--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOLCF/Tools/Domain/domain_induction.ML Tue Oct 19 15:01:51 2010 -0700
@@ -0,0 +1,438 @@
+(* Title: HOLCF/Tools/Domain/domain_induction.ML
+ Author: David von Oheimb
+ Author: Brian Huffman
+
+Proofs of high-level (co)induction rules for domain command.
+*)
+
+signature DOMAIN_INDUCTION =
+sig
+ val comp_theorems :
+ binding -> binding 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_Induction :> DOMAIN_INDUCTION =
+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 HOLCF_Library;
+
+(******************************************************************************)
+(***************************** proofs about take ******************************)
+(******************************************************************************)
+
+fun take_theorems
+ (dbinds : binding list)
+ (take_info : Domain_Take_Proofs.take_induct_info)
+ (constr_infos : Domain_Constructors.constr_info list)
+ (thy : theory) : thm list list * theory =
+let
+ 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, take_const), constr_info) thy =
+ let
+ val {iso_info, con_specs, con_betas, ...} = constr_info;
+ val {abs_inverse, ...} = iso_info;
+ fun prove_take_app (con_const, args) =
+ let
+ val Ts = map snd 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 = map prove_take_app con_specs;
+ 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
+ (dbinds ~~ take_consts ~~ constr_infos) thy
+end;
+
+(******************************************************************************)
+(****************************** induction rules *******************************)
+(******************************************************************************)
+
+val case_UU_allI =
+ @{lemma "(!!x. x ~= UU ==> P x) ==> P UU ==> ALL x. P x" by metis};
+
+fun prove_induction
+ (comp_dbind : binding)
+ (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 = Binding.name_of comp_dbind;
+
+ val iso_infos = map #iso_info constr_infos;
+ val exhausts = map #exhaust constr_infos;
+ val con_rews = maps #con_rews constr_infos;
+ val {take_consts, take_induct_thms, ...} = take_info;
+
+ val newTs = map #absT iso_infos;
+ val P_names = Datatype_Prop.indexify_names (map (K "P") newTs);
+ val x_names = Datatype_Prop.indexify_names (map (K "x") newTs);
+ val P_types = map (fn T => T --> HOLogic.boolT) newTs;
+ val Ps = map Free (P_names ~~ P_types);
+ val xs = map Free (x_names ~~ newTs);
+ val n = Free ("n", HOLogic.natT);
+
+ fun con_assm defined p (con, args) =
+ let
+ val Ts = map snd args;
+ val ns = Name.variant_list P_names (Datatype_Prop.make_tnames Ts);
+ val vs = map Free (ns ~~ Ts);
+ val nonlazy = map snd (filter_out (fst o fst) (args ~~ vs));
+ fun ind_hyp (v, T) t =
+ case AList.lookup (op =) (newTs ~~ Ps) T of NONE => t
+ | SOME p' => Logic.mk_implies (mk_trp (p' $ v), t);
+ val t1 = mk_trp (p $ list_ccomb (con, vs));
+ val t2 = fold_rev ind_hyp (vs ~~ Ts) t1;
+ val t3 = Logic.list_implies (map (mk_trp o mk_defined) nonlazy, t2);
+ in fold_rev Logic.all vs (if defined then t3 else t2) end;
+ fun eq_assms ((p, T), cons) =
+ mk_trp (p $ HOLCF_Library.mk_bottom T) :: map (con_assm true p) cons;
+ val assms = maps eq_assms (Ps ~~ newTs ~~ map #con_specs constr_infos);
+
+ val take_ss = HOL_ss addsimps (@{thm Rep_CFun_strict1} :: take_rews);
+ fun quant_tac ctxt i = EVERY
+ (map (fn name => res_inst_tac ctxt [(("x", 0), name)] spec i) x_names);
+
+ (* FIXME: move this message to domain_take_proofs.ML *)
+ val is_finite = #is_finite take_info;
+ val _ = if is_finite
+ then message ("Proving finiteness rule for domain "^comp_dname^" ...")
+ else ();
+
+ val _ = trace " Proving finite_ind...";
+ val finite_ind =
+ let
+ val concls =
+ map (fn ((P, t), x) => P $ mk_capply (t $ n, x))
+ (Ps ~~ take_consts ~~ xs);
+ val goal = mk_trp (foldr1 mk_conj concls);
+
+ fun tacf {prems, context} =
+ let
+ (* Prove stronger prems, without definedness side conditions *)
+ fun con_thm p (con, args) =
+ let
+ val subgoal = con_assm false p (con, args);
+ val rules = prems @ con_rews @ simp_thms;
+ val simplify = asm_simp_tac (HOL_basic_ss addsimps rules);
+ fun arg_tac (lazy, _) =
+ rtac (if lazy then allI else case_UU_allI) 1;
+ val tacs =
+ rewrite_goals_tac @{thms atomize_all atomize_imp} ::
+ map arg_tac args @
+ [REPEAT (rtac impI 1), ALLGOALS simplify];
+ in
+ Goal.prove context [] [] subgoal (K (EVERY tacs))
+ end;
+ fun eq_thms (p, cons) = map (con_thm p) cons;
+ val conss = map #con_specs constr_infos;
+ val prems' = maps eq_thms (Ps ~~ conss);
+
+ 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 con_tac _ =
+ asm_simp_tac take_ss 1 THEN
+ (resolve_tac prems' THEN_ALL_NEW etac spec) 1;
+ fun cases_tacs (cons, exhaust) =
+ res_inst_tac context [(("y", 0), "x")] exhaust 1 ::
+ asm_simp_tac (take_ss addsimps prems) 1 ::
+ map con_tac cons;
+ val tacs = tacs1 @ maps cases_tacs (conss ~~ exhausts)
+ in
+ EVERY (map DETERM tacs)
+ end;
+ in Goal.prove_global thy [] assms goal tacf end;
+
+ val _ = trace " Proving ind...";
+ val ind =
+ let
+ val concls = map (op $) (Ps ~~ xs);
+ val goal = mk_trp (foldr1 mk_conj concls);
+ val adms = if is_finite then [] else map (mk_trp o mk_adm) Ps;
+ fun tacf {prems, context} =
+ let
+ fun finite_tac (take_induct, fin_ind) =
+ rtac take_induct 1 THEN
+ (if is_finite then all_tac else resolve_tac prems 1) THEN
+ (rtac fin_ind THEN_ALL_NEW solve_tac prems) 1;
+ val fin_inds = Project_Rule.projections context finite_ind;
+ in
+ TRY (safe_tac HOL_cs) THEN
+ EVERY (map finite_tac (take_induct_thms ~~ fin_inds))
+ end;
+ in Goal.prove_global thy [] (adms @ assms) goal tacf end
+
+ (* case names for induction rules *)
+ val dnames = map (fst o dest_Type) newTs;
+ 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 constr_info =
+ let fun name_of (c, args) = Long_Name.base_name (fst (dest_Const c));
+ in bot :: map name_of (#con_specs constr_info) end;
+ in adms @ flat (map2 one_eq bottoms constr_infos) 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_thms [
+ ((Binding.qualified true "finite_induct" comp_dbind, finite_ind), []),
+ ((Binding.qualified true "induct" comp_dbind, ind ), [])]
+ |> (snd o Global_Theory.add_thms (map ind_rule (dnames ~~ inducts)))
+end; (* prove_induction *)
+
+(******************************************************************************)
+(************************ bisimulation and coinduction ************************)
+(******************************************************************************)
+
+fun prove_coinduction
+ (comp_dbind : binding, dbinds : binding list)
+ (constr_infos : Domain_Constructors.constr_info list)
+ (take_info : Domain_Take_Proofs.take_induct_info)
+ (take_rews : thm list list)
+ (thy : theory) : theory =
+let
+ val iso_infos = map #iso_info constr_infos;
+ val newTs = map #absT iso_infos;
+
+ val {take_consts, take_0_thms, take_lemma_thms, ...} = take_info;
+
+ val R_names = Datatype_Prop.indexify_names (map (K "R") newTs);
+ val R_types = map (fn T => T --> T --> boolT) newTs;
+ val Rs = map Free (R_names ~~ R_types);
+ val n = Free ("n", natT);
+ val reserved = "x" :: "y" :: R_names;
+
+ (* declare bisimulation predicate *)
+ val bisim_bind = Binding.suffix_name "_bisim" comp_dbind;
+ val bisim_type = R_types ---> boolT;
+ val (bisim_const, thy) =
+ Sign.declare_const ((bisim_bind, bisim_type), NoSyn) thy;
+
+ (* define bisimulation predicate *)
+ local
+ fun one_con T (con, args) =
+ let
+ val Ts = map snd args;
+ val ns1 = Name.variant_list reserved (Datatype_Prop.make_tnames Ts);
+ val ns2 = map (fn n => n^"'") ns1;
+ val vs1 = map Free (ns1 ~~ Ts);
+ val vs2 = map Free (ns2 ~~ Ts);
+ val eq1 = mk_eq (Free ("x", T), list_ccomb (con, vs1));
+ val eq2 = mk_eq (Free ("y", T), list_ccomb (con, vs2));
+ fun rel ((v1, v2), T) =
+ case AList.lookup (op =) (newTs ~~ Rs) T of
+ NONE => mk_eq (v1, v2) | SOME r => r $ v1 $ v2;
+ val eqs = foldr1 mk_conj (map rel (vs1 ~~ vs2 ~~ Ts) @ [eq1, eq2]);
+ in
+ Library.foldr mk_ex (vs1 @ vs2, eqs)
+ end;
+ fun one_eq ((T, R), cons) =
+ let
+ val x = Free ("x", T);
+ val y = Free ("y", T);
+ val disj1 = mk_conj (mk_eq (x, mk_bottom T), mk_eq (y, mk_bottom T));
+ val disjs = disj1 :: map (one_con T) cons;
+ in
+ mk_all (x, mk_all (y, mk_imp (R $ x $ y, foldr1 mk_disj disjs)))
+ end;
+ val conjs = map one_eq (newTs ~~ Rs ~~ map #con_specs constr_infos);
+ val bisim_rhs = lambdas Rs (Library.foldr1 mk_conj conjs);
+ val bisim_eqn = Logic.mk_equals (bisim_const, bisim_rhs);
+ in
+ val (bisim_def_thm, thy) = thy |>
+ yield_singleton (Global_Theory.add_defs false)
+ ((Binding.qualified true "bisim_def" comp_dbind, bisim_eqn), []);
+ end (* local *)
+
+ (* prove coinduction lemma *)
+ val coind_lemma =
+ let
+ val assm = mk_trp (list_comb (bisim_const, Rs));
+ fun one ((T, R), take_const) =
+ let
+ val x = Free ("x", T);
+ val y = Free ("y", T);
+ val lhs = mk_capply (take_const $ n, x);
+ val rhs = mk_capply (take_const $ n, y);
+ in
+ mk_all (x, mk_all (y, mk_imp (R $ x $ y, mk_eq (lhs, rhs))))
+ end;
+ val goal =
+ mk_trp (foldr1 mk_conj (map one (newTs ~~ Rs ~~ take_consts)));
+ val rules = @{thm Rep_CFun_strict1} :: take_0_thms;
+ fun tacf {prems, context} =
+ let
+ val prem' = rewrite_rule [bisim_def_thm] (hd prems);
+ val prems' = Project_Rule.projections context prem';
+ val dests = map (fn th => th RS spec RS spec RS mp) prems';
+ fun one_tac (dest, rews) =
+ dtac dest 1 THEN safe_tac HOL_cs THEN
+ ALLGOALS (asm_simp_tac (HOL_basic_ss addsimps rews));
+ in
+ rtac @{thm nat.induct} 1 THEN
+ simp_tac (HOL_ss addsimps rules) 1 THEN
+ safe_tac HOL_cs THEN
+ EVERY (map one_tac (dests ~~ take_rews))
+ end
+ in
+ Goal.prove_global thy [] [assm] goal tacf
+ end;
+
+ (* prove individual coinduction rules *)
+ fun prove_coind ((T, R), take_lemma) =
+ let
+ val x = Free ("x", T);
+ val y = Free ("y", T);
+ val assm1 = mk_trp (list_comb (bisim_const, Rs));
+ val assm2 = mk_trp (R $ x $ y);
+ val goal = mk_trp (mk_eq (x, y));
+ fun tacf {prems, context} =
+ let
+ val rule = hd prems RS coind_lemma;
+ in
+ rtac take_lemma 1 THEN
+ asm_simp_tac (HOL_basic_ss addsimps (rule :: prems)) 1
+ end;
+ in
+ Goal.prove_global thy [] [assm1, assm2] goal tacf
+ end;
+ val coinds = map prove_coind (newTs ~~ Rs ~~ take_lemma_thms);
+ val coind_binds = map (Binding.qualified true "coinduct") dbinds;
+
+in
+ thy |> snd o Global_Theory.add_thms
+ (map Thm.no_attributes (coind_binds ~~ coinds))
+end; (* let *)
+
+(******************************************************************************)
+(******************************* main function ********************************)
+(******************************************************************************)
+
+fun comp_theorems
+ (comp_dbind : binding)
+ (dbinds : binding list)
+ (take_info : Domain_Take_Proofs.take_induct_info)
+ (constr_infos : Domain_Constructors.constr_info list)
+ (thy : theory) =
+let
+val comp_dname = Binding.name_of comp_dbind;
+
+(* Test for emptiness *)
+(* FIXME: reimplement emptiness test
+local
+ open Domain_Library;
+ val dnames = map (fst o fst) eqs;
+ val conss = map snd eqs;
+ 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;
+end;
+*)
+
+(* Test for indirect recursion *)
+local
+ val newTs = map (#absT o #iso_info) constr_infos;
+ fun indirect_typ (Type (_, Ts)) =
+ exists (fn T => member (op =) newTs T orelse indirect_typ T) Ts
+ | indirect_typ _ = false;
+ fun indirect_arg (_, T) = indirect_typ T;
+ fun indirect_con (_, args) = exists indirect_arg args;
+ fun indirect_eq cons = exists indirect_con cons;
+in
+ val is_indirect = exists indirect_eq (map #con_specs constr_infos);
+ 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 dbinds 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 constr_infos take_info take_rews thy;
+
+val thy =
+ if is_indirect then thy else
+ prove_coinduction (comp_dbind, dbinds) constr_infos take_info take_rewss thy;
+
+in
+ (take_rews, thy)
+end; (* let *)
+end; (* struct *)