src/HOLCF/Tools/Domain/domain_theorems.ML
author huffman
Mon Feb 22 09:43:36 2010 -0800 (2010-02-22)
changeset 35287 978a936faace
parent 35117 eeec2a320a77
child 35288 aa7da51ae1ef
permissions -rw-r--r--
remove unnecessary local
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(*  Title:      HOLCF/Tools/Domain/domain_theorems.ML
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    Author:     David von Oheimb
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    Author:     Brian Huffman
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Proof generator for domain command.
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*)
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val HOLCF_ss = @{simpset};
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signature DOMAIN_THEOREMS =
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sig
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  val theorems: Domain_Library.eq * Domain_Library.eq list -> theory -> thm list * theory;
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  val comp_theorems: bstring * Domain_Library.eq list -> theory -> thm list * theory;
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  val quiet_mode: bool Unsynchronized.ref;
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  val trace_domain: bool Unsynchronized.ref;
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end;
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structure Domain_Theorems :> DOMAIN_THEOREMS =
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struct
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val quiet_mode = Unsynchronized.ref false;
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val trace_domain = Unsynchronized.ref false;
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fun message s = if !quiet_mode then () else writeln s;
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fun trace s = if !trace_domain then tracing s else ();
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val adm_impl_admw = @{thm adm_impl_admw};
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val adm_all = @{thm adm_all};
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val adm_conj = @{thm adm_conj};
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val adm_subst = @{thm adm_subst};
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val antisym_less_inverse = @{thm below_antisym_inverse};
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val beta_cfun = @{thm beta_cfun};
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val cfun_arg_cong = @{thm cfun_arg_cong};
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val ch2ch_fst = @{thm ch2ch_fst};
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val ch2ch_snd = @{thm ch2ch_snd};
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val ch2ch_Rep_CFunL = @{thm ch2ch_Rep_CFunL};
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val ch2ch_Rep_CFunR = @{thm ch2ch_Rep_CFunR};
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val chain_iterate = @{thm chain_iterate};
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val compact_ONE = @{thm compact_ONE};
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val compact_sinl = @{thm compact_sinl};
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val compact_sinr = @{thm compact_sinr};
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val compact_spair = @{thm compact_spair};
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val compact_up = @{thm compact_up};
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val contlub_cfun_arg = @{thm contlub_cfun_arg};
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val contlub_cfun_fun = @{thm contlub_cfun_fun};
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val contlub_fst = @{thm contlub_fst};
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val contlub_snd = @{thm contlub_snd};
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val contlubE = @{thm contlubE};
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val cont_const = @{thm cont_const};
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val cont_id = @{thm cont_id};
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val cont2cont_fst = @{thm cont2cont_fst};
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val cont2cont_snd = @{thm cont2cont_snd};
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val cont2cont_Rep_CFun = @{thm cont2cont_Rep_CFun};
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val fix_def2 = @{thm fix_def2};
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val injection_eq = @{thm injection_eq};
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val injection_less = @{thm injection_below};
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val lub_equal = @{thm lub_equal};
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val monofun_cfun_arg = @{thm monofun_cfun_arg};
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val retraction_strict = @{thm retraction_strict};
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val spair_eq = @{thm spair_eq};
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val spair_less = @{thm spair_below};
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val sscase1 = @{thm sscase1};
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val ssplit1 = @{thm ssplit1};
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val strictify1 = @{thm strictify1};
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val wfix_ind = @{thm wfix_ind};
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val iso_intro       = @{thm iso.intro};
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val iso_abs_iso     = @{thm iso.abs_iso};
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val iso_rep_iso     = @{thm iso.rep_iso};
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val iso_abs_strict  = @{thm iso.abs_strict};
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val iso_rep_strict  = @{thm iso.rep_strict};
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val iso_abs_defin'  = @{thm iso.abs_defin'};
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val iso_rep_defin'  = @{thm iso.rep_defin'};
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val iso_abs_defined = @{thm iso.abs_defined};
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val iso_rep_defined = @{thm iso.rep_defined};
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val iso_compact_abs = @{thm iso.compact_abs};
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val iso_compact_rep = @{thm iso.compact_rep};
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val iso_iso_swap    = @{thm iso.iso_swap};
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val exh_start = @{thm exh_start};
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val ex_defined_iffs = @{thms ex_defined_iffs};
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val exh_casedist0 = @{thm exh_casedist0};
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val exh_casedists = @{thms exh_casedists};
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open Domain_Library;
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infixr 0 ===>;
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infixr 0 ==>;
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infix 0 == ; 
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infix 1 ===;
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infix 1 ~= ;
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infix 1 <<;
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infix 1 ~<<;
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infix 9 `   ;
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infix 9 `% ;
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infix 9 `%%;
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infixr 9 oo;
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(* ----- general proof facilities ------------------------------------------- *)
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fun legacy_infer_term thy t =
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  let val ctxt = ProofContext.set_mode ProofContext.mode_schematic (ProofContext.init thy)
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  in singleton (Syntax.check_terms ctxt) (Sign.intern_term thy t) end;
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fun pg'' thy defs t tacs =
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  let
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    val t' = legacy_infer_term thy t;
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    val asms = Logic.strip_imp_prems t';
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    val prop = Logic.strip_imp_concl t';
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    fun tac {prems, context} =
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      rewrite_goals_tac defs THEN
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      EVERY (tacs {prems = map (rewrite_rule defs) prems, context = context})
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  in Goal.prove_global thy [] asms prop tac end;
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fun pg' thy defs t tacsf =
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  let
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    fun tacs {prems, context} =
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      if null prems then tacsf context
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      else cut_facts_tac prems 1 :: tacsf context;
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  in pg'' thy defs t tacs end;
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fun case_UU_tac ctxt rews i v =
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  InductTacs.case_tac ctxt (v^"=UU") i THEN
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  asm_simp_tac (HOLCF_ss addsimps rews) i;
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val chain_tac =
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  REPEAT_DETERM o resolve_tac 
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    [chain_iterate, ch2ch_Rep_CFunR, ch2ch_Rep_CFunL, ch2ch_fst, ch2ch_snd];
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(* ----- general proofs ----------------------------------------------------- *)
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val all2E = @{lemma "!x y . P x y ==> (P x y ==> R) ==> R" by simp}
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val dist_eqI = @{lemma "!!x::'a::po. ~ x << y ==> x ~= y" by (blast dest!: below_antisym_inverse)}
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fun theorems (((dname, _), cons) : eq, eqs : eq list) thy =
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let
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val _ = message ("Proving isomorphism properties of domain "^dname^" ...");
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val pg = pg' thy;
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val map_tab = Domain_Isomorphism.get_map_tab thy;
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(* ----- getting the axioms and definitions --------------------------------- *)
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local
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  fun ga s dn = PureThy.get_thm thy (dn ^ "." ^ s);
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in
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  val ax_abs_iso  = ga "abs_iso"  dname;
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  val ax_rep_iso  = ga "rep_iso"  dname;
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  val ax_when_def = ga "when_def" dname;
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  fun get_def mk_name (con,_) = ga (mk_name con^"_def") dname;
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  val axs_con_def = map (get_def extern_name) cons;
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  val axs_dis_def = map (get_def dis_name) cons;
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  val axs_mat_def = map (get_def mat_name) cons;
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  val axs_pat_def = map (get_def pat_name) cons;
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  val axs_sel_def =
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    let
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      fun def_of_sel sel = ga (sel^"_def") dname;
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      fun def_of_arg arg = Option.map def_of_sel (sel_of arg);
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      fun defs_of_con (_, args) = map_filter def_of_arg args;
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    in
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      maps defs_of_con cons
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    end;
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  val ax_copy_def = ga "copy_def" dname;
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end; (* local *)
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(* ----- theorems concerning the isomorphism -------------------------------- *)
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val dc_abs  = %%:(dname^"_abs");
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val dc_rep  = %%:(dname^"_rep");
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val dc_copy = %%:(dname^"_copy");
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val x_name = "x";
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val iso_locale = iso_intro OF [ax_abs_iso, ax_rep_iso];
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val abs_strict = ax_rep_iso RS (allI RS retraction_strict);
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val rep_strict = ax_abs_iso RS (allI RS retraction_strict);
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val abs_defin' = iso_locale RS iso_abs_defin';
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val rep_defin' = iso_locale RS iso_rep_defin';
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val iso_rews = map Drule.export_without_context [ax_abs_iso, ax_rep_iso, abs_strict, rep_strict];
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(* ----- generating beta reduction rules from definitions-------------------- *)
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val _ = trace " Proving beta reduction rules...";
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local
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  fun arglist (Const _ $ Abs (s, _, t)) =
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    let
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      val (vars,body) = arglist t;
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    in (s :: vars, body) end
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    | arglist t = ([], t);
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  fun bind_fun vars t = Library.foldr mk_All (vars, t);
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  fun bound_vars 0 = []
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    | bound_vars i = Bound (i-1) :: bound_vars (i - 1);
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in
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  fun appl_of_def def =
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    let
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      val (_ $ con $ lam) = concl_of def;
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      val (vars, rhs) = arglist lam;
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      val lhs = list_ccomb (con, bound_vars (length vars));
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      val appl = bind_fun vars (lhs == rhs);
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      val cs = ContProc.cont_thms lam;
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      val betas = map (fn c => mk_meta_eq (c RS beta_cfun)) cs;
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    in pg (def::betas) appl (K [rtac reflexive_thm 1]) end;
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end;
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val _ = trace "Proving when_appl...";
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val when_appl = appl_of_def ax_when_def;
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val _ = trace "Proving con_appls...";
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val con_appls = map appl_of_def axs_con_def;
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local
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  fun arg2typ n arg =
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    let val t = TVar (("'a", n), pcpoS)
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    in (n + 1, if is_lazy arg then mk_uT t else t) end;
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  fun args2typ n [] = (n, oneT)
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    | args2typ n [arg] = arg2typ n arg
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    | args2typ n (arg::args) =
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    let
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      val (n1, t1) = arg2typ n arg;
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      val (n2, t2) = args2typ n1 args
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    in (n2, mk_sprodT (t1, t2)) end;
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  fun cons2typ n [] = (n,oneT)
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    | cons2typ n [con] = args2typ n (snd con)
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    | cons2typ n (con::cons) =
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    let
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      val (n1, t1) = args2typ n (snd con);
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      val (n2, t2) = cons2typ n1 cons
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    in (n2, mk_ssumT (t1, t2)) end;
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in
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  fun cons2ctyp cons = ctyp_of thy (snd (cons2typ 1 cons));
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end;
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local 
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  val iso_swap = iso_locale RS iso_iso_swap;
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  fun one_con (con, args) =
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    let
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      val vns = map vname args;
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      val eqn = %:x_name === con_app2 con %: vns;
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      val conj = foldr1 mk_conj (eqn :: map (defined o %:) (nonlazy args));
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    in Library.foldr mk_ex (vns, conj) end;
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  val conj_assoc = @{thm conj_assoc};
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  val exh = foldr1 mk_disj ((%:x_name === UU) :: map one_con cons);
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  val thm1 = instantiate' [SOME (cons2ctyp cons)] [] exh_start;
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  val thm2 = rewrite_rule (map mk_meta_eq ex_defined_iffs) thm1;
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  val thm3 = rewrite_rule [mk_meta_eq @{thm conj_assoc}] thm2;
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  (* first 3 rules replace "x = UU \/ P" with "rep$x = UU \/ P" *)
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  val tacs = [
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    rtac disjE 1,
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    etac (rep_defin' RS disjI1) 2,
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    etac disjI2 2,
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    rewrite_goals_tac [mk_meta_eq iso_swap],
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    rtac thm3 1];
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in
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  val _ = trace " Proving exhaust...";
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  val exhaust = pg con_appls (mk_trp exh) (K tacs);
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  val _ = trace " Proving casedist...";
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  val casedist =
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    Drule.export_without_context (rewrite_rule exh_casedists (exhaust RS exh_casedist0));
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end;
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local 
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  fun bind_fun t = Library.foldr mk_All (when_funs cons, t);
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  fun bound_fun i _ = Bound (length cons - i);
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  val when_app = list_ccomb (%%:(dname^"_when"), mapn bound_fun 1 cons);
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in
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  val _ = trace " Proving when_strict...";
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  val when_strict =
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    let
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      val axs = [when_appl, mk_meta_eq rep_strict];
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      val goal = bind_fun (mk_trp (strict when_app));
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      val tacs = [resolve_tac [sscase1, ssplit1, strictify1] 1];
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    in pg axs goal (K tacs) end;
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  val _ = trace " Proving when_apps...";
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  val when_apps =
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    let
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      fun one_when n (con,args) =
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        let
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          val axs = when_appl :: con_appls;
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          val goal = bind_fun (lift_defined %: (nonlazy args, 
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                mk_trp (when_app`(con_app con args) ===
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                       list_ccomb (bound_fun n 0, map %# args))));
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          val tacs = [asm_simp_tac (HOLCF_ss addsimps [ax_abs_iso]) 1];
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        in pg axs goal (K tacs) end;
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    in mapn one_when 1 cons end;
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end;
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val when_rews = when_strict :: when_apps;
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(* ----- theorems concerning the constructors, discriminators and selectors - *)
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local
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  fun dis_strict (con, _) =
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    let
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      val goal = mk_trp (strict (%%:(dis_name con)));
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    in pg axs_dis_def goal (K [rtac when_strict 1]) end;
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  fun dis_app c (con, args) =
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    let
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      val lhs = %%:(dis_name c) ` con_app con args;
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      val rhs = if con = c then TT else FF;
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      val goal = lift_defined %: (nonlazy args, mk_trp (lhs === rhs));
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      val tacs = [asm_simp_tac (HOLCF_ss addsimps when_rews) 1];
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    in pg axs_dis_def goal (K tacs) end;
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  val _ = trace " Proving dis_apps...";
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  val dis_apps = maps (fn (c,_) => map (dis_app c) cons) cons;
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  fun dis_defin (con, args) =
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    let
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      val goal = defined (%:x_name) ==> defined (%%:(dis_name con) `% x_name);
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   315
      val tacs =
wenzelm@23152
   316
        [rtac casedist 1,
wenzelm@23152
   317
         contr_tac 1,
wenzelm@23152
   318
         DETERM_UNTIL_SOLVED (CHANGED
wenzelm@23152
   319
          (asm_simp_tac (HOLCF_ss addsimps dis_apps) 1))];
wenzelm@27208
   320
    in pg [] goal (K tacs) end;
wenzelm@23152
   321
huffman@29402
   322
  val _ = trace " Proving dis_stricts...";
wenzelm@23152
   323
  val dis_stricts = map dis_strict cons;
huffman@29402
   324
  val _ = trace " Proving dis_defins...";
wenzelm@23152
   325
  val dis_defins = map dis_defin cons;
wenzelm@23152
   326
in
wenzelm@23152
   327
  val dis_rews = dis_stricts @ dis_defins @ dis_apps;
wenzelm@23152
   328
end;
wenzelm@23152
   329
wenzelm@23152
   330
local
wenzelm@23152
   331
  fun mat_strict (con, _) =
wenzelm@23152
   332
    let
huffman@30912
   333
      val goal = mk_trp (%%:(mat_name con) ` UU ` %:"rhs" === UU);
huffman@30912
   334
      val tacs = [asm_simp_tac (HOLCF_ss addsimps [when_strict]) 1];
wenzelm@27208
   335
    in pg axs_mat_def goal (K tacs) end;
wenzelm@23152
   336
huffman@29402
   337
  val _ = trace " Proving mat_stricts...";
wenzelm@23152
   338
  val mat_stricts = map mat_strict cons;
wenzelm@23152
   339
wenzelm@23152
   340
  fun one_mat c (con, args) =
wenzelm@23152
   341
    let
huffman@30912
   342
      val lhs = %%:(mat_name c) ` con_app con args ` %:"rhs";
wenzelm@23152
   343
      val rhs =
wenzelm@23152
   344
        if con = c
huffman@30912
   345
        then list_ccomb (%:"rhs", map %# args)
huffman@26012
   346
        else mk_fail;
wenzelm@23152
   347
      val goal = lift_defined %: (nonlazy args, mk_trp (lhs === rhs));
wenzelm@23152
   348
      val tacs = [asm_simp_tac (HOLCF_ss addsimps when_rews) 1];
wenzelm@27208
   349
    in pg axs_mat_def goal (K tacs) end;
wenzelm@23152
   350
huffman@29402
   351
  val _ = trace " Proving mat_apps...";
wenzelm@23152
   352
  val mat_apps =
wenzelm@26336
   353
    maps (fn (c,_) => map (one_mat c) cons) cons;
wenzelm@23152
   354
in
wenzelm@23152
   355
  val mat_rews = mat_stricts @ mat_apps;
wenzelm@23152
   356
end;
wenzelm@23152
   357
wenzelm@23152
   358
local
wenzelm@23152
   359
  fun ps args = mapn (fn n => fn _ => %:("pat" ^ string_of_int n)) 1 args;
wenzelm@23152
   360
huffman@26012
   361
  fun pat_lhs (con,args) = mk_branch (list_comb (%%:(pat_name con), ps args));
wenzelm@23152
   362
huffman@26012
   363
  fun pat_rhs (con,[]) = mk_return ((%:"rhs") ` HOLogic.unit)
wenzelm@23152
   364
    | pat_rhs (con,args) =
huffman@26012
   365
        (mk_branch (mk_ctuple_pat (ps args)))
wenzelm@23152
   366
          `(%:"rhs")`(mk_ctuple (map %# args));
wenzelm@23152
   367
wenzelm@23152
   368
  fun pat_strict c =
wenzelm@23152
   369
    let
wenzelm@25132
   370
      val axs = @{thm branch_def} :: axs_pat_def;
wenzelm@23152
   371
      val goal = mk_trp (strict (pat_lhs c ` (%:"rhs")));
wenzelm@23152
   372
      val tacs = [simp_tac (HOLCF_ss addsimps [when_strict]) 1];
wenzelm@27208
   373
    in pg axs goal (K tacs) end;
wenzelm@23152
   374
wenzelm@23152
   375
  fun pat_app c (con, args) =
wenzelm@23152
   376
    let
wenzelm@25132
   377
      val axs = @{thm branch_def} :: axs_pat_def;
wenzelm@23152
   378
      val lhs = (pat_lhs c)`(%:"rhs")`(con_app con args);
huffman@26012
   379
      val rhs = if con = fst c then pat_rhs c else mk_fail;
wenzelm@23152
   380
      val goal = lift_defined %: (nonlazy args, mk_trp (lhs === rhs));
wenzelm@23152
   381
      val tacs = [asm_simp_tac (HOLCF_ss addsimps when_rews) 1];
wenzelm@27208
   382
    in pg axs goal (K tacs) end;
wenzelm@23152
   383
huffman@29402
   384
  val _ = trace " Proving pat_stricts...";
wenzelm@23152
   385
  val pat_stricts = map pat_strict cons;
huffman@29402
   386
  val _ = trace " Proving pat_apps...";
wenzelm@26336
   387
  val pat_apps = maps (fn c => map (pat_app c) cons) cons;
wenzelm@23152
   388
in
wenzelm@23152
   389
  val pat_rews = pat_stricts @ pat_apps;
wenzelm@23152
   390
end;
wenzelm@23152
   391
wenzelm@23152
   392
local
wenzelm@23152
   393
  fun con_strict (con, args) = 
wenzelm@23152
   394
    let
huffman@30911
   395
      val rules = abs_strict :: @{thms con_strict_rules};
wenzelm@23152
   396
      fun one_strict vn =
wenzelm@23152
   397
        let
wenzelm@23152
   398
          fun f arg = if vname arg = vn then UU else %# arg;
wenzelm@23152
   399
          val goal = mk_trp (con_app2 con f args === UU);
huffman@30911
   400
          val tacs = [simp_tac (HOL_basic_ss addsimps rules) 1];
wenzelm@27208
   401
        in pg con_appls goal (K tacs) end;
wenzelm@23152
   402
    in map one_strict (nonlazy args) end;
wenzelm@23152
   403
wenzelm@23152
   404
  fun con_defin (con, args) =
wenzelm@23152
   405
    let
huffman@30913
   406
      fun iff_disj (t, []) = HOLogic.mk_not t
huffman@30913
   407
        | iff_disj (t, ts) = t === foldr1 HOLogic.mk_disj ts;
huffman@30913
   408
      val lhs = con_app con args === UU;
huffman@30913
   409
      val rhss = map (fn x => %:x === UU) (nonlazy args);
huffman@30913
   410
      val goal = mk_trp (iff_disj (lhs, rhss));
huffman@30913
   411
      val rule1 = iso_locale RS @{thm iso.abs_defined_iff};
huffman@30913
   412
      val rules = rule1 :: @{thms con_defined_iff_rules};
huffman@30913
   413
      val tacs = [simp_tac (HOL_ss addsimps rules) 1];
huffman@30911
   414
    in pg con_appls goal (K tacs) end;
wenzelm@23152
   415
in
huffman@29402
   416
  val _ = trace " Proving con_stricts...";
wenzelm@26336
   417
  val con_stricts = maps con_strict cons;
huffman@30911
   418
  val _ = trace " Proving con_defins...";
wenzelm@23152
   419
  val con_defins = map con_defin cons;
wenzelm@23152
   420
  val con_rews = con_stricts @ con_defins;
wenzelm@23152
   421
end;
wenzelm@23152
   422
wenzelm@23152
   423
local
wenzelm@23152
   424
  val rules =
wenzelm@23152
   425
    [compact_sinl, compact_sinr, compact_spair, compact_up, compact_ONE];
wenzelm@23152
   426
  fun con_compact (con, args) =
wenzelm@23152
   427
    let
huffman@26012
   428
      val concl = mk_trp (mk_compact (con_app con args));
huffman@26012
   429
      val goal = lift (fn x => mk_compact (%#x)) (args, concl);
wenzelm@23152
   430
      val tacs = [
wenzelm@23152
   431
        rtac (iso_locale RS iso_compact_abs) 1,
wenzelm@23152
   432
        REPEAT (resolve_tac rules 1 ORELSE atac 1)];
wenzelm@27208
   433
    in pg con_appls goal (K tacs) end;
wenzelm@23152
   434
in
huffman@29402
   435
  val _ = trace " Proving con_compacts...";
wenzelm@23152
   436
  val con_compacts = map con_compact cons;
wenzelm@23152
   437
end;
wenzelm@23152
   438
wenzelm@23152
   439
local
wenzelm@23152
   440
  fun one_sel sel =
wenzelm@23152
   441
    pg axs_sel_def (mk_trp (strict (%%:sel)))
wenzelm@27208
   442
      (K [simp_tac (HOLCF_ss addsimps when_rews) 1]);
wenzelm@23152
   443
wenzelm@23152
   444
  fun sel_strict (_, args) =
wenzelm@32952
   445
    map_filter (Option.map one_sel o sel_of) args;
wenzelm@23152
   446
in
huffman@29402
   447
  val _ = trace " Proving sel_stricts...";
wenzelm@26336
   448
  val sel_stricts = maps sel_strict cons;
wenzelm@23152
   449
end;
wenzelm@23152
   450
wenzelm@23152
   451
local
wenzelm@23152
   452
  fun sel_app_same c n sel (con, args) =
wenzelm@23152
   453
    let
wenzelm@23152
   454
      val nlas = nonlazy args;
wenzelm@23152
   455
      val vns = map vname args;
wenzelm@23152
   456
      val vnn = List.nth (vns, n);
wenzelm@33317
   457
      val nlas' = filter (fn v => v <> vnn) nlas;
wenzelm@23152
   458
      val lhs = (%%:sel)`(con_app con args);
wenzelm@23152
   459
      val goal = lift_defined %: (nlas', mk_trp (lhs === %:vnn));
wenzelm@27208
   460
      fun tacs1 ctxt =
wenzelm@23152
   461
        if vnn mem nlas
wenzelm@27208
   462
        then [case_UU_tac ctxt (when_rews @ con_stricts) 1 vnn]
wenzelm@23152
   463
        else [];
wenzelm@23152
   464
      val tacs2 = [asm_simp_tac (HOLCF_ss addsimps when_rews) 1];
wenzelm@27208
   465
    in pg axs_sel_def goal (fn ctxt => (tacs1 ctxt @ tacs2)) end;
wenzelm@23152
   466
wenzelm@23152
   467
  fun sel_app_diff c n sel (con, args) =
wenzelm@23152
   468
    let
wenzelm@23152
   469
      val nlas = nonlazy args;
wenzelm@23152
   470
      val goal = mk_trp (%%:sel ` con_app con args === UU);
wenzelm@27208
   471
      fun tacs1 ctxt = map (case_UU_tac ctxt (when_rews @ con_stricts) 1) nlas;
wenzelm@23152
   472
      val tacs2 = [asm_simp_tac (HOLCF_ss addsimps when_rews) 1];
wenzelm@27208
   473
    in pg axs_sel_def goal (fn ctxt => (tacs1 ctxt @ tacs2)) end;
wenzelm@23152
   474
wenzelm@23152
   475
  fun sel_app c n sel (con, args) =
wenzelm@23152
   476
    if con = c
wenzelm@23152
   477
    then sel_app_same c n sel (con, args)
wenzelm@23152
   478
    else sel_app_diff c n sel (con, args);
wenzelm@23152
   479
wenzelm@23152
   480
  fun one_sel c n sel = map (sel_app c n sel) cons;
wenzelm@23152
   481
  fun one_sel' c n arg = Option.map (one_sel c n) (sel_of arg);
wenzelm@23152
   482
  fun one_con (c, args) =
wenzelm@26336
   483
    flat (map_filter I (mapn (one_sel' c) 0 args));
wenzelm@23152
   484
in
huffman@29402
   485
  val _ = trace " Proving sel_apps...";
wenzelm@26336
   486
  val sel_apps = maps one_con cons;
wenzelm@23152
   487
end;
wenzelm@23152
   488
wenzelm@23152
   489
local
wenzelm@23152
   490
  fun sel_defin sel =
wenzelm@23152
   491
    let
wenzelm@23152
   492
      val goal = defined (%:x_name) ==> defined (%%:sel`%x_name);
wenzelm@23152
   493
      val tacs = [
wenzelm@23152
   494
        rtac casedist 1,
wenzelm@23152
   495
        contr_tac 1,
wenzelm@23152
   496
        DETERM_UNTIL_SOLVED (CHANGED
wenzelm@23152
   497
          (asm_simp_tac (HOLCF_ss addsimps sel_apps) 1))];
wenzelm@27208
   498
    in pg [] goal (K tacs) end;
wenzelm@23152
   499
in
huffman@29402
   500
  val _ = trace " Proving sel_defins...";
wenzelm@23152
   501
  val sel_defins =
wenzelm@23152
   502
    if length cons = 1
wenzelm@32952
   503
    then map_filter (fn arg => Option.map sel_defin (sel_of arg))
wenzelm@23152
   504
                 (filter_out is_lazy (snd (hd cons)))
wenzelm@23152
   505
    else [];
wenzelm@23152
   506
end;
wenzelm@23152
   507
wenzelm@23152
   508
val sel_rews = sel_stricts @ sel_defins @ sel_apps;
wenzelm@23152
   509
huffman@29402
   510
val _ = trace " Proving dist_les...";
huffman@35117
   511
val dist_les =
wenzelm@23152
   512
  let
wenzelm@23152
   513
    fun dist (con1, args1) (con2, args2) =
wenzelm@23152
   514
      let
huffman@35117
   515
        fun iff_disj (t, []) = HOLogic.mk_not t
huffman@35117
   516
          | iff_disj (t, ts) = t === foldr1 HOLogic.mk_disj ts;
huffman@35117
   517
        val lhs = con_app con1 args1 << con_app con2 args2;
huffman@35117
   518
        val rhss = map (fn x => %:x === UU) (nonlazy args1);
huffman@35117
   519
        val goal = mk_trp (iff_disj (lhs, rhss));
huffman@35117
   520
        val rule1 = iso_locale RS @{thm iso.abs_below};
huffman@35117
   521
        val rules = rule1 :: @{thms con_below_iff_rules};
huffman@35117
   522
        val tacs = [simp_tac (HOL_ss addsimps rules) 1];
huffman@35117
   523
      in pg con_appls goal (K tacs) end;
wenzelm@23152
   524
wenzelm@23152
   525
    fun distinct (con1, args1) (con2, args2) =
wenzelm@23152
   526
        let
wenzelm@23152
   527
          val arg1 = (con1, args1);
wenzelm@23152
   528
          val arg2 =
wenzelm@23152
   529
            (con2, ListPair.map (fn (arg,vn) => upd_vname (K vn) arg)
wenzelm@23152
   530
              (args2, Name.variant_list (map vname args1) (map vname args2)));
wenzelm@23152
   531
        in [dist arg1 arg2, dist arg2 arg1] end;
wenzelm@23152
   532
    fun distincts []      = []
huffman@35117
   533
      | distincts (c::cs) = maps (distinct c) cs @ distincts cs;
wenzelm@23152
   534
  in distincts cons end;
huffman@29402
   535
huffman@29402
   536
val _ = trace " Proving dist_eqs...";
wenzelm@23152
   537
val dist_eqs =
wenzelm@23152
   538
  let
huffman@35117
   539
    fun dist (con1, args1) (con2, args2) =
wenzelm@23152
   540
      let
huffman@35117
   541
        fun iff_disj (t, [], us) = HOLogic.mk_not t
huffman@35117
   542
          | iff_disj (t, ts, []) = HOLogic.mk_not t
huffman@35117
   543
          | iff_disj (t, ts, us) =
huffman@35117
   544
            let
huffman@35117
   545
              val disj1 = foldr1 HOLogic.mk_disj ts;
huffman@35117
   546
              val disj2 = foldr1 HOLogic.mk_disj us;
huffman@35117
   547
            in t === HOLogic.mk_conj (disj1, disj2) end;
huffman@35117
   548
        val lhs = con_app con1 args1 === con_app con2 args2;
huffman@35117
   549
        val rhss1 = map (fn x => %:x === UU) (nonlazy args1);
huffman@35117
   550
        val rhss2 = map (fn x => %:x === UU) (nonlazy args2);
huffman@35117
   551
        val goal = mk_trp (iff_disj (lhs, rhss1, rhss2));
huffman@35117
   552
        val rule1 = iso_locale RS @{thm iso.abs_eq};
huffman@35117
   553
        val rules = rule1 :: @{thms con_eq_iff_rules};
huffman@35117
   554
        val tacs = [simp_tac (HOL_ss addsimps rules) 1];
huffman@35117
   555
      in pg con_appls goal (K tacs) end;
huffman@35117
   556
huffman@35117
   557
    fun distinct (con1, args1) (con2, args2) =
huffman@35117
   558
        let
huffman@35117
   559
          val arg1 = (con1, args1);
huffman@35117
   560
          val arg2 =
huffman@35117
   561
            (con2, ListPair.map (fn (arg,vn) => upd_vname (K vn) arg)
huffman@35117
   562
              (args2, Name.variant_list (map vname args1) (map vname args2)));
huffman@35117
   563
        in [dist arg1 arg2, dist arg2 arg1] end;
wenzelm@23152
   564
    fun distincts []      = []
huffman@35117
   565
      | distincts (c::cs) = maps (distinct c) cs @ distincts cs;
huffman@35117
   566
  in distincts cons end;
wenzelm@23152
   567
wenzelm@23152
   568
local 
wenzelm@23152
   569
  fun pgterm rel con args =
wenzelm@23152
   570
    let
wenzelm@23152
   571
      fun append s = upd_vname (fn v => v^s);
wenzelm@23152
   572
      val (largs, rargs) = (args, map (append "'") args);
wenzelm@23152
   573
      val concl =
wenzelm@23152
   574
        foldr1 mk_conj (ListPair.map rel (map %# largs, map %# rargs));
wenzelm@23152
   575
      val prem = rel (con_app con largs, con_app con rargs);
wenzelm@23152
   576
      val sargs = case largs of [_] => [] | _ => nonlazy args;
wenzelm@23152
   577
      val prop = lift_defined %: (sargs, mk_trp (prem === concl));
wenzelm@23152
   578
    in pg con_appls prop end;
wenzelm@33317
   579
  val cons' = filter (fn (_,args) => args<>[]) cons;
wenzelm@23152
   580
in
huffman@29402
   581
  val _ = trace " Proving inverts...";
wenzelm@23152
   582
  val inverts =
wenzelm@23152
   583
    let
wenzelm@23152
   584
      val abs_less = ax_abs_iso RS (allI RS injection_less);
wenzelm@23152
   585
      val tacs =
wenzelm@23152
   586
        [asm_full_simp_tac (HOLCF_ss addsimps [abs_less, spair_less]) 1];
wenzelm@27208
   587
    in map (fn (con, args) => pgterm (op <<) con args (K tacs)) cons' end;
wenzelm@23152
   588
huffman@29402
   589
  val _ = trace " Proving injects...";
wenzelm@23152
   590
  val injects =
wenzelm@23152
   591
    let
wenzelm@23152
   592
      val abs_eq = ax_abs_iso RS (allI RS injection_eq);
wenzelm@23152
   593
      val tacs = [asm_full_simp_tac (HOLCF_ss addsimps [abs_eq, spair_eq]) 1];
wenzelm@27208
   594
    in map (fn (con, args) => pgterm (op ===) con args (K tacs)) cons' end;
wenzelm@23152
   595
end;
wenzelm@23152
   596
wenzelm@23152
   597
(* ----- theorems concerning one induction step ----------------------------- *)
wenzelm@23152
   598
wenzelm@23152
   599
val copy_strict =
wenzelm@23152
   600
  let
huffman@31232
   601
    val _ = trace " Proving copy_strict...";
wenzelm@23152
   602
    val goal = mk_trp (strict (dc_copy `% "f"));
huffman@33504
   603
    val rules = [abs_strict, rep_strict] @ @{thms domain_map_stricts};
huffman@31232
   604
    val tacs = [asm_simp_tac (HOLCF_ss addsimps rules) 1];
huffman@35058
   605
  in
huffman@35058
   606
    SOME (pg [ax_copy_def] goal (K tacs))
huffman@35058
   607
    handle
huffman@35058
   608
      THM (s, _, _) => (trace s; NONE)
huffman@35058
   609
    | ERROR s => (trace s; NONE)
huffman@35058
   610
  end;
wenzelm@23152
   611
wenzelm@23152
   612
local
wenzelm@23152
   613
  fun copy_app (con, args) =
wenzelm@23152
   614
    let
wenzelm@23152
   615
      val lhs = dc_copy`%"f"`(con_app con args);
huffman@31232
   616
      fun one_rhs arg =
haftmann@33971
   617
          if Datatype_Aux.is_rec_type (dtyp_of arg)
huffman@33801
   618
          then Domain_Axioms.copy_of_dtyp map_tab
huffman@33801
   619
                 (proj (%:"f") eqs) (dtyp_of arg) ` (%# arg)
huffman@31232
   620
          else (%# arg);
huffman@31232
   621
      val rhs = con_app2 con one_rhs args;
huffman@35059
   622
      fun is_rec arg = Datatype_Aux.is_rec_type (dtyp_of arg);
huffman@35059
   623
      fun is_nonlazy_rec arg = is_rec arg andalso not (is_lazy arg);
huffman@35059
   624
      fun nonlazy_rec args = map vname (filter is_nonlazy_rec args);
wenzelm@23152
   625
      val goal = lift_defined %: (nonlazy_rec args, mk_trp (lhs === rhs));
wenzelm@33317
   626
      val args' = filter_out (fn a => is_rec a orelse is_lazy a) args;
huffman@33504
   627
      val stricts = abs_strict :: rep_strict :: @{thms domain_map_stricts};
wenzelm@27208
   628
      fun tacs1 ctxt = map (case_UU_tac ctxt stricts 1 o vname) args';
huffman@33504
   629
      val rules = [ax_abs_iso] @ @{thms domain_map_simps};
huffman@31232
   630
      val tacs2 = [asm_simp_tac (HOLCF_ss addsimps rules) 1];
huffman@31232
   631
    in pg (ax_copy_def::con_appls) goal (fn ctxt => (tacs1 ctxt @ tacs2)) end;
wenzelm@23152
   632
in
huffman@29402
   633
  val _ = trace " Proving copy_apps...";
wenzelm@23152
   634
  val copy_apps = map copy_app cons;
wenzelm@23152
   635
end;
wenzelm@23152
   636
wenzelm@23152
   637
local
wenzelm@23152
   638
  fun one_strict (con, args) = 
wenzelm@23152
   639
    let
wenzelm@23152
   640
      val goal = mk_trp (dc_copy`UU`(con_app con args) === UU);
huffman@35058
   641
      val rews = the_list copy_strict @ copy_apps @ con_rews;
wenzelm@27208
   642
      fun tacs ctxt = map (case_UU_tac ctxt rews 1) (nonlazy args) @
wenzelm@23152
   643
        [asm_simp_tac (HOLCF_ss addsimps rews) 1];
huffman@35058
   644
    in
huffman@35058
   645
      SOME (pg [] goal tacs)
huffman@35058
   646
      handle
huffman@35058
   647
        THM (s, _, _) => (trace s; NONE)
huffman@35058
   648
      | ERROR s => (trace s; NONE)
huffman@35058
   649
    end;
wenzelm@23152
   650
wenzelm@23152
   651
  fun has_nonlazy_rec (_, args) = exists is_nonlazy_rec args;
wenzelm@23152
   652
in
huffman@29402
   653
  val _ = trace " Proving copy_stricts...";
huffman@35058
   654
  val copy_stricts = map_filter one_strict (filter has_nonlazy_rec cons);
wenzelm@23152
   655
end;
wenzelm@23152
   656
huffman@35058
   657
val copy_rews = the_list copy_strict @ copy_apps @ copy_stricts;
wenzelm@23152
   658
wenzelm@23152
   659
in
wenzelm@23152
   660
  thy
wenzelm@30364
   661
    |> Sign.add_path (Long_Name.base_name dname)
huffman@31004
   662
    |> snd o PureThy.add_thmss [
huffman@31004
   663
        ((Binding.name "iso_rews"  , iso_rews    ), [Simplifier.simp_add]),
huffman@31004
   664
        ((Binding.name "exhaust"   , [exhaust]   ), []),
huffman@31004
   665
        ((Binding.name "casedist"  , [casedist]  ), [Induct.cases_type dname]),
huffman@31004
   666
        ((Binding.name "when_rews" , when_rews   ), [Simplifier.simp_add]),
huffman@31004
   667
        ((Binding.name "compacts"  , con_compacts), [Simplifier.simp_add]),
huffman@33427
   668
        ((Binding.name "con_rews"  , con_rews    ),
huffman@33427
   669
         [Simplifier.simp_add, Fixrec.fixrec_simp_add]),
huffman@31004
   670
        ((Binding.name "sel_rews"  , sel_rews    ), [Simplifier.simp_add]),
huffman@31004
   671
        ((Binding.name "dis_rews"  , dis_rews    ), [Simplifier.simp_add]),
huffman@31004
   672
        ((Binding.name "pat_rews"  , pat_rews    ), [Simplifier.simp_add]),
huffman@31004
   673
        ((Binding.name "dist_les"  , dist_les    ), [Simplifier.simp_add]),
huffman@31004
   674
        ((Binding.name "dist_eqs"  , dist_eqs    ), [Simplifier.simp_add]),
huffman@31004
   675
        ((Binding.name "inverts"   , inverts     ), [Simplifier.simp_add]),
huffman@31004
   676
        ((Binding.name "injects"   , injects     ), [Simplifier.simp_add]),
huffman@31004
   677
        ((Binding.name "copy_rews" , copy_rews   ), [Simplifier.simp_add]),
huffman@33427
   678
        ((Binding.name "match_rews", mat_rews    ),
huffman@33427
   679
         [Simplifier.simp_add, Fixrec.fixrec_simp_add])]
wenzelm@24712
   680
    |> Sign.parent_path
haftmann@28536
   681
    |> pair (iso_rews @ when_rews @ con_rews @ sel_rews @ dis_rews @
wenzelm@23152
   682
        pat_rews @ dist_les @ dist_eqs @ copy_rews)
wenzelm@23152
   683
end; (* let *)
wenzelm@23152
   684
wenzelm@23152
   685
fun comp_theorems (comp_dnam, eqs: eq list) thy =
wenzelm@23152
   686
let
wenzelm@27232
   687
val global_ctxt = ProofContext.init thy;
huffman@33801
   688
val map_tab = Domain_Isomorphism.get_map_tab thy;
wenzelm@27232
   689
wenzelm@23152
   690
val dnames = map (fst o fst) eqs;
wenzelm@23152
   691
val conss  = map  snd        eqs;
haftmann@28965
   692
val comp_dname = Sign.full_bname thy comp_dnam;
wenzelm@23152
   693
huffman@29402
   694
val _ = message ("Proving induction properties of domain "^comp_dname^" ...");
wenzelm@23152
   695
val pg = pg' thy;
wenzelm@23152
   696
wenzelm@23152
   697
(* ----- getting the composite axiom and definitions ------------------------ *)
wenzelm@23152
   698
wenzelm@23152
   699
local
wenzelm@26343
   700
  fun ga s dn = PureThy.get_thm thy (dn ^ "." ^ s);
wenzelm@23152
   701
in
wenzelm@23152
   702
  val axs_reach      = map (ga "reach"     ) dnames;
wenzelm@23152
   703
  val axs_take_def   = map (ga "take_def"  ) dnames;
wenzelm@23152
   704
  val axs_finite_def = map (ga "finite_def") dnames;
wenzelm@23152
   705
  val ax_copy2_def   =      ga "copy_def"  comp_dnam;
wenzelm@23152
   706
  val ax_bisim_def   =      ga "bisim_def" comp_dnam;
wenzelm@23152
   707
end;
wenzelm@23152
   708
wenzelm@23152
   709
local
wenzelm@26343
   710
  fun gt  s dn = PureThy.get_thm  thy (dn ^ "." ^ s);
wenzelm@26343
   711
  fun gts s dn = PureThy.get_thms thy (dn ^ "." ^ s);
wenzelm@23152
   712
in
wenzelm@23152
   713
  val cases = map (gt  "casedist" ) dnames;
wenzelm@26336
   714
  val con_rews  = maps (gts "con_rews" ) dnames;
wenzelm@26336
   715
  val copy_rews = maps (gts "copy_rews") dnames;
wenzelm@23152
   716
end;
wenzelm@23152
   717
wenzelm@23152
   718
fun dc_take dn = %%:(dn^"_take");
wenzelm@23152
   719
val x_name = idx_name dnames "x"; 
wenzelm@23152
   720
val P_name = idx_name dnames "P";
wenzelm@23152
   721
val n_eqs = length eqs;
wenzelm@23152
   722
wenzelm@23152
   723
(* ----- theorems concerning finite approximation and finite induction ------ *)
wenzelm@23152
   724
wenzelm@23152
   725
local
wenzelm@32149
   726
  val iterate_Cprod_ss = global_simpset_of @{theory Fix};
wenzelm@23152
   727
  val copy_con_rews  = copy_rews @ con_rews;
wenzelm@23152
   728
  val copy_take_defs =
wenzelm@23152
   729
    (if n_eqs = 1 then [] else [ax_copy2_def]) @ axs_take_def;
huffman@29402
   730
  val _ = trace " Proving take_stricts...";
huffman@35057
   731
  fun one_take_strict ((dn, args), _) =
wenzelm@23152
   732
    let
huffman@35057
   733
      val goal = mk_trp (strict (dc_take dn $ %:"n"));
huffman@35057
   734
      val rules = [
huffman@35057
   735
        @{thm monofun_fst [THEN monofunE]},
huffman@35057
   736
        @{thm monofun_snd [THEN monofunE]}];
huffman@35057
   737
      val tacs = [
huffman@35057
   738
        rtac @{thm UU_I} 1,
huffman@35057
   739
        rtac @{thm below_eq_trans} 1,
huffman@35057
   740
        resolve_tac axs_reach 2,
huffman@35057
   741
        rtac @{thm monofun_cfun_fun} 1,
huffman@35057
   742
        REPEAT (resolve_tac rules 1),
huffman@35057
   743
        rtac @{thm iterate_below_fix} 1];
huffman@35057
   744
    in pg axs_take_def goal (K tacs) end;
huffman@35057
   745
  val take_stricts = map one_take_strict eqs;
wenzelm@23152
   746
  fun take_0 n dn =
wenzelm@23152
   747
    let
huffman@35058
   748
      val goal = mk_trp ((dc_take dn $ @{term "0::nat"}) `% x_name n === UU);
wenzelm@27208
   749
    in pg axs_take_def goal (K [simp_tac iterate_Cprod_ss 1]) end;
wenzelm@23152
   750
  val take_0s = mapn take_0 1 dnames;
huffman@29402
   751
  val _ = trace " Proving take_apps...";
huffman@35058
   752
  fun one_take_app dn (con, args) =
wenzelm@23152
   753
    let
huffman@35058
   754
      fun mk_take n = dc_take (List.nth (dnames, n)) $ %:"n";
huffman@35058
   755
      fun one_rhs arg =
huffman@35058
   756
          if Datatype_Aux.is_rec_type (dtyp_of arg)
huffman@35058
   757
          then Domain_Axioms.copy_of_dtyp map_tab
huffman@35058
   758
                 mk_take (dtyp_of arg) ` (%# arg)
huffman@35058
   759
          else (%# arg);
huffman@35058
   760
      val lhs = (dc_take dn $ (%%:"Suc" $ %:"n"))`(con_app con args);
huffman@35058
   761
      val rhs = con_app2 con one_rhs args;
huffman@35059
   762
      fun is_rec arg = Datatype_Aux.is_rec_type (dtyp_of arg);
huffman@35059
   763
      fun is_nonlazy_rec arg = is_rec arg andalso not (is_lazy arg);
huffman@35059
   764
      fun nonlazy_rec args = map vname (filter is_nonlazy_rec args);
huffman@35059
   765
      val goal = lift_defined %: (nonlazy_rec args, mk_trp (lhs === rhs));
huffman@35059
   766
      val tacs = [asm_simp_tac (HOLCF_ss addsimps copy_con_rews) 1];
huffman@35059
   767
    in pg copy_take_defs goal (K tacs) end;
huffman@35058
   768
  fun one_take_apps ((dn, _), cons) = map (one_take_app dn) cons;
huffman@35058
   769
  val take_apps = maps one_take_apps eqs;
wenzelm@23152
   770
in
wenzelm@35021
   771
  val take_rews = map Drule.export_without_context
huffman@35058
   772
    (take_stricts @ take_0s @ take_apps);
wenzelm@23152
   773
end; (* local *)
wenzelm@23152
   774
wenzelm@23152
   775
local
wenzelm@23152
   776
  fun one_con p (con,args) =
wenzelm@23152
   777
    let
wenzelm@23152
   778
      fun ind_hyp arg = %:(P_name (1 + rec_of arg)) $ bound_arg args arg;
wenzelm@23152
   779
      val t1 = mk_trp (%:p $ con_app2 con (bound_arg args) args);
wenzelm@33317
   780
      val t2 = lift ind_hyp (filter is_rec args, t1);
wenzelm@23152
   781
      val t3 = lift_defined (bound_arg (map vname args)) (nonlazy args, t2);
wenzelm@23152
   782
    in Library.foldr mk_All (map vname args, t3) end;
wenzelm@23152
   783
wenzelm@23152
   784
  fun one_eq ((p, cons), concl) =
wenzelm@23152
   785
    mk_trp (%:p $ UU) ===> Logic.list_implies (map (one_con p) cons, concl);
wenzelm@23152
   786
wenzelm@23152
   787
  fun ind_term concf = Library.foldr one_eq
wenzelm@23152
   788
    (mapn (fn n => fn x => (P_name n, x)) 1 conss,
wenzelm@23152
   789
     mk_trp (foldr1 mk_conj (mapn concf 1 dnames)));
wenzelm@23152
   790
  val take_ss = HOL_ss addsimps take_rews;
wenzelm@27208
   791
  fun quant_tac ctxt i = EVERY
wenzelm@27239
   792
    (mapn (fn n => fn _ => res_inst_tac ctxt [(("x", 0), x_name n)] spec i) 1 dnames);
wenzelm@23152
   793
wenzelm@23152
   794
  fun ind_prems_tac prems = EVERY
wenzelm@26336
   795
    (maps (fn cons =>
wenzelm@23152
   796
      (resolve_tac prems 1 ::
wenzelm@26336
   797
        maps (fn (_,args) => 
wenzelm@23152
   798
          resolve_tac prems 1 ::
wenzelm@23152
   799
          map (K(atac 1)) (nonlazy args) @
wenzelm@33317
   800
          map (K(atac 1)) (filter is_rec args))
wenzelm@26336
   801
        cons))
wenzelm@26336
   802
      conss);
wenzelm@23152
   803
  local 
wenzelm@23152
   804
    (* check whether every/exists constructor of the n-th part of the equation:
wenzelm@23152
   805
       it has a possibly indirectly recursive argument that isn't/is possibly 
wenzelm@23152
   806
       indirectly lazy *)
wenzelm@23152
   807
    fun rec_to quant nfn rfn ns lazy_rec (n,cons) = quant (exists (fn arg => 
wenzelm@23152
   808
          is_rec arg andalso not(rec_of arg mem ns) andalso
wenzelm@23152
   809
          ((rec_of arg =  n andalso nfn(lazy_rec orelse is_lazy arg)) orelse 
wenzelm@23152
   810
            rec_of arg <> n andalso rec_to quant nfn rfn (rec_of arg::ns) 
wenzelm@23152
   811
              (lazy_rec orelse is_lazy arg) (n, (List.nth(conss,rec_of arg))))
wenzelm@23152
   812
          ) o snd) cons;
wenzelm@23152
   813
    fun all_rec_to ns  = rec_to forall not all_rec_to  ns;
wenzelm@23152
   814
    fun warn (n,cons) =
wenzelm@23152
   815
      if all_rec_to [] false (n,cons)
wenzelm@23152
   816
      then (warning ("domain "^List.nth(dnames,n)^" is empty!"); true)
wenzelm@23152
   817
      else false;
wenzelm@23152
   818
    fun lazy_rec_to ns = rec_to exists I  lazy_rec_to ns;
wenzelm@23152
   819
wenzelm@23152
   820
  in
wenzelm@23152
   821
    val n__eqs = mapn (fn n => fn (_,cons) => (n,cons)) 0 eqs;
wenzelm@23152
   822
    val is_emptys = map warn n__eqs;
wenzelm@23152
   823
    val is_finite = forall (not o lazy_rec_to [] false) n__eqs;
wenzelm@23152
   824
  end;
wenzelm@23152
   825
in (* local *)
huffman@29402
   826
  val _ = trace " Proving finite_ind...";
wenzelm@23152
   827
  val finite_ind =
wenzelm@23152
   828
    let
wenzelm@23152
   829
      fun concf n dn = %:(P_name n) $ (dc_take dn $ %:"n" `%(x_name n));
wenzelm@23152
   830
      val goal = ind_term concf;
wenzelm@23152
   831
wenzelm@27208
   832
      fun tacf {prems, context} =
wenzelm@23152
   833
        let
wenzelm@23152
   834
          val tacs1 = [
wenzelm@27208
   835
            quant_tac context 1,
wenzelm@23152
   836
            simp_tac HOL_ss 1,
wenzelm@27208
   837
            InductTacs.induct_tac context [[SOME "n"]] 1,
wenzelm@23152
   838
            simp_tac (take_ss addsimps prems) 1,
wenzelm@23152
   839
            TRY (safe_tac HOL_cs)];
wenzelm@23152
   840
          fun arg_tac arg =
wenzelm@27208
   841
            case_UU_tac context (prems @ con_rews) 1
wenzelm@23152
   842
              (List.nth (dnames, rec_of arg) ^ "_take n$" ^ vname arg);
wenzelm@23152
   843
          fun con_tacs (con, args) = 
wenzelm@23152
   844
            asm_simp_tac take_ss 1 ::
wenzelm@33317
   845
            map arg_tac (filter is_nonlazy_rec args) @
wenzelm@23152
   846
            [resolve_tac prems 1] @
wenzelm@33317
   847
            map (K (atac 1)) (nonlazy args) @
wenzelm@33317
   848
            map (K (etac spec 1)) (filter is_rec args);
wenzelm@23152
   849
          fun cases_tacs (cons, cases) =
wenzelm@27239
   850
            res_inst_tac context [(("x", 0), "x")] cases 1 ::
wenzelm@23152
   851
            asm_simp_tac (take_ss addsimps prems) 1 ::
wenzelm@26336
   852
            maps con_tacs cons;
wenzelm@23152
   853
        in
wenzelm@26336
   854
          tacs1 @ maps cases_tacs (conss ~~ cases)
wenzelm@23152
   855
        end;
huffman@31232
   856
    in pg'' thy [] goal tacf
huffman@31232
   857
       handle ERROR _ => (warning "Proof of finite_ind failed."; TrueI)
huffman@31232
   858
    end;
wenzelm@23152
   859
huffman@29402
   860
  val _ = trace " Proving take_lemmas...";
wenzelm@23152
   861
  val take_lemmas =
wenzelm@23152
   862
    let
wenzelm@23152
   863
      fun take_lemma n (dn, ax_reach) =
wenzelm@23152
   864
        let
wenzelm@23152
   865
          val lhs = dc_take dn $ Bound 0 `%(x_name n);
wenzelm@23152
   866
          val rhs = dc_take dn $ Bound 0 `%(x_name n^"'");
wenzelm@23152
   867
          val concl = mk_trp (%:(x_name n) === %:(x_name n^"'"));
wenzelm@23152
   868
          val goal = mk_All ("n", mk_trp (lhs === rhs)) ===> concl;
huffman@33396
   869
          val rules = [contlub_fst RS contlubE RS ssubst,
huffman@33396
   870
                       contlub_snd RS contlubE RS ssubst];
wenzelm@27208
   871
          fun tacf {prems, context} = [
wenzelm@27239
   872
            res_inst_tac context [(("t", 0), x_name n    )] (ax_reach RS subst) 1,
wenzelm@27239
   873
            res_inst_tac context [(("t", 0), x_name n^"'")] (ax_reach RS subst) 1,
wenzelm@23152
   874
            stac fix_def2 1,
wenzelm@23152
   875
            REPEAT (CHANGED
huffman@33396
   876
              (resolve_tac rules 1 THEN chain_tac 1)),
wenzelm@23152
   877
            stac contlub_cfun_fun 1,
wenzelm@23152
   878
            stac contlub_cfun_fun 2,
wenzelm@23152
   879
            rtac lub_equal 3,
wenzelm@23152
   880
            chain_tac 1,
wenzelm@23152
   881
            rtac allI 1,
wenzelm@23152
   882
            resolve_tac prems 1];
wenzelm@23152
   883
        in pg'' thy axs_take_def goal tacf end;
wenzelm@23152
   884
    in mapn take_lemma 1 (dnames ~~ axs_reach) end;
wenzelm@23152
   885
wenzelm@23152
   886
(* ----- theorems concerning finiteness and induction ----------------------- *)
wenzelm@23152
   887
huffman@29402
   888
  val _ = trace " Proving finites, ind...";
wenzelm@23152
   889
  val (finites, ind) =
huffman@31232
   890
  (
wenzelm@23152
   891
    if is_finite
wenzelm@23152
   892
    then (* finite case *)
wenzelm@23152
   893
      let 
wenzelm@23152
   894
        fun take_enough dn = mk_ex ("n",dc_take dn $ Bound 0 ` %:"x" === %:"x");
wenzelm@23152
   895
        fun dname_lemma dn =
wenzelm@23152
   896
          let
wenzelm@23152
   897
            val prem1 = mk_trp (defined (%:"x"));
wenzelm@23152
   898
            val disj1 = mk_all ("n", dc_take dn $ Bound 0 ` %:"x" === UU);
wenzelm@23152
   899
            val prem2 = mk_trp (mk_disj (disj1, take_enough dn));
wenzelm@23152
   900
            val concl = mk_trp (take_enough dn);
wenzelm@23152
   901
            val goal = prem1 ===> prem2 ===> concl;
wenzelm@23152
   902
            val tacs = [
wenzelm@23152
   903
              etac disjE 1,
wenzelm@23152
   904
              etac notE 1,
wenzelm@23152
   905
              resolve_tac take_lemmas 1,
wenzelm@23152
   906
              asm_simp_tac take_ss 1,
wenzelm@23152
   907
              atac 1];
wenzelm@27208
   908
          in pg [] goal (K tacs) end;
huffman@31232
   909
        val _ = trace " Proving finite_lemmas1a";
wenzelm@23152
   910
        val finite_lemmas1a = map dname_lemma dnames;
wenzelm@23152
   911
 
huffman@31232
   912
        val _ = trace " Proving finite_lemma1b";
wenzelm@23152
   913
        val finite_lemma1b =
wenzelm@23152
   914
          let
wenzelm@23152
   915
            fun mk_eqn n ((dn, args), _) =
wenzelm@23152
   916
              let
wenzelm@23152
   917
                val disj1 = dc_take dn $ Bound 1 ` Bound 0 === UU;
wenzelm@23152
   918
                val disj2 = dc_take dn $ Bound 1 ` Bound 0 === Bound 0;
wenzelm@23152
   919
              in
wenzelm@23152
   920
                mk_constrainall
wenzelm@23152
   921
                  (x_name n, Type (dn,args), mk_disj (disj1, disj2))
wenzelm@23152
   922
              end;
wenzelm@23152
   923
            val goal =
wenzelm@23152
   924
              mk_trp (mk_all ("n", foldr1 mk_conj (mapn mk_eqn 1 eqs)));
wenzelm@27208
   925
            fun arg_tacs ctxt vn = [
wenzelm@27239
   926
              eres_inst_tac ctxt [(("x", 0), vn)] all_dupE 1,
wenzelm@23152
   927
              etac disjE 1,
wenzelm@23152
   928
              asm_simp_tac (HOL_ss addsimps con_rews) 1,
wenzelm@23152
   929
              asm_simp_tac take_ss 1];
wenzelm@27208
   930
            fun con_tacs ctxt (con, args) =
wenzelm@23152
   931
              asm_simp_tac take_ss 1 ::
wenzelm@27208
   932
              maps (arg_tacs ctxt) (nonlazy_rec args);
wenzelm@27208
   933
            fun foo_tacs ctxt n (cons, cases) =
wenzelm@23152
   934
              simp_tac take_ss 1 ::
wenzelm@23152
   935
              rtac allI 1 ::
wenzelm@27239
   936
              res_inst_tac ctxt [(("x", 0), x_name n)] cases 1 ::
wenzelm@23152
   937
              asm_simp_tac take_ss 1 ::
wenzelm@27208
   938
              maps (con_tacs ctxt) cons;
wenzelm@27208
   939
            fun tacs ctxt =
wenzelm@23152
   940
              rtac allI 1 ::
wenzelm@27208
   941
              InductTacs.induct_tac ctxt [[SOME "n"]] 1 ::
wenzelm@23152
   942
              simp_tac take_ss 1 ::
wenzelm@23152
   943
              TRY (safe_tac (empty_cs addSEs [conjE] addSIs [conjI])) ::
wenzelm@27208
   944
              flat (mapn (foo_tacs ctxt) 1 (conss ~~ cases));
wenzelm@23152
   945
          in pg [] goal tacs end;
wenzelm@23152
   946
wenzelm@23152
   947
        fun one_finite (dn, l1b) =
wenzelm@23152
   948
          let
wenzelm@23152
   949
            val goal = mk_trp (%%:(dn^"_finite") $ %:"x");
wenzelm@27208
   950
            fun tacs ctxt = [
wenzelm@27208
   951
              case_UU_tac ctxt take_rews 1 "x",
wenzelm@23152
   952
              eresolve_tac finite_lemmas1a 1,
wenzelm@23152
   953
              step_tac HOL_cs 1,
wenzelm@23152
   954
              step_tac HOL_cs 1,
wenzelm@23152
   955
              cut_facts_tac [l1b] 1,
wenzelm@23152
   956
              fast_tac HOL_cs 1];
wenzelm@23152
   957
          in pg axs_finite_def goal tacs end;
wenzelm@23152
   958
huffman@31232
   959
        val _ = trace " Proving finites";
wenzelm@27232
   960
        val finites = map one_finite (dnames ~~ atomize global_ctxt finite_lemma1b);
huffman@31232
   961
        val _ = trace " Proving ind";
wenzelm@23152
   962
        val ind =
wenzelm@23152
   963
          let
wenzelm@23152
   964
            fun concf n dn = %:(P_name n) $ %:(x_name n);
wenzelm@27208
   965
            fun tacf {prems, context} =
wenzelm@23152
   966
              let
wenzelm@23152
   967
                fun finite_tacs (finite, fin_ind) = [
wenzelm@23152
   968
                  rtac(rewrite_rule axs_finite_def finite RS exE)1,
wenzelm@23152
   969
                  etac subst 1,
wenzelm@23152
   970
                  rtac fin_ind 1,
wenzelm@23152
   971
                  ind_prems_tac prems];
wenzelm@23152
   972
              in
wenzelm@23152
   973
                TRY (safe_tac HOL_cs) ::
wenzelm@27232
   974
                maps finite_tacs (finites ~~ atomize global_ctxt finite_ind)
wenzelm@23152
   975
              end;
wenzelm@23152
   976
          in pg'' thy [] (ind_term concf) tacf end;
wenzelm@23152
   977
      in (finites, ind) end (* let *)
wenzelm@23152
   978
wenzelm@23152
   979
    else (* infinite case *)
wenzelm@23152
   980
      let
wenzelm@23152
   981
        fun one_finite n dn =
wenzelm@27239
   982
          read_instantiate global_ctxt [(("P", 0), dn ^ "_finite " ^ x_name n)] excluded_middle;
wenzelm@23152
   983
        val finites = mapn one_finite 1 dnames;
wenzelm@23152
   984
wenzelm@23152
   985
        val goal =
wenzelm@23152
   986
          let
huffman@26012
   987
            fun one_adm n _ = mk_trp (mk_adm (%:(P_name n)));
wenzelm@23152
   988
            fun concf n dn = %:(P_name n) $ %:(x_name n);
wenzelm@23152
   989
          in Logic.list_implies (mapn one_adm 1 dnames, ind_term concf) end;
huffman@33396
   990
        val cont_rules =
huffman@33396
   991
            [cont_id, cont_const, cont2cont_Rep_CFun,
huffman@33396
   992
             cont2cont_fst, cont2cont_snd];
wenzelm@27208
   993
        fun tacf {prems, context} =
wenzelm@23152
   994
          map (fn ax_reach => rtac (ax_reach RS subst) 1) axs_reach @ [
wenzelm@27208
   995
          quant_tac context 1,
wenzelm@23152
   996
          rtac (adm_impl_admw RS wfix_ind) 1,
huffman@25895
   997
          REPEAT_DETERM (rtac adm_all 1),
wenzelm@23152
   998
          REPEAT_DETERM (
wenzelm@23152
   999
            TRY (rtac adm_conj 1) THEN 
wenzelm@23152
  1000
            rtac adm_subst 1 THEN 
huffman@33396
  1001
            REPEAT (resolve_tac cont_rules 1) THEN
huffman@33396
  1002
            resolve_tac prems 1),
wenzelm@23152
  1003
          strip_tac 1,
wenzelm@23152
  1004
          rtac (rewrite_rule axs_take_def finite_ind) 1,
wenzelm@23152
  1005
          ind_prems_tac prems];
wenzelm@23152
  1006
        val ind = (pg'' thy [] goal tacf
wenzelm@23152
  1007
          handle ERROR _ =>
huffman@33396
  1008
            (warning "Cannot prove infinite induction rule"; TrueI));
huffman@31232
  1009
      in (finites, ind) end
huffman@31232
  1010
  )
huffman@31232
  1011
      handle THM _ =>
huffman@31232
  1012
             (warning "Induction proofs failed (THM raised)."; ([], TrueI))
huffman@31232
  1013
           | ERROR _ =>
huffman@33810
  1014
             (warning "Cannot prove induction rule"; ([], TrueI));
huffman@31232
  1015
huffman@31232
  1016
wenzelm@23152
  1017
end; (* local *)
wenzelm@23152
  1018
wenzelm@23152
  1019
(* ----- theorem concerning coinduction ------------------------------------- *)
wenzelm@23152
  1020
wenzelm@23152
  1021
local
wenzelm@23152
  1022
  val xs = mapn (fn n => K (x_name n)) 1 dnames;
wenzelm@23152
  1023
  fun bnd_arg n i = Bound(2*(n_eqs - n)-i-1);
wenzelm@23152
  1024
  val take_ss = HOL_ss addsimps take_rews;
wenzelm@23152
  1025
  val sproj = prj (fn s => K("fst("^s^")")) (fn s => K("snd("^s^")"));
huffman@29402
  1026
  val _ = trace " Proving coind_lemma...";
wenzelm@23152
  1027
  val coind_lemma =
wenzelm@23152
  1028
    let
wenzelm@23152
  1029
      fun mk_prj n _ = proj (%:"R") eqs n $ bnd_arg n 0 $ bnd_arg n 1;
wenzelm@23152
  1030
      fun mk_eqn n dn =
wenzelm@23152
  1031
        (dc_take dn $ %:"n" ` bnd_arg n 0) ===
wenzelm@23152
  1032
        (dc_take dn $ %:"n" ` bnd_arg n 1);
wenzelm@23152
  1033
      fun mk_all2 (x,t) = mk_all (x, mk_all (x^"'", t));
wenzelm@23152
  1034
      val goal =
wenzelm@23152
  1035
        mk_trp (mk_imp (%%:(comp_dname^"_bisim") $ %:"R",
wenzelm@23152
  1036
          Library.foldr mk_all2 (xs,
wenzelm@23152
  1037
            Library.foldr mk_imp (mapn mk_prj 0 dnames,
wenzelm@23152
  1038
              foldr1 mk_conj (mapn mk_eqn 0 dnames)))));
wenzelm@27208
  1039
      fun x_tacs ctxt n x = [
wenzelm@23152
  1040
        rotate_tac (n+1) 1,
wenzelm@23152
  1041
        etac all2E 1,
wenzelm@27239
  1042
        eres_inst_tac ctxt [(("P", 1), sproj "R" eqs n^" "^x^" "^x^"'")] (mp RS disjE) 1,
wenzelm@23152
  1043
        TRY (safe_tac HOL_cs),
wenzelm@23152
  1044
        REPEAT (CHANGED (asm_simp_tac take_ss 1))];
wenzelm@27208
  1045
      fun tacs ctxt = [
wenzelm@23152
  1046
        rtac impI 1,
wenzelm@27208
  1047
        InductTacs.induct_tac ctxt [[SOME "n"]] 1,
wenzelm@23152
  1048
        simp_tac take_ss 1,
wenzelm@23152
  1049
        safe_tac HOL_cs] @
wenzelm@27208
  1050
        flat (mapn (x_tacs ctxt) 0 xs);
wenzelm@23152
  1051
    in pg [ax_bisim_def] goal tacs end;
wenzelm@23152
  1052
in
huffman@29402
  1053
  val _ = trace " Proving coind...";
wenzelm@23152
  1054
  val coind = 
wenzelm@23152
  1055
    let
wenzelm@23152
  1056
      fun mk_prj n x = mk_trp (proj (%:"R") eqs n $ %:x $ %:(x^"'"));
wenzelm@23152
  1057
      fun mk_eqn x = %:x === %:(x^"'");
wenzelm@23152
  1058
      val goal =
wenzelm@23152
  1059
        mk_trp (%%:(comp_dname^"_bisim") $ %:"R") ===>
wenzelm@23152
  1060
          Logic.list_implies (mapn mk_prj 0 xs,
wenzelm@23152
  1061
            mk_trp (foldr1 mk_conj (map mk_eqn xs)));
wenzelm@23152
  1062
      val tacs =
wenzelm@23152
  1063
        TRY (safe_tac HOL_cs) ::
wenzelm@26336
  1064
        maps (fn take_lemma => [
wenzelm@23152
  1065
          rtac take_lemma 1,
wenzelm@23152
  1066
          cut_facts_tac [coind_lemma] 1,
wenzelm@23152
  1067
          fast_tac HOL_cs 1])
wenzelm@26336
  1068
        take_lemmas;
wenzelm@27208
  1069
    in pg [] goal (K tacs) end;
wenzelm@23152
  1070
end; (* local *)
wenzelm@23152
  1071
wenzelm@32172
  1072
val inducts = Project_Rule.projections (ProofContext.init thy) ind;
huffman@30829
  1073
fun ind_rule (dname, rule) = ((Binding.empty, [rule]), [Induct.induct_type dname]);
huffman@31232
  1074
val induct_failed = (Thm.prop_of ind = Thm.prop_of TrueI);
huffman@30829
  1075
wenzelm@24712
  1076
in thy |> Sign.add_path comp_dnam
huffman@31004
  1077
       |> snd o PureThy.add_thmss [
huffman@31004
  1078
           ((Binding.name "take_rews"  , take_rews   ), [Simplifier.simp_add]),
huffman@31004
  1079
           ((Binding.name "take_lemmas", take_lemmas ), []),
huffman@31004
  1080
           ((Binding.name "finites"    , finites     ), []),
huffman@31004
  1081
           ((Binding.name "finite_ind" , [finite_ind]), []),
huffman@31004
  1082
           ((Binding.name "ind"        , [ind]       ), []),
huffman@31004
  1083
           ((Binding.name "coind"      , [coind]     ), [])]
huffman@31232
  1084
       |> (if induct_failed then I
huffman@31232
  1085
           else snd o PureThy.add_thmss (map ind_rule (dnames ~~ inducts)))
haftmann@28536
  1086
       |> Sign.parent_path |> pair take_rews
wenzelm@23152
  1087
end; (* let *)
wenzelm@23152
  1088
end; (* struct *)