src/HOL/Tools/inductive.ML
author bulwahn
Mon Aug 23 16:47:57 2010 +0200 (2010-08-23)
changeset 38665 e92223c886f8
parent 38388 94d5624dd1f7
child 38864 4abe644fcea5
permissions -rw-r--r--
introducing simplification equations for inductive sets; added data structure for storing equations; rewriting retrieval of simplification equation for inductive predicates and sets
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(*  Title:      HOL/Tools/inductive.ML
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Author:     Stefan Berghofer and Markus Wenzel, TU Muenchen
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(Co)Inductive Definition module for HOL.
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Features:
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  * least or greatest fixedpoints
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  * mutually recursive definitions
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  * definitions involving arbitrary monotone operators
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  * automatically proves introduction and elimination rules
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  Introduction rules have the form
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  [| M Pj ti, ..., Q x, ... |] ==> Pk t
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  where M is some monotone operator (usually the identity)
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  Q x is any side condition on the free variables
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  ti, t are any terms
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  Pj, Pk are two of the predicates being defined in mutual recursion
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*)
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signature BASIC_INDUCTIVE =
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sig
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  type inductive_result =
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    {preds: term list, elims: thm list, raw_induct: thm,
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     induct: thm, inducts: thm list, intrs: thm list, eqs: thm list}
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  val morph_result: morphism -> inductive_result -> inductive_result
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  type inductive_info = {names: string list, coind: bool} * inductive_result
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  val the_inductive: Proof.context -> string -> inductive_info
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  val print_inductives: Proof.context -> unit
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  val mono_add: attribute
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  val mono_del: attribute
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  val get_monos: Proof.context -> thm list
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  val mk_cases: Proof.context -> term -> thm
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  val inductive_forall_name: string
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  val inductive_forall_def: thm
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  val rulify: thm -> thm
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  val inductive_cases: (Attrib.binding * string list) list -> local_theory ->
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    thm list list * local_theory
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  val inductive_cases_i: (Attrib.binding * term list) list -> local_theory ->
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    thm list list * local_theory
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  type inductive_flags =
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    {quiet_mode: bool, verbose: bool, alt_name: binding, coind: bool,
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      no_elim: bool, no_ind: bool, skip_mono: bool, fork_mono: bool}
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  val add_inductive_i:
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    inductive_flags -> ((binding * typ) * mixfix) list ->
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    (string * typ) list -> (Attrib.binding * term) list -> thm list -> local_theory ->
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    inductive_result * local_theory
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  val add_inductive: bool -> bool ->
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    (binding * string option * mixfix) list ->
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    (binding * string option * mixfix) list ->
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    (Attrib.binding * string) list ->
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    (Facts.ref * Attrib.src list) list ->
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    bool -> local_theory -> inductive_result * local_theory
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  val add_inductive_global: inductive_flags ->
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    ((binding * typ) * mixfix) list -> (string * typ) list -> (Attrib.binding * term) list ->
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    thm list -> theory -> inductive_result * theory
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  val arities_of: thm -> (string * int) list
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  val params_of: thm -> term list
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  val partition_rules: thm -> thm list -> (string * thm list) list
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  val partition_rules': thm -> (thm * 'a) list -> (string * (thm * 'a) list) list
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  val unpartition_rules: thm list -> (string * 'a list) list -> 'a list
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  val infer_intro_vars: thm -> int -> thm list -> term list list
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  val setup: theory -> theory
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end;
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signature INDUCTIVE =
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sig
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  include BASIC_INDUCTIVE
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  type add_ind_def =
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    inductive_flags ->
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    term list -> (Attrib.binding * term) list -> thm list ->
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    term list -> (binding * mixfix) list ->
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    local_theory -> inductive_result * local_theory
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  val declare_rules: binding -> bool -> bool -> string list -> term list ->
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    thm list -> binding list -> Attrib.src list list -> (thm * string list * int) list ->
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    thm list -> thm -> local_theory -> thm list * thm list * thm list * thm * thm list * local_theory
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  val add_ind_def: add_ind_def
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  val gen_add_inductive_i: add_ind_def -> inductive_flags ->
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    ((binding * typ) * mixfix) list -> (string * typ) list -> (Attrib.binding * term) list ->
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    thm list -> local_theory -> inductive_result * local_theory
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  val gen_add_inductive: add_ind_def -> bool -> bool ->
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    (binding * string option * mixfix) list ->
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    (binding * string option * mixfix) list ->
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    (Attrib.binding * string) list -> (Facts.ref * Attrib.src list) list ->
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    bool -> local_theory -> inductive_result * local_theory
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  val gen_ind_decl: add_ind_def -> bool -> (bool -> local_theory -> local_theory) parser
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end;
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structure Inductive: INDUCTIVE =
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struct
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(** theory context references **)
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val inductive_forall_name = "HOL.induct_forall";
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val inductive_forall_def = @{thm induct_forall_def};
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val inductive_conj_name = "HOL.induct_conj";
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val inductive_conj_def = @{thm induct_conj_def};
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val inductive_conj = @{thms induct_conj};
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val inductive_atomize = @{thms induct_atomize};
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val inductive_rulify = @{thms induct_rulify};
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val inductive_rulify_fallback = @{thms induct_rulify_fallback};
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val notTrueE = TrueI RSN (2, notE);
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val notFalseI = Seq.hd (atac 1 notI);
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val simp_thms' = map mk_meta_eq
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  @{lemma "(~True) = False" "(~False) = True"
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      "(True --> P) = P" "(False --> P) = True"
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      "(P & True) = P" "(True & P) = P"
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    by (fact simp_thms)+};
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val simp_thms'' = map mk_meta_eq [@{thm inf_fun_eq}, @{thm inf_bool_eq}] @ simp_thms';
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val simp_thms''' = map mk_meta_eq
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  [@{thm le_fun_def}, @{thm le_bool_def}, @{thm sup_fun_eq}, @{thm sup_bool_eq}];
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(** context data **)
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type inductive_result =
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  {preds: term list, elims: thm list, raw_induct: thm,
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   induct: thm, inducts: thm list, intrs: thm list, eqs: thm list};
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fun morph_result phi {preds, elims, raw_induct: thm, induct, inducts, intrs, eqs} =
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  let
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    val term = Morphism.term phi;
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    val thm = Morphism.thm phi;
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    val fact = Morphism.fact phi;
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  in
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   {preds = map term preds, elims = fact elims, raw_induct = thm raw_induct,
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    induct = thm induct, inducts = fact inducts, intrs = fact intrs, eqs = fact eqs}
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  end;
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type inductive_info =
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  {names: string list, coind: bool} * inductive_result;
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structure InductiveData = Generic_Data
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(
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  type T = inductive_info Symtab.table * thm list;
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  val empty = (Symtab.empty, []);
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  val extend = I;
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  fun merge ((tab1, monos1), (tab2, monos2)) : T =
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    (Symtab.merge (K true) (tab1, tab2), Thm.merge_thms (monos1, monos2));
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);
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val get_inductives = InductiveData.get o Context.Proof;
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fun print_inductives ctxt =
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  let
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    val (tab, monos) = get_inductives ctxt;
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    val space = Consts.space_of (ProofContext.consts_of ctxt);
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  in
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    [Pretty.strs ("(co)inductives:" :: map #1 (Name_Space.extern_table (space, tab))),
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     Pretty.big_list "monotonicity rules:" (map (Display.pretty_thm ctxt) monos)]
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    |> Pretty.chunks |> Pretty.writeln
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  end;
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(* get and put data *)
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fun the_inductive ctxt name =
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  (case Symtab.lookup (#1 (get_inductives ctxt)) name of
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    NONE => error ("Unknown (co)inductive predicate " ^ quote name)
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  | SOME info => info);
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fun put_inductives names info = InductiveData.map
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  (apfst (fold (fn name => Symtab.update (name, info)) names));
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(** monotonicity rules **)
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val get_monos = #2 o get_inductives;
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val map_monos = InductiveData.map o apsnd;
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fun mk_mono thm =
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  let
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    fun eq2mono thm' = thm' RS (thm' RS eq_to_mono);
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    fun dest_less_concl thm = dest_less_concl (thm RS @{thm le_funD})
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      handle THM _ => thm RS @{thm le_boolD}
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  in
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    case concl_of thm of
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      Const ("==", _) $ _ $ _ => eq2mono (thm RS meta_eq_to_obj_eq)
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    | _ $ (Const (@{const_name "op ="}, _) $ _ $ _) => eq2mono thm
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    | _ $ (Const (@{const_name Orderings.less_eq}, _) $ _ $ _) =>
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      dest_less_concl (Seq.hd (REPEAT (FIRSTGOAL
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        (resolve_tac [@{thm le_funI}, @{thm le_boolI'}])) thm))
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    | _ => thm
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  end handle THM _ =>
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    error ("Bad monotonicity theorem:\n" ^ Display.string_of_thm_without_context thm);
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val mono_add = Thm.declaration_attribute (map_monos o Thm.add_thm o mk_mono);
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val mono_del = Thm.declaration_attribute (map_monos o Thm.del_thm o mk_mono);
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(** equations **)
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structure Equation_Data = Generic_Data
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(
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  type T = thm Item_Net.T;
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  val empty = Item_Net.init (op aconv o pairself Thm.prop_of)
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    (single o fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o Thm.prop_of);
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  val extend = I;
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  val merge = Item_Net.merge;
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);
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val add_equation = Thm.declaration_attribute (Equation_Data.map o Item_Net.update)
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(** misc utilities **)
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fun message quiet_mode s = if quiet_mode then () else writeln s;
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fun clean_message quiet_mode s = if ! quick_and_dirty then () else message quiet_mode s;
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fun coind_prefix true = "co"
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  | coind_prefix false = "";
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fun log (b:int) m n = if m >= n then 0 else 1 + log b (b * m) n;
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fun make_bool_args f g [] i = []
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  | make_bool_args f g (x :: xs) i =
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      (if i mod 2 = 0 then f x else g x) :: make_bool_args f g xs (i div 2);
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fun make_bool_args' xs =
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  make_bool_args (K HOLogic.false_const) (K HOLogic.true_const) xs;
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fun arg_types_of k c = drop k (binder_types (fastype_of c));
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fun find_arg T x [] = sys_error "find_arg"
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  | find_arg T x ((p as (_, (SOME _, _))) :: ps) =
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      apsnd (cons p) (find_arg T x ps)
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  | find_arg T x ((p as (U, (NONE, y))) :: ps) =
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      if (T: typ) = U then (y, (U, (SOME x, y)) :: ps)
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      else apsnd (cons p) (find_arg T x ps);
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fun make_args Ts xs =
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  map (fn (T, (NONE, ())) => Const (@{const_name undefined}, T) | (_, (SOME t, ())) => t)
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    (fold (fn (t, T) => snd o find_arg T t) xs (map (rpair (NONE, ())) Ts));
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fun make_args' Ts xs Us =
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  fst (fold_map (fn T => find_arg T ()) Us (Ts ~~ map (pair NONE) xs));
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fun dest_predicate cs params t =
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  let
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    val k = length params;
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    val (c, ts) = strip_comb t;
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    val (xs, ys) = chop k ts;
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    val i = find_index (fn c' => c' = c) cs;
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  in
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    if xs = params andalso i >= 0 then
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      SOME (c, i, ys, chop (length ys) (arg_types_of k c))
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    else NONE
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  end;
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fun mk_names a 0 = []
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  | mk_names a 1 = [a]
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  | mk_names a n = map (fn i => a ^ string_of_int i) (1 upto n);
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fun select_disj 1 1 = []
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  | select_disj _ 1 = [rtac disjI1]
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  | select_disj n i = (rtac disjI2)::(select_disj (n - 1) (i - 1));
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(** process rules **)
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local
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fun err_in_rule ctxt name t msg =
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  error (cat_lines ["Ill-formed introduction rule " ^ quote name,
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    Syntax.string_of_term ctxt t, msg]);
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fun err_in_prem ctxt name t p msg =
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  error (cat_lines ["Ill-formed premise", Syntax.string_of_term ctxt p,
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    "in introduction rule " ^ quote name, Syntax.string_of_term ctxt t, msg]);
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val bad_concl = "Conclusion of introduction rule must be an inductive predicate";
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val bad_ind_occ = "Inductive predicate occurs in argument of inductive predicate";
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val bad_app = "Inductive predicate must be applied to parameter(s) ";
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fun atomize_term thy = MetaSimplifier.rewrite_term thy inductive_atomize [];
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in
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fun check_rule ctxt cs params ((binding, att), rule) =
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  let
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    val err_name = Binding.str_of binding;
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    val params' = Term.variant_frees rule (Logic.strip_params rule);
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    val frees = rev (map Free params');
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    val concl = subst_bounds (frees, Logic.strip_assums_concl rule);
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    val prems = map (curry subst_bounds frees) (Logic.strip_assums_hyp rule);
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    val rule' = Logic.list_implies (prems, concl);
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    val aprems = map (atomize_term (ProofContext.theory_of ctxt)) prems;
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    val arule = list_all_free (params', Logic.list_implies (aprems, concl));
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    fun check_ind err t = case dest_predicate cs params t of
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        NONE => err (bad_app ^
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          commas (map (Syntax.string_of_term ctxt) params))
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      | SOME (_, _, ys, _) =>
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          if exists (fn c => exists (fn t => Logic.occs (c, t)) ys) cs
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          then err bad_ind_occ else ();
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    fun check_prem' prem t =
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      if member (op =) cs (head_of t) then
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        check_ind (err_in_prem ctxt err_name rule prem) t
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      else (case t of
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          Abs (_, _, t) => check_prem' prem t
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        | t $ u => (check_prem' prem t; check_prem' prem u)
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        | _ => ());
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    fun check_prem (prem, aprem) =
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      if can HOLogic.dest_Trueprop aprem then check_prem' prem prem
haftmann@29006
   317
      else err_in_prem ctxt err_name rule prem "Non-atomic premise";
wenzelm@10729
   318
  in
paulson@11358
   319
    (case concl of
wenzelm@35364
   320
       Const (@{const_name Trueprop}, _) $ t =>
haftmann@36692
   321
         if member (op =) cs (head_of t) then
haftmann@29006
   322
           (check_ind (err_in_rule ctxt err_name rule') t;
berghofe@21024
   323
            List.app check_prem (prems ~~ aprems))
haftmann@29006
   324
         else err_in_rule ctxt err_name rule' bad_concl
haftmann@29006
   325
     | _ => err_in_rule ctxt err_name rule' bad_concl);
wenzelm@28083
   326
    ((binding, att), arule)
wenzelm@10729
   327
  end;
berghofe@5094
   328
berghofe@24744
   329
val rulify =
wenzelm@18222
   330
  hol_simplify inductive_conj
wenzelm@18463
   331
  #> hol_simplify inductive_rulify
wenzelm@18463
   332
  #> hol_simplify inductive_rulify_fallback
wenzelm@30552
   333
  #> Simplifier.norm_hhf;
wenzelm@10729
   334
wenzelm@10729
   335
end;
wenzelm@10729
   336
berghofe@5094
   337
wenzelm@6424
   338
berghofe@21024
   339
(** proofs for (co)inductive predicates **)
wenzelm@6424
   340
berghofe@26534
   341
(* prove monotonicity *)
berghofe@5094
   342
berghofe@36642
   343
fun prove_mono quiet_mode skip_mono fork_mono predT fp_fun monos ctxt =
wenzelm@29388
   344
 (message (quiet_mode orelse skip_mono andalso !quick_and_dirty orelse fork_mono)
berghofe@26534
   345
    "  Proving monotonicity ...";
wenzelm@32970
   346
  (if skip_mono then Skip_Proof.prove else if fork_mono then Goal.prove_future else Goal.prove) ctxt
berghofe@36642
   347
    [] []
wenzelm@17985
   348
    (HOLogic.mk_Trueprop
wenzelm@24815
   349
      (Const (@{const_name Orderings.mono}, (predT --> predT) --> HOLogic.boolT) $ fp_fun))
wenzelm@25380
   350
    (fn _ => EVERY [rtac @{thm monoI} 1,
haftmann@32652
   351
      REPEAT (resolve_tac [@{thm le_funI}, @{thm le_boolI'}] 1),
berghofe@21024
   352
      REPEAT (FIRST
berghofe@21024
   353
        [atac 1,
berghofe@33933
   354
         resolve_tac (map mk_mono monos @ get_monos ctxt) 1,
haftmann@32652
   355
         etac @{thm le_funE} 1, dtac @{thm le_boolD} 1])]));
berghofe@5094
   356
wenzelm@6424
   357
wenzelm@10735
   358
(* prove introduction rules *)
berghofe@5094
   359
berghofe@36642
   360
fun prove_intrs quiet_mode coind mono fp_def k intr_ts rec_preds_defs ctxt ctxt' =
berghofe@5094
   361
  let
wenzelm@26477
   362
    val _ = clean_message quiet_mode "  Proving the introduction rules ...";
berghofe@5094
   363
berghofe@21024
   364
    val unfold = funpow k (fn th => th RS fun_cong)
berghofe@21024
   365
      (mono RS (fp_def RS
haftmann@32652
   366
        (if coind then @{thm def_gfp_unfold} else @{thm def_lfp_unfold})));
berghofe@5094
   367
berghofe@21024
   368
    val rules = [refl, TrueI, notFalseI, exI, conjI];
berghofe@21024
   369
berghofe@36642
   370
    val intrs = map_index (fn (i, intr) =>
berghofe@36642
   371
      Skip_Proof.prove ctxt [] [] intr (fn _ => EVERY
berghofe@21024
   372
       [rewrite_goals_tac rec_preds_defs,
berghofe@21024
   373
        rtac (unfold RS iffD2) 1,
berghofe@21024
   374
        EVERY1 (select_disj (length intr_ts) (i + 1)),
wenzelm@17985
   375
        (*Not ares_tac, since refl must be tried before any equality assumptions;
wenzelm@17985
   376
          backtracking may occur if the premises have extra variables!*)
berghofe@36642
   377
        DEPTH_SOLVE_1 (resolve_tac rules 1 APPEND assume_tac 1)])
berghofe@36642
   378
       |> singleton (ProofContext.export ctxt ctxt')) intr_ts
berghofe@5094
   379
berghofe@5094
   380
  in (intrs, unfold) end;
berghofe@5094
   381
wenzelm@6424
   382
wenzelm@10735
   383
(* prove elimination rules *)
berghofe@5094
   384
berghofe@36642
   385
fun prove_elims quiet_mode cs params intr_ts intr_names unfold rec_preds_defs ctxt ctxt''' =
berghofe@5094
   386
  let
wenzelm@26477
   387
    val _ = clean_message quiet_mode "  Proving the elimination rules ...";
berghofe@5094
   388
berghofe@36642
   389
    val ([pname], ctxt') = Variable.variant_fixes ["P"] ctxt;
berghofe@21024
   390
    val P = HOLogic.mk_Trueprop (Free (pname, HOLogic.boolT));
berghofe@21024
   391
berghofe@21024
   392
    fun dest_intr r =
berghofe@21024
   393
      (the (dest_predicate cs params (HOLogic.dest_Trueprop (Logic.strip_assums_concl r))),
berghofe@21024
   394
       Logic.strip_assums_hyp r, Logic.strip_params r);
berghofe@21024
   395
berghofe@21024
   396
    val intrs = map dest_intr intr_ts ~~ intr_names;
berghofe@21024
   397
berghofe@21024
   398
    val rules1 = [disjE, exE, FalseE];
berghofe@21024
   399
    val rules2 = [conjE, FalseE, notTrueE];
berghofe@21024
   400
berghofe@21024
   401
    fun prove_elim c =
berghofe@21024
   402
      let
haftmann@33077
   403
        val Ts = arg_types_of (length params) c;
berghofe@21024
   404
        val (anames, ctxt'') = Variable.variant_fixes (mk_names "a" (length Ts)) ctxt';
berghofe@21024
   405
        val frees = map Free (anames ~~ Ts);
berghofe@21024
   406
berghofe@21024
   407
        fun mk_elim_prem ((_, _, us, _), ts, params') =
berghofe@21024
   408
          list_all (params',
berghofe@21024
   409
            Logic.list_implies (map (HOLogic.mk_Trueprop o HOLogic.mk_eq)
berghofe@21024
   410
              (frees ~~ us) @ ts, P));
wenzelm@33317
   411
        val c_intrs = filter (equal c o #1 o #1 o #1) intrs;
berghofe@21024
   412
        val prems = HOLogic.mk_Trueprop (list_comb (c, params @ frees)) ::
berghofe@21024
   413
           map mk_elim_prem (map #1 c_intrs)
berghofe@21024
   414
      in
wenzelm@32970
   415
        (Skip_Proof.prove ctxt'' [] prems P
berghofe@21024
   416
          (fn {prems, ...} => EVERY
berghofe@21024
   417
            [cut_facts_tac [hd prems] 1,
berghofe@21024
   418
             rewrite_goals_tac rec_preds_defs,
berghofe@21024
   419
             dtac (unfold RS iffD1) 1,
berghofe@21024
   420
             REPEAT (FIRSTGOAL (eresolve_tac rules1)),
berghofe@21024
   421
             REPEAT (FIRSTGOAL (eresolve_tac rules2)),
berghofe@21024
   422
             EVERY (map (fn prem =>
berghofe@21024
   423
               DEPTH_SOLVE_1 (ares_tac [rewrite_rule rec_preds_defs prem, conjI] 1)) (tl prems))])
berghofe@36642
   424
          |> singleton (ProofContext.export ctxt'' ctxt'''),
berghofe@34986
   425
         map #2 c_intrs, length Ts)
berghofe@21024
   426
      end
berghofe@21024
   427
berghofe@21024
   428
   in map prove_elim cs end;
berghofe@5094
   429
bulwahn@37734
   430
(* prove simplification equations *)
wenzelm@6424
   431
wenzelm@37901
   432
fun prove_eqs quiet_mode cs params intr_ts intrs (elims: (thm * bstring list * int) list) ctxt ctxt'' =
bulwahn@37734
   433
  let
bulwahn@37734
   434
    val _ = clean_message quiet_mode "  Proving the simplification rules ...";
bulwahn@37734
   435
    
bulwahn@37734
   436
    fun dest_intr r =
bulwahn@37734
   437
      (the (dest_predicate cs params (HOLogic.dest_Trueprop (Logic.strip_assums_concl r))),
bulwahn@37734
   438
       Logic.strip_assums_hyp r, Logic.strip_params r);
bulwahn@37734
   439
    val intr_ts' = map dest_intr intr_ts;
wenzelm@37901
   440
    fun prove_eq c (elim: thm * 'a * 'b) =
bulwahn@37734
   441
      let
bulwahn@37734
   442
        val Ts = arg_types_of (length params) c;
bulwahn@37734
   443
        val (anames, ctxt') = Variable.variant_fixes (mk_names "a" (length Ts)) ctxt;
bulwahn@37734
   444
        val frees = map Free (anames ~~ Ts);
bulwahn@37734
   445
        val c_intrs = filter (equal c o #1 o #1 o #1) (intr_ts' ~~ intrs);
bulwahn@37734
   446
        fun mk_intr_conj (((_, _, us, _), ts, params'), _) =
bulwahn@37734
   447
          let
bulwahn@37734
   448
            fun list_ex ([], t) = t
bulwahn@37734
   449
              | list_ex ((a,T)::vars, t) =
bulwahn@37734
   450
                 (HOLogic.exists_const T) $ (Abs(a, T, list_ex(vars,t)));
bulwahn@37734
   451
            val conjs = map2 (curry HOLogic.mk_eq) frees us @ (map HOLogic.dest_Trueprop ts)
bulwahn@37734
   452
          in
bulwahn@37734
   453
            list_ex (params', if null conjs then @{term True} else foldr1 HOLogic.mk_conj conjs)
bulwahn@37734
   454
          end;
bulwahn@37734
   455
        val lhs = list_comb (c, params @ frees)
bulwahn@37734
   456
        val rhs =
bulwahn@37734
   457
          if null c_intrs then @{term False} else foldr1 HOLogic.mk_disj (map mk_intr_conj c_intrs)
bulwahn@37734
   458
        val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
bulwahn@37734
   459
        fun prove_intr1 (i, _) = Subgoal.FOCUS_PREMS (fn {params, prems, ...} =>
bulwahn@37734
   460
            let
bulwahn@37734
   461
              val (prems', last_prem) = split_last prems
bulwahn@37734
   462
            in
bulwahn@37734
   463
              EVERY1 (select_disj (length c_intrs) (i + 1))
bulwahn@37734
   464
              THEN EVERY (replicate (length params) (rtac @{thm exI} 1))
bulwahn@37734
   465
              THEN EVERY (map (fn prem => (rtac @{thm conjI} 1 THEN rtac prem 1)) prems')
bulwahn@37734
   466
              THEN rtac last_prem 1
bulwahn@37734
   467
            end) ctxt' 1
bulwahn@37734
   468
        fun prove_intr2 (((_, _, us, _), ts, params'), intr) =
bulwahn@37734
   469
          EVERY (replicate (length params') (etac @{thm exE} 1))
bulwahn@37734
   470
          THEN EVERY (replicate (length ts + length us - 1) (etac @{thm conjE} 1))
bulwahn@37734
   471
          THEN Subgoal.FOCUS_PREMS (fn {params, prems, ...} =>
bulwahn@37734
   472
            let
bulwahn@37734
   473
              val (eqs, prems') = chop (length us) prems
bulwahn@37734
   474
              val rew_thms = map (fn th => th RS @{thm eq_reflection}) eqs
bulwahn@37734
   475
            in
bulwahn@37734
   476
              rewrite_goal_tac rew_thms 1
bulwahn@37734
   477
              THEN rtac intr 1
bulwahn@37734
   478
              THEN (EVERY (map (fn p => rtac p 1) prems'))              
bulwahn@37734
   479
            end) ctxt' 1 
bulwahn@37734
   480
      in
bulwahn@37734
   481
        Skip_Proof.prove ctxt' [] [] eq (fn {...} =>
bulwahn@37734
   482
          rtac @{thm iffI} 1 THEN etac (#1 elim) 1
bulwahn@37734
   483
          THEN EVERY (map_index prove_intr1 c_intrs)
bulwahn@37734
   484
          THEN (if null c_intrs then etac @{thm FalseE} 1 else
bulwahn@37734
   485
            let val (c_intrs', last_c_intr) = split_last c_intrs in
bulwahn@37734
   486
              EVERY (map (fn ci => etac @{thm disjE} 1 THEN prove_intr2 ci) c_intrs')
bulwahn@37734
   487
              THEN prove_intr2 last_c_intr
bulwahn@37734
   488
            end))
bulwahn@37734
   489
        |> rulify
bulwahn@37734
   490
        |> singleton (ProofContext.export ctxt' ctxt'')
bulwahn@37734
   491
      end;  
bulwahn@37734
   492
  in
bulwahn@37734
   493
    map2 prove_eq cs elims
bulwahn@37734
   494
  end;
bulwahn@37734
   495
  
wenzelm@10735
   496
(* derivation of simplified elimination rules *)
berghofe@5094
   497
wenzelm@11682
   498
local
wenzelm@11682
   499
wenzelm@11682
   500
(*delete needless equality assumptions*)
wenzelm@29064
   501
val refl_thin = Goal.prove_global @{theory HOL} [] [] @{prop "!!P. a = a ==> P ==> P"}
haftmann@22838
   502
  (fn _ => assume_tac 1);
berghofe@21024
   503
val elim_rls = [asm_rl, FalseE, refl_thin, conjE, exE];
wenzelm@11682
   504
val elim_tac = REPEAT o Tactic.eresolve_tac elim_rls;
wenzelm@11682
   505
berghofe@23762
   506
fun simp_case_tac ss i =
berghofe@23762
   507
  EVERY' [elim_tac, asm_full_simp_tac ss, elim_tac, REPEAT o bound_hyp_subst_tac] i;
wenzelm@21367
   508
wenzelm@11682
   509
in
wenzelm@9598
   510
wenzelm@21367
   511
fun mk_cases ctxt prop =
wenzelm@7107
   512
  let
wenzelm@21367
   513
    val thy = ProofContext.theory_of ctxt;
wenzelm@32149
   514
    val ss = simpset_of ctxt;
wenzelm@21367
   515
wenzelm@21526
   516
    fun err msg =
wenzelm@21526
   517
      error (Pretty.string_of (Pretty.block
wenzelm@24920
   518
        [Pretty.str msg, Pretty.fbrk, Syntax.pretty_term ctxt prop]));
wenzelm@21526
   519
wenzelm@24861
   520
    val elims = Induct.find_casesP ctxt prop;
wenzelm@21367
   521
wenzelm@21367
   522
    val cprop = Thm.cterm_of thy prop;
berghofe@23762
   523
    val tac = ALLGOALS (simp_case_tac ss) THEN prune_params_tac;
wenzelm@21367
   524
    fun mk_elim rl =
wenzelm@36546
   525
      Thm.implies_intr cprop (Tactic.rule_by_tactic ctxt tac (Thm.assume cprop RS rl))
wenzelm@21367
   526
      |> singleton (Variable.export (Variable.auto_fixes prop ctxt) ctxt);
wenzelm@7107
   527
  in
wenzelm@7107
   528
    (case get_first (try mk_elim) elims of
skalberg@15531
   529
      SOME r => r
wenzelm@21526
   530
    | NONE => err "Proposition not an inductive predicate:")
wenzelm@7107
   531
  end;
wenzelm@7107
   532
wenzelm@11682
   533
end;
wenzelm@11682
   534
wenzelm@21367
   535
(* inductive_cases *)
wenzelm@7107
   536
wenzelm@21367
   537
fun gen_inductive_cases prep_att prep_prop args lthy =
wenzelm@9598
   538
  let
wenzelm@21367
   539
    val thy = ProofContext.theory_of lthy;
wenzelm@37957
   540
    val facts = args |> Par_List.map (fn ((a, atts), props) =>
wenzelm@21367
   541
      ((a, map (prep_att thy) atts),
wenzelm@37957
   542
        Par_List.map (Thm.no_attributes o single o mk_cases lthy o prep_prop lthy) props));
wenzelm@33671
   543
  in lthy |> Local_Theory.notes facts |>> map snd end;
berghofe@5094
   544
wenzelm@24509
   545
val inductive_cases = gen_inductive_cases Attrib.intern_src Syntax.read_prop;
wenzelm@24509
   546
val inductive_cases_i = gen_inductive_cases (K I) Syntax.check_prop;
wenzelm@7107
   547
wenzelm@6424
   548
wenzelm@30722
   549
val ind_cases_setup =
wenzelm@30722
   550
  Method.setup @{binding ind_cases}
wenzelm@30722
   551
    (Scan.lift (Scan.repeat1 Args.name_source --
wenzelm@30722
   552
      Scan.optional (Args.$$$ "for" |-- Scan.repeat1 Args.name) []) >>
wenzelm@30722
   553
      (fn (raw_props, fixes) => fn ctxt =>
wenzelm@30722
   554
        let
wenzelm@30722
   555
          val (_, ctxt') = Variable.add_fixes fixes ctxt;
wenzelm@30722
   556
          val props = Syntax.read_props ctxt' raw_props;
wenzelm@30722
   557
          val ctxt'' = fold Variable.declare_term props ctxt';
wenzelm@30722
   558
          val rules = ProofContext.export ctxt'' ctxt (map (mk_cases ctxt'') props)
wenzelm@30722
   559
        in Method.erule 0 rules end))
wenzelm@30722
   560
    "dynamic case analysis on predicates";
wenzelm@9598
   561
bulwahn@37734
   562
(* derivation of simplified equation *)
wenzelm@9598
   563
bulwahn@37734
   564
fun mk_simp_eq ctxt prop =
bulwahn@37734
   565
  let
bulwahn@38665
   566
    val thy = ProofContext.theory_of ctxt
bulwahn@37734
   567
    val ctxt' = Variable.auto_fixes prop ctxt
bulwahn@38665
   568
    val lhs_of = fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o Thm.prop_of
bulwahn@38665
   569
    val substs = Item_Net.retrieve (Equation_Data.get (Context.Proof ctxt)) (HOLogic.dest_Trueprop prop) 
bulwahn@38665
   570
      |> map_filter
bulwahn@38665
   571
        (fn eq => SOME (Pattern.match thy (lhs_of eq, HOLogic.dest_Trueprop prop)
bulwahn@38665
   572
            (Vartab.empty, Vartab.empty), eq)
bulwahn@38665
   573
          handle Pattern.MATCH => NONE)
bulwahn@38665
   574
    val (subst, eq) = case substs of
bulwahn@38665
   575
        [s] => s
bulwahn@38665
   576
      | _ => error
bulwahn@38665
   577
        ("equations matching pattern " ^ Syntax.string_of_term ctxt prop ^ " is not unique")
bulwahn@38665
   578
    val inst = map (fn v => (cterm_of thy (Var v), cterm_of thy (Envir.subst_term subst (Var v))))
bulwahn@38665
   579
      (Term.add_vars (lhs_of eq) [])
bulwahn@38118
   580
   in
bulwahn@38118
   581
    cterm_instantiate inst eq
bulwahn@37734
   582
    |> Conv.fconv_rule (Conv.arg_conv (Conv.arg_conv
bulwahn@37734
   583
      (Simplifier.full_rewrite (simpset_of ctxt))))
bulwahn@37734
   584
    |> singleton (Variable.export ctxt' ctxt)
bulwahn@37734
   585
  end
bulwahn@37734
   586
bulwahn@37734
   587
(* inductive simps *)
bulwahn@37734
   588
bulwahn@37734
   589
fun gen_inductive_simps prep_att prep_prop args lthy =
bulwahn@37734
   590
  let
bulwahn@37734
   591
    val thy = ProofContext.theory_of lthy;
bulwahn@37734
   592
    val facts = args |> map (fn ((a, atts), props) =>
bulwahn@37734
   593
      ((a, map (prep_att thy) atts),
bulwahn@37734
   594
        map (Thm.no_attributes o single o mk_simp_eq lthy o prep_prop lthy) props));
bulwahn@37734
   595
  in lthy |> Local_Theory.notes facts |>> map snd end;
bulwahn@37734
   596
bulwahn@37734
   597
val inductive_simps = gen_inductive_simps Attrib.intern_src Syntax.read_prop;
bulwahn@37734
   598
val inductive_simps_i = gen_inductive_simps (K I) Syntax.check_prop;
bulwahn@37734
   599
    
wenzelm@10735
   600
(* prove induction rule *)
berghofe@5094
   601
wenzelm@26477
   602
fun prove_indrule quiet_mode cs argTs bs xs rec_const params intr_ts mono
berghofe@36642
   603
    fp_def rec_preds_defs ctxt ctxt''' =
berghofe@5094
   604
  let
wenzelm@26477
   605
    val _ = clean_message quiet_mode "  Proving the induction rule ...";
wenzelm@20047
   606
    val thy = ProofContext.theory_of ctxt;
berghofe@5094
   607
berghofe@21024
   608
    (* predicates for induction rule *)
berghofe@21024
   609
berghofe@36642
   610
    val (pnames, ctxt') = Variable.variant_fixes (mk_names "P" (length cs)) ctxt;
haftmann@33077
   611
    val preds = map2 (curry Free) pnames
haftmann@33077
   612
      (map (fn c => arg_types_of (length params) c ---> HOLogic.boolT) cs);
berghofe@21024
   613
berghofe@21024
   614
    (* transform an introduction rule into a premise for induction rule *)
berghofe@21024
   615
berghofe@21024
   616
    fun mk_ind_prem r =
berghofe@21024
   617
      let
wenzelm@33669
   618
        fun subst s =
wenzelm@33669
   619
          (case dest_predicate cs params s of
berghofe@21024
   620
            SOME (_, i, ys, (_, Ts)) =>
berghofe@21024
   621
              let
berghofe@21024
   622
                val k = length Ts;
berghofe@21024
   623
                val bs = map Bound (k - 1 downto 0);
berghofe@23762
   624
                val P = list_comb (List.nth (preds, i),
berghofe@23762
   625
                  map (incr_boundvars k) ys @ bs);
berghofe@21024
   626
                val Q = list_abs (mk_names "x" k ~~ Ts,
berghofe@23762
   627
                  HOLogic.mk_binop inductive_conj_name
berghofe@23762
   628
                    (list_comb (incr_boundvars k s, bs), P))
berghofe@21024
   629
              in (Q, case Ts of [] => SOME (s, P) | _ => NONE) end
wenzelm@33669
   630
          | NONE =>
wenzelm@33669
   631
              (case s of
wenzelm@33669
   632
                (t $ u) => (fst (subst t) $ fst (subst u), NONE)
wenzelm@33669
   633
              | (Abs (a, T, t)) => (Abs (a, T, fst (subst t)), NONE)
wenzelm@33669
   634
              | _ => (s, NONE)));
berghofe@7293
   635
wenzelm@33338
   636
        fun mk_prem s prems =
wenzelm@33338
   637
          (case subst s of
wenzelm@33338
   638
            (_, SOME (t, u)) => t :: u :: prems
wenzelm@33338
   639
          | (t, _) => t :: prems);
berghofe@21024
   640
berghofe@21024
   641
        val SOME (_, i, ys, _) = dest_predicate cs params
berghofe@21024
   642
          (HOLogic.dest_Trueprop (Logic.strip_assums_concl r))
berghofe@21024
   643
berghofe@21024
   644
      in list_all_free (Logic.strip_params r,
wenzelm@33338
   645
        Logic.list_implies (map HOLogic.mk_Trueprop (fold_rev mk_prem
wenzelm@33338
   646
          (map HOLogic.dest_Trueprop (Logic.strip_assums_hyp r)) []),
berghofe@21024
   647
            HOLogic.mk_Trueprop (list_comb (List.nth (preds, i), ys))))
berghofe@21024
   648
      end;
berghofe@21024
   649
berghofe@21024
   650
    val ind_prems = map mk_ind_prem intr_ts;
berghofe@21024
   651
wenzelm@21526
   652
berghofe@21024
   653
    (* make conclusions for induction rules *)
berghofe@21024
   654
berghofe@21024
   655
    val Tss = map (binder_types o fastype_of) preds;
berghofe@21024
   656
    val (xnames, ctxt'') =
berghofe@21024
   657
      Variable.variant_fixes (mk_names "x" (length (flat Tss))) ctxt';
berghofe@21024
   658
    val mutual_ind_concl = HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
berghofe@21024
   659
        (map (fn (((xnames, Ts), c), P) =>
berghofe@21024
   660
           let val frees = map Free (xnames ~~ Ts)
berghofe@21024
   661
           in HOLogic.mk_imp
berghofe@21024
   662
             (list_comb (c, params @ frees), list_comb (P, frees))
berghofe@21024
   663
           end) (unflat Tss xnames ~~ Tss ~~ cs ~~ preds)));
berghofe@5094
   664
paulson@13626
   665
berghofe@5094
   666
    (* make predicate for instantiation of abstract induction rule *)
berghofe@5094
   667
berghofe@21024
   668
    val ind_pred = fold_rev lambda (bs @ xs) (foldr1 HOLogic.mk_conj
wenzelm@33338
   669
      (map_index (fn (i, P) => fold_rev (curry HOLogic.mk_imp)
wenzelm@33338
   670
         (make_bool_args HOLogic.mk_not I bs i)
wenzelm@33338
   671
         (list_comb (P, make_args' argTs xs (binder_types (fastype_of P))))) preds));
berghofe@5094
   672
berghofe@5094
   673
    val ind_concl = HOLogic.mk_Trueprop
haftmann@35092
   674
      (HOLogic.mk_binrel @{const_name Orderings.less_eq} (rec_const, ind_pred));
berghofe@5094
   675
haftmann@32652
   676
    val raw_fp_induct = (mono RS (fp_def RS @{thm def_lfp_induct}));
paulson@13626
   677
wenzelm@32970
   678
    val induct = Skip_Proof.prove ctxt'' [] ind_prems ind_concl
wenzelm@20248
   679
      (fn {prems, ...} => EVERY
wenzelm@17985
   680
        [rewrite_goals_tac [inductive_conj_def],
berghofe@21024
   681
         DETERM (rtac raw_fp_induct 1),
haftmann@32652
   682
         REPEAT (resolve_tac [@{thm le_funI}, @{thm le_boolI}] 1),
haftmann@32610
   683
         rewrite_goals_tac simp_thms'',
berghofe@21024
   684
         (*This disjE separates out the introduction rules*)
berghofe@21024
   685
         REPEAT (FIRSTGOAL (eresolve_tac [disjE, exE, FalseE])),
berghofe@5094
   686
         (*Now break down the individual cases.  No disjE here in case
berghofe@5094
   687
           some premise involves disjunction.*)
paulson@13747
   688
         REPEAT (FIRSTGOAL (etac conjE ORELSE' bound_hyp_subst_tac)),
berghofe@21024
   689
         REPEAT (FIRSTGOAL
berghofe@21024
   690
           (resolve_tac [conjI, impI] ORELSE' (etac notE THEN' atac))),
berghofe@21024
   691
         EVERY (map (fn prem => DEPTH_SOLVE_1 (ares_tac [rewrite_rule
haftmann@32610
   692
             (inductive_conj_def :: rec_preds_defs @ simp_thms'') prem,
berghofe@22980
   693
           conjI, refl] 1)) prems)]);
berghofe@5094
   694
wenzelm@32970
   695
    val lemma = Skip_Proof.prove ctxt'' [] []
wenzelm@17985
   696
      (Logic.mk_implies (ind_concl, mutual_ind_concl)) (fn _ => EVERY
berghofe@21024
   697
        [rewrite_goals_tac rec_preds_defs,
berghofe@5094
   698
         REPEAT (EVERY
berghofe@5094
   699
           [REPEAT (resolve_tac [conjI, impI] 1),
haftmann@32652
   700
            REPEAT (eresolve_tac [@{thm le_funE}, @{thm le_boolE}] 1),
berghofe@21024
   701
            atac 1,
berghofe@21024
   702
            rewrite_goals_tac simp_thms',
berghofe@21024
   703
            atac 1])])
berghofe@5094
   704
berghofe@36642
   705
  in singleton (ProofContext.export ctxt'' ctxt''') (induct RS lemma) end;
berghofe@5094
   706
wenzelm@6424
   707
wenzelm@6424
   708
berghofe@21024
   709
(** specification of (co)inductive predicates **)
wenzelm@10729
   710
wenzelm@33458
   711
fun mk_ind_def quiet_mode skip_mono fork_mono alt_name coind
wenzelm@33458
   712
    cs intr_ts monos params cnames_syn lthy =
wenzelm@33458
   713
  let
haftmann@24915
   714
    val fp_name = if coind then @{const_name Inductive.gfp} else @{const_name Inductive.lfp};
berghofe@5094
   715
haftmann@33077
   716
    val argTs = fold (combine (op =) o arg_types_of (length params)) cs [];
berghofe@21024
   717
    val k = log 2 1 (length cs);
berghofe@21024
   718
    val predT = replicate k HOLogic.boolT ---> argTs ---> HOLogic.boolT;
wenzelm@33458
   719
    val p :: xs = map Free (Variable.variant_frees lthy intr_ts
berghofe@21024
   720
      (("p", predT) :: (mk_names "x" (length argTs) ~~ argTs)));
wenzelm@33458
   721
    val bs = map Free (Variable.variant_frees lthy (p :: xs @ intr_ts)
berghofe@21024
   722
      (map (rpair HOLogic.boolT) (mk_names "b" k)));
berghofe@21024
   723
wenzelm@33458
   724
    fun subst t =
wenzelm@33458
   725
      (case dest_predicate cs params t of
berghofe@21024
   726
        SOME (_, i, ts, (Ts, Us)) =>
berghofe@23762
   727
          let
berghofe@23762
   728
            val l = length Us;
wenzelm@33669
   729
            val zs = map Bound (l - 1 downto 0);
berghofe@21024
   730
          in
berghofe@21024
   731
            list_abs (map (pair "z") Us, list_comb (p,
berghofe@23762
   732
              make_bool_args' bs i @ make_args argTs
berghofe@23762
   733
                ((map (incr_boundvars l) ts ~~ Ts) @ (zs ~~ Us))))
berghofe@21024
   734
          end
wenzelm@33669
   735
      | NONE =>
wenzelm@33669
   736
          (case t of
wenzelm@33669
   737
            t1 $ t2 => subst t1 $ subst t2
wenzelm@33669
   738
          | Abs (x, T, u) => Abs (x, T, subst u)
wenzelm@33669
   739
          | _ => t));
berghofe@5149
   740
berghofe@5094
   741
    (* transform an introduction rule into a conjunction  *)
berghofe@21024
   742
    (*   [| p_i t; ... |] ==> p_j u                       *)
berghofe@5094
   743
    (* is transformed into                                *)
berghofe@21024
   744
    (*   b_j & x_j = u & p b_j t & ...                    *)
berghofe@5094
   745
berghofe@5094
   746
    fun transform_rule r =
berghofe@5094
   747
      let
berghofe@21024
   748
        val SOME (_, i, ts, (Ts, _)) = dest_predicate cs params
berghofe@21048
   749
          (HOLogic.dest_Trueprop (Logic.strip_assums_concl r));
berghofe@21048
   750
        val ps = make_bool_args HOLogic.mk_not I bs i @
berghofe@21048
   751
          map HOLogic.mk_eq (make_args' argTs xs Ts ~~ ts) @
berghofe@21048
   752
          map (subst o HOLogic.dest_Trueprop)
berghofe@21048
   753
            (Logic.strip_assums_hyp r)
wenzelm@33338
   754
      in
wenzelm@33338
   755
        fold_rev (fn (x, T) => fn P => HOLogic.exists_const T $ Abs (x, T, P))
wenzelm@33338
   756
          (Logic.strip_params r)
wenzelm@33338
   757
          (if null ps then HOLogic.true_const else foldr1 HOLogic.mk_conj ps)
berghofe@5094
   758
      end
berghofe@5094
   759
berghofe@5094
   760
    (* make a disjunction of all introduction rules *)
berghofe@5094
   761
berghofe@21024
   762
    val fp_fun = fold_rev lambda (p :: bs @ xs)
berghofe@21024
   763
      (if null intr_ts then HOLogic.false_const
berghofe@21024
   764
       else foldr1 HOLogic.mk_disj (map transform_rule intr_ts));
berghofe@5094
   765
berghofe@21024
   766
    (* add definiton of recursive predicates to theory *)
berghofe@5094
   767
wenzelm@28083
   768
    val rec_name =
haftmann@28965
   769
      if Binding.is_empty alt_name then
wenzelm@30223
   770
        Binding.name (space_implode "_" (map (Binding.name_of o fst) cnames_syn))
wenzelm@28083
   771
      else alt_name;
berghofe@5094
   772
wenzelm@33458
   773
    val ((rec_const, (_, fp_def)), lthy') = lthy
wenzelm@33671
   774
      |> Local_Theory.conceal
wenzelm@33766
   775
      |> Local_Theory.define
berghofe@21024
   776
        ((rec_name, case cnames_syn of [(_, syn)] => syn | _ => NoSyn),
blanchet@33577
   777
         ((Binding.empty, [Attrib.internal (K Nitpick_Defs.add)]),
blanchet@33577
   778
         fold_rev lambda params
wenzelm@33278
   779
           (Const (fp_name, (predT --> predT) --> predT) $ fp_fun)))
wenzelm@33671
   780
      ||> Local_Theory.restore_naming lthy;
berghofe@21024
   781
    val fp_def' = Simplifier.rewrite (HOL_basic_ss addsimps [fp_def])
wenzelm@33458
   782
      (cterm_of (ProofContext.theory_of lthy') (list_comb (rec_const, params)));
wenzelm@33278
   783
    val specs =
wenzelm@33278
   784
      if length cs < 2 then []
wenzelm@33278
   785
      else
wenzelm@33278
   786
        map_index (fn (i, (name_mx, c)) =>
wenzelm@33278
   787
          let
wenzelm@33278
   788
            val Ts = arg_types_of (length params) c;
wenzelm@33458
   789
            val xs = map Free (Variable.variant_frees lthy intr_ts
wenzelm@33278
   790
              (mk_names "x" (length Ts) ~~ Ts))
wenzelm@33278
   791
          in
wenzelm@33278
   792
            (name_mx, (Attrib.empty_binding, fold_rev lambda (params @ xs)
wenzelm@33278
   793
              (list_comb (rec_const, params @ make_bool_args' bs i @
wenzelm@33278
   794
                make_args argTs (xs ~~ Ts)))))
wenzelm@33278
   795
          end) (cnames_syn ~~ cs);
wenzelm@33458
   796
    val (consts_defs, lthy'') = lthy'
wenzelm@33671
   797
      |> Local_Theory.conceal
wenzelm@33766
   798
      |> fold_map Local_Theory.define specs
wenzelm@33671
   799
      ||> Local_Theory.restore_naming lthy';
berghofe@21024
   800
    val preds = (case cs of [_] => [rec_const] | _ => map #1 consts_defs);
berghofe@5094
   801
berghofe@36642
   802
    val (_, lthy''') = Variable.add_fixes (map (fst o dest_Free) params) lthy'';
berghofe@36642
   803
    val mono = prove_mono quiet_mode skip_mono fork_mono predT fp_fun monos lthy''';
berghofe@36642
   804
    val (_, lthy'''') =
berghofe@36642
   805
      Local_Theory.note (apfst Binding.conceal Attrib.empty_binding,
berghofe@36642
   806
        ProofContext.export lthy''' lthy'' [mono]) lthy'';
berghofe@5094
   807
berghofe@36642
   808
  in (lthy'''', lthy''', rec_name, mono, fp_def', map (#2 o #2) consts_defs,
berghofe@21024
   809
    list_comb (rec_const, params), preds, argTs, bs, xs)
berghofe@21024
   810
  end;
berghofe@5094
   811
wenzelm@33669
   812
fun declare_rules rec_binding coind no_ind cnames
bulwahn@37734
   813
    preds intrs intr_bindings intr_atts elims eqs raw_induct lthy =
berghofe@23762
   814
  let
wenzelm@30223
   815
    val rec_name = Binding.name_of rec_binding;
haftmann@32773
   816
    fun rec_qualified qualified = Binding.qualify qualified rec_name;
wenzelm@30223
   817
    val intr_names = map Binding.name_of intr_bindings;
wenzelm@33368
   818
    val ind_case_names = Rule_Cases.case_names intr_names;
berghofe@23762
   819
    val induct =
berghofe@23762
   820
      if coind then
wenzelm@33368
   821
        (raw_induct, [Rule_Cases.case_names [rec_name],
wenzelm@33368
   822
          Rule_Cases.case_conclusion (rec_name, intr_names),
wenzelm@33368
   823
          Rule_Cases.consumes 1, Induct.coinduct_pred (hd cnames)])
berghofe@23762
   824
      else if no_ind orelse length cnames > 1 then
wenzelm@33368
   825
        (raw_induct, [ind_case_names, Rule_Cases.consumes 0])
wenzelm@33368
   826
      else (raw_induct RSN (2, rev_mp), [ind_case_names, Rule_Cases.consumes 1]);
berghofe@23762
   827
wenzelm@33458
   828
    val (intrs', lthy1) =
wenzelm@33458
   829
      lthy |>
bulwahn@35757
   830
      Spec_Rules.add
bulwahn@35757
   831
        (if coind then Spec_Rules.Co_Inductive else Spec_Rules.Inductive) (preds, intrs) |>
wenzelm@33671
   832
      Local_Theory.notes
wenzelm@33278
   833
        (map (rec_qualified false) intr_bindings ~~ intr_atts ~~
wenzelm@33278
   834
          map (fn th => [([th],
blanchet@37264
   835
           [Attrib.internal (K (Context_Rules.intro_query NONE))])]) intrs) |>>
berghofe@24744
   836
      map (hd o snd);
wenzelm@33458
   837
    val (((_, elims'), (_, [induct'])), lthy2) =
wenzelm@33458
   838
      lthy1 |>
wenzelm@33671
   839
      Local_Theory.note ((rec_qualified true (Binding.name "intros"), []), intrs') ||>>
berghofe@34986
   840
      fold_map (fn (name, (elim, cases, k)) =>
wenzelm@33671
   841
        Local_Theory.note
wenzelm@33458
   842
          ((Binding.qualify true (Long_Name.base_name name) (Binding.name "cases"),
wenzelm@33458
   843
            [Attrib.internal (K (Rule_Cases.case_names cases)),
wenzelm@33458
   844
             Attrib.internal (K (Rule_Cases.consumes 1)),
berghofe@34986
   845
             Attrib.internal (K (Rule_Cases.constraints k)),
wenzelm@33458
   846
             Attrib.internal (K (Induct.cases_pred name)),
wenzelm@33458
   847
             Attrib.internal (K (Context_Rules.elim_query NONE))]), [elim]) #>
berghofe@23762
   848
        apfst (hd o snd)) (if null elims then [] else cnames ~~ elims) ||>>
wenzelm@33671
   849
      Local_Theory.note
haftmann@32773
   850
        ((rec_qualified true (Binding.name (coind_prefix coind ^ "induct")),
wenzelm@28107
   851
          map (Attrib.internal o K) (#2 induct)), [rulify (#1 induct)]);
berghofe@23762
   852
bulwahn@37734
   853
    val (eqs', lthy3) = lthy2 |> 
bulwahn@37734
   854
      fold_map (fn (name, eq) => Local_Theory.note
bulwahn@38665
   855
          ((Binding.qualify true (Long_Name.base_name name) (Binding.name "simps"),
bulwahn@38665
   856
            [Attrib.internal (K add_equation)]), [eq])
bulwahn@37734
   857
          #> apfst (hd o snd))
bulwahn@37734
   858
        (if null eqs then [] else (cnames ~~ eqs))
bulwahn@37734
   859
    val (inducts, lthy4) =
bulwahn@37734
   860
      if no_ind orelse coind then ([], lthy3)
wenzelm@33458
   861
      else
bulwahn@37734
   862
        let val inducts = cnames ~~ Project_Rule.projects lthy3 (1 upto length cnames) induct' in
bulwahn@37734
   863
          lthy3 |>
wenzelm@33671
   864
          Local_Theory.notes [((rec_qualified true (Binding.name "inducts"), []),
wenzelm@33458
   865
            inducts |> map (fn (name, th) => ([th],
wenzelm@33458
   866
              [Attrib.internal (K ind_case_names),
wenzelm@33458
   867
               Attrib.internal (K (Rule_Cases.consumes 1)),
berghofe@35646
   868
               Attrib.internal (K (Induct.induct_pred name))])))] |>> snd o hd
wenzelm@33458
   869
        end;
bulwahn@37734
   870
  in (intrs', elims', eqs', induct', inducts, lthy4) end;
berghofe@23762
   871
berghofe@26534
   872
type inductive_flags =
wenzelm@33669
   873
  {quiet_mode: bool, verbose: bool, alt_name: binding, coind: bool,
wenzelm@33669
   874
    no_elim: bool, no_ind: bool, skip_mono: bool, fork_mono: bool};
berghofe@26534
   875
berghofe@26534
   876
type add_ind_def =
berghofe@26534
   877
  inductive_flags ->
wenzelm@28084
   878
  term list -> (Attrib.binding * term) list -> thm list ->
haftmann@29581
   879
  term list -> (binding * mixfix) list ->
wenzelm@33458
   880
  local_theory -> inductive_result * local_theory;
berghofe@23762
   881
wenzelm@33669
   882
fun add_ind_def {quiet_mode, verbose, alt_name, coind, no_elim, no_ind, skip_mono, fork_mono}
wenzelm@33458
   883
    cs intros monos params cnames_syn lthy =
berghofe@9072
   884
  let
wenzelm@25288
   885
    val _ = null cnames_syn andalso error "No inductive predicates given";
wenzelm@30223
   886
    val names = map (Binding.name_of o fst) cnames_syn;
wenzelm@26477
   887
    val _ = message (quiet_mode andalso not verbose)
wenzelm@28083
   888
      ("Proofs for " ^ coind_prefix coind ^ "inductive predicate(s) " ^ commas_quote names);
berghofe@9072
   889
wenzelm@33671
   890
    val cnames = map (Local_Theory.full_name lthy o #1) cnames_syn;  (* FIXME *)
berghofe@23762
   891
    val ((intr_names, intr_atts), intr_ts) =
wenzelm@33458
   892
      apfst split_list (split_list (map (check_rule lthy cs params) intros));
berghofe@21024
   893
berghofe@36642
   894
    val (lthy1, lthy2, rec_name, mono, fp_def, rec_preds_defs, rec_const, preds,
wenzelm@29388
   895
      argTs, bs, xs) = mk_ind_def quiet_mode skip_mono fork_mono alt_name coind cs intr_ts
wenzelm@33458
   896
        monos params cnames_syn lthy;
berghofe@9072
   897
wenzelm@26477
   898
    val (intrs, unfold) = prove_intrs quiet_mode coind mono fp_def (length bs + length xs)
berghofe@36642
   899
      intr_ts rec_preds_defs lthy2 lthy1;
wenzelm@33459
   900
    val elims =
wenzelm@33459
   901
      if no_elim then []
wenzelm@33459
   902
      else
wenzelm@33459
   903
        prove_elims quiet_mode cs params intr_ts (map Binding.name_of intr_names)
berghofe@36642
   904
          unfold rec_preds_defs lthy2 lthy1;
berghofe@22605
   905
    val raw_induct = zero_var_indexes
wenzelm@33459
   906
      (if no_ind then Drule.asm_rl
wenzelm@33459
   907
       else if coind then
berghofe@36642
   908
         singleton (ProofContext.export lthy2 lthy1)
wenzelm@35625
   909
           (rotate_prems ~1 (Object_Logic.rulify
wenzelm@28839
   910
             (fold_rule rec_preds_defs
haftmann@32652
   911
               (rewrite_rule simp_thms'''
haftmann@32652
   912
                (mono RS (fp_def RS @{thm def_coinduct}))))))
berghofe@21024
   913
       else
wenzelm@26477
   914
         prove_indrule quiet_mode cs argTs bs xs rec_const params intr_ts mono fp_def
berghofe@36642
   915
           rec_preds_defs lthy2 lthy1);
bulwahn@37734
   916
    val eqs =
bulwahn@37734
   917
      if no_elim then [] else prove_eqs quiet_mode cs params intr_ts intrs elims lthy2 lthy1
berghofe@5094
   918
bulwahn@37734
   919
    val elims' = map (fn (th, ns, i) => (rulify th, ns, i)) elims
bulwahn@37734
   920
    val intrs' = map rulify intrs
bulwahn@37734
   921
bulwahn@37734
   922
    val (intrs'', elims'', eqs', induct, inducts, lthy3) = declare_rules rec_name coind no_ind
bulwahn@37734
   923
      cnames preds intrs' intr_names intr_atts elims' eqs raw_induct lthy1;
berghofe@21048
   924
berghofe@21048
   925
    val result =
berghofe@21048
   926
      {preds = preds,
bulwahn@37734
   927
       intrs = intrs'',
bulwahn@37734
   928
       elims = elims'',
berghofe@21048
   929
       raw_induct = rulify raw_induct,
berghofe@35646
   930
       induct = induct,
bulwahn@37734
   931
       inducts = inducts,
bulwahn@37734
   932
       eqs = eqs'};
wenzelm@21367
   933
berghofe@36642
   934
    val lthy4 = lthy3
wenzelm@33671
   935
      |> Local_Theory.declaration false (fn phi =>
wenzelm@25380
   936
        let val result' = morph_result phi result;
wenzelm@25380
   937
        in put_inductives cnames (*global names!?*) ({names = cnames, coind = coind}, result') end);
berghofe@36642
   938
  in (result, lthy4) end;
berghofe@5094
   939
wenzelm@6424
   940
wenzelm@10735
   941
(* external interfaces *)
berghofe@5094
   942
wenzelm@26477
   943
fun gen_add_inductive_i mk_def
wenzelm@33669
   944
    (flags as {quiet_mode, verbose, alt_name, coind, no_elim, no_ind, skip_mono, fork_mono})
wenzelm@25029
   945
    cnames_syn pnames spec monos lthy =
berghofe@5094
   946
  let
wenzelm@25029
   947
    val thy = ProofContext.theory_of lthy;
wenzelm@6424
   948
    val _ = Theory.requires thy "Inductive" (coind_prefix coind ^ "inductive definitions");
berghofe@5094
   949
berghofe@21766
   950
wenzelm@25029
   951
    (* abbrevs *)
wenzelm@25029
   952
wenzelm@30223
   953
    val (_, ctxt1) = Variable.add_fixes (map (Binding.name_of o fst o fst) cnames_syn) lthy;
berghofe@21766
   954
wenzelm@25029
   955
    fun get_abbrev ((name, atts), t) =
wenzelm@25029
   956
      if can (Logic.strip_assums_concl #> Logic.dest_equals) t then
wenzelm@25029
   957
        let
haftmann@29006
   958
          val _ = Binding.is_empty name andalso null atts orelse
wenzelm@25029
   959
            error "Abbreviations may not have names or attributes";
wenzelm@35624
   960
          val ((x, T), rhs) = Local_Defs.abs_def (snd (Local_Defs.cert_def ctxt1 t));
wenzelm@28083
   961
          val var =
wenzelm@30223
   962
            (case find_first (fn ((c, _), _) => Binding.name_of c = x) cnames_syn of
wenzelm@25029
   963
              NONE => error ("Undeclared head of abbreviation " ^ quote x)
wenzelm@28083
   964
            | SOME ((b, T'), mx) =>
wenzelm@25029
   965
                if T <> T' then error ("Bad type specification for abbreviation " ^ quote x)
wenzelm@28083
   966
                else (b, mx));
wenzelm@28083
   967
        in SOME (var, rhs) end
wenzelm@25029
   968
      else NONE;
berghofe@21766
   969
wenzelm@25029
   970
    val abbrevs = map_filter get_abbrev spec;
wenzelm@30223
   971
    val bs = map (Binding.name_of o fst o fst) abbrevs;
wenzelm@25029
   972
berghofe@21766
   973
wenzelm@25029
   974
    (* predicates *)
berghofe@21766
   975
wenzelm@25029
   976
    val pre_intros = filter_out (is_some o get_abbrev) spec;
wenzelm@30223
   977
    val cnames_syn' = filter_out (member (op =) bs o Binding.name_of o fst o fst) cnames_syn;
wenzelm@30223
   978
    val cs = map (Free o apfst Binding.name_of o fst) cnames_syn';
wenzelm@25029
   979
    val ps = map Free pnames;
berghofe@5094
   980
wenzelm@30223
   981
    val (_, ctxt2) = lthy |> Variable.add_fixes (map (Binding.name_of o fst o fst) cnames_syn');
wenzelm@35624
   982
    val _ = map (fn abbr => Local_Defs.fixed_abbrev abbr ctxt2) abbrevs;
wenzelm@35624
   983
    val ctxt3 = ctxt2 |> fold (snd oo Local_Defs.fixed_abbrev) abbrevs;
wenzelm@25143
   984
    val expand = Assumption.export_term ctxt3 lthy #> ProofContext.cert_term lthy;
wenzelm@25029
   985
wenzelm@25029
   986
    fun close_rule r = list_all_free (rev (fold_aterms
berghofe@21024
   987
      (fn t as Free (v as (s, _)) =>
wenzelm@25029
   988
          if Variable.is_fixed ctxt1 s orelse
wenzelm@25029
   989
            member (op =) ps t then I else insert (op =) v
wenzelm@25029
   990
        | _ => I) r []), r);
berghofe@5094
   991
haftmann@26736
   992
    val intros = map (apsnd (Syntax.check_term lthy #> close_rule #> expand)) pre_intros;
wenzelm@25029
   993
    val preds = map (fn ((c, _), mx) => (c, mx)) cnames_syn';
berghofe@21048
   994
  in
wenzelm@25029
   995
    lthy
wenzelm@25029
   996
    |> mk_def flags cs intros monos ps preds
wenzelm@33671
   997
    ||> fold (snd oo Local_Theory.abbrev Syntax.mode_default) abbrevs
berghofe@21048
   998
  end;
berghofe@5094
   999
wenzelm@29388
  1000
fun gen_add_inductive mk_def verbose coind cnames_syn pnames_syn intro_srcs raw_monos int lthy =
berghofe@5094
  1001
  let
wenzelm@30486
  1002
    val ((vars, intrs), _) = lthy
wenzelm@30486
  1003
      |> ProofContext.set_mode ProofContext.mode_abbrev
wenzelm@30486
  1004
      |> Specification.read_spec (cnames_syn @ pnames_syn) intro_srcs;
wenzelm@24721
  1005
    val (cs, ps) = chop (length cnames_syn) vars;
wenzelm@24721
  1006
    val monos = Attrib.eval_thms lthy raw_monos;
wenzelm@33669
  1007
    val flags = {quiet_mode = false, verbose = verbose, alt_name = Binding.empty,
wenzelm@33669
  1008
      coind = coind, no_elim = false, no_ind = false, skip_mono = false, fork_mono = not int};
wenzelm@26128
  1009
  in
wenzelm@26128
  1010
    lthy
wenzelm@30223
  1011
    |> gen_add_inductive_i mk_def flags cs (map (apfst Binding.name_of o fst) ps) intrs monos
wenzelm@26128
  1012
  end;
berghofe@5094
  1013
berghofe@23762
  1014
val add_inductive_i = gen_add_inductive_i add_ind_def;
berghofe@23762
  1015
val add_inductive = gen_add_inductive add_ind_def;
berghofe@23762
  1016
wenzelm@33726
  1017
fun add_inductive_global flags cnames_syn pnames pre_intros monos thy =
wenzelm@25380
  1018
  let
haftmann@29006
  1019
    val name = Sign.full_name thy (fst (fst (hd cnames_syn)));
wenzelm@25380
  1020
    val ctxt' = thy
haftmann@38388
  1021
      |> Named_Target.theory_init
wenzelm@25380
  1022
      |> add_inductive_i flags cnames_syn pnames pre_intros monos |> snd
wenzelm@33671
  1023
      |> Local_Theory.exit;
wenzelm@25380
  1024
    val info = #2 (the_inductive ctxt' name);
wenzelm@25380
  1025
  in (info, ProofContext.theory_of ctxt') end;
wenzelm@6424
  1026
wenzelm@6424
  1027
berghofe@22789
  1028
(* read off arities of inductive predicates from raw induction rule *)
berghofe@22789
  1029
fun arities_of induct =
berghofe@22789
  1030
  map (fn (_ $ t $ u) =>
berghofe@22789
  1031
      (fst (dest_Const (head_of t)), length (snd (strip_comb u))))
berghofe@22789
  1032
    (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct)));
berghofe@22789
  1033
berghofe@22789
  1034
(* read off parameters of inductive predicate from raw induction rule *)
berghofe@22789
  1035
fun params_of induct =
berghofe@22789
  1036
  let
berghofe@22789
  1037
    val (_ $ t $ u :: _) =
berghofe@22789
  1038
      HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct));
berghofe@22789
  1039
    val (_, ts) = strip_comb t;
berghofe@22789
  1040
    val (_, us) = strip_comb u
berghofe@22789
  1041
  in
berghofe@22789
  1042
    List.take (ts, length ts - length us)
berghofe@22789
  1043
  end;
berghofe@22789
  1044
berghofe@22789
  1045
val pname_of_intr =
berghofe@22789
  1046
  concl_of #> HOLogic.dest_Trueprop #> head_of #> dest_Const #> fst;
berghofe@22789
  1047
berghofe@22789
  1048
(* partition introduction rules according to predicate name *)
berghofe@25822
  1049
fun gen_partition_rules f induct intros =
berghofe@25822
  1050
  fold_rev (fn r => AList.map_entry op = (pname_of_intr (f r)) (cons r)) intros
berghofe@22789
  1051
    (map (rpair [] o fst) (arities_of induct));
berghofe@22789
  1052
berghofe@25822
  1053
val partition_rules = gen_partition_rules I;
berghofe@25822
  1054
fun partition_rules' induct = gen_partition_rules fst induct;
berghofe@25822
  1055
berghofe@22789
  1056
fun unpartition_rules intros xs =
berghofe@22789
  1057
  fold_map (fn r => AList.map_entry_yield op = (pname_of_intr r)
berghofe@22789
  1058
    (fn x :: xs => (x, xs)) #>> the) intros xs |> fst;
berghofe@22789
  1059
berghofe@22789
  1060
(* infer order of variables in intro rules from order of quantifiers in elim rule *)
berghofe@22789
  1061
fun infer_intro_vars elim arity intros =
berghofe@22789
  1062
  let
berghofe@22789
  1063
    val thy = theory_of_thm elim;
berghofe@22789
  1064
    val _ :: cases = prems_of elim;
berghofe@22789
  1065
    val used = map (fst o fst) (Term.add_vars (prop_of elim) []);
berghofe@22789
  1066
    fun mtch (t, u) =
berghofe@22789
  1067
      let
berghofe@22789
  1068
        val params = Logic.strip_params t;
berghofe@22789
  1069
        val vars = map (Var o apfst (rpair 0))
berghofe@22789
  1070
          (Name.variant_list used (map fst params) ~~ map snd params);
berghofe@22789
  1071
        val ts = map (curry subst_bounds (rev vars))
berghofe@22789
  1072
          (List.drop (Logic.strip_assums_hyp t, arity));
berghofe@22789
  1073
        val us = Logic.strip_imp_prems u;
berghofe@22789
  1074
        val tab = fold (Pattern.first_order_match thy) (ts ~~ us)
berghofe@22789
  1075
          (Vartab.empty, Vartab.empty);
berghofe@22789
  1076
      in
wenzelm@32035
  1077
        map (Envir.subst_term tab) vars
berghofe@22789
  1078
      end
berghofe@22789
  1079
  in
berghofe@22789
  1080
    map (mtch o apsnd prop_of) (cases ~~ intros)
berghofe@22789
  1081
  end;
berghofe@22789
  1082
berghofe@22789
  1083
wenzelm@25978
  1084
wenzelm@6437
  1085
(** package setup **)
wenzelm@6437
  1086
wenzelm@6437
  1087
(* setup theory *)
wenzelm@6437
  1088
wenzelm@8634
  1089
val setup =
wenzelm@30722
  1090
  ind_cases_setup #>
wenzelm@30528
  1091
  Attrib.setup @{binding mono} (Attrib.add_del mono_add mono_del)
wenzelm@30528
  1092
    "declaration of monotonicity rule";
wenzelm@6437
  1093
wenzelm@6437
  1094
wenzelm@6437
  1095
(* outer syntax *)
wenzelm@6424
  1096
wenzelm@36960
  1097
val _ = Keyword.keyword "monos";
wenzelm@24867
  1098
berghofe@23762
  1099
fun gen_ind_decl mk_def coind =
wenzelm@36960
  1100
  Parse.fixes -- Parse.for_fixes --
wenzelm@36954
  1101
  Scan.optional Parse_Spec.where_alt_specs [] --
wenzelm@36960
  1102
  Scan.optional (Parse.$$$ "monos" |-- Parse.!!! Parse_Spec.xthms1) []
wenzelm@26988
  1103
  >> (fn (((preds, params), specs), monos) =>
wenzelm@30486
  1104
      (snd oo gen_add_inductive mk_def true coind preds params specs monos));
berghofe@23762
  1105
berghofe@23762
  1106
val ind_decl = gen_ind_decl add_ind_def;
wenzelm@6424
  1107
wenzelm@33458
  1108
val _ =
wenzelm@36960
  1109
  Outer_Syntax.local_theory' "inductive" "define inductive predicates" Keyword.thy_decl
wenzelm@33458
  1110
    (ind_decl false);
wenzelm@33458
  1111
wenzelm@33458
  1112
val _ =
wenzelm@36960
  1113
  Outer_Syntax.local_theory' "coinductive" "define coinductive predicates" Keyword.thy_decl
wenzelm@33458
  1114
    (ind_decl true);
wenzelm@6723
  1115
wenzelm@24867
  1116
val _ =
wenzelm@36960
  1117
  Outer_Syntax.local_theory "inductive_cases"
wenzelm@36960
  1118
    "create simplified instances of elimination rules (improper)" Keyword.thy_script
wenzelm@36960
  1119
    (Parse.and_list1 Parse_Spec.specs >> (snd oo inductive_cases));
wenzelm@7107
  1120
bulwahn@37734
  1121
val _ =
bulwahn@37734
  1122
  Outer_Syntax.local_theory "inductive_simps"
bulwahn@37734
  1123
    "create simplification rules for inductive predicates" Keyword.thy_script
bulwahn@37734
  1124
    (Parse.and_list1 Parse_Spec.specs >> (snd oo inductive_simps));
bulwahn@37734
  1125
berghofe@5094
  1126
end;