(* Title: HOL/Tools/induct_method.ML
ID: $Id$
Author: Markus Wenzel, TU Muenchen
Proof by cases and induction on types (intro) and sets (elim).
*)
signature INDUCT_METHOD =
sig
val print_global_rules: theory -> unit
val print_local_rules: Proof.context -> unit
val cases_type_global: string -> theory attribute
val cases_set_global: string -> theory attribute
val cases_type_local: string -> Proof.context attribute
val cases_set_local: string -> Proof.context attribute
val induct_type_global: string -> theory attribute
val induct_set_global: string -> theory attribute
val induct_type_local: string -> Proof.context attribute
val induct_set_local: string -> Proof.context attribute
val setup: (theory -> theory) list
end;
structure InductMethod: INDUCT_METHOD =
struct
(** global and local induct data **)
(* rules *)
type rules = (string * thm) NetRules.T;
fun eq_rule ((s1:string, th1), (s2, th2)) = s1 = s2 andalso Thm.eq_thm (th1, th2);
val type_rules = NetRules.init eq_rule (Thm.concl_of o #2);
val set_rules = NetRules.init eq_rule (Thm.major_prem_of o #2);
fun lookup_rule (rs:rules) name = Library.assoc (NetRules.rules rs, name);
fun print_rules kind rs =
let val thms = map snd (NetRules.rules rs)
in Pretty.writeln (Pretty.big_list kind (map Display.pretty_thm thms)) end;
(* theory data kind 'HOL/induct_method' *)
structure GlobalInductArgs =
struct
val name = "HOL/induct_method";
type T = (rules * rules) * (rules * rules);
val empty = ((type_rules, set_rules), (type_rules, set_rules));
val copy = I;
val prep_ext = I;
fun merge (((casesT1, casesS1), (inductT1, inductS1)),
((casesT2, casesS2), (inductT2, inductS2))) =
((NetRules.merge (casesT1, casesT2), NetRules.merge (casesS1, casesS2)),
(NetRules.merge (inductT1, inductT2), NetRules.merge (inductS1, inductS2)));
fun print _ ((casesT, casesS), (inductT, inductS)) =
(print_rules "type cases:" casesT;
print_rules "set cases:" casesS;
print_rules "type induct:" inductT;
print_rules "set induct:" inductS);
end;
structure GlobalInduct = TheoryDataFun(GlobalInductArgs);
val print_global_rules = GlobalInduct.print;
(* proof data kind 'HOL/induct_method' *)
structure LocalInductArgs =
struct
val name = "HOL/induct_method";
type T = GlobalInductArgs.T;
fun init thy = GlobalInduct.get thy;
fun print x = GlobalInductArgs.print x;
end;
structure LocalInduct = ProofDataFun(LocalInductArgs);
val print_local_rules = LocalInduct.print;
(* access rules *)
val get_cases = #1 o LocalInduct.get;
val get_induct = #2 o LocalInduct.get;
val lookup_casesT = lookup_rule o #1 o get_cases;
val lookup_casesS = lookup_rule o #2 o get_cases;
val lookup_inductT = lookup_rule o #1 o get_induct;
val lookup_inductS = lookup_rule o #2 o get_induct;
(** attributes **)
local
fun mk_att f g name (x, thm) = (f (g (name, thm)) x, thm);
fun add_casesT rule x = apfst (apfst (NetRules.insert rule)) x;
fun add_casesS rule x = apfst (apsnd (NetRules.insert rule)) x;
fun add_inductT rule x = apsnd (apfst (NetRules.insert rule)) x;
fun add_inductS rule x = apsnd (apsnd (NetRules.insert rule)) x;
in
val cases_type_global = mk_att GlobalInduct.map add_casesT;
val cases_set_global = mk_att GlobalInduct.map add_casesS;
val induct_type_global = mk_att GlobalInduct.map add_inductT;
val induct_set_global = mk_att GlobalInduct.map add_inductS;
val cases_type_local = mk_att LocalInduct.map add_casesT;
val cases_set_local = mk_att LocalInduct.map add_casesS;
val induct_type_local = mk_att LocalInduct.map add_inductT;
val induct_set_local = mk_att LocalInduct.map add_inductS;
end;
(** misc utils **)
fun vars_of tm = (*ordered left-to-right, preferring right!*)
Term.foldl_aterms (fn (ts, t as Var _) => t :: ts | (ts, _) => ts) ([], tm)
|> Library.distinct |> rev;
fun type_name t =
#1 (Term.dest_Type (Term.type_of t))
handle TYPE _ => raise TERM ("Bad type of term argument", [t]);
(** cases method **)
(*
rule selection:
cases - classical case split
<x:A> cases - set elimination
... cases t - datatype exhaustion
... cases ... r - explicit rule
*)
fun cases_var thm =
(case try (hd o vars_of o hd o Logic.strip_assums_hyp o Library.last_elem o Thm.prems_of) thm of
None => raise THM ("Malformed cases rule", 0, [thm])
| Some x => x);
fun cases_tac (ctxt, args) facts =
let
val sg = ProofContext.sign_of ctxt;
val cert = Thm.cterm_of sg;
fun inst_rule t thm =
Drule.cterm_instantiate [(cert (cases_var thm), cert t)] thm;
val thms =
(case (args, facts) of
((None, None), []) => [case_split_thm]
| ((None, None), th :: _) =>
NetRules.may_unify (#2 (get_cases ctxt))
(Logic.strip_assums_concl (#prop (Thm.rep_thm th)))
|> map #2
| ((Some t, None), _) =>
let val name = type_name t in
(case lookup_casesT ctxt name of
None => error ("No cases rule for type: " ^ quote name)
| Some thm => [inst_rule t thm])
end
| ((None, Some thm), _) => [thm]
| ((Some t, Some thm), _) => [inst_rule t thm]);
in Method.rule_tac thms facts end;
val cases_meth = Method.METHOD o (FINDGOAL oo cases_tac);
(** induct method **)
(*
rule selection:
induct - mathematical induction
<x:A> induct - set induction
... induct x - datatype induction
... induct ... r - explicit rule
*)
fun induct_tac (ctxt, args) facts =
let
val sg = ProofContext.sign_of ctxt;
val cert = Thm.cterm_of sg;
fun prep_inst (concl, ts) =
let val xs = vars_of concl; val n = length xs - length ts in
if n < 0 then error "More arguments given than in induction rule"
else map cert (Library.drop (n, xs)) ~~ map cert ts
end;
fun inst_rule insts thm =
Drule.cterm_instantiate (flat (map2 prep_inst
(HOLogic.dest_conj (HOLogic.dest_Trueprop (Thm.concl_of thm)), insts))) thm;
val thms =
(case (args, facts) of
(([], None), []) => [nat_induct]
| (([], None), th :: _) =>
NetRules.may_unify (#2 (get_induct ctxt))
(Logic.strip_assums_concl (#prop (Thm.rep_thm th)))
|> map #2
| ((insts, None), _) =>
let val name = type_name (last_elem (hd insts)) in
(case lookup_inductT ctxt name of
None => error ("No induct rule for type: " ^ quote name)
| Some thm => [inst_rule insts thm])
end
| (([], Some thm), _) => [thm]
| ((insts, Some thm), _) => [inst_rule insts thm]);
in Method.rule_tac thms facts end;
val induct_meth = Method.METHOD o (FINDGOAL oo induct_tac);
(** concrete syntax **)
val casesN = "cases";
val inductN = "induct";
val typeN = "type";
val setN = "set";
val ruleN = "rule";
(* attributes *)
fun spec k = (Args.$$$ k -- Args.$$$ ":") |-- Args.!!! Args.name;
fun attrib sign_of add_type add_set = Scan.depend (fn x =>
let val sg = sign_of x in
spec typeN >> (add_type o Sign.intern_tycon sg) ||
spec setN >> (add_set o Sign.intern_const sg)
end >> pair x);
val cases_attr =
(Attrib.syntax (attrib Theory.sign_of cases_type_global cases_set_global),
Attrib.syntax (attrib ProofContext.sign_of cases_type_local cases_set_local));
val induct_attr =
(Attrib.syntax (attrib Theory.sign_of induct_type_global induct_set_global),
Attrib.syntax (attrib ProofContext.sign_of induct_type_local induct_set_local));
(* methods *)
local
fun err k get name =
(case get name of Some x => x
| None => error ("No rule for " ^ k ^ " " ^ quote name));
fun rule get_type get_set =
Scan.depend (fn ctxt =>
let val sg = ProofContext.sign_of ctxt in
spec typeN >> (err typeN (get_type ctxt) o Sign.intern_tycon sg) ||
spec setN >> (err setN (get_set ctxt) o Sign.intern_const sg)
end >> pair ctxt) ||
Scan.lift (Args.$$$ ruleN -- Args.$$$ ":") |-- Attrib.local_thm;
val cases_rule = rule lookup_casesT lookup_casesS;
val induct_rule = rule lookup_inductT lookup_inductS;
val kind = (Args.$$$ typeN || Args.$$$ setN || Args.$$$ ruleN) -- Args.$$$ ":";
val term = Scan.unless (Scan.lift kind) Args.local_term;
in
val cases_args = Method.syntax (Scan.option term -- Scan.option cases_rule);
val induct_args = Method.syntax (Args.and_list (Scan.repeat1 term) -- Scan.option induct_rule);
end;
(** theory setup **)
val setup =
[GlobalInduct.init, LocalInduct.init,
Attrib.add_attributes
[(casesN, cases_attr, "cases rule for type or set"),
(inductN, induct_attr, "induction rule for type or set")],
Method.add_methods
[("cases", cases_meth oo cases_args, "case analysis on types or sets"),
("induct", induct_meth oo induct_args, "induction on types or sets")]];
end;