(* Title: HOL/Tools/inductive_package.ML
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Author: Stefan Berghofer and Markus Wenzel, TU Muenchen
(Co)Inductive Definition module for HOL.
Features:
* least or greatest fixedpoints
* mutually recursive definitions
* definitions involving arbitrary monotone operators
* automatically proves introduction and elimination rules
Introduction rules have the form
[| M Pj ti, ..., Q x, ... |] ==> Pk t
where M is some monotone operator (usually the identity)
Q x is any side condition on the free variables
ti, t are any terms
Pj, Pk are two of the predicates being defined in mutual recursion
*)
signature BASIC_INDUCTIVE_PACKAGE =
sig
type inductive_result
val morph_result: morphism -> inductive_result -> inductive_result
type inductive_info
val the_inductive: Proof.context -> string -> inductive_info
val print_inductives: Proof.context -> unit
val mono_add: attribute
val mono_del: attribute
val get_monos: Proof.context -> thm list
val mk_cases: Proof.context -> term -> thm
val inductive_forall_name: string
val inductive_forall_def: thm
val rulify: thm -> thm
val inductive_cases: (Attrib.binding * string list) list -> local_theory ->
thm list list * local_theory
val inductive_cases_i: (Attrib.binding * term list) list -> local_theory ->
thm list list * local_theory
type inductive_flags
val add_inductive_i:
inductive_flags -> ((binding * typ) * mixfix) list ->
(string * typ) list -> (Attrib.binding * term) list -> thm list -> local_theory ->
inductive_result * local_theory
val add_inductive: bool -> bool ->
(binding * string option * mixfix) list ->
(binding * string option * mixfix) list ->
(Attrib.binding * string) list ->
(Facts.ref * Attrib.src list) list ->
bool -> local_theory -> inductive_result * local_theory
val add_inductive_global: string -> inductive_flags ->
((binding * typ) * mixfix) list -> (string * typ) list -> (Attrib.binding * term) list ->
thm list -> theory -> inductive_result * theory
val arities_of: thm -> (string * int) list
val params_of: thm -> term list
val partition_rules: thm -> thm list -> (string * thm list) list
val partition_rules': thm -> (thm * 'a) list -> (string * (thm * 'a) list) list
val unpartition_rules: thm list -> (string * 'a list) list -> 'a list
val infer_intro_vars: thm -> int -> thm list -> term list list
val setup: theory -> theory
end;
signature INDUCTIVE_PACKAGE =
sig
include BASIC_INDUCTIVE_PACKAGE
type add_ind_def
val declare_rules: string -> binding -> bool -> bool -> string list ->
thm list -> binding list -> Attrib.src list list -> (thm * string list) list ->
thm -> local_theory -> thm list * thm list * thm * local_theory
val add_ind_def: add_ind_def
val gen_add_inductive_i: add_ind_def -> inductive_flags ->
((binding * typ) * mixfix) list -> (string * typ) list -> (Attrib.binding * term) list ->
thm list -> local_theory -> inductive_result * local_theory
val gen_add_inductive: add_ind_def -> bool -> bool ->
(binding * string option * mixfix) list ->
(binding * string option * mixfix) list ->
(Attrib.binding * string) list -> (Facts.ref * Attrib.src list) list ->
bool -> local_theory -> inductive_result * local_theory
val gen_ind_decl: add_ind_def -> bool ->
OuterParse.token list -> (bool -> local_theory -> local_theory) * OuterParse.token list
end;
structure InductivePackage: INDUCTIVE_PACKAGE =
struct
(** theory context references **)
val inductive_forall_name = "HOL.induct_forall";
val inductive_forall_def = thm "induct_forall_def";
val inductive_conj_name = "HOL.induct_conj";
val inductive_conj_def = thm "induct_conj_def";
val inductive_conj = thms "induct_conj";
val inductive_atomize = thms "induct_atomize";
val inductive_rulify = thms "induct_rulify";
val inductive_rulify_fallback = thms "induct_rulify_fallback";
val notTrueE = TrueI RSN (2, notE);
val notFalseI = Seq.hd (atac 1 notI);
val simp_thms' = map (fn s => mk_meta_eq (the (find_first
(equal (OldGoals.read_prop @{theory HOL} s) o prop_of) simp_thms)))
["(~True) = False", "(~False) = True",
"(True --> ?P) = ?P", "(False --> ?P) = True",
"(?P & True) = ?P", "(True & ?P) = ?P"];
(** context data **)
type inductive_result =
{preds: term list, elims: thm list, raw_induct: thm,
induct: thm, intrs: thm list};
fun morph_result phi {preds, elims, raw_induct: thm, induct, intrs} =
let
val term = Morphism.term phi;
val thm = Morphism.thm phi;
val fact = Morphism.fact phi;
in
{preds = map term preds, elims = fact elims, raw_induct = thm raw_induct,
induct = thm induct, intrs = fact intrs}
end;
type inductive_info =
{names: string list, coind: bool} * inductive_result;
structure InductiveData = GenericDataFun
(
type T = inductive_info Symtab.table * thm list;
val empty = (Symtab.empty, []);
val extend = I;
fun merge _ ((tab1, monos1), (tab2, monos2)) =
(Symtab.merge (K true) (tab1, tab2), Thm.merge_thms (monos1, monos2));
);
val get_inductives = InductiveData.get o Context.Proof;
fun print_inductives ctxt =
let
val (tab, monos) = get_inductives ctxt;
val space = Consts.space_of (ProofContext.consts_of ctxt);
in
[Pretty.strs ("(co)inductives:" :: map #1 (NameSpace.extern_table (space, tab))),
Pretty.big_list "monotonicity rules:" (map (ProofContext.pretty_thm ctxt) monos)]
|> Pretty.chunks |> Pretty.writeln
end;
(* get and put data *)
fun the_inductive ctxt name =
(case Symtab.lookup (#1 (get_inductives ctxt)) name of
NONE => error ("Unknown (co)inductive predicate " ^ quote name)
| SOME info => info);
fun put_inductives names info = InductiveData.map
(apfst (fold (fn name => Symtab.update (name, info)) names));
(** monotonicity rules **)
val get_monos = #2 o get_inductives;
val map_monos = InductiveData.map o apsnd;
fun mk_mono thm =
let
val concl = concl_of thm;
fun eq2mono thm' = [thm' RS (thm' RS eq_to_mono)] @
(case concl of
(_ $ (_ $ (Const ("Not", _) $ _) $ _)) => []
| _ => [thm' RS (thm' RS eq_to_mono2)]);
fun dest_less_concl thm = dest_less_concl (thm RS le_funD)
handle THM _ => thm RS le_boolD
in
case concl of
Const ("==", _) $ _ $ _ => eq2mono (thm RS meta_eq_to_obj_eq)
| _ $ (Const ("op =", _) $ _ $ _) => eq2mono thm
| _ $ (Const ("HOL.ord_class.less_eq", _) $ _ $ _) =>
[dest_less_concl (Seq.hd (REPEAT (FIRSTGOAL
(resolve_tac [le_funI, le_boolI'])) thm))]
| _ => [thm]
end handle THM _ => error ("Bad monotonicity theorem:\n" ^ Display.string_of_thm thm);
val mono_add = Thm.declaration_attribute (map_monos o fold Thm.add_thm o mk_mono);
val mono_del = Thm.declaration_attribute (map_monos o fold Thm.del_thm o mk_mono);
(** misc utilities **)
fun message quiet_mode s = if quiet_mode then () else writeln s;
fun clean_message quiet_mode s = if ! quick_and_dirty then () else message quiet_mode s;
fun coind_prefix true = "co"
| coind_prefix false = "";
fun log (b:int) m n = if m >= n then 0 else 1 + log b (b * m) n;
fun make_bool_args f g [] i = []
| make_bool_args f g (x :: xs) i =
(if i mod 2 = 0 then f x else g x) :: make_bool_args f g xs (i div 2);
fun make_bool_args' xs =
make_bool_args (K HOLogic.false_const) (K HOLogic.true_const) xs;
fun find_arg T x [] = sys_error "find_arg"
| find_arg T x ((p as (_, (SOME _, _))) :: ps) =
apsnd (cons p) (find_arg T x ps)
| find_arg T x ((p as (U, (NONE, y))) :: ps) =
if (T: typ) = U then (y, (U, (SOME x, y)) :: ps)
else apsnd (cons p) (find_arg T x ps);
fun make_args Ts xs =
map (fn (T, (NONE, ())) => Const (@{const_name undefined}, T) | (_, (SOME t, ())) => t)
(fold (fn (t, T) => snd o find_arg T t) xs (map (rpair (NONE, ())) Ts));
fun make_args' Ts xs Us =
fst (fold_map (fn T => find_arg T ()) Us (Ts ~~ map (pair NONE) xs));
fun dest_predicate cs params t =
let
val k = length params;
val (c, ts) = strip_comb t;
val (xs, ys) = chop k ts;
val i = find_index_eq c cs;
in
if xs = params andalso i >= 0 then
SOME (c, i, ys, chop (length ys)
(List.drop (binder_types (fastype_of c), k)))
else NONE
end;
fun mk_names a 0 = []
| mk_names a 1 = [a]
| mk_names a n = map (fn i => a ^ string_of_int i) (1 upto n);
(** process rules **)
local
fun err_in_rule ctxt name t msg =
error (cat_lines ["Ill-formed introduction rule " ^ quote name,
Syntax.string_of_term ctxt t, msg]);
fun err_in_prem ctxt name t p msg =
error (cat_lines ["Ill-formed premise", Syntax.string_of_term ctxt p,
"in introduction rule " ^ quote name, Syntax.string_of_term ctxt t, msg]);
val bad_concl = "Conclusion of introduction rule must be an inductive predicate";
val bad_ind_occ = "Inductive predicate occurs in argument of inductive predicate";
val bad_app = "Inductive predicate must be applied to parameter(s) ";
fun atomize_term thy = MetaSimplifier.rewrite_term thy inductive_atomize [];
in
fun check_rule ctxt cs params ((binding, att), rule) =
let
val err_name = Binding.str_of binding;
val params' = Term.variant_frees rule (Logic.strip_params rule);
val frees = rev (map Free params');
val concl = subst_bounds (frees, Logic.strip_assums_concl rule);
val prems = map (curry subst_bounds frees) (Logic.strip_assums_hyp rule);
val rule' = Logic.list_implies (prems, concl);
val aprems = map (atomize_term (ProofContext.theory_of ctxt)) prems;
val arule = list_all_free (params', Logic.list_implies (aprems, concl));
fun check_ind err t = case dest_predicate cs params t of
NONE => err (bad_app ^
commas (map (Syntax.string_of_term ctxt) params))
| SOME (_, _, ys, _) =>
if exists (fn c => exists (fn t => Logic.occs (c, t)) ys) cs
then err bad_ind_occ else ();
fun check_prem' prem t =
if head_of t mem cs then
check_ind (err_in_prem ctxt err_name rule prem) t
else (case t of
Abs (_, _, t) => check_prem' prem t
| t $ u => (check_prem' prem t; check_prem' prem u)
| _ => ());
fun check_prem (prem, aprem) =
if can HOLogic.dest_Trueprop aprem then check_prem' prem prem
else err_in_prem ctxt err_name rule prem "Non-atomic premise";
in
(case concl of
Const ("Trueprop", _) $ t =>
if head_of t mem cs then
(check_ind (err_in_rule ctxt err_name rule') t;
List.app check_prem (prems ~~ aprems))
else err_in_rule ctxt err_name rule' bad_concl
| _ => err_in_rule ctxt err_name rule' bad_concl);
((binding, att), arule)
end;
val rulify =
hol_simplify inductive_conj
#> hol_simplify inductive_rulify
#> hol_simplify inductive_rulify_fallback
#> Simplifier.norm_hhf;
end;
(** proofs for (co)inductive predicates **)
(* prove monotonicity *)
fun prove_mono quiet_mode skip_mono fork_mono predT fp_fun monos ctxt =
(message (quiet_mode orelse skip_mono andalso !quick_and_dirty orelse fork_mono)
" Proving monotonicity ...";
(if skip_mono then SkipProof.prove else if fork_mono then Goal.prove_future else Goal.prove) ctxt
[] []
(HOLogic.mk_Trueprop
(Const (@{const_name Orderings.mono}, (predT --> predT) --> HOLogic.boolT) $ fp_fun))
(fn _ => EVERY [rtac @{thm monoI} 1,
REPEAT (resolve_tac [le_funI, le_boolI'] 1),
REPEAT (FIRST
[atac 1,
resolve_tac (List.concat (map mk_mono monos) @ get_monos ctxt) 1,
etac le_funE 1, dtac le_boolD 1])]));
(* prove introduction rules *)
fun prove_intrs quiet_mode coind mono fp_def k params intr_ts rec_preds_defs ctxt =
let
val _ = clean_message quiet_mode " Proving the introduction rules ...";
val unfold = funpow k (fn th => th RS fun_cong)
(mono RS (fp_def RS
(if coind then def_gfp_unfold else def_lfp_unfold)));
fun select_disj 1 1 = []
| select_disj _ 1 = [rtac disjI1]
| select_disj n i = (rtac disjI2)::(select_disj (n - 1) (i - 1));
val rules = [refl, TrueI, notFalseI, exI, conjI];
val intrs = map_index (fn (i, intr) => rulify
(SkipProof.prove ctxt (map (fst o dest_Free) params) [] intr (fn _ => EVERY
[rewrite_goals_tac rec_preds_defs,
rtac (unfold RS iffD2) 1,
EVERY1 (select_disj (length intr_ts) (i + 1)),
(*Not ares_tac, since refl must be tried before any equality assumptions;
backtracking may occur if the premises have extra variables!*)
DEPTH_SOLVE_1 (resolve_tac rules 1 APPEND assume_tac 1)]))) intr_ts
in (intrs, unfold) end;
(* prove elimination rules *)
fun prove_elims quiet_mode cs params intr_ts intr_names unfold rec_preds_defs ctxt =
let
val _ = clean_message quiet_mode " Proving the elimination rules ...";
val ([pname], ctxt') = ctxt |>
Variable.add_fixes (map (fst o dest_Free) params) |> snd |>
Variable.variant_fixes ["P"];
val P = HOLogic.mk_Trueprop (Free (pname, HOLogic.boolT));
fun dest_intr r =
(the (dest_predicate cs params (HOLogic.dest_Trueprop (Logic.strip_assums_concl r))),
Logic.strip_assums_hyp r, Logic.strip_params r);
val intrs = map dest_intr intr_ts ~~ intr_names;
val rules1 = [disjE, exE, FalseE];
val rules2 = [conjE, FalseE, notTrueE];
fun prove_elim c =
let
val Ts = List.drop (binder_types (fastype_of c), length params);
val (anames, ctxt'') = Variable.variant_fixes (mk_names "a" (length Ts)) ctxt';
val frees = map Free (anames ~~ Ts);
fun mk_elim_prem ((_, _, us, _), ts, params') =
list_all (params',
Logic.list_implies (map (HOLogic.mk_Trueprop o HOLogic.mk_eq)
(frees ~~ us) @ ts, P));
val c_intrs = (List.filter (equal c o #1 o #1 o #1) intrs);
val prems = HOLogic.mk_Trueprop (list_comb (c, params @ frees)) ::
map mk_elim_prem (map #1 c_intrs)
in
(SkipProof.prove ctxt'' [] prems P
(fn {prems, ...} => EVERY
[cut_facts_tac [hd prems] 1,
rewrite_goals_tac rec_preds_defs,
dtac (unfold RS iffD1) 1,
REPEAT (FIRSTGOAL (eresolve_tac rules1)),
REPEAT (FIRSTGOAL (eresolve_tac rules2)),
EVERY (map (fn prem =>
DEPTH_SOLVE_1 (ares_tac [rewrite_rule rec_preds_defs prem, conjI] 1)) (tl prems))])
|> rulify
|> singleton (ProofContext.export ctxt'' ctxt),
map #2 c_intrs)
end
in map prove_elim cs end;
(* derivation of simplified elimination rules *)
local
(*delete needless equality assumptions*)
val refl_thin = Goal.prove_global @{theory HOL} [] [] @{prop "!!P. a = a ==> P ==> P"}
(fn _ => assume_tac 1);
val elim_rls = [asm_rl, FalseE, refl_thin, conjE, exE];
val elim_tac = REPEAT o Tactic.eresolve_tac elim_rls;
fun simp_case_tac ss i =
EVERY' [elim_tac, asm_full_simp_tac ss, elim_tac, REPEAT o bound_hyp_subst_tac] i;
in
fun mk_cases ctxt prop =
let
val thy = ProofContext.theory_of ctxt;
val ss = Simplifier.local_simpset_of ctxt;
fun err msg =
error (Pretty.string_of (Pretty.block
[Pretty.str msg, Pretty.fbrk, Syntax.pretty_term ctxt prop]));
val elims = Induct.find_casesP ctxt prop;
val cprop = Thm.cterm_of thy prop;
val tac = ALLGOALS (simp_case_tac ss) THEN prune_params_tac;
fun mk_elim rl =
Thm.implies_intr cprop (Tactic.rule_by_tactic tac (Thm.assume cprop RS rl))
|> singleton (Variable.export (Variable.auto_fixes prop ctxt) ctxt);
in
(case get_first (try mk_elim) elims of
SOME r => r
| NONE => err "Proposition not an inductive predicate:")
end;
end;
(* inductive_cases *)
fun gen_inductive_cases prep_att prep_prop args lthy =
let
val thy = ProofContext.theory_of lthy;
val facts = args |> map (fn ((a, atts), props) =>
((a, map (prep_att thy) atts),
map (Thm.no_attributes o single o mk_cases lthy o prep_prop lthy) props));
in lthy |> LocalTheory.notes Thm.theoremK facts |>> map snd end;
val inductive_cases = gen_inductive_cases Attrib.intern_src Syntax.read_prop;
val inductive_cases_i = gen_inductive_cases (K I) Syntax.check_prop;
val ind_cases_setup =
Method.setup @{binding ind_cases}
(Scan.lift (Scan.repeat1 Args.name_source --
Scan.optional (Args.$$$ "for" |-- Scan.repeat1 Args.name) []) >>
(fn (raw_props, fixes) => fn ctxt =>
let
val (_, ctxt') = Variable.add_fixes fixes ctxt;
val props = Syntax.read_props ctxt' raw_props;
val ctxt'' = fold Variable.declare_term props ctxt';
val rules = ProofContext.export ctxt'' ctxt (map (mk_cases ctxt'') props)
in Method.erule 0 rules end))
"dynamic case analysis on predicates";
(* prove induction rule *)
fun prove_indrule quiet_mode cs argTs bs xs rec_const params intr_ts mono
fp_def rec_preds_defs ctxt =
let
val _ = clean_message quiet_mode " Proving the induction rule ...";
val thy = ProofContext.theory_of ctxt;
(* predicates for induction rule *)
val (pnames, ctxt') = ctxt |>
Variable.add_fixes (map (fst o dest_Free) params) |> snd |>
Variable.variant_fixes (mk_names "P" (length cs));
val preds = map Free (pnames ~~
map (fn c => List.drop (binder_types (fastype_of c), length params) --->
HOLogic.boolT) cs);
(* transform an introduction rule into a premise for induction rule *)
fun mk_ind_prem r =
let
fun subst s = (case dest_predicate cs params s of
SOME (_, i, ys, (_, Ts)) =>
let
val k = length Ts;
val bs = map Bound (k - 1 downto 0);
val P = list_comb (List.nth (preds, i),
map (incr_boundvars k) ys @ bs);
val Q = list_abs (mk_names "x" k ~~ Ts,
HOLogic.mk_binop inductive_conj_name
(list_comb (incr_boundvars k s, bs), P))
in (Q, case Ts of [] => SOME (s, P) | _ => NONE) end
| NONE => (case s of
(t $ u) => (fst (subst t) $ fst (subst u), NONE)
| (Abs (a, T, t)) => (Abs (a, T, fst (subst t)), NONE)
| _ => (s, NONE)));
fun mk_prem (s, prems) = (case subst s of
(_, SOME (t, u)) => t :: u :: prems
| (t, _) => t :: prems);
val SOME (_, i, ys, _) = dest_predicate cs params
(HOLogic.dest_Trueprop (Logic.strip_assums_concl r))
in list_all_free (Logic.strip_params r,
Logic.list_implies (map HOLogic.mk_Trueprop (List.foldr mk_prem
[] (map HOLogic.dest_Trueprop (Logic.strip_assums_hyp r))),
HOLogic.mk_Trueprop (list_comb (List.nth (preds, i), ys))))
end;
val ind_prems = map mk_ind_prem intr_ts;
(* make conclusions for induction rules *)
val Tss = map (binder_types o fastype_of) preds;
val (xnames, ctxt'') =
Variable.variant_fixes (mk_names "x" (length (flat Tss))) ctxt';
val mutual_ind_concl = HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
(map (fn (((xnames, Ts), c), P) =>
let val frees = map Free (xnames ~~ Ts)
in HOLogic.mk_imp
(list_comb (c, params @ frees), list_comb (P, frees))
end) (unflat Tss xnames ~~ Tss ~~ cs ~~ preds)));
(* make predicate for instantiation of abstract induction rule *)
val ind_pred = fold_rev lambda (bs @ xs) (foldr1 HOLogic.mk_conj
(map_index (fn (i, P) => List.foldr HOLogic.mk_imp
(list_comb (P, make_args' argTs xs (binder_types (fastype_of P))))
(make_bool_args HOLogic.mk_not I bs i)) preds));
val ind_concl = HOLogic.mk_Trueprop
(HOLogic.mk_binrel "HOL.ord_class.less_eq" (rec_const, ind_pred));
val raw_fp_induct = (mono RS (fp_def RS def_lfp_induct));
val induct = SkipProof.prove ctxt'' [] ind_prems ind_concl
(fn {prems, ...} => EVERY
[rewrite_goals_tac [inductive_conj_def],
DETERM (rtac raw_fp_induct 1),
REPEAT (resolve_tac [le_funI, le_boolI] 1),
rewrite_goals_tac (inf_fun_eq :: inf_bool_eq :: simp_thms'),
(*This disjE separates out the introduction rules*)
REPEAT (FIRSTGOAL (eresolve_tac [disjE, exE, FalseE])),
(*Now break down the individual cases. No disjE here in case
some premise involves disjunction.*)
REPEAT (FIRSTGOAL (etac conjE ORELSE' bound_hyp_subst_tac)),
REPEAT (FIRSTGOAL
(resolve_tac [conjI, impI] ORELSE' (etac notE THEN' atac))),
EVERY (map (fn prem => DEPTH_SOLVE_1 (ares_tac [rewrite_rule
(inductive_conj_def :: rec_preds_defs @ simp_thms') prem,
conjI, refl] 1)) prems)]);
val lemma = SkipProof.prove ctxt'' [] []
(Logic.mk_implies (ind_concl, mutual_ind_concl)) (fn _ => EVERY
[rewrite_goals_tac rec_preds_defs,
REPEAT (EVERY
[REPEAT (resolve_tac [conjI, impI] 1),
REPEAT (eresolve_tac [le_funE, le_boolE] 1),
atac 1,
rewrite_goals_tac simp_thms',
atac 1])])
in singleton (ProofContext.export ctxt'' ctxt) (induct RS lemma) end;
(** specification of (co)inductive predicates **)
fun mk_ind_def quiet_mode skip_mono fork_mono alt_name coind cs intr_ts monos params cnames_syn ctxt =
let
val fp_name = if coind then @{const_name Inductive.gfp} else @{const_name Inductive.lfp};
val argTs = fold (fn c => fn Ts => Ts @
(List.drop (binder_types (fastype_of c), length params) \\ Ts)) cs [];
val k = log 2 1 (length cs);
val predT = replicate k HOLogic.boolT ---> argTs ---> HOLogic.boolT;
val p :: xs = map Free (Variable.variant_frees ctxt intr_ts
(("p", predT) :: (mk_names "x" (length argTs) ~~ argTs)));
val bs = map Free (Variable.variant_frees ctxt (p :: xs @ intr_ts)
(map (rpair HOLogic.boolT) (mk_names "b" k)));
fun subst t = (case dest_predicate cs params t of
SOME (_, i, ts, (Ts, Us)) =>
let
val l = length Us;
val zs = map Bound (l - 1 downto 0)
in
list_abs (map (pair "z") Us, list_comb (p,
make_bool_args' bs i @ make_args argTs
((map (incr_boundvars l) ts ~~ Ts) @ (zs ~~ Us))))
end
| NONE => (case t of
t1 $ t2 => subst t1 $ subst t2
| Abs (x, T, u) => Abs (x, T, subst u)
| _ => t));
(* transform an introduction rule into a conjunction *)
(* [| p_i t; ... |] ==> p_j u *)
(* is transformed into *)
(* b_j & x_j = u & p b_j t & ... *)
fun transform_rule r =
let
val SOME (_, i, ts, (Ts, _)) = dest_predicate cs params
(HOLogic.dest_Trueprop (Logic.strip_assums_concl r));
val ps = make_bool_args HOLogic.mk_not I bs i @
map HOLogic.mk_eq (make_args' argTs xs Ts ~~ ts) @
map (subst o HOLogic.dest_Trueprop)
(Logic.strip_assums_hyp r)
in List.foldr (fn ((x, T), P) => HOLogic.exists_const T $ (Abs (x, T, P)))
(if null ps then HOLogic.true_const else foldr1 HOLogic.mk_conj ps)
(Logic.strip_params r)
end
(* make a disjunction of all introduction rules *)
val fp_fun = fold_rev lambda (p :: bs @ xs)
(if null intr_ts then HOLogic.false_const
else foldr1 HOLogic.mk_disj (map transform_rule intr_ts));
(* add definiton of recursive predicates to theory *)
val rec_name =
if Binding.is_empty alt_name then
Binding.name (space_implode "_" (map (Binding.name_of o fst) cnames_syn))
else alt_name;
val ((rec_const, (_, fp_def)), ctxt') = ctxt |>
LocalTheory.define Thm.internalK
((rec_name, case cnames_syn of [(_, syn)] => syn | _ => NoSyn),
(Attrib.empty_binding, fold_rev lambda params
(Const (fp_name, (predT --> predT) --> predT) $ fp_fun)));
val fp_def' = Simplifier.rewrite (HOL_basic_ss addsimps [fp_def])
(cterm_of (ProofContext.theory_of ctxt') (list_comb (rec_const, params)));
val specs = if length cs < 2 then [] else
map_index (fn (i, (name_mx, c)) =>
let
val Ts = List.drop (binder_types (fastype_of c), length params);
val xs = map Free (Variable.variant_frees ctxt intr_ts
(mk_names "x" (length Ts) ~~ Ts))
in
(name_mx, (Attrib.empty_binding, fold_rev lambda (params @ xs)
(list_comb (rec_const, params @ make_bool_args' bs i @
make_args argTs (xs ~~ Ts)))))
end) (cnames_syn ~~ cs);
val (consts_defs, ctxt'') = fold_map (LocalTheory.define Thm.internalK) specs ctxt';
val preds = (case cs of [_] => [rec_const] | _ => map #1 consts_defs);
val mono = prove_mono quiet_mode skip_mono fork_mono predT fp_fun monos ctxt'';
val ((_, [mono']), ctxt''') =
LocalTheory.note Thm.internalK (Attrib.empty_binding, [mono]) ctxt'';
in (ctxt''', rec_name, mono', fp_def', map (#2 o #2) consts_defs,
list_comb (rec_const, params), preds, argTs, bs, xs)
end;
fun declare_rules kind rec_binding coind no_ind cnames intrs intr_bindings intr_atts
elims raw_induct ctxt =
let
val rec_name = Binding.name_of rec_binding;
val rec_qualified = Binding.qualify false rec_name;
val intr_names = map Binding.name_of intr_bindings;
val ind_case_names = RuleCases.case_names intr_names;
val induct =
if coind then
(raw_induct, [RuleCases.case_names [rec_name],
RuleCases.case_conclusion (rec_name, intr_names),
RuleCases.consumes 1, Induct.coinduct_pred (hd cnames)])
else if no_ind orelse length cnames > 1 then
(raw_induct, [ind_case_names, RuleCases.consumes 0])
else (raw_induct RSN (2, rev_mp), [ind_case_names, RuleCases.consumes 1]);
val (intrs', ctxt1) =
ctxt |>
LocalTheory.notes kind
(map rec_qualified intr_bindings ~~ intr_atts ~~ map (fn th => [([th],
[Attrib.internal (K (ContextRules.intro_query NONE)),
Attrib.internal (K Nitpick_Ind_Intro_Thms.add)])]) intrs) |>>
map (hd o snd);
val (((_, elims'), (_, [induct'])), ctxt2) =
ctxt1 |>
LocalTheory.note kind ((rec_qualified (Binding.name "intros"), []), intrs') ||>>
fold_map (fn (name, (elim, cases)) =>
LocalTheory.note kind ((Binding.qualified_name (Long_Name.qualify (Long_Name.base_name name) "cases"),
[Attrib.internal (K (RuleCases.case_names cases)),
Attrib.internal (K (RuleCases.consumes 1)),
Attrib.internal (K (Induct.cases_pred name)),
Attrib.internal (K (ContextRules.elim_query NONE))]), [elim]) #>
apfst (hd o snd)) (if null elims then [] else cnames ~~ elims) ||>>
LocalTheory.note kind
((rec_qualified (Binding.name (coind_prefix coind ^ "induct")),
map (Attrib.internal o K) (#2 induct)), [rulify (#1 induct)]);
val ctxt3 = if no_ind orelse coind then ctxt2 else
let val inducts = cnames ~~ ProjectRule.projects ctxt2 (1 upto length cnames) induct'
in
ctxt2 |>
LocalTheory.notes kind [((rec_qualified (Binding.name "inducts"), []),
inducts |> map (fn (name, th) => ([th],
[Attrib.internal (K ind_case_names),
Attrib.internal (K (RuleCases.consumes 1)),
Attrib.internal (K (Induct.induct_pred name))])))] |> snd
end
in (intrs', elims', induct', ctxt3) end;
type inductive_flags =
{quiet_mode: bool, verbose: bool, kind: string, alt_name: binding,
coind: bool, no_elim: bool, no_ind: bool, skip_mono: bool, fork_mono: bool}
type add_ind_def =
inductive_flags ->
term list -> (Attrib.binding * term) list -> thm list ->
term list -> (binding * mixfix) list ->
local_theory -> inductive_result * local_theory
fun add_ind_def {quiet_mode, verbose, kind, alt_name, coind, no_elim, no_ind, skip_mono, fork_mono}
cs intros monos params cnames_syn ctxt =
let
val _ = null cnames_syn andalso error "No inductive predicates given";
val names = map (Binding.name_of o fst) cnames_syn;
val _ = message (quiet_mode andalso not verbose)
("Proofs for " ^ coind_prefix coind ^ "inductive predicate(s) " ^ commas_quote names);
val cnames = map (LocalTheory.full_name ctxt o #1) cnames_syn; (* FIXME *)
val ((intr_names, intr_atts), intr_ts) =
apfst split_list (split_list (map (check_rule ctxt cs params) intros));
val (ctxt1, rec_name, mono, fp_def, rec_preds_defs, rec_const, preds,
argTs, bs, xs) = mk_ind_def quiet_mode skip_mono fork_mono alt_name coind cs intr_ts
monos params cnames_syn ctxt;
val (intrs, unfold) = prove_intrs quiet_mode coind mono fp_def (length bs + length xs)
params intr_ts rec_preds_defs ctxt1;
val elims = if no_elim then [] else
prove_elims quiet_mode cs params intr_ts (map Binding.name_of intr_names)
unfold rec_preds_defs ctxt1;
val raw_induct = zero_var_indexes
(if no_ind then Drule.asm_rl else
if coind then
singleton (ProofContext.export
(snd (Variable.add_fixes (map (fst o dest_Free) params) ctxt1)) ctxt1)
(rotate_prems ~1 (ObjectLogic.rulify
(fold_rule rec_preds_defs
(rewrite_rule [le_fun_def, le_bool_def, sup_fun_eq, sup_bool_eq]
(mono RS (fp_def RS def_coinduct))))))
else
prove_indrule quiet_mode cs argTs bs xs rec_const params intr_ts mono fp_def
rec_preds_defs ctxt1);
val (intrs', elims', induct, ctxt2) = declare_rules kind rec_name coind no_ind
cnames intrs intr_names intr_atts elims raw_induct ctxt1;
val result =
{preds = preds,
intrs = intrs',
elims = elims',
raw_induct = rulify raw_induct,
induct = induct};
val ctxt3 = ctxt2
|> LocalTheory.declaration (fn phi =>
let val result' = morph_result phi result;
in put_inductives cnames (*global names!?*) ({names = cnames, coind = coind}, result') end);
in (result, ctxt3) end;
(* external interfaces *)
fun gen_add_inductive_i mk_def
(flags as {quiet_mode, verbose, kind, alt_name, coind, no_elim, no_ind, skip_mono, fork_mono})
cnames_syn pnames spec monos lthy =
let
val thy = ProofContext.theory_of lthy;
val _ = Theory.requires thy "Inductive" (coind_prefix coind ^ "inductive definitions");
(* abbrevs *)
val (_, ctxt1) = Variable.add_fixes (map (Binding.name_of o fst o fst) cnames_syn) lthy;
fun get_abbrev ((name, atts), t) =
if can (Logic.strip_assums_concl #> Logic.dest_equals) t then
let
val _ = Binding.is_empty name andalso null atts orelse
error "Abbreviations may not have names or attributes";
val ((x, T), rhs) = LocalDefs.abs_def (snd (LocalDefs.cert_def ctxt1 t));
val var =
(case find_first (fn ((c, _), _) => Binding.name_of c = x) cnames_syn of
NONE => error ("Undeclared head of abbreviation " ^ quote x)
| SOME ((b, T'), mx) =>
if T <> T' then error ("Bad type specification for abbreviation " ^ quote x)
else (b, mx));
in SOME (var, rhs) end
else NONE;
val abbrevs = map_filter get_abbrev spec;
val bs = map (Binding.name_of o fst o fst) abbrevs;
(* predicates *)
val pre_intros = filter_out (is_some o get_abbrev) spec;
val cnames_syn' = filter_out (member (op =) bs o Binding.name_of o fst o fst) cnames_syn;
val cs = map (Free o apfst Binding.name_of o fst) cnames_syn';
val ps = map Free pnames;
val (_, ctxt2) = lthy |> Variable.add_fixes (map (Binding.name_of o fst o fst) cnames_syn');
val _ = map (fn abbr => LocalDefs.fixed_abbrev abbr ctxt2) abbrevs;
val ctxt3 = ctxt2 |> fold (snd oo LocalDefs.fixed_abbrev) abbrevs;
val expand = Assumption.export_term ctxt3 lthy #> ProofContext.cert_term lthy;
fun close_rule r = list_all_free (rev (fold_aterms
(fn t as Free (v as (s, _)) =>
if Variable.is_fixed ctxt1 s orelse
member (op =) ps t then I else insert (op =) v
| _ => I) r []), r);
val intros = map (apsnd (Syntax.check_term lthy #> close_rule #> expand)) pre_intros;
val preds = map (fn ((c, _), mx) => (c, mx)) cnames_syn';
in
lthy
|> mk_def flags cs intros monos ps preds
||> fold (snd oo LocalTheory.abbrev Syntax.mode_default) abbrevs
end;
fun gen_add_inductive mk_def verbose coind cnames_syn pnames_syn intro_srcs raw_monos int lthy =
let
val ((vars, intrs), _) = lthy
|> ProofContext.set_mode ProofContext.mode_abbrev
|> Specification.read_spec (cnames_syn @ pnames_syn) intro_srcs;
val (cs, ps) = chop (length cnames_syn) vars;
val monos = Attrib.eval_thms lthy raw_monos;
val flags = {quiet_mode = false, verbose = verbose, kind = Thm.theoremK,
alt_name = Binding.empty, coind = coind, no_elim = false, no_ind = false,
skip_mono = false, fork_mono = not int};
in
lthy
|> LocalTheory.set_group (serial_string ())
|> gen_add_inductive_i mk_def flags cs (map (apfst Binding.name_of o fst) ps) intrs monos
end;
val add_inductive_i = gen_add_inductive_i add_ind_def;
val add_inductive = gen_add_inductive add_ind_def;
fun add_inductive_global group flags cnames_syn pnames pre_intros monos thy =
let
val name = Sign.full_name thy (fst (fst (hd cnames_syn)));
val ctxt' = thy
|> TheoryTarget.init NONE
|> LocalTheory.set_group group
|> add_inductive_i flags cnames_syn pnames pre_intros monos |> snd
|> LocalTheory.exit;
val info = #2 (the_inductive ctxt' name);
in (info, ProofContext.theory_of ctxt') end;
(* read off arities of inductive predicates from raw induction rule *)
fun arities_of induct =
map (fn (_ $ t $ u) =>
(fst (dest_Const (head_of t)), length (snd (strip_comb u))))
(HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct)));
(* read off parameters of inductive predicate from raw induction rule *)
fun params_of induct =
let
val (_ $ t $ u :: _) =
HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct));
val (_, ts) = strip_comb t;
val (_, us) = strip_comb u
in
List.take (ts, length ts - length us)
end;
val pname_of_intr =
concl_of #> HOLogic.dest_Trueprop #> head_of #> dest_Const #> fst;
(* partition introduction rules according to predicate name *)
fun gen_partition_rules f induct intros =
fold_rev (fn r => AList.map_entry op = (pname_of_intr (f r)) (cons r)) intros
(map (rpair [] o fst) (arities_of induct));
val partition_rules = gen_partition_rules I;
fun partition_rules' induct = gen_partition_rules fst induct;
fun unpartition_rules intros xs =
fold_map (fn r => AList.map_entry_yield op = (pname_of_intr r)
(fn x :: xs => (x, xs)) #>> the) intros xs |> fst;
(* infer order of variables in intro rules from order of quantifiers in elim rule *)
fun infer_intro_vars elim arity intros =
let
val thy = theory_of_thm elim;
val _ :: cases = prems_of elim;
val used = map (fst o fst) (Term.add_vars (prop_of elim) []);
fun mtch (t, u) =
let
val params = Logic.strip_params t;
val vars = map (Var o apfst (rpair 0))
(Name.variant_list used (map fst params) ~~ map snd params);
val ts = map (curry subst_bounds (rev vars))
(List.drop (Logic.strip_assums_hyp t, arity));
val us = Logic.strip_imp_prems u;
val tab = fold (Pattern.first_order_match thy) (ts ~~ us)
(Vartab.empty, Vartab.empty);
in
map (Envir.subst_vars tab) vars
end
in
map (mtch o apsnd prop_of) (cases ~~ intros)
end;
(** package setup **)
(* setup theory *)
val setup =
ind_cases_setup #>
Attrib.setup @{binding mono} (Attrib.add_del mono_add mono_del)
"declaration of monotonicity rule";
(* outer syntax *)
local structure P = OuterParse and K = OuterKeyword in
val _ = OuterKeyword.keyword "monos";
fun gen_ind_decl mk_def coind =
P.fixes -- P.for_fixes --
Scan.optional SpecParse.where_alt_specs [] --
Scan.optional (P.$$$ "monos" |-- P.!!! SpecParse.xthms1) []
>> (fn (((preds, params), specs), monos) =>
(snd oo gen_add_inductive mk_def true coind preds params specs monos));
val ind_decl = gen_ind_decl add_ind_def;
val _ = OuterSyntax.local_theory' "inductive" "define inductive predicates" K.thy_decl (ind_decl false);
val _ = OuterSyntax.local_theory' "coinductive" "define coinductive predicates" K.thy_decl (ind_decl true);
val _ =
OuterSyntax.local_theory "inductive_cases"
"create simplified instances of elimination rules (improper)" K.thy_script
(P.and_list1 SpecParse.specs >> (snd oo inductive_cases));
end;
end;