(* Title: HOL/Tools/induct_method.ML
ID: $Id$
Author: Markus Wenzel, TU Muenchen
Proof by cases and induction on types and sets.
*)
signature INDUCT_METHOD =
sig
val dest_global_rules: theory ->
{type_cases: (string * thm) list, set_cases: (string * thm) list,
type_induct: (string * thm) list, set_induct: (string * thm) list}
val print_global_rules: theory -> unit
val dest_local_rules: Proof.context ->
{type_cases: (string * thm) list, set_cases: (string * thm) list,
type_induct: (string * thm) list, set_induct: (string * thm) list}
val print_local_rules: Proof.context -> unit
val vars_of: term -> term list
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 con_elim_tac: simpset -> tactic
val con_elim_solved_tac: simpset -> tactic
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);
fun dest ((casesT, casesS), (inductT, inductS)) =
{type_cases = NetRules.rules casesT,
set_cases = NetRules.rules casesS,
type_induct = NetRules.rules inductT,
set_induct = NetRules.rules inductS};
end;
structure GlobalInduct = TheoryDataFun(GlobalInductArgs);
val print_global_rules = GlobalInduct.print;
val dest_global_rules = GlobalInductArgs.dest o GlobalInduct.get;
(* 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;
val dest_local_rules = GlobalInductArgs.dest o LocalInduct.get;
(* 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 **)
(* terms *)
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]);
(* simplifying cases rules *)
local
(*delete needless equality assumptions*)
val refl_thin = prove_goal HOL.thy "!!P. [| a=a; P |] ==> P"
(fn _ => [assume_tac 1]);
val elim_rls = [asm_rl, FalseE, refl_thin, conjE, exE, Pair_inject];
val elim_tac = REPEAT o Tactic.eresolve_tac elim_rls;
fun simp_case_tac ss =
EVERY' [elim_tac, asm_full_simp_tac ss, elim_tac, REPEAT o bound_hyp_subst_tac];
in
fun con_elim_tac ss = ALLGOALS (simp_case_tac ss) THEN prune_params_tac;
fun con_elim_solved_tac ss =
ALLGOALS (fn i => TRY (simp_case_tac ss i THEN_MAYBE no_tac)) THEN prune_params_tac;
end;
(** cases method **)
(*
rule selection:
cases - classical case split
cases t - datatype exhaustion
<x:A> cases ... - set elimination
... cases ... R - explicit rule
*)
local
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 simplify_cases ctxt =
Tactic.rule_by_tactic (con_elim_solved_tac (Simplifier.get_local_simpset ctxt));
fun cases_tac (ctxt, (simplified, 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 cond_simp = if simplified then simplify_cases ctxt else I;
fun find_cases th =
NetRules.may_unify (#2 (get_cases ctxt))
(Logic.strip_assums_concl (#prop (Thm.rep_thm th)));
val rules =
(case (args, facts) of
((None, None), []) => [(case_split_thm, ["True", "False"])]
| ((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, RuleCases.get thm)])
end
| ((None, None), th :: _) => map (RuleCases.add o #2) (find_cases th)
| ((Some t, None), th :: _) =>
(case find_cases th of (*may instantiate first rule only!*)
(_, thm) :: _ => [(inst_rule t thm, RuleCases.get thm)]
| [] => [])
| ((None, Some thm), _) => [RuleCases.add thm]
| ((Some t, Some thm), _) => [(inst_rule t thm, RuleCases.get thm)]);
fun prep_rule (thm, cases) =
Seq.map (rpair cases o cond_simp) (Method.multi_resolves facts [thm]);
in Method.resolveq_cases_tac (Seq.flat (Seq.map prep_rule (Seq.of_list rules))) end;
in
val cases_meth = Method.METHOD_CASES o (HEADGOAL oo cases_tac);
end;
(** induct method **)
(*
rule selection:
induct - mathematical induction
induct x - datatype induction
<x:A> induct ... - set induction
... induct ... R - explicit rule
*)
local
infix 1 THEN_ALL_NEW_CASES;
fun (tac1 THEN_ALL_NEW_CASES tac2) i st =
st |> Seq.THEN (tac1 i, (fn (st', cases) =>
Seq.map (rpair cases) (Seq.INTERVAL tac2 i (i + nprems_of st' - nprems_of st) st')));
fun induct_rule ctxt t =
let val name = type_name t in
(case lookup_inductT ctxt name of
None => error ("No induct rule for type: " ^ quote name)
| Some thm => (name, thm))
end;
fun join_rules [(_, thm)] = thm
| join_rules raw_thms =
let
val thms = (map (apsnd Drule.freeze_all) raw_thms);
fun eq_prems ((_, th1), (_, th2)) =
Term.aconvs (Thm.prems_of th1, Thm.prems_of th2);
in
(case Library.gen_distinct eq_prems thms of
[(_, thm)] =>
let
val cprems = Drule.cprems_of thm;
val asms = map Thm.assume cprems;
fun strip (_, th) = Drule.implies_elim_list th asms;
in
foldr1 (fn (th, th') => [th, th'] MRS conjI) (map strip thms)
|> Drule.implies_intr_list cprems
|> Drule.standard
end
| [] => error "No rule given"
| bads => error ("Incompatible rules for " ^ commas_quote (map #1 bads)))
end;
fun induct_tac (ctxt, (stripped, 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;
fun find_induct th =
NetRules.may_unify (#2 (get_induct ctxt))
(Logic.strip_assums_concl (#prop (Thm.rep_thm th)));
val rules =
(case (args, facts) of
(([], None), []) => []
| ((insts, None), []) =>
let val thms = map (induct_rule ctxt o last_elem) insts
in [(inst_rule insts (join_rules thms), RuleCases.get (#2 (hd thms)))] end
| (([], None), th :: _) => map (RuleCases.add o #2) (find_induct th)
| ((insts, None), th :: _) =>
(case find_induct th of (*may instantiate first rule only!*)
(_, thm) :: _ => [(inst_rule insts thm, RuleCases.get thm)]
| [] => [])
| (([], Some thm), _) => [RuleCases.add thm]
| ((insts, Some thm), _) => [(inst_rule insts thm, RuleCases.get thm)]);
fun prep_rule (thm, cases) =
Seq.map (rpair cases) (Method.multi_resolves facts [thm]);
val tac = Method.resolveq_cases_tac (Seq.flat (Seq.map prep_rule (Seq.of_list rules)));
in
if stripped then tac THEN_ALL_NEW_CASES (REPEAT o Tactic.match_tac [impI, allI, ballI])
else tac
end;
in
val induct_meth = Method.METHOD_CASES o (HEADGOAL oo induct_tac);
end;
(** concrete syntax **)
val casesN = "cases";
val inductN = "induct";
val simplifiedN = "simplified";
val strippedN = "stripped";
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;
fun mode name =
Scan.lift (Scan.optional (Args.$$$ name -- Args.$$$ ":" >> K true) false);
in
val cases_args =
Method.syntax (mode simplifiedN -- (Scan.option term -- Scan.option cases_rule));
val induct_args =
Method.syntax (mode strippedN -- (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;