1 (* Title: Tools/induct.ML
2 Author: Markus Wenzel, TU Muenchen
4 Proof by cases, induction, and coinduction.
7 signature INDUCT_ARGS =
12 val rulify_fallback: thm list
14 val dest_def: term -> (term * term) option
15 val trivial_tac: int -> tactic
21 val vars_of: term -> term list
22 val dest_rules: Proof.context ->
23 {type_cases: (string * thm) list, pred_cases: (string * thm) list,
24 type_induct: (string * thm) list, pred_induct: (string * thm) list,
25 type_coinduct: (string * thm) list, pred_coinduct: (string * thm) list}
26 val print_rules: Proof.context -> unit
27 val lookup_casesT: Proof.context -> string -> thm option
28 val lookup_casesP: Proof.context -> string -> thm option
29 val lookup_inductT: Proof.context -> string -> thm option
30 val lookup_inductP: Proof.context -> string -> thm option
31 val lookup_coinductT: Proof.context -> string -> thm option
32 val lookup_coinductP: Proof.context -> string -> thm option
33 val find_casesT: Proof.context -> typ -> thm list
34 val find_casesP: Proof.context -> term -> thm list
35 val find_inductT: Proof.context -> typ -> thm list
36 val find_inductP: Proof.context -> term -> thm list
37 val find_coinductT: Proof.context -> typ -> thm list
38 val find_coinductP: Proof.context -> term -> thm list
39 val cases_type: string -> attribute
40 val cases_pred: string -> attribute
41 val cases_del: attribute
42 val induct_type: string -> attribute
43 val induct_pred: string -> attribute
44 val induct_del: attribute
45 val coinduct_type: string -> attribute
46 val coinduct_pred: string -> attribute
47 val coinduct_del: attribute
48 val map_simpset: (simpset -> simpset) -> Context.generic -> Context.generic
49 val induct_simp_add: attribute
50 val induct_simp_del: attribute
59 val fix_tac: Proof.context -> int -> (string * typ) list -> int -> tactic
60 val add_defs: (binding option * (term * bool)) option list -> Proof.context ->
61 (term option list * thm list) * Proof.context
62 val atomize_term: theory -> term -> term
63 val atomize_cterm: conv
64 val atomize_tac: int -> tactic
65 val inner_atomize_tac: int -> tactic
66 val rulified_term: thm -> theory * term
67 val rulify_tac: int -> tactic
68 val simplified_rule: Proof.context -> thm -> thm
69 val simplify_tac: Proof.context -> int -> tactic
70 val trivial_tac: int -> tactic
71 val rotate_tac: int -> int -> int -> tactic
72 val internalize: int -> thm -> thm
73 val guess_instance: Proof.context -> thm -> int -> thm -> thm Seq.seq
74 val cases_tac: Proof.context -> bool -> term option list list -> thm option ->
75 thm list -> int -> cases_tactic
76 val get_inductT: Proof.context -> term option list list -> thm list list
77 val induct_tac: Proof.context -> bool -> (binding option * (term * bool)) option list list ->
78 (string * typ) list list -> term option list -> thm list option ->
79 thm list -> int -> cases_tactic
80 val coinduct_tac: Proof.context -> term option list -> term option list -> thm option ->
81 thm list -> int -> cases_tactic
82 val setup: theory -> theory
85 functor Induct(Induct_Args: INDUCT_ARGS): INDUCT =
88 (** variables -- ordered left-to-right, preferring right **)
91 rev (distinct (op =) (Term.fold_aterms (fn (t as Var _) => cons t | _ => I) tm []));
95 val mk_var = Net.encode_type o #2 o Term.dest_Var;
97 fun concl_var which thm = mk_var (which (vars_of (Thm.concl_of thm))) handle Empty =>
98 raise THM ("No variables in conclusion of rule", 0, [thm]);
102 fun left_var_prem thm = mk_var (hd (vars_of (hd (Thm.prems_of thm)))) handle Empty =>
103 raise THM ("No variables in major premise of rule", 0, [thm]);
105 val left_var_concl = concl_var hd;
106 val right_var_concl = concl_var List.last;
112 (** constraint simplification **)
114 (* rearrange parameters and premises to allow application of one-point-rules *)
116 fun swap_params_conv ctxt i j cv =
118 fun conv1 0 ctxt = Conv.forall_conv (cv o snd) ctxt
120 Conv.rewr_conv @{thm swap_params} then_conv
121 Conv.forall_conv (conv1 (k - 1) o snd) ctxt
122 fun conv2 0 ctxt = conv1 j ctxt
123 | conv2 k ctxt = Conv.forall_conv (conv2 (k - 1) o snd) ctxt
126 fun swap_prems_conv 0 = Conv.all_conv
127 | swap_prems_conv i =
128 Conv.implies_concl_conv (swap_prems_conv (i - 1)) then_conv
129 Conv.rewr_conv Drule.swap_prems_eq
131 fun drop_judgment ctxt = Object_Logic.drop_judgment (Proof_Context.theory_of ctxt);
135 val l = length (Logic.strip_params t);
136 val Hs = Logic.strip_assums_hyp t;
138 (case Induct_Args.dest_def (drop_judgment ctxt t) of
139 SOME (Bound j, _) => SOME (i, j)
140 | SOME (_, Bound j) => SOME (i, j)
143 (case get_first find (map_index I Hs) of
145 | SOME (0, 0) => NONE
146 | SOME (i, j) => SOME (i, l - j - 1, j))
149 fun mk_swap_rrule ctxt ct =
150 (case find_eq ctxt (term_of ct) of
152 | SOME (i, k, j) => SOME (swap_params_conv ctxt k j (K (swap_prems_conv i)) ct));
154 val rearrange_eqs_simproc =
155 Simplifier.simproc_global
156 (Thm.theory_of_thm Drule.swap_prems_eq) "rearrange_eqs" ["all t"]
157 (fn thy => fn ss => fn t =>
158 mk_swap_rrule (Simplifier.the_context ss) (cterm_of thy t));
161 (* rotate k premises to the left by j, skipping over first j premises *)
163 fun rotate_conv 0 j 0 = Conv.all_conv
164 | rotate_conv 0 j k = swap_prems_conv j then_conv rotate_conv 1 j (k - 1)
165 | rotate_conv i j k = Conv.implies_concl_conv (rotate_conv (i - 1) j k);
167 fun rotate_tac j 0 = K all_tac
168 | rotate_tac j k = SUBGOAL (fn (goal, i) =>
169 CONVERSION (rotate_conv
170 j (length (Logic.strip_assums_hyp goal) - j - k) k) i);
173 (* rulify operators around definition *)
175 fun rulify_defs_conv ctxt ct =
176 if exists_subterm (is_some o Induct_Args.dest_def) (term_of ct) andalso
177 not (is_some (Induct_Args.dest_def (drop_judgment ctxt (term_of ct))))
179 (Conv.forall_conv (rulify_defs_conv o snd) ctxt else_conv
180 Conv.implies_conv (Conv.try_conv (rulify_defs_conv ctxt))
181 (Conv.try_conv (rulify_defs_conv ctxt)) else_conv
182 Conv.first_conv (map Conv.rewr_conv Induct_Args.rulify) then_conv
183 Conv.try_conv (rulify_defs_conv ctxt)) ct
184 else Conv.no_conv ct;
192 type rules = (string * thm) Item_Net.T;
194 fun init_rules index : rules =
196 (fn ((s1, th1), (s2, th2)) => s1 = s2 andalso Thm.eq_thm_prop (th1, th2))
199 fun filter_rules (rs: rules) th =
200 filter (fn (_, th') => Thm.eq_thm_prop (th, th')) (Item_Net.content rs);
202 fun lookup_rule (rs: rules) = AList.lookup (op =) (Item_Net.content rs);
204 fun pretty_rules ctxt kind rs =
205 let val thms = map snd (Item_Net.content rs)
206 in Pretty.big_list kind (map (Display.pretty_thm ctxt) thms) end;
211 structure Data = Generic_Data
213 type T = (rules * rules) * (rules * rules) * (rules * rules) * simpset;
215 ((init_rules (left_var_prem o #2), init_rules (Thm.major_prem_of o #2)),
216 (init_rules (right_var_concl o #2), init_rules (Thm.major_prem_of o #2)),
217 (init_rules (left_var_concl o #2), init_rules (Thm.concl_of o #2)),
218 empty_ss addsimprocs [rearrange_eqs_simproc] addsimps [Drule.norm_hhf_eq]);
220 fun merge (((casesT1, casesP1), (inductT1, inductP1), (coinductT1, coinductP1), simpset1),
221 ((casesT2, casesP2), (inductT2, inductP2), (coinductT2, coinductP2), simpset2)) =
222 ((Item_Net.merge (casesT1, casesT2), Item_Net.merge (casesP1, casesP2)),
223 (Item_Net.merge (inductT1, inductT2), Item_Net.merge (inductP1, inductP2)),
224 (Item_Net.merge (coinductT1, coinductT2), Item_Net.merge (coinductP1, coinductP2)),
225 merge_ss (simpset1, simpset2));
228 val get_local = Data.get o Context.Proof;
230 fun dest_rules ctxt =
231 let val ((casesT, casesP), (inductT, inductP), (coinductT, coinductP), _) = get_local ctxt in
232 {type_cases = Item_Net.content casesT,
233 pred_cases = Item_Net.content casesP,
234 type_induct = Item_Net.content inductT,
235 pred_induct = Item_Net.content inductP,
236 type_coinduct = Item_Net.content coinductT,
237 pred_coinduct = Item_Net.content coinductP}
240 fun print_rules ctxt =
241 let val ((casesT, casesP), (inductT, inductP), (coinductT, coinductP), _) = get_local ctxt in
242 [pretty_rules ctxt "coinduct type:" coinductT,
243 pretty_rules ctxt "coinduct pred:" coinductP,
244 pretty_rules ctxt "induct type:" inductT,
245 pretty_rules ctxt "induct pred:" inductP,
246 pretty_rules ctxt "cases type:" casesT,
247 pretty_rules ctxt "cases pred:" casesP]
248 |> Pretty.chunks |> Pretty.writeln
252 Outer_Syntax.improper_command "print_induct_rules" "print induction and cases rules"
253 Keyword.diag (Scan.succeed (Toplevel.no_timing o Toplevel.unknown_context o
254 Toplevel.keep (print_rules o Toplevel.context_of)));
259 val lookup_casesT = lookup_rule o #1 o #1 o get_local;
260 val lookup_casesP = lookup_rule o #2 o #1 o get_local;
261 val lookup_inductT = lookup_rule o #1 o #2 o get_local;
262 val lookup_inductP = lookup_rule o #2 o #2 o get_local;
263 val lookup_coinductT = lookup_rule o #1 o #3 o get_local;
264 val lookup_coinductP = lookup_rule o #2 o #3 o get_local;
267 fun find_rules which how ctxt x =
268 map snd (Item_Net.retrieve (which (get_local ctxt)) (how x));
270 val find_casesT = find_rules (#1 o #1) Net.encode_type;
271 val find_casesP = find_rules (#2 o #1) I;
272 val find_inductT = find_rules (#1 o #2) Net.encode_type;
273 val find_inductP = find_rules (#2 o #2) I;
274 val find_coinductT = find_rules (#1 o #3) Net.encode_type;
275 val find_coinductP = find_rules (#2 o #3) I;
283 fun mk_att f g name arg =
284 let val (x, thm) = g arg in (Data.map (f (name, thm)) x, thm) end;
287 Thm.declaration_attribute (fn th => Data.map (which (pairself (fn rs =>
288 fold Item_Net.remove (filter_rules rs th) rs))));
290 fun map1 f (x, y, z, s) = (f x, y, z, s);
291 fun map2 f (x, y, z, s) = (x, f y, z, s);
292 fun map3 f (x, y, z, s) = (x, y, f z, s);
293 fun map4 f (x, y, z, s) = (x, y, z, f s);
295 fun add_casesT rule x = map1 (apfst (Item_Net.update rule)) x;
296 fun add_casesP rule x = map1 (apsnd (Item_Net.update rule)) x;
297 fun add_inductT rule x = map2 (apfst (Item_Net.update rule)) x;
298 fun add_inductP rule x = map2 (apsnd (Item_Net.update rule)) x;
299 fun add_coinductT rule x = map3 (apfst (Item_Net.update rule)) x;
300 fun add_coinductP rule x = map3 (apsnd (Item_Net.update rule)) x;
302 val consumes0 = Rule_Cases.consumes_default 0;
303 val consumes1 = Rule_Cases.consumes_default 1;
307 val cases_type = mk_att add_casesT consumes0;
308 val cases_pred = mk_att add_casesP consumes1;
309 val cases_del = del_att map1;
311 val induct_type = mk_att add_inductT consumes0;
312 val induct_pred = mk_att add_inductP consumes1;
313 val induct_del = del_att map2;
315 val coinduct_type = mk_att add_coinductT consumes0;
316 val coinduct_pred = mk_att add_coinductP consumes1;
317 val coinduct_del = del_att map3;
319 fun map_simpset f = Data.map (map4 f);
322 Thm.declaration_attribute (fn thm => fn context =>
324 (Simplifier.with_context (Context.proof_of context) (fn ss => f (ss, [thm]))) context);
326 val induct_simp_add = induct_simp (op addsimps);
327 val induct_simp_del = induct_simp (op delsimps);
333 (** attribute syntax **)
335 val no_simpN = "no_simp";
336 val casesN = "cases";
337 val inductN = "induct";
338 val coinductN = "coinduct";
347 Scan.lift (Args.$$$ k -- Args.colon) |-- arg ||
348 Scan.lift (Args.$$$ k) >> K "";
350 fun attrib add_type add_pred del =
351 spec typeN (Args.type_name false) >> add_type ||
352 spec predN (Args.const false) >> add_pred ||
353 spec setN (Args.const false) >> add_pred ||
354 Scan.lift Args.del >> K del;
359 Attrib.setup @{binding cases} (attrib cases_type cases_pred cases_del)
360 "declaration of cases rule" #>
361 Attrib.setup @{binding induct} (attrib induct_type induct_pred induct_del)
362 "declaration of induction rule" #>
363 Attrib.setup @{binding coinduct} (attrib coinduct_type coinduct_pred coinduct_del)
364 "declaration of coinduction rule" #>
365 Attrib.setup @{binding induct_simp} (Attrib.add_del induct_simp_add induct_simp_del)
366 "declaration of rules for simplifying induction or cases rules";
376 fun align_left msg xs ys =
377 let val m = length xs and n = length ys
378 in if m < n then error msg else (take n xs ~~ ys) end;
380 fun align_right msg xs ys =
381 let val m = length xs and n = length ys
382 in if m < n then error msg else (drop (m - n) xs ~~ ys) end;
387 fun prep_inst ctxt align tune (tm, ts) =
389 val cert = Thm.cterm_of (Proof_Context.theory_of ctxt);
390 fun prep_var (x, SOME t) =
393 val xT = #T (Thm.rep_cterm cx);
394 val ct = cert (tune t);
395 val tT = #T (Thm.rep_cterm ct);
397 if Type.could_unify (tT, xT) then SOME (cx, ct)
398 else error (Pretty.string_of (Pretty.block
399 [Pretty.str "Ill-typed instantiation:", Pretty.fbrk,
400 Syntax.pretty_term ctxt (Thm.term_of ct), Pretty.str " ::", Pretty.brk 1,
401 Syntax.pretty_typ ctxt tT]))
403 | prep_var (_, NONE) = NONE;
406 align "Rule has fewer variables than instantiations given" xs ts
407 |> map_filter prep_var
413 fun trace_rules _ kind [] = error ("Unable to figure out " ^ kind ^ " rule")
414 | trace_rules ctxt _ rules = Method.trace ctxt rules;
417 (* mark equality constraints in cases rule *)
419 val equal_def' = Thm.symmetric Induct_Args.equal_def;
421 fun mark_constraints n ctxt = Conv.fconv_rule
422 (Conv.prems_conv (~1) (Conv.params_conv ~1 (K (Conv.prems_conv n
423 (Raw_Simplifier.rewrite false [equal_def']))) ctxt));
425 val unmark_constraints = Conv.fconv_rule
426 (Raw_Simplifier.rewrite true [Induct_Args.equal_def]);
431 fun simplify_conv' ctxt =
432 Simplifier.full_rewrite (Simplifier.context ctxt (#4 (get_local ctxt)));
434 fun simplify_conv ctxt ct =
435 if exists_subterm (is_some o Induct_Args.dest_def) (term_of ct) then
436 (Conv.try_conv (rulify_defs_conv ctxt) then_conv simplify_conv' ctxt) ct
437 else Conv.all_conv ct;
439 fun gen_simplified_rule cv ctxt =
440 Conv.fconv_rule (Conv.prems_conv ~1 (cv ctxt));
442 val simplified_rule' = gen_simplified_rule simplify_conv';
443 val simplified_rule = gen_simplified_rule simplify_conv;
445 fun simplify_tac ctxt = CONVERSION (simplify_conv ctxt);
447 val trivial_tac = Induct_Args.trivial_tac;
454 rule selection scheme:
455 cases - default case split
456 `A t` cases ... - predicate/set cases
458 ... cases ... r - explicit rule
463 fun get_casesT ctxt ((SOME t :: _) :: _) = find_casesT ctxt (Term.fastype_of t)
464 | get_casesT _ _ = [];
466 fun get_casesP ctxt (fact :: _) = find_casesP ctxt (Thm.concl_of fact)
467 | get_casesP _ _ = [];
471 fun cases_tac ctxt simp insts opt_rule facts =
473 val thy = Proof_Context.theory_of ctxt;
476 (if null insts then r
477 else (align_left "Rule has fewer premises than arguments given" (Thm.prems_of r) insts
478 |> maps (prep_inst ctxt align_left I)
479 |> Drule.cterm_instantiate) r) |>
480 (if simp then mark_constraints (Rule_Cases.get_constraints r) ctxt else I) |>
481 pair (Rule_Cases.get r);
485 SOME r => Seq.single (inst_rule r)
487 (get_casesP ctxt facts @ get_casesT ctxt insts @ [Induct_Args.cases_default])
488 |> tap (trace_rules ctxt casesN)
489 |> Seq.of_list |> Seq.maps (Seq.try inst_rule));
493 |> Seq.maps (Rule_Cases.consume [] facts)
494 |> Seq.maps (fn ((cases, (_, more_facts)), rule) =>
496 (if simp then simplified_rule' ctxt #> unmark_constraints else I) rule
498 CASES (Rule_Cases.make_common (thy,
499 Thm.prop_of (Rule_Cases.internalize_params rule')) cases)
500 ((Method.insert_tac more_facts THEN' Tactic.rtac rule' THEN_ALL_NEW
501 (if simp then TRY o trivial_tac else K all_tac)) i) st
509 (** induct method **)
511 val conjunction_congs = [@{thm Pure.all_conjunction}, @{thm imp_conjunction}];
516 fun atomize_term thy =
517 Raw_Simplifier.rewrite_term thy Induct_Args.atomize []
518 #> Object_Logic.drop_judgment thy;
520 val atomize_cterm = Raw_Simplifier.rewrite true Induct_Args.atomize;
522 val atomize_tac = Simplifier.rewrite_goal_tac Induct_Args.atomize;
524 val inner_atomize_tac =
525 Simplifier.rewrite_goal_tac (map Thm.symmetric conjunction_congs) THEN' atomize_tac;
530 fun rulify_term thy =
531 Raw_Simplifier.rewrite_term thy (Induct_Args.rulify @ conjunction_congs) [] #>
532 Raw_Simplifier.rewrite_term thy Induct_Args.rulify_fallback [];
534 fun rulified_term thm =
536 val thy = Thm.theory_of_thm thm;
537 val rulify = rulify_term thy;
538 val (As, B) = Logic.strip_horn (Thm.prop_of thm);
539 in (thy, Logic.list_implies (map rulify As, rulify B)) end;
542 Simplifier.rewrite_goal_tac (Induct_Args.rulify @ conjunction_congs) THEN'
543 Simplifier.rewrite_goal_tac Induct_Args.rulify_fallback THEN'
544 Goal.conjunction_tac THEN_ALL_NEW
545 (Simplifier.rewrite_goal_tac [@{thm Pure.conjunction_imp}] THEN' Goal.norm_hhf_tac);
550 fun rule_instance ctxt inst rule =
551 Drule.cterm_instantiate (prep_inst ctxt align_left I (Thm.prop_of rule, inst)) rule;
553 fun internalize k th =
554 th |> Thm.permute_prems 0 k
555 |> Conv.fconv_rule (Conv.concl_conv (Thm.nprems_of th - k) atomize_cterm);
558 (* guess rule instantiation -- cannot handle pending goal parameters *)
562 fun dest_env thy env =
564 val cert = Thm.cterm_of thy;
565 val certT = Thm.ctyp_of thy;
566 val pairs = Vartab.dest (Envir.term_env env);
567 val types = Vartab.dest (Envir.type_env env);
568 val ts = map (cert o Envir.norm_term env o #2 o #2) pairs;
569 val xs = map2 (curry (cert o Var)) (map #1 pairs) (map (#T o Thm.rep_cterm) ts);
570 in (map (fn (xi, (S, T)) => (certT (TVar (xi, S)), certT T)) types, xs ~~ ts) end;
574 fun guess_instance ctxt rule i st =
576 val thy = Proof_Context.theory_of ctxt;
577 val maxidx = Thm.maxidx_of st;
578 val goal = Thm.term_of (Thm.cprem_of st i); (*exception Subscript*)
579 val params = rev (Term.rename_wrt_term goal (Logic.strip_params goal));
581 if not (null params) then
582 (warning ("Cannot determine rule instantiation due to pending parameter(s): " ^
583 commas_quote (map (Syntax.string_of_term ctxt o Syntax_Trans.mark_boundT) params));
587 val rule' = Thm.incr_indexes (maxidx + 1) rule;
588 val concl = Logic.strip_assums_concl goal;
590 Unify.smash_unifiers thy [(Thm.concl_of rule', concl)] (Envir.empty (Thm.maxidx_of rule'))
591 |> Seq.map (fn env => Drule.instantiate (dest_env thy env) rule')
593 end handle Subscript => Seq.empty;
598 (* special renaming of rule parameters *)
600 fun special_rename_params ctxt [[SOME (Free (z, Type (T, _)))]] [thm] =
602 val x = Name.clean (Variable.revert_fixed ctxt z);
604 | index i (y :: ys) =
605 if x = y then x ^ string_of_int i :: index (i + 1) ys
606 else y :: index i ys;
607 fun rename_params [] = []
608 | rename_params ((y, Type (U, _)) :: ys) =
609 (if U = T then x else y) :: rename_params ys
610 | rename_params ((y, _) :: ys) = y :: rename_params ys;
613 val xs = rename_params (Logic.strip_params A);
615 (case filter (fn x' => x' = x) xs of
616 [] => xs | [_] => xs | _ => index 1 xs);
617 in Logic.list_rename_params (xs', A) end;
619 let val (As, C) = Logic.strip_horn p
620 in Logic.list_implies (map rename_asm As, C) end;
621 val cp' = cterm_fun rename_prop (Thm.cprop_of thm);
622 val thm' = Thm.equal_elim (Thm.reflexive cp') thm;
623 in [Rule_Cases.save thm thm'] end
624 | special_rename_params _ _ ths = ths;
631 fun goal_prefix k ((c as Const ("all", _)) $ Abs (a, T, B)) = c $ Abs (a, T, goal_prefix k B)
632 | goal_prefix 0 _ = Term.dummy_pattern propT
633 | goal_prefix k ((c as Const ("==>", _)) $ A $ B) = c $ A $ goal_prefix (k - 1) B
634 | goal_prefix _ _ = Term.dummy_pattern propT;
636 fun goal_params k (Const ("all", _) $ Abs (_, _, B)) = goal_params k B + 1
637 | goal_params 0 _ = 0
638 | goal_params k (Const ("==>", _) $ _ $ B) = goal_params (k - 1) B
639 | goal_params _ _ = 0;
641 fun meta_spec_tac ctxt n (x, T) = SUBGOAL (fn (goal, i) =>
643 val thy = Proof_Context.theory_of ctxt;
644 val cert = Thm.cterm_of thy;
647 fun spec_rule prfx (xs, body) =
648 @{thm Pure.meta_spec}
649 |> Thm.rename_params_rule ([Name.clean (Variable.revert_fixed ctxt x)], 1)
650 |> Thm.lift_rule (cert prfx)
651 |> `(Thm.prop_of #> Logic.strip_assums_concl)
652 |-> (fn pred $ arg =>
653 Drule.cterm_instantiate
654 [(cert (Term.head_of pred), cert (Logic.rlist_abs (xs, body))),
655 (cert (Term.head_of arg), cert (Logic.rlist_abs (xs, v)))]);
657 fun goal_concl k xs (Const ("all", _) $ Abs (a, T, B)) = goal_concl k ((a, T) :: xs) B
658 | goal_concl 0 xs B =
659 if not (Term.exists_subterm (fn t => t aconv v) B) then NONE
660 else SOME (xs, Term.absfree (x, T, Term.incr_boundvars 1 B))
661 | goal_concl k xs (Const ("==>", _) $ _ $ B) = goal_concl (k - 1) xs B
662 | goal_concl _ _ _ = NONE;
664 (case goal_concl n [] goal of
666 (compose_tac (false, spec_rule (goal_prefix n goal) concl, 1) THEN' rtac asm_rl) i
670 fun miniscope_tac p = CONVERSION o
671 Conv.params_conv p (K (Raw_Simplifier.rewrite true [Thm.symmetric Drule.norm_hhf_eq]));
675 fun fix_tac _ _ [] = K all_tac
676 | fix_tac ctxt n xs = SUBGOAL (fn (goal, i) =>
677 (EVERY' (map (meta_spec_tac ctxt n) xs) THEN'
678 (miniscope_tac (goal_params n goal) ctxt)) i);
685 fun add_defs def_insts =
687 fun add (SOME (_, (t, true))) ctxt = ((SOME t, []), ctxt)
688 | add (SOME (SOME x, (t, _))) ctxt =
689 let val ([(lhs, (_, th))], ctxt') =
690 Local_Defs.add_defs [((x, NoSyn), (Thm.empty_binding, t))] ctxt
691 in ((SOME lhs, [th]), ctxt') end
692 | add (SOME (NONE, (t as Free _, _))) ctxt = ((SOME t, []), ctxt)
693 | add (SOME (NONE, (t, _))) ctxt =
695 val ([s], _) = Name.variants ["x"] (Variable.names_of ctxt);
696 val ([(lhs, (_, th))], ctxt') =
697 Local_Defs.add_defs [((Binding.name s, NoSyn),
698 (Thm.empty_binding, t))] ctxt
699 in ((SOME lhs, [th]), ctxt') end
700 | add NONE ctxt = ((NONE, []), ctxt);
701 in fold_map add def_insts #> apfst (split_list #> apsnd flat) end;
707 rule selection scheme:
708 `A x` induct ... - predicate/set induction
709 induct x - type induction
710 ... induct ... r - explicit rule
713 fun get_inductT ctxt insts =
714 fold_rev (map_product cons) (insts |> map
715 ((fn [] => NONE | ts => List.last ts) #>
716 (fn NONE => TVar (("'a", 0), []) | SOME t => Term.fastype_of t) #>
717 find_inductT ctxt)) [[]]
718 |> filter_out (forall Rule_Cases.is_inner_rule);
720 fun get_inductP ctxt (fact :: _) = map single (find_inductP ctxt (Thm.concl_of fact))
721 | get_inductP _ _ = [];
723 fun induct_tac ctxt simp def_insts arbitrary taking opt_rule facts =
725 val thy = Proof_Context.theory_of ctxt;
727 val ((insts, defs), defs_ctxt) = fold_map add_defs def_insts ctxt |>> split_list;
728 val atomized_defs = map (map (Conv.fconv_rule atomize_cterm)) defs;
730 fun inst_rule (concls, r) =
731 (if null insts then `Rule_Cases.get r
732 else (align_left "Rule has fewer conclusions than arguments given"
733 (map Logic.strip_imp_concl (Logic.dest_conjunctions (Thm.concl_of r))) insts
734 |> maps (prep_inst ctxt align_right (atomize_term thy))
735 |> Drule.cterm_instantiate) r |> pair (Rule_Cases.get r))
736 |> (fn ((cases, consumes), th) => (((cases, concls), consumes), th));
740 SOME rs => Seq.single (inst_rule (Rule_Cases.strict_mutual_rule ctxt rs))
742 (get_inductP ctxt facts @
743 map (special_rename_params defs_ctxt insts) (get_inductT ctxt insts))
744 |> map_filter (Rule_Cases.mutual_rule ctxt)
745 |> tap (trace_rules ctxt inductN o map #2)
746 |> Seq.of_list |> Seq.maps (Seq.try inst_rule));
748 fun rule_cases ctxt rule =
749 let val rule' = (if simp then simplified_rule ctxt else I)
750 (Rule_Cases.internalize_params rule);
751 in Rule_Cases.make_nested (Thm.prop_of rule') (rulified_term rule') end;
755 |> Seq.maps (Rule_Cases.consume (flat defs) facts)
756 |> Seq.maps (fn (((cases, concls), (more_consumes, more_facts)), rule) =>
757 (PRECISE_CONJUNCTS (length concls) (ALLGOALS (fn j =>
760 val adefs = nth_list atomized_defs (j - 1);
761 val frees = fold (Term.add_frees o prop_of) adefs [];
762 val xs = nth_list arbitrary (j - 1);
763 val k = nth concls (j - 1) + more_consumes
765 Method.insert_tac (more_facts @ adefs) THEN'
767 rotate_tac k (length adefs) THEN'
769 (List.partition (member op = frees) xs |> op @)
771 fix_tac defs_ctxt k xs)
773 THEN' inner_atomize_tac) j))
774 THEN' atomize_tac) i st |> Seq.maps (fn st' =>
775 guess_instance ctxt (internalize more_consumes rule) i st'
776 |> Seq.map (rule_instance ctxt (burrow_options (Variable.polymorphic ctxt) taking))
777 |> Seq.maps (fn rule' =>
778 CASES (rule_cases ctxt rule' cases)
779 (Tactic.rtac rule' i THEN
780 PRIMITIVE (singleton (Proof_Context.export defs_ctxt ctxt))) st'))))
782 ((if simp then simplify_tac ctxt THEN' (TRY o trivial_tac)
784 THEN_ALL_NEW rulify_tac)
789 (** coinduct method **)
792 rule selection scheme:
793 goal "A x" coinduct ... - predicate/set coinduction
794 coinduct x - type coinduction
795 coinduct ... r - explicit rule
800 fun get_coinductT ctxt (SOME t :: _) = find_coinductT ctxt (Term.fastype_of t)
801 | get_coinductT _ _ = [];
803 fun get_coinductP ctxt goal = find_coinductP ctxt (Logic.strip_assums_concl goal);
805 fun main_prop_of th =
806 if Rule_Cases.get_consumes th > 0 then Thm.major_prem_of th else Thm.concl_of th;
810 fun coinduct_tac ctxt inst taking opt_rule facts =
812 val thy = Proof_Context.theory_of ctxt;
815 if null inst then `Rule_Cases.get r
816 else Drule.cterm_instantiate (prep_inst ctxt align_right I (main_prop_of r, inst)) r
817 |> pair (Rule_Cases.get r);
821 SOME r => Seq.single (inst_rule r)
823 (get_coinductP ctxt goal @ get_coinductT ctxt inst)
824 |> tap (trace_rules ctxt coinductN)
825 |> Seq.of_list |> Seq.maps (Seq.try inst_rule));
827 SUBGOAL_CASES (fn (goal, i) => fn st =>
829 |> Seq.maps (Rule_Cases.consume [] facts)
830 |> Seq.maps (fn ((cases, (_, more_facts)), rule) =>
831 guess_instance ctxt rule i st
832 |> Seq.map (rule_instance ctxt (burrow_options (Variable.polymorphic ctxt) taking))
833 |> Seq.maps (fn rule' =>
834 CASES (Rule_Cases.make_common (thy,
835 Thm.prop_of (Rule_Cases.internalize_params rule')) cases)
836 (Method.insert_tac more_facts i THEN Tactic.rtac rule' i) st)))
843 (** concrete syntax **)
845 val arbitraryN = "arbitrary";
846 val takingN = "taking";
851 fun single_rule [rule] = rule
852 | single_rule _ = error "Single rule expected";
854 fun named_rule k arg get =
855 Scan.lift (Args.$$$ k -- Args.colon) |-- Scan.repeat arg :|--
856 (fn names => Scan.peek (fn context => Scan.succeed (names |> map (fn name =>
857 (case get (Context.proof_of context) name of SOME x => x
858 | NONE => error ("No rule for " ^ k ^ " " ^ quote name))))));
860 fun rule get_type get_pred =
861 named_rule typeN (Args.type_name false) get_type ||
862 named_rule predN (Args.const false) get_pred ||
863 named_rule setN (Args.const false) get_pred ||
864 Scan.lift (Args.$$$ ruleN -- Args.colon) |-- Attrib.thms;
866 val cases_rule = rule lookup_casesT lookup_casesP >> single_rule;
867 val induct_rule = rule lookup_inductT lookup_inductP;
868 val coinduct_rule = rule lookup_coinductT lookup_coinductP >> single_rule;
870 val inst = Scan.lift (Args.$$$ "_") >> K NONE || Args.term >> SOME;
872 val inst' = Scan.lift (Args.$$$ "_") >> K NONE ||
873 Args.term >> (SOME o rpair false) ||
874 Scan.lift (Args.$$$ "(") |-- (Args.term >> (SOME o rpair true)) --|
875 Scan.lift (Args.$$$ ")");
878 ((Scan.lift (Args.binding --| (Args.$$$ "\<equiv>" || Args.$$$ "==")) >> SOME)
879 -- (Args.term >> rpair false)) >> SOME ||
880 inst' >> Option.map (pair NONE);
882 val free = Args.context -- Args.term >> (fn (_, Free v) => v | (ctxt, t) =>
883 error ("Bad free variable: " ^ Syntax.string_of_term ctxt t));
885 fun unless_more_args scan = Scan.unless (Scan.lift
886 ((Args.$$$ arbitraryN || Args.$$$ takingN || Args.$$$ typeN ||
887 Args.$$$ predN || Args.$$$ setN || Args.$$$ ruleN) -- Args.colon)) scan;
889 val arbitrary = Scan.optional (Scan.lift (Args.$$$ arbitraryN -- Args.colon) |--
890 Parse.and_list1' (Scan.repeat (unless_more_args free))) [];
892 val taking = Scan.optional (Scan.lift (Args.$$$ takingN -- Args.colon) |--
893 Scan.repeat1 (unless_more_args inst)) [];
898 Method.setup @{binding cases}
899 (Args.mode no_simpN --
900 (Parse.and_list' (Scan.repeat (unless_more_args inst)) -- Scan.option cases_rule) >>
901 (fn (no_simp, (insts, opt_rule)) => fn ctxt =>
902 METHOD_CASES (fn facts => Seq.DETERM (HEADGOAL
903 (cases_tac ctxt (not no_simp) insts opt_rule facts)))))
904 "case analysis on types or predicates/sets";
907 Method.setup @{binding induct}
908 (Args.mode no_simpN -- (Parse.and_list' (Scan.repeat (unless_more_args def_inst)) --
909 (arbitrary -- taking -- Scan.option induct_rule)) >>
910 (fn (no_simp, (insts, ((arbitrary, taking), opt_rule))) => fn ctxt =>
911 RAW_METHOD_CASES (fn facts =>
913 (HEADGOAL (induct_tac ctxt (not no_simp) insts arbitrary taking opt_rule facts)))))
914 "induction on types or predicates/sets";
917 Method.setup @{binding coinduct}
918 (Scan.repeat (unless_more_args inst) -- taking -- Scan.option coinduct_rule >>
919 (fn ((insts, taking), opt_rule) => fn ctxt =>
920 RAW_METHOD_CASES (fn facts =>
921 Seq.DETERM (HEADGOAL (coinduct_tac ctxt insts taking opt_rule facts)))))
922 "coinduction on types or predicates/sets";
930 val setup = attrib_setup #> cases_setup #> induct_setup #> coinduct_setup;