(* Title: HOLCF/Tools/domain/domain_theorems.ML ID: $Id: domain_theorems.ML,v 1.5 2007/10/21 12:21:53 wenzelm Exp $ Author: David von Oheimb New proofs/tactics by Brian Huffman Proof generator for domain command. *) val HOLCF_ss = simpset(); structure Domain_Theorems = struct local val adm_impl_admw = thm "adm_impl_admw"; val antisym_less_inverse = thm "antisym_less_inverse"; val beta_cfun = thm "beta_cfun"; val cfun_arg_cong = thm "cfun_arg_cong"; val ch2ch_Rep_CFunL = thm "ch2ch_Rep_CFunL"; val ch2ch_Rep_CFunR = thm "ch2ch_Rep_CFunR"; val chain_iterate = thm "chain_iterate"; val compact_ONE = thm "compact_ONE"; val compact_sinl = thm "compact_sinl"; val compact_sinr = thm "compact_sinr"; val compact_spair = thm "compact_spair"; val compact_up = thm "compact_up"; val contlub_cfun_arg = thm "contlub_cfun_arg"; val contlub_cfun_fun = thm "contlub_cfun_fun"; val fix_def2 = thm "fix_def2"; val injection_eq = thm "injection_eq"; val injection_less = thm "injection_less"; val lub_equal = thm "lub_equal"; val monofun_cfun_arg = thm "monofun_cfun_arg"; val retraction_strict = thm "retraction_strict"; val spair_eq = thm "spair_eq"; val spair_less = thm "spair_less"; val sscase1 = thm "sscase1"; val ssplit1 = thm "ssplit1"; val strictify1 = thm "strictify1"; val wfix_ind = thm "wfix_ind"; 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 = rewrite_goals_tac defs THEN EVERY (tacs (map (rewrite_rule defs) prems)); in Goal.prove_global thy [] asms prop tac end; fun pg' thy defs t tacsf = let fun tacs [] = tacsf | tacs prems = cut_facts_tac prems 1 :: tacsf; in pg'' thy defs t tacs end; fun case_UU_tac rews i v = case_tac (v^"=UU") i THEN asm_simp_tac (HOLCF_ss addsimps rews) i; val chain_tac = REPEAT_DETERM o resolve_tac [chain_iterate, ch2ch_Rep_CFunR, ch2ch_Rep_CFunL]; (* ----- general proofs ----------------------------------------------------- *) val all2E = prove_goal HOL.thy "[| !x y . P x y; P x y ==> R |] ==> R" (fn prems =>[ resolve_tac prems 1, cut_facts_tac prems 1, fast_tac HOL_cs 1]); val dist_eqI = prove_goal (the_context ()) "!!x::'a::po. ~ x << y ==> x ~= y" (fn prems => [blast_tac (claset () addDs [antisym_less_inverse]) 1]); in fun theorems (((dname, _), cons) : eq, eqs : eq list) thy = let val dummy = writeln ("Proving isomorphism properties of domain "^dname^" ..."); val pg = pg' thy; (* ----- getting the axioms and definitions --------------------------------- *) local fun ga s dn = get_thm thy (Name (dn ^ "." ^ s)); in val ax_abs_iso = ga "abs_iso" dname; val ax_rep_iso = ga "rep_iso" dname; val ax_when_def = ga "when_def" dname; fun get_def mk_name (con,_) = ga (mk_name con^"_def") dname; val axs_con_def = map (get_def extern_name) cons; val axs_dis_def = map (get_def dis_name) cons; val axs_mat_def = map (get_def mat_name) cons; val axs_pat_def = map (get_def pat_name) cons; val axs_sel_def = let fun def_of_sel sel = ga (sel^"_def") dname; fun def_of_arg arg = Option.map def_of_sel (sel_of arg); fun defs_of_con (_, args) = List.mapPartial def_of_arg args; in List.concat (map defs_of_con cons) end; val ax_copy_def = ga "copy_def" dname; end; (* local *) (* ----- theorems concerning the isomorphism -------------------------------- *) val dc_abs = %%:(dname^"_abs"); val dc_rep = %%:(dname^"_rep"); val dc_copy = %%:(dname^"_copy"); val x_name = "x"; val iso_locale = iso_intro OF [ax_abs_iso, ax_rep_iso]; val abs_strict = ax_rep_iso RS (allI RS retraction_strict); val rep_strict = ax_abs_iso RS (allI RS retraction_strict); val abs_defin' = iso_locale RS iso_abs_defin'; val rep_defin' = iso_locale RS iso_rep_defin'; val iso_rews = map standard [ax_abs_iso,ax_rep_iso,abs_strict,rep_strict]; (* ----- generating beta reduction rules from definitions-------------------- *) local fun arglist (Const _ $ Abs (s, _, t)) = let val (vars,body) = arglist t; in (s :: vars, body) end | arglist t = ([], t); fun bind_fun vars t = Library.foldr mk_All (vars, t); fun bound_vars 0 = [] | bound_vars i = Bound (i-1) :: bound_vars (i - 1); in fun appl_of_def def = let val (_ $ con $ lam) = concl_of def; val (vars, rhs) = arglist lam; val lhs = list_ccomb (con, bound_vars (length vars)); val appl = bind_fun vars (lhs == rhs); val cs = ContProc.cont_thms lam; val betas = map (fn c => mk_meta_eq (c RS beta_cfun)) cs; in pg (def::betas) appl [rtac reflexive_thm 1] end; end; val when_appl = appl_of_def ax_when_def; val con_appls = map appl_of_def axs_con_def; local fun arg2typ n arg = let val t = TVar (("'a", n), pcpoS) in (n + 1, if is_lazy arg then mk_uT t else t) end; fun args2typ n [] = (n, oneT) | args2typ n [arg] = arg2typ n arg | args2typ n (arg::args) = let val (n1, t1) = arg2typ n arg; val (n2, t2) = args2typ n1 args in (n2, mk_sprodT (t1, t2)) end; fun cons2typ n [] = (n,oneT) | cons2typ n [con] = args2typ n (snd con) | cons2typ n (con::cons) = let val (n1, t1) = args2typ n (snd con); val (n2, t2) = cons2typ n1 cons in (n2, mk_ssumT (t1, t2)) end; in fun cons2ctyp cons = ctyp_of thy (snd (cons2typ 1 cons)); end; local val iso_swap = iso_locale RS iso_iso_swap; fun one_con (con, args) = let val vns = map vname args; val eqn = %:x_name === con_app2 con %: vns; val conj = foldr1 mk_conj (eqn :: map (defined o %:) (nonlazy args)); in Library.foldr mk_ex (vns, conj) end; val conj_assoc = @{thm conj_assoc}; val exh = foldr1 mk_disj ((%:x_name === UU) :: map one_con cons); val thm1 = instantiate' [SOME (cons2ctyp cons)] [] exh_start; val thm2 = rewrite_rule (map mk_meta_eq ex_defined_iffs) thm1; val thm3 = rewrite_rule [mk_meta_eq conj_assoc] thm2; (* first 3 rules replace "x = UU \/ P" with "rep$x = UU \/ P" *) val tacs = [ rtac disjE 1, etac (rep_defin' RS disjI1) 2, etac disjI2 2, rewrite_goals_tac [mk_meta_eq iso_swap], rtac thm3 1]; in val exhaust = pg con_appls (mk_trp exh) tacs; val casedist = standard (rewrite_rule exh_casedists (exhaust RS exh_casedist0)); end; local fun bind_fun t = Library.foldr mk_All (when_funs cons, t); fun bound_fun i _ = Bound (length cons - i); val when_app = list_ccomb (%%:(dname^"_when"), mapn bound_fun 1 cons); in val when_strict = let val axs = [when_appl, mk_meta_eq rep_strict]; val goal = bind_fun (mk_trp (strict when_app)); val tacs = [resolve_tac [sscase1, ssplit1, strictify1] 1]; in pg axs goal tacs end; val when_apps = let fun one_when n (con,args) = let val axs = when_appl :: con_appls; val goal = bind_fun (lift_defined %: (nonlazy args, mk_trp (when_app`(con_app con args) === list_ccomb (bound_fun n 0, map %# args)))); val tacs = [asm_simp_tac (HOLCF_ss addsimps [ax_abs_iso]) 1]; in pg axs goal tacs end; in mapn one_when 1 cons end; end; val when_rews = when_strict :: when_apps; (* ----- theorems concerning the constructors, discriminators and selectors - *) local fun dis_strict (con, _) = let val goal = mk_trp (strict (%%:(dis_name con))); in pg axs_dis_def goal [rtac when_strict 1] end; fun dis_app c (con, args) = let val lhs = %%:(dis_name c) ` con_app con args; val rhs = %%:(if con = c then TT_N else FF_N); val goal = lift_defined %: (nonlazy args, mk_trp (lhs === rhs)); val tacs = [asm_simp_tac (HOLCF_ss addsimps when_rews) 1]; in pg axs_dis_def goal tacs end; val dis_apps = List.concat (map (fn (c,_) => map (dis_app c) cons) cons); fun dis_defin (con, args) = let val goal = defined (%:x_name) ==> defined (%%:(dis_name con) `% x_name); val tacs = [rtac casedist 1, contr_tac 1, DETERM_UNTIL_SOLVED (CHANGED (asm_simp_tac (HOLCF_ss addsimps dis_apps) 1))]; in pg [] goal tacs end; val dis_stricts = map dis_strict cons; val dis_defins = map dis_defin cons; in val dis_rews = dis_stricts @ dis_defins @ dis_apps; end; local fun mat_strict (con, _) = let val goal = mk_trp (strict (%%:(mat_name con))); val tacs = [rtac when_strict 1]; in pg axs_mat_def goal tacs end; val mat_stricts = map mat_strict cons; fun one_mat c (con, args) = let val lhs = %%:(mat_name c) ` con_app con args; val rhs = if con = c then %%:returnN ` mk_ctuple (map %# args) else %%:failN; val goal = lift_defined %: (nonlazy args, mk_trp (lhs === rhs)); val tacs = [asm_simp_tac (HOLCF_ss addsimps when_rews) 1]; in pg axs_mat_def goal tacs end; val mat_apps = List.concat (map (fn (c,_) => map (one_mat c) cons) cons); in val mat_rews = mat_stricts @ mat_apps; end; local fun ps args = mapn (fn n => fn _ => %:("pat" ^ string_of_int n)) 1 args; fun pat_lhs (con,args) = %%:branchN $ list_comb (%%:(pat_name con), ps args); fun pat_rhs (con,[]) = %%:returnN ` ((%:"rhs") ` HOLogic.unit) | pat_rhs (con,args) = (%%:branchN $ foldr1 cpair_pat (ps args)) `(%:"rhs")`(mk_ctuple (map %# args)); fun pat_strict c = let val axs = @{thm branch_def} :: axs_pat_def; val goal = mk_trp (strict (pat_lhs c ` (%:"rhs"))); val tacs = [simp_tac (HOLCF_ss addsimps [when_strict]) 1]; in pg axs goal tacs end; fun pat_app c (con, args) = let val axs = @{thm branch_def} :: axs_pat_def; val lhs = (pat_lhs c)`(%:"rhs")`(con_app con args); val rhs = if con = fst c then pat_rhs c else %%:failN; val goal = lift_defined %: (nonlazy args, mk_trp (lhs === rhs)); val tacs = [asm_simp_tac (HOLCF_ss addsimps when_rews) 1]; in pg axs goal tacs end; val pat_stricts = map pat_strict cons; val pat_apps = List.concat (map (fn c => map (pat_app c) cons) cons); in val pat_rews = pat_stricts @ pat_apps; end; local val rev_contrapos = @{thm rev_contrapos}; fun con_strict (con, args) = let fun one_strict vn = let fun f arg = if vname arg = vn then UU else %# arg; val goal = mk_trp (con_app2 con f args === UU); val tacs = [asm_simp_tac (HOLCF_ss addsimps [abs_strict]) 1]; in pg con_appls goal tacs end; in map one_strict (nonlazy args) end; fun con_defin (con, args) = let val concl = mk_trp (defined (con_app con args)); val goal = lift_defined %: (nonlazy args, concl); val tacs = [ rtac rev_contrapos 1, eres_inst_tac [("f",dis_name con)] cfun_arg_cong 1, asm_simp_tac (HOLCF_ss addsimps dis_rews) 1]; in pg [] goal tacs end; in val con_stricts = List.concat (map con_strict cons); val con_defins = map con_defin cons; val con_rews = con_stricts @ con_defins; end; local val rules = [compact_sinl, compact_sinr, compact_spair, compact_up, compact_ONE]; fun con_compact (con, args) = let val concl = mk_trp (%%:compactN $ con_app con args); val goal = lift (fn x => %%:compactN $ %#x) (args, concl); val tacs = [ rtac (iso_locale RS iso_compact_abs) 1, REPEAT (resolve_tac rules 1 ORELSE atac 1)]; in pg con_appls goal tacs end; in val con_compacts = map con_compact cons; end; local fun one_sel sel = pg axs_sel_def (mk_trp (strict (%%:sel))) [simp_tac (HOLCF_ss addsimps when_rews) 1]; fun sel_strict (_, args) = List.mapPartial (Option.map one_sel o sel_of) args; in val sel_stricts = List.concat (map sel_strict cons); end; local fun sel_app_same c n sel (con, args) = let val nlas = nonlazy args; val vns = map vname args; val vnn = List.nth (vns, n); val nlas' = List.filter (fn v => v <> vnn) nlas; val lhs = (%%:sel)`(con_app con args); val goal = lift_defined %: (nlas', mk_trp (lhs === %:vnn)); val tacs1 = if vnn mem nlas then [case_UU_tac (when_rews @ con_stricts) 1 vnn] else []; val tacs2 = [asm_simp_tac (HOLCF_ss addsimps when_rews) 1]; in pg axs_sel_def goal (tacs1 @ tacs2) end; fun sel_app_diff c n sel (con, args) = let val nlas = nonlazy args; val goal = mk_trp (%%:sel ` con_app con args === UU); val tacs1 = map (case_UU_tac (when_rews @ con_stricts) 1) nlas; val tacs2 = [asm_simp_tac (HOLCF_ss addsimps when_rews) 1]; in pg axs_sel_def goal (tacs1 @ tacs2) end; fun sel_app c n sel (con, args) = if con = c then sel_app_same c n sel (con, args) else sel_app_diff c n sel (con, args); fun one_sel c n sel = map (sel_app c n sel) cons; fun one_sel' c n arg = Option.map (one_sel c n) (sel_of arg); fun one_con (c, args) = List.concat (List.mapPartial I (mapn (one_sel' c) 0 args)); in val sel_apps = List.concat (map one_con cons); end; local fun sel_defin sel = let val goal = defined (%:x_name) ==> defined (%%:sel`%x_name); val tacs = [ rtac casedist 1, contr_tac 1, DETERM_UNTIL_SOLVED (CHANGED (asm_simp_tac (HOLCF_ss addsimps sel_apps) 1))]; in pg [] goal tacs end; in val sel_defins = if length cons = 1 then List.mapPartial (fn arg => Option.map sel_defin (sel_of arg)) (filter_out is_lazy (snd (hd cons))) else []; end; val sel_rews = sel_stricts @ sel_defins @ sel_apps; val rev_contrapos = @{thm rev_contrapos}; val distincts_le = let fun dist (con1, args1) (con2, args2) = let val goal = lift_defined %: (nonlazy args1, mk_trp (con_app con1 args1 ~<< con_app con2 args2)); val tacs = [ rtac rev_contrapos 1, eres_inst_tac [("f", dis_name con1)] monofun_cfun_arg 1] @ map (case_UU_tac (con_stricts @ dis_rews) 1) (nonlazy args2) @ [asm_simp_tac (HOLCF_ss addsimps dis_rews) 1]; in pg [] goal tacs end; fun distinct (con1, args1) (con2, args2) = let val arg1 = (con1, args1); val arg2 = (con2, ListPair.map (fn (arg,vn) => upd_vname (K vn) arg) (args2, Name.variant_list (map vname args1) (map vname args2))); in [dist arg1 arg2, dist arg2 arg1] end; fun distincts [] = [] | distincts (c::cs) = (map (distinct c) cs) :: distincts cs; in distincts cons end; val dist_les = List.concat (List.concat distincts_le); val dist_eqs = let fun distinct (_,args1) ((_,args2), leqs) = let val (le1,le2) = (hd leqs, hd(tl leqs)); val (eq1,eq2) = (le1 RS dist_eqI, le2 RS dist_eqI) in if nonlazy args1 = [] then [eq1, eq1 RS not_sym] else if nonlazy args2 = [] then [eq2, eq2 RS not_sym] else [eq1, eq2] end; fun distincts [] = [] | distincts ((c,leqs)::cs) = List.concat (ListPair.map (distinct c) ((map #1 cs),leqs)) @ distincts cs; in map standard (distincts (cons ~~ distincts_le)) end; local fun pgterm rel con args = let fun append s = upd_vname (fn v => v^s); val (largs, rargs) = (args, map (append "'") args); val concl = foldr1 mk_conj (ListPair.map rel (map %# largs, map %# rargs)); val prem = rel (con_app con largs, con_app con rargs); val sargs = case largs of [_] => [] | _ => nonlazy args; val prop = lift_defined %: (sargs, mk_trp (prem === concl)); in pg con_appls prop end; val cons' = List.filter (fn (_,args) => args<>[]) cons; in val inverts = let val abs_less = ax_abs_iso RS (allI RS injection_less); val tacs = [asm_full_simp_tac (HOLCF_ss addsimps [abs_less, spair_less]) 1]; in map (fn (con, args) => pgterm (op <<) con args tacs) cons' end; val injects = let val abs_eq = ax_abs_iso RS (allI RS injection_eq); val tacs = [asm_full_simp_tac (HOLCF_ss addsimps [abs_eq, spair_eq]) 1]; in map (fn (con, args) => pgterm (op ===) con args tacs) cons' end; end; (* ----- theorems concerning one induction step ----------------------------- *) val copy_strict = let val goal = mk_trp (strict (dc_copy `% "f")); val tacs = [asm_simp_tac (HOLCF_ss addsimps [abs_strict, when_strict]) 1]; in pg [ax_copy_def] goal tacs end; local fun copy_app (con, args) = let val lhs = dc_copy`%"f"`(con_app con args); val rhs = con_app2 con (app_rec_arg (cproj (%:"f") eqs)) args; val goal = lift_defined %: (nonlazy_rec args, mk_trp (lhs === rhs)); val args' = List.filter (fn a => not (is_rec a orelse is_lazy a)) args; val stricts = abs_strict::when_strict::con_stricts; val tacs1 = map (case_UU_tac stricts 1 o vname) args'; val tacs2 = [asm_simp_tac (HOLCF_ss addsimps when_apps) 1]; in pg [ax_copy_def] goal (tacs1 @ tacs2) end; in val copy_apps = map copy_app cons; end; local fun one_strict (con, args) = let val goal = mk_trp (dc_copy`UU`(con_app con args) === UU); val rews = copy_strict :: copy_apps @ con_rews; val tacs = map (case_UU_tac rews 1) (nonlazy args) @ [asm_simp_tac (HOLCF_ss addsimps rews) 1]; in pg [] goal tacs end; fun has_nonlazy_rec (_, args) = exists is_nonlazy_rec args; in val copy_stricts = map one_strict (List.filter has_nonlazy_rec cons); end; val copy_rews = copy_strict :: copy_apps @ copy_stricts; in thy |> Sign.add_path (Sign.base_name dname) |> (snd o (PureThy.add_thmss (map Thm.no_attributes [ ("iso_rews" , iso_rews ), ("exhaust" , [exhaust] ), ("casedist" , [casedist]), ("when_rews", when_rews ), ("compacts", con_compacts), ("con_rews", con_rews), ("sel_rews", sel_rews), ("dis_rews", dis_rews), ("pat_rews", pat_rews), ("dist_les", dist_les), ("dist_eqs", dist_eqs), ("inverts" , inverts ), ("injects" , injects ), ("copy_rews", copy_rews)]))) |> (snd o PureThy.add_thmss [(("match_rews", mat_rews), [Simplifier.simp_add])]) |> Sign.parent_path |> rpair (iso_rews @ when_rews @ con_rews @ sel_rews @ dis_rews @ pat_rews @ dist_les @ dist_eqs @ copy_rews) end; (* let *) fun comp_theorems (comp_dnam, eqs: eq list) thy = let val dnames = map (fst o fst) eqs; val conss = map snd eqs; val comp_dname = Sign.full_name thy comp_dnam; val d = writeln("Proving induction properties of domain "^comp_dname^" ..."); val pg = pg' thy; (* ----- getting the composite axiom and definitions ------------------------ *) local fun ga s dn = get_thm thy (Name (dn ^ "." ^ s)); in val axs_reach = map (ga "reach" ) dnames; val axs_take_def = map (ga "take_def" ) dnames; val axs_finite_def = map (ga "finite_def") dnames; val ax_copy2_def = ga "copy_def" comp_dnam; val ax_bisim_def = ga "bisim_def" comp_dnam; end; local fun gt s dn = get_thm thy (Name (dn ^ "." ^ s)); fun gts s dn = get_thms thy (Name (dn ^ "." ^ s)); in val cases = map (gt "casedist" ) dnames; val con_rews = List.concat (map (gts "con_rews" ) dnames); val copy_rews = List.concat (map (gts "copy_rews") dnames); end; fun dc_take dn = %%:(dn^"_take"); val x_name = idx_name dnames "x"; val P_name = idx_name dnames "P"; val n_eqs = length eqs; (* ----- theorems concerning finite approximation and finite induction ------ *) local val iterate_Cprod_ss = simpset_of (theory "Fix"); val copy_con_rews = copy_rews @ con_rews; val copy_take_defs = (if n_eqs = 1 then [] else [ax_copy2_def]) @ axs_take_def; val take_stricts = let fun one_eq ((dn, args), _) = strict (dc_take dn $ %:"n"); val goal = mk_trp (foldr1 mk_conj (map one_eq eqs)); val tacs = [ induct_tac "n" 1, simp_tac iterate_Cprod_ss 1, asm_simp_tac (iterate_Cprod_ss addsimps copy_rews) 1]; in pg copy_take_defs goal tacs end; val take_stricts' = rewrite_rule copy_take_defs take_stricts; fun take_0 n dn = let val goal = mk_trp ((dc_take dn $ %%:"HOL.zero") `% x_name n === UU); in pg axs_take_def goal [simp_tac iterate_Cprod_ss 1] end; val take_0s = mapn take_0 1 dnames; val c_UU_tac = case_UU_tac (take_stricts'::copy_con_rews) 1; val take_apps = let fun mk_eqn dn (con, args) = let fun mk_take n = dc_take (List.nth (dnames, n)) $ %:"n"; val lhs = (dc_take dn $ (%%:"Suc" $ %:"n"))`(con_app con args); val rhs = con_app2 con (app_rec_arg mk_take) args; in Library.foldr mk_all (map vname args, lhs === rhs) end; fun mk_eqns ((dn, _), cons) = map (mk_eqn dn) cons; val goal = mk_trp (foldr1 mk_conj (List.concat (map mk_eqns eqs))); val simps = List.filter (has_fewer_prems 1) copy_rews; fun con_tac (con, args) = if nonlazy_rec args = [] then all_tac else EVERY (map c_UU_tac (nonlazy_rec args)) THEN asm_full_simp_tac (HOLCF_ss addsimps copy_rews) 1; fun eq_tacs ((dn, _), cons) = map con_tac cons; val tacs = simp_tac iterate_Cprod_ss 1 :: induct_tac "n" 1 :: simp_tac (iterate_Cprod_ss addsimps copy_con_rews) 1 :: asm_full_simp_tac (HOLCF_ss addsimps simps) 1 :: TRY (safe_tac HOL_cs) :: List.concat (map eq_tacs eqs); in pg copy_take_defs goal tacs end; in val take_rews = map standard (atomize take_stricts @ take_0s @ atomize take_apps); end; (* local *) local fun one_con p (con,args) = let 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 (List.filter is_rec args, t1); val t3 = lift_defined (bound_arg (map vname args)) (nonlazy args, t2); in Library.foldr mk_All (map vname args, 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 take_rews; fun quant_tac i = EVERY (mapn (fn n => fn _ => res_inst_tac [("x", x_name n)] spec i) 1 dnames); fun ind_prems_tac prems = EVERY (List.concat (map (fn cons => (resolve_tac prems 1 :: List.concat (map (fn (_,args) => resolve_tac prems 1 :: map (K(atac 1)) (nonlazy args) @ map (K(atac 1)) (List.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; end; in (* local *) 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 = let val tacs1 = [ quant_tac 1, simp_tac HOL_ss 1, induct_tac "n" 1, simp_tac (take_ss addsimps prems) 1, TRY (safe_tac HOL_cs)]; fun arg_tac arg = case_UU_tac (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 (List.filter is_nonlazy_rec args) @ [resolve_tac prems 1] @ map (K (atac 1)) (nonlazy args) @ map (K (etac spec 1)) (List.filter is_rec args); fun cases_tacs (cons, cases) = res_inst_tac [("x","x")] cases 1 :: asm_simp_tac (take_ss addsimps prems) 1 :: List.concat (map con_tacs cons); in tacs1 @ List.concat (map cases_tacs (conss ~~ cases)) end; in pg'' thy [] goal tacf end; val take_lemmas = let fun take_lemma n (dn, ax_reach) = let val lhs = dc_take dn $ Bound 0 `%(x_name n); val rhs = dc_take dn $ Bound 0 `%(x_name n^"'"); val concl = mk_trp (%:(x_name n) === %:(x_name n^"'")); val goal = mk_All ("n", mk_trp (lhs === rhs)) ===> concl; fun tacf prems = [ res_inst_tac [("t", x_name n )] (ax_reach RS subst) 1, res_inst_tac [("t", x_name n^"'")] (ax_reach RS subst) 1, stac fix_def2 1, REPEAT (CHANGED (rtac (contlub_cfun_arg RS ssubst) 1 THEN chain_tac 1)), stac contlub_cfun_fun 1, stac contlub_cfun_fun 2, rtac lub_equal 3, chain_tac 1, rtac allI 1, resolve_tac prems 1]; in pg'' thy axs_take_def goal tacf end; in mapn take_lemma 1 (dnames ~~ axs_reach) end; (* ----- theorems concerning finiteness and induction ----------------------- *) val (finites, ind) = if is_finite then (* finite case *) let fun take_enough dn = mk_ex ("n",dc_take dn $ Bound 0 ` %:"x" === %:"x"); fun dname_lemma dn = let val prem1 = mk_trp (defined (%:"x")); val disj1 = mk_all ("n", dc_take dn $ Bound 0 ` %:"x" === UU); val prem2 = mk_trp (mk_disj (disj1, take_enough dn)); val concl = mk_trp (take_enough dn); val goal = prem1 ===> prem2 ===> concl; val tacs = [ etac disjE 1, etac notE 1, resolve_tac take_lemmas 1, asm_simp_tac take_ss 1, atac 1]; in pg [] goal tacs end; val finite_lemmas1a = map dname_lemma dnames; val finite_lemma1b = let fun mk_eqn n ((dn, args), _) = let val disj1 = dc_take dn $ Bound 1 ` Bound 0 === UU; val disj2 = dc_take dn $ Bound 1 ` Bound 0 === Bound 0; in mk_constrainall (x_name n, Type (dn,args), mk_disj (disj1, disj2)) end; val goal = mk_trp (mk_all ("n", foldr1 mk_conj (mapn mk_eqn 1 eqs))); fun arg_tacs vn = [ eres_inst_tac [("x", vn)] all_dupE 1, etac disjE 1, asm_simp_tac (HOL_ss addsimps con_rews) 1, asm_simp_tac take_ss 1]; fun con_tacs (con, args) = asm_simp_tac take_ss 1 :: List.concat (map arg_tacs (nonlazy_rec args)); fun foo_tacs n (cons, cases) = simp_tac take_ss 1 :: rtac allI 1 :: res_inst_tac [("x",x_name n)] cases 1 :: asm_simp_tac take_ss 1 :: List.concat (map con_tacs cons); val tacs = rtac allI 1 :: induct_tac "n" 1 :: simp_tac take_ss 1 :: TRY (safe_tac (empty_cs addSEs [conjE] addSIs [conjI])) :: List.concat (mapn foo_tacs 1 (conss ~~ cases)); in pg [] goal tacs end; fun one_finite (dn, l1b) = let val goal = mk_trp (%%:(dn^"_finite") $ %:"x"); val tacs = [ case_UU_tac take_rews 1 "x", eresolve_tac finite_lemmas1a 1, step_tac HOL_cs 1, step_tac HOL_cs 1, cut_facts_tac [l1b] 1, fast_tac HOL_cs 1]; in pg axs_finite_def goal tacs end; val finites = map one_finite (dnames ~~ atomize finite_lemma1b); val ind = let fun concf n dn = %:(P_name n) $ %:(x_name n); fun tacf prems = 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) :: List.concat (map finite_tacs (finites ~~ atomize 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_sg thy [("P",dn^"_finite "^x_name n)] excluded_middle; val finites = mapn one_finite 1 dnames; val goal = let fun one_adm n _ = mk_trp (%%:admN $ %:(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; fun tacf prems = map (fn ax_reach => rtac (ax_reach RS subst) 1) axs_reach @ [ quant_tac 1, rtac (adm_impl_admw RS wfix_ind) 1, REPEAT_DETERM (rtac adm_all2 1), REPEAT_DETERM ( TRY (rtac adm_conj 1) THEN rtac adm_subst 1 THEN cont_tacR 1 THEN resolve_tac prems 1), strip_tac 1, rtac (rewrite_rule axs_take_def finite_ind) 1, ind_prems_tac prems]; val ind = (pg'' thy [] goal tacf handle ERROR _ => (warning "Cannot prove infinite induction rule"; refl)); in (finites, ind) end; end; (* local *) (* ----- theorem concerning coinduction ------------------------------------- *) local 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 take_rews; val sproj = prj (fn s => K("fst("^s^")")) (fn s => K("snd("^s^")")); 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 n x = [ rotate_tac (n+1) 1, etac all2E 1, eres_inst_tac [("P1", sproj "R" eqs n^" "^x^" "^x^"'")] (mp RS disjE) 1, TRY (safe_tac HOL_cs), REPEAT (CHANGED (asm_simp_tac take_ss 1))]; val tacs = [ rtac impI 1, induct_tac "n" 1, simp_tac take_ss 1, safe_tac HOL_cs] @ List.concat (mapn x_tacs 0 xs); in pg [ax_bisim_def] goal tacs end; in 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) :: List.concat (map (fn take_lemma => [ rtac take_lemma 1, cut_facts_tac [coind_lemma] 1, fast_tac HOL_cs 1]) take_lemmas); in pg [] goal tacs end; end; (* local *) in thy |> Sign.add_path comp_dnam |> (snd o (PureThy.add_thmss (map Thm.no_attributes [ ("take_rews" , take_rews ), ("take_lemmas", take_lemmas), ("finites" , finites ), ("finite_ind", [finite_ind]), ("ind" , [ind ]), ("coind" , [coind ])]))) |> Sign.parent_path |> rpair take_rews end; (* let *) end; (* local *) end; (* struct *)