(* 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 comp_theorems :
binding * Domain_Library.eq list ->
(binding * (binding * (bool * binding option * typ) list * mixfix) list) list ->
Domain_Take_Proofs.take_induct_info ->
Domain_Constructors.constr_info 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 ------------------------------------------- *)
local
fun map_typ f g (Type (c, Ts)) = Type (g c, map (map_typ f g) Ts)
| map_typ f _ (TFree (x, S)) = TFree (x, map f S)
| map_typ f _ (TVar (xi, S)) = TVar (xi, map f S);
fun map_term f g h (Const (c, T)) = Const (h c, map_typ f g T)
| map_term f g _ (Free (x, T)) = Free (x, map_typ f g T)
| map_term f g _ (Var (xi, T)) = Var (xi, map_typ f g T)
| map_term _ _ _ (t as Bound _) = t
| map_term f g h (Abs (x, T, t)) = Abs (x, map_typ f g T, map_term f g h t)
| map_term f g h (t $ u) = map_term f g h t $ map_term f g h u;
in
fun intern_term thy =
map_term (Sign.intern_class thy) (Sign.intern_type thy) (Sign.intern_const thy);
end;
fun legacy_infer_term thy t =
let val ctxt = ProofContext.set_mode ProofContext.mode_schematic (ProofContext.init_global thy)
in singleton (Syntax.check_terms ctxt) (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;
(******************************************************************************)
(***************************** proofs about take ******************************)
(******************************************************************************)
fun take_theorems
(specs : (binding * (binding * (bool * binding option * typ) list * mixfix) list) list)
(take_info : Domain_Take_Proofs.take_induct_info)
(constr_infos : Domain_Constructors.constr_info list)
(thy : theory) : thm list list * theory =
let
open HOLCF_Library;
val {take_consts, take_Suc_thms, deflation_take_thms, ...} = take_info;
val deflation_thms = Domain_Take_Proofs.get_deflation_thms thy;
val n = Free ("n", @{typ nat});
val n' = @{const Suc} $ n;
local
val newTs = map (#absT o #iso_info) constr_infos;
val subs = newTs ~~ map (fn t => t $ n) take_consts;
fun is_ID (Const (c, _)) = (c = @{const_name ID})
| is_ID _ = false;
in
fun map_of_arg v T =
let val m = Domain_Take_Proofs.map_of_typ thy subs T;
in if is_ID m then v else mk_capply (m, v) end;
end
fun prove_take_apps
(((dbind, spec), take_const), constr_info) thy =
let
val {iso_info, con_consts, con_betas, ...} = constr_info;
val {abs_inverse, ...} = iso_info;
fun prove_take_app (con_const : term) (bind, args, mx) =
let
val Ts = map (fn (_, _, T) => T) args;
val ns = Name.variant_list ["n"] (Datatype_Prop.make_tnames Ts);
val vs = map Free (ns ~~ Ts);
val lhs = mk_capply (take_const $ n', list_ccomb (con_const, vs));
val rhs = list_ccomb (con_const, map2 map_of_arg vs Ts);
val goal = mk_trp (mk_eq (lhs, rhs));
val rules =
[abs_inverse] @ con_betas @ @{thms take_con_rules}
@ take_Suc_thms @ deflation_thms @ deflation_take_thms;
val tac = simp_tac (HOL_basic_ss addsimps rules) 1;
in
Goal.prove_global thy [] [] goal (K tac)
end;
val take_apps = map2 prove_take_app con_consts spec;
in
yield_singleton Global_Theory.add_thmss
((Binding.qualified true "take_rews" dbind, take_apps),
[Simplifier.simp_add]) thy
end;
in
fold_map prove_take_apps
(specs ~~ take_consts ~~ constr_infos) thy
end;
(* ----- general proofs ----------------------------------------------------- *)
val all2E = @{lemma "!x y . P x y ==> (P x y ==> R) ==> R" by simp}
(******************************************************************************)
(****************************** induction rules *******************************)
(******************************************************************************)
fun prove_induction
(comp_dbind : binding, eqs : eq list)
(constr_infos : Domain_Constructors.constr_info list)
(take_info : Domain_Take_Proofs.take_induct_info)
(take_rews : thm list)
(thy : theory) =
let
val comp_dname = Sign.full_name thy comp_dbind;
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;
val iso_infos = map #iso_info constr_infos;
val axs_rep_iso = map #rep_inverse iso_infos;
val axs_abs_iso = map #abs_inverse iso_infos;
val exhausts = map #exhaust constr_infos;
val con_rews = maps #con_rews constr_infos;
val {take_consts, ...} = take_info;
val {take_0_thms, take_Suc_thms, chain_take_thms, ...} = take_info;
val {lub_take_thms, finite_defs, reach_thms, ...} = take_info;
val {take_induct_thms, ...} = take_info;
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
fun rec_to ns lazy_rec (n,cons) = forall (exists (fn arg =>
is_rec arg andalso not (member (op =) ns (rec_of arg)) andalso
((rec_of arg = n andalso not (lazy_rec orelse is_lazy arg)) orelse
rec_of arg <> n andalso rec_to (rec_of arg::ns)
(lazy_rec orelse is_lazy arg) (n, (List.nth(conss,rec_of arg))))
) o snd) cons;
fun warn (n,cons) =
if rec_to [] false (n,cons)
then (warning ("domain "^List.nth(dnames,n)^" is empty!"); true)
else false;
in
val n__eqs = mapn (fn n => fn (_,cons) => (n,cons)) 0 eqs;
val is_emptys = map warn n__eqs;
val is_finite = #is_finite take_info;
val _ = if is_finite
then message ("Proving finiteness rule for domain "^comp_dname^" ...")
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, exhaust) =
res_inst_tac context [(("y", 0), "x")] exhaust 1 ::
asm_simp_tac (take_ss addsimps prems) 1 ::
maps con_tacs cons;
in
tacs1 @ maps cases_tacs (conss ~~ exhausts)
end;
in pg'' thy [] goal tacf end;
(* ----- theorems concerning finiteness and induction ----------------------- *)
val global_ctxt = ProofContext.init_global thy;
val _ = trace " Proving ind...";
val ind =
if is_finite
then (* finite case *)
let
fun concf n dn = %:(P_name n) $ %:(x_name n);
fun tacf {prems, context} =
let
fun finite_tacs (take_induct, fin_ind) = [
rtac take_induct 1,
rtac fin_ind 1,
ind_prems_tac prems];
in
TRY (safe_tac HOL_cs) ::
maps finite_tacs (take_induct_thms ~~ atomize global_ctxt finite_ind)
end;
in pg'' thy [] (ind_term concf) tacf end
else (* infinite case *)
let
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
val Ts = map (Type o fst) eqs;
val P_names = Datatype_Prop.indexify_names (map (K "P") dnames);
val x_names = Datatype_Prop.indexify_names (map (K "x") dnames);
val P_types = map (fn T => T --> HOLogic.boolT) Ts;
val Ps = map Free (P_names ~~ P_types);
val xs = map Free (x_names ~~ Ts);
val n = Free ("n", HOLogic.natT);
val goals =
map (fn ((P,t),x) => P $ HOLCF_Library.mk_capply (t $ n, x))
(Ps ~~ take_consts ~~ xs);
in
HOLogic.mk_Trueprop
(HOLogic.mk_all ("n", HOLogic.natT, foldr1 HOLogic.mk_conj goals))
end;
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) reach_thms @ [
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 chain_take_thms 1,
asm_simp_tac HOL_basic_ss 1])
]
end;
in pg'' thy [] goal tacf end;
val case_ns =
let
val adms =
if is_finite then [] else
if length dnames = 1 then ["adm"] else
map (fn s => "adm_" ^ Long_Name.base_name s) dnames;
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 adms @ flat (map2 one_eq bottoms eqs) end;
val inducts = Project_Rule.projections (ProofContext.init_global thy) ind;
fun ind_rule (dname, rule) =
((Binding.empty, [rule]),
[Rule_Cases.case_names case_ns, Induct.induct_type dname]);
in
thy
|> snd o Global_Theory.add_thmss [
((Binding.qualified true "finite_induct" comp_dbind, [finite_ind]), []),
((Binding.qualified true "induct" comp_dbind, [ind] ), [])]
|> (snd o Global_Theory.add_thmss (map ind_rule (dnames ~~ inducts)))
end; (* prove_induction *)
(******************************************************************************)
(************************ bisimulation and coinduction ************************)
(******************************************************************************)
fun prove_coinduction
(comp_dbind : binding, eqs : eq list)
(take_rews : thm list)
(take_lemmas : thm list)
(thy : theory) : theory =
let
val dnames = map (fst o fst) eqs;
val comp_dname = Sign.full_name thy comp_dbind;
fun dc_take dn = %%:(dn^"_take");
val x_name = idx_name dnames "x";
val n_eqs = length eqs;
(* ----- 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.suffix_name "_bisim" comp_dbind;
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_global thy)) (intern_term thy t);
fun legacy_infer_prop thy t = legacy_infer_term thy (Type.constraint propT t);
fun infer_props thy = map (apsnd (legacy_infer_prop thy));
fun add_defs_i x = Global_Theory.add_defs false (map Thm.no_attributes x);
fun add_defs_infer defs thy = add_defs_i (infer_props thy defs) thy;
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) =
yield_singleton add_defs_infer
(Binding.qualified true "bisim_def" comp_dbind, bisim_eqn) thy;
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 |> snd o Global_Theory.add_thmss
[((Binding.qualified true "coinduct" comp_dbind, [coind]), [])]
end; (* let *)
(******************************************************************************)
(******************************* main function ********************************)
(******************************************************************************)
fun comp_theorems
(comp_dbind : binding, eqs : eq list)
(specs : (binding * (binding * (bool * binding option * typ) list * mixfix) list) list)
(take_info : Domain_Take_Proofs.take_induct_info)
(constr_infos : Domain_Constructors.constr_info list)
(thy : theory) =
let
val map_tab = Domain_Take_Proofs.get_map_tab thy;
val dnames = map (fst o fst) eqs;
val comp_dname = Sign.full_name thy comp_dbind;
(* ----- 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 *)
val (take_rewss, thy) =
take_theorems specs take_info constr_infos thy;
val {take_lemma_thms, take_0_thms, take_strict_thms, ...} = take_info;
val take_rews = take_0_thms @ take_strict_thms @ flat take_rewss;
(* prove induction rules, unless definition is indirect recursive *)
val thy =
if is_indirect then thy else
prove_induction (comp_dbind, eqs) constr_infos take_info take_rews thy;
val thy =
if is_indirect then thy else
prove_coinduction (comp_dbind, eqs) take_rews take_lemma_thms thy;
in
(take_rews, thy)
end; (* let *)
end; (* struct *)