(* Title: HOLCF/Tools/domain/domain_theorems.ML
ID: $Id$
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 *)