--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/Tools/induct.ML Thu Oct 04 14:42:47 2007 +0200
@@ -0,0 +1,762 @@
+(* Title: Tools/induct.ML
+ ID: $Id$
+ Author: Markus Wenzel, TU Muenchen
+
+Proof by cases and induction.
+*)
+
+signature INDUCT_DATA =
+sig
+ val cases_default: thm
+ val atomize: thm list
+ val rulify: thm list
+ val rulify_fallback: thm list
+end;
+
+signature INDUCT =
+sig
+ (*rule declarations*)
+ val vars_of: term -> term list
+ val dest_rules: Proof.context ->
+ {type_cases: (string * thm) list, set_cases: (string * thm) list,
+ type_induct: (string * thm) list, set_induct: (string * thm) list,
+ type_coinduct: (string * thm) list, set_coinduct: (string * thm) list}
+ val print_rules: Proof.context -> unit
+ val lookup_casesT: Proof.context -> string -> thm option
+ val lookup_casesS: Proof.context -> string -> thm option
+ val lookup_inductT: Proof.context -> string -> thm option
+ val lookup_inductS: Proof.context -> string -> thm option
+ val lookup_coinductT: Proof.context -> string -> thm option
+ val lookup_coinductS: Proof.context -> string -> thm option
+ val find_casesT: Proof.context -> typ -> thm list
+ val find_casesS: Proof.context -> term -> thm list
+ val find_inductT: Proof.context -> typ -> thm list
+ val find_inductS: Proof.context -> term -> thm list
+ val find_coinductT: Proof.context -> typ -> thm list
+ val find_coinductS: Proof.context -> term -> thm list
+ val cases_type: string -> attribute
+ val cases_set: string -> attribute
+ val induct_type: string -> attribute
+ val induct_set: string -> attribute
+ val coinduct_type: string -> attribute
+ val coinduct_set: string -> attribute
+ val casesN: string
+ val inductN: string
+ val coinductN: string
+ val typeN: string
+ val setN: string
+ (*proof methods*)
+ val fix_tac: Proof.context -> int -> (string * typ) list -> int -> tactic
+ val add_defs: (string option * term) option list -> Proof.context ->
+ (term option list * thm list) * Proof.context
+ val atomize_term: theory -> term -> term
+ val atomize_tac: int -> tactic
+ val inner_atomize_tac: int -> tactic
+ val rulified_term: thm -> theory * term
+ val rulify_tac: int -> tactic
+ val internalize: int -> thm -> thm
+ val guess_instance: thm -> int -> thm -> thm Seq.seq
+ val cases_tac: Proof.context -> bool -> term option list list -> thm option ->
+ thm list -> int -> cases_tactic
+ val induct_tac: Proof.context -> bool -> (string option * term) option list list ->
+ (string * typ) list list -> term option list -> thm list option -> thm list -> int ->
+ cases_tactic
+ val coinduct_tac: Proof.context -> bool -> term option list -> term option list ->
+ thm option -> thm list -> int -> cases_tactic
+ val setup: theory -> theory
+end;
+
+functor InductFun(Data: INDUCT_DATA): INDUCT =
+struct
+
+
+(** misc utils **)
+
+(* encode_type -- for indexing purposes *)
+
+fun encode_type (Type (c, Ts)) = Term.list_comb (Const (c, dummyT), map encode_type Ts)
+ | encode_type (TFree (a, _)) = Free (a, dummyT)
+ | encode_type (TVar (a, _)) = Var (a, dummyT);
+
+
+(* variables -- ordered left-to-right, preferring right *)
+
+fun vars_of tm =
+ rev (distinct (op =) (Term.fold_aterms (fn (t as Var _) => cons t | _ => I) tm []));
+
+local
+
+val mk_var = encode_type o #2 o Term.dest_Var;
+
+fun concl_var which thm = mk_var (which (vars_of (Thm.concl_of thm))) handle Empty =>
+ raise THM ("No variables in conclusion of rule", 0, [thm]);
+
+in
+
+fun left_var_prem thm = mk_var (hd (vars_of (hd (Thm.prems_of thm)))) handle Empty =>
+ raise THM ("No variables in major premise of rule", 0, [thm]);
+
+val left_var_concl = concl_var hd;
+val right_var_concl = concl_var List.last;
+
+end;
+
+
+
+(** induct data **)
+
+(* rules *)
+
+type rules = (string * thm) NetRules.T;
+
+val init_rules =
+ NetRules.init (fn ((s1: string, th1), (s2, th2)) => s1 = s2 andalso
+ Thm.eq_thm_prop (th1, th2));
+
+fun lookup_rule (rs: rules) = AList.lookup (op =) (NetRules.rules rs);
+
+fun pretty_rules ctxt kind rs =
+ let val thms = map snd (NetRules.rules rs)
+ in Pretty.big_list kind (map (ProofContext.pretty_thm ctxt) thms) end;
+
+
+(* context data *)
+
+structure Induct = GenericDataFun
+(
+ type T = (rules * rules) * (rules * rules) * (rules * rules);
+ val empty =
+ ((init_rules (left_var_prem o #2), init_rules (Thm.major_prem_of o #2)),
+ (init_rules (right_var_concl o #2), init_rules (Thm.major_prem_of o #2)),
+ (init_rules (left_var_concl o #2), init_rules (Thm.concl_of o #2)));
+ val extend = I;
+ fun merge _ (((casesT1, casesS1), (inductT1, inductS1), (coinductT1, coinductS1)),
+ ((casesT2, casesS2), (inductT2, inductS2), (coinductT2, coinductS2))) =
+ ((NetRules.merge (casesT1, casesT2), NetRules.merge (casesS1, casesS2)),
+ (NetRules.merge (inductT1, inductT2), NetRules.merge (inductS1, inductS2)),
+ (NetRules.merge (coinductT1, coinductT2), NetRules.merge (coinductS1, coinductS2)));
+);
+
+val get_local = Induct.get o Context.Proof;
+
+fun dest_rules ctxt =
+ let val ((casesT, casesS), (inductT, inductS), (coinductT, coinductS)) = get_local ctxt in
+ {type_cases = NetRules.rules casesT,
+ set_cases = NetRules.rules casesS,
+ type_induct = NetRules.rules inductT,
+ set_induct = NetRules.rules inductS,
+ type_coinduct = NetRules.rules coinductT,
+ set_coinduct = NetRules.rules coinductS}
+ end;
+
+fun print_rules ctxt =
+ let val ((casesT, casesS), (inductT, inductS), (coinductT, coinductS)) = get_local ctxt in
+ [pretty_rules ctxt "coinduct type:" coinductT,
+ pretty_rules ctxt "coinduct set:" coinductS,
+ pretty_rules ctxt "induct type:" inductT,
+ pretty_rules ctxt "induct set:" inductS,
+ pretty_rules ctxt "cases type:" casesT,
+ pretty_rules ctxt "cases set:" casesS]
+ |> Pretty.chunks |> Pretty.writeln
+ end;
+
+val _ = OuterSyntax.add_parsers [
+ OuterSyntax.improper_command "print_induct_rules" "print induction and cases rules"
+ OuterKeyword.diag (Scan.succeed (Toplevel.no_timing o Toplevel.unknown_context o
+ Toplevel.keep (print_rules o Toplevel.context_of)))];
+
+
+(* access rules *)
+
+val lookup_casesT = lookup_rule o #1 o #1 o get_local;
+val lookup_casesS = lookup_rule o #2 o #1 o get_local;
+val lookup_inductT = lookup_rule o #1 o #2 o get_local;
+val lookup_inductS = lookup_rule o #2 o #2 o get_local;
+val lookup_coinductT = lookup_rule o #1 o #3 o get_local;
+val lookup_coinductS = lookup_rule o #2 o #3 o get_local;
+
+
+fun find_rules which how ctxt x =
+ map snd (NetRules.retrieve (which (get_local ctxt)) (how x));
+
+val find_casesT = find_rules (#1 o #1) encode_type;
+val find_casesS = find_rules (#2 o #1) I;
+val find_inductT = find_rules (#1 o #2) encode_type;
+val find_inductS = find_rules (#2 o #2) I;
+val find_coinductT = find_rules (#1 o #3) encode_type;
+val find_coinductS = find_rules (#2 o #3) I;
+
+
+
+(** attributes **)
+
+local
+
+fun mk_att f g name arg =
+ let val (x, thm) = g arg in (Induct.map (f (name, thm)) x, thm) end;
+
+fun map1 f (x, y, z) = (f x, y, z);
+fun map2 f (x, y, z) = (x, f y, z);
+fun map3 f (x, y, z) = (x, y, f z);
+
+fun add_casesT rule x = map1 (apfst (NetRules.insert rule)) x;
+fun add_casesS rule x = map1 (apsnd (NetRules.insert rule)) x;
+fun add_inductT rule x = map2 (apfst (NetRules.insert rule)) x;
+fun add_inductS rule x = map2 (apsnd (NetRules.insert rule)) x;
+fun add_coinductT rule x = map3 (apfst (NetRules.insert rule)) x;
+fun add_coinductS rule x = map3 (apsnd (NetRules.insert rule)) x;
+
+fun consumes0 x = RuleCases.consumes_default 0 x;
+fun consumes1 x = RuleCases.consumes_default 1 x;
+
+in
+
+val cases_type = mk_att add_casesT consumes0;
+val cases_set = mk_att add_casesS consumes1;
+val induct_type = mk_att add_inductT consumes0;
+val induct_set = mk_att add_inductS consumes1;
+val coinduct_type = mk_att add_coinductT consumes0;
+val coinduct_set = mk_att add_coinductS consumes1;
+
+end;
+
+
+
+(** attribute syntax **)
+
+val casesN = "cases";
+val inductN = "induct";
+val coinductN = "coinduct";
+
+val typeN = "type";
+val setN = "set";
+
+local
+
+fun spec k arg =
+ Scan.lift (Args.$$$ k -- Args.colon) |-- arg ||
+ Scan.lift (Args.$$$ k) >> K "";
+
+fun attrib add_type add_set =
+ Attrib.syntax (spec typeN Args.tyname >> add_type || spec setN Args.const >> add_set);
+
+val cases_att = attrib cases_type cases_set;
+val induct_att = attrib induct_type induct_set;
+val coinduct_att = attrib coinduct_type coinduct_set;
+
+in
+
+val attrib_setup = Attrib.add_attributes
+ [(casesN, cases_att, "declaration of cases rule for type or set"),
+ (inductN, induct_att, "declaration of induction rule for type or set"),
+ (coinductN, coinduct_att, "declaration of coinduction rule for type or set")];
+
+end;
+
+
+
+(** method utils **)
+
+(* alignment *)
+
+fun align_left msg xs ys =
+ let val m = length xs and n = length ys
+ in if m < n then error msg else (Library.take (n, xs) ~~ ys) end;
+
+fun align_right msg xs ys =
+ let val m = length xs and n = length ys
+ in if m < n then error msg else (Library.drop (m - n, xs) ~~ ys) end;
+
+
+(* prep_inst *)
+
+fun prep_inst thy align tune (tm, ts) =
+ let
+ val cert = Thm.cterm_of thy;
+ fun prep_var (x, SOME t) =
+ let
+ val cx = cert x;
+ val {T = xT, thy, ...} = Thm.rep_cterm cx;
+ val ct = cert (tune t);
+ in
+ if Sign.typ_instance thy (#T (Thm.rep_cterm ct), xT) then SOME (cx, ct)
+ else error (Pretty.string_of (Pretty.block
+ [Pretty.str "Ill-typed instantiation:", Pretty.fbrk,
+ Display.pretty_cterm ct, Pretty.str " ::", Pretty.brk 1,
+ Display.pretty_ctyp (#T (Thm.crep_cterm ct))]))
+ end
+ | prep_var (_, NONE) = NONE;
+ val xs = vars_of tm;
+ in
+ align "Rule has fewer variables than instantiations given" xs ts
+ |> map_filter prep_var
+ end;
+
+
+(* trace_rules *)
+
+fun trace_rules _ kind [] = error ("Unable to figure out " ^ kind ^ " rule")
+ | trace_rules ctxt _ rules = Method.trace ctxt rules;
+
+
+(* make_cases *)
+
+fun make_cases is_open rule =
+ RuleCases.make_common is_open (Thm.theory_of_thm rule, Thm.prop_of rule);
+
+fun warn_open true = legacy_feature "open rule cases in proof method"
+ | warn_open false = ();
+
+
+
+(** cases method **)
+
+(*
+ rule selection scheme:
+ cases - default case split
+ `x:A` cases ... - set cases
+ cases t - type cases
+ ... cases ... r - explicit rule
+*)
+
+local
+
+fun get_casesT ctxt ((SOME t :: _) :: _) = find_casesT ctxt (Term.fastype_of t)
+ | get_casesT _ _ = [];
+
+fun get_casesS ctxt (fact :: _) = find_casesS ctxt (Thm.concl_of fact)
+ | get_casesS _ _ = [];
+
+in
+
+fun cases_tac ctxt is_open insts opt_rule facts =
+ let
+ val _ = warn_open is_open;
+ val thy = ProofContext.theory_of ctxt;
+ val cert = Thm.cterm_of thy;
+
+ fun inst_rule r =
+ if null insts then `RuleCases.get r
+ else (align_left "Rule has fewer premises than arguments given" (Thm.prems_of r) insts
+ |> maps (prep_inst thy align_left I)
+ |> Drule.cterm_instantiate) r |> pair (RuleCases.get r);
+
+ val ruleq =
+ (case opt_rule of
+ SOME r => Seq.single (inst_rule r)
+ | NONE =>
+ (get_casesS ctxt facts @ get_casesT ctxt insts @ [Data.cases_default])
+ |> tap (trace_rules ctxt casesN)
+ |> Seq.of_list |> Seq.maps (Seq.try inst_rule));
+ in
+ fn i => fn st =>
+ ruleq
+ |> Seq.maps (RuleCases.consume [] facts)
+ |> Seq.maps (fn ((cases, (_, more_facts)), rule) =>
+ CASES (make_cases is_open rule cases)
+ (Method.insert_tac more_facts i THEN Tactic.rtac rule i) st)
+ end;
+
+end;
+
+
+
+(** induct method **)
+
+val conjunction_congs = [@{thm Pure.all_conjunction}, @{thm imp_conjunction}];
+
+
+(* atomize *)
+
+fun atomize_term thy =
+ MetaSimplifier.rewrite_term thy Data.atomize []
+ #> ObjectLogic.drop_judgment thy;
+
+val atomize_cterm = MetaSimplifier.rewrite true Data.atomize;
+
+val atomize_tac = Simplifier.rewrite_goal_tac Data.atomize;
+
+val inner_atomize_tac =
+ Simplifier.rewrite_goal_tac (map Thm.symmetric conjunction_congs) THEN' atomize_tac;
+
+
+(* rulify *)
+
+fun rulify_term thy =
+ MetaSimplifier.rewrite_term thy (Data.rulify @ conjunction_congs) [] #>
+ MetaSimplifier.rewrite_term thy Data.rulify_fallback [];
+
+fun rulified_term thm =
+ let
+ val thy = Thm.theory_of_thm thm;
+ val rulify = rulify_term thy;
+ val (As, B) = Logic.strip_horn (Thm.prop_of thm);
+ in (thy, Logic.list_implies (map rulify As, rulify B)) end;
+
+val rulify_tac =
+ Simplifier.rewrite_goal_tac (Data.rulify @ conjunction_congs) THEN'
+ Simplifier.rewrite_goal_tac Data.rulify_fallback THEN'
+ Goal.conjunction_tac THEN_ALL_NEW
+ (Simplifier.rewrite_goal_tac [@{thm Pure.conjunction_imp}] THEN' Goal.norm_hhf_tac);
+
+
+(* prepare rule *)
+
+fun rule_instance thy inst rule =
+ Drule.cterm_instantiate (prep_inst thy align_left I (Thm.prop_of rule, inst)) rule;
+
+fun internalize k th =
+ th |> Thm.permute_prems 0 k
+ |> Conv.fconv_rule (Conv.concl_conv (Thm.nprems_of th - k) atomize_cterm);
+
+
+(* guess rule instantiation -- cannot handle pending goal parameters *)
+
+local
+
+fun dest_env thy (env as Envir.Envir {iTs, ...}) =
+ let
+ val cert = Thm.cterm_of thy;
+ val certT = Thm.ctyp_of thy;
+ val pairs = Envir.alist_of env;
+ val ts = map (cert o Envir.norm_term env o #2 o #2) pairs;
+ val xs = map2 (curry (cert o Var)) (map #1 pairs) (map (#T o Thm.rep_cterm) ts);
+ in (map (fn (xi, (S, T)) => (certT (TVar (xi, S)), certT T)) (Vartab.dest iTs), xs ~~ ts) end;
+
+in
+
+fun guess_instance rule i st =
+ let
+ val {thy, maxidx, ...} = Thm.rep_thm st;
+ val goal = Thm.term_of (Thm.cprem_of st i); (*exception Subscript*)
+ val params = rev (rename_wrt_term goal (Logic.strip_params goal));
+ in
+ if not (null params) then
+ (warning ("Cannot determine rule instantiation due to pending parameter(s): " ^
+ commas_quote (map (Sign.string_of_term thy o Syntax.mark_boundT) params));
+ Seq.single rule)
+ else
+ let
+ val rule' = Thm.incr_indexes (maxidx + 1) rule;
+ val concl = Logic.strip_assums_concl goal;
+ in
+ Unify.smash_unifiers thy [(Thm.concl_of rule', concl)]
+ (Envir.empty (#maxidx (Thm.rep_thm rule')))
+ |> Seq.map (fn env => Drule.instantiate (dest_env thy env) rule')
+ end
+ end handle Subscript => Seq.empty;
+
+end;
+
+
+(* special renaming of rule parameters *)
+
+fun special_rename_params ctxt [[SOME (Free (z, Type (T, _)))]] [thm] =
+ let
+ val x = ProofContext.revert_skolem ctxt z;
+ fun index i [] = []
+ | index i (y :: ys) =
+ if x = y then x ^ string_of_int i :: index (i + 1) ys
+ else y :: index i ys;
+ fun rename_params [] = []
+ | rename_params ((y, Type (U, _)) :: ys) =
+ (if U = T then x else y) :: rename_params ys
+ | rename_params ((y, _) :: ys) = y :: rename_params ys;
+ fun rename_asm A =
+ let
+ val xs = rename_params (Logic.strip_params A);
+ val xs' =
+ (case List.filter (equal x) xs of
+ [] => xs | [_] => xs | _ => index 1 xs);
+ in Logic.list_rename_params (xs', A) end;
+ fun rename_prop p =
+ let val (As, C) = Logic.strip_horn p
+ in Logic.list_implies (map rename_asm As, C) end;
+ val cp' = cterm_fun rename_prop (Thm.cprop_of thm);
+ val thm' = Thm.equal_elim (Thm.reflexive cp') thm;
+ in [RuleCases.save thm thm'] end
+ | special_rename_params _ _ ths = ths;
+
+
+(* fix_tac *)
+
+local
+
+fun goal_prefix k ((c as Const ("all", _)) $ Abs (a, T, B)) = c $ Abs (a, T, goal_prefix k B)
+ | goal_prefix 0 _ = Term.dummy_pattern propT
+ | goal_prefix k ((c as Const ("==>", _)) $ A $ B) = c $ A $ goal_prefix (k - 1) B
+ | goal_prefix _ _ = Term.dummy_pattern propT;
+
+fun goal_params k (Const ("all", _) $ Abs (_, _, B)) = goal_params k B + 1
+ | goal_params 0 _ = 0
+ | goal_params k (Const ("==>", _) $ _ $ B) = goal_params (k - 1) B
+ | goal_params _ _ = 0;
+
+fun meta_spec_tac ctxt n (x, T) = SUBGOAL (fn (goal, i) =>
+ let
+ val thy = ProofContext.theory_of ctxt;
+ val cert = Thm.cterm_of thy;
+ val certT = Thm.ctyp_of thy;
+
+ val v = Free (x, T);
+ fun spec_rule prfx (xs, body) =
+ @{thm Pure.meta_spec}
+ |> Thm.rename_params_rule ([ProofContext.revert_skolem ctxt x], 1)
+ |> Thm.lift_rule (cert prfx)
+ |> `(Thm.prop_of #> Logic.strip_assums_concl)
+ |-> (fn pred $ arg =>
+ Drule.cterm_instantiate
+ [(cert (Term.head_of pred), cert (Logic.rlist_abs (xs, body))),
+ (cert (Term.head_of arg), cert (Logic.rlist_abs (xs, v)))]);
+
+ fun goal_concl k xs (Const ("all", _) $ Abs (a, T, B)) = goal_concl k ((a, T) :: xs) B
+ | goal_concl 0 xs B =
+ if not (Term.exists_subterm (fn t => t aconv v) B) then NONE
+ else SOME (xs, Term.absfree (x, T, Term.incr_boundvars 1 B))
+ | goal_concl k xs (Const ("==>", _) $ _ $ B) = goal_concl (k - 1) xs B
+ | goal_concl _ _ _ = NONE;
+ in
+ (case goal_concl n [] goal of
+ SOME concl =>
+ (compose_tac (false, spec_rule (goal_prefix n goal) concl, 1) THEN' rtac asm_rl) i
+ | NONE => all_tac)
+ end);
+
+fun miniscope_tac p =
+ CONVERSION (Conv.forall_conv p (MetaSimplifier.rewrite true [Thm.symmetric Drule.norm_hhf_eq]));
+
+in
+
+fun fix_tac _ _ [] = K all_tac
+ | fix_tac ctxt n xs = SUBGOAL (fn (goal, i) =>
+ (EVERY' (map (meta_spec_tac ctxt n) xs) THEN'
+ (miniscope_tac (goal_params n goal))) i);
+
+end;
+
+
+(* add_defs *)
+
+fun add_defs def_insts =
+ let
+ fun add (SOME (SOME x, t)) ctxt =
+ let val ([(lhs, (_, th))], ctxt') = LocalDefs.add_defs [((x, NoSyn), (("", []), t))] ctxt
+ in ((SOME lhs, [th]), ctxt') end
+ | add (SOME (NONE, t)) ctxt = ((SOME t, []), ctxt)
+ | add NONE ctxt = ((NONE, []), ctxt);
+ in fold_map add def_insts #> apfst (split_list #> apsnd flat) end;
+
+
+(* induct_tac *)
+
+(*
+ rule selection scheme:
+ `x:A` induct ... - set induction
+ induct x - type induction
+ ... induct ... r - explicit rule
+*)
+
+local
+
+fun get_inductT ctxt insts =
+ fold_rev multiply (insts |> map_filter (fn [] => NONE | ts => List.last ts)
+ |> map (find_inductT ctxt o Term.fastype_of)) [[]]
+ |> filter_out (forall PureThy.is_internal);
+
+fun get_inductS ctxt (fact :: _) = map single (find_inductS ctxt (Thm.concl_of fact))
+ | get_inductS _ _ = [];
+
+in
+
+fun induct_tac ctxt is_open def_insts arbitrary taking opt_rule facts =
+ let
+ val _ = warn_open is_open;
+ val thy = ProofContext.theory_of ctxt;
+ val cert = Thm.cterm_of thy;
+
+ val ((insts, defs), defs_ctxt) = fold_map add_defs def_insts ctxt |>> split_list;
+ val atomized_defs = map (map (Conv.fconv_rule ObjectLogic.atomize)) defs;
+
+ fun inst_rule (concls, r) =
+ (if null insts then `RuleCases.get r
+ else (align_left "Rule has fewer conclusions than arguments given"
+ (map Logic.strip_imp_concl (Logic.dest_conjunctions (Thm.concl_of r))) insts
+ |> maps (prep_inst thy align_right (atomize_term thy))
+ |> Drule.cterm_instantiate) r |> pair (RuleCases.get r))
+ |> (fn ((cases, consumes), th) => (((cases, concls), consumes), th));
+
+ val ruleq =
+ (case opt_rule of
+ SOME rs => Seq.single (inst_rule (RuleCases.strict_mutual_rule ctxt rs))
+ | NONE =>
+ (get_inductS ctxt facts @
+ map (special_rename_params defs_ctxt insts) (get_inductT ctxt insts))
+ |> map_filter (RuleCases.mutual_rule ctxt)
+ |> tap (trace_rules ctxt inductN o map #2)
+ |> Seq.of_list |> Seq.maps (Seq.try inst_rule));
+
+ fun rule_cases rule =
+ RuleCases.make_nested is_open (Thm.prop_of rule) (rulified_term rule);
+ in
+ (fn i => fn st =>
+ ruleq
+ |> Seq.maps (RuleCases.consume (flat defs) facts)
+ |> Seq.maps (fn (((cases, concls), (more_consumes, more_facts)), rule) =>
+ (PRECISE_CONJUNCTS (length concls) (ALLGOALS (fn j =>
+ (CONJUNCTS (ALLGOALS
+ (Method.insert_tac (more_facts @ nth_list atomized_defs (j - 1))
+ THEN' fix_tac defs_ctxt
+ (nth concls (j - 1) + more_consumes)
+ (nth_list arbitrary (j - 1))))
+ THEN' inner_atomize_tac) j))
+ THEN' atomize_tac) i st |> Seq.maps (fn st' =>
+ guess_instance (internalize more_consumes rule) i st'
+ |> Seq.map (rule_instance thy taking)
+ |> Seq.maps (fn rule' =>
+ CASES (rule_cases rule' cases)
+ (Tactic.rtac rule' i THEN
+ PRIMITIVE (singleton (ProofContext.export defs_ctxt ctxt))) st'))))
+ THEN_ALL_NEW_CASES rulify_tac
+ end;
+
+end;
+
+
+
+(** coinduct method **)
+
+(*
+ rule selection scheme:
+ goal "x:A" coinduct ... - set coinduction
+ coinduct x - type coinduction
+ coinduct ... r - explicit rule
+*)
+
+local
+
+fun get_coinductT ctxt (SOME t :: _) = find_coinductT ctxt (Term.fastype_of t)
+ | get_coinductT _ _ = [];
+
+fun get_coinductS ctxt goal = find_coinductS ctxt (Logic.strip_assums_concl goal);
+
+in
+
+fun coinduct_tac ctxt is_open inst taking opt_rule facts =
+ let
+ val _ = warn_open is_open;
+ val thy = ProofContext.theory_of ctxt;
+ val cert = Thm.cterm_of thy;
+
+ fun inst_rule r =
+ if null inst then `RuleCases.get r
+ else Drule.cterm_instantiate (prep_inst thy align_left I (Thm.concl_of r, inst)) r
+ |> pair (RuleCases.get r);
+
+ fun ruleq goal =
+ (case opt_rule of
+ SOME r => Seq.single (inst_rule r)
+ | NONE =>
+ (get_coinductS ctxt goal @ get_coinductT ctxt inst)
+ |> tap (trace_rules ctxt coinductN)
+ |> Seq.of_list |> Seq.maps (Seq.try inst_rule));
+ in
+ SUBGOAL_CASES (fn (goal, i) => fn st =>
+ ruleq goal
+ |> Seq.maps (RuleCases.consume [] facts)
+ |> Seq.maps (fn ((cases, (_, more_facts)), rule) =>
+ guess_instance rule i st
+ |> Seq.map (rule_instance thy taking)
+ |> Seq.maps (fn rule' =>
+ CASES (make_cases is_open rule' cases)
+ (Method.insert_tac more_facts i THEN Tactic.rtac rule' i) st)))
+ end;
+
+end;
+
+
+
+(** concrete syntax **)
+
+val openN = "open";
+val arbitraryN = "arbitrary";
+val takingN = "taking";
+val ruleN = "rule";
+
+local
+
+fun single_rule [rule] = rule
+ | single_rule _ = error "Single rule expected";
+
+fun named_rule k arg get =
+ Scan.lift (Args.$$$ k -- Args.colon) |-- Scan.repeat arg :|--
+ (fn names => Scan.peek (fn context => Scan.succeed (names |> map (fn name =>
+ (case get (Context.proof_of context) name of SOME x => x
+ | NONE => error ("No rule for " ^ k ^ " " ^ quote name))))));
+
+fun rule get_type get_set =
+ named_rule typeN Args.tyname get_type ||
+ named_rule setN Args.const get_set ||
+ Scan.lift (Args.$$$ ruleN -- Args.colon) |-- Attrib.thms;
+
+val cases_rule = rule lookup_casesT lookup_casesS >> single_rule;
+val induct_rule = rule lookup_inductT lookup_inductS;
+val coinduct_rule = rule lookup_coinductT lookup_coinductS >> single_rule;
+
+val inst = Scan.lift (Args.$$$ "_") >> K NONE || Args.term >> SOME;
+
+val def_inst =
+ ((Scan.lift (Args.name --| (Args.$$$ "\\<equiv>" || Args.$$$ "==")) >> SOME)
+ -- Args.term) >> SOME ||
+ inst >> Option.map (pair NONE);
+
+val free = Scan.state -- Args.term >> (fn (_, Free v) => v | (context, t) =>
+ error ("Bad free variable: " ^ ProofContext.string_of_term (Context.proof_of context) t));
+
+fun unless_more_args scan = Scan.unless (Scan.lift
+ ((Args.$$$ arbitraryN || Args.$$$ takingN || Args.$$$ typeN ||
+ Args.$$$ setN || Args.$$$ ruleN) -- Args.colon)) scan;
+
+val arbitrary = Scan.optional (Scan.lift (Args.$$$ arbitraryN -- Args.colon) |--
+ Args.and_list1 (Scan.repeat (unless_more_args free))) [];
+
+val taking = Scan.optional (Scan.lift (Args.$$$ takingN -- Args.colon) |--
+ Scan.repeat1 (unless_more_args inst)) [];
+
+in
+
+fun cases_meth src =
+ Method.syntax (Args.mode openN --
+ (Args.and_list (Scan.repeat (unless_more_args inst)) -- Scan.option cases_rule)) src
+ #> (fn ((is_open, (insts, opt_rule)), ctxt) =>
+ Method.METHOD_CASES (fn facts =>
+ Seq.DETERM (HEADGOAL (cases_tac ctxt is_open insts opt_rule facts))));
+
+fun induct_meth src =
+ Method.syntax (Args.mode openN --
+ (Args.and_list (Scan.repeat (unless_more_args def_inst)) --
+ (arbitrary -- taking -- Scan.option induct_rule))) src
+ #> (fn ((is_open, (insts, ((arbitrary, taking), opt_rule))), ctxt) =>
+ Method.RAW_METHOD_CASES (fn facts =>
+ Seq.DETERM (HEADGOAL (induct_tac ctxt is_open insts arbitrary taking opt_rule facts))));
+
+fun coinduct_meth src =
+ Method.syntax (Args.mode openN --
+ (Scan.repeat (unless_more_args inst) -- taking -- Scan.option coinduct_rule)) src
+ #> (fn ((is_open, ((insts, taking), opt_rule)), ctxt) =>
+ Method.RAW_METHOD_CASES (fn facts =>
+ Seq.DETERM (HEADGOAL (coinduct_tac ctxt is_open insts taking opt_rule facts))));
+
+end;
+
+
+
+(** theory setup **)
+
+val setup =
+ attrib_setup #>
+ Method.add_methods
+ [(casesN, cases_meth, "case analysis on types or sets"),
+ (inductN, induct_meth, "induction on types or sets"),
+ (coinductN, coinduct_meth, "coinduction on types or sets")];
+
+end;