(* Title: HOLCF/domain/theorems.ML ID: $Id: theorems.ML,v 1.55 2005/09/17 16:11:25 wenzelm Exp $ Author: David von Oheimb New proofs/tactics by Brian Huffman Proof generator for domain section. *) val HOLCF_ss = simpset(); structure Domain_Theorems = struct local 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 inferT sg pre_tm = #1 (Sign.infer_types (Sign.pp sg) sg (K NONE) (K NONE) [] true ([pre_tm],propT)); fun pg'' thy defs t = let val sg = sign_of thy; val ct = Thm.cterm_of sg (inferT sg t); in prove_goalw_cterm defs ct end; fun pg' thy defs t tacsf=pg'' thy defs t (fn [] => tacsf | prems=> (cut_facts_tac prems 1)::tacsf); 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 Porder.thy "!!x::'a::po. ~ x << y ==> x ~= y" (fn prems => [ (blast_tac (claset() addDs [antisym_less_inverse]) 1)]); (* infixr 0 y; val b = 0; fun _ y t = by t; fun g defs t = let val sg = sign_of thy; val ct = Thm.cterm_of sg (inferT sg t); in goalw_cterm defs ct end; *) 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; val axs_con_def = map (fn (con,_) => ga (extern_name con^"_def") dname) cons; val axs_dis_def = map (fn (con,_) => ga ( dis_name con^"_def") dname) cons; val axs_mat_def = map (fn (con,_) => ga ( mat_name con^"_def") dname) cons; val axs_sel_def = List.concat(map (fn (_,args) => List.mapPartial (fn arg => Option.map (fn sel => ga (sel^"_def") dname) (sel_of arg)) args) cons); 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 = mk_cRep_CFun (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 (sign_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 in Library.foldr mk_ex (vns, foldr' mk_conj ((%:x_name === con_app2 con %: vns):: map (defined o %:) (nonlazy args))) end; val exh = foldr' mk_disj ((%:x_name===UU)::map one_con cons); val my_ctyp = cons2ctyp cons; val thm1 = instantiate' [SOME my_ctyp] [] exh_start; val thm2 = rewrite_rule (map mk_meta_eq ex_defined_iffs) thm1; val thm3 = rewrite_rule [mk_meta_eq conj_assoc] thm2; in val exhaust = pg con_appls (mk_trp exh)[ (* first 3 rules replace "x = UU \/ P" with "rep$x = UU \/ P" *) rtac disjE 1, etac (rep_defin' RS disjI1) 2, etac disjI2 2, rewrite_goals_tac [mk_meta_eq iso_swap], rtac thm3 1]; 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 = Library.foldl (op `) (%%:(dname^"_when"), mapn bound_fun 1 cons); in val when_strict = pg [when_appl, mk_meta_eq rep_strict] (bind_fun(mk_trp(strict when_app))) [resolve_tac [sscase1,ssplit1,strictify1] 1]; val when_apps = let fun one_when n (con,args) = pg (when_appl :: con_appls) (bind_fun (lift_defined %: (nonlazy args, mk_trp(when_app`(con_app con args) === mk_cRep_CFun(bound_fun n 0,map %# args)))))[ asm_simp_tac (HOLCF_ss addsimps [ax_abs_iso]) 1]; in mapn one_when 1 cons end; end; val when_rews = when_strict::when_apps; (* ----- theorems concerning the constructors, discriminators and selectors - *) val dis_rews = let val dis_stricts = map (fn (con,_) => pg axs_dis_def (mk_trp( strict(%%:(dis_name con)))) [ rtac when_strict 1]) cons; val dis_apps = let fun one_dis c (con,args)= pg axs_dis_def (lift_defined %: (nonlazy args, (mk_trp((%%:(dis_name c))`(con_app con args) === %%:(if con=c then TT_N else FF_N))))) [ asm_simp_tac (HOLCF_ss addsimps when_rews) 1]; in List.concat(map (fn (c,_) => map (one_dis c) cons) cons) end; val dis_defins = map (fn (con,args) => pg [] (defined(%:x_name) ==> defined(%%:(dis_name con)`%x_name)) [ rtac casedist 1, contr_tac 1, DETERM_UNTIL_SOLVED (CHANGED(asm_simp_tac (HOLCF_ss addsimps dis_apps) 1))]) cons; in dis_stricts @ dis_defins @ dis_apps end; val mat_rews = let val mat_stricts = map (fn (con,_) => pg axs_mat_def (mk_trp( strict(%%:(mat_name con)))) [ rtac when_strict 1]) cons; val mat_apps = let fun one_mat c (con,args)= pg axs_mat_def (lift_defined %: (nonlazy args, (mk_trp((%%:(mat_name c))`(con_app con args) === (if con=c then %%:returnN`(mk_ctuple (map %# args)) else %%:failN))))) [asm_simp_tac (HOLCF_ss addsimps when_rews) 1]; in List.concat(map (fn (c,_) => map (one_mat c) cons) cons) end; in mat_stricts @ mat_apps end; val con_stricts = List.concat(map (fn (con,args) => map (fn vn => pg con_appls (mk_trp(con_app2 con (fn arg => if vname arg = vn then UU else %# arg) args === UU))[ asm_simp_tac (HOLCF_ss addsimps [abs_strict]) 1] ) (nonlazy args)) cons); val con_defins = map (fn (con,args) => pg [] (lift_defined %: (nonlazy args, mk_trp(defined(con_app con args)))) ([ rtac rev_contrapos 1, eres_inst_tac [("f",dis_name con)] cfun_arg_cong 1, asm_simp_tac (HOLCF_ss addsimps dis_rews) 1] )) cons; val con_rews = con_stricts @ con_defins; val sel_stricts = let fun one_sel sel = pg axs_sel_def (mk_trp(strict(%%:sel))) [ simp_tac (HOLCF_ss addsimps when_rews) 1]; in List.concat(map (fn (_,args) => List.mapPartial (fn arg => Option.map one_sel (sel_of arg)) args) cons) end; val sel_apps = let fun one_sel c n sel = map (fn (con,args) => let val nlas = nonlazy args; val vns = map vname args; in pg axs_sel_def (lift_defined %: (List.filter (fn v => con=c andalso (v<>List.nth(vns,n))) nlas, mk_trp((%%:sel)`(con_app con args) === (if con=c then %:(List.nth(vns,n)) else UU)))) ( (if con=c then [] else map(case_UU_tac(when_rews@con_stricts)1) nlas) @(if con=c andalso ((List.nth(vns,n)) mem nlas) then[case_UU_tac (when_rews @ con_stricts) 1 (List.nth(vns,n))] else []) @ [asm_simp_tac(HOLCF_ss addsimps when_rews)1])end) cons; in List.concat(map (fn (c,args) => List.concat(List.mapPartial I (mapn (fn n => fn arg => Option.map (one_sel c n) (sel_of arg)) 0 args))) cons) end; val sel_defins = if length cons=1 then List.mapPartial (fn arg => Option.map (fn sel => pg [](defined(%:x_name)==> defined(%%:sel`%x_name)) [ rtac casedist 1, contr_tac 1, DETERM_UNTIL_SOLVED (CHANGED(asm_simp_tac (HOLCF_ss addsimps sel_apps) 1))])(sel_of arg)) (filter_out is_lazy (snd(hd cons))) else []; val sel_rews = sel_stricts @ sel_defins @ sel_apps; val distincts_le = let fun dist (con1, args1) (con2, args2) = pg [] (lift_defined %: ((nonlazy args1), (mk_trp (con_app con1 args1 ~<< con_app con2 args2))))([ 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]); 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, variantlist(map vname args2,map vname args1))) 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 = mk_trp (foldr' mk_conj (ListPair.map rel (map %# largs, map %# rargs))); val prem = mk_trp (rel(con_app con largs,con_app con rargs)); val prop = prem ===> lift_defined %: (nonlazy largs, 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) RS iffD1; val tacs = [ dtac abs_less 1, REPEAT (dresolve_tac [sinl_less RS iffD1, sinr_less RS iffD1] 1), asm_full_simp_tac (HOLCF_ss addsimps [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) RS iffD1; val tacs = [ dtac abs_eq 1, REPEAT (dresolve_tac [sinl_inject, sinr_inject] 1), asm_full_simp_tac (HOLCF_ss addsimps [spair_eq]) 1]; in map (fn (con,args) => pgterm (op ===) con args tacs) cons' end; end; (* ----- theorems concerning one induction step ----------------------------- *) val copy_strict = pg[ax_copy_def](mk_trp(strict(dc_copy`%"f"))) [ asm_simp_tac(HOLCF_ss addsimps [abs_strict, when_strict]) 1]; val copy_apps = map (fn (con,args) => pg [ax_copy_def] (lift_defined %: (nonlazy_rec args, mk_trp(dc_copy`%"f"`(con_app con args) === (con_app2 con (app_rec_arg (cproj (%:"f") eqs)) args)))) (map (case_UU_tac (abs_strict::when_strict::con_stricts) 1 o vname) (List.filter (fn a => not (is_rec a orelse is_lazy a)) args) @[asm_simp_tac (HOLCF_ss addsimps when_apps) 1, simp_tac (HOLCF_ss addsimps con_appls) 1]))cons; val copy_stricts = map (fn (con,args) => pg [] (mk_trp(dc_copy`UU` (con_app con args) ===UU)) (let val rews = copy_strict::copy_apps@con_rews in map (case_UU_tac rews 1) (nonlazy args) @ [ asm_simp_tac (HOLCF_ss addsimps rews) 1] end)) (List.filter (fn (_,args)=>exists is_nonlazy_rec args) cons); val copy_rews = copy_strict::copy_apps @ copy_stricts; in thy |> Theory.add_path (Sign.base_name dname) |> (#1 o (PureThy.add_thmss (map Thm.no_attributes [ ("iso_rews" , iso_rews ), ("exhaust" , [exhaust] ), ("casedist" , [casedist]), ("when_rews", when_rews ), ("con_rews", con_rews), ("sel_rews", sel_rews), ("dis_rews", dis_rews), ("dist_les", dist_les), ("dist_eqs", dist_eqs), ("inverts" , inverts ), ("injects" , injects ), ("copy_rews", copy_rews)]))) |> (#1 o PureThy.add_thmss [(("match_rews", mat_rews), [Simplifier.simp_add_global])]) |> Theory.parent_path |> rpair (iso_rews @ when_rews @ con_rews @ sel_rews @ dis_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 (sign_of 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 *) 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; (* local *) 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 Fix.thy; 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=pg copy_take_defs(mk_trp(foldr' mk_conj(map(fn((dn,args),_)=> strict(dc_take dn $ %:"n")) eqs))) ([ induct_tac "n" 1, simp_tac iterate_Cprod_ss 1, asm_simp_tac (iterate_Cprod_ss addsimps copy_rews)1]); val take_stricts' = rewrite_rule copy_take_defs take_stricts; val take_0s = mapn(fn n=> fn dn => pg axs_take_def(mk_trp((dc_take dn $ %%:"0") `%x_name n === UU))[ simp_tac iterate_Cprod_ss 1]) 1 dnames; val c_UU_tac = case_UU_tac (take_stricts'::copy_con_rews) 1; val take_apps = pg copy_take_defs (mk_trp(foldr' mk_conj (List.concat(map (fn ((dn,_),cons) => map (fn (con,args) => Library.foldr mk_all (map vname args,(dc_take dn $ (%%:"Suc" $ %:"n"))`(con_app con args) === con_app2 con (app_rec_arg (fn n=>dc_take (List.nth(dnames,n))$ %:"n")) args)) cons) eqs)))) ([ 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 (List.filter (has_fewer_prems 1) copy_rews)) 1, TRY(safe_tac HOL_cs)] @ (List.concat(map (fn ((dn,_),cons) => map (fn (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 ) cons) eqs))); in val take_rews = map standard (atomize take_stricts @ take_0s @ atomize take_apps); end; (* local *) local fun one_con p (con,args) = Library.foldr mk_All (map vname args, lift_defined (bound_arg (map vname args)) (nonlazy args, lift (fn arg => %:(P_name (1+rec_of arg)) $ bound_arg args arg) (List.filter is_rec args,mk_trp(%:p $ con_app2 con (bound_arg args) args)))); fun one_eq ((p,cons),concl) = (mk_trp(%:p $ UU) ===> Library.foldr (op ===>) (map (one_con p) cons,concl)); fun ind_term concf = Library.foldr one_eq (mapn (fn n => fn x => (P_name n, x))1conss, mk_trp(foldr' 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 = pg'' thy [] (ind_term (fn n => fn dn => %:(P_name n)$ (dc_take dn $ %:"n" `%(x_name n)))) (fn prems => [ quant_tac 1, simp_tac HOL_ss 1, induct_tac "n" 1, simp_tac (take_ss addsimps prems) 1, TRY(safe_tac HOL_cs)] @ List.concat(map (fn (cons,cases) => [ res_inst_tac [("x","x")] cases 1, asm_simp_tac (take_ss addsimps prems) 1] @ List.concat(map (fn (con,args) => asm_simp_tac take_ss 1 :: map (fn arg => case_UU_tac (prems@con_rews) 1 ( List.nth(dnames,rec_of arg)^"_take n$"^vname arg)) (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)) cons)) (conss~~cases))); val take_lemmas =mapn(fn n=> fn(dn,ax_reach)=> pg'' thy axs_take_def(mk_All("n", mk_trp(dc_take dn $ Bound 0 `%(x_name n) === dc_take dn $ Bound 0 `%(x_name n^"'"))) ===> mk_trp(%:(x_name n) === %:(x_name n^"'"))) (fn 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])) 1 (dnames~~axs_reach); (* ----- theorems concerning finiteness and induction ----------------------- *) val (finites,ind) = if is_finite then let fun take_enough dn = mk_ex ("n",dc_take dn $ Bound 0 ` %:"x" === %:"x"); val finite_lemmas1a = map (fn dn => pg [] (mk_trp(defined (%:"x")) ===> mk_trp(mk_disj(mk_all("n",dc_take dn $ Bound 0 ` %:"x" === UU), take_enough dn)) ===> mk_trp(take_enough dn)) [ etac disjE 1, etac notE 1, resolve_tac take_lemmas 1, asm_simp_tac take_ss 1, atac 1]) dnames; val finite_lemma1b = pg [] (mk_trp (mk_all("n",foldr' mk_conj (mapn (fn n => fn ((dn,args),_) => mk_constrainall(x_name n,Type(dn,args), mk_disj(dc_take dn $ Bound 1 ` Bound 0 === UU, dc_take dn $ Bound 1 ` Bound 0 === Bound 0))) 1 eqs)))) ([ rtac allI 1, induct_tac "n" 1, simp_tac take_ss 1, TRY(safe_tac(empty_cs addSEs[conjE] addSIs[conjI]))] @ List.concat(mapn (fn n => fn (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 (fn (con,args) => asm_simp_tac take_ss 1 :: List.concat(map (fn 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]) (nonlazy_rec args))) cons)) 1 (conss~~cases))); val finites = map (fn (dn,l1b) => pg axs_finite_def (mk_trp( %%:(dn^"_finite") $ %:"x"))[ 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]) (dnames~~atomize finite_lemma1b); in (finites, pg'' thy[](ind_term (fn n => fn dn => %:(P_name n) $ %:(x_name n)))(fn prems => TRY(safe_tac HOL_cs) :: List.concat (map (fn (finite,fin_ind) => [ rtac(rewrite_rule axs_finite_def finite RS exE)1, etac subst 1, rtac fin_ind 1, ind_prems_tac prems]) (finites~~(atomize finite_ind)) )) ) end (* let *) else (mapn (fn n => fn dn => read_instantiate_sg (sign_of thy) [("P",dn^"_finite "^x_name n)] excluded_middle) 1 dnames, pg'' thy [] (Library.foldr (op ===>) (mapn (fn n => K(mk_trp(%%:admN $ %:(P_name n)))) 1 dnames, ind_term (fn n => fn dn => %:(P_name n) $ %:(x_name n)))) (fn 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]) handle ERROR => (warning "Cannot prove infinite induction rule"; refl)) 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=pg[ax_bisim_def](mk_trp(mk_imp(%%:(comp_dname^"_bisim") $ %:"R", Library.foldr (fn (x,t)=> mk_all(x,mk_all(x^"'",t))) (xs, Library.foldr mk_imp (mapn (fn n => K(proj (%:"R") eqs n $ bnd_arg n 0 $ bnd_arg n 1)) 0 dnames, foldr' mk_conj (mapn (fn n => fn dn => (dc_take dn $ %:"n" `bnd_arg n 0 === (dc_take dn $ %:"n" `bnd_arg n 1)))0 dnames)))))) ([ rtac impI 1, induct_tac "n" 1, simp_tac take_ss 1, safe_tac HOL_cs] @ List.concat(mapn (fn n => fn 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))]) 0 xs)); in val coind = pg [] (mk_trp(%%:(comp_dname^"_bisim") $ %:"R") ===> Library.foldr (op ===>) (mapn (fn n => fn x => mk_trp(proj (%:"R") eqs n $ %:x $ %:(x^"'"))) 0 xs, mk_trp(foldr' mk_conj (map (fn x => %:x === %:(x^"'")) xs)))) ([ 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)); end; (* local *) in thy |> Theory.add_path comp_dnam |> (#1 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 ])]))) |> Theory.parent_path |> rpair take_rews end; (* let *) end; (* local *) end; (* struct *)