(* Title: HOLCF/Tools/Domain/domain_theorems.ML
Author: David von Oheimb
Author: Brian Huffman
Proof generator for domain command.
*)
val HOLCF_ss = @{simpset};
signature DOMAIN_THEOREMS =
sig
val theorems:
Domain_Library.eq * Domain_Library.eq list
-> typ * (binding * (bool * binding option * typ) list * mixfix) list
-> Domain_Take_Proofs.iso_info
-> theory -> thm list * theory;
val comp_theorems: bstring * Domain_Library.eq list -> theory -> thm list * theory;
val quiet_mode: bool Unsynchronized.ref;
val trace_domain: bool Unsynchronized.ref;
end;
structure Domain_Theorems :> DOMAIN_THEOREMS =
struct
val quiet_mode = Unsynchronized.ref false;
val trace_domain = Unsynchronized.ref false;
fun message s = if !quiet_mode then () else writeln s;
fun trace s = if !trace_domain then tracing s else ();
open Domain_Library;
infixr 0 ===>;
infixr 0 ==>;
infix 0 == ;
infix 1 ===;
infix 1 ~= ;
infix 1 <<;
infix 1 ~<<;
infix 9 ` ;
infix 9 `% ;
infix 9 `%%;
infixr 9 oo;
(* ----- general proof facilities ------------------------------------------- *)
fun legacy_infer_term thy t =
let val ctxt = ProofContext.set_mode ProofContext.mode_schematic (ProofContext.init thy)
in singleton (Syntax.check_terms ctxt) (Sign.intern_term thy t) end;
fun pg'' thy defs t tacs =
let
val t' = legacy_infer_term thy t;
val asms = Logic.strip_imp_prems t';
val prop = Logic.strip_imp_concl t';
fun tac {prems, context} =
rewrite_goals_tac defs THEN
EVERY (tacs {prems = map (rewrite_rule defs) prems, context = context})
in Goal.prove_global thy [] asms prop tac end;
fun pg' thy defs t tacsf =
let
fun tacs {prems, context} =
if null prems then tacsf context
else cut_facts_tac prems 1 :: tacsf context;
in pg'' thy defs t tacs end;
(* FIXME!!!!!!!!! *)
(* We should NEVER re-parse variable names as strings! *)
(* The names can conflict with existing constants or other syntax! *)
fun case_UU_tac ctxt rews i v =
InductTacs.case_tac ctxt (v^"=UU") i THEN
asm_simp_tac (HOLCF_ss addsimps rews) i;
(* ----- general proofs ----------------------------------------------------- *)
val all2E = @{lemma "!x y . P x y ==> (P x y ==> R) ==> R" by simp}
fun theorems
(((dname, _), cons) : eq, eqs : eq list)
(dom_eqn : typ * (binding * (bool * binding option * typ) list * mixfix) list)
(iso_info : Domain_Take_Proofs.iso_info)
(thy : theory) =
let
val _ = message ("Proving isomorphism properties of domain "^dname^" ...");
val map_tab = Domain_Take_Proofs.get_map_tab thy;
(* ----- getting the axioms and definitions --------------------------------- *)
val ax_abs_iso = #abs_inverse iso_info;
val ax_rep_iso = #rep_inverse iso_info;
val abs_const = #abs_const iso_info;
val rep_const = #rep_const iso_info;
local
fun ga s dn = PureThy.get_thm thy (dn ^ "." ^ s);
in
val ax_take_0 = ga "take_0" dname;
val ax_take_Suc = ga "take_Suc" dname;
val ax_take_strict = ga "take_strict" dname;
end; (* local *)
(* ----- define constructors ------------------------------------------------ *)
val (result, thy) =
Domain_Constructors.add_domain_constructors
(Long_Name.base_name dname) (snd dom_eqn) iso_info thy;
val con_appls = #con_betas result;
val {exhaust, casedist, ...} = result;
val {con_compacts, con_rews, inverts, injects, dist_les, dist_eqs, ...} = result;
val {sel_rews, ...} = result;
val when_rews = #cases result;
val when_strict = hd when_rews;
val dis_rews = #dis_rews result;
val mat_rews = #match_rews result;
val pat_rews = #pat_rews result;
(* ----- theorems concerning the isomorphism -------------------------------- *)
val pg = pg' thy;
val retraction_strict = @{thm retraction_strict};
val abs_strict = ax_rep_iso RS (allI RS retraction_strict);
val rep_strict = ax_abs_iso RS (allI RS retraction_strict);
val iso_rews = map Drule.export_without_context [ax_abs_iso, ax_rep_iso, abs_strict, rep_strict];
(* ----- theorems concerning one induction step ----------------------------- *)
local
fun dc_take dn = %%:(dn^"_take");
val dnames = map (fst o fst) eqs;
val deflation_thms = Domain_Take_Proofs.get_deflation_thms thy;
fun get_deflation_take dn = PureThy.get_thm thy (dn ^ ".deflation_take");
val axs_deflation_take = map get_deflation_take dnames;
fun copy_of_dtyp tab r dt =
if Datatype_Aux.is_rec_type dt then copy tab r dt else ID
and copy tab r (Datatype_Aux.DtRec i) = r i
| copy tab r (Datatype_Aux.DtTFree a) = ID
| copy tab r (Datatype_Aux.DtType (c, ds)) =
case Symtab.lookup tab c of
SOME f => list_ccomb (%%:f, map (copy_of_dtyp tab r) ds)
| NONE => (warning ("copy_of_dtyp: unknown type constructor " ^ c); ID);
fun one_take_app (con, args) =
let
fun mk_take n = dc_take (List.nth (dnames, n)) $ %:"n";
fun one_rhs arg =
if Datatype_Aux.is_rec_type (dtyp_of arg)
then copy_of_dtyp map_tab
mk_take (dtyp_of arg) ` (%# arg)
else (%# arg);
val lhs = (dc_take dname $ (%%:"Suc" $ %:"n"))`(con_app con args);
val rhs = con_app2 con one_rhs args;
val goal = mk_trp (lhs === rhs);
val rules =
[ax_take_Suc, ax_abs_iso, @{thm cfcomp2}]
@ @{thms take_con_rules ID1 deflation_strict}
@ deflation_thms @ axs_deflation_take;
val tacs = [simp_tac (HOL_basic_ss addsimps rules) 1];
in pg con_appls goal (K tacs) end;
val take_apps = map one_take_app cons;
in
val take_rews = ax_take_0 :: ax_take_strict :: take_apps;
end;
val case_ns =
"bottom" :: map (fn (b,_,_) => Binding.name_of b) (snd dom_eqn);
in
thy
|> Sign.add_path (Long_Name.base_name dname)
|> snd o PureThy.add_thmss [
((Binding.name "iso_rews" , iso_rews ), [Simplifier.simp_add]),
((Binding.name "exhaust" , [exhaust] ), []),
((Binding.name "casedist" , [casedist] ),
[Rule_Cases.case_names case_ns, Induct.cases_type dname]),
((Binding.name "when_rews" , when_rews ), [Simplifier.simp_add]),
((Binding.name "compacts" , con_compacts), [Simplifier.simp_add]),
((Binding.name "con_rews" , con_rews ),
[Simplifier.simp_add, Fixrec.fixrec_simp_add]),
((Binding.name "sel_rews" , sel_rews ), [Simplifier.simp_add]),
((Binding.name "dis_rews" , dis_rews ), [Simplifier.simp_add]),
((Binding.name "pat_rews" , pat_rews ), [Simplifier.simp_add]),
((Binding.name "dist_les" , dist_les ), [Simplifier.simp_add]),
((Binding.name "dist_eqs" , dist_eqs ), [Simplifier.simp_add]),
((Binding.name "inverts" , inverts ), [Simplifier.simp_add]),
((Binding.name "injects" , injects ), [Simplifier.simp_add]),
((Binding.name "take_rews" , take_rews ), [Simplifier.simp_add]),
((Binding.name "match_rews", mat_rews ),
[Simplifier.simp_add, Fixrec.fixrec_simp_add])]
|> Sign.parent_path
|> pair (iso_rews @ when_rews @ con_rews @ sel_rews @ dis_rews @
pat_rews @ dist_les @ dist_eqs)
end; (* let *)
(******************************************************************************)
(****************************** induction rules *******************************)
(******************************************************************************)
fun prove_induction
(comp_dnam, eqs : eq list)
(take_lemmas : thm list)
(axs_reach : thm list)
(take_rews : thm list)
(thy : theory) =
let
val dnames = map (fst o fst) eqs;
val conss = map snd eqs;
fun dc_take dn = %%:(dn^"_take");
val x_name = idx_name dnames "x";
val P_name = idx_name dnames "P";
val pg = pg' thy;
local
fun ga s dn = PureThy.get_thm thy (dn ^ "." ^ s);
fun gts s dn = PureThy.get_thms thy (dn ^ "." ^ s);
in
val axs_rep_iso = map (ga "rep_iso") dnames;
val axs_abs_iso = map (ga "abs_iso") dnames;
val axs_chain_take = map (ga "chain_take") dnames;
val lub_take_thms = map (ga "lub_take") dnames;
val axs_finite_def = map (ga "finite_def") dnames;
val take_0_thms = map (ga "take_0") dnames;
val take_Suc_thms = map (ga "take_Suc") dnames;
val cases = map (ga "casedist" ) dnames;
val con_rews = maps (gts "con_rews" ) dnames;
end;
fun one_con p (con, args) =
let
val P_names = map P_name (1 upto (length dnames));
val vns = Name.variant_list P_names (map vname args);
val nonlazy_vns = map snd (filter_out (is_lazy o fst) (args ~~ vns));
fun ind_hyp arg = %:(P_name (1 + rec_of arg)) $ bound_arg args arg;
val t1 = mk_trp (%:p $ con_app2 con (bound_arg args) args);
val t2 = lift ind_hyp (filter is_rec args, t1);
val t3 = lift_defined (bound_arg vns) (nonlazy_vns, t2);
in Library.foldr mk_All (vns, t3) end;
fun one_eq ((p, cons), concl) =
mk_trp (%:p $ UU) ===> Logic.list_implies (map (one_con p) cons, concl);
fun ind_term concf = Library.foldr one_eq
(mapn (fn n => fn x => (P_name n, x)) 1 conss,
mk_trp (foldr1 mk_conj (mapn concf 1 dnames)));
val take_ss = HOL_ss addsimps (@{thm Rep_CFun_strict1} :: take_rews);
fun quant_tac ctxt i = EVERY
(mapn (fn n => fn _ => res_inst_tac ctxt [(("x", 0), x_name n)] spec i) 1 dnames);
fun ind_prems_tac prems = EVERY
(maps (fn cons =>
(resolve_tac prems 1 ::
maps (fn (_,args) =>
resolve_tac prems 1 ::
map (K(atac 1)) (nonlazy args) @
map (K(atac 1)) (filter is_rec args))
cons))
conss);
local
(* check whether every/exists constructor of the n-th part of the equation:
it has a possibly indirectly recursive argument that isn't/is possibly
indirectly lazy *)
fun rec_to quant nfn rfn ns lazy_rec (n,cons) = quant (exists (fn arg =>
is_rec arg andalso not(rec_of arg mem ns) andalso
((rec_of arg = n andalso nfn(lazy_rec orelse is_lazy arg)) orelse
rec_of arg <> n andalso rec_to quant nfn rfn (rec_of arg::ns)
(lazy_rec orelse is_lazy arg) (n, (List.nth(conss,rec_of arg))))
) o snd) cons;
fun all_rec_to ns = rec_to forall not all_rec_to ns;
fun warn (n,cons) =
if all_rec_to [] false (n,cons)
then (warning ("domain "^List.nth(dnames,n)^" is empty!"); true)
else false;
fun lazy_rec_to ns = rec_to exists I lazy_rec_to ns;
in
val n__eqs = mapn (fn n => fn (_,cons) => (n,cons)) 0 eqs;
val is_emptys = map warn n__eqs;
val is_finite = forall (not o lazy_rec_to [] false) n__eqs;
val _ = if is_finite
then message ("Proving finiteness rule for domain "^comp_dnam^" ...")
else ();
end;
val _ = trace " Proving finite_ind...";
val finite_ind =
let
fun concf n dn = %:(P_name n) $ (dc_take dn $ %:"n" `%(x_name n));
val goal = ind_term concf;
fun tacf {prems, context} =
let
val tacs1 = [
quant_tac context 1,
simp_tac HOL_ss 1,
InductTacs.induct_tac context [[SOME "n"]] 1,
simp_tac (take_ss addsimps prems) 1,
TRY (safe_tac HOL_cs)];
fun arg_tac arg =
(* FIXME! case_UU_tac *)
case_UU_tac context (prems @ con_rews) 1
(List.nth (dnames, rec_of arg) ^ "_take n$" ^ vname arg);
fun con_tacs (con, args) =
asm_simp_tac take_ss 1 ::
map arg_tac (filter is_nonlazy_rec args) @
[resolve_tac prems 1] @
map (K (atac 1)) (nonlazy args) @
map (K (etac spec 1)) (filter is_rec args);
fun cases_tacs (cons, cases) =
res_inst_tac context [(("y", 0), "x")] cases 1 ::
asm_simp_tac (take_ss addsimps prems) 1 ::
maps con_tacs cons;
in
tacs1 @ maps cases_tacs (conss ~~ cases)
end;
in pg'' thy [] goal tacf
handle ERROR _ => (warning "Proof of finite_ind failed."; TrueI)
end;
(* ----- theorems concerning finiteness and induction ----------------------- *)
val global_ctxt = ProofContext.init thy;
val _ = trace " Proving finites, ind...";
val (finites, ind) =
(
if is_finite
then (* finite case *)
let
val decisive_lemma =
let
val iso_locale_thms =
map2 (fn x => fn y => @{thm iso.intro} OF [x, y])
axs_abs_iso axs_rep_iso;
val decisive_abs_rep_thms =
map (fn x => @{thm decisive_abs_rep} OF [x])
iso_locale_thms;
val n = Free ("n", @{typ nat});
fun mk_decisive t = %%: @{const_name decisive} $ t;
fun f dn = mk_decisive (dc_take dn $ n);
val goal = mk_trp (foldr1 mk_conj (map f dnames));
val rules0 = @{thm decisive_bottom} :: take_0_thms;
val rules1 =
take_Suc_thms @ decisive_abs_rep_thms
@ @{thms decisive_ID decisive_ssum_map decisive_sprod_map};
val tacs = [
rtac @{thm nat.induct} 1,
simp_tac (HOL_ss addsimps rules0) 1,
asm_simp_tac (HOL_ss addsimps rules1) 1];
in pg [] goal (K tacs) end;
fun take_enough dn = mk_ex ("n",dc_take dn $ Bound 0 ` %:"x" === %:"x");
fun one_finite (dn, decisive_thm) =
let
val goal = mk_trp (%%:(dn^"_finite") $ %:"x");
val tacs = [
rtac @{thm lub_ID_finite} 1,
resolve_tac axs_chain_take 1,
resolve_tac lub_take_thms 1,
rtac decisive_thm 1];
in pg axs_finite_def goal (K tacs) end;
val _ = trace " Proving finites";
val finites = map one_finite (dnames ~~ atomize global_ctxt decisive_lemma);
val _ = trace " Proving ind";
val ind =
let
fun concf n dn = %:(P_name n) $ %:(x_name n);
fun tacf {prems, context} =
let
fun finite_tacs (finite, fin_ind) = [
rtac(rewrite_rule axs_finite_def finite RS exE)1,
etac subst 1,
rtac fin_ind 1,
ind_prems_tac prems];
in
TRY (safe_tac HOL_cs) ::
maps finite_tacs (finites ~~ atomize global_ctxt finite_ind)
end;
in pg'' thy [] (ind_term concf) tacf end;
in (finites, ind) end (* let *)
else (* infinite case *)
let
fun one_finite n dn =
read_instantiate global_ctxt [(("P", 0), dn ^ "_finite " ^ x_name n)] excluded_middle;
val finites = mapn one_finite 1 dnames;
val goal =
let
fun one_adm n _ = mk_trp (mk_adm (%:(P_name n)));
fun concf n dn = %:(P_name n) $ %:(x_name n);
in Logic.list_implies (mapn one_adm 1 dnames, ind_term concf) end;
val cont_rules =
@{thms cont_id cont_const cont2cont_Rep_CFun
cont2cont_fst cont2cont_snd};
val subgoal =
let fun p n dn = %:(P_name n) $ (dc_take dn $ Bound 0 `%(x_name n));
in mk_trp (mk_all ("n", foldr1 mk_conj (mapn p 1 dnames))) end;
val subgoal' = legacy_infer_term thy subgoal;
fun tacf {prems, context} =
let
val subtac =
EVERY [rtac allI 1, rtac finite_ind 1, ind_prems_tac prems];
val subthm = Goal.prove context [] [] subgoal' (K subtac);
in
map (fn ax_reach => rtac (ax_reach RS subst) 1) axs_reach @ [
cut_facts_tac (subthm :: take (length dnames) prems) 1,
REPEAT (rtac @{thm conjI} 1 ORELSE
EVERY [etac @{thm admD [OF _ ch2ch_Rep_CFunL]} 1,
resolve_tac axs_chain_take 1,
asm_simp_tac HOL_basic_ss 1])
]
end;
val ind = (pg'' thy [] goal tacf
handle ERROR _ =>
(warning "Cannot prove infinite induction rule"; TrueI)
);
in (finites, ind) end
)
handle THM _ =>
(warning "Induction proofs failed (THM raised)."; ([], TrueI))
| ERROR _ =>
(warning "Cannot prove induction rule"; ([], TrueI));
val case_ns =
let
val bottoms =
if length dnames = 1 then ["bottom"] else
map (fn s => "bottom_" ^ Long_Name.base_name s) dnames;
fun one_eq bot (_,cons) =
bot :: map (fn (c,_) => Long_Name.base_name c) cons;
in flat (map2 one_eq bottoms eqs) end;
val inducts = Project_Rule.projections (ProofContext.init thy) ind;
fun ind_rule (dname, rule) =
((Binding.empty, [rule]),
[Rule_Cases.case_names case_ns, Induct.induct_type dname]);
val induct_failed = (Thm.prop_of ind = Thm.prop_of TrueI);
in thy |> Sign.add_path comp_dnam
|> snd o PureThy.add_thmss [
((Binding.name "finites" , finites ), []),
((Binding.name "finite_ind" , [finite_ind]), []),
((Binding.name "ind" , [ind] ), [])]
|> (if induct_failed then I
else snd o PureThy.add_thmss (map ind_rule (dnames ~~ inducts)))
|> Sign.parent_path
end; (* prove_induction *)
(******************************************************************************)
(************************ bisimulation and coinduction ************************)
(******************************************************************************)
fun prove_coinduction
(comp_dnam, eqs : eq list)
(take_lemmas : thm list)
(thy : theory) : theory =
let
val dnames = map (fst o fst) eqs;
val comp_dname = Sign.full_bname thy comp_dnam;
fun dc_take dn = %%:(dn^"_take");
val x_name = idx_name dnames "x";
val n_eqs = length eqs;
val take_rews =
maps (fn dn => PureThy.get_thms thy (dn ^ ".take_rews")) dnames;
(* ----- define bisimulation predicate -------------------------------------- *)
local
open HOLCF_Library
val dtypes = map (Type o fst) eqs;
val relprod = mk_tupleT (map (fn tp => tp --> tp --> boolT) dtypes);
val bisim_bind = Binding.name (comp_dnam ^ "_bisim");
val bisim_type = relprod --> boolT;
in
val (bisim_const, thy) =
Sign.declare_const ((bisim_bind, bisim_type), NoSyn) thy;
end;
local
fun legacy_infer_term thy t =
singleton (Syntax.check_terms (ProofContext.init thy)) (Sign.intern_term thy t);
fun legacy_infer_prop thy t = legacy_infer_term thy (TypeInfer.constrain propT t);
fun infer_props thy = map (apsnd (legacy_infer_prop thy));
fun add_defs_i x = PureThy.add_defs false (map Thm.no_attributes x);
fun add_defs_infer defs thy = add_defs_i (infer_props thy defs) thy;
val comp_dname = Sign.full_bname thy comp_dnam;
val dnames = map (fst o fst) eqs;
val x_name = idx_name dnames "x";
fun one_con (con, args) =
let
val nonrec_args = filter_out is_rec args;
val rec_args = filter is_rec args;
val recs_cnt = length rec_args;
val allargs = nonrec_args @ rec_args
@ map (upd_vname (fn s=> s^"'")) rec_args;
val allvns = map vname allargs;
fun vname_arg s arg = if is_rec arg then vname arg^s else vname arg;
val vns1 = map (vname_arg "" ) args;
val vns2 = map (vname_arg "'") args;
val allargs_cnt = length nonrec_args + 2*recs_cnt;
val rec_idxs = (recs_cnt-1) downto 0;
val nonlazy_idxs = map snd (filter_out (fn (arg,_) => is_lazy arg)
(allargs~~((allargs_cnt-1) downto 0)));
fun rel_app i ra = proj (Bound(allargs_cnt+2)) eqs (rec_of ra) $
Bound (2*recs_cnt-i) $ Bound (recs_cnt-i);
val capps =
List.foldr
mk_conj
(mk_conj(
Bound(allargs_cnt+1)===list_ccomb(%%:con,map (bound_arg allvns) vns1),
Bound(allargs_cnt+0)===list_ccomb(%%:con,map (bound_arg allvns) vns2)))
(mapn rel_app 1 rec_args);
in
List.foldr
mk_ex
(Library.foldr mk_conj
(map (defined o Bound) nonlazy_idxs,capps)) allvns
end;
fun one_comp n (_,cons) =
mk_all (x_name(n+1),
mk_all (x_name(n+1)^"'",
mk_imp (proj (Bound 2) eqs n $ Bound 1 $ Bound 0,
foldr1 mk_disj (mk_conj(Bound 1 === UU,Bound 0 === UU)
::map one_con cons))));
val bisim_eqn =
%%:(comp_dname^"_bisim") ==
mk_lam("R", foldr1 mk_conj (mapn one_comp 0 eqs));
in
val ([ax_bisim_def], thy) =
thy
|> Sign.add_path comp_dnam
|> add_defs_infer [(Binding.name "bisim_def", bisim_eqn)]
||> Sign.parent_path;
end; (* local *)
(* ----- theorem concerning coinduction ------------------------------------- *)
local
val pg = pg' thy;
val xs = mapn (fn n => K (x_name n)) 1 dnames;
fun bnd_arg n i = Bound(2*(n_eqs - n)-i-1);
val take_ss = HOL_ss addsimps (@{thm Rep_CFun_strict1} :: take_rews);
val sproj = prj (fn s => K("fst("^s^")")) (fn s => K("snd("^s^")"));
val _ = trace " Proving coind_lemma...";
val coind_lemma =
let
fun mk_prj n _ = proj (%:"R") eqs n $ bnd_arg n 0 $ bnd_arg n 1;
fun mk_eqn n dn =
(dc_take dn $ %:"n" ` bnd_arg n 0) ===
(dc_take dn $ %:"n" ` bnd_arg n 1);
fun mk_all2 (x,t) = mk_all (x, mk_all (x^"'", t));
val goal =
mk_trp (mk_imp (%%:(comp_dname^"_bisim") $ %:"R",
Library.foldr mk_all2 (xs,
Library.foldr mk_imp (mapn mk_prj 0 dnames,
foldr1 mk_conj (mapn mk_eqn 0 dnames)))));
fun x_tacs ctxt n x = [
rotate_tac (n+1) 1,
etac all2E 1,
eres_inst_tac ctxt [(("P", 1), sproj "R" eqs n^" "^x^" "^x^"'")] (mp RS disjE) 1,
TRY (safe_tac HOL_cs),
REPEAT (CHANGED (asm_simp_tac take_ss 1))];
fun tacs ctxt = [
rtac impI 1,
InductTacs.induct_tac ctxt [[SOME "n"]] 1,
simp_tac take_ss 1,
safe_tac HOL_cs] @
flat (mapn (x_tacs ctxt) 0 xs);
in pg [ax_bisim_def] goal tacs end;
in
val _ = trace " Proving coind...";
val coind =
let
fun mk_prj n x = mk_trp (proj (%:"R") eqs n $ %:x $ %:(x^"'"));
fun mk_eqn x = %:x === %:(x^"'");
val goal =
mk_trp (%%:(comp_dname^"_bisim") $ %:"R") ===>
Logic.list_implies (mapn mk_prj 0 xs,
mk_trp (foldr1 mk_conj (map mk_eqn xs)));
val tacs =
TRY (safe_tac HOL_cs) ::
maps (fn take_lemma => [
rtac take_lemma 1,
cut_facts_tac [coind_lemma] 1,
fast_tac HOL_cs 1])
take_lemmas;
in pg [] goal (K tacs) end;
end; (* local *)
in thy |> Sign.add_path comp_dnam
|> snd o PureThy.add_thmss [((Binding.name "coind", [coind]), [])]
|> Sign.parent_path
end; (* let *)
fun comp_theorems (comp_dnam, eqs: eq list) thy =
let
val map_tab = Domain_Take_Proofs.get_map_tab thy;
val dnames = map (fst o fst) eqs;
val comp_dname = Sign.full_bname thy comp_dnam;
(* ----- getting the composite axiom and definitions ------------------------ *)
(* Test for indirect recursion *)
local
fun indirect_arg arg =
rec_of arg = ~1 andalso Datatype_Aux.is_rec_type (dtyp_of arg);
fun indirect_con (_, args) = exists indirect_arg args;
fun indirect_eq (_, cons) = exists indirect_con cons;
in
val is_indirect = exists indirect_eq eqs;
val _ =
if is_indirect
then message "Indirect recursion detected, skipping proofs of (co)induction rules"
else message ("Proving induction properties of domain "^comp_dname^" ...");
end;
(* theorems about take *)
local
fun ga s dn = PureThy.get_thm thy (dn ^ "." ^ s);
val axs_chain_take = map (ga "chain_take") dnames;
val axs_lub_take = map (ga "lub_take" ) dnames;
in
val _ = trace " Proving take_lemmas...";
val take_lemmas =
let
fun take_lemma (ax_chain_take, ax_lub_take) =
Drule.export_without_context
(@{thm lub_ID_take_lemma} OF [ax_chain_take, ax_lub_take]);
in map take_lemma (axs_chain_take ~~ axs_lub_take) end;
val axs_reach =
let
fun reach (ax_chain_take, ax_lub_take) =
Drule.export_without_context
(@{thm lub_ID_reach} OF [ax_chain_take, ax_lub_take]);
in map reach (axs_chain_take ~~ axs_lub_take) end;
end;
val take_rews =
maps (fn dn => PureThy.get_thms thy (dn ^ ".take_rews")) dnames;
(* prove induction rules, unless definition is indirect recursive *)
val thy =
if is_indirect then thy else
prove_induction (comp_dnam, eqs) take_lemmas axs_reach take_rews thy;
val thy =
if is_indirect then thy else
prove_coinduction (comp_dnam, eqs) take_lemmas thy;
in thy |> Sign.add_path comp_dnam
|> snd o PureThy.add_thmss [
((Binding.name "take_lemmas", take_lemmas ), []),
((Binding.name "reach" , axs_reach ), [])]
|> Sign.parent_path |> pair take_rews
end; (* let *)
end; (* struct *)