(* Title: HOL/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 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
val quiet_mode = Unsynchronized.reffalse val trace_domain = Unsynchronized.reffalse
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 listlist * 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>\<open>nat\<close>) val n' = \<^Const>\Suc for 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_>\<open>ID _\<close> = true
| is_ID _ = false in fun map_of_arg thy v T = letval m = Domain_Take_Proofs.map_of_typ thy subs T inif 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, ...} : Domain_Constructors.constr_info = 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"] (Case_Translation.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 thy) 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 fun tac ctxt = simp_tac (put_simpset HOL_basic_ss ctxt addsimps rules) 1 in
Goal.prove_global thy [] [] goal (tac o #context) end val take_apps = map prove_take_app con_specs in
yield_singleton Global_Theory.add_thmss
((Binding.qualify_name true dbind "take_rews", take_apps),
[Simplifier.simp_add]) thy end in
fold_map prove_take_apps
(dbinds ~~ take_consts ~~ constr_infos) thy end
val case_UU_allI =
@{lemma "(\x. x \ UU \ P x) \ P UU \ \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 = Case_Translation.indexify_names (map (K "P") newTs) val x_names = Case_Translation.indexify_names (map (K "x") newTs) val P_types = map (fn T => T --> \<^Type>\<open>bool\<close>) newTs val Ps = map Free (P_names ~~ P_types) val xs = map Free (x_names ~~ newTs) val n = Free ("n", \<^Type>\<open>nat\<close>)
fun con_assm defined p (con, args) = let val Ts = map snd args val ns = Name.variant_list P_names (Case_Translation.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)
fun quant_tac ctxt i = EVERY
(map (fn name =>
Rule_Insts.res_inst_tac ctxt [((("x", 0), Position.none), 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 = ctxt} = let val take_ctxt = put_simpset HOL_ss ctxt addsimps (@{thm Rep_cfun_strict1} :: take_rews)
(* 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 @ @{thms simp_thms} val simplify = asm_simp_tac (put_simpset HOL_basic_ss ctxt addsimps rules) fun arg_tac (lazy, _) =
resolve_tac ctxt [if lazy then allI else case_UU_allI] 1 fun tacs ctxt =
rewrite_goals_tac ctxt @{thms atomize_all atomize_imp} :: map arg_tac args @
[REPEAT (resolve_tac ctxt [impI] 1), ALLGOALS simplify] in
Goal.prove ctxt [] [] subgoal (EVERY o tacs o #context) 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 ctxt 1,
simp_tac (put_simpset HOL_ss ctxt) 1,
Induct_Tacs.induct_tac ctxt [[SOME "n"]] NONE 1,
simp_tac (take_ctxt addsimps prems) 1, TRY (safe_tac (put_claset HOL_cs ctxt))] fun con_tac _ =
asm_simp_tac take_ctxt 1 THEN
(resolve_tac ctxt prems' THEN_ALL_NEW eresolve_tac ctxt [spec]) 1 fun cases_tacs (cons, exhaust) =
Rule_Insts.res_inst_tac ctxt [((("y", 0), Position.none), "x")] [] exhaust 1 ::
asm_simp_tac (take_ctxt 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 [] elsemap (mk_trp o mk_adm) Ps fun tacf {prems, context = ctxt} = let fun finite_tac (take_induct, fin_ind) =
resolve_tac ctxt [take_induct] 1 THEN
(if is_finite then all_tac else resolve_tac ctxt prems 1) THEN
(resolve_tac ctxt [fin_ind] THEN_ALL_NEW solve_tac ctxt prems) 1 val fin_inds = Project_Rule.projections ctxt finite_ind in TRY (safe_tac (put_claset HOL_cs ctxt)) 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 dest_Type_name 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 : Domain_Constructors.constr_info) = letfun name_of (c, _) = Long_Name.base_name (dest_Const_name 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 (Proof_Context.init_global thy) ind fun ind_rule (dname, rule) =
((Binding.empty, rule),
[Rule_Cases.case_names case_ns, Induct.induct_type dname])
(******************************************************************************) (************************ 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 listlist)
(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 = Case_Translation.indexify_names (map (K "R") newTs) val R_types = map (fn T => T --> T --> \<^Type>\<open>bool\<close>) newTs val Rs = map Free (R_names ~~ R_types) val n = Free ("n", \<^Type>\<open>nat\<close>) val reserved = "x" :: "y" :: R_names
(* declare bisimulation predicate *) val bisim_bind = Binding.suffix_name "_bisim" comp_dbind val bisim_type = R_types ---> \<^Type>\<open>bool\<close> val (bisim_const, thy) =
Sign.declare_const_global ((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 (Case_Translation.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 |>
Global_Theory.add_def (Binding.qualify_name true comp_dbind "bisim_def", 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 = ctxt} = let val prem' = rewrite_rule ctxt [bisim_def_thm] (hd prems) val prems' = Project_Rule.projections ctxt prem' val dests = map (fn th => th RS spec RS spec RS mp) prems' fun one_tac (dest, rews) =
dresolve_tac ctxt [dest] 1 THEN safe_tac (put_claset HOL_cs ctxt) THEN
ALLGOALS (asm_simp_tac (put_simpset HOL_basic_ss ctxt addsimps rews)) in
resolve_tac ctxt @{thms nat.induct} 1 THEN
simp_tac (put_simpset HOL_ss ctxt addsimps rules) 1 THEN
safe_tac (put_claset HOL_cs ctxt) 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 = ctxt} = let val rule = hd prems RS coind_lemma in
resolve_tac ctxt [take_lemma] 1 THEN
asm_simp_tac (put_simpset HOL_basic_ss ctxt 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 (fn b => Binding.qualify_name true b "coinduct") dbinds
in
thy |> snd o Global_Theory.add_thms
(map Thm.no_attributes (coind_binds ~~ coinds)) end(* let *)
(******************************************************************************) (******************************* main function ********************************) (******************************************************************************)
fun comp_theorems
(dbinds : binding list)
(take_info : Domain_Take_Proofs.take_induct_info)
(constr_infos : Domain_Constructors.constr_info list)
(thy : theory) = let
val comp_dbind = Binding.conglomerate dbinds 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, nth conss (rec_of arg))) ) o snd) cons fun warn (n,cons) = if rec_to [] false (n,cons) then (warning ("domain " ^ 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_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 *)
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