src/HOLCF/Tools/Domain/domain_theorems.ML
author huffman
Sun Feb 28 20:56:28 2010 -0800 (2010-02-28)
changeset 35481 7bb9157507a9
parent 35468 09bc6a2e2296
child 35482 d756837b708d
permissions -rw-r--r--
add_domain_constructors takes iso_info record as argument
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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 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 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 lub_equal = @{thm lub_equal};
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val retraction_strict = @{thm retraction_strict};
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val wfix_ind = @{thm wfix_ind};
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val iso_intro = @{thm iso.intro};
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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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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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  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 iso_info : Domain_Isomorphism.iso_info =
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  {
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    absT = lhsT,
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    repT = rhsT,
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    abs_const = abs_const,
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    rep_const = rep_const,
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    abs_inverse = ax_abs_iso,
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    rep_inverse = ax_rep_iso
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  };
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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) (snd dom_eqn) iso_info ax_when_def thy;
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val con_appls = #con_betas result;
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val {exhaust, casedist, ...} = result;
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val {con_compacts, con_rews, inverts, injects, dist_les, dist_eqs, ...} = result;
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val {sel_rews, ...} = result;
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val when_rews = #cases result;
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val when_strict = hd when_rews;
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val dis_rews = #dis_rews result;
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val mat_rews = #match_rews result;
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val axs_pat_def = #pat_rews result;
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(* ----- theorems concerning the isomorphism -------------------------------- *)
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val pg = pg' thy;
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val dc_copy = %%:(dname^"_copy");
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val abs_strict = ax_rep_iso RS (allI RS retraction_strict);
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val rep_strict = ax_abs_iso RS (allI RS retraction_strict);
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val iso_rews = map Drule.export_without_context [ax_abs_iso, ax_rep_iso, abs_strict, rep_strict];
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(* ----- theorems concerning the constructors, discriminators and selectors - *)
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local
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  fun ps args = mapn (fn n => fn _ => %:("pat" ^ string_of_int n)) 1 args;
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  fun pat_lhs (con,_,args) = mk_branch (list_comb (%%:(pat_name con), ps args));
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  fun pat_rhs (con,_,[]) = mk_return ((%:"rhs") ` HOLogic.unit)
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    | pat_rhs (con,_,args) =
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        (mk_branch (mk_ctuple_pat (ps args)))
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          `(%:"rhs")`(mk_ctuple (map %# args));
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  fun pat_strict c =
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    let
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      val axs = @{thm branch_def} :: axs_pat_def;
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      val goal = mk_trp (strict (pat_lhs c ` (%:"rhs")));
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      val tacs = [simp_tac (HOLCF_ss addsimps [when_strict]) 1];
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    in pg axs goal (K tacs) end;
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  fun pat_app c (con, _, args) =
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    let
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      val axs = @{thm branch_def} :: axs_pat_def;
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      val lhs = (pat_lhs c)`(%:"rhs")`(con_app con args);
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      val rhs = if con = first c then pat_rhs c else mk_fail;
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      val goal = lift_defined %: (nonlazy args, mk_trp (lhs === rhs));
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      val tacs = [asm_simp_tac (HOLCF_ss addsimps when_rews) 1];
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    in pg axs goal (K tacs) end;
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  val _ = trace " Proving pat_stricts...";
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  val pat_stricts = map pat_strict cons;
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  val _ = trace " Proving pat_apps...";
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  val pat_apps = maps (fn c => map (pat_app c) cons) cons;
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in
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  val pat_rews = pat_stricts @ pat_apps;
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end;
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(* ----- theorems concerning one induction step ----------------------------- *)
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val copy_strict =
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  let
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    val _ = trace " Proving copy_strict...";
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    val goal = mk_trp (strict (dc_copy `% "f"));
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    val rules = [abs_strict, rep_strict] @ @{thms domain_map_stricts};
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    val tacs = [asm_simp_tac (HOLCF_ss addsimps rules) 1];
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  in
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    SOME (pg [ax_copy_def] goal (K tacs))
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    handle
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      THM (s, _, _) => (trace s; NONE)
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    | ERROR s => (trace s; NONE)
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  end;
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local
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  fun copy_app (con, _, args) =
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    let
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      val lhs = dc_copy`%"f"`(con_app con args);
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      fun one_rhs arg =
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          if Datatype_Aux.is_rec_type (dtyp_of arg)
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          then Domain_Axioms.copy_of_dtyp map_tab
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                 (proj (%:"f") eqs) (dtyp_of arg) ` (%# arg)
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          else (%# arg);
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      val rhs = con_app2 con one_rhs args;
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      fun is_rec arg = Datatype_Aux.is_rec_type (dtyp_of arg);
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      fun is_nonlazy_rec arg = is_rec arg andalso not (is_lazy arg);
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      fun nonlazy_rec args = map vname (filter is_nonlazy_rec args);
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      val goal = lift_defined %: (nonlazy_rec args, mk_trp (lhs === rhs));
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      val args' = filter_out (fn a => is_rec a orelse is_lazy a) args;
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      val stricts = abs_strict :: rep_strict :: @{thms domain_map_stricts};
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                        (* FIXME! case_UU_tac *)
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      fun tacs1 ctxt = map (case_UU_tac ctxt stricts 1 o vname) args';
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      val rules = [ax_abs_iso] @ @{thms domain_map_simps};
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      val tacs2 = [asm_simp_tac (HOLCF_ss addsimps rules) 1];
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    in pg (ax_copy_def::con_appls) goal (fn ctxt => (tacs1 ctxt @ tacs2)) end;
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in
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  val _ = trace " Proving copy_apps...";
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  val copy_apps = map copy_app cons;
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end;
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local
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  fun one_strict (con, _, args) = 
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    let
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      val goal = mk_trp (dc_copy`UU`(con_app con args) === UU);
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      val rews = the_list copy_strict @ copy_apps @ con_rews;
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                        (* FIXME! case_UU_tac *)
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      fun tacs ctxt = map (case_UU_tac ctxt rews 1) (nonlazy args) @
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        [asm_simp_tac (HOLCF_ss addsimps rews) 1];
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    in
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      SOME (pg [] goal tacs)
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      handle
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        THM (s, _, _) => (trace s; NONE)
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      | ERROR s => (trace s; NONE)
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    end;
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  fun has_nonlazy_rec (_, _, args) = exists is_nonlazy_rec args;
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in
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  val _ = trace " Proving copy_stricts...";
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  val copy_stricts = map_filter one_strict (filter has_nonlazy_rec cons);
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end;
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val copy_rews = the_list copy_strict @ copy_apps @ copy_stricts;
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in
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  thy
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    |> Sign.add_path (Long_Name.base_name dname)
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    |> snd o PureThy.add_thmss [
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        ((Binding.name "iso_rews"  , iso_rews    ), [Simplifier.simp_add]),
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        ((Binding.name "exhaust"   , [exhaust]   ), []),
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        ((Binding.name "casedist"  , [casedist]  ), [Induct.cases_type dname]),
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        ((Binding.name "when_rews" , when_rews   ), [Simplifier.simp_add]),
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        ((Binding.name "compacts"  , con_compacts), [Simplifier.simp_add]),
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        ((Binding.name "con_rews"  , con_rews    ),
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         [Simplifier.simp_add, Fixrec.fixrec_simp_add]),
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        ((Binding.name "sel_rews"  , sel_rews    ), [Simplifier.simp_add]),
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        ((Binding.name "dis_rews"  , dis_rews    ), [Simplifier.simp_add]),
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        ((Binding.name "pat_rews"  , pat_rews    ), [Simplifier.simp_add]),
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        ((Binding.name "dist_les"  , dist_les    ), [Simplifier.simp_add]),
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        ((Binding.name "dist_eqs"  , dist_eqs    ), [Simplifier.simp_add]),
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        ((Binding.name "inverts"   , inverts     ), [Simplifier.simp_add]),
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        ((Binding.name "injects"   , injects     ), [Simplifier.simp_add]),
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        ((Binding.name "copy_rews" , copy_rews   ), [Simplifier.simp_add]),
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        ((Binding.name "match_rews", mat_rews    ),
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         [Simplifier.simp_add, Fixrec.fixrec_simp_add])]
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    |> Sign.parent_path
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    |> pair (iso_rews @ when_rews @ con_rews @ sel_rews @ dis_rews @
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        pat_rews @ dist_les @ dist_eqs @ copy_rews)
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end; (* let *)
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fun comp_theorems (comp_dnam, eqs: eq list) thy =
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let
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val global_ctxt = ProofContext.init thy;
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val map_tab = Domain_Isomorphism.get_map_tab thy;
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val dnames = map (fst o fst) eqs;
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val conss  = map  snd        eqs;
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val comp_dname = Sign.full_bname thy comp_dnam;
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val _ = message ("Proving induction properties of domain "^comp_dname^" ...");
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   315
val pg = pg' thy;
wenzelm@23152
   316
wenzelm@23152
   317
(* ----- getting the composite axiom and definitions ------------------------ *)
wenzelm@23152
   318
wenzelm@23152
   319
local
wenzelm@26343
   320
  fun ga s dn = PureThy.get_thm thy (dn ^ "." ^ s);
wenzelm@23152
   321
in
wenzelm@23152
   322
  val axs_reach      = map (ga "reach"     ) dnames;
wenzelm@23152
   323
  val axs_take_def   = map (ga "take_def"  ) dnames;
wenzelm@23152
   324
  val axs_finite_def = map (ga "finite_def") dnames;
wenzelm@23152
   325
  val ax_copy2_def   =      ga "copy_def"  comp_dnam;
huffman@35444
   326
(* TEMPORARILY DISABLED
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   327
  val ax_bisim_def   =      ga "bisim_def" comp_dnam;
huffman@35444
   328
TEMPORARILY DISABLED *)
wenzelm@23152
   329
end;
wenzelm@23152
   330
wenzelm@23152
   331
local
wenzelm@26343
   332
  fun gt  s dn = PureThy.get_thm  thy (dn ^ "." ^ s);
wenzelm@26343
   333
  fun gts s dn = PureThy.get_thms thy (dn ^ "." ^ s);
wenzelm@23152
   334
in
wenzelm@23152
   335
  val cases = map (gt  "casedist" ) dnames;
wenzelm@26336
   336
  val con_rews  = maps (gts "con_rews" ) dnames;
wenzelm@26336
   337
  val copy_rews = maps (gts "copy_rews") dnames;
wenzelm@23152
   338
end;
wenzelm@23152
   339
wenzelm@23152
   340
fun dc_take dn = %%:(dn^"_take");
wenzelm@23152
   341
val x_name = idx_name dnames "x"; 
wenzelm@23152
   342
val P_name = idx_name dnames "P";
wenzelm@23152
   343
val n_eqs = length eqs;
wenzelm@23152
   344
wenzelm@23152
   345
(* ----- theorems concerning finite approximation and finite induction ------ *)
wenzelm@23152
   346
wenzelm@23152
   347
local
wenzelm@32149
   348
  val iterate_Cprod_ss = global_simpset_of @{theory Fix};
wenzelm@23152
   349
  val copy_con_rews  = copy_rews @ con_rews;
wenzelm@23152
   350
  val copy_take_defs =
wenzelm@23152
   351
    (if n_eqs = 1 then [] else [ax_copy2_def]) @ axs_take_def;
huffman@29402
   352
  val _ = trace " Proving take_stricts...";
huffman@35057
   353
  fun one_take_strict ((dn, args), _) =
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   354
    let
huffman@35057
   355
      val goal = mk_trp (strict (dc_take dn $ %:"n"));
huffman@35057
   356
      val rules = [
huffman@35057
   357
        @{thm monofun_fst [THEN monofunE]},
huffman@35057
   358
        @{thm monofun_snd [THEN monofunE]}];
huffman@35057
   359
      val tacs = [
huffman@35057
   360
        rtac @{thm UU_I} 1,
huffman@35057
   361
        rtac @{thm below_eq_trans} 1,
huffman@35057
   362
        resolve_tac axs_reach 2,
huffman@35057
   363
        rtac @{thm monofun_cfun_fun} 1,
huffman@35057
   364
        REPEAT (resolve_tac rules 1),
huffman@35057
   365
        rtac @{thm iterate_below_fix} 1];
huffman@35057
   366
    in pg axs_take_def goal (K tacs) end;
huffman@35057
   367
  val take_stricts = map one_take_strict eqs;
wenzelm@23152
   368
  fun take_0 n dn =
wenzelm@23152
   369
    let
huffman@35058
   370
      val goal = mk_trp ((dc_take dn $ @{term "0::nat"}) `% x_name n === UU);
wenzelm@27208
   371
    in pg axs_take_def goal (K [simp_tac iterate_Cprod_ss 1]) end;
wenzelm@23152
   372
  val take_0s = mapn take_0 1 dnames;
huffman@29402
   373
  val _ = trace " Proving take_apps...";
huffman@35288
   374
  fun one_take_app dn (con, _, args) =
wenzelm@23152
   375
    let
huffman@35058
   376
      fun mk_take n = dc_take (List.nth (dnames, n)) $ %:"n";
huffman@35058
   377
      fun one_rhs arg =
huffman@35058
   378
          if Datatype_Aux.is_rec_type (dtyp_of arg)
huffman@35058
   379
          then Domain_Axioms.copy_of_dtyp map_tab
huffman@35058
   380
                 mk_take (dtyp_of arg) ` (%# arg)
huffman@35058
   381
          else (%# arg);
huffman@35058
   382
      val lhs = (dc_take dn $ (%%:"Suc" $ %:"n"))`(con_app con args);
huffman@35058
   383
      val rhs = con_app2 con one_rhs args;
huffman@35059
   384
      fun is_rec arg = Datatype_Aux.is_rec_type (dtyp_of arg);
huffman@35059
   385
      fun is_nonlazy_rec arg = is_rec arg andalso not (is_lazy arg);
huffman@35059
   386
      fun nonlazy_rec args = map vname (filter is_nonlazy_rec args);
huffman@35059
   387
      val goal = lift_defined %: (nonlazy_rec args, mk_trp (lhs === rhs));
huffman@35059
   388
      val tacs = [asm_simp_tac (HOLCF_ss addsimps copy_con_rews) 1];
huffman@35059
   389
    in pg copy_take_defs goal (K tacs) end;
huffman@35058
   390
  fun one_take_apps ((dn, _), cons) = map (one_take_app dn) cons;
huffman@35058
   391
  val take_apps = maps one_take_apps eqs;
wenzelm@23152
   392
in
wenzelm@35021
   393
  val take_rews = map Drule.export_without_context
huffman@35058
   394
    (take_stricts @ take_0s @ take_apps);
wenzelm@23152
   395
end; (* local *)
wenzelm@23152
   396
wenzelm@23152
   397
local
huffman@35288
   398
  fun one_con p (con, _, args) =
wenzelm@23152
   399
    let
huffman@35443
   400
      val P_names = map P_name (1 upto (length dnames));
huffman@35443
   401
      val vns = Name.variant_list P_names (map vname args);
huffman@35443
   402
      val nonlazy_vns = map snd (filter_out (is_lazy o fst) (args ~~ vns));
wenzelm@23152
   403
      fun ind_hyp arg = %:(P_name (1 + rec_of arg)) $ bound_arg args arg;
wenzelm@23152
   404
      val t1 = mk_trp (%:p $ con_app2 con (bound_arg args) args);
wenzelm@33317
   405
      val t2 = lift ind_hyp (filter is_rec args, t1);
huffman@35443
   406
      val t3 = lift_defined (bound_arg vns) (nonlazy_vns, t2);
huffman@35443
   407
    in Library.foldr mk_All (vns, t3) end;
wenzelm@23152
   408
wenzelm@23152
   409
  fun one_eq ((p, cons), concl) =
wenzelm@23152
   410
    mk_trp (%:p $ UU) ===> Logic.list_implies (map (one_con p) cons, concl);
wenzelm@23152
   411
wenzelm@23152
   412
  fun ind_term concf = Library.foldr one_eq
wenzelm@23152
   413
    (mapn (fn n => fn x => (P_name n, x)) 1 conss,
wenzelm@23152
   414
     mk_trp (foldr1 mk_conj (mapn concf 1 dnames)));
wenzelm@23152
   415
  val take_ss = HOL_ss addsimps take_rews;
wenzelm@27208
   416
  fun quant_tac ctxt i = EVERY
wenzelm@27239
   417
    (mapn (fn n => fn _ => res_inst_tac ctxt [(("x", 0), x_name n)] spec i) 1 dnames);
wenzelm@23152
   418
wenzelm@23152
   419
  fun ind_prems_tac prems = EVERY
wenzelm@26336
   420
    (maps (fn cons =>
wenzelm@23152
   421
      (resolve_tac prems 1 ::
huffman@35288
   422
        maps (fn (_,_,args) => 
wenzelm@23152
   423
          resolve_tac prems 1 ::
wenzelm@23152
   424
          map (K(atac 1)) (nonlazy args) @
wenzelm@33317
   425
          map (K(atac 1)) (filter is_rec args))
wenzelm@26336
   426
        cons))
wenzelm@26336
   427
      conss);
wenzelm@23152
   428
  local 
wenzelm@23152
   429
    (* check whether every/exists constructor of the n-th part of the equation:
wenzelm@23152
   430
       it has a possibly indirectly recursive argument that isn't/is possibly 
wenzelm@23152
   431
       indirectly lazy *)
wenzelm@23152
   432
    fun rec_to quant nfn rfn ns lazy_rec (n,cons) = quant (exists (fn arg => 
wenzelm@23152
   433
          is_rec arg andalso not(rec_of arg mem ns) andalso
wenzelm@23152
   434
          ((rec_of arg =  n andalso nfn(lazy_rec orelse is_lazy arg)) orelse 
wenzelm@23152
   435
            rec_of arg <> n andalso rec_to quant nfn rfn (rec_of arg::ns) 
wenzelm@23152
   436
              (lazy_rec orelse is_lazy arg) (n, (List.nth(conss,rec_of arg))))
huffman@35288
   437
          ) o third) cons;
wenzelm@23152
   438
    fun all_rec_to ns  = rec_to forall not all_rec_to  ns;
wenzelm@23152
   439
    fun warn (n,cons) =
wenzelm@23152
   440
      if all_rec_to [] false (n,cons)
wenzelm@23152
   441
      then (warning ("domain "^List.nth(dnames,n)^" is empty!"); true)
wenzelm@23152
   442
      else false;
wenzelm@23152
   443
    fun lazy_rec_to ns = rec_to exists I  lazy_rec_to ns;
wenzelm@23152
   444
wenzelm@23152
   445
  in
wenzelm@23152
   446
    val n__eqs = mapn (fn n => fn (_,cons) => (n,cons)) 0 eqs;
wenzelm@23152
   447
    val is_emptys = map warn n__eqs;
wenzelm@23152
   448
    val is_finite = forall (not o lazy_rec_to [] false) n__eqs;
wenzelm@23152
   449
  end;
wenzelm@23152
   450
in (* local *)
huffman@29402
   451
  val _ = trace " Proving finite_ind...";
wenzelm@23152
   452
  val finite_ind =
wenzelm@23152
   453
    let
wenzelm@23152
   454
      fun concf n dn = %:(P_name n) $ (dc_take dn $ %:"n" `%(x_name n));
wenzelm@23152
   455
      val goal = ind_term concf;
wenzelm@23152
   456
wenzelm@27208
   457
      fun tacf {prems, context} =
wenzelm@23152
   458
        let
wenzelm@23152
   459
          val tacs1 = [
wenzelm@27208
   460
            quant_tac context 1,
wenzelm@23152
   461
            simp_tac HOL_ss 1,
wenzelm@27208
   462
            InductTacs.induct_tac context [[SOME "n"]] 1,
wenzelm@23152
   463
            simp_tac (take_ss addsimps prems) 1,
wenzelm@23152
   464
            TRY (safe_tac HOL_cs)];
wenzelm@23152
   465
          fun arg_tac arg =
huffman@35443
   466
                        (* FIXME! case_UU_tac *)
wenzelm@27208
   467
            case_UU_tac context (prems @ con_rews) 1
wenzelm@23152
   468
              (List.nth (dnames, rec_of arg) ^ "_take n$" ^ vname arg);
huffman@35288
   469
          fun con_tacs (con, _, args) = 
wenzelm@23152
   470
            asm_simp_tac take_ss 1 ::
wenzelm@33317
   471
            map arg_tac (filter is_nonlazy_rec args) @
wenzelm@23152
   472
            [resolve_tac prems 1] @
wenzelm@33317
   473
            map (K (atac 1)) (nonlazy args) @
wenzelm@33317
   474
            map (K (etac spec 1)) (filter is_rec args);
wenzelm@23152
   475
          fun cases_tacs (cons, cases) =
wenzelm@27239
   476
            res_inst_tac context [(("x", 0), "x")] cases 1 ::
wenzelm@23152
   477
            asm_simp_tac (take_ss addsimps prems) 1 ::
wenzelm@26336
   478
            maps con_tacs cons;
wenzelm@23152
   479
        in
wenzelm@26336
   480
          tacs1 @ maps cases_tacs (conss ~~ cases)
wenzelm@23152
   481
        end;
huffman@31232
   482
    in pg'' thy [] goal tacf
huffman@31232
   483
       handle ERROR _ => (warning "Proof of finite_ind failed."; TrueI)
huffman@31232
   484
    end;
wenzelm@23152
   485
huffman@29402
   486
  val _ = trace " Proving take_lemmas...";
wenzelm@23152
   487
  val take_lemmas =
wenzelm@23152
   488
    let
wenzelm@23152
   489
      fun take_lemma n (dn, ax_reach) =
wenzelm@23152
   490
        let
wenzelm@23152
   491
          val lhs = dc_take dn $ Bound 0 `%(x_name n);
wenzelm@23152
   492
          val rhs = dc_take dn $ Bound 0 `%(x_name n^"'");
wenzelm@23152
   493
          val concl = mk_trp (%:(x_name n) === %:(x_name n^"'"));
wenzelm@23152
   494
          val goal = mk_All ("n", mk_trp (lhs === rhs)) ===> concl;
huffman@33396
   495
          val rules = [contlub_fst RS contlubE RS ssubst,
huffman@33396
   496
                       contlub_snd RS contlubE RS ssubst];
wenzelm@27208
   497
          fun tacf {prems, context} = [
wenzelm@27239
   498
            res_inst_tac context [(("t", 0), x_name n    )] (ax_reach RS subst) 1,
wenzelm@27239
   499
            res_inst_tac context [(("t", 0), x_name n^"'")] (ax_reach RS subst) 1,
wenzelm@23152
   500
            stac fix_def2 1,
wenzelm@23152
   501
            REPEAT (CHANGED
huffman@33396
   502
              (resolve_tac rules 1 THEN chain_tac 1)),
wenzelm@23152
   503
            stac contlub_cfun_fun 1,
wenzelm@23152
   504
            stac contlub_cfun_fun 2,
wenzelm@23152
   505
            rtac lub_equal 3,
wenzelm@23152
   506
            chain_tac 1,
wenzelm@23152
   507
            rtac allI 1,
wenzelm@23152
   508
            resolve_tac prems 1];
wenzelm@23152
   509
        in pg'' thy axs_take_def goal tacf end;
wenzelm@23152
   510
    in mapn take_lemma 1 (dnames ~~ axs_reach) end;
wenzelm@23152
   511
wenzelm@23152
   512
(* ----- theorems concerning finiteness and induction ----------------------- *)
wenzelm@23152
   513
huffman@29402
   514
  val _ = trace " Proving finites, ind...";
wenzelm@23152
   515
  val (finites, ind) =
huffman@31232
   516
  (
wenzelm@23152
   517
    if is_finite
wenzelm@23152
   518
    then (* finite case *)
wenzelm@23152
   519
      let 
wenzelm@23152
   520
        fun take_enough dn = mk_ex ("n",dc_take dn $ Bound 0 ` %:"x" === %:"x");
wenzelm@23152
   521
        fun dname_lemma dn =
wenzelm@23152
   522
          let
wenzelm@23152
   523
            val prem1 = mk_trp (defined (%:"x"));
wenzelm@23152
   524
            val disj1 = mk_all ("n", dc_take dn $ Bound 0 ` %:"x" === UU);
wenzelm@23152
   525
            val prem2 = mk_trp (mk_disj (disj1, take_enough dn));
wenzelm@23152
   526
            val concl = mk_trp (take_enough dn);
wenzelm@23152
   527
            val goal = prem1 ===> prem2 ===> concl;
wenzelm@23152
   528
            val tacs = [
wenzelm@23152
   529
              etac disjE 1,
wenzelm@23152
   530
              etac notE 1,
wenzelm@23152
   531
              resolve_tac take_lemmas 1,
wenzelm@23152
   532
              asm_simp_tac take_ss 1,
wenzelm@23152
   533
              atac 1];
wenzelm@27208
   534
          in pg [] goal (K tacs) end;
huffman@31232
   535
        val _ = trace " Proving finite_lemmas1a";
wenzelm@23152
   536
        val finite_lemmas1a = map dname_lemma dnames;
wenzelm@23152
   537
 
huffman@31232
   538
        val _ = trace " Proving finite_lemma1b";
wenzelm@23152
   539
        val finite_lemma1b =
wenzelm@23152
   540
          let
wenzelm@23152
   541
            fun mk_eqn n ((dn, args), _) =
wenzelm@23152
   542
              let
wenzelm@23152
   543
                val disj1 = dc_take dn $ Bound 1 ` Bound 0 === UU;
wenzelm@23152
   544
                val disj2 = dc_take dn $ Bound 1 ` Bound 0 === Bound 0;
wenzelm@23152
   545
              in
wenzelm@23152
   546
                mk_constrainall
wenzelm@23152
   547
                  (x_name n, Type (dn,args), mk_disj (disj1, disj2))
wenzelm@23152
   548
              end;
wenzelm@23152
   549
            val goal =
wenzelm@23152
   550
              mk_trp (mk_all ("n", foldr1 mk_conj (mapn mk_eqn 1 eqs)));
wenzelm@27208
   551
            fun arg_tacs ctxt vn = [
wenzelm@27239
   552
              eres_inst_tac ctxt [(("x", 0), vn)] all_dupE 1,
wenzelm@23152
   553
              etac disjE 1,
wenzelm@23152
   554
              asm_simp_tac (HOL_ss addsimps con_rews) 1,
wenzelm@23152
   555
              asm_simp_tac take_ss 1];
huffman@35288
   556
            fun con_tacs ctxt (con, _, args) =
wenzelm@23152
   557
              asm_simp_tac take_ss 1 ::
wenzelm@27208
   558
              maps (arg_tacs ctxt) (nonlazy_rec args);
wenzelm@27208
   559
            fun foo_tacs ctxt n (cons, cases) =
wenzelm@23152
   560
              simp_tac take_ss 1 ::
wenzelm@23152
   561
              rtac allI 1 ::
wenzelm@27239
   562
              res_inst_tac ctxt [(("x", 0), x_name n)] cases 1 ::
wenzelm@23152
   563
              asm_simp_tac take_ss 1 ::
wenzelm@27208
   564
              maps (con_tacs ctxt) cons;
wenzelm@27208
   565
            fun tacs ctxt =
wenzelm@23152
   566
              rtac allI 1 ::
wenzelm@27208
   567
              InductTacs.induct_tac ctxt [[SOME "n"]] 1 ::
wenzelm@23152
   568
              simp_tac take_ss 1 ::
wenzelm@23152
   569
              TRY (safe_tac (empty_cs addSEs [conjE] addSIs [conjI])) ::
wenzelm@27208
   570
              flat (mapn (foo_tacs ctxt) 1 (conss ~~ cases));
wenzelm@23152
   571
          in pg [] goal tacs end;
wenzelm@23152
   572
wenzelm@23152
   573
        fun one_finite (dn, l1b) =
wenzelm@23152
   574
          let
wenzelm@23152
   575
            val goal = mk_trp (%%:(dn^"_finite") $ %:"x");
wenzelm@27208
   576
            fun tacs ctxt = [
huffman@35443
   577
                        (* FIXME! case_UU_tac *)
wenzelm@27208
   578
              case_UU_tac ctxt take_rews 1 "x",
wenzelm@23152
   579
              eresolve_tac finite_lemmas1a 1,
wenzelm@23152
   580
              step_tac HOL_cs 1,
wenzelm@23152
   581
              step_tac HOL_cs 1,
wenzelm@23152
   582
              cut_facts_tac [l1b] 1,
wenzelm@23152
   583
              fast_tac HOL_cs 1];
wenzelm@23152
   584
          in pg axs_finite_def goal tacs end;
wenzelm@23152
   585
huffman@31232
   586
        val _ = trace " Proving finites";
wenzelm@27232
   587
        val finites = map one_finite (dnames ~~ atomize global_ctxt finite_lemma1b);
huffman@31232
   588
        val _ = trace " Proving ind";
wenzelm@23152
   589
        val ind =
wenzelm@23152
   590
          let
wenzelm@23152
   591
            fun concf n dn = %:(P_name n) $ %:(x_name n);
wenzelm@27208
   592
            fun tacf {prems, context} =
wenzelm@23152
   593
              let
wenzelm@23152
   594
                fun finite_tacs (finite, fin_ind) = [
wenzelm@23152
   595
                  rtac(rewrite_rule axs_finite_def finite RS exE)1,
wenzelm@23152
   596
                  etac subst 1,
wenzelm@23152
   597
                  rtac fin_ind 1,
wenzelm@23152
   598
                  ind_prems_tac prems];
wenzelm@23152
   599
              in
wenzelm@23152
   600
                TRY (safe_tac HOL_cs) ::
wenzelm@27232
   601
                maps finite_tacs (finites ~~ atomize global_ctxt finite_ind)
wenzelm@23152
   602
              end;
wenzelm@23152
   603
          in pg'' thy [] (ind_term concf) tacf end;
wenzelm@23152
   604
      in (finites, ind) end (* let *)
wenzelm@23152
   605
wenzelm@23152
   606
    else (* infinite case *)
wenzelm@23152
   607
      let
wenzelm@23152
   608
        fun one_finite n dn =
wenzelm@27239
   609
          read_instantiate global_ctxt [(("P", 0), dn ^ "_finite " ^ x_name n)] excluded_middle;
wenzelm@23152
   610
        val finites = mapn one_finite 1 dnames;
wenzelm@23152
   611
wenzelm@23152
   612
        val goal =
wenzelm@23152
   613
          let
huffman@26012
   614
            fun one_adm n _ = mk_trp (mk_adm (%:(P_name n)));
wenzelm@23152
   615
            fun concf n dn = %:(P_name n) $ %:(x_name n);
wenzelm@23152
   616
          in Logic.list_implies (mapn one_adm 1 dnames, ind_term concf) end;
huffman@33396
   617
        val cont_rules =
huffman@33396
   618
            [cont_id, cont_const, cont2cont_Rep_CFun,
huffman@33396
   619
             cont2cont_fst, cont2cont_snd];
wenzelm@27208
   620
        fun tacf {prems, context} =
wenzelm@23152
   621
          map (fn ax_reach => rtac (ax_reach RS subst) 1) axs_reach @ [
wenzelm@27208
   622
          quant_tac context 1,
wenzelm@23152
   623
          rtac (adm_impl_admw RS wfix_ind) 1,
huffman@25895
   624
          REPEAT_DETERM (rtac adm_all 1),
wenzelm@23152
   625
          REPEAT_DETERM (
wenzelm@23152
   626
            TRY (rtac adm_conj 1) THEN 
wenzelm@23152
   627
            rtac adm_subst 1 THEN 
huffman@33396
   628
            REPEAT (resolve_tac cont_rules 1) THEN
huffman@33396
   629
            resolve_tac prems 1),
wenzelm@23152
   630
          strip_tac 1,
wenzelm@23152
   631
          rtac (rewrite_rule axs_take_def finite_ind) 1,
wenzelm@23152
   632
          ind_prems_tac prems];
wenzelm@23152
   633
        val ind = (pg'' thy [] goal tacf
wenzelm@23152
   634
          handle ERROR _ =>
huffman@33396
   635
            (warning "Cannot prove infinite induction rule"; TrueI));
huffman@31232
   636
      in (finites, ind) end
huffman@31232
   637
  )
huffman@31232
   638
      handle THM _ =>
huffman@31232
   639
             (warning "Induction proofs failed (THM raised)."; ([], TrueI))
huffman@31232
   640
           | ERROR _ =>
huffman@33810
   641
             (warning "Cannot prove induction rule"; ([], TrueI));
huffman@31232
   642
huffman@31232
   643
wenzelm@23152
   644
end; (* local *)
wenzelm@23152
   645
wenzelm@23152
   646
(* ----- theorem concerning coinduction ------------------------------------- *)
wenzelm@23152
   647
huffman@35444
   648
(* COINDUCTION TEMPORARILY DISABLED
wenzelm@23152
   649
local
wenzelm@23152
   650
  val xs = mapn (fn n => K (x_name n)) 1 dnames;
wenzelm@23152
   651
  fun bnd_arg n i = Bound(2*(n_eqs - n)-i-1);
wenzelm@23152
   652
  val take_ss = HOL_ss addsimps take_rews;
wenzelm@23152
   653
  val sproj = prj (fn s => K("fst("^s^")")) (fn s => K("snd("^s^")"));
huffman@29402
   654
  val _ = trace " Proving coind_lemma...";
wenzelm@23152
   655
  val coind_lemma =
wenzelm@23152
   656
    let
wenzelm@23152
   657
      fun mk_prj n _ = proj (%:"R") eqs n $ bnd_arg n 0 $ bnd_arg n 1;
wenzelm@23152
   658
      fun mk_eqn n dn =
wenzelm@23152
   659
        (dc_take dn $ %:"n" ` bnd_arg n 0) ===
wenzelm@23152
   660
        (dc_take dn $ %:"n" ` bnd_arg n 1);
wenzelm@23152
   661
      fun mk_all2 (x,t) = mk_all (x, mk_all (x^"'", t));
wenzelm@23152
   662
      val goal =
wenzelm@23152
   663
        mk_trp (mk_imp (%%:(comp_dname^"_bisim") $ %:"R",
wenzelm@23152
   664
          Library.foldr mk_all2 (xs,
wenzelm@23152
   665
            Library.foldr mk_imp (mapn mk_prj 0 dnames,
wenzelm@23152
   666
              foldr1 mk_conj (mapn mk_eqn 0 dnames)))));
wenzelm@27208
   667
      fun x_tacs ctxt n x = [
wenzelm@23152
   668
        rotate_tac (n+1) 1,
wenzelm@23152
   669
        etac all2E 1,
wenzelm@27239
   670
        eres_inst_tac ctxt [(("P", 1), sproj "R" eqs n^" "^x^" "^x^"'")] (mp RS disjE) 1,
wenzelm@23152
   671
        TRY (safe_tac HOL_cs),
wenzelm@23152
   672
        REPEAT (CHANGED (asm_simp_tac take_ss 1))];
wenzelm@27208
   673
      fun tacs ctxt = [
wenzelm@23152
   674
        rtac impI 1,
wenzelm@27208
   675
        InductTacs.induct_tac ctxt [[SOME "n"]] 1,
wenzelm@23152
   676
        simp_tac take_ss 1,
wenzelm@23152
   677
        safe_tac HOL_cs] @
wenzelm@27208
   678
        flat (mapn (x_tacs ctxt) 0 xs);
wenzelm@23152
   679
    in pg [ax_bisim_def] goal tacs end;
wenzelm@23152
   680
in
huffman@29402
   681
  val _ = trace " Proving coind...";
wenzelm@23152
   682
  val coind = 
wenzelm@23152
   683
    let
wenzelm@23152
   684
      fun mk_prj n x = mk_trp (proj (%:"R") eqs n $ %:x $ %:(x^"'"));
wenzelm@23152
   685
      fun mk_eqn x = %:x === %:(x^"'");
wenzelm@23152
   686
      val goal =
wenzelm@23152
   687
        mk_trp (%%:(comp_dname^"_bisim") $ %:"R") ===>
wenzelm@23152
   688
          Logic.list_implies (mapn mk_prj 0 xs,
wenzelm@23152
   689
            mk_trp (foldr1 mk_conj (map mk_eqn xs)));
wenzelm@23152
   690
      val tacs =
wenzelm@23152
   691
        TRY (safe_tac HOL_cs) ::
wenzelm@26336
   692
        maps (fn take_lemma => [
wenzelm@23152
   693
          rtac take_lemma 1,
wenzelm@23152
   694
          cut_facts_tac [coind_lemma] 1,
wenzelm@23152
   695
          fast_tac HOL_cs 1])
wenzelm@26336
   696
        take_lemmas;
wenzelm@27208
   697
    in pg [] goal (K tacs) end;
wenzelm@23152
   698
end; (* local *)
huffman@35444
   699
COINDUCTION TEMPORARILY DISABLED *)
wenzelm@23152
   700
wenzelm@32172
   701
val inducts = Project_Rule.projections (ProofContext.init thy) ind;
huffman@30829
   702
fun ind_rule (dname, rule) = ((Binding.empty, [rule]), [Induct.induct_type dname]);
huffman@31232
   703
val induct_failed = (Thm.prop_of ind = Thm.prop_of TrueI);
huffman@30829
   704
wenzelm@24712
   705
in thy |> Sign.add_path comp_dnam
huffman@31004
   706
       |> snd o PureThy.add_thmss [
huffman@31004
   707
           ((Binding.name "take_rews"  , take_rews   ), [Simplifier.simp_add]),
huffman@31004
   708
           ((Binding.name "take_lemmas", take_lemmas ), []),
huffman@31004
   709
           ((Binding.name "finites"    , finites     ), []),
huffman@31004
   710
           ((Binding.name "finite_ind" , [finite_ind]), []),
huffman@35444
   711
           ((Binding.name "ind"        , [ind]       ), [])(*,
huffman@35444
   712
           ((Binding.name "coind"      , [coind]     ), [])*)]
huffman@31232
   713
       |> (if induct_failed then I
huffman@31232
   714
           else snd o PureThy.add_thmss (map ind_rule (dnames ~~ inducts)))
haftmann@28536
   715
       |> Sign.parent_path |> pair take_rews
wenzelm@23152
   716
end; (* let *)
wenzelm@23152
   717
end; (* struct *)