(* Title: ZF/Tools/inductive_package.ML
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Fixedpoint definition module -- for Inductive/Coinductive Definitions
The functor will be instantiated for normal sums/products (inductive defs)
and non-standard sums/products (coinductive defs)
Sums are used only for mutual recursion;
Products are used only to derive "streamlined" induction rules for relations
*)
type inductive_result =
{defs : thm list, (*definitions made in thy*)
bnd_mono : thm, (*monotonicity for the lfp definition*)
dom_subset : thm, (*inclusion of recursive set in dom*)
intrs : thm list, (*introduction rules*)
elim : thm, (*case analysis theorem*)
induct : thm, (*main induction rule*)
mutual_induct : thm}; (*mutual induction rule*)
(*Functor's result signature*)
signature INDUCTIVE_PACKAGE =
sig
(*Insert definitions for the recursive sets, which
must *already* be declared as constants in parent theory!*)
val add_inductive_i: bool -> term list * term ->
((binding * term) * attribute list) list ->
thm list * thm list * thm list * thm list -> theory -> theory * inductive_result
val add_inductive: string list * string ->
((binding * string) * Attrib.src list) list ->
(Facts.ref * Attrib.src list) list * (Facts.ref * Attrib.src list) list *
(Facts.ref * Attrib.src list) list * (Facts.ref * Attrib.src list) list ->
theory -> theory * inductive_result
end;
(*Declares functions to add fixedpoint/constructor defs to a theory.
Recursive sets must *already* be declared as constants.*)
functor Add_inductive_def_Fun
(structure Fp: FP and Pr : PR and CP: CARTPROD and Su : SU val coind: bool)
: INDUCTIVE_PACKAGE =
struct
val co_prefix = if coind then "co" else "";
(* utils *)
(*make distinct individual variables a1, a2, a3, ..., an. *)
fun mk_frees a [] = []
| mk_frees a (T::Ts) = Free(a,T) :: mk_frees (Symbol.bump_string a) Ts;
(* add_inductive(_i) *)
(*internal version, accepting terms*)
fun add_inductive_i verbose (rec_tms, dom_sum)
raw_intr_specs (monos, con_defs, type_intrs, type_elims) thy =
let
val _ = Theory.requires thy "Inductive_ZF" "(co)inductive definitions";
val ctxt = Proof_Context.init_global thy;
val intr_specs = map (apfst (apfst Binding.name_of)) raw_intr_specs;
val (intr_names, intr_tms) = split_list (map fst intr_specs);
val case_names = Rule_Cases.case_names intr_names;
(*recT and rec_params should agree for all mutually recursive components*)
val rec_hds = map head_of rec_tms;
val dummy = assert_all is_Const rec_hds
(fn t => "Recursive set not previously declared as constant: " ^
Syntax.string_of_term ctxt t);
(*Now we know they are all Consts, so get their names, type and params*)
val rec_names = map (#1 o dest_Const) rec_hds
and (Const(_,recT),rec_params) = strip_comb (hd rec_tms);
val rec_base_names = map Long_Name.base_name rec_names;
val dummy = assert_all Lexicon.is_identifier rec_base_names
(fn a => "Base name of recursive set not an identifier: " ^ a);
local (*Checking the introduction rules*)
val intr_sets = map (#2 o Ind_Syntax.rule_concl_msg thy) intr_tms;
fun intr_ok set =
case head_of set of Const(a,recT) => member (op =) rec_names a | _ => false;
in
val dummy = assert_all intr_ok intr_sets
(fn t => "Conclusion of rule does not name a recursive set: " ^
Syntax.string_of_term ctxt t);
end;
val dummy = assert_all is_Free rec_params
(fn t => "Param in recursion term not a free variable: " ^
Syntax.string_of_term ctxt t);
(*** Construct the fixedpoint definition ***)
val mk_variant = singleton (Name.variant_list (List.foldr Misc_Legacy.add_term_names [] intr_tms));
val z' = mk_variant"z" and X' = mk_variant"X" and w' = mk_variant"w";
fun dest_tprop (Const(@{const_name Trueprop},_) $ P) = P
| dest_tprop Q = error ("Ill-formed premise of introduction rule: " ^
Syntax.string_of_term ctxt Q);
(*Makes a disjunct from an introduction rule*)
fun fp_part intr = (*quantify over rule's free vars except parameters*)
let val prems = map dest_tprop (Logic.strip_imp_prems intr)
val dummy = List.app (fn rec_hd => List.app (Ind_Syntax.chk_prem rec_hd) prems) rec_hds
val exfrees = subtract (op =) rec_params (Misc_Legacy.term_frees intr)
val zeq = FOLogic.mk_eq (Free(z', Ind_Syntax.iT), #1 (Ind_Syntax.rule_concl intr))
in List.foldr FOLogic.mk_exists
(Balanced_Tree.make FOLogic.mk_conj (zeq::prems)) exfrees
end;
(*The Part(A,h) terms -- compose injections to make h*)
fun mk_Part (Bound 0) = Free(X', Ind_Syntax.iT) (*no mutual rec, no Part needed*)
| mk_Part h = @{const Part} $ Free(X', Ind_Syntax.iT) $ Abs (w', Ind_Syntax.iT, h);
(*Access to balanced disjoint sums via injections*)
val parts = map mk_Part
(Balanced_Tree.accesses {left = fn t => Su.inl $ t, right = fn t => Su.inr $ t, init = Bound 0}
(length rec_tms));
(*replace each set by the corresponding Part(A,h)*)
val part_intrs = map (subst_free (rec_tms ~~ parts) o fp_part) intr_tms;
val fp_abs =
absfree (X', Ind_Syntax.iT)
(Ind_Syntax.mk_Collect (z', dom_sum,
Balanced_Tree.make FOLogic.mk_disj part_intrs));
val fp_rhs = Fp.oper $ dom_sum $ fp_abs
val dummy = List.app (fn rec_hd => (Logic.occs (rec_hd, fp_rhs) andalso
error "Illegal occurrence of recursion operator"; ()))
rec_hds;
(*** Make the new theory ***)
(*A key definition:
If no mutual recursion then it equals the one recursive set.
If mutual recursion then it differs from all the recursive sets. *)
val big_rec_base_name = space_implode "_" rec_base_names;
val big_rec_name = Proof_Context.intern_const ctxt big_rec_base_name;
val _ =
if verbose then
writeln ((if coind then "Coind" else "Ind") ^ "uctive definition " ^ quote big_rec_name)
else ();
(*Big_rec... is the union of the mutually recursive sets*)
val big_rec_tm = list_comb(Const(big_rec_name,recT), rec_params);
(*The individual sets must already be declared*)
val axpairs = map Misc_Legacy.mk_defpair
((big_rec_tm, fp_rhs) ::
(case parts of
[_] => [] (*no mutual recursion*)
| _ => rec_tms ~~ (*define the sets as Parts*)
map (subst_atomic [(Free (X', Ind_Syntax.iT), big_rec_tm)]) parts));
(*tracing: print the fixedpoint definition*)
val dummy = if !Ind_Syntax.trace then
writeln (cat_lines (map (Syntax.string_of_term ctxt o #2) axpairs))
else ()
(*add definitions of the inductive sets*)
val (_, thy1) =
thy
|> Sign.add_path big_rec_base_name
|> Global_Theory.add_defs false (map (Thm.no_attributes o apfst Binding.name) axpairs);
val ctxt1 = Proof_Context.init_global thy1;
(*fetch fp definitions from the theory*)
val big_rec_def::part_rec_defs =
map (Misc_Legacy.get_def thy1)
(case rec_names of [_] => rec_names
| _ => big_rec_base_name::rec_names);
(********)
val dummy = writeln " Proving monotonicity...";
val bnd_mono =
Goal.prove_global thy1 [] [] (FOLogic.mk_Trueprop (Fp.bnd_mono $ dom_sum $ fp_abs))
(fn _ => EVERY
[rtac (@{thm Collect_subset} RS @{thm bnd_monoI}) 1,
REPEAT (ares_tac (@{thms basic_monos} @ monos) 1)]);
val dom_subset = Drule.export_without_context (big_rec_def RS Fp.subs);
val unfold = Drule.export_without_context ([big_rec_def, bnd_mono] MRS Fp.Tarski);
(********)
val dummy = writeln " Proving the introduction rules...";
(*Mutual recursion? Helps to derive subset rules for the
individual sets.*)
val Part_trans =
case rec_names of
[_] => asm_rl
| _ => Drule.export_without_context (@{thm Part_subset} RS @{thm subset_trans});
(*To type-check recursive occurrences of the inductive sets, possibly
enclosed in some monotonic operator M.*)
val rec_typechecks =
[dom_subset] RL (asm_rl :: ([Part_trans] RL monos))
RL [@{thm subsetD}];
(*Type-checking is hardest aspect of proof;
disjIn selects the correct disjunct after unfolding*)
fun intro_tacsf disjIn =
[DETERM (stac unfold 1),
REPEAT (resolve_tac [@{thm Part_eqI}, @{thm CollectI}] 1),
(*Now 2-3 subgoals: typechecking, the disjunction, perhaps equality.*)
rtac disjIn 2,
(*Not ares_tac, since refl must be tried before equality assumptions;
backtracking may occur if the premises have extra variables!*)
DEPTH_SOLVE_1 (resolve_tac [@{thm refl}, @{thm exI}, @{thm conjI}] 2 APPEND assume_tac 2),
(*Now solve the equations like Tcons(a,f) = Inl(?b4)*)
rewrite_goals_tac con_defs,
REPEAT (rtac @{thm refl} 2),
(*Typechecking; this can fail*)
if !Ind_Syntax.trace then print_tac "The type-checking subgoal:"
else all_tac,
REPEAT (FIRSTGOAL ( dresolve_tac rec_typechecks
ORELSE' eresolve_tac (asm_rl :: @{thm PartE} :: @{thm SigmaE2} ::
type_elims)
ORELSE' hyp_subst_tac)),
if !Ind_Syntax.trace then print_tac "The subgoal after monos, type_elims:"
else all_tac,
DEPTH_SOLVE (swap_res_tac (@{thm SigmaI} :: @{thm subsetI} :: type_intrs) 1)];
(*combines disjI1 and disjI2 to get the corresponding nested disjunct...*)
val mk_disj_rls = Balanced_Tree.accesses
{left = fn rl => rl RS @{thm disjI1},
right = fn rl => rl RS @{thm disjI2},
init = @{thm asm_rl}};
val intrs =
(intr_tms, map intro_tacsf (mk_disj_rls (length intr_tms)))
|> ListPair.map (fn (t, tacs) =>
Goal.prove_global thy1 [] [] t
(fn _ => EVERY (rewrite_goals_tac part_rec_defs :: tacs)));
(********)
val dummy = writeln " Proving the elimination rule...";
(*Breaks down logical connectives in the monotonic function*)
val basic_elim_tac =
REPEAT (SOMEGOAL (eresolve_tac (Ind_Syntax.elim_rls @ Su.free_SEs)
ORELSE' bound_hyp_subst_tac))
THEN prune_params_tac
(*Mutual recursion: collapse references to Part(D,h)*)
THEN (PRIMITIVE (fold_rule part_rec_defs));
(*Elimination*)
val elim = rule_by_tactic (Proof_Context.init_global thy1) basic_elim_tac
(unfold RS Ind_Syntax.equals_CollectD)
(*Applies freeness of the given constructors, which *must* be unfolded by
the given defs. Cannot simply use the local con_defs because
con_defs=[] for inference systems.
Proposition A should have the form t:Si where Si is an inductive set*)
fun make_cases ctxt A =
rule_by_tactic ctxt
(basic_elim_tac THEN ALLGOALS (asm_full_simp_tac (simpset_of ctxt)) THEN basic_elim_tac)
(Thm.assume A RS elim)
|> Drule.export_without_context_open;
fun induction_rules raw_induct thy =
let
val dummy = writeln " Proving the induction rule...";
(*** Prove the main induction rule ***)
val pred_name = "P"; (*name for predicate variables*)
(*Used to make induction rules;
ind_alist = [(rec_tm1,pred1),...] associates predicates with rec ops
prem is a premise of an intr rule*)
fun add_induct_prem ind_alist (prem as Const (@{const_name Trueprop}, _) $
(Const (@{const_name mem}, _) $ t $ X), iprems) =
(case AList.lookup (op aconv) ind_alist X of
SOME pred => prem :: FOLogic.mk_Trueprop (pred $ t) :: iprems
| NONE => (*possibly membership in M(rec_tm), for M monotone*)
let fun mk_sb (rec_tm,pred) =
(rec_tm, @{const Collect} $ rec_tm $ pred)
in subst_free (map mk_sb ind_alist) prem :: iprems end)
| add_induct_prem ind_alist (prem,iprems) = prem :: iprems;
(*Make a premise of the induction rule.*)
fun induct_prem ind_alist intr =
let val xs = subtract (op =) rec_params (Misc_Legacy.term_frees intr)
val iprems = List.foldr (add_induct_prem ind_alist) []
(Logic.strip_imp_prems intr)
val (t,X) = Ind_Syntax.rule_concl intr
val (SOME pred) = AList.lookup (op aconv) ind_alist X
val concl = FOLogic.mk_Trueprop (pred $ t)
in fold_rev Logic.all xs (Logic.list_implies (iprems,concl)) end
handle Bind => error"Recursion term not found in conclusion";
(*Minimizes backtracking by delivering the correct premise to each goal.
Intro rules with extra Vars in premises still cause some backtracking *)
fun ind_tac [] 0 = all_tac
| ind_tac(prem::prems) i =
DEPTH_SOLVE_1 (ares_tac [prem, @{thm refl}] i) THEN ind_tac prems (i-1);
val pred = Free(pred_name, Ind_Syntax.iT --> FOLogic.oT);
val ind_prems = map (induct_prem (map (rpair pred) rec_tms))
intr_tms;
val dummy =
if ! Ind_Syntax.trace then
writeln (cat_lines
(["ind_prems:"] @ map (Syntax.string_of_term ctxt1) ind_prems @
["raw_induct:", Display.string_of_thm ctxt1 raw_induct]))
else ();
(*We use a MINIMAL simpset. Even FOL_ss contains too many simpules.
If the premises get simplified, then the proofs could fail.*)
val min_ss =
(Simplifier.global_context thy empty_ss
|> Simplifier.set_mksimps (K (map mk_eq o ZF_atomize o gen_all)))
setSolver (mk_solver "minimal"
(fn ss => resolve_tac (triv_rls @ Simplifier.prems_of ss)
ORELSE' assume_tac
ORELSE' etac @{thm FalseE}));
val quant_induct =
Goal.prove_global thy1 [] ind_prems
(FOLogic.mk_Trueprop (Ind_Syntax.mk_all_imp (big_rec_tm, pred)))
(fn {prems, ...} => EVERY
[rewrite_goals_tac part_rec_defs,
rtac (@{thm impI} RS @{thm allI}) 1,
DETERM (etac raw_induct 1),
(*Push Part inside Collect*)
full_simp_tac (min_ss addsimps [@{thm Part_Collect}]) 1,
(*This CollectE and disjE separates out the introduction rules*)
REPEAT (FIRSTGOAL (eresolve_tac [@{thm CollectE}, @{thm disjE}])),
(*Now break down the individual cases. No disjE here in case
some premise involves disjunction.*)
REPEAT (FIRSTGOAL (eresolve_tac [@{thm CollectE}, @{thm exE}, @{thm conjE}]
ORELSE' bound_hyp_subst_tac)),
ind_tac (rev (map (rewrite_rule part_rec_defs) prems)) (length prems)]);
val dummy =
if ! Ind_Syntax.trace then
writeln ("quant_induct:\n" ^ Display.string_of_thm ctxt1 quant_induct)
else ();
(*** Prove the simultaneous induction rule ***)
(*Make distinct predicates for each inductive set*)
(*The components of the element type, several if it is a product*)
val elem_type = CP.pseudo_type dom_sum;
val elem_factors = CP.factors elem_type;
val elem_frees = mk_frees "za" elem_factors;
val elem_tuple = CP.mk_tuple Pr.pair elem_type elem_frees;
(*Given a recursive set and its domain, return the "fsplit" predicate
and a conclusion for the simultaneous induction rule.
NOTE. This will not work for mutually recursive predicates. Previously
a summand 'domt' was also an argument, but this required the domain of
mutual recursion to invariably be a disjoint sum.*)
fun mk_predpair rec_tm =
let val rec_name = (#1 o dest_Const o head_of) rec_tm
val pfree = Free(pred_name ^ "_" ^ Long_Name.base_name rec_name,
elem_factors ---> FOLogic.oT)
val qconcl =
List.foldr FOLogic.mk_all
(FOLogic.imp $
(@{const mem} $ elem_tuple $ rec_tm)
$ (list_comb (pfree, elem_frees))) elem_frees
in (CP.ap_split elem_type FOLogic.oT pfree,
qconcl)
end;
val (preds,qconcls) = split_list (map mk_predpair rec_tms);
(*Used to form simultaneous induction lemma*)
fun mk_rec_imp (rec_tm,pred) =
FOLogic.imp $ (@{const mem} $ Bound 0 $ rec_tm) $
(pred $ Bound 0);
(*To instantiate the main induction rule*)
val induct_concl =
FOLogic.mk_Trueprop
(Ind_Syntax.mk_all_imp
(big_rec_tm,
Abs("z", Ind_Syntax.iT,
Balanced_Tree.make FOLogic.mk_conj
(ListPair.map mk_rec_imp (rec_tms, preds)))))
and mutual_induct_concl =
FOLogic.mk_Trueprop (Balanced_Tree.make FOLogic.mk_conj qconcls);
val dummy = if !Ind_Syntax.trace then
(writeln ("induct_concl = " ^
Syntax.string_of_term ctxt1 induct_concl);
writeln ("mutual_induct_concl = " ^
Syntax.string_of_term ctxt1 mutual_induct_concl))
else ();
val lemma_tac = FIRST' [eresolve_tac [@{thm asm_rl}, @{thm conjE}, @{thm PartE}, @{thm mp}],
resolve_tac [@{thm allI}, @{thm impI}, @{thm conjI}, @{thm Part_eqI}],
dresolve_tac [@{thm spec}, @{thm mp}, Pr.fsplitD]];
val need_mutual = length rec_names > 1;
val lemma = (*makes the link between the two induction rules*)
if need_mutual then
(writeln " Proving the mutual induction rule...";
Goal.prove_global thy1 [] []
(Logic.mk_implies (induct_concl, mutual_induct_concl))
(fn _ => EVERY
[rewrite_goals_tac part_rec_defs,
REPEAT (rewrite_goals_tac [Pr.split_eq] THEN lemma_tac 1)]))
else (writeln " [ No mutual induction rule needed ]"; @{thm TrueI});
val dummy =
if ! Ind_Syntax.trace then
writeln ("lemma: " ^ Display.string_of_thm ctxt1 lemma)
else ();
(*Mutual induction follows by freeness of Inl/Inr.*)
(*Simplification largely reduces the mutual induction rule to the
standard rule*)
val mut_ss =
min_ss addsimps [Su.distinct, Su.distinct', Su.inl_iff, Su.inr_iff];
val all_defs = con_defs @ part_rec_defs;
(*Removes Collects caused by M-operators in the intro rules. It is very
hard to simplify
list({v: tf. (v : t --> P_t(v)) & (v : f --> P_f(v))})
where t==Part(tf,Inl) and f==Part(tf,Inr) to list({v: tf. P_t(v)}).
Instead the following rules extract the relevant conjunct.
*)
val cmonos = [@{thm subset_refl} RS @{thm Collect_mono}] RL monos
RLN (2,[@{thm rev_subsetD}]);
(*Minimizes backtracking by delivering the correct premise to each goal*)
fun mutual_ind_tac [] 0 = all_tac
| mutual_ind_tac(prem::prems) i =
DETERM
(SELECT_GOAL
(
(*Simplify the assumptions and goal by unfolding Part and
using freeness of the Sum constructors; proves all but one
conjunct by contradiction*)
rewrite_goals_tac all_defs THEN
simp_tac (mut_ss addsimps [@{thm Part_iff}]) 1 THEN
IF_UNSOLVED (*simp_tac may have finished it off!*)
((*simplify assumptions*)
(*some risk of excessive simplification here -- might have
to identify the bare minimum set of rewrites*)
full_simp_tac
(mut_ss addsimps @{thms conj_simps} @ @{thms imp_simps} @ @{thms quant_simps}) 1
THEN
(*unpackage and use "prem" in the corresponding place*)
REPEAT (rtac @{thm impI} 1) THEN
rtac (rewrite_rule all_defs prem) 1 THEN
(*prem must not be REPEATed below: could loop!*)
DEPTH_SOLVE (FIRSTGOAL (ares_tac [@{thm impI}] ORELSE'
eresolve_tac (@{thm conjE} :: @{thm mp} :: cmonos))))
) i)
THEN mutual_ind_tac prems (i-1);
val mutual_induct_fsplit =
if need_mutual then
Goal.prove_global thy1 [] (map (induct_prem (rec_tms~~preds)) intr_tms)
mutual_induct_concl
(fn {prems, ...} => EVERY
[rtac (quant_induct RS lemma) 1,
mutual_ind_tac (rev prems) (length prems)])
else @{thm TrueI};
(** Uncurrying the predicate in the ordinary induction rule **)
(*instantiate the variable to a tuple, if it is non-trivial, in order to
allow the predicate to be "opened up".
The name "x.1" comes from the "RS spec" !*)
val inst =
case elem_frees of [_] => I
| _ => Drule.instantiate_normalize ([], [(cterm_of thy1 (Var(("x",1), Ind_Syntax.iT)),
cterm_of thy1 elem_tuple)]);
(*strip quantifier and the implication*)
val induct0 = inst (quant_induct RS @{thm spec} RSN (2, @{thm rev_mp}));
val Const (@{const_name Trueprop}, _) $ (pred_var $ _) = concl_of induct0
val induct = CP.split_rule_var(pred_var, elem_type-->FOLogic.oT, induct0)
|> Drule.export_without_context
and mutual_induct = CP.remove_split mutual_induct_fsplit
val ([induct', mutual_induct'], thy') =
thy
|> Global_Theory.add_thms [((Binding.name (co_prefix ^ "induct"), induct),
[case_names, Induct.induct_pred big_rec_name]),
((Binding.name "mutual_induct", mutual_induct), [case_names])];
in ((thy', induct'), mutual_induct')
end; (*of induction_rules*)
val raw_induct = Drule.export_without_context ([big_rec_def, bnd_mono] MRS Fp.induct)
val ((thy2, induct), mutual_induct) =
if not coind then induction_rules raw_induct thy1
else
(thy1
|> Global_Theory.add_thms [((Binding.name (co_prefix ^ "induct"), raw_induct), [])]
|> apfst hd |> Library.swap, @{thm TrueI})
and defs = big_rec_def :: part_rec_defs
val (([bnd_mono', dom_subset', elim'], [defs', intrs']), thy3) =
thy2
|> IndCases.declare big_rec_name make_cases
|> Global_Theory.add_thms
[((Binding.name "bnd_mono", bnd_mono), []),
((Binding.name "dom_subset", dom_subset), []),
((Binding.name "cases", elim), [case_names, Induct.cases_pred big_rec_name])]
||>> (Global_Theory.add_thmss o map Thm.no_attributes)
[(Binding.name "defs", defs),
(Binding.name "intros", intrs)];
val (intrs'', thy4) =
thy3
|> Global_Theory.add_thms ((map Binding.name intr_names ~~ intrs') ~~ map #2 intr_specs)
||> Sign.parent_path;
in
(thy4,
{defs = defs',
bnd_mono = bnd_mono',
dom_subset = dom_subset',
intrs = intrs'',
elim = elim',
induct = induct,
mutual_induct = mutual_induct})
end;
(*source version*)
fun add_inductive (srec_tms, sdom_sum) intr_srcs
(raw_monos, raw_con_defs, raw_type_intrs, raw_type_elims) thy =
let
val ctxt = Proof_Context.init_global thy;
val read_terms = map (Syntax.parse_term ctxt #> Type.constraint Ind_Syntax.iT)
#> Syntax.check_terms ctxt;
val intr_atts = map (map (Attrib.attribute_cmd_global thy) o snd) intr_srcs;
val sintrs = map fst intr_srcs ~~ intr_atts;
val rec_tms = read_terms srec_tms;
val dom_sum = singleton read_terms sdom_sum;
val intr_tms = Syntax.read_props ctxt (map (snd o fst) sintrs);
val intr_specs = (map (fst o fst) sintrs ~~ intr_tms) ~~ map snd sintrs;
val monos = Attrib.eval_thms ctxt raw_monos;
val con_defs = Attrib.eval_thms ctxt raw_con_defs;
val type_intrs = Attrib.eval_thms ctxt raw_type_intrs;
val type_elims = Attrib.eval_thms ctxt raw_type_elims;
in
thy
|> add_inductive_i true (rec_tms, dom_sum) intr_specs (monos, con_defs, type_intrs, type_elims)
end;
(* outer syntax *)
fun mk_ind (((((doms, intrs), monos), con_defs), type_intrs), type_elims) =
#1 o add_inductive doms (map Parse.triple_swap intrs) (monos, con_defs, type_intrs, type_elims);
val ind_decl =
(@{keyword "domains"} |-- Parse.!!! (Parse.enum1 "+" Parse.term --
((@{keyword "\<subseteq>"} || @{keyword "<="}) |-- Parse.term))) --
(@{keyword "intros"} |--
Parse.!!! (Scan.repeat1 (Parse_Spec.opt_thm_name ":" -- Parse.prop))) --
Scan.optional (@{keyword "monos"} |-- Parse.!!! Parse_Spec.xthms1) [] --
Scan.optional (@{keyword "con_defs"} |-- Parse.!!! Parse_Spec.xthms1) [] --
Scan.optional (@{keyword "type_intros"} |-- Parse.!!! Parse_Spec.xthms1) [] --
Scan.optional (@{keyword "type_elims"} |-- Parse.!!! Parse_Spec.xthms1) []
>> (Toplevel.theory o mk_ind);
val _ =
Outer_Syntax.command
(if coind then @{command_spec "coinductive"} else @{command_spec "inductive"})
("define " ^ co_prefix ^ "inductive sets") ind_decl;
end;