(* Title: Provers/induct_method.ML
ID: $Id$
Author: Markus Wenzel, TU Muenchen
Proof by cases and induction on sets and types.
*)
signature INDUCT_METHOD_DATA =
sig
val dest_concls: term -> term list
val cases_default: thm
val local_impI: thm
val conjI: thm
val atomize: thm list
val rulify1: thm list
val rulify2: thm list
val localize: thm list
end;
signature INDUCT_METHOD =
sig
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 rulified_term: thm -> theory * term
val rulify_tac: int -> tactic
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 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) list
end;
functor InductMethodFun(Data: INDUCT_METHOD_DATA): INDUCT_METHOD =
struct
(** misc utils **)
(* alignment *)
fun align_left msg xs ys =
let val m = length xs and n = length ys
in if m < n then raise ERROR_MESSAGE 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 raise ERROR_MESSAGE 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 raise ERROR_MESSAGE (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 = InductAttrib.vars_of tm;
in
align "Rule has fewer variables than instantiations given" xs ts
|> List.mapPartial 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 is_open NONE (Thm.theory_of_thm rule, Thm.prop_of rule);
fun warn_open true = warning "Encountered open rule cases -- deprecated"
| 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 find_casesT ctxt ((SOME t :: _) :: _) = InductAttrib.find_casesT ctxt (Term.fastype_of t)
| find_casesT _ _ = [];
fun find_casesS ctxt (fact :: _) = InductAttrib.find_casesS ctxt (Thm.concl_of fact)
| find_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
|> (List.concat o map (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 =>
(find_casesS ctxt facts @ find_casesT ctxt insts @ [Data.cases_default])
|> tap (trace_rules ctxt InductAttrib.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 **)
(* 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;
val meta_spec = PureThy.get_thm Pure.thy (Name "meta_spec");
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) =
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 (Unify.rlist_abs (xs, body))),
(cert (Term.head_of arg), cert (Unify.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 => (warning ("Induction: no variable " ^ ProofContext.string_of_term ctxt v ^
" to be fixed -- ignored"); all_tac))
end);
fun miniscope_tac n i = PRIMITIVE (Drule.fconv_rule
(Drule.goals_conv (Library.equal i)
(Drule.forall_conv n (Tactic.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, def), ctxt') = ProofContext.add_def (x, t) ctxt
in ((SOME (Free lhs), [def]), 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 List.concat) end;
(* atomize and rulify *)
fun atomize_term thy =
MetaSimplifier.rewrite_term thy Data.atomize []
#> ObjectLogic.drop_judgment thy;
val atomize_tac =
Tactic.rewrite_goal_tac Data.atomize;
fun rulified_term thm =
let val thy = Thm.theory_of_thm thm in
Thm.prop_of thm
|> MetaSimplifier.rewrite_term thy Data.rulify1 []
|> MetaSimplifier.rewrite_term thy Data.rulify2 []
|> pair thy
end;
val rulify_tac =
Tactic.rewrite_goal_tac Data.rulify1 THEN'
Tactic.rewrite_goal_tac Data.rulify2 THEN'
Tactic.norm_hhf_tac;
(* internalize/localize rules -- pseudo-elimination *)
local
fun imp_intr i raw_th =
let
val cert = Thm.cterm_of (Thm.theory_of_thm raw_th);
val th = Thm.permute_prems (i - 1) 1 raw_th;
val prems = Thm.prems_of th;
val As = Library.take (length prems - 1, prems);
val C = Term.dummy_pattern propT;
in th COMP Thm.lift_rule (cert (Logic.list_implies (As, C))) Data.local_impI end;
in
fun internalize k th = if k > 0 then internalize (k - 1) (imp_intr k th) else th;
val localize = Goal.norm_hhf o Tactic.simplify false Data.localize;
end;
(* join multi-rules *)
val eq_prems = curry (Term.aconvs o pairself Thm.prems_of);
fun join_rules [] = []
| join_rules [th] = [th]
| join_rules (rules as r :: rs) =
if not (forall (eq_prems r) rs) then []
else
let
val th :: ths = map Drule.freeze_all rules;
val cprems = Drule.cprems_of th;
val asms = map Thm.assume cprems;
in
[foldr1 (fn (x, x') => [x, x'] MRS Data.conjI)
(map (fn x => Drule.implies_elim_list x asms) (th :: ths))
|> Drule.implies_intr_list cprems
|> Drule.standard'
|> RuleCases.save r]
end;
(* 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 (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)); (*as they are printed :-*)
in
if not (null params) then
(warning ("Cannot determine rule instantiation due to pending parameter(s): " ^
commas (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, Envir.empty (#maxidx (Thm.rep_thm rule')),
[(Thm.concl_of rule', concl)])
|> 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 _ _ thm = thm;
(* rule_versions *)
fun rule_versions rule = Seq.cons (rule,
(Seq.make (fn () =>
SOME (localize rule, Seq.empty)))
|> Seq.filter (not o curry Thm.eq_thm rule))
|> Seq.map (pair (RuleCases.get rule));
fun rule_instance thy inst rule =
Drule.cterm_instantiate (prep_inst thy align_left I (Thm.prop_of rule, inst)) rule;
(* induct_tac *)
(*
rule selection scheme:
`x:A` induct ... - set induction
induct x - type induction
... induct ... r - explicit rule
*)
local
fun find_inductT ctxt insts =
fold_rev multiply (insts |> List.mapPartial (fn [] => NONE | ts => List.last ts)
|> map (InductAttrib.find_inductT ctxt o Term.fastype_of)) [[]]
|> map join_rules |> List.concat;
fun find_inductS ctxt (fact :: _) = InductAttrib.find_inductS ctxt (Thm.concl_of fact)
| find_inductS _ _ = [];
in
fun induct_tac ctxt is_open def_insts fixing 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 ObjectLogic.atomize_thm) defs;
val inst_rule = apsnd (fn r =>
if null insts then r
else (align_right "Rule has fewer conclusions than arguments given"
(Data.dest_concls (Thm.concl_of r)) insts
|> (List.concat o map (prep_inst thy align_right (atomize_term thy)))
|> Drule.cterm_instantiate) r);
val ruleq =
(case opt_rule of
SOME r => r |> rule_versions |> Seq.map inst_rule
| NONE =>
(find_inductS ctxt facts @
map (special_rename_params defs_ctxt insts) (find_inductT ctxt insts))
|> tap (trace_rules ctxt InductAttrib.inductN)
|> Seq.of_list |> Seq.maps rule_versions |> Seq.maps (Seq.try inst_rule));
fun rule_cases rule = RuleCases.make is_open (SOME (Thm.prop_of rule)) (rulified_term rule);
in
(fn i => fn st =>
ruleq
|> Seq.maps (RuleCases.consume (List.concat defs) facts)
|> Seq.maps (fn ((cases, (k, more_facts)), rule) =>
(CONJUNCTS (ALLGOALS (fn j =>
Method.insert_tac (more_facts @ nth_list atomized_defs (j - 1)) j
THEN fix_tac defs_ctxt k (nth_list fixing (j - 1)) j))
THEN' atomize_tac) i st |> Seq.maps (fn st' =>
guess_instance (internalize k rule) i st'
|> Seq.map (rule_instance thy taking)
|> Seq.maps (fn rule' =>
CASES (rule_cases rule' cases)
(Tactic.rtac rule' i THEN
PRIMSEQ (ProofContext.exports 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 find_coinductT ctxt (SOME t :: _) = InductAttrib.find_coinductT ctxt (Term.fastype_of t)
| find_coinductT _ _ = [];
fun find_coinductS ctxt goal = InductAttrib.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;
val inst_rule = apsnd (fn r =>
if null inst then r
else Drule.cterm_instantiate (prep_inst thy align_left I (Thm.concl_of r, inst)) r);
fun ruleq goal =
(case opt_rule of
SOME r => r |> rule_versions |> Seq.map inst_rule
| NONE =>
(find_coinductS ctxt goal @ find_coinductT ctxt inst)
|> tap (trace_rules ctxt InductAttrib.coinductN)
|> Seq.of_list |> Seq.maps rule_versions |> 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 fixingN = "fixing";
val takingN = "taking";
val ruleN = "rule";
local
fun named_rule k arg get =
Scan.lift (Args.$$$ k -- Args.colon) |-- arg :-- (fn name => Scan.peek (fn ctxt =>
(case get ctxt name of SOME x => Scan.succeed x
| NONE => error ("No rule for " ^ k ^ " " ^ quote name)))) >> #2;
fun rule get_type get_set =
named_rule InductAttrib.typeN Args.local_tyname get_type ||
named_rule InductAttrib.setN Args.local_const get_set ||
Scan.lift (Args.$$$ 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 coinduct_rule = rule InductAttrib.lookup_coinductT InductAttrib.lookup_coinductS;
val inst = Scan.lift (Args.$$$ "_") >> K NONE || Args.local_term >> SOME;
val def_inst =
((Scan.lift (Args.name --| (Args.$$$ "\\<equiv>" || Args.$$$ "==")) >> SOME)
-- Args.local_term) >> SOME ||
inst >> Option.map (pair NONE);
val free = Scan.state -- Args.local_term >> (fn (_, Free v) => v | (ctxt, t) =>
error ("Bad free variable: " ^ ProofContext.string_of_term ctxt t));
fun unless_more_args scan = Scan.unless (Scan.lift
((Args.$$$ fixingN || Args.$$$ takingN || Args.$$$ InductAttrib.typeN ||
Args.$$$ InductAttrib.setN || Args.$$$ ruleN) -- Args.colon)) scan;
val fixing = Scan.optional (Scan.lift (Args.$$$ fixingN -- Args.colon) |--
Args.and_list1 (Scan.repeat (unless_more_args free))) [];
val taking = Scan.optional (Scan.lift (Args.$$$ 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 (ctxt, (is_open, (insts, opt_rule))) =>
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)) --
(fixing -- taking -- Scan.option induct_rule))) src
#> (fn (ctxt, (is_open, (insts, ((fixing, taking), opt_rule)))) =>
Method.RAW_METHOD_CASES (fn facts =>
Seq.DETERM (HEADGOAL (induct_tac ctxt is_open insts fixing 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 (ctxt, (is_open, ((insts, taking), opt_rule))) =>
Method.RAW_METHOD_CASES (fn facts =>
Seq.DETERM (HEADGOAL (coinduct_tac ctxt is_open insts taking opt_rule facts))));
end;
(** theory setup **)
val setup =
[Method.add_methods
[(InductAttrib.casesN, cases_meth, "case analysis on types or sets"),
(InductAttrib.inductN, induct_meth, "induction on types or sets"),
(InductAttrib.coinductN, coinduct_meth, "coinduction on types or sets")]];
end;