src/HOLCF/Tools/Domain/domain_theorems.ML
author huffman
Fri Feb 26 09:13:29 2010 -0800 (2010-02-26)
changeset 35451 a726a033b313
parent 35448 f9f73f0475eb
child 35452 cf8c5a751a9a
permissions -rw-r--r--
don't bother returning con_defs
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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_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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(* 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_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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(*
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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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*)
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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) thy;
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val con_appls = #con_betas result;
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val con_compacts = #con_compacts result;
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val sel_rews = #sel_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 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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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 (third con)
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    | cons2typ n (con::cons) =
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    let
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      val (n1, t1) = args2typ n (third 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 = Name.variant_list ["x"] (map vname args);
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      val nonlazy_vns = map snd (filter_out (is_lazy o fst) (args ~~ vns));
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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_vns);
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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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   319
      fun one_when n (con, _, args) =
wenzelm@23152
   320
        let
wenzelm@23152
   321
          val axs = when_appl :: con_appls;
wenzelm@23152
   322
          val goal = bind_fun (lift_defined %: (nonlazy args, 
wenzelm@23152
   323
                mk_trp (when_app`(con_app con args) ===
wenzelm@23152
   324
                       list_ccomb (bound_fun n 0, map %# args))));
wenzelm@23152
   325
          val tacs = [asm_simp_tac (HOLCF_ss addsimps [ax_abs_iso]) 1];
wenzelm@27208
   326
        in pg axs goal (K tacs) end;
wenzelm@23152
   327
    in mapn one_when 1 cons end;
wenzelm@23152
   328
end;
wenzelm@23152
   329
val when_rews = when_strict :: when_apps;
wenzelm@23152
   330
wenzelm@23152
   331
(* ----- theorems concerning the constructors, discriminators and selectors - *)
wenzelm@23152
   332
wenzelm@23152
   333
local
huffman@35288
   334
  fun dis_strict (con, _, _) =
wenzelm@23152
   335
    let
wenzelm@23152
   336
      val goal = mk_trp (strict (%%:(dis_name con)));
wenzelm@27208
   337
    in pg axs_dis_def goal (K [rtac when_strict 1]) end;
wenzelm@23152
   338
huffman@35288
   339
  fun dis_app c (con, _, args) =
wenzelm@23152
   340
    let
wenzelm@23152
   341
      val lhs = %%:(dis_name c) ` con_app con args;
huffman@26012
   342
      val rhs = if con = c then TT else FF;
wenzelm@23152
   343
      val goal = lift_defined %: (nonlazy args, mk_trp (lhs === rhs));
wenzelm@23152
   344
      val tacs = [asm_simp_tac (HOLCF_ss addsimps when_rews) 1];
wenzelm@27208
   345
    in pg axs_dis_def goal (K tacs) end;
wenzelm@23152
   346
huffman@29402
   347
  val _ = trace " Proving dis_apps...";
huffman@35288
   348
  val dis_apps = maps (fn (c,_,_) => map (dis_app c) cons) cons;
wenzelm@23152
   349
huffman@35288
   350
  fun dis_defin (con, _, args) =
wenzelm@23152
   351
    let
wenzelm@23152
   352
      val goal = defined (%:x_name) ==> defined (%%:(dis_name con) `% x_name);
wenzelm@23152
   353
      val tacs =
wenzelm@23152
   354
        [rtac casedist 1,
wenzelm@23152
   355
         contr_tac 1,
wenzelm@23152
   356
         DETERM_UNTIL_SOLVED (CHANGED
wenzelm@23152
   357
          (asm_simp_tac (HOLCF_ss addsimps dis_apps) 1))];
wenzelm@27208
   358
    in pg [] goal (K tacs) end;
wenzelm@23152
   359
huffman@29402
   360
  val _ = trace " Proving dis_stricts...";
wenzelm@23152
   361
  val dis_stricts = map dis_strict cons;
huffman@29402
   362
  val _ = trace " Proving dis_defins...";
wenzelm@23152
   363
  val dis_defins = map dis_defin cons;
wenzelm@23152
   364
in
wenzelm@23152
   365
  val dis_rews = dis_stricts @ dis_defins @ dis_apps;
wenzelm@23152
   366
end;
wenzelm@23152
   367
wenzelm@23152
   368
local
huffman@35288
   369
  fun mat_strict (con, _, _) =
wenzelm@23152
   370
    let
huffman@30912
   371
      val goal = mk_trp (%%:(mat_name con) ` UU ` %:"rhs" === UU);
huffman@30912
   372
      val tacs = [asm_simp_tac (HOLCF_ss addsimps [when_strict]) 1];
wenzelm@27208
   373
    in pg axs_mat_def goal (K tacs) end;
wenzelm@23152
   374
huffman@29402
   375
  val _ = trace " Proving mat_stricts...";
wenzelm@23152
   376
  val mat_stricts = map mat_strict cons;
wenzelm@23152
   377
huffman@35288
   378
  fun one_mat c (con, _, args) =
wenzelm@23152
   379
    let
huffman@30912
   380
      val lhs = %%:(mat_name c) ` con_app con args ` %:"rhs";
wenzelm@23152
   381
      val rhs =
wenzelm@23152
   382
        if con = c
huffman@30912
   383
        then list_ccomb (%:"rhs", map %# args)
huffman@26012
   384
        else mk_fail;
wenzelm@23152
   385
      val goal = lift_defined %: (nonlazy args, mk_trp (lhs === rhs));
wenzelm@23152
   386
      val tacs = [asm_simp_tac (HOLCF_ss addsimps when_rews) 1];
wenzelm@27208
   387
    in pg axs_mat_def goal (K tacs) end;
wenzelm@23152
   388
huffman@29402
   389
  val _ = trace " Proving mat_apps...";
wenzelm@23152
   390
  val mat_apps =
huffman@35288
   391
    maps (fn (c,_,_) => map (one_mat c) cons) cons;
wenzelm@23152
   392
in
wenzelm@23152
   393
  val mat_rews = mat_stricts @ mat_apps;
wenzelm@23152
   394
end;
wenzelm@23152
   395
wenzelm@23152
   396
local
wenzelm@23152
   397
  fun ps args = mapn (fn n => fn _ => %:("pat" ^ string_of_int n)) 1 args;
wenzelm@23152
   398
huffman@35288
   399
  fun pat_lhs (con,_,args) = mk_branch (list_comb (%%:(pat_name con), ps args));
wenzelm@23152
   400
huffman@35288
   401
  fun pat_rhs (con,_,[]) = mk_return ((%:"rhs") ` HOLogic.unit)
huffman@35288
   402
    | pat_rhs (con,_,args) =
huffman@26012
   403
        (mk_branch (mk_ctuple_pat (ps args)))
wenzelm@23152
   404
          `(%:"rhs")`(mk_ctuple (map %# args));
wenzelm@23152
   405
wenzelm@23152
   406
  fun pat_strict c =
wenzelm@23152
   407
    let
wenzelm@25132
   408
      val axs = @{thm branch_def} :: axs_pat_def;
wenzelm@23152
   409
      val goal = mk_trp (strict (pat_lhs c ` (%:"rhs")));
wenzelm@23152
   410
      val tacs = [simp_tac (HOLCF_ss addsimps [when_strict]) 1];
wenzelm@27208
   411
    in pg axs goal (K tacs) end;
wenzelm@23152
   412
huffman@35288
   413
  fun pat_app c (con, _, args) =
wenzelm@23152
   414
    let
wenzelm@25132
   415
      val axs = @{thm branch_def} :: axs_pat_def;
wenzelm@23152
   416
      val lhs = (pat_lhs c)`(%:"rhs")`(con_app con args);
huffman@35288
   417
      val rhs = if con = first c then pat_rhs c else mk_fail;
wenzelm@23152
   418
      val goal = lift_defined %: (nonlazy args, mk_trp (lhs === rhs));
wenzelm@23152
   419
      val tacs = [asm_simp_tac (HOLCF_ss addsimps when_rews) 1];
wenzelm@27208
   420
    in pg axs goal (K tacs) end;
wenzelm@23152
   421
huffman@29402
   422
  val _ = trace " Proving pat_stricts...";
wenzelm@23152
   423
  val pat_stricts = map pat_strict cons;
huffman@29402
   424
  val _ = trace " Proving pat_apps...";
wenzelm@26336
   425
  val pat_apps = maps (fn c => map (pat_app c) cons) cons;
wenzelm@23152
   426
in
wenzelm@23152
   427
  val pat_rews = pat_stricts @ pat_apps;
wenzelm@23152
   428
end;
wenzelm@23152
   429
wenzelm@23152
   430
local
huffman@35288
   431
  fun con_strict (con, _, args) = 
wenzelm@23152
   432
    let
huffman@30911
   433
      val rules = abs_strict :: @{thms con_strict_rules};
wenzelm@23152
   434
      fun one_strict vn =
wenzelm@23152
   435
        let
wenzelm@23152
   436
          fun f arg = if vname arg = vn then UU else %# arg;
wenzelm@23152
   437
          val goal = mk_trp (con_app2 con f args === UU);
huffman@30911
   438
          val tacs = [simp_tac (HOL_basic_ss addsimps rules) 1];
wenzelm@27208
   439
        in pg con_appls goal (K tacs) end;
wenzelm@23152
   440
    in map one_strict (nonlazy args) end;
wenzelm@23152
   441
huffman@35288
   442
  fun con_defin (con, _, args) =
wenzelm@23152
   443
    let
huffman@30913
   444
      fun iff_disj (t, []) = HOLogic.mk_not t
huffman@30913
   445
        | iff_disj (t, ts) = t === foldr1 HOLogic.mk_disj ts;
huffman@30913
   446
      val lhs = con_app con args === UU;
huffman@30913
   447
      val rhss = map (fn x => %:x === UU) (nonlazy args);
huffman@30913
   448
      val goal = mk_trp (iff_disj (lhs, rhss));
huffman@30913
   449
      val rule1 = iso_locale RS @{thm iso.abs_defined_iff};
huffman@30913
   450
      val rules = rule1 :: @{thms con_defined_iff_rules};
huffman@30913
   451
      val tacs = [simp_tac (HOL_ss addsimps rules) 1];
huffman@30911
   452
    in pg con_appls goal (K tacs) end;
wenzelm@23152
   453
in
huffman@29402
   454
  val _ = trace " Proving con_stricts...";
wenzelm@26336
   455
  val con_stricts = maps con_strict cons;
huffman@30911
   456
  val _ = trace " Proving con_defins...";
wenzelm@23152
   457
  val con_defins = map con_defin cons;
wenzelm@23152
   458
  val con_rews = con_stricts @ con_defins;
wenzelm@23152
   459
end;
wenzelm@23152
   460
huffman@29402
   461
val _ = trace " Proving dist_les...";
huffman@35117
   462
val dist_les =
wenzelm@23152
   463
  let
wenzelm@23152
   464
    fun dist (con1, args1) (con2, args2) =
wenzelm@23152
   465
      let
huffman@35117
   466
        fun iff_disj (t, []) = HOLogic.mk_not t
huffman@35117
   467
          | iff_disj (t, ts) = t === foldr1 HOLogic.mk_disj ts;
huffman@35117
   468
        val lhs = con_app con1 args1 << con_app con2 args2;
huffman@35117
   469
        val rhss = map (fn x => %:x === UU) (nonlazy args1);
huffman@35117
   470
        val goal = mk_trp (iff_disj (lhs, rhss));
huffman@35117
   471
        val rule1 = iso_locale RS @{thm iso.abs_below};
huffman@35117
   472
        val rules = rule1 :: @{thms con_below_iff_rules};
huffman@35117
   473
        val tacs = [simp_tac (HOL_ss addsimps rules) 1];
huffman@35117
   474
      in pg con_appls goal (K tacs) end;
wenzelm@23152
   475
huffman@35288
   476
    fun distinct (con1, _, args1) (con2, _, args2) =
wenzelm@23152
   477
        let
wenzelm@23152
   478
          val arg1 = (con1, args1);
wenzelm@23152
   479
          val arg2 =
wenzelm@23152
   480
            (con2, ListPair.map (fn (arg,vn) => upd_vname (K vn) arg)
wenzelm@23152
   481
              (args2, Name.variant_list (map vname args1) (map vname args2)));
wenzelm@23152
   482
        in [dist arg1 arg2, dist arg2 arg1] end;
wenzelm@23152
   483
    fun distincts []      = []
huffman@35117
   484
      | distincts (c::cs) = maps (distinct c) cs @ distincts cs;
wenzelm@23152
   485
  in distincts cons end;
huffman@29402
   486
huffman@29402
   487
val _ = trace " Proving dist_eqs...";
wenzelm@23152
   488
val dist_eqs =
wenzelm@23152
   489
  let
huffman@35117
   490
    fun dist (con1, args1) (con2, args2) =
wenzelm@23152
   491
      let
huffman@35117
   492
        fun iff_disj (t, [], us) = HOLogic.mk_not t
huffman@35117
   493
          | iff_disj (t, ts, []) = HOLogic.mk_not t
huffman@35117
   494
          | iff_disj (t, ts, us) =
huffman@35117
   495
            let
huffman@35117
   496
              val disj1 = foldr1 HOLogic.mk_disj ts;
huffman@35117
   497
              val disj2 = foldr1 HOLogic.mk_disj us;
huffman@35117
   498
            in t === HOLogic.mk_conj (disj1, disj2) end;
huffman@35117
   499
        val lhs = con_app con1 args1 === con_app con2 args2;
huffman@35117
   500
        val rhss1 = map (fn x => %:x === UU) (nonlazy args1);
huffman@35117
   501
        val rhss2 = map (fn x => %:x === UU) (nonlazy args2);
huffman@35117
   502
        val goal = mk_trp (iff_disj (lhs, rhss1, rhss2));
huffman@35117
   503
        val rule1 = iso_locale RS @{thm iso.abs_eq};
huffman@35117
   504
        val rules = rule1 :: @{thms con_eq_iff_rules};
huffman@35117
   505
        val tacs = [simp_tac (HOL_ss addsimps rules) 1];
huffman@35117
   506
      in pg con_appls goal (K tacs) end;
huffman@35117
   507
huffman@35288
   508
    fun distinct (con1, _, args1) (con2, _, args2) =
huffman@35117
   509
        let
huffman@35117
   510
          val arg1 = (con1, args1);
huffman@35117
   511
          val arg2 =
huffman@35117
   512
            (con2, ListPair.map (fn (arg,vn) => upd_vname (K vn) arg)
huffman@35117
   513
              (args2, Name.variant_list (map vname args1) (map vname args2)));
huffman@35117
   514
        in [dist arg1 arg2, dist arg2 arg1] end;
wenzelm@23152
   515
    fun distincts []      = []
huffman@35117
   516
      | distincts (c::cs) = maps (distinct c) cs @ distincts cs;
huffman@35117
   517
  in distincts cons end;
wenzelm@23152
   518
wenzelm@23152
   519
local 
wenzelm@23152
   520
  fun pgterm rel con args =
wenzelm@23152
   521
    let
wenzelm@23152
   522
      fun append s = upd_vname (fn v => v^s);
wenzelm@23152
   523
      val (largs, rargs) = (args, map (append "'") args);
wenzelm@23152
   524
      val concl =
wenzelm@23152
   525
        foldr1 mk_conj (ListPair.map rel (map %# largs, map %# rargs));
wenzelm@23152
   526
      val prem = rel (con_app con largs, con_app con rargs);
wenzelm@23152
   527
      val sargs = case largs of [_] => [] | _ => nonlazy args;
wenzelm@23152
   528
      val prop = lift_defined %: (sargs, mk_trp (prem === concl));
wenzelm@23152
   529
    in pg con_appls prop end;
huffman@35288
   530
  val cons' = filter (fn (_, _, args) => args<>[]) cons;
wenzelm@23152
   531
in
huffman@29402
   532
  val _ = trace " Proving inverts...";
wenzelm@23152
   533
  val inverts =
wenzelm@23152
   534
    let
wenzelm@23152
   535
      val abs_less = ax_abs_iso RS (allI RS injection_less);
wenzelm@23152
   536
      val tacs =
wenzelm@23152
   537
        [asm_full_simp_tac (HOLCF_ss addsimps [abs_less, spair_less]) 1];
huffman@35288
   538
    in map (fn (con, _, args) => pgterm (op <<) con args (K tacs)) cons' end;
wenzelm@23152
   539
huffman@29402
   540
  val _ = trace " Proving injects...";
wenzelm@23152
   541
  val injects =
wenzelm@23152
   542
    let
wenzelm@23152
   543
      val abs_eq = ax_abs_iso RS (allI RS injection_eq);
wenzelm@23152
   544
      val tacs = [asm_full_simp_tac (HOLCF_ss addsimps [abs_eq, spair_eq]) 1];
huffman@35288
   545
    in map (fn (con, _, args) => pgterm (op ===) con args (K tacs)) cons' end;
wenzelm@23152
   546
end;
wenzelm@23152
   547
wenzelm@23152
   548
(* ----- theorems concerning one induction step ----------------------------- *)
wenzelm@23152
   549
wenzelm@23152
   550
val copy_strict =
wenzelm@23152
   551
  let
huffman@31232
   552
    val _ = trace " Proving copy_strict...";
wenzelm@23152
   553
    val goal = mk_trp (strict (dc_copy `% "f"));
huffman@33504
   554
    val rules = [abs_strict, rep_strict] @ @{thms domain_map_stricts};
huffman@31232
   555
    val tacs = [asm_simp_tac (HOLCF_ss addsimps rules) 1];
huffman@35058
   556
  in
huffman@35058
   557
    SOME (pg [ax_copy_def] goal (K tacs))
huffman@35058
   558
    handle
huffman@35058
   559
      THM (s, _, _) => (trace s; NONE)
huffman@35058
   560
    | ERROR s => (trace s; NONE)
huffman@35058
   561
  end;
wenzelm@23152
   562
wenzelm@23152
   563
local
huffman@35288
   564
  fun copy_app (con, _, args) =
wenzelm@23152
   565
    let
wenzelm@23152
   566
      val lhs = dc_copy`%"f"`(con_app con args);
huffman@31232
   567
      fun one_rhs arg =
haftmann@33971
   568
          if Datatype_Aux.is_rec_type (dtyp_of arg)
huffman@33801
   569
          then Domain_Axioms.copy_of_dtyp map_tab
huffman@33801
   570
                 (proj (%:"f") eqs) (dtyp_of arg) ` (%# arg)
huffman@31232
   571
          else (%# arg);
huffman@31232
   572
      val rhs = con_app2 con one_rhs args;
huffman@35059
   573
      fun is_rec arg = Datatype_Aux.is_rec_type (dtyp_of arg);
huffman@35059
   574
      fun is_nonlazy_rec arg = is_rec arg andalso not (is_lazy arg);
huffman@35059
   575
      fun nonlazy_rec args = map vname (filter is_nonlazy_rec args);
wenzelm@23152
   576
      val goal = lift_defined %: (nonlazy_rec args, mk_trp (lhs === rhs));
wenzelm@33317
   577
      val args' = filter_out (fn a => is_rec a orelse is_lazy a) args;
huffman@33504
   578
      val stricts = abs_strict :: rep_strict :: @{thms domain_map_stricts};
huffman@35443
   579
                        (* FIXME! case_UU_tac *)
wenzelm@27208
   580
      fun tacs1 ctxt = map (case_UU_tac ctxt stricts 1 o vname) args';
huffman@33504
   581
      val rules = [ax_abs_iso] @ @{thms domain_map_simps};
huffman@31232
   582
      val tacs2 = [asm_simp_tac (HOLCF_ss addsimps rules) 1];
huffman@31232
   583
    in pg (ax_copy_def::con_appls) goal (fn ctxt => (tacs1 ctxt @ tacs2)) end;
wenzelm@23152
   584
in
huffman@29402
   585
  val _ = trace " Proving copy_apps...";
wenzelm@23152
   586
  val copy_apps = map copy_app cons;
wenzelm@23152
   587
end;
wenzelm@23152
   588
wenzelm@23152
   589
local
huffman@35288
   590
  fun one_strict (con, _, args) = 
wenzelm@23152
   591
    let
wenzelm@23152
   592
      val goal = mk_trp (dc_copy`UU`(con_app con args) === UU);
huffman@35058
   593
      val rews = the_list copy_strict @ copy_apps @ con_rews;
huffman@35443
   594
                        (* FIXME! case_UU_tac *)
wenzelm@27208
   595
      fun tacs ctxt = map (case_UU_tac ctxt rews 1) (nonlazy args) @
wenzelm@23152
   596
        [asm_simp_tac (HOLCF_ss addsimps rews) 1];
huffman@35058
   597
    in
huffman@35058
   598
      SOME (pg [] goal tacs)
huffman@35058
   599
      handle
huffman@35058
   600
        THM (s, _, _) => (trace s; NONE)
huffman@35058
   601
      | ERROR s => (trace s; NONE)
huffman@35058
   602
    end;
wenzelm@23152
   603
huffman@35288
   604
  fun has_nonlazy_rec (_, _, args) = exists is_nonlazy_rec args;
wenzelm@23152
   605
in
huffman@29402
   606
  val _ = trace " Proving copy_stricts...";
huffman@35058
   607
  val copy_stricts = map_filter one_strict (filter has_nonlazy_rec cons);
wenzelm@23152
   608
end;
wenzelm@23152
   609
huffman@35058
   610
val copy_rews = the_list copy_strict @ copy_apps @ copy_stricts;
wenzelm@23152
   611
wenzelm@23152
   612
in
wenzelm@23152
   613
  thy
wenzelm@30364
   614
    |> Sign.add_path (Long_Name.base_name dname)
huffman@31004
   615
    |> snd o PureThy.add_thmss [
huffman@31004
   616
        ((Binding.name "iso_rews"  , iso_rews    ), [Simplifier.simp_add]),
huffman@31004
   617
        ((Binding.name "exhaust"   , [exhaust]   ), []),
huffman@31004
   618
        ((Binding.name "casedist"  , [casedist]  ), [Induct.cases_type dname]),
huffman@31004
   619
        ((Binding.name "when_rews" , when_rews   ), [Simplifier.simp_add]),
huffman@31004
   620
        ((Binding.name "compacts"  , con_compacts), [Simplifier.simp_add]),
huffman@33427
   621
        ((Binding.name "con_rews"  , con_rews    ),
huffman@33427
   622
         [Simplifier.simp_add, Fixrec.fixrec_simp_add]),
huffman@31004
   623
        ((Binding.name "sel_rews"  , sel_rews    ), [Simplifier.simp_add]),
huffman@31004
   624
        ((Binding.name "dis_rews"  , dis_rews    ), [Simplifier.simp_add]),
huffman@31004
   625
        ((Binding.name "pat_rews"  , pat_rews    ), [Simplifier.simp_add]),
huffman@31004
   626
        ((Binding.name "dist_les"  , dist_les    ), [Simplifier.simp_add]),
huffman@31004
   627
        ((Binding.name "dist_eqs"  , dist_eqs    ), [Simplifier.simp_add]),
huffman@31004
   628
        ((Binding.name "inverts"   , inverts     ), [Simplifier.simp_add]),
huffman@31004
   629
        ((Binding.name "injects"   , injects     ), [Simplifier.simp_add]),
huffman@31004
   630
        ((Binding.name "copy_rews" , copy_rews   ), [Simplifier.simp_add]),
huffman@33427
   631
        ((Binding.name "match_rews", mat_rews    ),
huffman@33427
   632
         [Simplifier.simp_add, Fixrec.fixrec_simp_add])]
wenzelm@24712
   633
    |> Sign.parent_path
haftmann@28536
   634
    |> pair (iso_rews @ when_rews @ con_rews @ sel_rews @ dis_rews @
wenzelm@23152
   635
        pat_rews @ dist_les @ dist_eqs @ copy_rews)
wenzelm@23152
   636
end; (* let *)
wenzelm@23152
   637
wenzelm@23152
   638
fun comp_theorems (comp_dnam, eqs: eq list) thy =
wenzelm@23152
   639
let
wenzelm@27232
   640
val global_ctxt = ProofContext.init thy;
huffman@33801
   641
val map_tab = Domain_Isomorphism.get_map_tab thy;
wenzelm@27232
   642
wenzelm@23152
   643
val dnames = map (fst o fst) eqs;
wenzelm@23152
   644
val conss  = map  snd        eqs;
haftmann@28965
   645
val comp_dname = Sign.full_bname thy comp_dnam;
wenzelm@23152
   646
huffman@29402
   647
val _ = message ("Proving induction properties of domain "^comp_dname^" ...");
wenzelm@23152
   648
val pg = pg' thy;
wenzelm@23152
   649
wenzelm@23152
   650
(* ----- getting the composite axiom and definitions ------------------------ *)
wenzelm@23152
   651
wenzelm@23152
   652
local
wenzelm@26343
   653
  fun ga s dn = PureThy.get_thm thy (dn ^ "." ^ s);
wenzelm@23152
   654
in
wenzelm@23152
   655
  val axs_reach      = map (ga "reach"     ) dnames;
wenzelm@23152
   656
  val axs_take_def   = map (ga "take_def"  ) dnames;
wenzelm@23152
   657
  val axs_finite_def = map (ga "finite_def") dnames;
wenzelm@23152
   658
  val ax_copy2_def   =      ga "copy_def"  comp_dnam;
huffman@35444
   659
(* TEMPORARILY DISABLED
wenzelm@23152
   660
  val ax_bisim_def   =      ga "bisim_def" comp_dnam;
huffman@35444
   661
TEMPORARILY DISABLED *)
wenzelm@23152
   662
end;
wenzelm@23152
   663
wenzelm@23152
   664
local
wenzelm@26343
   665
  fun gt  s dn = PureThy.get_thm  thy (dn ^ "." ^ s);
wenzelm@26343
   666
  fun gts s dn = PureThy.get_thms thy (dn ^ "." ^ s);
wenzelm@23152
   667
in
wenzelm@23152
   668
  val cases = map (gt  "casedist" ) dnames;
wenzelm@26336
   669
  val con_rews  = maps (gts "con_rews" ) dnames;
wenzelm@26336
   670
  val copy_rews = maps (gts "copy_rews") dnames;
wenzelm@23152
   671
end;
wenzelm@23152
   672
wenzelm@23152
   673
fun dc_take dn = %%:(dn^"_take");
wenzelm@23152
   674
val x_name = idx_name dnames "x"; 
wenzelm@23152
   675
val P_name = idx_name dnames "P";
wenzelm@23152
   676
val n_eqs = length eqs;
wenzelm@23152
   677
wenzelm@23152
   678
(* ----- theorems concerning finite approximation and finite induction ------ *)
wenzelm@23152
   679
wenzelm@23152
   680
local
wenzelm@32149
   681
  val iterate_Cprod_ss = global_simpset_of @{theory Fix};
wenzelm@23152
   682
  val copy_con_rews  = copy_rews @ con_rews;
wenzelm@23152
   683
  val copy_take_defs =
wenzelm@23152
   684
    (if n_eqs = 1 then [] else [ax_copy2_def]) @ axs_take_def;
huffman@29402
   685
  val _ = trace " Proving take_stricts...";
huffman@35057
   686
  fun one_take_strict ((dn, args), _) =
wenzelm@23152
   687
    let
huffman@35057
   688
      val goal = mk_trp (strict (dc_take dn $ %:"n"));
huffman@35057
   689
      val rules = [
huffman@35057
   690
        @{thm monofun_fst [THEN monofunE]},
huffman@35057
   691
        @{thm monofun_snd [THEN monofunE]}];
huffman@35057
   692
      val tacs = [
huffman@35057
   693
        rtac @{thm UU_I} 1,
huffman@35057
   694
        rtac @{thm below_eq_trans} 1,
huffman@35057
   695
        resolve_tac axs_reach 2,
huffman@35057
   696
        rtac @{thm monofun_cfun_fun} 1,
huffman@35057
   697
        REPEAT (resolve_tac rules 1),
huffman@35057
   698
        rtac @{thm iterate_below_fix} 1];
huffman@35057
   699
    in pg axs_take_def goal (K tacs) end;
huffman@35057
   700
  val take_stricts = map one_take_strict eqs;
wenzelm@23152
   701
  fun take_0 n dn =
wenzelm@23152
   702
    let
huffman@35058
   703
      val goal = mk_trp ((dc_take dn $ @{term "0::nat"}) `% x_name n === UU);
wenzelm@27208
   704
    in pg axs_take_def goal (K [simp_tac iterate_Cprod_ss 1]) end;
wenzelm@23152
   705
  val take_0s = mapn take_0 1 dnames;
huffman@29402
   706
  val _ = trace " Proving take_apps...";
huffman@35288
   707
  fun one_take_app dn (con, _, args) =
wenzelm@23152
   708
    let
huffman@35058
   709
      fun mk_take n = dc_take (List.nth (dnames, n)) $ %:"n";
huffman@35058
   710
      fun one_rhs arg =
huffman@35058
   711
          if Datatype_Aux.is_rec_type (dtyp_of arg)
huffman@35058
   712
          then Domain_Axioms.copy_of_dtyp map_tab
huffman@35058
   713
                 mk_take (dtyp_of arg) ` (%# arg)
huffman@35058
   714
          else (%# arg);
huffman@35058
   715
      val lhs = (dc_take dn $ (%%:"Suc" $ %:"n"))`(con_app con args);
huffman@35058
   716
      val rhs = con_app2 con one_rhs args;
huffman@35059
   717
      fun is_rec arg = Datatype_Aux.is_rec_type (dtyp_of arg);
huffman@35059
   718
      fun is_nonlazy_rec arg = is_rec arg andalso not (is_lazy arg);
huffman@35059
   719
      fun nonlazy_rec args = map vname (filter is_nonlazy_rec args);
huffman@35059
   720
      val goal = lift_defined %: (nonlazy_rec args, mk_trp (lhs === rhs));
huffman@35059
   721
      val tacs = [asm_simp_tac (HOLCF_ss addsimps copy_con_rews) 1];
huffman@35059
   722
    in pg copy_take_defs goal (K tacs) end;
huffman@35058
   723
  fun one_take_apps ((dn, _), cons) = map (one_take_app dn) cons;
huffman@35058
   724
  val take_apps = maps one_take_apps eqs;
wenzelm@23152
   725
in
wenzelm@35021
   726
  val take_rews = map Drule.export_without_context
huffman@35058
   727
    (take_stricts @ take_0s @ take_apps);
wenzelm@23152
   728
end; (* local *)
wenzelm@23152
   729
wenzelm@23152
   730
local
huffman@35288
   731
  fun one_con p (con, _, args) =
wenzelm@23152
   732
    let
huffman@35443
   733
      val P_names = map P_name (1 upto (length dnames));
huffman@35443
   734
      val vns = Name.variant_list P_names (map vname args);
huffman@35443
   735
      val nonlazy_vns = map snd (filter_out (is_lazy o fst) (args ~~ vns));
wenzelm@23152
   736
      fun ind_hyp arg = %:(P_name (1 + rec_of arg)) $ bound_arg args arg;
wenzelm@23152
   737
      val t1 = mk_trp (%:p $ con_app2 con (bound_arg args) args);
wenzelm@33317
   738
      val t2 = lift ind_hyp (filter is_rec args, t1);
huffman@35443
   739
      val t3 = lift_defined (bound_arg vns) (nonlazy_vns, t2);
huffman@35443
   740
    in Library.foldr mk_All (vns, t3) end;
wenzelm@23152
   741
wenzelm@23152
   742
  fun one_eq ((p, cons), concl) =
wenzelm@23152
   743
    mk_trp (%:p $ UU) ===> Logic.list_implies (map (one_con p) cons, concl);
wenzelm@23152
   744
wenzelm@23152
   745
  fun ind_term concf = Library.foldr one_eq
wenzelm@23152
   746
    (mapn (fn n => fn x => (P_name n, x)) 1 conss,
wenzelm@23152
   747
     mk_trp (foldr1 mk_conj (mapn concf 1 dnames)));
wenzelm@23152
   748
  val take_ss = HOL_ss addsimps take_rews;
wenzelm@27208
   749
  fun quant_tac ctxt i = EVERY
wenzelm@27239
   750
    (mapn (fn n => fn _ => res_inst_tac ctxt [(("x", 0), x_name n)] spec i) 1 dnames);
wenzelm@23152
   751
wenzelm@23152
   752
  fun ind_prems_tac prems = EVERY
wenzelm@26336
   753
    (maps (fn cons =>
wenzelm@23152
   754
      (resolve_tac prems 1 ::
huffman@35288
   755
        maps (fn (_,_,args) => 
wenzelm@23152
   756
          resolve_tac prems 1 ::
wenzelm@23152
   757
          map (K(atac 1)) (nonlazy args) @
wenzelm@33317
   758
          map (K(atac 1)) (filter is_rec args))
wenzelm@26336
   759
        cons))
wenzelm@26336
   760
      conss);
wenzelm@23152
   761
  local 
wenzelm@23152
   762
    (* check whether every/exists constructor of the n-th part of the equation:
wenzelm@23152
   763
       it has a possibly indirectly recursive argument that isn't/is possibly 
wenzelm@23152
   764
       indirectly lazy *)
wenzelm@23152
   765
    fun rec_to quant nfn rfn ns lazy_rec (n,cons) = quant (exists (fn arg => 
wenzelm@23152
   766
          is_rec arg andalso not(rec_of arg mem ns) andalso
wenzelm@23152
   767
          ((rec_of arg =  n andalso nfn(lazy_rec orelse is_lazy arg)) orelse 
wenzelm@23152
   768
            rec_of arg <> n andalso rec_to quant nfn rfn (rec_of arg::ns) 
wenzelm@23152
   769
              (lazy_rec orelse is_lazy arg) (n, (List.nth(conss,rec_of arg))))
huffman@35288
   770
          ) o third) cons;
wenzelm@23152
   771
    fun all_rec_to ns  = rec_to forall not all_rec_to  ns;
wenzelm@23152
   772
    fun warn (n,cons) =
wenzelm@23152
   773
      if all_rec_to [] false (n,cons)
wenzelm@23152
   774
      then (warning ("domain "^List.nth(dnames,n)^" is empty!"); true)
wenzelm@23152
   775
      else false;
wenzelm@23152
   776
    fun lazy_rec_to ns = rec_to exists I  lazy_rec_to ns;
wenzelm@23152
   777
wenzelm@23152
   778
  in
wenzelm@23152
   779
    val n__eqs = mapn (fn n => fn (_,cons) => (n,cons)) 0 eqs;
wenzelm@23152
   780
    val is_emptys = map warn n__eqs;
wenzelm@23152
   781
    val is_finite = forall (not o lazy_rec_to [] false) n__eqs;
wenzelm@23152
   782
  end;
wenzelm@23152
   783
in (* local *)
huffman@29402
   784
  val _ = trace " Proving finite_ind...";
wenzelm@23152
   785
  val finite_ind =
wenzelm@23152
   786
    let
wenzelm@23152
   787
      fun concf n dn = %:(P_name n) $ (dc_take dn $ %:"n" `%(x_name n));
wenzelm@23152
   788
      val goal = ind_term concf;
wenzelm@23152
   789
wenzelm@27208
   790
      fun tacf {prems, context} =
wenzelm@23152
   791
        let
wenzelm@23152
   792
          val tacs1 = [
wenzelm@27208
   793
            quant_tac context 1,
wenzelm@23152
   794
            simp_tac HOL_ss 1,
wenzelm@27208
   795
            InductTacs.induct_tac context [[SOME "n"]] 1,
wenzelm@23152
   796
            simp_tac (take_ss addsimps prems) 1,
wenzelm@23152
   797
            TRY (safe_tac HOL_cs)];
wenzelm@23152
   798
          fun arg_tac arg =
huffman@35443
   799
                        (* FIXME! case_UU_tac *)
wenzelm@27208
   800
            case_UU_tac context (prems @ con_rews) 1
wenzelm@23152
   801
              (List.nth (dnames, rec_of arg) ^ "_take n$" ^ vname arg);
huffman@35288
   802
          fun con_tacs (con, _, args) = 
wenzelm@23152
   803
            asm_simp_tac take_ss 1 ::
wenzelm@33317
   804
            map arg_tac (filter is_nonlazy_rec args) @
wenzelm@23152
   805
            [resolve_tac prems 1] @
wenzelm@33317
   806
            map (K (atac 1)) (nonlazy args) @
wenzelm@33317
   807
            map (K (etac spec 1)) (filter is_rec args);
wenzelm@23152
   808
          fun cases_tacs (cons, cases) =
wenzelm@27239
   809
            res_inst_tac context [(("x", 0), "x")] cases 1 ::
wenzelm@23152
   810
            asm_simp_tac (take_ss addsimps prems) 1 ::
wenzelm@26336
   811
            maps con_tacs cons;
wenzelm@23152
   812
        in
wenzelm@26336
   813
          tacs1 @ maps cases_tacs (conss ~~ cases)
wenzelm@23152
   814
        end;
huffman@31232
   815
    in pg'' thy [] goal tacf
huffman@31232
   816
       handle ERROR _ => (warning "Proof of finite_ind failed."; TrueI)
huffman@31232
   817
    end;
wenzelm@23152
   818
huffman@29402
   819
  val _ = trace " Proving take_lemmas...";
wenzelm@23152
   820
  val take_lemmas =
wenzelm@23152
   821
    let
wenzelm@23152
   822
      fun take_lemma n (dn, ax_reach) =
wenzelm@23152
   823
        let
wenzelm@23152
   824
          val lhs = dc_take dn $ Bound 0 `%(x_name n);
wenzelm@23152
   825
          val rhs = dc_take dn $ Bound 0 `%(x_name n^"'");
wenzelm@23152
   826
          val concl = mk_trp (%:(x_name n) === %:(x_name n^"'"));
wenzelm@23152
   827
          val goal = mk_All ("n", mk_trp (lhs === rhs)) ===> concl;
huffman@33396
   828
          val rules = [contlub_fst RS contlubE RS ssubst,
huffman@33396
   829
                       contlub_snd RS contlubE RS ssubst];
wenzelm@27208
   830
          fun tacf {prems, context} = [
wenzelm@27239
   831
            res_inst_tac context [(("t", 0), x_name n    )] (ax_reach RS subst) 1,
wenzelm@27239
   832
            res_inst_tac context [(("t", 0), x_name n^"'")] (ax_reach RS subst) 1,
wenzelm@23152
   833
            stac fix_def2 1,
wenzelm@23152
   834
            REPEAT (CHANGED
huffman@33396
   835
              (resolve_tac rules 1 THEN chain_tac 1)),
wenzelm@23152
   836
            stac contlub_cfun_fun 1,
wenzelm@23152
   837
            stac contlub_cfun_fun 2,
wenzelm@23152
   838
            rtac lub_equal 3,
wenzelm@23152
   839
            chain_tac 1,
wenzelm@23152
   840
            rtac allI 1,
wenzelm@23152
   841
            resolve_tac prems 1];
wenzelm@23152
   842
        in pg'' thy axs_take_def goal tacf end;
wenzelm@23152
   843
    in mapn take_lemma 1 (dnames ~~ axs_reach) end;
wenzelm@23152
   844
wenzelm@23152
   845
(* ----- theorems concerning finiteness and induction ----------------------- *)
wenzelm@23152
   846
huffman@29402
   847
  val _ = trace " Proving finites, ind...";
wenzelm@23152
   848
  val (finites, ind) =
huffman@31232
   849
  (
wenzelm@23152
   850
    if is_finite
wenzelm@23152
   851
    then (* finite case *)
wenzelm@23152
   852
      let 
wenzelm@23152
   853
        fun take_enough dn = mk_ex ("n",dc_take dn $ Bound 0 ` %:"x" === %:"x");
wenzelm@23152
   854
        fun dname_lemma dn =
wenzelm@23152
   855
          let
wenzelm@23152
   856
            val prem1 = mk_trp (defined (%:"x"));
wenzelm@23152
   857
            val disj1 = mk_all ("n", dc_take dn $ Bound 0 ` %:"x" === UU);
wenzelm@23152
   858
            val prem2 = mk_trp (mk_disj (disj1, take_enough dn));
wenzelm@23152
   859
            val concl = mk_trp (take_enough dn);
wenzelm@23152
   860
            val goal = prem1 ===> prem2 ===> concl;
wenzelm@23152
   861
            val tacs = [
wenzelm@23152
   862
              etac disjE 1,
wenzelm@23152
   863
              etac notE 1,
wenzelm@23152
   864
              resolve_tac take_lemmas 1,
wenzelm@23152
   865
              asm_simp_tac take_ss 1,
wenzelm@23152
   866
              atac 1];
wenzelm@27208
   867
          in pg [] goal (K tacs) end;
huffman@31232
   868
        val _ = trace " Proving finite_lemmas1a";
wenzelm@23152
   869
        val finite_lemmas1a = map dname_lemma dnames;
wenzelm@23152
   870
 
huffman@31232
   871
        val _ = trace " Proving finite_lemma1b";
wenzelm@23152
   872
        val finite_lemma1b =
wenzelm@23152
   873
          let
wenzelm@23152
   874
            fun mk_eqn n ((dn, args), _) =
wenzelm@23152
   875
              let
wenzelm@23152
   876
                val disj1 = dc_take dn $ Bound 1 ` Bound 0 === UU;
wenzelm@23152
   877
                val disj2 = dc_take dn $ Bound 1 ` Bound 0 === Bound 0;
wenzelm@23152
   878
              in
wenzelm@23152
   879
                mk_constrainall
wenzelm@23152
   880
                  (x_name n, Type (dn,args), mk_disj (disj1, disj2))
wenzelm@23152
   881
              end;
wenzelm@23152
   882
            val goal =
wenzelm@23152
   883
              mk_trp (mk_all ("n", foldr1 mk_conj (mapn mk_eqn 1 eqs)));
wenzelm@27208
   884
            fun arg_tacs ctxt vn = [
wenzelm@27239
   885
              eres_inst_tac ctxt [(("x", 0), vn)] all_dupE 1,
wenzelm@23152
   886
              etac disjE 1,
wenzelm@23152
   887
              asm_simp_tac (HOL_ss addsimps con_rews) 1,
wenzelm@23152
   888
              asm_simp_tac take_ss 1];
huffman@35288
   889
            fun con_tacs ctxt (con, _, args) =
wenzelm@23152
   890
              asm_simp_tac take_ss 1 ::
wenzelm@27208
   891
              maps (arg_tacs ctxt) (nonlazy_rec args);
wenzelm@27208
   892
            fun foo_tacs ctxt n (cons, cases) =
wenzelm@23152
   893
              simp_tac take_ss 1 ::
wenzelm@23152
   894
              rtac allI 1 ::
wenzelm@27239
   895
              res_inst_tac ctxt [(("x", 0), x_name n)] cases 1 ::
wenzelm@23152
   896
              asm_simp_tac take_ss 1 ::
wenzelm@27208
   897
              maps (con_tacs ctxt) cons;
wenzelm@27208
   898
            fun tacs ctxt =
wenzelm@23152
   899
              rtac allI 1 ::
wenzelm@27208
   900
              InductTacs.induct_tac ctxt [[SOME "n"]] 1 ::
wenzelm@23152
   901
              simp_tac take_ss 1 ::
wenzelm@23152
   902
              TRY (safe_tac (empty_cs addSEs [conjE] addSIs [conjI])) ::
wenzelm@27208
   903
              flat (mapn (foo_tacs ctxt) 1 (conss ~~ cases));
wenzelm@23152
   904
          in pg [] goal tacs end;
wenzelm@23152
   905
wenzelm@23152
   906
        fun one_finite (dn, l1b) =
wenzelm@23152
   907
          let
wenzelm@23152
   908
            val goal = mk_trp (%%:(dn^"_finite") $ %:"x");
wenzelm@27208
   909
            fun tacs ctxt = [
huffman@35443
   910
                        (* FIXME! case_UU_tac *)
wenzelm@27208
   911
              case_UU_tac ctxt take_rews 1 "x",
wenzelm@23152
   912
              eresolve_tac finite_lemmas1a 1,
wenzelm@23152
   913
              step_tac HOL_cs 1,
wenzelm@23152
   914
              step_tac HOL_cs 1,
wenzelm@23152
   915
              cut_facts_tac [l1b] 1,
wenzelm@23152
   916
              fast_tac HOL_cs 1];
wenzelm@23152
   917
          in pg axs_finite_def goal tacs end;
wenzelm@23152
   918
huffman@31232
   919
        val _ = trace " Proving finites";
wenzelm@27232
   920
        val finites = map one_finite (dnames ~~ atomize global_ctxt finite_lemma1b);
huffman@31232
   921
        val _ = trace " Proving ind";
wenzelm@23152
   922
        val ind =
wenzelm@23152
   923
          let
wenzelm@23152
   924
            fun concf n dn = %:(P_name n) $ %:(x_name n);
wenzelm@27208
   925
            fun tacf {prems, context} =
wenzelm@23152
   926
              let
wenzelm@23152
   927
                fun finite_tacs (finite, fin_ind) = [
wenzelm@23152
   928
                  rtac(rewrite_rule axs_finite_def finite RS exE)1,
wenzelm@23152
   929
                  etac subst 1,
wenzelm@23152
   930
                  rtac fin_ind 1,
wenzelm@23152
   931
                  ind_prems_tac prems];
wenzelm@23152
   932
              in
wenzelm@23152
   933
                TRY (safe_tac HOL_cs) ::
wenzelm@27232
   934
                maps finite_tacs (finites ~~ atomize global_ctxt finite_ind)
wenzelm@23152
   935
              end;
wenzelm@23152
   936
          in pg'' thy [] (ind_term concf) tacf end;
wenzelm@23152
   937
      in (finites, ind) end (* let *)
wenzelm@23152
   938
wenzelm@23152
   939
    else (* infinite case *)
wenzelm@23152
   940
      let
wenzelm@23152
   941
        fun one_finite n dn =
wenzelm@27239
   942
          read_instantiate global_ctxt [(("P", 0), dn ^ "_finite " ^ x_name n)] excluded_middle;
wenzelm@23152
   943
        val finites = mapn one_finite 1 dnames;
wenzelm@23152
   944
wenzelm@23152
   945
        val goal =
wenzelm@23152
   946
          let
huffman@26012
   947
            fun one_adm n _ = mk_trp (mk_adm (%:(P_name n)));
wenzelm@23152
   948
            fun concf n dn = %:(P_name n) $ %:(x_name n);
wenzelm@23152
   949
          in Logic.list_implies (mapn one_adm 1 dnames, ind_term concf) end;
huffman@33396
   950
        val cont_rules =
huffman@33396
   951
            [cont_id, cont_const, cont2cont_Rep_CFun,
huffman@33396
   952
             cont2cont_fst, cont2cont_snd];
wenzelm@27208
   953
        fun tacf {prems, context} =
wenzelm@23152
   954
          map (fn ax_reach => rtac (ax_reach RS subst) 1) axs_reach @ [
wenzelm@27208
   955
          quant_tac context 1,
wenzelm@23152
   956
          rtac (adm_impl_admw RS wfix_ind) 1,
huffman@25895
   957
          REPEAT_DETERM (rtac adm_all 1),
wenzelm@23152
   958
          REPEAT_DETERM (
wenzelm@23152
   959
            TRY (rtac adm_conj 1) THEN 
wenzelm@23152
   960
            rtac adm_subst 1 THEN 
huffman@33396
   961
            REPEAT (resolve_tac cont_rules 1) THEN
huffman@33396
   962
            resolve_tac prems 1),
wenzelm@23152
   963
          strip_tac 1,
wenzelm@23152
   964
          rtac (rewrite_rule axs_take_def finite_ind) 1,
wenzelm@23152
   965
          ind_prems_tac prems];
wenzelm@23152
   966
        val ind = (pg'' thy [] goal tacf
wenzelm@23152
   967
          handle ERROR _ =>
huffman@33396
   968
            (warning "Cannot prove infinite induction rule"; TrueI));
huffman@31232
   969
      in (finites, ind) end
huffman@31232
   970
  )
huffman@31232
   971
      handle THM _ =>
huffman@31232
   972
             (warning "Induction proofs failed (THM raised)."; ([], TrueI))
huffman@31232
   973
           | ERROR _ =>
huffman@33810
   974
             (warning "Cannot prove induction rule"; ([], TrueI));
huffman@31232
   975
huffman@31232
   976
wenzelm@23152
   977
end; (* local *)
wenzelm@23152
   978
wenzelm@23152
   979
(* ----- theorem concerning coinduction ------------------------------------- *)
wenzelm@23152
   980
huffman@35444
   981
(* COINDUCTION TEMPORARILY DISABLED
wenzelm@23152
   982
local
wenzelm@23152
   983
  val xs = mapn (fn n => K (x_name n)) 1 dnames;
wenzelm@23152
   984
  fun bnd_arg n i = Bound(2*(n_eqs - n)-i-1);
wenzelm@23152
   985
  val take_ss = HOL_ss addsimps take_rews;
wenzelm@23152
   986
  val sproj = prj (fn s => K("fst("^s^")")) (fn s => K("snd("^s^")"));
huffman@29402
   987
  val _ = trace " Proving coind_lemma...";
wenzelm@23152
   988
  val coind_lemma =
wenzelm@23152
   989
    let
wenzelm@23152
   990
      fun mk_prj n _ = proj (%:"R") eqs n $ bnd_arg n 0 $ bnd_arg n 1;
wenzelm@23152
   991
      fun mk_eqn n dn =
wenzelm@23152
   992
        (dc_take dn $ %:"n" ` bnd_arg n 0) ===
wenzelm@23152
   993
        (dc_take dn $ %:"n" ` bnd_arg n 1);
wenzelm@23152
   994
      fun mk_all2 (x,t) = mk_all (x, mk_all (x^"'", t));
wenzelm@23152
   995
      val goal =
wenzelm@23152
   996
        mk_trp (mk_imp (%%:(comp_dname^"_bisim") $ %:"R",
wenzelm@23152
   997
          Library.foldr mk_all2 (xs,
wenzelm@23152
   998
            Library.foldr mk_imp (mapn mk_prj 0 dnames,
wenzelm@23152
   999
              foldr1 mk_conj (mapn mk_eqn 0 dnames)))));
wenzelm@27208
  1000
      fun x_tacs ctxt n x = [
wenzelm@23152
  1001
        rotate_tac (n+1) 1,
wenzelm@23152
  1002
        etac all2E 1,
wenzelm@27239
  1003
        eres_inst_tac ctxt [(("P", 1), sproj "R" eqs n^" "^x^" "^x^"'")] (mp RS disjE) 1,
wenzelm@23152
  1004
        TRY (safe_tac HOL_cs),
wenzelm@23152
  1005
        REPEAT (CHANGED (asm_simp_tac take_ss 1))];
wenzelm@27208
  1006
      fun tacs ctxt = [
wenzelm@23152
  1007
        rtac impI 1,
wenzelm@27208
  1008
        InductTacs.induct_tac ctxt [[SOME "n"]] 1,
wenzelm@23152
  1009
        simp_tac take_ss 1,
wenzelm@23152
  1010
        safe_tac HOL_cs] @
wenzelm@27208
  1011
        flat (mapn (x_tacs ctxt) 0 xs);
wenzelm@23152
  1012
    in pg [ax_bisim_def] goal tacs end;
wenzelm@23152
  1013
in
huffman@29402
  1014
  val _ = trace " Proving coind...";
wenzelm@23152
  1015
  val coind = 
wenzelm@23152
  1016
    let
wenzelm@23152
  1017
      fun mk_prj n x = mk_trp (proj (%:"R") eqs n $ %:x $ %:(x^"'"));
wenzelm@23152
  1018
      fun mk_eqn x = %:x === %:(x^"'");
wenzelm@23152
  1019
      val goal =
wenzelm@23152
  1020
        mk_trp (%%:(comp_dname^"_bisim") $ %:"R") ===>
wenzelm@23152
  1021
          Logic.list_implies (mapn mk_prj 0 xs,
wenzelm@23152
  1022
            mk_trp (foldr1 mk_conj (map mk_eqn xs)));
wenzelm@23152
  1023
      val tacs =
wenzelm@23152
  1024
        TRY (safe_tac HOL_cs) ::
wenzelm@26336
  1025
        maps (fn take_lemma => [
wenzelm@23152
  1026
          rtac take_lemma 1,
wenzelm@23152
  1027
          cut_facts_tac [coind_lemma] 1,
wenzelm@23152
  1028
          fast_tac HOL_cs 1])
wenzelm@26336
  1029
        take_lemmas;
wenzelm@27208
  1030
    in pg [] goal (K tacs) end;
wenzelm@23152
  1031
end; (* local *)
huffman@35444
  1032
COINDUCTION TEMPORARILY DISABLED *)
wenzelm@23152
  1033
wenzelm@32172
  1034
val inducts = Project_Rule.projections (ProofContext.init thy) ind;
huffman@30829
  1035
fun ind_rule (dname, rule) = ((Binding.empty, [rule]), [Induct.induct_type dname]);
huffman@31232
  1036
val induct_failed = (Thm.prop_of ind = Thm.prop_of TrueI);
huffman@30829
  1037
wenzelm@24712
  1038
in thy |> Sign.add_path comp_dnam
huffman@31004
  1039
       |> snd o PureThy.add_thmss [
huffman@31004
  1040
           ((Binding.name "take_rews"  , take_rews   ), [Simplifier.simp_add]),
huffman@31004
  1041
           ((Binding.name "take_lemmas", take_lemmas ), []),
huffman@31004
  1042
           ((Binding.name "finites"    , finites     ), []),
huffman@31004
  1043
           ((Binding.name "finite_ind" , [finite_ind]), []),
huffman@35444
  1044
           ((Binding.name "ind"        , [ind]       ), [])(*,
huffman@35444
  1045
           ((Binding.name "coind"      , [coind]     ), [])*)]
huffman@31232
  1046
       |> (if induct_failed then I
huffman@31232
  1047
           else snd o PureThy.add_thmss (map ind_rule (dnames ~~ inducts)))
haftmann@28536
  1048
       |> Sign.parent_path |> pair take_rews
wenzelm@23152
  1049
end; (* let *)
wenzelm@23152
  1050
end; (* struct *)