src/HOL/Tools/inductive_package.ML
 author wenzelm Fri Feb 03 23:12:28 2006 +0100 (2006-02-03 ago) changeset 18921 f47c46d7d654 parent 18799 f137c5e971f5 child 19359 5d523a1b6ddc permissions -rw-r--r--
canonical member/insert/merge;
1 (*  Title:      HOL/Tools/inductive_package.ML
2     ID:         \$Id\$
3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
4     Author:     Stefan Berghofer, TU Muenchen
5     Author:     Markus Wenzel, TU Muenchen
7 (Co)Inductive Definition module for HOL.
9 Features:
10   * least or greatest fixedpoints
11   * user-specified product and sum constructions
12   * mutually recursive definitions
13   * definitions involving arbitrary monotone operators
14   * automatically proves introduction and elimination rules
16 The recursive sets must *already* be declared as constants in the
17 current theory!
19   Introduction rules have the form
20   [| ti:M(Sj), ..., P(x), ... |] ==> t: Sk
21   where M is some monotone operator (usually the identity)
22   P(x) is any side condition on the free variables
23   ti, t are any terms
24   Sj, Sk are two of the sets being defined in mutual recursion
26 Sums are used only for mutual recursion.  Products are used only to
27 derive "streamlined" induction rules for relations.
28 *)
30 signature INDUCTIVE_PACKAGE =
31 sig
32   val quiet_mode: bool ref
33   val trace: bool ref
34   val unify_consts: theory -> term list -> term list -> term list * term list
35   val split_rule_vars: term list -> thm -> thm
36   val get_inductive: theory -> string -> ({names: string list, coind: bool} *
37     {defs: thm list, elims: thm list, raw_induct: thm, induct: thm,
38      intrs: thm list, mk_cases: string -> thm, mono: thm, unfold: thm}) option
39   val the_mk_cases: theory -> string -> string -> thm
40   val print_inductives: theory -> unit
41   val mono_add: attribute
42   val mono_del: attribute
43   val get_monos: theory -> thm list
44   val inductive_forall_name: string
45   val inductive_forall_def: thm
46   val rulify: thm -> thm
47   val inductive_cases: ((bstring * Attrib.src list) * string list) list -> theory -> theory
48   val inductive_cases_i: ((bstring * attribute list) * term list) list -> theory -> theory
49   val add_inductive_i: bool -> bool -> bstring -> bool -> bool -> bool -> term list ->
50     ((bstring * term) * attribute list) list -> thm list -> theory -> theory *
51       {defs: thm list, elims: thm list, raw_induct: thm, induct: thm,
52        intrs: thm list, mk_cases: string -> thm, mono: thm, unfold: thm}
53   val add_inductive: bool -> bool -> string list ->
54     ((bstring * string) * Attrib.src list) list -> (thmref * Attrib.src list) list ->
55     theory -> theory *
56       {defs: thm list, elims: thm list, raw_induct: thm, induct: thm,
57        intrs: thm list, mk_cases: string -> thm, mono: thm, unfold: thm}
58   val setup: theory -> theory
59 end;
61 structure InductivePackage: INDUCTIVE_PACKAGE =
62 struct
65 (** theory context references **)
67 val mono_name = "Orderings.mono";
68 val gfp_name = "FixedPoint.gfp";
69 val lfp_name = "FixedPoint.lfp";
70 val vimage_name = "Set.vimage";
71 val Const _ \$ (vimage_f \$ _) \$ _ = HOLogic.dest_Trueprop (Thm.concl_of vimageD);
73 val inductive_forall_name = "HOL.induct_forall";
74 val inductive_forall_def = thm "induct_forall_def";
75 val inductive_conj_name = "HOL.induct_conj";
76 val inductive_conj_def = thm "induct_conj_def";
77 val inductive_conj = thms "induct_conj";
78 val inductive_atomize = thms "induct_atomize";
79 val inductive_rulify = thms "induct_rulify";
80 val inductive_rulify_fallback = thms "induct_rulify_fallback";
84 (** theory data **)
86 type inductive_info =
87   {names: string list, coind: bool} * {defs: thm list, elims: thm list, raw_induct: thm,
88     induct: thm, intrs: thm list, mk_cases: string -> thm, mono: thm, unfold: thm};
90 structure InductiveData = TheoryDataFun
91 (struct
92   val name = "HOL/inductive";
93   type T = inductive_info Symtab.table * thm list;
95   val empty = (Symtab.empty, []);
96   val copy = I;
97   val extend = I;
98   fun merge _ ((tab1, monos1), (tab2, monos2)) =
99     (Symtab.merge (K true) (tab1, tab2), Drule.merge_rules (monos1, monos2));
101   fun print thy (tab, monos) =
102     [Pretty.strs ("(co)inductives:" ::
103       map #1 (NameSpace.extern_table (Sign.const_space thy, tab))),
104      Pretty.big_list "monotonicity rules:" (map (Display.pretty_thm_sg thy) monos)]
105     |> Pretty.chunks |> Pretty.writeln;
106 end);
108 val print_inductives = InductiveData.print;
111 (* get and put data *)
113 val get_inductive = Symtab.lookup o #1 o InductiveData.get;
115 fun the_inductive thy name =
116   (case get_inductive thy name of
117     NONE => error ("Unknown (co)inductive set " ^ quote name)
118   | SOME info => info);
120 val the_mk_cases = (#mk_cases o #2) oo the_inductive;
122 fun put_inductives names info = InductiveData.map (apfst (fn tab =>
123   fold (fn name => Symtab.update_new (name, info)) names tab
124     handle Symtab.DUP dup => error ("Duplicate definition of (co)inductive set " ^ quote dup)));
128 (** monotonicity rules **)
130 val get_monos = #2 o InductiveData.get;
131 val map_monos = Context.map_theory o InductiveData.map o Library.apsnd;
133 fun mk_mono thm =
134   let
135     fun eq2mono thm' = [standard (thm' RS (thm' RS eq_to_mono))] @
136       (case concl_of thm of
137           (_ \$ (_ \$ (Const ("Not", _) \$ _) \$ _)) => []
138         | _ => [standard (thm' RS (thm' RS eq_to_mono2))]);
139     val concl = concl_of thm
140   in
141     if Logic.is_equals concl then
142       eq2mono (thm RS meta_eq_to_obj_eq)
143     else if can (HOLogic.dest_eq o HOLogic.dest_Trueprop) concl then
144       eq2mono thm
145     else [thm]
146   end;
149 (* attributes *)
151 val mono_add = Thm.declaration_attribute (map_monos o fold Drule.add_rule o mk_mono);
152 val mono_del = Thm.declaration_attribute (map_monos o fold Drule.del_rule o mk_mono);
156 (** misc utilities **)
158 val quiet_mode = ref false;
159 val trace = ref false;  (*for debugging*)
160 fun message s = if ! quiet_mode then () else writeln s;
161 fun clean_message s = if ! quick_and_dirty then () else message s;
163 fun coind_prefix true = "co"
164   | coind_prefix false = "";
167 (*the following code ensures that each recursive set always has the
168   same type in all introduction rules*)
169 fun unify_consts thy cs intr_ts =
170   (let
171     val add_term_consts_2 = fold_aterms (fn Const c => insert (op =) c | _ => I);
172     fun varify (t, (i, ts)) =
173       let val t' = map_term_types (Logic.incr_tvar (i + 1)) (#1 (Type.varify (t, [])))
174       in (maxidx_of_term t', t'::ts) end;
175     val (i, cs') = foldr varify (~1, []) cs;
176     val (i', intr_ts') = foldr varify (i, []) intr_ts;
177     val rec_consts = fold add_term_consts_2 cs' [];
178     val intr_consts = fold add_term_consts_2 intr_ts' [];
179     fun unify (cname, cT) =
180       let val consts = map snd (List.filter (fn c => fst c = cname) intr_consts)
181       in fold (Sign.typ_unify thy) ((replicate (length consts) cT) ~~ consts) end;
182     val (env, _) = fold unify rec_consts (Vartab.empty, i');
183     val subst = Type.freeze o map_term_types (Envir.norm_type env)
185   in (map subst cs', map subst intr_ts')
186   end) handle Type.TUNIFY =>
187     (warning "Occurrences of recursive constant have non-unifiable types"; (cs, intr_ts));
190 (*make injections used in mutually recursive definitions*)
191 fun mk_inj cs sumT c x =
192   let
193     fun mk_inj' T n i =
194       if n = 1 then x else
195       let val n2 = n div 2;
196           val Type (_, [T1, T2]) = T
197       in
198         if i <= n2 then
199           Const ("Sum_Type.Inl", T1 --> T) \$ (mk_inj' T1 n2 i)
200         else
201           Const ("Sum_Type.Inr", T2 --> T) \$ (mk_inj' T2 (n - n2) (i - n2))
202       end
203   in mk_inj' sumT (length cs) (1 + find_index_eq c cs)
204   end;
206 (*make "vimage" terms for selecting out components of mutually rec.def*)
207 fun mk_vimage cs sumT t c = if length cs < 2 then t else
208   let
209     val cT = HOLogic.dest_setT (fastype_of c);
210     val vimageT = [cT --> sumT, HOLogic.mk_setT sumT] ---> HOLogic.mk_setT cT
211   in
212     Const (vimage_name, vimageT) \$
213       Abs ("y", cT, mk_inj cs sumT c (Bound 0)) \$ t
214   end;
216 (** proper splitting **)
218 fun prod_factors p (Const ("Pair", _) \$ t \$ u) =
219       p :: prod_factors (1::p) t @ prod_factors (2::p) u
220   | prod_factors p _ = [];
222 fun mg_prod_factors ts (t \$ u) fs = if t mem ts then
223         let val f = prod_factors [] u
224         in AList.update (op =) (t, f inter (AList.lookup (op =) fs t) |> the_default f) fs end
225       else mg_prod_factors ts u (mg_prod_factors ts t fs)
226   | mg_prod_factors ts (Abs (_, _, t)) fs = mg_prod_factors ts t fs
227   | mg_prod_factors ts _ fs = fs;
229 fun prodT_factors p ps (T as Type ("*", [T1, T2])) =
230       if p mem ps then prodT_factors (1::p) ps T1 @ prodT_factors (2::p) ps T2
231       else [T]
232   | prodT_factors _ _ T = [T];
234 fun ap_split p ps (Type ("*", [T1, T2])) T3 u =
235       if p mem ps then HOLogic.split_const (T1, T2, T3) \$
236         Abs ("v", T1, ap_split (2::p) ps T2 T3 (ap_split (1::p) ps T1
237           (prodT_factors (2::p) ps T2 ---> T3) (incr_boundvars 1 u) \$ Bound 0))
238       else u
239   | ap_split _ _ _ _ u =  u;
241 fun mk_tuple p ps (Type ("*", [T1, T2])) (tms as t::_) =
242       if p mem ps then HOLogic.mk_prod (mk_tuple (1::p) ps T1 tms,
243         mk_tuple (2::p) ps T2 (Library.drop (length (prodT_factors (1::p) ps T1), tms)))
244       else t
245   | mk_tuple _ _ _ (t::_) = t;
247 fun split_rule_var' ((t as Var (v, Type ("fun", [T1, T2])), ps), rl) =
248       let val T' = prodT_factors [] ps T1 ---> T2
249           val newt = ap_split [] ps T1 T2 (Var (v, T'))
250           val cterm = Thm.cterm_of (Thm.theory_of_thm rl)
251       in
252           instantiate ([], [(cterm t, cterm newt)]) rl
253       end
254   | split_rule_var' (_, rl) = rl;
256 val remove_split = rewrite_rule [split_conv RS eq_reflection];
258 fun split_rule_vars vs rl = standard (remove_split (foldr split_rule_var'
259   rl (mg_prod_factors vs (Thm.prop_of rl) [])));
261 fun split_rule vs rl = standard (remove_split (foldr split_rule_var'
262   rl (List.mapPartial (fn (t as Var ((a, _), _)) =>
263       Option.map (pair t) (AList.lookup (op =) vs a)) (term_vars (Thm.prop_of rl)))));
266 (** process rules **)
268 local
270 fun err_in_rule thy name t msg =
271   error (cat_lines ["Ill-formed introduction rule " ^ quote name,
272     Sign.string_of_term thy t, msg]);
274 fun err_in_prem thy name t p msg =
275   error (cat_lines ["Ill-formed premise", Sign.string_of_term thy p,
276     "in introduction rule " ^ quote name, Sign.string_of_term thy t, msg]);
278 val bad_concl = "Conclusion of introduction rule must have form \"t : S_i\"";
280 val all_not_allowed =
281     "Introduction rule must not have a leading \"!!\" quantifier";
283 fun atomize_term thy = MetaSimplifier.rewrite_term thy inductive_atomize [];
285 in
287 fun check_rule thy cs ((name, rule), att) =
288   let
289     val concl = Logic.strip_imp_concl rule;
290     val prems = Logic.strip_imp_prems rule;
291     val aprems = map (atomize_term thy) prems;
292     val arule = Logic.list_implies (aprems, concl);
294     fun check_prem (prem, aprem) =
295       if can HOLogic.dest_Trueprop aprem then ()
296       else err_in_prem thy name rule prem "Non-atomic premise";
297   in
298     (case concl of
299       Const ("Trueprop", _) \$ (Const ("op :", _) \$ t \$ u) =>
300         if u mem cs then
301           if exists (Logic.occs o rpair t) cs then
302             err_in_rule thy name rule "Recursion term on left of member symbol"
303           else List.app check_prem (prems ~~ aprems)
304         else err_in_rule thy name rule bad_concl
305       | Const ("all", _) \$ _ => err_in_rule thy name rule all_not_allowed
306       | _ => err_in_rule thy name rule bad_concl);
307     ((name, arule), att)
308   end;
310 val rulify =  (* FIXME norm_hhf *)
311   hol_simplify inductive_conj
312   #> hol_simplify inductive_rulify
313   #> hol_simplify inductive_rulify_fallback
314   #> standard;
316 end;
320 (** properties of (co)inductive sets **)
322 (* elimination rules *)
324 fun mk_elims cs cTs params intr_ts intr_names =
325   let
326     val used = foldr add_term_names [] intr_ts;
327     val [aname, pname] = variantlist (["a", "P"], used);
328     val P = HOLogic.mk_Trueprop (Free (pname, HOLogic.boolT));
330     fun dest_intr r =
331       let val Const ("op :", _) \$ t \$ u =
332         HOLogic.dest_Trueprop (Logic.strip_imp_concl r)
333       in (u, t, Logic.strip_imp_prems r) end;
335     val intrs = map dest_intr intr_ts ~~ intr_names;
337     fun mk_elim (c, T) =
338       let
339         val a = Free (aname, T);
341         fun mk_elim_prem (_, t, ts) =
342           list_all_free (map dest_Free ((foldr add_term_frees [] (t::ts)) \\ params),
343             Logic.list_implies (HOLogic.mk_Trueprop (HOLogic.mk_eq (a, t)) :: ts, P));
344         val c_intrs = (List.filter (equal c o #1 o #1) intrs);
345       in
346         (Logic.list_implies (HOLogic.mk_Trueprop (HOLogic.mk_mem (a, c)) ::
347           map mk_elim_prem (map #1 c_intrs), P), map #2 c_intrs)
348       end
349   in
350     map mk_elim (cs ~~ cTs)
351   end;
354 (* premises and conclusions of induction rules *)
356 fun mk_indrule cs cTs params intr_ts =
357   let
358     val used = foldr add_term_names [] intr_ts;
360     (* predicates for induction rule *)
362     val preds = map Free (variantlist (if length cs < 2 then ["P"] else
363       map (fn i => "P" ^ string_of_int i) (1 upto length cs), used) ~~
364         map (fn T => T --> HOLogic.boolT) cTs);
366     (* transform an introduction rule into a premise for induction rule *)
368     fun mk_ind_prem r =
369       let
370         val frees = map dest_Free ((add_term_frees (r, [])) \\ params);
372         val pred_of = AList.lookup (op aconv) (cs ~~ preds);
374         fun subst (s as ((m as Const ("op :", T)) \$ t \$ u)) =
375               (case pred_of u of
376                   NONE => (m \$ fst (subst t) \$ fst (subst u), NONE)
377                 | SOME P => (HOLogic.mk_binop inductive_conj_name (s, P \$ t), SOME (s, P \$ t)))
378           | subst s =
379               (case pred_of s of
380                   SOME P => (HOLogic.mk_binop "op Int"
381                     (s, HOLogic.Collect_const (HOLogic.dest_setT
382                       (fastype_of s)) \$ P), NONE)
383                 | NONE => (case s of
384                      (t \$ u) => (fst (subst t) \$ fst (subst u), NONE)
385                    | (Abs (a, T, t)) => (Abs (a, T, fst (subst t)), NONE)
386                    | _ => (s, NONE)));
388         fun mk_prem (s, prems) = (case subst s of
389               (_, SOME (t, u)) => t :: u :: prems
390             | (t, _) => t :: prems);
392         val Const ("op :", _) \$ t \$ u =
393           HOLogic.dest_Trueprop (Logic.strip_imp_concl r)
395       in list_all_free (frees,
396            Logic.list_implies (map HOLogic.mk_Trueprop (foldr mk_prem
397              [] (map HOLogic.dest_Trueprop (Logic.strip_imp_prems r))),
398                HOLogic.mk_Trueprop (valOf (pred_of u) \$ t)))
399       end;
401     val ind_prems = map mk_ind_prem intr_ts;
403     val factors = Library.fold (mg_prod_factors preds) ind_prems [];
405     (* make conclusions for induction rules *)
407     fun mk_ind_concl ((c, P), (ts, x)) =
408       let val T = HOLogic.dest_setT (fastype_of c);
409           val ps = AList.lookup (op =) factors P |> the_default [];
410           val Ts = prodT_factors [] ps T;
411           val (frees, x') = foldr (fn (T', (fs, s)) =>
412             ((Free (s, T'))::fs, Symbol.bump_string s)) ([], x) Ts;
413           val tuple = mk_tuple [] ps T frees;
414       in ((HOLogic.mk_binop "op -->"
415         (HOLogic.mk_mem (tuple, c), P \$ tuple))::ts, x')
416       end;
418     val mutual_ind_concl = HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
419         (fst (foldr mk_ind_concl ([], "xa") (cs ~~ preds))))
421   in (preds, ind_prems, mutual_ind_concl,
422     map (apfst (fst o dest_Free)) factors)
423   end;
426 (* prepare cases and induct rules *)
428 fun add_cases_induct no_elim no_induct coind names elims induct =
429   let
430     fun cases_spec name elim thy =
431       thy
432       |> Theory.parent_path
433       |> Theory.add_path (Sign.base_name name)
434       |> PureThy.add_thms [(("cases", elim), [InductAttrib.cases_set name])] |> snd
435       |> Theory.restore_naming thy;
436     val cases_specs = if no_elim then [] else map2 cases_spec names elims;
438     val induct_att = if coind then InductAttrib.coinduct_set else InductAttrib.induct_set;
439     val induct_specs =
440       if no_induct then I
441       else
442         let
443           val rules = names ~~ map (ProjectRule.project induct) (1 upto length names);
444           val inducts = map (RuleCases.save induct o standard o #2) rules;
445         in
446           PureThy.add_thms (rules |> map (fn (name, th) =>
447             (("", th), [RuleCases.consumes 1, induct_att name]))) #> snd #>
449             [((coind_prefix coind ^ "inducts", inducts), [RuleCases.consumes 1])] #> snd
450         end;
451   in Library.apply cases_specs #> induct_specs end;
455 (** proofs for (co)inductive sets **)
457 (* prove monotonicity -- NOT subject to quick_and_dirty! *)
459 fun prove_mono setT fp_fun monos thy =
460  (message "  Proving monotonicity ...";
461   standard (Goal.prove thy [] []   (*NO quick_and_dirty here!*)
462     (HOLogic.mk_Trueprop
463       (Const (mono_name, (setT --> setT) --> HOLogic.boolT) \$ fp_fun))
464     (fn _ => EVERY [rtac monoI 1,
465       REPEAT (ares_tac (List.concat (map mk_mono monos) @ get_monos thy) 1)])));
468 (* prove introduction rules *)
470 fun prove_intrs coind mono fp_def intr_ts rec_sets_defs thy =
471   let
472     val _ = clean_message "  Proving the introduction rules ...";
474     val unfold = standard' (mono RS (fp_def RS
475       (if coind then def_gfp_unfold else def_lfp_unfold)));
477     fun select_disj 1 1 = []
478       | select_disj _ 1 = [rtac disjI1]
479       | select_disj n i = (rtac disjI2)::(select_disj (n - 1) (i - 1));
481     val intrs = (1 upto (length intr_ts) ~~ intr_ts) |> map (fn (i, intr) =>
482       rulify (SkipProof.prove thy [] [] intr (fn _ => EVERY
483        [rewrite_goals_tac rec_sets_defs,
484         stac unfold 1,
485         REPEAT (resolve_tac [vimageI2, CollectI] 1),
486         (*Now 1-2 subgoals: the disjunction, perhaps equality.*)
487         EVERY1 (select_disj (length intr_ts) i),
488         (*Not ares_tac, since refl must be tried before any equality assumptions;
489           backtracking may occur if the premises have extra variables!*)
490         DEPTH_SOLVE_1 (resolve_tac [refl, exI, conjI] 1 APPEND assume_tac 1),
491         (*Now solve the equations like Inl 0 = Inl ?b2*)
492         REPEAT (rtac refl 1)])))
494   in (intrs, unfold) end;
497 (* prove elimination rules *)
499 fun prove_elims cs cTs params intr_ts intr_names unfold rec_sets_defs thy =
500   let
501     val _ = clean_message "  Proving the elimination rules ...";
503     val rules1 = [CollectE, disjE, make_elim vimageD, exE, FalseE];
504     val rules2 = [conjE, Inl_neq_Inr, Inr_neq_Inl] @ map make_elim [Inl_inject, Inr_inject];
505   in
506     mk_elims cs cTs params intr_ts intr_names |> map (fn (t, cases) =>
507       SkipProof.prove thy [] (Logic.strip_imp_prems t) (Logic.strip_imp_concl t)
508         (fn prems => EVERY
509           [cut_facts_tac [hd prems] 1,
510            rewrite_goals_tac rec_sets_defs,
511            dtac (unfold RS subst) 1,
512            REPEAT (FIRSTGOAL (eresolve_tac rules1)),
513            REPEAT (FIRSTGOAL (eresolve_tac rules2)),
514            EVERY (map (fn prem =>
515              DEPTH_SOLVE_1 (ares_tac [rewrite_rule rec_sets_defs prem, conjI] 1)) (tl prems))])
516         |> rulify
517         |> RuleCases.name cases)
518   end;
521 (* derivation of simplified elimination rules *)
523 local
525 (*cprop should have the form t:Si where Si is an inductive set*)
526 val mk_cases_err = "mk_cases: proposition not of form \"t : S_i\"";
528 (*delete needless equality assumptions*)
529 val refl_thin = prove_goal HOL.thy "!!P. a = a ==> P ==> P" (fn _ => [assume_tac 1]);
530 val elim_rls = [asm_rl, FalseE, refl_thin, conjE, exE, Pair_inject];
531 val elim_tac = REPEAT o Tactic.eresolve_tac elim_rls;
533 fun simp_case_tac solved ss i =
534   EVERY' [elim_tac, asm_full_simp_tac ss, elim_tac, REPEAT o bound_hyp_subst_tac] i
535   THEN_MAYBE (if solved then no_tac else all_tac);
537 in
539 fun mk_cases_i elims ss cprop =
540   let
541     val prem = Thm.assume cprop;
542     val tac = ALLGOALS (simp_case_tac false ss) THEN prune_params_tac;
543     fun mk_elim rl = Drule.standard (Tactic.rule_by_tactic tac (prem RS rl));
544   in
545     (case get_first (try mk_elim) elims of
546       SOME r => r
547     | NONE => error (Pretty.string_of (Pretty.block
548         [Pretty.str mk_cases_err, Pretty.fbrk, Display.pretty_cterm cprop])))
549   end;
551 fun mk_cases elims s =
552   mk_cases_i elims (simpset()) (Thm.read_cterm (Thm.theory_of_thm (hd elims)) (s, propT));
554 fun smart_mk_cases thy ss cprop =
555   let
556     val c = #1 (Term.dest_Const (Term.head_of (#2 (HOLogic.dest_mem (HOLogic.dest_Trueprop
557       (Logic.strip_imp_concl (Thm.term_of cprop))))))) handle TERM _ => error mk_cases_err;
558     val (_, {elims, ...}) = the_inductive thy c;
559   in mk_cases_i elims ss cprop end;
561 end;
564 (* inductive_cases(_i) *)
566 fun gen_inductive_cases prep_att prep_prop args thy =
567   let
568     val cert_prop = Thm.cterm_of thy o prep_prop (ProofContext.init thy);
569     val mk_cases = smart_mk_cases thy (Simplifier.simpset_of thy) o cert_prop;
571     val facts = args |> map (fn ((a, atts), props) =>
572      ((a, map (prep_att thy) atts), map (Thm.no_attributes o single o mk_cases) props));
573   in thy |> IsarThy.theorems_i PureThy.lemmaK facts |> snd end;
575 val inductive_cases = gen_inductive_cases Attrib.attribute ProofContext.read_prop;
576 val inductive_cases_i = gen_inductive_cases (K I) ProofContext.cert_prop;
579 (* mk_cases_meth *)
581 fun mk_cases_meth (ctxt, raw_props) =
582   let
583     val thy = ProofContext.theory_of ctxt;
584     val ss = local_simpset_of ctxt;
585     val cprops = map (Thm.cterm_of thy o ProofContext.read_prop ctxt) raw_props;
586   in Method.erule 0 (map (smart_mk_cases thy ss) cprops) end;
588 val mk_cases_args = Method.syntax (Scan.lift (Scan.repeat1 Args.name));
591 (* prove induction rule *)
593 fun prove_indrule cs cTs sumT rec_const params intr_ts mono
594     fp_def rec_sets_defs thy =
595   let
596     val _ = clean_message "  Proving the induction rule ...";
598     val sum_case_rewrites =
599       (if Context.theory_name thy = "Datatype" then
600         PureThy.get_thms thy (Name "sum.cases")
601       else
602         (case ThyInfo.lookup_theory "Datatype" of
603           NONE => []
604         | SOME thy' =>
605             if Theory.subthy (thy', thy) then
606               PureThy.get_thms thy' (Name "sum.cases")
607             else []))
608       |> map mk_meta_eq;
610     val (preds, ind_prems, mutual_ind_concl, factors) =
611       mk_indrule cs cTs params intr_ts;
613     val dummy = if !trace then
614                 (writeln "ind_prems = ";
615                  List.app (writeln o Sign.string_of_term thy) ind_prems)
616             else ();
618     (* make predicate for instantiation of abstract induction rule *)
620     fun mk_ind_pred _ [P] = P
621       | mk_ind_pred T Ps =
622          let val n = (length Ps) div 2;
623              val Type (_, [T1, T2]) = T
624          in Const ("Datatype.sum.sum_case",
625            [T1 --> HOLogic.boolT, T2 --> HOLogic.boolT, T] ---> HOLogic.boolT) \$
626              mk_ind_pred T1 (Library.take (n, Ps)) \$ mk_ind_pred T2 (Library.drop (n, Ps))
627          end;
629     val ind_pred = mk_ind_pred sumT preds;
631     val ind_concl = HOLogic.mk_Trueprop
632       (HOLogic.all_const sumT \$ Abs ("x", sumT, HOLogic.mk_binop "op -->"
633         (HOLogic.mk_mem (Bound 0, rec_const), ind_pred \$ Bound 0)));
635     (* simplification rules for vimage and Collect *)
637     val vimage_simps = if length cs < 2 then [] else
638       map (fn c => standard (SkipProof.prove thy [] []
639         (HOLogic.mk_Trueprop (HOLogic.mk_eq
640           (mk_vimage cs sumT (HOLogic.Collect_const sumT \$ ind_pred) c,
641            HOLogic.Collect_const (HOLogic.dest_setT (fastype_of c)) \$
642              List.nth (preds, find_index_eq c cs))))
643         (fn _ => EVERY
644           [rtac vimage_Collect 1, rewrite_goals_tac sum_case_rewrites, rtac refl 1]))) cs;
646     val raw_fp_induct = (mono RS (fp_def RS def_lfp_induct));
648     val dummy = if !trace then
649                 (writeln "raw_fp_induct = "; print_thm raw_fp_induct)
650             else ();
652     val induct = standard (SkipProof.prove thy [] ind_prems ind_concl
653       (fn prems => EVERY
654         [rewrite_goals_tac [inductive_conj_def],
655          rtac (impI RS allI) 1,
656          DETERM (etac raw_fp_induct 1),
657          rewrite_goals_tac (map mk_meta_eq (vimage_Int::Int_Collect::vimage_simps)),
658          fold_goals_tac rec_sets_defs,
659          (*This CollectE and disjE separates out the introduction rules*)
660          REPEAT (FIRSTGOAL (eresolve_tac [CollectE, disjE, exE, FalseE])),
661          (*Now break down the individual cases.  No disjE here in case
662            some premise involves disjunction.*)
663          REPEAT (FIRSTGOAL (etac conjE ORELSE' bound_hyp_subst_tac)),
664          ALLGOALS (simp_tac (HOL_basic_ss addsimps sum_case_rewrites)),
665          EVERY (map (fn prem =>
666            DEPTH_SOLVE_1 (ares_tac [rewrite_rule [inductive_conj_def] prem, conjI, refl] 1)) prems)]));
668     val lemma = standard (SkipProof.prove thy [] []
669       (Logic.mk_implies (ind_concl, mutual_ind_concl)) (fn _ => EVERY
670         [rewrite_goals_tac rec_sets_defs,
671          REPEAT (EVERY
672            [REPEAT (resolve_tac [conjI, impI] 1),
673             TRY (dtac vimageD 1), etac allE 1, dtac mp 1, atac 1,
674             rewrite_goals_tac sum_case_rewrites,
675             atac 1])]))
677   in standard (split_rule factors (induct RS lemma)) end;
681 (** specification of (co)inductive sets **)
683 fun cond_declare_consts declare_consts cs paramTs cnames =
684   if declare_consts then
685     Theory.add_consts_i (map (fn (c, n) => (Sign.base_name n, paramTs ---> fastype_of c, NoSyn)) (cs ~~ cnames))
686   else I;
688 fun mk_ind_def declare_consts alt_name coind cs intr_ts monos thy
689       params paramTs cTs cnames =
690   let
691     val sumT = fold_bal (fn (T, U) => Type ("+", [T, U])) cTs;
692     val setT = HOLogic.mk_setT sumT;
694     val fp_name = if coind then gfp_name else lfp_name;
696     val used = foldr add_term_names [] intr_ts;
697     val [sname, xname] = variantlist (["S", "x"], used);
699     (* transform an introduction rule into a conjunction  *)
700     (*   [| t : ... S_i ... ; ... |] ==> u : S_j          *)
701     (* is transformed into                                *)
702     (*   x = Inj_j u & t : ... Inj_i -`` S ... & ...      *)
704     fun transform_rule r =
705       let
706         val frees = map dest_Free ((add_term_frees (r, [])) \\ params);
707         val subst = subst_free
708           (cs ~~ (map (mk_vimage cs sumT (Free (sname, setT))) cs));
709         val Const ("op :", _) \$ t \$ u =
710           HOLogic.dest_Trueprop (Logic.strip_imp_concl r)
712       in foldr (fn ((x, T), P) => HOLogic.mk_exists (x, T, P))
713         (foldr1 HOLogic.mk_conj
714           (((HOLogic.eq_const sumT) \$ Free (xname, sumT) \$ (mk_inj cs sumT u t))::
715             (map (subst o HOLogic.dest_Trueprop)
716               (Logic.strip_imp_prems r)))) frees
717       end
719     (* make a disjunction of all introduction rules *)
721     val fp_fun = absfree (sname, setT, (HOLogic.Collect_const sumT) \$
722       absfree (xname, sumT, if null intr_ts then HOLogic.false_const
723         else foldr1 HOLogic.mk_disj (map transform_rule intr_ts)));
725     (* add definiton of recursive sets to theory *)
727     val rec_name = if alt_name = "" then
728       space_implode "_" (map Sign.base_name cnames) else alt_name;
729     val full_rec_name = if length cs < 2 then hd cnames
730       else Sign.full_name thy rec_name;
732     val rec_const = list_comb
733       (Const (full_rec_name, paramTs ---> setT), params);
735     val fp_def_term = Logic.mk_equals (rec_const,
736       Const (fp_name, (setT --> setT) --> setT) \$ fp_fun);
738     val def_terms = fp_def_term :: (if length cs < 2 then [] else
739       map (fn c => Logic.mk_equals (c, mk_vimage cs sumT rec_const c)) cs);
741     val ([fp_def :: rec_sets_defs], thy') =
742       thy
743       |> cond_declare_consts declare_consts cs paramTs cnames
744       |> (if length cs < 2 then I
745           else Theory.add_consts_i [(rec_name, paramTs ---> setT, NoSyn)])
746       |> Theory.add_path rec_name
747       |> PureThy.add_defss_i false [(("defs", def_terms), [])];
749     val mono = prove_mono setT fp_fun monos thy'
751   in (thy', rec_name, mono, fp_def, rec_sets_defs, rec_const, sumT) end;
753 fun add_ind_def verbose declare_consts alt_name coind no_elim no_ind cs
754     intros monos thy params paramTs cTs cnames induct_cases =
755   let
756     val _ =
757       if verbose then message ("Proofs for " ^ coind_prefix coind ^ "inductive set(s) " ^
758         commas_quote (map Sign.base_name cnames)) else ();
760     val ((intr_names, intr_ts), intr_atts) = apfst split_list (split_list intros);
762     val (thy1, rec_name, mono, fp_def, rec_sets_defs, rec_const, sumT) =
763       mk_ind_def declare_consts alt_name coind cs intr_ts monos thy
764         params paramTs cTs cnames;
766     val (intrs, unfold) = prove_intrs coind mono fp_def intr_ts rec_sets_defs thy1;
767     val elims = if no_elim then [] else
768       prove_elims cs cTs params intr_ts intr_names unfold rec_sets_defs thy1;
769     val raw_induct = if no_ind then Drule.asm_rl else
770       if coind then standard (rule_by_tactic
771         (rewrite_tac [mk_meta_eq vimage_Un] THEN
772           fold_tac rec_sets_defs) (mono RS (fp_def RS def_Collect_coinduct)))
773       else
774         prove_indrule cs cTs sumT rec_const params intr_ts mono fp_def
775           rec_sets_defs thy1;
776     val induct =
777       if coind then
778         (raw_induct, [RuleCases.case_names [rec_name],
779           RuleCases.case_conclusion (rec_name, induct_cases),
780           RuleCases.consumes 1])
781       else if no_ind orelse length cs > 1 then
782         (raw_induct, [RuleCases.case_names induct_cases, RuleCases.consumes 0])
783       else (raw_induct RSN (2, rev_mp), [RuleCases.case_names induct_cases, RuleCases.consumes 1]);
785     val (intrs', thy2) =
786       thy1
787       |> PureThy.add_thms ((intr_names ~~ intrs) ~~ intr_atts);
788     val (([_, elims'], [induct']), thy3) =
789       thy2
791         [(("intros", intrs'), []),
792           (("elims", elims), [RuleCases.consumes 1])]
794         [((coind_prefix coind ^ "induct", rulify (#1 induct)), #2 induct)];
795   in (thy3,
796     {defs = fp_def :: rec_sets_defs,
797      mono = mono,
798      unfold = unfold,
799      intrs = intrs',
800      elims = elims',
801      mk_cases = mk_cases elims',
802      raw_induct = rulify raw_induct,
803      induct = induct'})
804   end;
807 (* external interfaces *)
809 fun try_term f msg thy t =
810   (case Library.try f t of
811     SOME x => x
812   | NONE => error (msg ^ Sign.string_of_term thy t));
814 fun add_inductive_i verbose declare_consts alt_name coind no_elim no_ind cs pre_intros monos thy =
815   let
816     val _ = Theory.requires thy "Inductive" (coind_prefix coind ^ "inductive definitions");
818     (*parameters should agree for all mutually recursive components*)
819     val (_, params) = strip_comb (hd cs);
820     val paramTs = map (try_term (snd o dest_Free) "Parameter in recursive\
821       \ component is not a free variable: " thy) params;
823     val cTs = map (try_term (HOLogic.dest_setT o fastype_of)
824       "Recursive component not of type set: " thy) cs;
826     val cnames = map (try_term (fst o dest_Const o head_of)
827       "Recursive set not previously declared as constant: " thy) cs;
829     val save_thy = thy
830       |> Theory.copy |> cond_declare_consts declare_consts cs paramTs cnames;
831     val intros = map (check_rule save_thy cs) pre_intros;
832     val induct_cases = map (#1 o #1) intros;
834     val (thy1, result as {elims, induct, ...}) =
835       add_ind_def verbose declare_consts alt_name coind no_elim no_ind cs intros monos
836         thy params paramTs cTs cnames induct_cases;
837     val thy2 = thy1
838       |> put_inductives cnames ({names = cnames, coind = coind}, result)
839       |> add_cases_induct no_elim no_ind coind cnames elims induct
840       |> Theory.parent_path;
841   in (thy2, result) end;
843 fun add_inductive verbose coind c_strings intro_srcs raw_monos thy =
844   let
845     val cs = map (Sign.read_term thy) c_strings;
847     val intr_names = map (fst o fst) intro_srcs;
848     fun read_rule s = Thm.read_cterm thy (s, propT)
849       handle ERROR msg => cat_error msg ("The error(s) above occurred for " ^ s);
850     val intr_ts = map (Thm.term_of o read_rule o snd o fst) intro_srcs;
851     val intr_atts = map (map (Attrib.attribute thy) o snd) intro_srcs;
852     val (cs', intr_ts') = unify_consts thy cs intr_ts;
854     val (monos, thy') = thy |> IsarThy.apply_theorems raw_monos;
855   in
856     add_inductive_i verbose false "" coind false false cs'
857       ((intr_names ~~ intr_ts') ~~ intr_atts) monos thy'
858   end;
862 (** package setup **)
864 (* setup theory *)
866 val setup =
867   InductiveData.init #>
868   Method.add_methods [("ind_cases", mk_cases_meth oo mk_cases_args,
869     "dynamic case analysis on sets")] #>
871     "declaration of monotonicity rule")];
874 (* outer syntax *)
876 local structure P = OuterParse and K = OuterKeyword in
878 fun mk_ind coind ((sets, intrs), monos) =
879   #1 o add_inductive true coind sets (map P.triple_swap intrs) monos;
881 fun ind_decl coind =
882   Scan.repeat1 P.term --
883   (P.\$\$\$ "intros" |--
884     P.!!! (Scan.repeat (P.opt_thm_name ":" -- P.prop))) --
885   Scan.optional (P.\$\$\$ "monos" |-- P.!!! P.xthms1) []
886   >> (Toplevel.theory o mk_ind coind);
888 val inductiveP =
889   OuterSyntax.command "inductive" "define inductive sets" K.thy_decl (ind_decl false);
891 val coinductiveP =
892   OuterSyntax.command "coinductive" "define coinductive sets" K.thy_decl (ind_decl true);
895 val ind_cases =
896   P.and_list1 (P.opt_thm_name ":" -- Scan.repeat1 P.prop)
897   >> (Toplevel.theory o inductive_cases);
899 val inductive_casesP =
900   OuterSyntax.command "inductive_cases"
901     "create simplified instances of elimination rules (improper)" K.thy_script ind_cases;
903 val _ = OuterSyntax.add_keywords ["intros", "monos"];
904 val _ = OuterSyntax.add_parsers [inductiveP, coinductiveP, inductive_casesP];
906 end;
908 end;