [(term_of_rule thy' prfx types pfuns ids rl, [])]))
1024 Element.Notes ("", [((Binding.name "defns", []), map (rpair [] o single o snd) defs')])]
1027 set_env {ctxt = ctxt, defs = defs', types = types, funs = funs,
1028 ids = ids, proving = false, vcs = vcs', path = path, prefix = prfx} thy'