src/HOL/Tools/inductive.ML
author wenzelm
Sun Nov 27 14:40:08 2011 +0100 (2011-11-27)
changeset 45649 2d995773da1a
parent 45648 7654f750fb43
child 45651 172aa230ce69
permissions -rw-r--r--
tuned;
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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 transform_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_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_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 simp_thms1 =
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  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_thms2 =
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  map mk_meta_eq [@{thm inf_fun_def}, @{thm inf_bool_def}] @ simp_thms1;
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val simp_thms3 =
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  map mk_meta_eq [@{thm le_fun_def}, @{thm le_bool_def}, @{thm sup_fun_def}, @{thm sup_bool_def}];
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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 transform_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 Data = 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 = Data.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 (Proof_Context.consts_of ctxt);
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  in
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    [Pretty.strs ("(co)inductives:" :: map #1 (Name_Space.extern_table ctxt (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 =
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  Data.map (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 = Data.map o apsnd;
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fun mk_mono ctxt 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 HOL.eq}, _) $ _ $ _) => 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 _ => error ("Bad monotonicity theorem:\n" ^ Display.string_of_thm ctxt thm);
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val mono_add =
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  Thm.declaration_attribute (fn thm => fn context =>
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    map_monos (Thm.add_thm (mk_mono (Context.proof_of context) thm)) context);
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val mono_del =
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  Thm.declaration_attribute (fn thm => fn context =>
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    map_monos (Thm.del_thm (mk_mono (Context.proof_of context) thm)) context);
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(** equations **)
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structure Equation_Data = Generic_Data   (* FIXME just one data slot per module *)
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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);  (* FIXME fragile wrt. morphisms *)
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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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val get_equations = Equation_Data.get o Context.Proof;
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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 [] = raise Fail "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 " ^ Binding.print 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 " ^ Binding.print 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 = Raw_Simplifier.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 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 (Proof_Context.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 =
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      (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 binding rule prem) t
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      else
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        (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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        | _ => ());
berghofe@5094
   320
wenzelm@10729
   321
    fun check_prem (prem, aprem) =
berghofe@21024
   322
      if can HOLogic.dest_Trueprop aprem then check_prem' prem prem
wenzelm@42381
   323
      else err_in_prem ctxt binding rule prem "Non-atomic premise";
wenzelm@45647
   324
wenzelm@45647
   325
    val _ =
wenzelm@45647
   326
      (case concl of
wenzelm@45647
   327
        Const (@{const_name Trueprop}, _) $ t =>
wenzelm@45647
   328
          if member (op =) cs (head_of t) then
wenzelm@42381
   329
           (check_ind (err_in_rule ctxt binding rule') t;
berghofe@21024
   330
            List.app check_prem (prems ~~ aprems))
wenzelm@45647
   331
          else err_in_rule ctxt binding rule' bad_concl
wenzelm@45647
   332
       | _ => err_in_rule ctxt binding rule' bad_concl);
wenzelm@45647
   333
  in
wenzelm@28083
   334
    ((binding, att), arule)
wenzelm@10729
   335
  end;
berghofe@5094
   336
berghofe@24744
   337
val rulify =
wenzelm@18222
   338
  hol_simplify inductive_conj
wenzelm@18463
   339
  #> hol_simplify inductive_rulify
wenzelm@18463
   340
  #> hol_simplify inductive_rulify_fallback
wenzelm@30552
   341
  #> Simplifier.norm_hhf;
wenzelm@10729
   342
wenzelm@10729
   343
end;
wenzelm@10729
   344
berghofe@5094
   345
wenzelm@6424
   346
berghofe@21024
   347
(** proofs for (co)inductive predicates **)
wenzelm@6424
   348
berghofe@26534
   349
(* prove monotonicity *)
berghofe@5094
   350
berghofe@36642
   351
fun prove_mono quiet_mode skip_mono fork_mono predT fp_fun monos ctxt =
wenzelm@29388
   352
 (message (quiet_mode orelse skip_mono andalso !quick_and_dirty orelse fork_mono)
berghofe@26534
   353
    "  Proving monotonicity ...";
wenzelm@32970
   354
  (if skip_mono then Skip_Proof.prove else if fork_mono then Goal.prove_future else Goal.prove) ctxt
berghofe@36642
   355
    [] []
wenzelm@17985
   356
    (HOLogic.mk_Trueprop
wenzelm@24815
   357
      (Const (@{const_name Orderings.mono}, (predT --> predT) --> HOLogic.boolT) $ fp_fun))
wenzelm@25380
   358
    (fn _ => EVERY [rtac @{thm monoI} 1,
haftmann@32652
   359
      REPEAT (resolve_tac [@{thm le_funI}, @{thm le_boolI'}] 1),
berghofe@21024
   360
      REPEAT (FIRST
berghofe@21024
   361
        [atac 1,
wenzelm@42439
   362
         resolve_tac (map (mk_mono ctxt) monos @ get_monos ctxt) 1,
haftmann@32652
   363
         etac @{thm le_funE} 1, dtac @{thm le_boolD} 1])]));
berghofe@5094
   364
wenzelm@6424
   365
wenzelm@10735
   366
(* prove introduction rules *)
berghofe@5094
   367
berghofe@36642
   368
fun prove_intrs quiet_mode coind mono fp_def k intr_ts rec_preds_defs ctxt ctxt' =
berghofe@5094
   369
  let
wenzelm@26477
   370
    val _ = clean_message quiet_mode "  Proving the introduction rules ...";
berghofe@5094
   371
berghofe@21024
   372
    val unfold = funpow k (fn th => th RS fun_cong)
berghofe@21024
   373
      (mono RS (fp_def RS
haftmann@32652
   374
        (if coind then @{thm def_gfp_unfold} else @{thm def_lfp_unfold})));
berghofe@5094
   375
wenzelm@45648
   376
    val rules = [refl, TrueI, @{lemma "~ False" by (rule notI)}, exI, conjI];
berghofe@21024
   377
berghofe@36642
   378
    val intrs = map_index (fn (i, intr) =>
berghofe@36642
   379
      Skip_Proof.prove ctxt [] [] intr (fn _ => EVERY
berghofe@21024
   380
       [rewrite_goals_tac rec_preds_defs,
berghofe@21024
   381
        rtac (unfold RS iffD2) 1,
berghofe@21024
   382
        EVERY1 (select_disj (length intr_ts) (i + 1)),
wenzelm@17985
   383
        (*Not ares_tac, since refl must be tried before any equality assumptions;
wenzelm@17985
   384
          backtracking may occur if the premises have extra variables!*)
berghofe@36642
   385
        DEPTH_SOLVE_1 (resolve_tac rules 1 APPEND assume_tac 1)])
wenzelm@42361
   386
       |> singleton (Proof_Context.export ctxt ctxt')) intr_ts
berghofe@5094
   387
berghofe@5094
   388
  in (intrs, unfold) end;
berghofe@5094
   389
wenzelm@6424
   390
wenzelm@10735
   391
(* prove elimination rules *)
berghofe@5094
   392
berghofe@36642
   393
fun prove_elims quiet_mode cs params intr_ts intr_names unfold rec_preds_defs ctxt ctxt''' =
berghofe@5094
   394
  let
wenzelm@26477
   395
    val _ = clean_message quiet_mode "  Proving the elimination rules ...";
berghofe@5094
   396
berghofe@36642
   397
    val ([pname], ctxt') = Variable.variant_fixes ["P"] ctxt;
berghofe@21024
   398
    val P = HOLogic.mk_Trueprop (Free (pname, HOLogic.boolT));
berghofe@21024
   399
berghofe@21024
   400
    fun dest_intr r =
berghofe@21024
   401
      (the (dest_predicate cs params (HOLogic.dest_Trueprop (Logic.strip_assums_concl r))),
berghofe@21024
   402
       Logic.strip_assums_hyp r, Logic.strip_params r);
berghofe@21024
   403
berghofe@21024
   404
    val intrs = map dest_intr intr_ts ~~ intr_names;
berghofe@21024
   405
berghofe@21024
   406
    val rules1 = [disjE, exE, FalseE];
wenzelm@45648
   407
    val rules2 = [conjE, FalseE, @{lemma "~ True ==> R" by (rule notE [OF _ TrueI])}];
berghofe@21024
   408
berghofe@21024
   409
    fun prove_elim c =
berghofe@21024
   410
      let
haftmann@33077
   411
        val Ts = arg_types_of (length params) c;
berghofe@21024
   412
        val (anames, ctxt'') = Variable.variant_fixes (mk_names "a" (length Ts)) ctxt';
berghofe@21024
   413
        val frees = map Free (anames ~~ Ts);
berghofe@21024
   414
berghofe@21024
   415
        fun mk_elim_prem ((_, _, us, _), ts, params') =
berghofe@21024
   416
          list_all (params',
berghofe@21024
   417
            Logic.list_implies (map (HOLogic.mk_Trueprop o HOLogic.mk_eq)
berghofe@21024
   418
              (frees ~~ us) @ ts, P));
wenzelm@33317
   419
        val c_intrs = filter (equal c o #1 o #1 o #1) intrs;
berghofe@21024
   420
        val prems = HOLogic.mk_Trueprop (list_comb (c, params @ frees)) ::
berghofe@21024
   421
           map mk_elim_prem (map #1 c_intrs)
berghofe@21024
   422
      in
wenzelm@32970
   423
        (Skip_Proof.prove ctxt'' [] prems P
berghofe@21024
   424
          (fn {prems, ...} => EVERY
berghofe@21024
   425
            [cut_facts_tac [hd prems] 1,
berghofe@21024
   426
             rewrite_goals_tac rec_preds_defs,
berghofe@21024
   427
             dtac (unfold RS iffD1) 1,
berghofe@21024
   428
             REPEAT (FIRSTGOAL (eresolve_tac rules1)),
berghofe@21024
   429
             REPEAT (FIRSTGOAL (eresolve_tac rules2)),
berghofe@21024
   430
             EVERY (map (fn prem =>
berghofe@21024
   431
               DEPTH_SOLVE_1 (ares_tac [rewrite_rule rec_preds_defs prem, conjI] 1)) (tl prems))])
wenzelm@42361
   432
          |> singleton (Proof_Context.export ctxt'' ctxt'''),
berghofe@34986
   433
         map #2 c_intrs, length Ts)
berghofe@21024
   434
      end
berghofe@21024
   435
berghofe@21024
   436
   in map prove_elim cs end;
berghofe@5094
   437
wenzelm@45647
   438
bulwahn@37734
   439
(* prove simplification equations *)
wenzelm@6424
   440
wenzelm@45647
   441
fun prove_eqs quiet_mode cs params intr_ts intrs
wenzelm@45647
   442
    (elims: (thm * bstring list * int) list) ctxt ctxt'' =  (* FIXME ctxt'' ?? *)
bulwahn@37734
   443
  let
bulwahn@37734
   444
    val _ = clean_message quiet_mode "  Proving the simplification rules ...";
wenzelm@45647
   445
bulwahn@37734
   446
    fun dest_intr r =
bulwahn@37734
   447
      (the (dest_predicate cs params (HOLogic.dest_Trueprop (Logic.strip_assums_concl r))),
bulwahn@37734
   448
       Logic.strip_assums_hyp r, Logic.strip_params r);
bulwahn@37734
   449
    val intr_ts' = map dest_intr intr_ts;
wenzelm@45647
   450
wenzelm@37901
   451
    fun prove_eq c (elim: thm * 'a * 'b) =
bulwahn@37734
   452
      let
bulwahn@37734
   453
        val Ts = arg_types_of (length params) c;
bulwahn@37734
   454
        val (anames, ctxt') = Variable.variant_fixes (mk_names "a" (length Ts)) ctxt;
bulwahn@37734
   455
        val frees = map Free (anames ~~ Ts);
bulwahn@37734
   456
        val c_intrs = filter (equal c o #1 o #1 o #1) (intr_ts' ~~ intrs);
bulwahn@37734
   457
        fun mk_intr_conj (((_, _, us, _), ts, params'), _) =
bulwahn@37734
   458
          let
bulwahn@37734
   459
            fun list_ex ([], t) = t
wenzelm@45647
   460
              | list_ex ((a, T) :: vars, t) =
wenzelm@45647
   461
                  HOLogic.exists_const T $ Abs (a, T, list_ex (vars, t));
wenzelm@45647
   462
            val conjs = map2 (curry HOLogic.mk_eq) frees us @ (map HOLogic.dest_Trueprop ts);
bulwahn@37734
   463
          in
bulwahn@37734
   464
            list_ex (params', if null conjs then @{term True} else foldr1 HOLogic.mk_conj conjs)
bulwahn@37734
   465
          end;
wenzelm@45647
   466
        val lhs = list_comb (c, params @ frees);
bulwahn@37734
   467
        val rhs =
wenzelm@45647
   468
          if null c_intrs then @{term False}
wenzelm@45647
   469
          else foldr1 HOLogic.mk_disj (map mk_intr_conj c_intrs);
wenzelm@45647
   470
        val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs));
bulwahn@37734
   471
        fun prove_intr1 (i, _) = Subgoal.FOCUS_PREMS (fn {params, prems, ...} =>
bulwahn@37734
   472
            let
wenzelm@45647
   473
              val (prems', last_prem) = split_last prems;
bulwahn@37734
   474
            in
wenzelm@45647
   475
              EVERY1 (select_disj (length c_intrs) (i + 1)) THEN
wenzelm@45647
   476
              EVERY (replicate (length params) (rtac @{thm exI} 1)) THEN
wenzelm@45647
   477
              EVERY (map (fn prem => (rtac @{thm conjI} 1 THEN rtac prem 1)) prems') THEN
wenzelm@45647
   478
              rtac last_prem 1
wenzelm@45647
   479
            end) ctxt' 1;
bulwahn@37734
   480
        fun prove_intr2 (((_, _, us, _), ts, params'), intr) =
wenzelm@45647
   481
          EVERY (replicate (length params') (etac @{thm exE} 1)) THEN
wenzelm@45647
   482
          EVERY (replicate (length ts + length us - 1) (etac @{thm conjE} 1)) THEN
wenzelm@45647
   483
          Subgoal.FOCUS_PREMS (fn {params, prems, ...} =>
bulwahn@37734
   484
            let
wenzelm@45647
   485
              val (eqs, prems') = chop (length us) prems;
wenzelm@45647
   486
              val rew_thms = map (fn th => th RS @{thm eq_reflection}) eqs;
bulwahn@37734
   487
            in
wenzelm@45647
   488
              rewrite_goal_tac rew_thms 1 THEN
wenzelm@45647
   489
              rtac intr 1 THEN
wenzelm@45647
   490
              EVERY (map (fn p => rtac p 1) prems')
wenzelm@45647
   491
            end) ctxt' 1;
bulwahn@37734
   492
      in
wenzelm@45647
   493
        Skip_Proof.prove ctxt' [] [] eq (fn _ =>
wenzelm@45647
   494
          rtac @{thm iffI} 1 THEN etac (#1 elim) 1 THEN
wenzelm@45647
   495
          EVERY (map_index prove_intr1 c_intrs) THEN
wenzelm@45647
   496
          (if null c_intrs then etac @{thm FalseE} 1
wenzelm@45647
   497
           else
bulwahn@37734
   498
            let val (c_intrs', last_c_intr) = split_last c_intrs in
wenzelm@45647
   499
              EVERY (map (fn ci => etac @{thm disjE} 1 THEN prove_intr2 ci) c_intrs') THEN
wenzelm@45647
   500
              prove_intr2 last_c_intr
bulwahn@37734
   501
            end))
bulwahn@37734
   502
        |> rulify
wenzelm@42361
   503
        |> singleton (Proof_Context.export ctxt' ctxt'')
wenzelm@45647
   504
      end;
bulwahn@37734
   505
  in
bulwahn@37734
   506
    map2 prove_eq cs elims
bulwahn@37734
   507
  end;
wenzelm@45647
   508
wenzelm@45647
   509
wenzelm@10735
   510
(* derivation of simplified elimination rules *)
berghofe@5094
   511
wenzelm@11682
   512
local
wenzelm@11682
   513
wenzelm@11682
   514
(*delete needless equality assumptions*)
wenzelm@29064
   515
val refl_thin = Goal.prove_global @{theory HOL} [] [] @{prop "!!P. a = a ==> P ==> P"}
haftmann@22838
   516
  (fn _ => assume_tac 1);
berghofe@21024
   517
val elim_rls = [asm_rl, FalseE, refl_thin, conjE, exE];
wenzelm@11682
   518
val elim_tac = REPEAT o Tactic.eresolve_tac elim_rls;
wenzelm@11682
   519
berghofe@23762
   520
fun simp_case_tac ss i =
berghofe@23762
   521
  EVERY' [elim_tac, asm_full_simp_tac ss, elim_tac, REPEAT o bound_hyp_subst_tac] i;
wenzelm@21367
   522
wenzelm@11682
   523
in
wenzelm@9598
   524
wenzelm@21367
   525
fun mk_cases ctxt prop =
wenzelm@7107
   526
  let
wenzelm@42361
   527
    val thy = Proof_Context.theory_of ctxt;
wenzelm@32149
   528
    val ss = simpset_of ctxt;
wenzelm@21367
   529
wenzelm@21526
   530
    fun err msg =
wenzelm@21526
   531
      error (Pretty.string_of (Pretty.block
wenzelm@24920
   532
        [Pretty.str msg, Pretty.fbrk, Syntax.pretty_term ctxt prop]));
wenzelm@21526
   533
wenzelm@24861
   534
    val elims = Induct.find_casesP ctxt prop;
wenzelm@21367
   535
wenzelm@21367
   536
    val cprop = Thm.cterm_of thy prop;
berghofe@23762
   537
    val tac = ALLGOALS (simp_case_tac ss) THEN prune_params_tac;
wenzelm@21367
   538
    fun mk_elim rl =
wenzelm@36546
   539
      Thm.implies_intr cprop (Tactic.rule_by_tactic ctxt tac (Thm.assume cprop RS rl))
wenzelm@21367
   540
      |> singleton (Variable.export (Variable.auto_fixes prop ctxt) ctxt);
wenzelm@7107
   541
  in
wenzelm@7107
   542
    (case get_first (try mk_elim) elims of
skalberg@15531
   543
      SOME r => r
wenzelm@21526
   544
    | NONE => err "Proposition not an inductive predicate:")
wenzelm@7107
   545
  end;
wenzelm@7107
   546
wenzelm@11682
   547
end;
wenzelm@11682
   548
wenzelm@45647
   549
wenzelm@21367
   550
(* inductive_cases *)
wenzelm@7107
   551
wenzelm@21367
   552
fun gen_inductive_cases prep_att prep_prop args lthy =
wenzelm@9598
   553
  let
wenzelm@42361
   554
    val thy = Proof_Context.theory_of lthy;
wenzelm@37957
   555
    val facts = args |> Par_List.map (fn ((a, atts), props) =>
wenzelm@21367
   556
      ((a, map (prep_att thy) atts),
wenzelm@37957
   557
        Par_List.map (Thm.no_attributes o single o mk_cases lthy o prep_prop lthy) props));
wenzelm@33671
   558
  in lthy |> Local_Theory.notes facts |>> map snd end;
berghofe@5094
   559
wenzelm@24509
   560
val inductive_cases = gen_inductive_cases Attrib.intern_src Syntax.read_prop;
wenzelm@24509
   561
val inductive_cases_i = gen_inductive_cases (K I) Syntax.check_prop;
wenzelm@7107
   562
wenzelm@6424
   563
wenzelm@30722
   564
val ind_cases_setup =
wenzelm@30722
   565
  Method.setup @{binding ind_cases}
wenzelm@30722
   566
    (Scan.lift (Scan.repeat1 Args.name_source --
wenzelm@42491
   567
      Scan.optional (Args.$$$ "for" |-- Scan.repeat1 Args.binding) []) >>
wenzelm@30722
   568
      (fn (raw_props, fixes) => fn ctxt =>
wenzelm@30722
   569
        let
wenzelm@42491
   570
          val (_, ctxt') = Variable.add_fixes_binding fixes ctxt;
wenzelm@30722
   571
          val props = Syntax.read_props ctxt' raw_props;
wenzelm@30722
   572
          val ctxt'' = fold Variable.declare_term props ctxt';
wenzelm@42361
   573
          val rules = Proof_Context.export ctxt'' ctxt (map (mk_cases ctxt'') props)
wenzelm@30722
   574
        in Method.erule 0 rules end))
wenzelm@30722
   575
    "dynamic case analysis on predicates";
wenzelm@9598
   576
wenzelm@45647
   577
bulwahn@37734
   578
(* derivation of simplified equation *)
wenzelm@9598
   579
bulwahn@37734
   580
fun mk_simp_eq ctxt prop =
bulwahn@37734
   581
  let
wenzelm@45647
   582
    val thy = Proof_Context.theory_of ctxt;
wenzelm@45647
   583
    val ctxt' = Variable.auto_fixes prop ctxt;
wenzelm@45647
   584
    val lhs_of = fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o Thm.prop_of;
wenzelm@45647
   585
    val substs =
wenzelm@45649
   586
      Item_Net.retrieve (get_equations ctxt) (HOLogic.dest_Trueprop prop)
bulwahn@38665
   587
      |> map_filter
bulwahn@38665
   588
        (fn eq => SOME (Pattern.match thy (lhs_of eq, HOLogic.dest_Trueprop prop)
bulwahn@38665
   589
            (Vartab.empty, Vartab.empty), eq)
wenzelm@45647
   590
          handle Pattern.MATCH => NONE);
wenzelm@45647
   591
    val (subst, eq) =
wenzelm@45647
   592
      (case substs of
bulwahn@38665
   593
        [s] => s
bulwahn@38665
   594
      | _ => error
wenzelm@45647
   595
        ("equations matching pattern " ^ Syntax.string_of_term ctxt prop ^ " is not unique"));
wenzelm@45647
   596
    val inst =
wenzelm@45647
   597
      map (fn v => (cterm_of thy (Var v), cterm_of thy (Envir.subst_term subst (Var v))))
wenzelm@45647
   598
        (Term.add_vars (lhs_of eq) []);
wenzelm@45647
   599
  in
bulwahn@38118
   600
    cterm_instantiate inst eq
wenzelm@45647
   601
    |> Conv.fconv_rule (Conv.arg_conv (Conv.arg_conv (Simplifier.full_rewrite (simpset_of ctxt))))
bulwahn@37734
   602
    |> singleton (Variable.export ctxt' ctxt)
bulwahn@37734
   603
  end
bulwahn@37734
   604
wenzelm@45647
   605
bulwahn@37734
   606
(* inductive simps *)
bulwahn@37734
   607
bulwahn@37734
   608
fun gen_inductive_simps prep_att prep_prop args lthy =
bulwahn@37734
   609
  let
wenzelm@42361
   610
    val thy = Proof_Context.theory_of lthy;
bulwahn@37734
   611
    val facts = args |> map (fn ((a, atts), props) =>
bulwahn@37734
   612
      ((a, map (prep_att thy) atts),
bulwahn@37734
   613
        map (Thm.no_attributes o single o mk_simp_eq lthy o prep_prop lthy) props));
bulwahn@37734
   614
  in lthy |> Local_Theory.notes facts |>> map snd end;
bulwahn@37734
   615
bulwahn@37734
   616
val inductive_simps = gen_inductive_simps Attrib.intern_src Syntax.read_prop;
bulwahn@37734
   617
val inductive_simps_i = gen_inductive_simps (K I) Syntax.check_prop;
bulwahn@40902
   618
wenzelm@45647
   619
wenzelm@10735
   620
(* prove induction rule *)
berghofe@5094
   621
wenzelm@26477
   622
fun prove_indrule quiet_mode cs argTs bs xs rec_const params intr_ts mono
wenzelm@45647
   623
    fp_def rec_preds_defs ctxt ctxt''' =  (* FIXME ctxt''' ?? *)
berghofe@5094
   624
  let
wenzelm@26477
   625
    val _ = clean_message quiet_mode "  Proving the induction rule ...";
berghofe@5094
   626
berghofe@21024
   627
    (* predicates for induction rule *)
berghofe@21024
   628
berghofe@36642
   629
    val (pnames, ctxt') = Variable.variant_fixes (mk_names "P" (length cs)) ctxt;
wenzelm@45647
   630
    val preds =
wenzelm@45647
   631
      map2 (curry Free) pnames
wenzelm@45647
   632
        (map (fn c => arg_types_of (length params) c ---> HOLogic.boolT) cs);
berghofe@21024
   633
berghofe@21024
   634
    (* transform an introduction rule into a premise for induction rule *)
berghofe@21024
   635
berghofe@21024
   636
    fun mk_ind_prem r =
berghofe@21024
   637
      let
wenzelm@33669
   638
        fun subst s =
wenzelm@33669
   639
          (case dest_predicate cs params s of
berghofe@21024
   640
            SOME (_, i, ys, (_, Ts)) =>
berghofe@21024
   641
              let
berghofe@21024
   642
                val k = length Ts;
berghofe@21024
   643
                val bs = map Bound (k - 1 downto 0);
wenzelm@42364
   644
                val P = list_comb (nth preds i, map (incr_boundvars k) ys @ bs);
berghofe@21024
   645
                val Q = list_abs (mk_names "x" k ~~ Ts,
berghofe@23762
   646
                  HOLogic.mk_binop inductive_conj_name
wenzelm@45647
   647
                    (list_comb (incr_boundvars k s, bs), P));
berghofe@21024
   648
              in (Q, case Ts of [] => SOME (s, P) | _ => NONE) end
wenzelm@33669
   649
          | NONE =>
wenzelm@33669
   650
              (case s of
wenzelm@45647
   651
                t $ u => (fst (subst t) $ fst (subst u), NONE)
wenzelm@45647
   652
              | Abs (a, T, t) => (Abs (a, T, fst (subst t)), NONE)
wenzelm@33669
   653
              | _ => (s, NONE)));
berghofe@7293
   654
wenzelm@33338
   655
        fun mk_prem s prems =
wenzelm@33338
   656
          (case subst s of
wenzelm@33338
   657
            (_, SOME (t, u)) => t :: u :: prems
wenzelm@33338
   658
          | (t, _) => t :: prems);
berghofe@21024
   659
wenzelm@45647
   660
        val SOME (_, i, ys, _) =
wenzelm@45647
   661
          dest_predicate cs params (HOLogic.dest_Trueprop (Logic.strip_assums_concl r));
wenzelm@42364
   662
      in
wenzelm@42364
   663
        list_all_free (Logic.strip_params r,
wenzelm@42364
   664
          Logic.list_implies (map HOLogic.mk_Trueprop (fold_rev mk_prem
wenzelm@42364
   665
            (map HOLogic.dest_Trueprop (Logic.strip_assums_hyp r)) []),
wenzelm@42364
   666
              HOLogic.mk_Trueprop (list_comb (nth preds i, ys))))
berghofe@21024
   667
      end;
berghofe@21024
   668
berghofe@21024
   669
    val ind_prems = map mk_ind_prem intr_ts;
berghofe@21024
   670
wenzelm@21526
   671
berghofe@21024
   672
    (* make conclusions for induction rules *)
berghofe@21024
   673
berghofe@21024
   674
    val Tss = map (binder_types o fastype_of) preds;
wenzelm@45647
   675
    val (xnames, ctxt'') = Variable.variant_fixes (mk_names "x" (length (flat Tss))) ctxt';
wenzelm@45647
   676
    val mutual_ind_concl =
wenzelm@45647
   677
      HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
berghofe@21024
   678
        (map (fn (((xnames, Ts), c), P) =>
wenzelm@45647
   679
          let val frees = map Free (xnames ~~ Ts)
wenzelm@45647
   680
          in HOLogic.mk_imp (list_comb (c, params @ frees), list_comb (P, frees)) end)
wenzelm@45647
   681
        (unflat Tss xnames ~~ Tss ~~ cs ~~ preds)));
berghofe@5094
   682
paulson@13626
   683
berghofe@5094
   684
    (* make predicate for instantiation of abstract induction rule *)
berghofe@5094
   685
wenzelm@45647
   686
    val ind_pred =
wenzelm@45647
   687
      fold_rev lambda (bs @ xs) (foldr1 HOLogic.mk_conj
wenzelm@45647
   688
        (map_index (fn (i, P) => fold_rev (curry HOLogic.mk_imp)
wenzelm@45647
   689
           (make_bool_args HOLogic.mk_not I bs i)
wenzelm@45647
   690
           (list_comb (P, make_args' argTs xs (binder_types (fastype_of P))))) preds));
berghofe@5094
   691
wenzelm@45647
   692
    val ind_concl =
wenzelm@45647
   693
      HOLogic.mk_Trueprop
wenzelm@45647
   694
        (HOLogic.mk_binrel @{const_name Orderings.less_eq} (rec_const, ind_pred));
berghofe@5094
   695
wenzelm@45647
   696
    val raw_fp_induct = mono RS (fp_def RS @{thm def_lfp_induct});
paulson@13626
   697
wenzelm@32970
   698
    val induct = Skip_Proof.prove ctxt'' [] ind_prems ind_concl
wenzelm@20248
   699
      (fn {prems, ...} => EVERY
wenzelm@17985
   700
        [rewrite_goals_tac [inductive_conj_def],
berghofe@21024
   701
         DETERM (rtac raw_fp_induct 1),
haftmann@32652
   702
         REPEAT (resolve_tac [@{thm le_funI}, @{thm le_boolI}] 1),
wenzelm@45649
   703
         rewrite_goals_tac simp_thms2,
berghofe@21024
   704
         (*This disjE separates out the introduction rules*)
berghofe@21024
   705
         REPEAT (FIRSTGOAL (eresolve_tac [disjE, exE, FalseE])),
berghofe@5094
   706
         (*Now break down the individual cases.  No disjE here in case
berghofe@5094
   707
           some premise involves disjunction.*)
paulson@13747
   708
         REPEAT (FIRSTGOAL (etac conjE ORELSE' bound_hyp_subst_tac)),
berghofe@21024
   709
         REPEAT (FIRSTGOAL
berghofe@21024
   710
           (resolve_tac [conjI, impI] ORELSE' (etac notE THEN' atac))),
berghofe@21024
   711
         EVERY (map (fn prem => DEPTH_SOLVE_1 (ares_tac [rewrite_rule
wenzelm@45649
   712
             (inductive_conj_def :: rec_preds_defs @ simp_thms2) prem,
berghofe@22980
   713
           conjI, refl] 1)) prems)]);
berghofe@5094
   714
wenzelm@32970
   715
    val lemma = Skip_Proof.prove ctxt'' [] []
wenzelm@17985
   716
      (Logic.mk_implies (ind_concl, mutual_ind_concl)) (fn _ => EVERY
berghofe@21024
   717
        [rewrite_goals_tac rec_preds_defs,
berghofe@5094
   718
         REPEAT (EVERY
berghofe@5094
   719
           [REPEAT (resolve_tac [conjI, impI] 1),
haftmann@32652
   720
            REPEAT (eresolve_tac [@{thm le_funE}, @{thm le_boolE}] 1),
berghofe@21024
   721
            atac 1,
wenzelm@45649
   722
            rewrite_goals_tac simp_thms1,
wenzelm@45647
   723
            atac 1])]);
berghofe@5094
   724
wenzelm@42361
   725
  in singleton (Proof_Context.export ctxt'' ctxt''') (induct RS lemma) end;
berghofe@5094
   726
wenzelm@6424
   727
wenzelm@6424
   728
berghofe@21024
   729
(** specification of (co)inductive predicates **)
wenzelm@10729
   730
wenzelm@33458
   731
fun mk_ind_def quiet_mode skip_mono fork_mono alt_name coind
wenzelm@33458
   732
    cs intr_ts monos params cnames_syn lthy =
wenzelm@33458
   733
  let
haftmann@24915
   734
    val fp_name = if coind then @{const_name Inductive.gfp} else @{const_name Inductive.lfp};
berghofe@5094
   735
haftmann@33077
   736
    val argTs = fold (combine (op =) o arg_types_of (length params)) cs [];
berghofe@21024
   737
    val k = log 2 1 (length cs);
berghofe@21024
   738
    val predT = replicate k HOLogic.boolT ---> argTs ---> HOLogic.boolT;
wenzelm@45647
   739
    val p :: xs =
wenzelm@45647
   740
      map Free (Variable.variant_frees lthy intr_ts
wenzelm@45647
   741
        (("p", predT) :: (mk_names "x" (length argTs) ~~ argTs)));
wenzelm@45647
   742
    val bs =
wenzelm@45647
   743
      map Free (Variable.variant_frees lthy (p :: xs @ intr_ts)
wenzelm@45647
   744
        (map (rpair HOLogic.boolT) (mk_names "b" k)));
berghofe@21024
   745
wenzelm@33458
   746
    fun subst t =
wenzelm@33458
   747
      (case dest_predicate cs params t of
berghofe@21024
   748
        SOME (_, i, ts, (Ts, Us)) =>
berghofe@23762
   749
          let
berghofe@23762
   750
            val l = length Us;
wenzelm@33669
   751
            val zs = map Bound (l - 1 downto 0);
berghofe@21024
   752
          in
berghofe@21024
   753
            list_abs (map (pair "z") Us, list_comb (p,
berghofe@23762
   754
              make_bool_args' bs i @ make_args argTs
berghofe@23762
   755
                ((map (incr_boundvars l) ts ~~ Ts) @ (zs ~~ Us))))
berghofe@21024
   756
          end
wenzelm@33669
   757
      | NONE =>
wenzelm@33669
   758
          (case t of
wenzelm@33669
   759
            t1 $ t2 => subst t1 $ subst t2
wenzelm@33669
   760
          | Abs (x, T, u) => Abs (x, T, subst u)
wenzelm@33669
   761
          | _ => t));
berghofe@5149
   762
berghofe@5094
   763
    (* transform an introduction rule into a conjunction  *)
berghofe@21024
   764
    (*   [| p_i t; ... |] ==> p_j u                       *)
berghofe@5094
   765
    (* is transformed into                                *)
berghofe@21024
   766
    (*   b_j & x_j = u & p b_j t & ...                    *)
berghofe@5094
   767
berghofe@5094
   768
    fun transform_rule r =
berghofe@5094
   769
      let
wenzelm@45647
   770
        val SOME (_, i, ts, (Ts, _)) =
wenzelm@45647
   771
          dest_predicate cs params (HOLogic.dest_Trueprop (Logic.strip_assums_concl r));
wenzelm@45647
   772
        val ps =
wenzelm@45647
   773
          make_bool_args HOLogic.mk_not I bs i @
berghofe@21048
   774
          map HOLogic.mk_eq (make_args' argTs xs Ts ~~ ts) @
wenzelm@45647
   775
          map (subst o HOLogic.dest_Trueprop) (Logic.strip_assums_hyp r);
wenzelm@33338
   776
      in
wenzelm@33338
   777
        fold_rev (fn (x, T) => fn P => HOLogic.exists_const T $ Abs (x, T, P))
wenzelm@33338
   778
          (Logic.strip_params r)
wenzelm@33338
   779
          (if null ps then HOLogic.true_const else foldr1 HOLogic.mk_conj ps)
wenzelm@45647
   780
      end;
berghofe@5094
   781
berghofe@5094
   782
    (* make a disjunction of all introduction rules *)
berghofe@5094
   783
wenzelm@45647
   784
    val fp_fun =
wenzelm@45647
   785
      fold_rev lambda (p :: bs @ xs)
wenzelm@45647
   786
        (if null intr_ts then HOLogic.false_const
wenzelm@45647
   787
         else foldr1 HOLogic.mk_disj (map transform_rule intr_ts));
berghofe@5094
   788
berghofe@21024
   789
    (* add definiton of recursive predicates to theory *)
berghofe@5094
   790
wenzelm@28083
   791
    val rec_name =
haftmann@28965
   792
      if Binding.is_empty alt_name then
wenzelm@30223
   793
        Binding.name (space_implode "_" (map (Binding.name_of o fst) cnames_syn))
wenzelm@28083
   794
      else alt_name;
berghofe@5094
   795
wenzelm@33458
   796
    val ((rec_const, (_, fp_def)), lthy') = lthy
wenzelm@33671
   797
      |> Local_Theory.conceal
wenzelm@33766
   798
      |> Local_Theory.define
berghofe@21024
   799
        ((rec_name, case cnames_syn of [(_, syn)] => syn | _ => NoSyn),
wenzelm@45592
   800
         ((Binding.empty, @{attributes [nitpick_unfold]}),
wenzelm@45592
   801
           fold_rev lambda params
wenzelm@45592
   802
             (Const (fp_name, (predT --> predT) --> predT) $ fp_fun)))
wenzelm@33671
   803
      ||> Local_Theory.restore_naming lthy;
wenzelm@45647
   804
    val fp_def' =
wenzelm@45647
   805
      Simplifier.rewrite (HOL_basic_ss addsimps [fp_def])
wenzelm@45647
   806
        (cterm_of (Proof_Context.theory_of lthy') (list_comb (rec_const, params)));
wenzelm@33278
   807
    val specs =
wenzelm@33278
   808
      if length cs < 2 then []
wenzelm@33278
   809
      else
wenzelm@33278
   810
        map_index (fn (i, (name_mx, c)) =>
wenzelm@33278
   811
          let
wenzelm@33278
   812
            val Ts = arg_types_of (length params) c;
wenzelm@45647
   813
            val xs =
wenzelm@45647
   814
              map Free (Variable.variant_frees lthy intr_ts (mk_names "x" (length Ts) ~~ Ts));
wenzelm@33278
   815
          in
haftmann@39248
   816
            (name_mx, (apfst Binding.conceal Attrib.empty_binding, fold_rev lambda (params @ xs)
wenzelm@33278
   817
              (list_comb (rec_const, params @ make_bool_args' bs i @
wenzelm@33278
   818
                make_args argTs (xs ~~ Ts)))))
wenzelm@33278
   819
          end) (cnames_syn ~~ cs);
wenzelm@33458
   820
    val (consts_defs, lthy'') = lthy'
haftmann@39248
   821
      |> fold_map Local_Theory.define specs;
berghofe@21024
   822
    val preds = (case cs of [_] => [rec_const] | _ => map #1 consts_defs);
berghofe@5094
   823
berghofe@36642
   824
    val (_, lthy''') = Variable.add_fixes (map (fst o dest_Free) params) lthy'';
berghofe@36642
   825
    val mono = prove_mono quiet_mode skip_mono fork_mono predT fp_fun monos lthy''';
berghofe@36642
   826
    val (_, lthy'''') =
berghofe@36642
   827
      Local_Theory.note (apfst Binding.conceal Attrib.empty_binding,
wenzelm@42361
   828
        Proof_Context.export lthy''' lthy'' [mono]) lthy'';
berghofe@5094
   829
berghofe@36642
   830
  in (lthy'''', lthy''', rec_name, mono, fp_def', map (#2 o #2) consts_defs,
berghofe@21024
   831
    list_comb (rec_const, params), preds, argTs, bs, xs)
berghofe@21024
   832
  end;
berghofe@5094
   833
wenzelm@33669
   834
fun declare_rules rec_binding coind no_ind cnames
bulwahn@37734
   835
    preds intrs intr_bindings intr_atts elims eqs raw_induct lthy =
berghofe@23762
   836
  let
wenzelm@30223
   837
    val rec_name = Binding.name_of rec_binding;
haftmann@32773
   838
    fun rec_qualified qualified = Binding.qualify qualified rec_name;
wenzelm@30223
   839
    val intr_names = map Binding.name_of intr_bindings;
wenzelm@33368
   840
    val ind_case_names = Rule_Cases.case_names intr_names;
berghofe@23762
   841
    val induct =
berghofe@23762
   842
      if coind then
wenzelm@33368
   843
        (raw_induct, [Rule_Cases.case_names [rec_name],
wenzelm@33368
   844
          Rule_Cases.case_conclusion (rec_name, intr_names),
wenzelm@33368
   845
          Rule_Cases.consumes 1, Induct.coinduct_pred (hd cnames)])
berghofe@23762
   846
      else if no_ind orelse length cnames > 1 then
wenzelm@33368
   847
        (raw_induct, [ind_case_names, Rule_Cases.consumes 0])
wenzelm@33368
   848
      else (raw_induct RSN (2, rev_mp), [ind_case_names, Rule_Cases.consumes 1]);
berghofe@23762
   849
wenzelm@33458
   850
    val (intrs', lthy1) =
wenzelm@33458
   851
      lthy |>
bulwahn@35757
   852
      Spec_Rules.add
bulwahn@35757
   853
        (if coind then Spec_Rules.Co_Inductive else Spec_Rules.Inductive) (preds, intrs) |>
wenzelm@33671
   854
      Local_Theory.notes
wenzelm@33278
   855
        (map (rec_qualified false) intr_bindings ~~ intr_atts ~~
wenzelm@33278
   856
          map (fn th => [([th],
blanchet@37264
   857
           [Attrib.internal (K (Context_Rules.intro_query NONE))])]) intrs) |>>
berghofe@24744
   858
      map (hd o snd);
wenzelm@33458
   859
    val (((_, elims'), (_, [induct'])), lthy2) =
wenzelm@33458
   860
      lthy1 |>
wenzelm@33671
   861
      Local_Theory.note ((rec_qualified true (Binding.name "intros"), []), intrs') ||>>
berghofe@34986
   862
      fold_map (fn (name, (elim, cases, k)) =>
wenzelm@33671
   863
        Local_Theory.note
wenzelm@33458
   864
          ((Binding.qualify true (Long_Name.base_name name) (Binding.name "cases"),
wenzelm@33458
   865
            [Attrib.internal (K (Rule_Cases.case_names cases)),
wenzelm@33458
   866
             Attrib.internal (K (Rule_Cases.consumes 1)),
berghofe@34986
   867
             Attrib.internal (K (Rule_Cases.constraints k)),
wenzelm@33458
   868
             Attrib.internal (K (Induct.cases_pred name)),
wenzelm@33458
   869
             Attrib.internal (K (Context_Rules.elim_query NONE))]), [elim]) #>
berghofe@23762
   870
        apfst (hd o snd)) (if null elims then [] else cnames ~~ elims) ||>>
wenzelm@33671
   871
      Local_Theory.note
haftmann@32773
   872
        ((rec_qualified true (Binding.name (coind_prefix coind ^ "induct")),
wenzelm@28107
   873
          map (Attrib.internal o K) (#2 induct)), [rulify (#1 induct)]);
berghofe@23762
   874
wenzelm@45647
   875
    val (eqs', lthy3) = lthy2 |>
bulwahn@37734
   876
      fold_map (fn (name, eq) => Local_Theory.note
bulwahn@38665
   877
          ((Binding.qualify true (Long_Name.base_name name) (Binding.name "simps"),
bulwahn@38665
   878
            [Attrib.internal (K add_equation)]), [eq])
bulwahn@37734
   879
          #> apfst (hd o snd))
bulwahn@37734
   880
        (if null eqs then [] else (cnames ~~ eqs))
bulwahn@37734
   881
    val (inducts, lthy4) =
bulwahn@37734
   882
      if no_ind orelse coind then ([], lthy3)
wenzelm@33458
   883
      else
bulwahn@37734
   884
        let val inducts = cnames ~~ Project_Rule.projects lthy3 (1 upto length cnames) induct' in
bulwahn@37734
   885
          lthy3 |>
wenzelm@33671
   886
          Local_Theory.notes [((rec_qualified true (Binding.name "inducts"), []),
wenzelm@33458
   887
            inducts |> map (fn (name, th) => ([th],
wenzelm@33458
   888
              [Attrib.internal (K ind_case_names),
wenzelm@33458
   889
               Attrib.internal (K (Rule_Cases.consumes 1)),
berghofe@35646
   890
               Attrib.internal (K (Induct.induct_pred name))])))] |>> snd o hd
wenzelm@33458
   891
        end;
bulwahn@37734
   892
  in (intrs', elims', eqs', induct', inducts, lthy4) end;
berghofe@23762
   893
berghofe@26534
   894
type inductive_flags =
wenzelm@33669
   895
  {quiet_mode: bool, verbose: bool, alt_name: binding, coind: bool,
wenzelm@33669
   896
    no_elim: bool, no_ind: bool, skip_mono: bool, fork_mono: bool};
berghofe@26534
   897
berghofe@26534
   898
type add_ind_def =
berghofe@26534
   899
  inductive_flags ->
wenzelm@28084
   900
  term list -> (Attrib.binding * term) list -> thm list ->
haftmann@29581
   901
  term list -> (binding * mixfix) list ->
wenzelm@33458
   902
  local_theory -> inductive_result * local_theory;
berghofe@23762
   903
wenzelm@33669
   904
fun add_ind_def {quiet_mode, verbose, alt_name, coind, no_elim, no_ind, skip_mono, fork_mono}
wenzelm@33458
   905
    cs intros monos params cnames_syn lthy =
berghofe@9072
   906
  let
wenzelm@25288
   907
    val _ = null cnames_syn andalso error "No inductive predicates given";
wenzelm@30223
   908
    val names = map (Binding.name_of o fst) cnames_syn;
wenzelm@26477
   909
    val _ = message (quiet_mode andalso not verbose)
wenzelm@28083
   910
      ("Proofs for " ^ coind_prefix coind ^ "inductive predicate(s) " ^ commas_quote names);
berghofe@9072
   911
wenzelm@33671
   912
    val cnames = map (Local_Theory.full_name lthy o #1) cnames_syn;  (* FIXME *)
berghofe@23762
   913
    val ((intr_names, intr_atts), intr_ts) =
wenzelm@33458
   914
      apfst split_list (split_list (map (check_rule lthy cs params) intros));
berghofe@21024
   915
berghofe@36642
   916
    val (lthy1, lthy2, rec_name, mono, fp_def, rec_preds_defs, rec_const, preds,
wenzelm@29388
   917
      argTs, bs, xs) = mk_ind_def quiet_mode skip_mono fork_mono alt_name coind cs intr_ts
wenzelm@33458
   918
        monos params cnames_syn lthy;
berghofe@9072
   919
wenzelm@26477
   920
    val (intrs, unfold) = prove_intrs quiet_mode coind mono fp_def (length bs + length xs)
berghofe@36642
   921
      intr_ts rec_preds_defs lthy2 lthy1;
wenzelm@33459
   922
    val elims =
wenzelm@33459
   923
      if no_elim then []
wenzelm@33459
   924
      else
wenzelm@33459
   925
        prove_elims quiet_mode cs params intr_ts (map Binding.name_of intr_names)
berghofe@36642
   926
          unfold rec_preds_defs lthy2 lthy1;
berghofe@22605
   927
    val raw_induct = zero_var_indexes
wenzelm@33459
   928
      (if no_ind then Drule.asm_rl
wenzelm@33459
   929
       else if coind then
wenzelm@42361
   930
         singleton (Proof_Context.export lthy2 lthy1)
wenzelm@35625
   931
           (rotate_prems ~1 (Object_Logic.rulify
wenzelm@28839
   932
             (fold_rule rec_preds_defs
wenzelm@45649
   933
               (rewrite_rule simp_thms3
haftmann@32652
   934
                (mono RS (fp_def RS @{thm def_coinduct}))))))
berghofe@21024
   935
       else
wenzelm@26477
   936
         prove_indrule quiet_mode cs argTs bs xs rec_const params intr_ts mono fp_def
berghofe@36642
   937
           rec_preds_defs lthy2 lthy1);
bulwahn@37734
   938
    val eqs =
wenzelm@45647
   939
      if no_elim then [] else prove_eqs quiet_mode cs params intr_ts intrs elims lthy2 lthy1;
berghofe@5094
   940
wenzelm@45647
   941
    val elims' = map (fn (th, ns, i) => (rulify th, ns, i)) elims;
wenzelm@45647
   942
    val intrs' = map rulify intrs;
bulwahn@37734
   943
wenzelm@45647
   944
    val (intrs'', elims'', eqs', induct, inducts, lthy3) =
wenzelm@45647
   945
      declare_rules rec_name coind no_ind
wenzelm@45647
   946
        cnames preds intrs' intr_names intr_atts elims' eqs raw_induct lthy1;
berghofe@21048
   947
berghofe@21048
   948
    val result =
berghofe@21048
   949
      {preds = preds,
bulwahn@37734
   950
       intrs = intrs'',
bulwahn@37734
   951
       elims = elims'',
berghofe@21048
   952
       raw_induct = rulify raw_induct,
berghofe@35646
   953
       induct = induct,
bulwahn@37734
   954
       inducts = inducts,
bulwahn@37734
   955
       eqs = eqs'};
wenzelm@21367
   956
berghofe@36642
   957
    val lthy4 = lthy3
wenzelm@45291
   958
      |> Local_Theory.declaration {syntax = false, pervasive = false} (fn phi =>
wenzelm@45290
   959
        let val result' = transform_result phi result;
wenzelm@25380
   960
        in put_inductives cnames (*global names!?*) ({names = cnames, coind = coind}, result') end);
berghofe@36642
   961
  in (result, lthy4) end;
berghofe@5094
   962
wenzelm@6424
   963
wenzelm@10735
   964
(* external interfaces *)
berghofe@5094
   965
wenzelm@26477
   966
fun gen_add_inductive_i mk_def
wenzelm@33669
   967
    (flags as {quiet_mode, verbose, alt_name, coind, no_elim, no_ind, skip_mono, fork_mono})
wenzelm@25029
   968
    cnames_syn pnames spec monos lthy =
berghofe@5094
   969
  let
wenzelm@42361
   970
    val thy = Proof_Context.theory_of lthy;
wenzelm@6424
   971
    val _ = Theory.requires thy "Inductive" (coind_prefix coind ^ "inductive definitions");
berghofe@5094
   972
berghofe@21766
   973
wenzelm@25029
   974
    (* abbrevs *)
wenzelm@25029
   975
wenzelm@30223
   976
    val (_, ctxt1) = Variable.add_fixes (map (Binding.name_of o fst o fst) cnames_syn) lthy;
berghofe@21766
   977
wenzelm@25029
   978
    fun get_abbrev ((name, atts), t) =
wenzelm@25029
   979
      if can (Logic.strip_assums_concl #> Logic.dest_equals) t then
wenzelm@25029
   980
        let
haftmann@29006
   981
          val _ = Binding.is_empty name andalso null atts orelse
wenzelm@25029
   982
            error "Abbreviations may not have names or attributes";
wenzelm@35624
   983
          val ((x, T), rhs) = Local_Defs.abs_def (snd (Local_Defs.cert_def ctxt1 t));
wenzelm@28083
   984
          val var =
wenzelm@30223
   985
            (case find_first (fn ((c, _), _) => Binding.name_of c = x) cnames_syn of
wenzelm@25029
   986
              NONE => error ("Undeclared head of abbreviation " ^ quote x)
wenzelm@28083
   987
            | SOME ((b, T'), mx) =>
wenzelm@25029
   988
                if T <> T' then error ("Bad type specification for abbreviation " ^ quote x)
wenzelm@28083
   989
                else (b, mx));
wenzelm@28083
   990
        in SOME (var, rhs) end
wenzelm@25029
   991
      else NONE;
berghofe@21766
   992
wenzelm@25029
   993
    val abbrevs = map_filter get_abbrev spec;
wenzelm@30223
   994
    val bs = map (Binding.name_of o fst o fst) abbrevs;
wenzelm@25029
   995
berghofe@21766
   996
wenzelm@25029
   997
    (* predicates *)
berghofe@21766
   998
wenzelm@25029
   999
    val pre_intros = filter_out (is_some o get_abbrev) spec;
wenzelm@30223
  1000
    val cnames_syn' = filter_out (member (op =) bs o Binding.name_of o fst o fst) cnames_syn;
wenzelm@30223
  1001
    val cs = map (Free o apfst Binding.name_of o fst) cnames_syn';
wenzelm@25029
  1002
    val ps = map Free pnames;
berghofe@5094
  1003
wenzelm@30223
  1004
    val (_, ctxt2) = lthy |> Variable.add_fixes (map (Binding.name_of o fst o fst) cnames_syn');
wenzelm@35624
  1005
    val _ = map (fn abbr => Local_Defs.fixed_abbrev abbr ctxt2) abbrevs;
wenzelm@35624
  1006
    val ctxt3 = ctxt2 |> fold (snd oo Local_Defs.fixed_abbrev) abbrevs;
wenzelm@42361
  1007
    val expand = Assumption.export_term ctxt3 lthy #> Proof_Context.cert_term lthy;
wenzelm@25029
  1008
wenzelm@25029
  1009
    fun close_rule r = list_all_free (rev (fold_aterms
berghofe@21024
  1010
      (fn t as Free (v as (s, _)) =>
wenzelm@25029
  1011
          if Variable.is_fixed ctxt1 s orelse
wenzelm@25029
  1012
            member (op =) ps t then I else insert (op =) v
wenzelm@25029
  1013
        | _ => I) r []), r);
berghofe@5094
  1014
haftmann@26736
  1015
    val intros = map (apsnd (Syntax.check_term lthy #> close_rule #> expand)) pre_intros;
wenzelm@25029
  1016
    val preds = map (fn ((c, _), mx) => (c, mx)) cnames_syn';
berghofe@21048
  1017
  in
wenzelm@25029
  1018
    lthy
wenzelm@25029
  1019
    |> mk_def flags cs intros monos ps preds
wenzelm@33671
  1020
    ||> fold (snd oo Local_Theory.abbrev Syntax.mode_default) abbrevs
berghofe@21048
  1021
  end;
berghofe@5094
  1022
wenzelm@29388
  1023
fun gen_add_inductive mk_def verbose coind cnames_syn pnames_syn intro_srcs raw_monos int lthy =
berghofe@5094
  1024
  let
wenzelm@30486
  1025
    val ((vars, intrs), _) = lthy
wenzelm@42361
  1026
      |> Proof_Context.set_mode Proof_Context.mode_abbrev
wenzelm@30486
  1027
      |> Specification.read_spec (cnames_syn @ pnames_syn) intro_srcs;
wenzelm@24721
  1028
    val (cs, ps) = chop (length cnames_syn) vars;
wenzelm@24721
  1029
    val monos = Attrib.eval_thms lthy raw_monos;
wenzelm@33669
  1030
    val flags = {quiet_mode = false, verbose = verbose, alt_name = Binding.empty,
wenzelm@33669
  1031
      coind = coind, no_elim = false, no_ind = false, skip_mono = false, fork_mono = not int};
wenzelm@26128
  1032
  in
wenzelm@26128
  1033
    lthy
wenzelm@30223
  1034
    |> gen_add_inductive_i mk_def flags cs (map (apfst Binding.name_of o fst) ps) intrs monos
wenzelm@26128
  1035
  end;
berghofe@5094
  1036
berghofe@23762
  1037
val add_inductive_i = gen_add_inductive_i add_ind_def;
berghofe@23762
  1038
val add_inductive = gen_add_inductive add_ind_def;
berghofe@23762
  1039
wenzelm@33726
  1040
fun add_inductive_global flags cnames_syn pnames pre_intros monos thy =
wenzelm@25380
  1041
  let
haftmann@29006
  1042
    val name = Sign.full_name thy (fst (fst (hd cnames_syn)));
wenzelm@25380
  1043
    val ctxt' = thy
haftmann@38388
  1044
      |> Named_Target.theory_init
wenzelm@25380
  1045
      |> add_inductive_i flags cnames_syn pnames pre_intros monos |> snd
wenzelm@33671
  1046
      |> Local_Theory.exit;
wenzelm@25380
  1047
    val info = #2 (the_inductive ctxt' name);
wenzelm@42361
  1048
  in (info, Proof_Context.theory_of ctxt') end;
wenzelm@6424
  1049
wenzelm@6424
  1050
berghofe@22789
  1051
(* read off arities of inductive predicates from raw induction rule *)
berghofe@22789
  1052
fun arities_of induct =
berghofe@22789
  1053
  map (fn (_ $ t $ u) =>
berghofe@22789
  1054
      (fst (dest_Const (head_of t)), length (snd (strip_comb u))))
berghofe@22789
  1055
    (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct)));
berghofe@22789
  1056
berghofe@22789
  1057
(* read off parameters of inductive predicate from raw induction rule *)
berghofe@22789
  1058
fun params_of induct =
berghofe@22789
  1059
  let
wenzelm@45647
  1060
    val (_ $ t $ u :: _) = HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct));
berghofe@22789
  1061
    val (_, ts) = strip_comb t;
wenzelm@45647
  1062
    val (_, us) = strip_comb u;
berghofe@22789
  1063
  in
berghofe@22789
  1064
    List.take (ts, length ts - length us)
berghofe@22789
  1065
  end;
berghofe@22789
  1066
berghofe@22789
  1067
val pname_of_intr =
berghofe@22789
  1068
  concl_of #> HOLogic.dest_Trueprop #> head_of #> dest_Const #> fst;
berghofe@22789
  1069
berghofe@22789
  1070
(* partition introduction rules according to predicate name *)
berghofe@25822
  1071
fun gen_partition_rules f induct intros =
berghofe@25822
  1072
  fold_rev (fn r => AList.map_entry op = (pname_of_intr (f r)) (cons r)) intros
berghofe@22789
  1073
    (map (rpair [] o fst) (arities_of induct));
berghofe@22789
  1074
berghofe@25822
  1075
val partition_rules = gen_partition_rules I;
berghofe@25822
  1076
fun partition_rules' induct = gen_partition_rules fst induct;
berghofe@25822
  1077
berghofe@22789
  1078
fun unpartition_rules intros xs =
berghofe@22789
  1079
  fold_map (fn r => AList.map_entry_yield op = (pname_of_intr r)
berghofe@22789
  1080
    (fn x :: xs => (x, xs)) #>> the) intros xs |> fst;
berghofe@22789
  1081
berghofe@22789
  1082
(* infer order of variables in intro rules from order of quantifiers in elim rule *)
berghofe@22789
  1083
fun infer_intro_vars elim arity intros =
berghofe@22789
  1084
  let
berghofe@22789
  1085
    val thy = theory_of_thm elim;
berghofe@22789
  1086
    val _ :: cases = prems_of elim;
berghofe@22789
  1087
    val used = map (fst o fst) (Term.add_vars (prop_of elim) []);
berghofe@22789
  1088
    fun mtch (t, u) =
berghofe@22789
  1089
      let
berghofe@22789
  1090
        val params = Logic.strip_params t;
wenzelm@45647
  1091
        val vars =
wenzelm@45647
  1092
          map (Var o apfst (rpair 0))
wenzelm@45647
  1093
            (Name.variant_list used (map fst params) ~~ map snd params);
wenzelm@45647
  1094
        val ts =
wenzelm@45647
  1095
          map (curry subst_bounds (rev vars))
wenzelm@45647
  1096
            (List.drop (Logic.strip_assums_hyp t, arity));
berghofe@22789
  1097
        val us = Logic.strip_imp_prems u;
wenzelm@45647
  1098
        val tab =
wenzelm@45647
  1099
          fold (Pattern.first_order_match thy) (ts ~~ us) (Vartab.empty, Vartab.empty);
berghofe@22789
  1100
      in
wenzelm@32035
  1101
        map (Envir.subst_term tab) vars
berghofe@22789
  1102
      end
berghofe@22789
  1103
  in
berghofe@22789
  1104
    map (mtch o apsnd prop_of) (cases ~~ intros)
berghofe@22789
  1105
  end;
berghofe@22789
  1106
berghofe@22789
  1107
wenzelm@25978
  1108
wenzelm@6437
  1109
(** package setup **)
wenzelm@6437
  1110
wenzelm@6437
  1111
(* setup theory *)
wenzelm@6437
  1112
wenzelm@8634
  1113
val setup =
wenzelm@30722
  1114
  ind_cases_setup #>
wenzelm@30528
  1115
  Attrib.setup @{binding mono} (Attrib.add_del mono_add mono_del)
wenzelm@30528
  1116
    "declaration of monotonicity rule";
wenzelm@6437
  1117
wenzelm@6437
  1118
wenzelm@6437
  1119
(* outer syntax *)
wenzelm@6424
  1120
wenzelm@36960
  1121
val _ = Keyword.keyword "monos";
wenzelm@24867
  1122
berghofe@23762
  1123
fun gen_ind_decl mk_def coind =
wenzelm@36960
  1124
  Parse.fixes -- Parse.for_fixes --
wenzelm@36954
  1125
  Scan.optional Parse_Spec.where_alt_specs [] --
wenzelm@36960
  1126
  Scan.optional (Parse.$$$ "monos" |-- Parse.!!! Parse_Spec.xthms1) []
wenzelm@26988
  1127
  >> (fn (((preds, params), specs), monos) =>
wenzelm@30486
  1128
      (snd oo gen_add_inductive mk_def true coind preds params specs monos));
berghofe@23762
  1129
berghofe@23762
  1130
val ind_decl = gen_ind_decl add_ind_def;
wenzelm@6424
  1131
wenzelm@33458
  1132
val _ =
wenzelm@36960
  1133
  Outer_Syntax.local_theory' "inductive" "define inductive predicates" Keyword.thy_decl
wenzelm@33458
  1134
    (ind_decl false);
wenzelm@33458
  1135
wenzelm@33458
  1136
val _ =
wenzelm@36960
  1137
  Outer_Syntax.local_theory' "coinductive" "define coinductive predicates" Keyword.thy_decl
wenzelm@33458
  1138
    (ind_decl true);
wenzelm@6723
  1139
wenzelm@24867
  1140
val _ =
wenzelm@36960
  1141
  Outer_Syntax.local_theory "inductive_cases"
wenzelm@36960
  1142
    "create simplified instances of elimination rules (improper)" Keyword.thy_script
wenzelm@36960
  1143
    (Parse.and_list1 Parse_Spec.specs >> (snd oo inductive_cases));
wenzelm@7107
  1144
bulwahn@37734
  1145
val _ =
bulwahn@37734
  1146
  Outer_Syntax.local_theory "inductive_simps"
bulwahn@37734
  1147
    "create simplification rules for inductive predicates" Keyword.thy_script
bulwahn@37734
  1148
    (Parse.and_list1 Parse_Spec.specs >> (snd oo inductive_simps));
bulwahn@37734
  1149
berghofe@5094
  1150
end;