src/HOLCF/Tools/Domain/domain_theorems.ML
author huffman
Sat Feb 27 18:09:11 2010 -0800 (2010-02-27)
changeset 35462 f5461b02d754
parent 35461 34360a1e3537
child 35464 2ae3362ba8ee
permissions -rw-r--r--
move definition of match combinators to domain_constructors.ML
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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:
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    Domain_Library.eq * Domain_Library.eq list
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    -> typ * (binding * (bool * binding option * typ) list * mixfix) list
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    -> 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_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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(* FIXME!!!!!!!!! *)
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(* We should NEVER re-parse variable names as strings! *)
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(* The names can conflict with existing constants or other syntax! *)
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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
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    (((dname, _), cons) : eq, eqs : eq list)
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    (dom_eqn : typ * (binding * (bool * binding option * typ) list * mixfix) list)
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    (thy : theory) =
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let
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val _ = message ("Proving isomorphism properties of domain "^dname^" ...");
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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_pat_def = map (get_def pat_name) cons;
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  val ax_copy_def = ga "copy_def" dname;
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end; (* local *)
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(* ----- define constructors ------------------------------------------------ *)
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val lhsT = fst dom_eqn;
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val rhsT =
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  let
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    fun mk_arg_typ (lazy, sel, T) = if lazy then mk_uT T else T;
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    fun mk_con_typ (bind, args, mx) =
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        if null args then oneT else foldr1 mk_sprodT (map mk_arg_typ args);
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    fun mk_eq_typ (_, cons) = foldr1 mk_ssumT (map mk_con_typ cons);
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  in
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    mk_eq_typ dom_eqn
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  end;
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val rep_const = Const(dname^"_rep", lhsT ->> rhsT);
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val abs_const = Const(dname^"_abs", rhsT ->> lhsT);
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val (result, thy) =
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  Domain_Constructors.add_domain_constructors
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    (Long_Name.base_name dname) dom_eqn
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    (rep_const, abs_const) (ax_rep_iso, ax_abs_iso) ax_when_def thy;
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val con_appls = #con_betas result;
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val {exhaust, casedist, ...} = result;
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val {con_compacts, con_rews, inverts, injects, dist_les, dist_eqs, ...} = result;
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val {sel_rews, ...} = result;
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val when_rews = #cases result;
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val when_strict = hd when_rews;
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val dis_rews = #dis_rews result;
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val axs_mat_def = #match_rews result;
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(* ----- theorems concerning the isomorphism -------------------------------- *)
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val pg = pg' thy;
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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 iso_rews = map Drule.export_without_context [ax_abs_iso, ax_rep_iso, abs_strict, rep_strict];
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(* ----- theorems concerning the constructors, discriminators and selectors - *)
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local
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  fun mat_strict (con, _, _) =
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    let
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      val goal = mk_trp (%%:(mat_name con) ` UU ` %:"rhs" === UU);
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      val tacs = [asm_simp_tac (HOLCF_ss addsimps [when_strict]) 1];
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    in pg axs_mat_def goal (K tacs) end;
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  val _ = trace " Proving mat_stricts...";
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  val mat_stricts = map mat_strict cons;
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  fun one_mat c (con, _, args) =
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    let
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      val lhs = %%:(mat_name c) ` con_app con args ` %:"rhs";
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      val rhs =
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        if con = c
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        then list_ccomb (%:"rhs", map %# args)
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        else mk_fail;
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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_mat_def goal (K tacs) end;
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  val _ = trace " Proving mat_apps...";
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  val mat_apps =
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    maps (fn (c,_,_) => map (one_mat c) cons) cons;
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in
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  val mat_rews = mat_stricts @ mat_apps;
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end;
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local
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  fun ps args = mapn (fn n => fn _ => %:("pat" ^ string_of_int n)) 1 args;
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  fun pat_lhs (con,_,args) = mk_branch (list_comb (%%:(pat_name con), ps args));
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  fun pat_rhs (con,_,[]) = mk_return ((%:"rhs") ` HOLogic.unit)
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    | pat_rhs (con,_,args) =
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        (mk_branch (mk_ctuple_pat (ps args)))
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          `(%:"rhs")`(mk_ctuple (map %# args));
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  fun pat_strict c =
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    let
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      val axs = @{thm branch_def} :: axs_pat_def;
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      val goal = mk_trp (strict (pat_lhs c ` (%:"rhs")));
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      val tacs = [simp_tac (HOLCF_ss addsimps [when_strict]) 1];
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    in pg axs goal (K tacs) end;
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  fun pat_app c (con, _, args) =
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    let
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      val axs = @{thm branch_def} :: axs_pat_def;
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      val lhs = (pat_lhs c)`(%:"rhs")`(con_app con args);
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      val rhs = if con = first c then pat_rhs c else mk_fail;
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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 goal (K tacs) end;
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  val _ = trace " Proving pat_stricts...";
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  val pat_stricts = map pat_strict cons;
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  val _ = trace " Proving pat_apps...";
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  val pat_apps = maps (fn c => map (pat_app c) cons) cons;
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in
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  val pat_rews = pat_stricts @ pat_apps;
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end;
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(* ----- theorems concerning one induction step ----------------------------- *)
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val copy_strict =
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  let
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    val _ = trace " Proving copy_strict...";
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    val goal = mk_trp (strict (dc_copy `% "f"));
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    val rules = [abs_strict, rep_strict] @ @{thms domain_map_stricts};
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    val tacs = [asm_simp_tac (HOLCF_ss addsimps rules) 1];
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  in
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    SOME (pg [ax_copy_def] goal (K tacs))
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    handle
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      THM (s, _, _) => (trace s; NONE)
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    | ERROR s => (trace s; NONE)
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  end;
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local
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  fun copy_app (con, _, args) =
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    let
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      val lhs = dc_copy`%"f"`(con_app con args);
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      fun one_rhs arg =
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          if Datatype_Aux.is_rec_type (dtyp_of arg)
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          then Domain_Axioms.copy_of_dtyp map_tab
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                 (proj (%:"f") eqs) (dtyp_of arg) ` (%# arg)
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          else (%# arg);
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      val rhs = con_app2 con one_rhs args;
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      fun is_rec arg = Datatype_Aux.is_rec_type (dtyp_of arg);
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      fun is_nonlazy_rec arg = is_rec arg andalso not (is_lazy arg);
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      fun nonlazy_rec args = map vname (filter is_nonlazy_rec args);
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      val goal = lift_defined %: (nonlazy_rec args, mk_trp (lhs === rhs));
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      val args' = filter_out (fn a => is_rec a orelse is_lazy a) args;
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      val stricts = abs_strict :: rep_strict :: @{thms domain_map_stricts};
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                        (* FIXME! case_UU_tac *)
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      fun tacs1 ctxt = map (case_UU_tac ctxt stricts 1 o vname) args';
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      val rules = [ax_abs_iso] @ @{thms domain_map_simps};
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      val tacs2 = [asm_simp_tac (HOLCF_ss addsimps rules) 1];
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    in pg (ax_copy_def::con_appls) goal (fn ctxt => (tacs1 ctxt @ tacs2)) end;
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in
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  val _ = trace " Proving copy_apps...";
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  val copy_apps = map copy_app cons;
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end;
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local
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  fun one_strict (con, _, args) = 
wenzelm@23152
   316
    let
wenzelm@23152
   317
      val goal = mk_trp (dc_copy`UU`(con_app con args) === UU);
huffman@35058
   318
      val rews = the_list copy_strict @ copy_apps @ con_rews;
huffman@35443
   319
                        (* FIXME! case_UU_tac *)
wenzelm@27208
   320
      fun tacs ctxt = map (case_UU_tac ctxt rews 1) (nonlazy args) @
wenzelm@23152
   321
        [asm_simp_tac (HOLCF_ss addsimps rews) 1];
huffman@35058
   322
    in
huffman@35058
   323
      SOME (pg [] goal tacs)
huffman@35058
   324
      handle
huffman@35058
   325
        THM (s, _, _) => (trace s; NONE)
huffman@35058
   326
      | ERROR s => (trace s; NONE)
huffman@35058
   327
    end;
wenzelm@23152
   328
huffman@35288
   329
  fun has_nonlazy_rec (_, _, args) = exists is_nonlazy_rec args;
wenzelm@23152
   330
in
huffman@29402
   331
  val _ = trace " Proving copy_stricts...";
huffman@35058
   332
  val copy_stricts = map_filter one_strict (filter has_nonlazy_rec cons);
wenzelm@23152
   333
end;
wenzelm@23152
   334
huffman@35058
   335
val copy_rews = the_list copy_strict @ copy_apps @ copy_stricts;
wenzelm@23152
   336
wenzelm@23152
   337
in
wenzelm@23152
   338
  thy
wenzelm@30364
   339
    |> Sign.add_path (Long_Name.base_name dname)
huffman@31004
   340
    |> snd o PureThy.add_thmss [
huffman@31004
   341
        ((Binding.name "iso_rews"  , iso_rews    ), [Simplifier.simp_add]),
huffman@31004
   342
        ((Binding.name "exhaust"   , [exhaust]   ), []),
huffman@31004
   343
        ((Binding.name "casedist"  , [casedist]  ), [Induct.cases_type dname]),
huffman@31004
   344
        ((Binding.name "when_rews" , when_rews   ), [Simplifier.simp_add]),
huffman@31004
   345
        ((Binding.name "compacts"  , con_compacts), [Simplifier.simp_add]),
huffman@33427
   346
        ((Binding.name "con_rews"  , con_rews    ),
huffman@33427
   347
         [Simplifier.simp_add, Fixrec.fixrec_simp_add]),
huffman@31004
   348
        ((Binding.name "sel_rews"  , sel_rews    ), [Simplifier.simp_add]),
huffman@31004
   349
        ((Binding.name "dis_rews"  , dis_rews    ), [Simplifier.simp_add]),
huffman@31004
   350
        ((Binding.name "pat_rews"  , pat_rews    ), [Simplifier.simp_add]),
huffman@31004
   351
        ((Binding.name "dist_les"  , dist_les    ), [Simplifier.simp_add]),
huffman@31004
   352
        ((Binding.name "dist_eqs"  , dist_eqs    ), [Simplifier.simp_add]),
huffman@31004
   353
        ((Binding.name "inverts"   , inverts     ), [Simplifier.simp_add]),
huffman@31004
   354
        ((Binding.name "injects"   , injects     ), [Simplifier.simp_add]),
huffman@31004
   355
        ((Binding.name "copy_rews" , copy_rews   ), [Simplifier.simp_add]),
huffman@33427
   356
        ((Binding.name "match_rews", mat_rews    ),
huffman@33427
   357
         [Simplifier.simp_add, Fixrec.fixrec_simp_add])]
wenzelm@24712
   358
    |> Sign.parent_path
haftmann@28536
   359
    |> pair (iso_rews @ when_rews @ con_rews @ sel_rews @ dis_rews @
wenzelm@23152
   360
        pat_rews @ dist_les @ dist_eqs @ copy_rews)
wenzelm@23152
   361
end; (* let *)
wenzelm@23152
   362
wenzelm@23152
   363
fun comp_theorems (comp_dnam, eqs: eq list) thy =
wenzelm@23152
   364
let
wenzelm@27232
   365
val global_ctxt = ProofContext.init thy;
huffman@33801
   366
val map_tab = Domain_Isomorphism.get_map_tab thy;
wenzelm@27232
   367
wenzelm@23152
   368
val dnames = map (fst o fst) eqs;
wenzelm@23152
   369
val conss  = map  snd        eqs;
haftmann@28965
   370
val comp_dname = Sign.full_bname thy comp_dnam;
wenzelm@23152
   371
huffman@29402
   372
val _ = message ("Proving induction properties of domain "^comp_dname^" ...");
wenzelm@23152
   373
val pg = pg' thy;
wenzelm@23152
   374
wenzelm@23152
   375
(* ----- getting the composite axiom and definitions ------------------------ *)
wenzelm@23152
   376
wenzelm@23152
   377
local
wenzelm@26343
   378
  fun ga s dn = PureThy.get_thm thy (dn ^ "." ^ s);
wenzelm@23152
   379
in
wenzelm@23152
   380
  val axs_reach      = map (ga "reach"     ) dnames;
wenzelm@23152
   381
  val axs_take_def   = map (ga "take_def"  ) dnames;
wenzelm@23152
   382
  val axs_finite_def = map (ga "finite_def") dnames;
wenzelm@23152
   383
  val ax_copy2_def   =      ga "copy_def"  comp_dnam;
huffman@35444
   384
(* TEMPORARILY DISABLED
wenzelm@23152
   385
  val ax_bisim_def   =      ga "bisim_def" comp_dnam;
huffman@35444
   386
TEMPORARILY DISABLED *)
wenzelm@23152
   387
end;
wenzelm@23152
   388
wenzelm@23152
   389
local
wenzelm@26343
   390
  fun gt  s dn = PureThy.get_thm  thy (dn ^ "." ^ s);
wenzelm@26343
   391
  fun gts s dn = PureThy.get_thms thy (dn ^ "." ^ s);
wenzelm@23152
   392
in
wenzelm@23152
   393
  val cases = map (gt  "casedist" ) dnames;
wenzelm@26336
   394
  val con_rews  = maps (gts "con_rews" ) dnames;
wenzelm@26336
   395
  val copy_rews = maps (gts "copy_rews") dnames;
wenzelm@23152
   396
end;
wenzelm@23152
   397
wenzelm@23152
   398
fun dc_take dn = %%:(dn^"_take");
wenzelm@23152
   399
val x_name = idx_name dnames "x"; 
wenzelm@23152
   400
val P_name = idx_name dnames "P";
wenzelm@23152
   401
val n_eqs = length eqs;
wenzelm@23152
   402
wenzelm@23152
   403
(* ----- theorems concerning finite approximation and finite induction ------ *)
wenzelm@23152
   404
wenzelm@23152
   405
local
wenzelm@32149
   406
  val iterate_Cprod_ss = global_simpset_of @{theory Fix};
wenzelm@23152
   407
  val copy_con_rews  = copy_rews @ con_rews;
wenzelm@23152
   408
  val copy_take_defs =
wenzelm@23152
   409
    (if n_eqs = 1 then [] else [ax_copy2_def]) @ axs_take_def;
huffman@29402
   410
  val _ = trace " Proving take_stricts...";
huffman@35057
   411
  fun one_take_strict ((dn, args), _) =
wenzelm@23152
   412
    let
huffman@35057
   413
      val goal = mk_trp (strict (dc_take dn $ %:"n"));
huffman@35057
   414
      val rules = [
huffman@35057
   415
        @{thm monofun_fst [THEN monofunE]},
huffman@35057
   416
        @{thm monofun_snd [THEN monofunE]}];
huffman@35057
   417
      val tacs = [
huffman@35057
   418
        rtac @{thm UU_I} 1,
huffman@35057
   419
        rtac @{thm below_eq_trans} 1,
huffman@35057
   420
        resolve_tac axs_reach 2,
huffman@35057
   421
        rtac @{thm monofun_cfun_fun} 1,
huffman@35057
   422
        REPEAT (resolve_tac rules 1),
huffman@35057
   423
        rtac @{thm iterate_below_fix} 1];
huffman@35057
   424
    in pg axs_take_def goal (K tacs) end;
huffman@35057
   425
  val take_stricts = map one_take_strict eqs;
wenzelm@23152
   426
  fun take_0 n dn =
wenzelm@23152
   427
    let
huffman@35058
   428
      val goal = mk_trp ((dc_take dn $ @{term "0::nat"}) `% x_name n === UU);
wenzelm@27208
   429
    in pg axs_take_def goal (K [simp_tac iterate_Cprod_ss 1]) end;
wenzelm@23152
   430
  val take_0s = mapn take_0 1 dnames;
huffman@29402
   431
  val _ = trace " Proving take_apps...";
huffman@35288
   432
  fun one_take_app dn (con, _, args) =
wenzelm@23152
   433
    let
huffman@35058
   434
      fun mk_take n = dc_take (List.nth (dnames, n)) $ %:"n";
huffman@35058
   435
      fun one_rhs arg =
huffman@35058
   436
          if Datatype_Aux.is_rec_type (dtyp_of arg)
huffman@35058
   437
          then Domain_Axioms.copy_of_dtyp map_tab
huffman@35058
   438
                 mk_take (dtyp_of arg) ` (%# arg)
huffman@35058
   439
          else (%# arg);
huffman@35058
   440
      val lhs = (dc_take dn $ (%%:"Suc" $ %:"n"))`(con_app con args);
huffman@35058
   441
      val rhs = con_app2 con one_rhs args;
huffman@35059
   442
      fun is_rec arg = Datatype_Aux.is_rec_type (dtyp_of arg);
huffman@35059
   443
      fun is_nonlazy_rec arg = is_rec arg andalso not (is_lazy arg);
huffman@35059
   444
      fun nonlazy_rec args = map vname (filter is_nonlazy_rec args);
huffman@35059
   445
      val goal = lift_defined %: (nonlazy_rec args, mk_trp (lhs === rhs));
huffman@35059
   446
      val tacs = [asm_simp_tac (HOLCF_ss addsimps copy_con_rews) 1];
huffman@35059
   447
    in pg copy_take_defs goal (K tacs) end;
huffman@35058
   448
  fun one_take_apps ((dn, _), cons) = map (one_take_app dn) cons;
huffman@35058
   449
  val take_apps = maps one_take_apps eqs;
wenzelm@23152
   450
in
wenzelm@35021
   451
  val take_rews = map Drule.export_without_context
huffman@35058
   452
    (take_stricts @ take_0s @ take_apps);
wenzelm@23152
   453
end; (* local *)
wenzelm@23152
   454
wenzelm@23152
   455
local
huffman@35288
   456
  fun one_con p (con, _, args) =
wenzelm@23152
   457
    let
huffman@35443
   458
      val P_names = map P_name (1 upto (length dnames));
huffman@35443
   459
      val vns = Name.variant_list P_names (map vname args);
huffman@35443
   460
      val nonlazy_vns = map snd (filter_out (is_lazy o fst) (args ~~ vns));
wenzelm@23152
   461
      fun ind_hyp arg = %:(P_name (1 + rec_of arg)) $ bound_arg args arg;
wenzelm@23152
   462
      val t1 = mk_trp (%:p $ con_app2 con (bound_arg args) args);
wenzelm@33317
   463
      val t2 = lift ind_hyp (filter is_rec args, t1);
huffman@35443
   464
      val t3 = lift_defined (bound_arg vns) (nonlazy_vns, t2);
huffman@35443
   465
    in Library.foldr mk_All (vns, t3) end;
wenzelm@23152
   466
wenzelm@23152
   467
  fun one_eq ((p, cons), concl) =
wenzelm@23152
   468
    mk_trp (%:p $ UU) ===> Logic.list_implies (map (one_con p) cons, concl);
wenzelm@23152
   469
wenzelm@23152
   470
  fun ind_term concf = Library.foldr one_eq
wenzelm@23152
   471
    (mapn (fn n => fn x => (P_name n, x)) 1 conss,
wenzelm@23152
   472
     mk_trp (foldr1 mk_conj (mapn concf 1 dnames)));
wenzelm@23152
   473
  val take_ss = HOL_ss addsimps take_rews;
wenzelm@27208
   474
  fun quant_tac ctxt i = EVERY
wenzelm@27239
   475
    (mapn (fn n => fn _ => res_inst_tac ctxt [(("x", 0), x_name n)] spec i) 1 dnames);
wenzelm@23152
   476
wenzelm@23152
   477
  fun ind_prems_tac prems = EVERY
wenzelm@26336
   478
    (maps (fn cons =>
wenzelm@23152
   479
      (resolve_tac prems 1 ::
huffman@35288
   480
        maps (fn (_,_,args) => 
wenzelm@23152
   481
          resolve_tac prems 1 ::
wenzelm@23152
   482
          map (K(atac 1)) (nonlazy args) @
wenzelm@33317
   483
          map (K(atac 1)) (filter is_rec args))
wenzelm@26336
   484
        cons))
wenzelm@26336
   485
      conss);
wenzelm@23152
   486
  local 
wenzelm@23152
   487
    (* check whether every/exists constructor of the n-th part of the equation:
wenzelm@23152
   488
       it has a possibly indirectly recursive argument that isn't/is possibly 
wenzelm@23152
   489
       indirectly lazy *)
wenzelm@23152
   490
    fun rec_to quant nfn rfn ns lazy_rec (n,cons) = quant (exists (fn arg => 
wenzelm@23152
   491
          is_rec arg andalso not(rec_of arg mem ns) andalso
wenzelm@23152
   492
          ((rec_of arg =  n andalso nfn(lazy_rec orelse is_lazy arg)) orelse 
wenzelm@23152
   493
            rec_of arg <> n andalso rec_to quant nfn rfn (rec_of arg::ns) 
wenzelm@23152
   494
              (lazy_rec orelse is_lazy arg) (n, (List.nth(conss,rec_of arg))))
huffman@35288
   495
          ) o third) cons;
wenzelm@23152
   496
    fun all_rec_to ns  = rec_to forall not all_rec_to  ns;
wenzelm@23152
   497
    fun warn (n,cons) =
wenzelm@23152
   498
      if all_rec_to [] false (n,cons)
wenzelm@23152
   499
      then (warning ("domain "^List.nth(dnames,n)^" is empty!"); true)
wenzelm@23152
   500
      else false;
wenzelm@23152
   501
    fun lazy_rec_to ns = rec_to exists I  lazy_rec_to ns;
wenzelm@23152
   502
wenzelm@23152
   503
  in
wenzelm@23152
   504
    val n__eqs = mapn (fn n => fn (_,cons) => (n,cons)) 0 eqs;
wenzelm@23152
   505
    val is_emptys = map warn n__eqs;
wenzelm@23152
   506
    val is_finite = forall (not o lazy_rec_to [] false) n__eqs;
wenzelm@23152
   507
  end;
wenzelm@23152
   508
in (* local *)
huffman@29402
   509
  val _ = trace " Proving finite_ind...";
wenzelm@23152
   510
  val finite_ind =
wenzelm@23152
   511
    let
wenzelm@23152
   512
      fun concf n dn = %:(P_name n) $ (dc_take dn $ %:"n" `%(x_name n));
wenzelm@23152
   513
      val goal = ind_term concf;
wenzelm@23152
   514
wenzelm@27208
   515
      fun tacf {prems, context} =
wenzelm@23152
   516
        let
wenzelm@23152
   517
          val tacs1 = [
wenzelm@27208
   518
            quant_tac context 1,
wenzelm@23152
   519
            simp_tac HOL_ss 1,
wenzelm@27208
   520
            InductTacs.induct_tac context [[SOME "n"]] 1,
wenzelm@23152
   521
            simp_tac (take_ss addsimps prems) 1,
wenzelm@23152
   522
            TRY (safe_tac HOL_cs)];
wenzelm@23152
   523
          fun arg_tac arg =
huffman@35443
   524
                        (* FIXME! case_UU_tac *)
wenzelm@27208
   525
            case_UU_tac context (prems @ con_rews) 1
wenzelm@23152
   526
              (List.nth (dnames, rec_of arg) ^ "_take n$" ^ vname arg);
huffman@35288
   527
          fun con_tacs (con, _, args) = 
wenzelm@23152
   528
            asm_simp_tac take_ss 1 ::
wenzelm@33317
   529
            map arg_tac (filter is_nonlazy_rec args) @
wenzelm@23152
   530
            [resolve_tac prems 1] @
wenzelm@33317
   531
            map (K (atac 1)) (nonlazy args) @
wenzelm@33317
   532
            map (K (etac spec 1)) (filter is_rec args);
wenzelm@23152
   533
          fun cases_tacs (cons, cases) =
wenzelm@27239
   534
            res_inst_tac context [(("x", 0), "x")] cases 1 ::
wenzelm@23152
   535
            asm_simp_tac (take_ss addsimps prems) 1 ::
wenzelm@26336
   536
            maps con_tacs cons;
wenzelm@23152
   537
        in
wenzelm@26336
   538
          tacs1 @ maps cases_tacs (conss ~~ cases)
wenzelm@23152
   539
        end;
huffman@31232
   540
    in pg'' thy [] goal tacf
huffman@31232
   541
       handle ERROR _ => (warning "Proof of finite_ind failed."; TrueI)
huffman@31232
   542
    end;
wenzelm@23152
   543
huffman@29402
   544
  val _ = trace " Proving take_lemmas...";
wenzelm@23152
   545
  val take_lemmas =
wenzelm@23152
   546
    let
wenzelm@23152
   547
      fun take_lemma n (dn, ax_reach) =
wenzelm@23152
   548
        let
wenzelm@23152
   549
          val lhs = dc_take dn $ Bound 0 `%(x_name n);
wenzelm@23152
   550
          val rhs = dc_take dn $ Bound 0 `%(x_name n^"'");
wenzelm@23152
   551
          val concl = mk_trp (%:(x_name n) === %:(x_name n^"'"));
wenzelm@23152
   552
          val goal = mk_All ("n", mk_trp (lhs === rhs)) ===> concl;
huffman@33396
   553
          val rules = [contlub_fst RS contlubE RS ssubst,
huffman@33396
   554
                       contlub_snd RS contlubE RS ssubst];
wenzelm@27208
   555
          fun tacf {prems, context} = [
wenzelm@27239
   556
            res_inst_tac context [(("t", 0), x_name n    )] (ax_reach RS subst) 1,
wenzelm@27239
   557
            res_inst_tac context [(("t", 0), x_name n^"'")] (ax_reach RS subst) 1,
wenzelm@23152
   558
            stac fix_def2 1,
wenzelm@23152
   559
            REPEAT (CHANGED
huffman@33396
   560
              (resolve_tac rules 1 THEN chain_tac 1)),
wenzelm@23152
   561
            stac contlub_cfun_fun 1,
wenzelm@23152
   562
            stac contlub_cfun_fun 2,
wenzelm@23152
   563
            rtac lub_equal 3,
wenzelm@23152
   564
            chain_tac 1,
wenzelm@23152
   565
            rtac allI 1,
wenzelm@23152
   566
            resolve_tac prems 1];
wenzelm@23152
   567
        in pg'' thy axs_take_def goal tacf end;
wenzelm@23152
   568
    in mapn take_lemma 1 (dnames ~~ axs_reach) end;
wenzelm@23152
   569
wenzelm@23152
   570
(* ----- theorems concerning finiteness and induction ----------------------- *)
wenzelm@23152
   571
huffman@29402
   572
  val _ = trace " Proving finites, ind...";
wenzelm@23152
   573
  val (finites, ind) =
huffman@31232
   574
  (
wenzelm@23152
   575
    if is_finite
wenzelm@23152
   576
    then (* finite case *)
wenzelm@23152
   577
      let 
wenzelm@23152
   578
        fun take_enough dn = mk_ex ("n",dc_take dn $ Bound 0 ` %:"x" === %:"x");
wenzelm@23152
   579
        fun dname_lemma dn =
wenzelm@23152
   580
          let
wenzelm@23152
   581
            val prem1 = mk_trp (defined (%:"x"));
wenzelm@23152
   582
            val disj1 = mk_all ("n", dc_take dn $ Bound 0 ` %:"x" === UU);
wenzelm@23152
   583
            val prem2 = mk_trp (mk_disj (disj1, take_enough dn));
wenzelm@23152
   584
            val concl = mk_trp (take_enough dn);
wenzelm@23152
   585
            val goal = prem1 ===> prem2 ===> concl;
wenzelm@23152
   586
            val tacs = [
wenzelm@23152
   587
              etac disjE 1,
wenzelm@23152
   588
              etac notE 1,
wenzelm@23152
   589
              resolve_tac take_lemmas 1,
wenzelm@23152
   590
              asm_simp_tac take_ss 1,
wenzelm@23152
   591
              atac 1];
wenzelm@27208
   592
          in pg [] goal (K tacs) end;
huffman@31232
   593
        val _ = trace " Proving finite_lemmas1a";
wenzelm@23152
   594
        val finite_lemmas1a = map dname_lemma dnames;
wenzelm@23152
   595
 
huffman@31232
   596
        val _ = trace " Proving finite_lemma1b";
wenzelm@23152
   597
        val finite_lemma1b =
wenzelm@23152
   598
          let
wenzelm@23152
   599
            fun mk_eqn n ((dn, args), _) =
wenzelm@23152
   600
              let
wenzelm@23152
   601
                val disj1 = dc_take dn $ Bound 1 ` Bound 0 === UU;
wenzelm@23152
   602
                val disj2 = dc_take dn $ Bound 1 ` Bound 0 === Bound 0;
wenzelm@23152
   603
              in
wenzelm@23152
   604
                mk_constrainall
wenzelm@23152
   605
                  (x_name n, Type (dn,args), mk_disj (disj1, disj2))
wenzelm@23152
   606
              end;
wenzelm@23152
   607
            val goal =
wenzelm@23152
   608
              mk_trp (mk_all ("n", foldr1 mk_conj (mapn mk_eqn 1 eqs)));
wenzelm@27208
   609
            fun arg_tacs ctxt vn = [
wenzelm@27239
   610
              eres_inst_tac ctxt [(("x", 0), vn)] all_dupE 1,
wenzelm@23152
   611
              etac disjE 1,
wenzelm@23152
   612
              asm_simp_tac (HOL_ss addsimps con_rews) 1,
wenzelm@23152
   613
              asm_simp_tac take_ss 1];
huffman@35288
   614
            fun con_tacs ctxt (con, _, args) =
wenzelm@23152
   615
              asm_simp_tac take_ss 1 ::
wenzelm@27208
   616
              maps (arg_tacs ctxt) (nonlazy_rec args);
wenzelm@27208
   617
            fun foo_tacs ctxt n (cons, cases) =
wenzelm@23152
   618
              simp_tac take_ss 1 ::
wenzelm@23152
   619
              rtac allI 1 ::
wenzelm@27239
   620
              res_inst_tac ctxt [(("x", 0), x_name n)] cases 1 ::
wenzelm@23152
   621
              asm_simp_tac take_ss 1 ::
wenzelm@27208
   622
              maps (con_tacs ctxt) cons;
wenzelm@27208
   623
            fun tacs ctxt =
wenzelm@23152
   624
              rtac allI 1 ::
wenzelm@27208
   625
              InductTacs.induct_tac ctxt [[SOME "n"]] 1 ::
wenzelm@23152
   626
              simp_tac take_ss 1 ::
wenzelm@23152
   627
              TRY (safe_tac (empty_cs addSEs [conjE] addSIs [conjI])) ::
wenzelm@27208
   628
              flat (mapn (foo_tacs ctxt) 1 (conss ~~ cases));
wenzelm@23152
   629
          in pg [] goal tacs end;
wenzelm@23152
   630
wenzelm@23152
   631
        fun one_finite (dn, l1b) =
wenzelm@23152
   632
          let
wenzelm@23152
   633
            val goal = mk_trp (%%:(dn^"_finite") $ %:"x");
wenzelm@27208
   634
            fun tacs ctxt = [
huffman@35443
   635
                        (* FIXME! case_UU_tac *)
wenzelm@27208
   636
              case_UU_tac ctxt take_rews 1 "x",
wenzelm@23152
   637
              eresolve_tac finite_lemmas1a 1,
wenzelm@23152
   638
              step_tac HOL_cs 1,
wenzelm@23152
   639
              step_tac HOL_cs 1,
wenzelm@23152
   640
              cut_facts_tac [l1b] 1,
wenzelm@23152
   641
              fast_tac HOL_cs 1];
wenzelm@23152
   642
          in pg axs_finite_def goal tacs end;
wenzelm@23152
   643
huffman@31232
   644
        val _ = trace " Proving finites";
wenzelm@27232
   645
        val finites = map one_finite (dnames ~~ atomize global_ctxt finite_lemma1b);
huffman@31232
   646
        val _ = trace " Proving ind";
wenzelm@23152
   647
        val ind =
wenzelm@23152
   648
          let
wenzelm@23152
   649
            fun concf n dn = %:(P_name n) $ %:(x_name n);
wenzelm@27208
   650
            fun tacf {prems, context} =
wenzelm@23152
   651
              let
wenzelm@23152
   652
                fun finite_tacs (finite, fin_ind) = [
wenzelm@23152
   653
                  rtac(rewrite_rule axs_finite_def finite RS exE)1,
wenzelm@23152
   654
                  etac subst 1,
wenzelm@23152
   655
                  rtac fin_ind 1,
wenzelm@23152
   656
                  ind_prems_tac prems];
wenzelm@23152
   657
              in
wenzelm@23152
   658
                TRY (safe_tac HOL_cs) ::
wenzelm@27232
   659
                maps finite_tacs (finites ~~ atomize global_ctxt finite_ind)
wenzelm@23152
   660
              end;
wenzelm@23152
   661
          in pg'' thy [] (ind_term concf) tacf end;
wenzelm@23152
   662
      in (finites, ind) end (* let *)
wenzelm@23152
   663
wenzelm@23152
   664
    else (* infinite case *)
wenzelm@23152
   665
      let
wenzelm@23152
   666
        fun one_finite n dn =
wenzelm@27239
   667
          read_instantiate global_ctxt [(("P", 0), dn ^ "_finite " ^ x_name n)] excluded_middle;
wenzelm@23152
   668
        val finites = mapn one_finite 1 dnames;
wenzelm@23152
   669
wenzelm@23152
   670
        val goal =
wenzelm@23152
   671
          let
huffman@26012
   672
            fun one_adm n _ = mk_trp (mk_adm (%:(P_name n)));
wenzelm@23152
   673
            fun concf n dn = %:(P_name n) $ %:(x_name n);
wenzelm@23152
   674
          in Logic.list_implies (mapn one_adm 1 dnames, ind_term concf) end;
huffman@33396
   675
        val cont_rules =
huffman@33396
   676
            [cont_id, cont_const, cont2cont_Rep_CFun,
huffman@33396
   677
             cont2cont_fst, cont2cont_snd];
wenzelm@27208
   678
        fun tacf {prems, context} =
wenzelm@23152
   679
          map (fn ax_reach => rtac (ax_reach RS subst) 1) axs_reach @ [
wenzelm@27208
   680
          quant_tac context 1,
wenzelm@23152
   681
          rtac (adm_impl_admw RS wfix_ind) 1,
huffman@25895
   682
          REPEAT_DETERM (rtac adm_all 1),
wenzelm@23152
   683
          REPEAT_DETERM (
wenzelm@23152
   684
            TRY (rtac adm_conj 1) THEN 
wenzelm@23152
   685
            rtac adm_subst 1 THEN 
huffman@33396
   686
            REPEAT (resolve_tac cont_rules 1) THEN
huffman@33396
   687
            resolve_tac prems 1),
wenzelm@23152
   688
          strip_tac 1,
wenzelm@23152
   689
          rtac (rewrite_rule axs_take_def finite_ind) 1,
wenzelm@23152
   690
          ind_prems_tac prems];
wenzelm@23152
   691
        val ind = (pg'' thy [] goal tacf
wenzelm@23152
   692
          handle ERROR _ =>
huffman@33396
   693
            (warning "Cannot prove infinite induction rule"; TrueI));
huffman@31232
   694
      in (finites, ind) end
huffman@31232
   695
  )
huffman@31232
   696
      handle THM _ =>
huffman@31232
   697
             (warning "Induction proofs failed (THM raised)."; ([], TrueI))
huffman@31232
   698
           | ERROR _ =>
huffman@33810
   699
             (warning "Cannot prove induction rule"; ([], TrueI));
huffman@31232
   700
huffman@31232
   701
wenzelm@23152
   702
end; (* local *)
wenzelm@23152
   703
wenzelm@23152
   704
(* ----- theorem concerning coinduction ------------------------------------- *)
wenzelm@23152
   705
huffman@35444
   706
(* COINDUCTION TEMPORARILY DISABLED
wenzelm@23152
   707
local
wenzelm@23152
   708
  val xs = mapn (fn n => K (x_name n)) 1 dnames;
wenzelm@23152
   709
  fun bnd_arg n i = Bound(2*(n_eqs - n)-i-1);
wenzelm@23152
   710
  val take_ss = HOL_ss addsimps take_rews;
wenzelm@23152
   711
  val sproj = prj (fn s => K("fst("^s^")")) (fn s => K("snd("^s^")"));
huffman@29402
   712
  val _ = trace " Proving coind_lemma...";
wenzelm@23152
   713
  val coind_lemma =
wenzelm@23152
   714
    let
wenzelm@23152
   715
      fun mk_prj n _ = proj (%:"R") eqs n $ bnd_arg n 0 $ bnd_arg n 1;
wenzelm@23152
   716
      fun mk_eqn n dn =
wenzelm@23152
   717
        (dc_take dn $ %:"n" ` bnd_arg n 0) ===
wenzelm@23152
   718
        (dc_take dn $ %:"n" ` bnd_arg n 1);
wenzelm@23152
   719
      fun mk_all2 (x,t) = mk_all (x, mk_all (x^"'", t));
wenzelm@23152
   720
      val goal =
wenzelm@23152
   721
        mk_trp (mk_imp (%%:(comp_dname^"_bisim") $ %:"R",
wenzelm@23152
   722
          Library.foldr mk_all2 (xs,
wenzelm@23152
   723
            Library.foldr mk_imp (mapn mk_prj 0 dnames,
wenzelm@23152
   724
              foldr1 mk_conj (mapn mk_eqn 0 dnames)))));
wenzelm@27208
   725
      fun x_tacs ctxt n x = [
wenzelm@23152
   726
        rotate_tac (n+1) 1,
wenzelm@23152
   727
        etac all2E 1,
wenzelm@27239
   728
        eres_inst_tac ctxt [(("P", 1), sproj "R" eqs n^" "^x^" "^x^"'")] (mp RS disjE) 1,
wenzelm@23152
   729
        TRY (safe_tac HOL_cs),
wenzelm@23152
   730
        REPEAT (CHANGED (asm_simp_tac take_ss 1))];
wenzelm@27208
   731
      fun tacs ctxt = [
wenzelm@23152
   732
        rtac impI 1,
wenzelm@27208
   733
        InductTacs.induct_tac ctxt [[SOME "n"]] 1,
wenzelm@23152
   734
        simp_tac take_ss 1,
wenzelm@23152
   735
        safe_tac HOL_cs] @
wenzelm@27208
   736
        flat (mapn (x_tacs ctxt) 0 xs);
wenzelm@23152
   737
    in pg [ax_bisim_def] goal tacs end;
wenzelm@23152
   738
in
huffman@29402
   739
  val _ = trace " Proving coind...";
wenzelm@23152
   740
  val coind = 
wenzelm@23152
   741
    let
wenzelm@23152
   742
      fun mk_prj n x = mk_trp (proj (%:"R") eqs n $ %:x $ %:(x^"'"));
wenzelm@23152
   743
      fun mk_eqn x = %:x === %:(x^"'");
wenzelm@23152
   744
      val goal =
wenzelm@23152
   745
        mk_trp (%%:(comp_dname^"_bisim") $ %:"R") ===>
wenzelm@23152
   746
          Logic.list_implies (mapn mk_prj 0 xs,
wenzelm@23152
   747
            mk_trp (foldr1 mk_conj (map mk_eqn xs)));
wenzelm@23152
   748
      val tacs =
wenzelm@23152
   749
        TRY (safe_tac HOL_cs) ::
wenzelm@26336
   750
        maps (fn take_lemma => [
wenzelm@23152
   751
          rtac take_lemma 1,
wenzelm@23152
   752
          cut_facts_tac [coind_lemma] 1,
wenzelm@23152
   753
          fast_tac HOL_cs 1])
wenzelm@26336
   754
        take_lemmas;
wenzelm@27208
   755
    in pg [] goal (K tacs) end;
wenzelm@23152
   756
end; (* local *)
huffman@35444
   757
COINDUCTION TEMPORARILY DISABLED *)
wenzelm@23152
   758
wenzelm@32172
   759
val inducts = Project_Rule.projections (ProofContext.init thy) ind;
huffman@30829
   760
fun ind_rule (dname, rule) = ((Binding.empty, [rule]), [Induct.induct_type dname]);
huffman@31232
   761
val induct_failed = (Thm.prop_of ind = Thm.prop_of TrueI);
huffman@30829
   762
wenzelm@24712
   763
in thy |> Sign.add_path comp_dnam
huffman@31004
   764
       |> snd o PureThy.add_thmss [
huffman@31004
   765
           ((Binding.name "take_rews"  , take_rews   ), [Simplifier.simp_add]),
huffman@31004
   766
           ((Binding.name "take_lemmas", take_lemmas ), []),
huffman@31004
   767
           ((Binding.name "finites"    , finites     ), []),
huffman@31004
   768
           ((Binding.name "finite_ind" , [finite_ind]), []),
huffman@35444
   769
           ((Binding.name "ind"        , [ind]       ), [])(*,
huffman@35444
   770
           ((Binding.name "coind"      , [coind]     ), [])*)]
huffman@31232
   771
       |> (if induct_failed then I
huffman@31232
   772
           else snd o PureThy.add_thmss (map ind_rule (dnames ~~ inducts)))
haftmann@28536
   773
       |> Sign.parent_path |> pair take_rews
wenzelm@23152
   774
end; (* let *)
wenzelm@23152
   775
end; (* struct *)