(* Title: HOL/Tools/induct_method.ML
ID: $Id$
Author: Markus Wenzel, TU Muenchen
License: GPL (GNU GENERAL PUBLIC LICENSE)
Proof by cases and induction on types and sets.
*)
signature INDUCT_METHOD =
sig
val vars_of: term -> term list
val concls_of: thm -> term list
val simp_case_tac: bool -> simpset -> int -> tactic
val setup: (theory -> theory) list
end;
structure InductMethod: INDUCT_METHOD =
struct
(** theory context references **)
val inductive_atomize = thms "inductive_atomize";
val inductive_rulify1 = thms "inductive_rulify1";
val inductive_rulify2 = thms "inductive_rulify2";
(** misc utils **)
(* align lists *)
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;
(* thms and terms *)
fun imp_concl_of t = imp_concl_of (#2 (HOLogic.dest_imp t)) handle TERM _ => t;
val concls_of = map imp_concl_of o HOLogic.dest_conj o HOLogic.dest_Trueprop o Thm.concl_of;
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 ("Type of term argument is too general", [t]);
fun prep_inst align cert f (tm, ts) =
let
fun prep_var (x, Some t) =
let
val cx = cert x;
val {T = xT, sign, ...} = Thm.rep_cterm cx;
val orig_ct = cert t;
val ct = f orig_ct;
in
if Sign.typ_instance sign (#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 orig_ct, Pretty.str " ::", Pretty.brk 1,
Display.pretty_ctyp (#T (Thm.crep_cterm orig_ct))]))
end
| prep_var (_, None) = None;
in
align "Rule has fewer variables than instantiations given" (vars_of tm) ts
|> mapfilter prep_var
end;
(* 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;
in
fun simp_case_tac solved ss i =
EVERY' [elim_tac, asm_full_simp_tac ss, elim_tac, REPEAT o bound_hyp_subst_tac] i
THEN_MAYBE (if solved then no_tac else all_tac);
end;
(* resolution and cases *)
local
fun gen_resolveq_tac tac rules i st =
Seq.flat (Seq.map (fn rule => tac rule i st) rules);
in
fun resolveq_cases_tac make tac = gen_resolveq_tac (fn (rule, (cases, facts)) => fn i => fn st =>
Seq.map (rpair (make rule cases))
((Method.insert_tac facts THEN' tac THEN' Tactic.rtac rule) i st));
end;
(** cases method **)
(*
rule selection:
cases - classical case split
cases t - datatype exhaustion
<x:A> cases ... - set elimination
... cases ... R - explicit rule
*)
val case_split = RuleCases.name ["True", "False"] case_split_thm;
local
fun simplified_cases ctxt cases thm =
let
val nprems = Thm.nprems_of thm;
val opt_cases =
Library.replicate (nprems - Int.min (nprems, length cases)) None @
map Some (Library.take (nprems, cases));
val tac = simp_case_tac true (Simplifier.get_local_simpset ctxt);
fun simp ((i, c), (th, cs)) =
(case try (Tactic.rule_by_tactic (tac i)) th of
None => (th, c :: cs)
| Some th' => (th', None :: cs));
val (thm', opt_cases') = foldr simp (1 upto Thm.nprems_of thm ~~ opt_cases, (thm, []));
in (thm', mapfilter I opt_cases') end;
fun cases_tac (ctxt, ((simplified, open_parms), args)) facts =
let
val sg = ProofContext.sign_of ctxt;
val cert = Thm.cterm_of sg;
fun inst_rule insts thm =
(align_left "Rule has fewer premises than arguments given" (Thm.prems_of thm) insts
|> (flat o map (prep_inst align_left cert I))
|> Drule.cterm_instantiate) thm;
fun find_cases th =
NetRules.may_unify (#2 (InductAttrib.get_cases ctxt))
(Logic.strip_assums_concl (#prop (Thm.rep_thm th)));
val rules =
(case (fst args, facts) of
(([], None), []) => [RuleCases.add case_split]
| ((insts, None), []) =>
let
val name = type_name (hd (flat (map (mapfilter I) insts)))
handle Library.LIST _ => error "Unable to figure out type cases rule"
in
(case InductAttrib.lookup_casesT ctxt name of
None => error ("No cases rule for type: " ^ quote name)
| Some thm => [(inst_rule insts thm, RuleCases.get thm)])
end
| (([], None), th :: _) => map (RuleCases.add o #2) (find_cases th)
| ((insts, None), th :: _) =>
(case find_cases 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)])
|> map (Library.apfst (Attrib.read_inst' (snd args) ctxt));
val cond_simp = if simplified then simplified_cases ctxt else rpair;
fun prep_rule (thm, (cases, n)) = Seq.map (apsnd (rpair (drop (n, facts))) o cond_simp cases)
(Method.multi_resolves (take (n, facts)) [thm]);
in
resolveq_cases_tac (RuleCases.make open_parms) (K all_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 x - datatype induction
<x:A> induct ... - set induction
... induct ... R - explicit rule
*)
local
val atomize_cterm = Thm.cterm_fun AutoBind.drop_judgment o hol_rewrite_cterm inductive_atomize;
val atomize_tac = Tactic.rewrite_goal_tac inductive_atomize;
val rulify_cterm = hol_rewrite_cterm inductive_rulify2 o hol_rewrite_cterm inductive_rulify1;
val rulify_tac =
Tactic.rewrite_goal_tac inductive_rulify1 THEN'
Tactic.rewrite_goal_tac inductive_rulify2 THEN'
Tactic.norm_hhf_tac;
fun rulify_cases cert =
let
val ruly = Thm.term_of o rulify_cterm o cert;
fun ruly_case {fixes, assumes, binds} =
{fixes = fixes, assumes = map ruly assumes,
binds = map (apsnd (apsome (AutoBind.drop_judgment o ruly))) binds};
in map (apsnd ruly_case) ooo RuleCases.make_raw end;
val weak_strip_tac = REPEAT o Tactic.match_tac [impI, allI, ballI];
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 InductAttrib.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, open_parms), args)) facts =
let
val sg = ProofContext.sign_of ctxt;
val cert = Thm.cterm_of sg;
fun inst_rule insts thm =
(align_right "Rule has fewer conclusions than arguments given" (concls_of thm) insts
|> (flat o map (prep_inst align_right cert atomize_cterm))
|> Drule.cterm_instantiate) thm;
fun find_induct th =
NetRules.may_unify (#2 (InductAttrib.get_induct ctxt))
(Logic.strip_assums_concl (#prop (Thm.rep_thm th)));
val rules =
(case (fst args, facts) of
(([], None), []) => []
| ((insts, None), []) =>
let val thms = map (induct_rule ctxt o last_elem o mapfilter I) insts
handle Library.LIST _ => error "Unable to figure out type induction rule"
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)])
|> map (Library.apfst (Attrib.read_inst' (snd args) ctxt));
fun prep_rule (thm, (cases, n)) =
Seq.map (rpair (cases, drop (n, facts))) (Method.multi_resolves (take (n, facts)) [thm]);
val tac = resolveq_cases_tac (rulify_cases cert open_parms) atomize_tac
(Seq.flat (Seq.map prep_rule (Seq.of_list rules)));
in
tac THEN_ALL_NEW_CASES (rulify_tac THEN' (if stripped then weak_strip_tac else K all_tac))
end;
in
val induct_meth = Method.METHOD_CASES o (HEADGOAL oo induct_tac);
end;
(** concrete syntax **)
val simplifiedN = "simplified";
val strippedN = "stripped";
val openN = "open";
val ruleN = "rule";
val ofN = "of";
local
fun err k get name =
(case get name of Some x => x
| None => error ("No rule for " ^ k ^ " " ^ quote name));
fun spec k = (Args.$$$ k -- Args.colon) |-- Args.!!! Args.name;
fun rule get_type get_set =
Scan.depend (fn ctxt =>
let val sg = ProofContext.sign_of ctxt in
spec InductAttrib.typeN >> (err InductAttrib.typeN (get_type ctxt) o Sign.intern_tycon sg) ||
spec InductAttrib.setN >> (err InductAttrib.setN (get_set ctxt) o Sign.intern_const sg)
end >> pair ctxt) ||
Scan.lift (Args.$$$ ruleN -- Args.colon) |-- Attrib.local_thm;
val cases_rule = rule InductAttrib.lookup_casesT InductAttrib.lookup_casesS;
val induct_rule = rule InductAttrib.lookup_inductT InductAttrib.lookup_inductS;
val kind_inst =
(Args.$$$ InductAttrib.typeN || Args.$$$ InductAttrib.setN || Args.$$$ ruleN || Args.$$$ ofN)
-- Args.colon;
val term = Scan.unless (Scan.lift kind_inst) Args.local_term;
val term_dummy = Scan.unless (Scan.lift kind_inst)
(Scan.lift (Args.$$$ "_") >> K None || Args.local_term >> Some);
val instss = Args.and_list (Scan.repeat1 term_dummy);
(* FIXME Attrib.insts': better use actual term args *)
val rule_insts =
Scan.lift (Scan.optional ((Args.$$$ ofN -- Args.colon) |-- Args.!!! Attrib.insts') ([], []));
in
val cases_args = Method.syntax
(Args.mode simplifiedN -- Args.mode openN -- (instss -- Scan.option cases_rule -- rule_insts));
val induct_args = Method.syntax
(Args.mode strippedN -- Args.mode openN -- (instss -- Scan.option induct_rule -- rule_insts));
end;
(** theory setup **)
val setup =
[Method.add_methods
[(InductAttrib.casesN, cases_meth oo cases_args, "case analysis on types or sets"),
(InductAttrib.inductN, induct_meth oo induct_args, "induction on types or sets")],
(#1 o PureThy.add_thms [(("case_split", case_split), [])])];
end;