src/HOLCF/Tools/Domain/domain_theorems.ML
author haftmann
Thu Jan 28 11:48:49 2010 +0100 (2010-01-28)
changeset 34974 18b41bba42b5
parent 33971 9c7fa7f76950
child 35021 c839a4c670c6
child 35057 03d023236fcd
permissions -rw-r--r--
new theory Algebras.thy for generic algebraic structures
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(*  Title:      HOLCF/Tools/Domain/domain_theorems.ML
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    Author:     David von Oheimb
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    Author:     Brian Huffman
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Proof generator for domain command.
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*)
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val HOLCF_ss = @{simpset};
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signature DOMAIN_THEOREMS =
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sig
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  val theorems: Domain_Library.eq * Domain_Library.eq list -> theory -> thm list * theory;
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  val comp_theorems: bstring * Domain_Library.eq list -> theory -> thm list * theory;
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  val quiet_mode: bool Unsynchronized.ref;
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  val trace_domain: bool Unsynchronized.ref;
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end;
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structure Domain_Theorems :> DOMAIN_THEOREMS =
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struct
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val quiet_mode = Unsynchronized.ref false;
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val trace_domain = Unsynchronized.ref false;
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fun message s = if !quiet_mode then () else writeln s;
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fun trace s = if !trace_domain then tracing s else ();
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local
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val adm_impl_admw = @{thm adm_impl_admw};
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val adm_all = @{thm adm_all};
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val adm_conj = @{thm adm_conj};
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val adm_subst = @{thm adm_subst};
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val antisym_less_inverse = @{thm below_antisym_inverse};
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val beta_cfun = @{thm beta_cfun};
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val cfun_arg_cong = @{thm cfun_arg_cong};
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val ch2ch_fst = @{thm ch2ch_fst};
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val ch2ch_snd = @{thm ch2ch_snd};
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val ch2ch_Rep_CFunL = @{thm ch2ch_Rep_CFunL};
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val ch2ch_Rep_CFunR = @{thm ch2ch_Rep_CFunR};
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val chain_iterate = @{thm chain_iterate};
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val compact_ONE = @{thm compact_ONE};
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val compact_sinl = @{thm compact_sinl};
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val compact_sinr = @{thm compact_sinr};
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val compact_spair = @{thm compact_spair};
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val compact_up = @{thm compact_up};
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val contlub_cfun_arg = @{thm contlub_cfun_arg};
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val contlub_cfun_fun = @{thm contlub_cfun_fun};
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val contlub_fst = @{thm contlub_fst};
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val contlub_snd = @{thm contlub_snd};
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val contlubE = @{thm contlubE};
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val cont_const = @{thm cont_const};
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val cont_id = @{thm cont_id};
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val cont2cont_fst = @{thm cont2cont_fst};
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val cont2cont_snd = @{thm cont2cont_snd};
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val cont2cont_Rep_CFun = @{thm cont2cont_Rep_CFun};
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val fix_def2 = @{thm fix_def2};
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val injection_eq = @{thm injection_eq};
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val injection_less = @{thm injection_below};
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val lub_equal = @{thm lub_equal};
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val monofun_cfun_arg = @{thm monofun_cfun_arg};
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val retraction_strict = @{thm retraction_strict};
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val spair_eq = @{thm spair_eq};
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val spair_less = @{thm spair_below};
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val sscase1 = @{thm sscase1};
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val ssplit1 = @{thm ssplit1};
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val strictify1 = @{thm strictify1};
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val wfix_ind = @{thm wfix_ind};
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val iso_intro       = @{thm iso.intro};
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val iso_abs_iso     = @{thm iso.abs_iso};
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val iso_rep_iso     = @{thm iso.rep_iso};
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val iso_abs_strict  = @{thm iso.abs_strict};
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val iso_rep_strict  = @{thm iso.rep_strict};
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val iso_abs_defin'  = @{thm iso.abs_defin'};
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val iso_rep_defin'  = @{thm iso.rep_defin'};
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val iso_abs_defined = @{thm iso.abs_defined};
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val iso_rep_defined = @{thm iso.rep_defined};
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val iso_compact_abs = @{thm iso.compact_abs};
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val iso_compact_rep = @{thm iso.compact_rep};
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val iso_iso_swap    = @{thm iso.iso_swap};
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val exh_start = @{thm exh_start};
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val ex_defined_iffs = @{thms ex_defined_iffs};
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val exh_casedist0 = @{thm exh_casedist0};
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val exh_casedists = @{thms exh_casedists};
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open Domain_Library;
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infixr 0 ===>;
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infixr 0 ==>;
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infix 0 == ; 
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infix 1 ===;
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infix 1 ~= ;
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infix 1 <<;
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infix 1 ~<<;
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infix 9 `   ;
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infix 9 `% ;
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infix 9 `%%;
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infixr 9 oo;
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(* ----- general proof facilities ------------------------------------------- *)
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fun legacy_infer_term thy t =
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  let val ctxt = ProofContext.set_mode ProofContext.mode_schematic (ProofContext.init thy)
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  in singleton (Syntax.check_terms ctxt) (Sign.intern_term thy t) end;
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fun pg'' thy defs t tacs =
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  let
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    val t' = legacy_infer_term thy t;
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    val asms = Logic.strip_imp_prems t';
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    val prop = Logic.strip_imp_concl t';
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    fun tac {prems, context} =
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      rewrite_goals_tac defs THEN
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      EVERY (tacs {prems = map (rewrite_rule defs) prems, context = context})
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  in Goal.prove_global thy [] asms prop tac end;
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fun pg' thy defs t tacsf =
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  let
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    fun tacs {prems, context} =
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      if null prems then tacsf context
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      else cut_facts_tac prems 1 :: tacsf context;
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  in pg'' thy defs t tacs end;
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fun case_UU_tac ctxt rews i v =
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  InductTacs.case_tac ctxt (v^"=UU") i THEN
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  asm_simp_tac (HOLCF_ss addsimps rews) i;
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val chain_tac =
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  REPEAT_DETERM o resolve_tac 
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    [chain_iterate, ch2ch_Rep_CFunR, ch2ch_Rep_CFunL, ch2ch_fst, ch2ch_snd];
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(* ----- general proofs ----------------------------------------------------- *)
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val all2E = @{lemma "!x y . P x y ==> (P x y ==> R) ==> R" by simp}
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val dist_eqI = @{lemma "!!x::'a::po. ~ x << y ==> x ~= y" by (blast dest!: below_antisym_inverse)}
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in
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fun theorems (((dname, _), cons) : eq, eqs : eq list) thy =
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let
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val _ = message ("Proving isomorphism properties of domain "^dname^" ...");
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val pg = pg' thy;
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val map_tab = Domain_Isomorphism.get_map_tab thy;
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(* ----- getting the axioms and definitions --------------------------------- *)
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local
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  fun ga s dn = PureThy.get_thm thy (dn ^ "." ^ s);
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in
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  val ax_abs_iso  = ga "abs_iso"  dname;
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  val ax_rep_iso  = ga "rep_iso"  dname;
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  val ax_when_def = ga "when_def" dname;
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  fun get_def mk_name (con,_) = ga (mk_name con^"_def") dname;
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  val axs_con_def = map (get_def extern_name) cons;
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  val axs_dis_def = map (get_def dis_name) cons;
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  val axs_mat_def = map (get_def mat_name) cons;
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  val axs_pat_def = map (get_def pat_name) cons;
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  val axs_sel_def =
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    let
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      fun def_of_sel sel = ga (sel^"_def") dname;
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      fun def_of_arg arg = Option.map def_of_sel (sel_of arg);
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      fun defs_of_con (_, args) = map_filter def_of_arg args;
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    in
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      maps defs_of_con cons
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    end;
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  val ax_copy_def = ga "copy_def" dname;
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end; (* local *)
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(* ----- theorems concerning the isomorphism -------------------------------- *)
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val dc_abs  = %%:(dname^"_abs");
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val dc_rep  = %%:(dname^"_rep");
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val dc_copy = %%:(dname^"_copy");
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val x_name = "x";
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val iso_locale = iso_intro OF [ax_abs_iso, ax_rep_iso];
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val abs_strict = ax_rep_iso RS (allI RS retraction_strict);
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val rep_strict = ax_abs_iso RS (allI RS retraction_strict);
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val abs_defin' = iso_locale RS iso_abs_defin';
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val rep_defin' = iso_locale RS iso_rep_defin';
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val iso_rews = map Drule.standard [ax_abs_iso,ax_rep_iso,abs_strict,rep_strict];
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(* ----- generating beta reduction rules from definitions-------------------- *)
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val _ = trace " Proving beta reduction rules...";
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local
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  fun arglist (Const _ $ Abs (s, _, t)) =
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    let
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      val (vars,body) = arglist t;
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    in (s :: vars, body) end
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    | arglist t = ([], t);
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  fun bind_fun vars t = Library.foldr mk_All (vars, t);
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  fun bound_vars 0 = []
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    | bound_vars i = Bound (i-1) :: bound_vars (i - 1);
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in
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  fun appl_of_def def =
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    let
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      val (_ $ con $ lam) = concl_of def;
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      val (vars, rhs) = arglist lam;
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      val lhs = list_ccomb (con, bound_vars (length vars));
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      val appl = bind_fun vars (lhs == rhs);
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      val cs = ContProc.cont_thms lam;
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      val betas = map (fn c => mk_meta_eq (c RS beta_cfun)) cs;
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    in pg (def::betas) appl (K [rtac reflexive_thm 1]) end;
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end;
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val _ = trace "Proving when_appl...";
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val when_appl = appl_of_def ax_when_def;
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val _ = trace "Proving con_appls...";
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val con_appls = map appl_of_def axs_con_def;
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local
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  fun arg2typ n arg =
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    let val t = TVar (("'a", n), pcpoS)
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    in (n + 1, if is_lazy arg then mk_uT t else t) end;
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  fun args2typ n [] = (n, oneT)
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    | args2typ n [arg] = arg2typ n arg
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    | args2typ n (arg::args) =
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    let
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      val (n1, t1) = arg2typ n arg;
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      val (n2, t2) = args2typ n1 args
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    in (n2, mk_sprodT (t1, t2)) end;
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  fun cons2typ n [] = (n,oneT)
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    | cons2typ n [con] = args2typ n (snd con)
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    | cons2typ n (con::cons) =
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    let
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      val (n1, t1) = args2typ n (snd con);
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      val (n2, t2) = cons2typ n1 cons
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    in (n2, mk_ssumT (t1, t2)) end;
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in
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  fun cons2ctyp cons = ctyp_of thy (snd (cons2typ 1 cons));
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end;
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local 
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  val iso_swap = iso_locale RS iso_iso_swap;
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  fun one_con (con, args) =
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    let
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      val vns = map vname args;
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      val eqn = %:x_name === con_app2 con %: vns;
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      val conj = foldr1 mk_conj (eqn :: map (defined o %:) (nonlazy args));
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    in Library.foldr mk_ex (vns, conj) end;
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  val conj_assoc = @{thm conj_assoc};
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  val exh = foldr1 mk_disj ((%:x_name === UU) :: map one_con cons);
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  val thm1 = instantiate' [SOME (cons2ctyp cons)] [] exh_start;
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  val thm2 = rewrite_rule (map mk_meta_eq ex_defined_iffs) thm1;
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  val thm3 = rewrite_rule [mk_meta_eq @{thm conj_assoc}] thm2;
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  (* first 3 rules replace "x = UU \/ P" with "rep$x = UU \/ P" *)
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  val tacs = [
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    rtac disjE 1,
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    etac (rep_defin' RS disjI1) 2,
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    etac disjI2 2,
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    rewrite_goals_tac [mk_meta_eq iso_swap],
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    rtac thm3 1];
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in
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  val _ = trace " Proving exhaust...";
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  val exhaust = pg con_appls (mk_trp exh) (K tacs);
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  val _ = trace " Proving casedist...";
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  val casedist =
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    Drule.standard (rewrite_rule exh_casedists (exhaust RS exh_casedist0));
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end;
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local 
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  fun bind_fun t = Library.foldr mk_All (when_funs cons, t);
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  fun bound_fun i _ = Bound (length cons - i);
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  val when_app = list_ccomb (%%:(dname^"_when"), mapn bound_fun 1 cons);
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in
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  val _ = trace " Proving when_strict...";
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  val when_strict =
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    let
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      val axs = [when_appl, mk_meta_eq rep_strict];
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      val goal = bind_fun (mk_trp (strict when_app));
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      val tacs = [resolve_tac [sscase1, ssplit1, strictify1] 1];
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    in pg axs goal (K tacs) end;
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  val _ = trace " Proving when_apps...";
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  val when_apps =
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    let
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      fun one_when n (con,args) =
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        let
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          val axs = when_appl :: con_appls;
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          val goal = bind_fun (lift_defined %: (nonlazy args, 
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                mk_trp (when_app`(con_app con args) ===
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                       list_ccomb (bound_fun n 0, map %# args))));
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          val tacs = [asm_simp_tac (HOLCF_ss addsimps [ax_abs_iso]) 1];
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        in pg axs goal (K tacs) end;
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    in mapn one_when 1 cons end;
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end;
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val when_rews = when_strict :: when_apps;
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(* ----- theorems concerning the constructors, discriminators and selectors - *)
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local
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  fun dis_strict (con, _) =
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    let
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      val goal = mk_trp (strict (%%:(dis_name con)));
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    in pg axs_dis_def goal (K [rtac when_strict 1]) end;
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  fun dis_app c (con, args) =
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    let
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      val lhs = %%:(dis_name c) ` con_app con args;
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      val rhs = if con = c then TT else FF;
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      val goal = lift_defined %: (nonlazy args, mk_trp (lhs === rhs));
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      val tacs = [asm_simp_tac (HOLCF_ss addsimps when_rews) 1];
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    in pg axs_dis_def goal (K tacs) end;
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  val _ = trace " Proving dis_apps...";
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  val dis_apps = maps (fn (c,_) => map (dis_app c) cons) cons;
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  fun dis_defin (con, args) =
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   317
    let
wenzelm@23152
   318
      val goal = defined (%:x_name) ==> defined (%%:(dis_name con) `% x_name);
wenzelm@23152
   319
      val tacs =
wenzelm@23152
   320
        [rtac casedist 1,
wenzelm@23152
   321
         contr_tac 1,
wenzelm@23152
   322
         DETERM_UNTIL_SOLVED (CHANGED
wenzelm@23152
   323
          (asm_simp_tac (HOLCF_ss addsimps dis_apps) 1))];
wenzelm@27208
   324
    in pg [] goal (K tacs) end;
wenzelm@23152
   325
huffman@29402
   326
  val _ = trace " Proving dis_stricts...";
wenzelm@23152
   327
  val dis_stricts = map dis_strict cons;
huffman@29402
   328
  val _ = trace " Proving dis_defins...";
wenzelm@23152
   329
  val dis_defins = map dis_defin cons;
wenzelm@23152
   330
in
wenzelm@23152
   331
  val dis_rews = dis_stricts @ dis_defins @ dis_apps;
wenzelm@23152
   332
end;
wenzelm@23152
   333
wenzelm@23152
   334
local
wenzelm@23152
   335
  fun mat_strict (con, _) =
wenzelm@23152
   336
    let
huffman@30912
   337
      val goal = mk_trp (%%:(mat_name con) ` UU ` %:"rhs" === UU);
huffman@30912
   338
      val tacs = [asm_simp_tac (HOLCF_ss addsimps [when_strict]) 1];
wenzelm@27208
   339
    in pg axs_mat_def goal (K tacs) end;
wenzelm@23152
   340
huffman@29402
   341
  val _ = trace " Proving mat_stricts...";
wenzelm@23152
   342
  val mat_stricts = map mat_strict cons;
wenzelm@23152
   343
wenzelm@23152
   344
  fun one_mat c (con, args) =
wenzelm@23152
   345
    let
huffman@30912
   346
      val lhs = %%:(mat_name c) ` con_app con args ` %:"rhs";
wenzelm@23152
   347
      val rhs =
wenzelm@23152
   348
        if con = c
huffman@30912
   349
        then list_ccomb (%:"rhs", map %# args)
huffman@26012
   350
        else mk_fail;
wenzelm@23152
   351
      val goal = lift_defined %: (nonlazy args, mk_trp (lhs === rhs));
wenzelm@23152
   352
      val tacs = [asm_simp_tac (HOLCF_ss addsimps when_rews) 1];
wenzelm@27208
   353
    in pg axs_mat_def goal (K tacs) end;
wenzelm@23152
   354
huffman@29402
   355
  val _ = trace " Proving mat_apps...";
wenzelm@23152
   356
  val mat_apps =
wenzelm@26336
   357
    maps (fn (c,_) => map (one_mat c) cons) cons;
wenzelm@23152
   358
in
wenzelm@23152
   359
  val mat_rews = mat_stricts @ mat_apps;
wenzelm@23152
   360
end;
wenzelm@23152
   361
wenzelm@23152
   362
local
wenzelm@23152
   363
  fun ps args = mapn (fn n => fn _ => %:("pat" ^ string_of_int n)) 1 args;
wenzelm@23152
   364
huffman@26012
   365
  fun pat_lhs (con,args) = mk_branch (list_comb (%%:(pat_name con), ps args));
wenzelm@23152
   366
huffman@26012
   367
  fun pat_rhs (con,[]) = mk_return ((%:"rhs") ` HOLogic.unit)
wenzelm@23152
   368
    | pat_rhs (con,args) =
huffman@26012
   369
        (mk_branch (mk_ctuple_pat (ps args)))
wenzelm@23152
   370
          `(%:"rhs")`(mk_ctuple (map %# args));
wenzelm@23152
   371
wenzelm@23152
   372
  fun pat_strict c =
wenzelm@23152
   373
    let
wenzelm@25132
   374
      val axs = @{thm branch_def} :: axs_pat_def;
wenzelm@23152
   375
      val goal = mk_trp (strict (pat_lhs c ` (%:"rhs")));
wenzelm@23152
   376
      val tacs = [simp_tac (HOLCF_ss addsimps [when_strict]) 1];
wenzelm@27208
   377
    in pg axs goal (K tacs) end;
wenzelm@23152
   378
wenzelm@23152
   379
  fun pat_app c (con, args) =
wenzelm@23152
   380
    let
wenzelm@25132
   381
      val axs = @{thm branch_def} :: axs_pat_def;
wenzelm@23152
   382
      val lhs = (pat_lhs c)`(%:"rhs")`(con_app con args);
huffman@26012
   383
      val rhs = if con = fst c then pat_rhs c else mk_fail;
wenzelm@23152
   384
      val goal = lift_defined %: (nonlazy args, mk_trp (lhs === rhs));
wenzelm@23152
   385
      val tacs = [asm_simp_tac (HOLCF_ss addsimps when_rews) 1];
wenzelm@27208
   386
    in pg axs goal (K tacs) end;
wenzelm@23152
   387
huffman@29402
   388
  val _ = trace " Proving pat_stricts...";
wenzelm@23152
   389
  val pat_stricts = map pat_strict cons;
huffman@29402
   390
  val _ = trace " Proving pat_apps...";
wenzelm@26336
   391
  val pat_apps = maps (fn c => map (pat_app c) cons) cons;
wenzelm@23152
   392
in
wenzelm@23152
   393
  val pat_rews = pat_stricts @ pat_apps;
wenzelm@23152
   394
end;
wenzelm@23152
   395
wenzelm@23152
   396
local
wenzelm@23152
   397
  fun con_strict (con, args) = 
wenzelm@23152
   398
    let
huffman@30911
   399
      val rules = abs_strict :: @{thms con_strict_rules};
wenzelm@23152
   400
      fun one_strict vn =
wenzelm@23152
   401
        let
wenzelm@23152
   402
          fun f arg = if vname arg = vn then UU else %# arg;
wenzelm@23152
   403
          val goal = mk_trp (con_app2 con f args === UU);
huffman@30911
   404
          val tacs = [simp_tac (HOL_basic_ss addsimps rules) 1];
wenzelm@27208
   405
        in pg con_appls goal (K tacs) end;
wenzelm@23152
   406
    in map one_strict (nonlazy args) end;
wenzelm@23152
   407
wenzelm@23152
   408
  fun con_defin (con, args) =
wenzelm@23152
   409
    let
huffman@30913
   410
      fun iff_disj (t, []) = HOLogic.mk_not t
huffman@30913
   411
        | iff_disj (t, ts) = t === foldr1 HOLogic.mk_disj ts;
huffman@30913
   412
      val lhs = con_app con args === UU;
huffman@30913
   413
      val rhss = map (fn x => %:x === UU) (nonlazy args);
huffman@30913
   414
      val goal = mk_trp (iff_disj (lhs, rhss));
huffman@30913
   415
      val rule1 = iso_locale RS @{thm iso.abs_defined_iff};
huffman@30913
   416
      val rules = rule1 :: @{thms con_defined_iff_rules};
huffman@30913
   417
      val tacs = [simp_tac (HOL_ss addsimps rules) 1];
huffman@30911
   418
    in pg con_appls goal (K tacs) end;
wenzelm@23152
   419
in
huffman@29402
   420
  val _ = trace " Proving con_stricts...";
wenzelm@26336
   421
  val con_stricts = maps con_strict cons;
huffman@30911
   422
  val _ = trace " Proving con_defins...";
wenzelm@23152
   423
  val con_defins = map con_defin cons;
wenzelm@23152
   424
  val con_rews = con_stricts @ con_defins;
wenzelm@23152
   425
end;
wenzelm@23152
   426
wenzelm@23152
   427
local
wenzelm@23152
   428
  val rules =
wenzelm@23152
   429
    [compact_sinl, compact_sinr, compact_spair, compact_up, compact_ONE];
wenzelm@23152
   430
  fun con_compact (con, args) =
wenzelm@23152
   431
    let
huffman@26012
   432
      val concl = mk_trp (mk_compact (con_app con args));
huffman@26012
   433
      val goal = lift (fn x => mk_compact (%#x)) (args, concl);
wenzelm@23152
   434
      val tacs = [
wenzelm@23152
   435
        rtac (iso_locale RS iso_compact_abs) 1,
wenzelm@23152
   436
        REPEAT (resolve_tac rules 1 ORELSE atac 1)];
wenzelm@27208
   437
    in pg con_appls goal (K tacs) end;
wenzelm@23152
   438
in
huffman@29402
   439
  val _ = trace " Proving con_compacts...";
wenzelm@23152
   440
  val con_compacts = map con_compact cons;
wenzelm@23152
   441
end;
wenzelm@23152
   442
wenzelm@23152
   443
local
wenzelm@23152
   444
  fun one_sel sel =
wenzelm@23152
   445
    pg axs_sel_def (mk_trp (strict (%%:sel)))
wenzelm@27208
   446
      (K [simp_tac (HOLCF_ss addsimps when_rews) 1]);
wenzelm@23152
   447
wenzelm@23152
   448
  fun sel_strict (_, args) =
wenzelm@32952
   449
    map_filter (Option.map one_sel o sel_of) args;
wenzelm@23152
   450
in
huffman@29402
   451
  val _ = trace " Proving sel_stricts...";
wenzelm@26336
   452
  val sel_stricts = maps sel_strict cons;
wenzelm@23152
   453
end;
wenzelm@23152
   454
wenzelm@23152
   455
local
wenzelm@23152
   456
  fun sel_app_same c n sel (con, args) =
wenzelm@23152
   457
    let
wenzelm@23152
   458
      val nlas = nonlazy args;
wenzelm@23152
   459
      val vns = map vname args;
wenzelm@23152
   460
      val vnn = List.nth (vns, n);
wenzelm@33317
   461
      val nlas' = filter (fn v => v <> vnn) nlas;
wenzelm@23152
   462
      val lhs = (%%:sel)`(con_app con args);
wenzelm@23152
   463
      val goal = lift_defined %: (nlas', mk_trp (lhs === %:vnn));
wenzelm@27208
   464
      fun tacs1 ctxt =
wenzelm@23152
   465
        if vnn mem nlas
wenzelm@27208
   466
        then [case_UU_tac ctxt (when_rews @ con_stricts) 1 vnn]
wenzelm@23152
   467
        else [];
wenzelm@23152
   468
      val tacs2 = [asm_simp_tac (HOLCF_ss addsimps when_rews) 1];
wenzelm@27208
   469
    in pg axs_sel_def goal (fn ctxt => (tacs1 ctxt @ tacs2)) end;
wenzelm@23152
   470
wenzelm@23152
   471
  fun sel_app_diff c n sel (con, args) =
wenzelm@23152
   472
    let
wenzelm@23152
   473
      val nlas = nonlazy args;
wenzelm@23152
   474
      val goal = mk_trp (%%:sel ` con_app con args === UU);
wenzelm@27208
   475
      fun tacs1 ctxt = map (case_UU_tac ctxt (when_rews @ con_stricts) 1) nlas;
wenzelm@23152
   476
      val tacs2 = [asm_simp_tac (HOLCF_ss addsimps when_rews) 1];
wenzelm@27208
   477
    in pg axs_sel_def goal (fn ctxt => (tacs1 ctxt @ tacs2)) end;
wenzelm@23152
   478
wenzelm@23152
   479
  fun sel_app c n sel (con, args) =
wenzelm@23152
   480
    if con = c
wenzelm@23152
   481
    then sel_app_same c n sel (con, args)
wenzelm@23152
   482
    else sel_app_diff c n sel (con, args);
wenzelm@23152
   483
wenzelm@23152
   484
  fun one_sel c n sel = map (sel_app c n sel) cons;
wenzelm@23152
   485
  fun one_sel' c n arg = Option.map (one_sel c n) (sel_of arg);
wenzelm@23152
   486
  fun one_con (c, args) =
wenzelm@26336
   487
    flat (map_filter I (mapn (one_sel' c) 0 args));
wenzelm@23152
   488
in
huffman@29402
   489
  val _ = trace " Proving sel_apps...";
wenzelm@26336
   490
  val sel_apps = maps one_con cons;
wenzelm@23152
   491
end;
wenzelm@23152
   492
wenzelm@23152
   493
local
wenzelm@23152
   494
  fun sel_defin sel =
wenzelm@23152
   495
    let
wenzelm@23152
   496
      val goal = defined (%:x_name) ==> defined (%%:sel`%x_name);
wenzelm@23152
   497
      val tacs = [
wenzelm@23152
   498
        rtac casedist 1,
wenzelm@23152
   499
        contr_tac 1,
wenzelm@23152
   500
        DETERM_UNTIL_SOLVED (CHANGED
wenzelm@23152
   501
          (asm_simp_tac (HOLCF_ss addsimps sel_apps) 1))];
wenzelm@27208
   502
    in pg [] goal (K tacs) end;
wenzelm@23152
   503
in
huffman@29402
   504
  val _ = trace " Proving sel_defins...";
wenzelm@23152
   505
  val sel_defins =
wenzelm@23152
   506
    if length cons = 1
wenzelm@32952
   507
    then map_filter (fn arg => Option.map sel_defin (sel_of arg))
wenzelm@23152
   508
                 (filter_out is_lazy (snd (hd cons)))
wenzelm@23152
   509
    else [];
wenzelm@23152
   510
end;
wenzelm@23152
   511
wenzelm@23152
   512
val sel_rews = sel_stricts @ sel_defins @ sel_apps;
wenzelm@23152
   513
huffman@29402
   514
val _ = trace " Proving dist_les...";
wenzelm@23152
   515
val distincts_le =
wenzelm@23152
   516
  let
wenzelm@23152
   517
    fun dist (con1, args1) (con2, args2) =
wenzelm@23152
   518
      let
wenzelm@23152
   519
        val goal = lift_defined %: (nonlazy args1,
wenzelm@23152
   520
                        mk_trp (con_app con1 args1 ~<< con_app con2 args2));
wenzelm@27208
   521
        fun tacs ctxt = [
huffman@25805
   522
          rtac @{thm rev_contrapos} 1,
wenzelm@27239
   523
          eres_inst_tac ctxt [(("f", 0), dis_name con1)] monofun_cfun_arg 1]
wenzelm@27208
   524
          @ map (case_UU_tac ctxt (con_stricts @ dis_rews) 1) (nonlazy args2)
wenzelm@23152
   525
          @ [asm_simp_tac (HOLCF_ss addsimps dis_rews) 1];
wenzelm@23152
   526
      in pg [] goal tacs end;
wenzelm@23152
   527
wenzelm@23152
   528
    fun distinct (con1, args1) (con2, args2) =
wenzelm@23152
   529
        let
wenzelm@23152
   530
          val arg1 = (con1, args1);
wenzelm@23152
   531
          val arg2 =
wenzelm@23152
   532
            (con2, ListPair.map (fn (arg,vn) => upd_vname (K vn) arg)
wenzelm@23152
   533
              (args2, Name.variant_list (map vname args1) (map vname args2)));
wenzelm@23152
   534
        in [dist arg1 arg2, dist arg2 arg1] end;
wenzelm@23152
   535
    fun distincts []      = []
wenzelm@23152
   536
      | distincts (c::cs) = (map (distinct c) cs) :: distincts cs;
wenzelm@23152
   537
  in distincts cons end;
wenzelm@26336
   538
val dist_les = flat (flat distincts_le);
huffman@29402
   539
huffman@29402
   540
val _ = trace " Proving dist_eqs...";
wenzelm@23152
   541
val dist_eqs =
wenzelm@23152
   542
  let
wenzelm@23152
   543
    fun distinct (_,args1) ((_,args2), leqs) =
wenzelm@23152
   544
      let
wenzelm@23152
   545
        val (le1,le2) = (hd leqs, hd(tl leqs));
wenzelm@23152
   546
        val (eq1,eq2) = (le1 RS dist_eqI, le2 RS dist_eqI)
wenzelm@23152
   547
      in
wenzelm@23152
   548
        if nonlazy args1 = [] then [eq1, eq1 RS not_sym] else
wenzelm@23152
   549
        if nonlazy args2 = [] then [eq2, eq2 RS not_sym] else
wenzelm@23152
   550
          [eq1, eq2]
wenzelm@23152
   551
      end;
wenzelm@23152
   552
    fun distincts []      = []
huffman@31288
   553
      | distincts ((c,leqs)::cs) =
huffman@31288
   554
        flat
huffman@31288
   555
          (ListPair.map (distinct c) ((map #1 cs),leqs)) @
huffman@31288
   556
        distincts cs;
wenzelm@32957
   557
  in map Drule.standard (distincts (cons ~~ distincts_le)) end;
wenzelm@23152
   558
wenzelm@23152
   559
local 
wenzelm@23152
   560
  fun pgterm rel con args =
wenzelm@23152
   561
    let
wenzelm@23152
   562
      fun append s = upd_vname (fn v => v^s);
wenzelm@23152
   563
      val (largs, rargs) = (args, map (append "'") args);
wenzelm@23152
   564
      val concl =
wenzelm@23152
   565
        foldr1 mk_conj (ListPair.map rel (map %# largs, map %# rargs));
wenzelm@23152
   566
      val prem = rel (con_app con largs, con_app con rargs);
wenzelm@23152
   567
      val sargs = case largs of [_] => [] | _ => nonlazy args;
wenzelm@23152
   568
      val prop = lift_defined %: (sargs, mk_trp (prem === concl));
wenzelm@23152
   569
    in pg con_appls prop end;
wenzelm@33317
   570
  val cons' = filter (fn (_,args) => args<>[]) cons;
wenzelm@23152
   571
in
huffman@29402
   572
  val _ = trace " Proving inverts...";
wenzelm@23152
   573
  val inverts =
wenzelm@23152
   574
    let
wenzelm@23152
   575
      val abs_less = ax_abs_iso RS (allI RS injection_less);
wenzelm@23152
   576
      val tacs =
wenzelm@23152
   577
        [asm_full_simp_tac (HOLCF_ss addsimps [abs_less, spair_less]) 1];
wenzelm@27208
   578
    in map (fn (con, args) => pgterm (op <<) con args (K tacs)) cons' end;
wenzelm@23152
   579
huffman@29402
   580
  val _ = trace " Proving injects...";
wenzelm@23152
   581
  val injects =
wenzelm@23152
   582
    let
wenzelm@23152
   583
      val abs_eq = ax_abs_iso RS (allI RS injection_eq);
wenzelm@23152
   584
      val tacs = [asm_full_simp_tac (HOLCF_ss addsimps [abs_eq, spair_eq]) 1];
wenzelm@27208
   585
    in map (fn (con, args) => pgterm (op ===) con args (K tacs)) cons' end;
wenzelm@23152
   586
end;
wenzelm@23152
   587
wenzelm@23152
   588
(* ----- theorems concerning one induction step ----------------------------- *)
wenzelm@23152
   589
wenzelm@23152
   590
val copy_strict =
wenzelm@23152
   591
  let
huffman@31232
   592
    val _ = trace " Proving copy_strict...";
wenzelm@23152
   593
    val goal = mk_trp (strict (dc_copy `% "f"));
huffman@33504
   594
    val rules = [abs_strict, rep_strict] @ @{thms domain_map_stricts};
huffman@31232
   595
    val tacs = [asm_simp_tac (HOLCF_ss addsimps rules) 1];
wenzelm@27208
   596
  in pg [ax_copy_def] goal (K tacs) end;
wenzelm@23152
   597
wenzelm@23152
   598
local
wenzelm@23152
   599
  fun copy_app (con, args) =
wenzelm@23152
   600
    let
wenzelm@23152
   601
      val lhs = dc_copy`%"f"`(con_app con args);
huffman@31232
   602
      fun one_rhs arg =
haftmann@33971
   603
          if Datatype_Aux.is_rec_type (dtyp_of arg)
huffman@33801
   604
          then Domain_Axioms.copy_of_dtyp map_tab
huffman@33801
   605
                 (proj (%:"f") eqs) (dtyp_of arg) ` (%# arg)
huffman@31232
   606
          else (%# arg);
huffman@31232
   607
      val rhs = con_app2 con one_rhs args;
wenzelm@23152
   608
      val goal = lift_defined %: (nonlazy_rec args, mk_trp (lhs === rhs));
wenzelm@33317
   609
      val args' = filter_out (fn a => is_rec a orelse is_lazy a) args;
huffman@33504
   610
      val stricts = abs_strict :: rep_strict :: @{thms domain_map_stricts};
wenzelm@27208
   611
      fun tacs1 ctxt = map (case_UU_tac ctxt stricts 1 o vname) args';
huffman@33504
   612
      val rules = [ax_abs_iso] @ @{thms domain_map_simps};
huffman@31232
   613
      val tacs2 = [asm_simp_tac (HOLCF_ss addsimps rules) 1];
huffman@31232
   614
    in pg (ax_copy_def::con_appls) goal (fn ctxt => (tacs1 ctxt @ tacs2)) end;
wenzelm@23152
   615
in
huffman@29402
   616
  val _ = trace " Proving copy_apps...";
wenzelm@23152
   617
  val copy_apps = map copy_app cons;
wenzelm@23152
   618
end;
wenzelm@23152
   619
wenzelm@23152
   620
local
wenzelm@23152
   621
  fun one_strict (con, args) = 
wenzelm@23152
   622
    let
wenzelm@23152
   623
      val goal = mk_trp (dc_copy`UU`(con_app con args) === UU);
wenzelm@23152
   624
      val rews = copy_strict :: copy_apps @ con_rews;
wenzelm@27208
   625
      fun tacs ctxt = map (case_UU_tac ctxt rews 1) (nonlazy args) @
wenzelm@23152
   626
        [asm_simp_tac (HOLCF_ss addsimps rews) 1];
wenzelm@23152
   627
    in pg [] goal tacs end;
wenzelm@23152
   628
wenzelm@23152
   629
  fun has_nonlazy_rec (_, args) = exists is_nonlazy_rec args;
wenzelm@23152
   630
in
huffman@29402
   631
  val _ = trace " Proving copy_stricts...";
wenzelm@33317
   632
  val copy_stricts = map one_strict (filter has_nonlazy_rec cons);
wenzelm@23152
   633
end;
wenzelm@23152
   634
wenzelm@23152
   635
val copy_rews = copy_strict :: copy_apps @ copy_stricts;
wenzelm@23152
   636
wenzelm@23152
   637
in
wenzelm@23152
   638
  thy
wenzelm@30364
   639
    |> Sign.add_path (Long_Name.base_name dname)
huffman@31004
   640
    |> snd o PureThy.add_thmss [
huffman@31004
   641
        ((Binding.name "iso_rews"  , iso_rews    ), [Simplifier.simp_add]),
huffman@31004
   642
        ((Binding.name "exhaust"   , [exhaust]   ), []),
huffman@31004
   643
        ((Binding.name "casedist"  , [casedist]  ), [Induct.cases_type dname]),
huffman@31004
   644
        ((Binding.name "when_rews" , when_rews   ), [Simplifier.simp_add]),
huffman@31004
   645
        ((Binding.name "compacts"  , con_compacts), [Simplifier.simp_add]),
huffman@33427
   646
        ((Binding.name "con_rews"  , con_rews    ),
huffman@33427
   647
         [Simplifier.simp_add, Fixrec.fixrec_simp_add]),
huffman@31004
   648
        ((Binding.name "sel_rews"  , sel_rews    ), [Simplifier.simp_add]),
huffman@31004
   649
        ((Binding.name "dis_rews"  , dis_rews    ), [Simplifier.simp_add]),
huffman@31004
   650
        ((Binding.name "pat_rews"  , pat_rews    ), [Simplifier.simp_add]),
huffman@31004
   651
        ((Binding.name "dist_les"  , dist_les    ), [Simplifier.simp_add]),
huffman@31004
   652
        ((Binding.name "dist_eqs"  , dist_eqs    ), [Simplifier.simp_add]),
huffman@31004
   653
        ((Binding.name "inverts"   , inverts     ), [Simplifier.simp_add]),
huffman@31004
   654
        ((Binding.name "injects"   , injects     ), [Simplifier.simp_add]),
huffman@31004
   655
        ((Binding.name "copy_rews" , copy_rews   ), [Simplifier.simp_add]),
huffman@33427
   656
        ((Binding.name "match_rews", mat_rews    ),
huffman@33427
   657
         [Simplifier.simp_add, Fixrec.fixrec_simp_add])]
wenzelm@24712
   658
    |> Sign.parent_path
haftmann@28536
   659
    |> pair (iso_rews @ when_rews @ con_rews @ sel_rews @ dis_rews @
wenzelm@23152
   660
        pat_rews @ dist_les @ dist_eqs @ copy_rews)
wenzelm@23152
   661
end; (* let *)
wenzelm@23152
   662
wenzelm@23152
   663
fun comp_theorems (comp_dnam, eqs: eq list) thy =
wenzelm@23152
   664
let
wenzelm@27232
   665
val global_ctxt = ProofContext.init thy;
huffman@33801
   666
val map_tab = Domain_Isomorphism.get_map_tab thy;
wenzelm@27232
   667
wenzelm@23152
   668
val dnames = map (fst o fst) eqs;
wenzelm@23152
   669
val conss  = map  snd        eqs;
haftmann@28965
   670
val comp_dname = Sign.full_bname thy comp_dnam;
wenzelm@23152
   671
huffman@29402
   672
val _ = message ("Proving induction properties of domain "^comp_dname^" ...");
wenzelm@23152
   673
val pg = pg' thy;
wenzelm@23152
   674
wenzelm@23152
   675
(* ----- getting the composite axiom and definitions ------------------------ *)
wenzelm@23152
   676
wenzelm@23152
   677
local
wenzelm@26343
   678
  fun ga s dn = PureThy.get_thm thy (dn ^ "." ^ s);
wenzelm@23152
   679
in
wenzelm@23152
   680
  val axs_reach      = map (ga "reach"     ) dnames;
wenzelm@23152
   681
  val axs_take_def   = map (ga "take_def"  ) dnames;
wenzelm@23152
   682
  val axs_finite_def = map (ga "finite_def") dnames;
wenzelm@23152
   683
  val ax_copy2_def   =      ga "copy_def"  comp_dnam;
wenzelm@23152
   684
  val ax_bisim_def   =      ga "bisim_def" comp_dnam;
wenzelm@23152
   685
end;
wenzelm@23152
   686
wenzelm@23152
   687
local
wenzelm@26343
   688
  fun gt  s dn = PureThy.get_thm  thy (dn ^ "." ^ s);
wenzelm@26343
   689
  fun gts s dn = PureThy.get_thms thy (dn ^ "." ^ s);
wenzelm@23152
   690
in
wenzelm@23152
   691
  val cases = map (gt  "casedist" ) dnames;
wenzelm@26336
   692
  val con_rews  = maps (gts "con_rews" ) dnames;
wenzelm@26336
   693
  val copy_rews = maps (gts "copy_rews") dnames;
wenzelm@23152
   694
end;
wenzelm@23152
   695
wenzelm@23152
   696
fun dc_take dn = %%:(dn^"_take");
wenzelm@23152
   697
val x_name = idx_name dnames "x"; 
wenzelm@23152
   698
val P_name = idx_name dnames "P";
wenzelm@23152
   699
val n_eqs = length eqs;
wenzelm@23152
   700
wenzelm@23152
   701
(* ----- theorems concerning finite approximation and finite induction ------ *)
wenzelm@23152
   702
wenzelm@23152
   703
local
wenzelm@32149
   704
  val iterate_Cprod_ss = global_simpset_of @{theory Fix};
wenzelm@23152
   705
  val copy_con_rews  = copy_rews @ con_rews;
wenzelm@23152
   706
  val copy_take_defs =
wenzelm@23152
   707
    (if n_eqs = 1 then [] else [ax_copy2_def]) @ axs_take_def;
huffman@29402
   708
  val _ = trace " Proving take_stricts...";
wenzelm@23152
   709
  val take_stricts =
wenzelm@23152
   710
    let
wenzelm@23152
   711
      fun one_eq ((dn, args), _) = strict (dc_take dn $ %:"n");
wenzelm@23152
   712
      val goal = mk_trp (foldr1 mk_conj (map one_eq eqs));
wenzelm@27208
   713
      fun tacs ctxt = [
wenzelm@27208
   714
        InductTacs.induct_tac ctxt [[SOME "n"]] 1,
wenzelm@23152
   715
        simp_tac iterate_Cprod_ss 1,
wenzelm@23152
   716
        asm_simp_tac (iterate_Cprod_ss addsimps copy_rews) 1];
wenzelm@23152
   717
    in pg copy_take_defs goal tacs end;
wenzelm@23152
   718
wenzelm@23152
   719
  val take_stricts' = rewrite_rule copy_take_defs take_stricts;
wenzelm@23152
   720
  fun take_0 n dn =
wenzelm@23152
   721
    let
haftmann@34974
   722
      val goal = mk_trp ((dc_take dn $ %%: @{const_name Algebras.zero}) `% x_name n === UU);
wenzelm@27208
   723
    in pg axs_take_def goal (K [simp_tac iterate_Cprod_ss 1]) end;
wenzelm@23152
   724
  val take_0s = mapn take_0 1 dnames;
wenzelm@27208
   725
  fun c_UU_tac ctxt = case_UU_tac ctxt (take_stricts'::copy_con_rews) 1;
huffman@29402
   726
  val _ = trace " Proving take_apps...";
wenzelm@23152
   727
  val take_apps =
wenzelm@23152
   728
    let
wenzelm@23152
   729
      fun mk_eqn dn (con, args) =
wenzelm@23152
   730
        let
wenzelm@23152
   731
          fun mk_take n = dc_take (List.nth (dnames, n)) $ %:"n";
huffman@31232
   732
          fun one_rhs arg =
haftmann@33971
   733
              if Datatype_Aux.is_rec_type (dtyp_of arg)
huffman@33801
   734
              then Domain_Axioms.copy_of_dtyp map_tab
huffman@33801
   735
                     mk_take (dtyp_of arg) ` (%# arg)
huffman@31232
   736
              else (%# arg);
wenzelm@23152
   737
          val lhs = (dc_take dn $ (%%:"Suc" $ %:"n"))`(con_app con args);
huffman@31232
   738
          val rhs = con_app2 con one_rhs args;
wenzelm@23152
   739
        in Library.foldr mk_all (map vname args, lhs === rhs) end;
wenzelm@23152
   740
      fun mk_eqns ((dn, _), cons) = map (mk_eqn dn) cons;
wenzelm@26336
   741
      val goal = mk_trp (foldr1 mk_conj (maps mk_eqns eqs));
wenzelm@33317
   742
      val simps = filter (has_fewer_prems 1) copy_rews;
wenzelm@27208
   743
      fun con_tac ctxt (con, args) =
wenzelm@23152
   744
        if nonlazy_rec args = []
wenzelm@23152
   745
        then all_tac
wenzelm@27208
   746
        else EVERY (map (c_UU_tac ctxt) (nonlazy_rec args)) THEN
wenzelm@23152
   747
          asm_full_simp_tac (HOLCF_ss addsimps copy_rews) 1;
wenzelm@27208
   748
      fun eq_tacs ctxt ((dn, _), cons) = map (con_tac ctxt) cons;
wenzelm@27208
   749
      fun tacs ctxt =
wenzelm@23152
   750
        simp_tac iterate_Cprod_ss 1 ::
wenzelm@27208
   751
        InductTacs.induct_tac ctxt [[SOME "n"]] 1 ::
wenzelm@23152
   752
        simp_tac (iterate_Cprod_ss addsimps copy_con_rews) 1 ::
wenzelm@23152
   753
        asm_full_simp_tac (HOLCF_ss addsimps simps) 1 ::
wenzelm@23152
   754
        TRY (safe_tac HOL_cs) ::
wenzelm@27208
   755
        maps (eq_tacs ctxt) eqs;
wenzelm@23152
   756
    in pg copy_take_defs goal tacs end;
wenzelm@23152
   757
in
wenzelm@32957
   758
  val take_rews = map Drule.standard
wenzelm@27232
   759
    (atomize global_ctxt take_stricts @ take_0s @ atomize global_ctxt take_apps);
wenzelm@23152
   760
end; (* local *)
wenzelm@23152
   761
wenzelm@23152
   762
local
wenzelm@23152
   763
  fun one_con p (con,args) =
wenzelm@23152
   764
    let
wenzelm@23152
   765
      fun ind_hyp arg = %:(P_name (1 + rec_of arg)) $ bound_arg args arg;
wenzelm@23152
   766
      val t1 = mk_trp (%:p $ con_app2 con (bound_arg args) args);
wenzelm@33317
   767
      val t2 = lift ind_hyp (filter is_rec args, t1);
wenzelm@23152
   768
      val t3 = lift_defined (bound_arg (map vname args)) (nonlazy args, t2);
wenzelm@23152
   769
    in Library.foldr mk_All (map vname args, t3) end;
wenzelm@23152
   770
wenzelm@23152
   771
  fun one_eq ((p, cons), concl) =
wenzelm@23152
   772
    mk_trp (%:p $ UU) ===> Logic.list_implies (map (one_con p) cons, concl);
wenzelm@23152
   773
wenzelm@23152
   774
  fun ind_term concf = Library.foldr one_eq
wenzelm@23152
   775
    (mapn (fn n => fn x => (P_name n, x)) 1 conss,
wenzelm@23152
   776
     mk_trp (foldr1 mk_conj (mapn concf 1 dnames)));
wenzelm@23152
   777
  val take_ss = HOL_ss addsimps take_rews;
wenzelm@27208
   778
  fun quant_tac ctxt i = EVERY
wenzelm@27239
   779
    (mapn (fn n => fn _ => res_inst_tac ctxt [(("x", 0), x_name n)] spec i) 1 dnames);
wenzelm@23152
   780
wenzelm@23152
   781
  fun ind_prems_tac prems = EVERY
wenzelm@26336
   782
    (maps (fn cons =>
wenzelm@23152
   783
      (resolve_tac prems 1 ::
wenzelm@26336
   784
        maps (fn (_,args) => 
wenzelm@23152
   785
          resolve_tac prems 1 ::
wenzelm@23152
   786
          map (K(atac 1)) (nonlazy args) @
wenzelm@33317
   787
          map (K(atac 1)) (filter is_rec args))
wenzelm@26336
   788
        cons))
wenzelm@26336
   789
      conss);
wenzelm@23152
   790
  local 
wenzelm@23152
   791
    (* check whether every/exists constructor of the n-th part of the equation:
wenzelm@23152
   792
       it has a possibly indirectly recursive argument that isn't/is possibly 
wenzelm@23152
   793
       indirectly lazy *)
wenzelm@23152
   794
    fun rec_to quant nfn rfn ns lazy_rec (n,cons) = quant (exists (fn arg => 
wenzelm@23152
   795
          is_rec arg andalso not(rec_of arg mem ns) andalso
wenzelm@23152
   796
          ((rec_of arg =  n andalso nfn(lazy_rec orelse is_lazy arg)) orelse 
wenzelm@23152
   797
            rec_of arg <> n andalso rec_to quant nfn rfn (rec_of arg::ns) 
wenzelm@23152
   798
              (lazy_rec orelse is_lazy arg) (n, (List.nth(conss,rec_of arg))))
wenzelm@23152
   799
          ) o snd) cons;
wenzelm@23152
   800
    fun all_rec_to ns  = rec_to forall not all_rec_to  ns;
wenzelm@23152
   801
    fun warn (n,cons) =
wenzelm@23152
   802
      if all_rec_to [] false (n,cons)
wenzelm@23152
   803
      then (warning ("domain "^List.nth(dnames,n)^" is empty!"); true)
wenzelm@23152
   804
      else false;
wenzelm@23152
   805
    fun lazy_rec_to ns = rec_to exists I  lazy_rec_to ns;
wenzelm@23152
   806
wenzelm@23152
   807
  in
wenzelm@23152
   808
    val n__eqs = mapn (fn n => fn (_,cons) => (n,cons)) 0 eqs;
wenzelm@23152
   809
    val is_emptys = map warn n__eqs;
wenzelm@23152
   810
    val is_finite = forall (not o lazy_rec_to [] false) n__eqs;
wenzelm@23152
   811
  end;
wenzelm@23152
   812
in (* local *)
huffman@29402
   813
  val _ = trace " Proving finite_ind...";
wenzelm@23152
   814
  val finite_ind =
wenzelm@23152
   815
    let
wenzelm@23152
   816
      fun concf n dn = %:(P_name n) $ (dc_take dn $ %:"n" `%(x_name n));
wenzelm@23152
   817
      val goal = ind_term concf;
wenzelm@23152
   818
wenzelm@27208
   819
      fun tacf {prems, context} =
wenzelm@23152
   820
        let
wenzelm@23152
   821
          val tacs1 = [
wenzelm@27208
   822
            quant_tac context 1,
wenzelm@23152
   823
            simp_tac HOL_ss 1,
wenzelm@27208
   824
            InductTacs.induct_tac context [[SOME "n"]] 1,
wenzelm@23152
   825
            simp_tac (take_ss addsimps prems) 1,
wenzelm@23152
   826
            TRY (safe_tac HOL_cs)];
wenzelm@23152
   827
          fun arg_tac arg =
wenzelm@27208
   828
            case_UU_tac context (prems @ con_rews) 1
wenzelm@23152
   829
              (List.nth (dnames, rec_of arg) ^ "_take n$" ^ vname arg);
wenzelm@23152
   830
          fun con_tacs (con, args) = 
wenzelm@23152
   831
            asm_simp_tac take_ss 1 ::
wenzelm@33317
   832
            map arg_tac (filter is_nonlazy_rec args) @
wenzelm@23152
   833
            [resolve_tac prems 1] @
wenzelm@33317
   834
            map (K (atac 1)) (nonlazy args) @
wenzelm@33317
   835
            map (K (etac spec 1)) (filter is_rec args);
wenzelm@23152
   836
          fun cases_tacs (cons, cases) =
wenzelm@27239
   837
            res_inst_tac context [(("x", 0), "x")] cases 1 ::
wenzelm@23152
   838
            asm_simp_tac (take_ss addsimps prems) 1 ::
wenzelm@26336
   839
            maps con_tacs cons;
wenzelm@23152
   840
        in
wenzelm@26336
   841
          tacs1 @ maps cases_tacs (conss ~~ cases)
wenzelm@23152
   842
        end;
huffman@31232
   843
    in pg'' thy [] goal tacf
huffman@31232
   844
       handle ERROR _ => (warning "Proof of finite_ind failed."; TrueI)
huffman@31232
   845
    end;
wenzelm@23152
   846
huffman@29402
   847
  val _ = trace " Proving take_lemmas...";
wenzelm@23152
   848
  val take_lemmas =
wenzelm@23152
   849
    let
wenzelm@23152
   850
      fun take_lemma n (dn, ax_reach) =
wenzelm@23152
   851
        let
wenzelm@23152
   852
          val lhs = dc_take dn $ Bound 0 `%(x_name n);
wenzelm@23152
   853
          val rhs = dc_take dn $ Bound 0 `%(x_name n^"'");
wenzelm@23152
   854
          val concl = mk_trp (%:(x_name n) === %:(x_name n^"'"));
wenzelm@23152
   855
          val goal = mk_All ("n", mk_trp (lhs === rhs)) ===> concl;
huffman@33396
   856
          val rules = [contlub_fst RS contlubE RS ssubst,
huffman@33396
   857
                       contlub_snd RS contlubE RS ssubst];
wenzelm@27208
   858
          fun tacf {prems, context} = [
wenzelm@27239
   859
            res_inst_tac context [(("t", 0), x_name n    )] (ax_reach RS subst) 1,
wenzelm@27239
   860
            res_inst_tac context [(("t", 0), x_name n^"'")] (ax_reach RS subst) 1,
wenzelm@23152
   861
            stac fix_def2 1,
wenzelm@23152
   862
            REPEAT (CHANGED
huffman@33396
   863
              (resolve_tac rules 1 THEN chain_tac 1)),
wenzelm@23152
   864
            stac contlub_cfun_fun 1,
wenzelm@23152
   865
            stac contlub_cfun_fun 2,
wenzelm@23152
   866
            rtac lub_equal 3,
wenzelm@23152
   867
            chain_tac 1,
wenzelm@23152
   868
            rtac allI 1,
wenzelm@23152
   869
            resolve_tac prems 1];
wenzelm@23152
   870
        in pg'' thy axs_take_def goal tacf end;
wenzelm@23152
   871
    in mapn take_lemma 1 (dnames ~~ axs_reach) end;
wenzelm@23152
   872
wenzelm@23152
   873
(* ----- theorems concerning finiteness and induction ----------------------- *)
wenzelm@23152
   874
huffman@29402
   875
  val _ = trace " Proving finites, ind...";
wenzelm@23152
   876
  val (finites, ind) =
huffman@31232
   877
  (
wenzelm@23152
   878
    if is_finite
wenzelm@23152
   879
    then (* finite case *)
wenzelm@23152
   880
      let 
wenzelm@23152
   881
        fun take_enough dn = mk_ex ("n",dc_take dn $ Bound 0 ` %:"x" === %:"x");
wenzelm@23152
   882
        fun dname_lemma dn =
wenzelm@23152
   883
          let
wenzelm@23152
   884
            val prem1 = mk_trp (defined (%:"x"));
wenzelm@23152
   885
            val disj1 = mk_all ("n", dc_take dn $ Bound 0 ` %:"x" === UU);
wenzelm@23152
   886
            val prem2 = mk_trp (mk_disj (disj1, take_enough dn));
wenzelm@23152
   887
            val concl = mk_trp (take_enough dn);
wenzelm@23152
   888
            val goal = prem1 ===> prem2 ===> concl;
wenzelm@23152
   889
            val tacs = [
wenzelm@23152
   890
              etac disjE 1,
wenzelm@23152
   891
              etac notE 1,
wenzelm@23152
   892
              resolve_tac take_lemmas 1,
wenzelm@23152
   893
              asm_simp_tac take_ss 1,
wenzelm@23152
   894
              atac 1];
wenzelm@27208
   895
          in pg [] goal (K tacs) end;
huffman@31232
   896
        val _ = trace " Proving finite_lemmas1a";
wenzelm@23152
   897
        val finite_lemmas1a = map dname_lemma dnames;
wenzelm@23152
   898
 
huffman@31232
   899
        val _ = trace " Proving finite_lemma1b";
wenzelm@23152
   900
        val finite_lemma1b =
wenzelm@23152
   901
          let
wenzelm@23152
   902
            fun mk_eqn n ((dn, args), _) =
wenzelm@23152
   903
              let
wenzelm@23152
   904
                val disj1 = dc_take dn $ Bound 1 ` Bound 0 === UU;
wenzelm@23152
   905
                val disj2 = dc_take dn $ Bound 1 ` Bound 0 === Bound 0;
wenzelm@23152
   906
              in
wenzelm@23152
   907
                mk_constrainall
wenzelm@23152
   908
                  (x_name n, Type (dn,args), mk_disj (disj1, disj2))
wenzelm@23152
   909
              end;
wenzelm@23152
   910
            val goal =
wenzelm@23152
   911
              mk_trp (mk_all ("n", foldr1 mk_conj (mapn mk_eqn 1 eqs)));
wenzelm@27208
   912
            fun arg_tacs ctxt vn = [
wenzelm@27239
   913
              eres_inst_tac ctxt [(("x", 0), vn)] all_dupE 1,
wenzelm@23152
   914
              etac disjE 1,
wenzelm@23152
   915
              asm_simp_tac (HOL_ss addsimps con_rews) 1,
wenzelm@23152
   916
              asm_simp_tac take_ss 1];
wenzelm@27208
   917
            fun con_tacs ctxt (con, args) =
wenzelm@23152
   918
              asm_simp_tac take_ss 1 ::
wenzelm@27208
   919
              maps (arg_tacs ctxt) (nonlazy_rec args);
wenzelm@27208
   920
            fun foo_tacs ctxt n (cons, cases) =
wenzelm@23152
   921
              simp_tac take_ss 1 ::
wenzelm@23152
   922
              rtac allI 1 ::
wenzelm@27239
   923
              res_inst_tac ctxt [(("x", 0), x_name n)] cases 1 ::
wenzelm@23152
   924
              asm_simp_tac take_ss 1 ::
wenzelm@27208
   925
              maps (con_tacs ctxt) cons;
wenzelm@27208
   926
            fun tacs ctxt =
wenzelm@23152
   927
              rtac allI 1 ::
wenzelm@27208
   928
              InductTacs.induct_tac ctxt [[SOME "n"]] 1 ::
wenzelm@23152
   929
              simp_tac take_ss 1 ::
wenzelm@23152
   930
              TRY (safe_tac (empty_cs addSEs [conjE] addSIs [conjI])) ::
wenzelm@27208
   931
              flat (mapn (foo_tacs ctxt) 1 (conss ~~ cases));
wenzelm@23152
   932
          in pg [] goal tacs end;
wenzelm@23152
   933
wenzelm@23152
   934
        fun one_finite (dn, l1b) =
wenzelm@23152
   935
          let
wenzelm@23152
   936
            val goal = mk_trp (%%:(dn^"_finite") $ %:"x");
wenzelm@27208
   937
            fun tacs ctxt = [
wenzelm@27208
   938
              case_UU_tac ctxt take_rews 1 "x",
wenzelm@23152
   939
              eresolve_tac finite_lemmas1a 1,
wenzelm@23152
   940
              step_tac HOL_cs 1,
wenzelm@23152
   941
              step_tac HOL_cs 1,
wenzelm@23152
   942
              cut_facts_tac [l1b] 1,
wenzelm@23152
   943
              fast_tac HOL_cs 1];
wenzelm@23152
   944
          in pg axs_finite_def goal tacs end;
wenzelm@23152
   945
huffman@31232
   946
        val _ = trace " Proving finites";
wenzelm@27232
   947
        val finites = map one_finite (dnames ~~ atomize global_ctxt finite_lemma1b);
huffman@31232
   948
        val _ = trace " Proving ind";
wenzelm@23152
   949
        val ind =
wenzelm@23152
   950
          let
wenzelm@23152
   951
            fun concf n dn = %:(P_name n) $ %:(x_name n);
wenzelm@27208
   952
            fun tacf {prems, context} =
wenzelm@23152
   953
              let
wenzelm@23152
   954
                fun finite_tacs (finite, fin_ind) = [
wenzelm@23152
   955
                  rtac(rewrite_rule axs_finite_def finite RS exE)1,
wenzelm@23152
   956
                  etac subst 1,
wenzelm@23152
   957
                  rtac fin_ind 1,
wenzelm@23152
   958
                  ind_prems_tac prems];
wenzelm@23152
   959
              in
wenzelm@23152
   960
                TRY (safe_tac HOL_cs) ::
wenzelm@27232
   961
                maps finite_tacs (finites ~~ atomize global_ctxt finite_ind)
wenzelm@23152
   962
              end;
wenzelm@23152
   963
          in pg'' thy [] (ind_term concf) tacf end;
wenzelm@23152
   964
      in (finites, ind) end (* let *)
wenzelm@23152
   965
wenzelm@23152
   966
    else (* infinite case *)
wenzelm@23152
   967
      let
wenzelm@23152
   968
        fun one_finite n dn =
wenzelm@27239
   969
          read_instantiate global_ctxt [(("P", 0), dn ^ "_finite " ^ x_name n)] excluded_middle;
wenzelm@23152
   970
        val finites = mapn one_finite 1 dnames;
wenzelm@23152
   971
wenzelm@23152
   972
        val goal =
wenzelm@23152
   973
          let
huffman@26012
   974
            fun one_adm n _ = mk_trp (mk_adm (%:(P_name n)));
wenzelm@23152
   975
            fun concf n dn = %:(P_name n) $ %:(x_name n);
wenzelm@23152
   976
          in Logic.list_implies (mapn one_adm 1 dnames, ind_term concf) end;
huffman@33396
   977
        val cont_rules =
huffman@33396
   978
            [cont_id, cont_const, cont2cont_Rep_CFun,
huffman@33396
   979
             cont2cont_fst, cont2cont_snd];
wenzelm@27208
   980
        fun tacf {prems, context} =
wenzelm@23152
   981
          map (fn ax_reach => rtac (ax_reach RS subst) 1) axs_reach @ [
wenzelm@27208
   982
          quant_tac context 1,
wenzelm@23152
   983
          rtac (adm_impl_admw RS wfix_ind) 1,
huffman@25895
   984
          REPEAT_DETERM (rtac adm_all 1),
wenzelm@23152
   985
          REPEAT_DETERM (
wenzelm@23152
   986
            TRY (rtac adm_conj 1) THEN 
wenzelm@23152
   987
            rtac adm_subst 1 THEN 
huffman@33396
   988
            REPEAT (resolve_tac cont_rules 1) THEN
huffman@33396
   989
            resolve_tac prems 1),
wenzelm@23152
   990
          strip_tac 1,
wenzelm@23152
   991
          rtac (rewrite_rule axs_take_def finite_ind) 1,
wenzelm@23152
   992
          ind_prems_tac prems];
wenzelm@23152
   993
        val ind = (pg'' thy [] goal tacf
wenzelm@23152
   994
          handle ERROR _ =>
huffman@33396
   995
            (warning "Cannot prove infinite induction rule"; TrueI));
huffman@31232
   996
      in (finites, ind) end
huffman@31232
   997
  )
huffman@31232
   998
      handle THM _ =>
huffman@31232
   999
             (warning "Induction proofs failed (THM raised)."; ([], TrueI))
huffman@31232
  1000
           | ERROR _ =>
huffman@33810
  1001
             (warning "Cannot prove induction rule"; ([], TrueI));
huffman@31232
  1002
huffman@31232
  1003
wenzelm@23152
  1004
end; (* local *)
wenzelm@23152
  1005
wenzelm@23152
  1006
(* ----- theorem concerning coinduction ------------------------------------- *)
wenzelm@23152
  1007
wenzelm@23152
  1008
local
wenzelm@23152
  1009
  val xs = mapn (fn n => K (x_name n)) 1 dnames;
wenzelm@23152
  1010
  fun bnd_arg n i = Bound(2*(n_eqs - n)-i-1);
wenzelm@23152
  1011
  val take_ss = HOL_ss addsimps take_rews;
wenzelm@23152
  1012
  val sproj = prj (fn s => K("fst("^s^")")) (fn s => K("snd("^s^")"));
huffman@29402
  1013
  val _ = trace " Proving coind_lemma...";
wenzelm@23152
  1014
  val coind_lemma =
wenzelm@23152
  1015
    let
wenzelm@23152
  1016
      fun mk_prj n _ = proj (%:"R") eqs n $ bnd_arg n 0 $ bnd_arg n 1;
wenzelm@23152
  1017
      fun mk_eqn n dn =
wenzelm@23152
  1018
        (dc_take dn $ %:"n" ` bnd_arg n 0) ===
wenzelm@23152
  1019
        (dc_take dn $ %:"n" ` bnd_arg n 1);
wenzelm@23152
  1020
      fun mk_all2 (x,t) = mk_all (x, mk_all (x^"'", t));
wenzelm@23152
  1021
      val goal =
wenzelm@23152
  1022
        mk_trp (mk_imp (%%:(comp_dname^"_bisim") $ %:"R",
wenzelm@23152
  1023
          Library.foldr mk_all2 (xs,
wenzelm@23152
  1024
            Library.foldr mk_imp (mapn mk_prj 0 dnames,
wenzelm@23152
  1025
              foldr1 mk_conj (mapn mk_eqn 0 dnames)))));
wenzelm@27208
  1026
      fun x_tacs ctxt n x = [
wenzelm@23152
  1027
        rotate_tac (n+1) 1,
wenzelm@23152
  1028
        etac all2E 1,
wenzelm@27239
  1029
        eres_inst_tac ctxt [(("P", 1), sproj "R" eqs n^" "^x^" "^x^"'")] (mp RS disjE) 1,
wenzelm@23152
  1030
        TRY (safe_tac HOL_cs),
wenzelm@23152
  1031
        REPEAT (CHANGED (asm_simp_tac take_ss 1))];
wenzelm@27208
  1032
      fun tacs ctxt = [
wenzelm@23152
  1033
        rtac impI 1,
wenzelm@27208
  1034
        InductTacs.induct_tac ctxt [[SOME "n"]] 1,
wenzelm@23152
  1035
        simp_tac take_ss 1,
wenzelm@23152
  1036
        safe_tac HOL_cs] @
wenzelm@27208
  1037
        flat (mapn (x_tacs ctxt) 0 xs);
wenzelm@23152
  1038
    in pg [ax_bisim_def] goal tacs end;
wenzelm@23152
  1039
in
huffman@29402
  1040
  val _ = trace " Proving coind...";
wenzelm@23152
  1041
  val coind = 
wenzelm@23152
  1042
    let
wenzelm@23152
  1043
      fun mk_prj n x = mk_trp (proj (%:"R") eqs n $ %:x $ %:(x^"'"));
wenzelm@23152
  1044
      fun mk_eqn x = %:x === %:(x^"'");
wenzelm@23152
  1045
      val goal =
wenzelm@23152
  1046
        mk_trp (%%:(comp_dname^"_bisim") $ %:"R") ===>
wenzelm@23152
  1047
          Logic.list_implies (mapn mk_prj 0 xs,
wenzelm@23152
  1048
            mk_trp (foldr1 mk_conj (map mk_eqn xs)));
wenzelm@23152
  1049
      val tacs =
wenzelm@23152
  1050
        TRY (safe_tac HOL_cs) ::
wenzelm@26336
  1051
        maps (fn take_lemma => [
wenzelm@23152
  1052
          rtac take_lemma 1,
wenzelm@23152
  1053
          cut_facts_tac [coind_lemma] 1,
wenzelm@23152
  1054
          fast_tac HOL_cs 1])
wenzelm@26336
  1055
        take_lemmas;
wenzelm@27208
  1056
    in pg [] goal (K tacs) end;
wenzelm@23152
  1057
end; (* local *)
wenzelm@23152
  1058
wenzelm@32172
  1059
val inducts = Project_Rule.projections (ProofContext.init thy) ind;
huffman@30829
  1060
fun ind_rule (dname, rule) = ((Binding.empty, [rule]), [Induct.induct_type dname]);
huffman@31232
  1061
val induct_failed = (Thm.prop_of ind = Thm.prop_of TrueI);
huffman@30829
  1062
wenzelm@24712
  1063
in thy |> Sign.add_path comp_dnam
huffman@31004
  1064
       |> snd o PureThy.add_thmss [
huffman@31004
  1065
           ((Binding.name "take_rews"  , take_rews   ), [Simplifier.simp_add]),
huffman@31004
  1066
           ((Binding.name "take_lemmas", take_lemmas ), []),
huffman@31004
  1067
           ((Binding.name "finites"    , finites     ), []),
huffman@31004
  1068
           ((Binding.name "finite_ind" , [finite_ind]), []),
huffman@31004
  1069
           ((Binding.name "ind"        , [ind]       ), []),
huffman@31004
  1070
           ((Binding.name "coind"      , [coind]     ), [])]
huffman@31232
  1071
       |> (if induct_failed then I
huffman@31232
  1072
           else snd o PureThy.add_thmss (map ind_rule (dnames ~~ inducts)))
haftmann@28536
  1073
       |> Sign.parent_path |> pair take_rews
wenzelm@23152
  1074
end; (* let *)
wenzelm@23152
  1075
end; (* local *)
wenzelm@23152
  1076
end; (* struct *)