(* Title: HOL/Tools/typedef.ML
Author: Markus Wenzel and Stefan Berghofer, TU Muenchen
Gordon/HOL-style type definitions: create a new syntactic type
represented by a non-empty set.
*)
signature TYPEDEF =
sig
type info =
{rep_type: typ, abs_type: typ, Rep_name: string, Abs_name: string, axiom_name: string} *
{inhabited: thm, type_definition: thm, Rep: thm, Rep_inverse: thm, Abs_inverse: thm,
Rep_inject: thm, Abs_inject: thm, Rep_cases: thm, Abs_cases: thm,
Rep_induct: thm, Abs_induct: thm}
val transform_info: morphism -> info -> info
val get_info: Proof.context -> string -> info list
val get_info_global: theory -> string -> info list
val interpretation: (string -> local_theory -> local_theory) -> theory -> theory
type bindings = {Rep_name: binding, Abs_name: binding, type_definition_name: binding}
val default_bindings: binding -> bindings
val make_bindings: binding -> bindings option -> bindings
val make_morphisms: binding -> (binding * binding) option -> bindings
val overloaded: bool Config.T
val add_typedef: {overloaded: bool} -> binding * (string * sort) list * mixfix ->
term -> bindings option -> (Proof.context -> tactic) -> local_theory ->
(string * info) * local_theory
val add_typedef_global: {overloaded: bool} -> binding * (string * sort) list * mixfix ->
term -> bindings option -> (Proof.context -> tactic) -> theory ->
(string * info) * theory
val typedef: {overloaded: bool} -> binding * (string * sort) list * mixfix ->
term -> bindings option -> local_theory -> Proof.state
val typedef_cmd: {overloaded: bool} -> binding * (string * string option) list * mixfix ->
string -> bindings option -> local_theory -> Proof.state
end;
structure Typedef: TYPEDEF =
struct
(** type definitions **)
(* theory data *)
type info =
(*global part*)
{rep_type: typ, abs_type: typ, Rep_name: string, Abs_name: string, axiom_name: string} *
(*local part*)
{inhabited: thm, type_definition: thm, Rep: thm, Rep_inverse: thm, Abs_inverse: thm,
Rep_inject: thm, Abs_inject: thm, Rep_cases: thm, Abs_cases: thm,
Rep_induct: thm, Abs_induct: thm};
fun transform_info phi (info: info) =
let
val thm = Morphism.thm phi;
val (global_info, {inhabited, type_definition, Rep, Rep_inverse, Abs_inverse,
Rep_inject, Abs_inject, Rep_cases, Abs_cases, Rep_induct, Abs_induct}) = info;
in
(global_info,
{inhabited = thm inhabited, type_definition = thm type_definition,
Rep = thm Rep, Rep_inverse = thm Rep_inverse, Abs_inverse = thm Abs_inverse,
Rep_inject = thm Rep_inject, Abs_inject = thm Abs_inject,
Rep_cases = thm Rep_cases, Abs_cases = thm Abs_cases,
Rep_induct = thm Rep_induct, Abs_induct = thm Abs_induct})
end;
structure Data = Generic_Data
(
type T = info list Symtab.table;
val empty = Symtab.empty;
val extend = I;
fun merge data = Symtab.merge_list (K true) data;
);
fun get_info_generic context =
Symtab.lookup_list (Data.get context) #>
map (transform_info (Morphism.transfer_morphism (Context.theory_of context)));
val get_info = get_info_generic o Context.Proof;
val get_info_global = get_info_generic o Context.Theory;
fun put_info name info =
Data.map (Symtab.cons_list (name, transform_info Morphism.trim_context_morphism info));
(* global interpretation *)
structure Typedef_Plugin = Plugin(type T = string);
val typedef_plugin = Plugin_Name.declare_setup @{binding typedef};
fun interpretation f =
Typedef_Plugin.interpretation typedef_plugin
(fn name => fn lthy =>
lthy
|> Local_Theory.map_background_naming
(Name_Space.root_path #> Name_Space.add_path (Long_Name.qualifier name))
|> f name
|> Local_Theory.restore_background_naming lthy);
(* primitive typedef axiomatization -- for fresh typedecl *)
val typedef_overloaded = Attrib.setup_config_bool @{binding typedef_overloaded} (K false);
fun mk_inhabited A =
let val T = HOLogic.dest_setT (Term.fastype_of A)
in HOLogic.mk_Trueprop (HOLogic.exists_const T $ Abs ("x", T, HOLogic.mk_mem (Bound 0, A))) end;
fun mk_typedef newT oldT RepC AbsC A =
let
val typedefC =
Const (@{const_name type_definition},
(newT --> oldT) --> (oldT --> newT) --> HOLogic.mk_setT oldT --> HOLogic.boolT);
in Logic.mk_implies (mk_inhabited A, HOLogic.mk_Trueprop (typedefC $ RepC $ AbsC $ A)) end;
fun primitive_typedef {overloaded} type_definition_name newT oldT Rep_name Abs_name A lthy =
let
(* errors *)
fun show_names pairs = commas_quote (map fst pairs);
val lhs_tfrees = Term.add_tfreesT newT [];
val rhs_tfrees = Term.add_tfreesT oldT [];
val _ =
(case fold (remove (op =)) lhs_tfrees rhs_tfrees of
[] => ()
| extras => error ("Extra type variables in representing set: " ^ show_names extras));
val _ =
(case Term.add_frees A [] of [] =>
[]
| xs => error ("Illegal variables in representing set: " ^ show_names xs));
(* axiomatization *)
val ((RepC, AbsC), consts_lthy) = lthy
|> Local_Theory.background_theory_result
(Sign.declare_const lthy ((Rep_name, newT --> oldT), NoSyn) ##>>
Sign.declare_const lthy ((Abs_name, oldT --> newT), NoSyn));
val const_dep = Theory.const_dep (Proof_Context.theory_of consts_lthy);
val A_consts = fold_aterms (fn Const c => insert (op =) (const_dep c) | _ => I) A [];
val A_types =
(fold_types o fold_subtypes) (fn Type t => insert (op =) (Theory.type_dep t) | _ => I) A [];
val typedef_deps = A_consts @ A_types;
val newT_dep = Theory.type_dep (dest_Type newT);
val ((axiom_name, axiom), axiom_lthy) = consts_lthy
|> Local_Theory.background_theory_result
(Thm.add_axiom consts_lthy (type_definition_name, mk_typedef newT oldT RepC AbsC A) ##>
Theory.add_deps consts_lthy "" newT_dep typedef_deps ##>
Theory.add_deps consts_lthy "" (const_dep (dest_Const RepC)) [newT_dep] ##>
Theory.add_deps consts_lthy "" (const_dep (dest_Const AbsC)) [newT_dep]);
val axiom_defs = Theory.defs_of (Proof_Context.theory_of axiom_lthy);
val newT_deps = maps #2 (Defs.get_deps axiom_defs (#1 newT_dep));
val _ =
if null newT_deps orelse overloaded orelse Config.get lthy typedef_overloaded then ()
else
error (Pretty.string_of (Pretty.chunks
[Pretty.paragraph
(Pretty.text "Type definition with open dependencies, use" @
[Pretty.brk 1, Pretty.str "\"", Pretty.keyword1 "typedef", Pretty.brk 1,
Pretty.str "(", Pretty.keyword2 "overloaded", Pretty.str ")\"", Pretty.brk 1] @
Pretty.text "or enable configuration option \"typedef_overloaded\" in the context."),
Pretty.block [Pretty.str " Type:", Pretty.brk 2, Syntax.pretty_typ axiom_lthy newT],
Pretty.block (Pretty.str " Deps:" :: Pretty.brk 2 ::
Pretty.commas (map (Defs.pretty_entry axiom_lthy) newT_deps))]))
in ((RepC, AbsC, axiom_name, axiom), axiom_lthy) end;
(* derived bindings *)
type bindings = {Rep_name: binding, Abs_name: binding, type_definition_name: binding};
fun default_bindings name =
{Rep_name = Binding.prefix_name "Rep_" name,
Abs_name = Binding.prefix_name "Abs_" name,
type_definition_name = Binding.prefix_name "type_definition_" name};
fun make_bindings name NONE = default_bindings name
| make_bindings _ (SOME bindings) = bindings;
fun make_morphisms name NONE = default_bindings name
| make_morphisms name (SOME (Rep_name, Abs_name)) =
{Rep_name = Rep_name, Abs_name = Abs_name,
type_definition_name = #type_definition_name (default_bindings name)};
(* prepare_typedef *)
fun prepare_typedef prep_term overloaded (name, raw_args, mx) raw_set opt_bindings lthy =
let
val bname = Binding.name_of name;
(* rhs *)
val tmp_ctxt = lthy |> fold (Variable.declare_typ o TFree) raw_args;
val set = prep_term tmp_ctxt raw_set;
val tmp_ctxt' = tmp_ctxt |> Variable.declare_term set;
val setT = Term.fastype_of set;
val oldT = HOLogic.dest_setT setT handle TYPE _ =>
error ("Not a set type: " ^ quote (Syntax.string_of_typ lthy setT));
val goal = mk_inhabited set;
val goal_pat = mk_inhabited (Var (the_default (bname, 0) (Lexicon.read_variable bname), setT));
(* lhs *)
val args = map (Proof_Context.check_tfree tmp_ctxt') raw_args;
val (newT, typedecl_lthy) = lthy
|> Typedecl.typedecl {final = false} (name, args, mx)
||> Variable.declare_term set;
val Type (full_name, _) = newT;
(* axiomatization *)
val {Rep_name, Abs_name, type_definition_name} = make_bindings name opt_bindings;
val ((RepC, AbsC, axiom_name, typedef), typedef_lthy) = typedecl_lthy
|> primitive_typedef overloaded type_definition_name newT oldT Rep_name Abs_name set;
val alias_lthy = typedef_lthy
|> Local_Theory.const_alias Rep_name (#1 (Term.dest_Const RepC))
|> Local_Theory.const_alias Abs_name (#1 (Term.dest_Const AbsC));
(* result *)
fun note_qualify ((b, atts), th) =
Local_Theory.note ((Binding.qualify false bname b, map (Attrib.internal o K) atts), [th])
#>> (fn (_, [th']) => th');
fun typedef_result inhabited lthy1 =
let
val typedef' = inhabited RS typedef;
fun make th = Goal.norm_result lthy1 (typedef' RS th);
val (((((((((((_, [type_definition]), Rep), Rep_inverse), Abs_inverse), Rep_inject),
Abs_inject), Rep_cases), Abs_cases), Rep_induct), Abs_induct), lthy2) = lthy1
|> Local_Theory.note ((type_definition_name, []), [typedef'])
||>> note_qualify ((Rep_name, []), make @{thm type_definition.Rep})
||>> note_qualify ((Binding.suffix_name "_inverse" Rep_name, []),
make @{thm type_definition.Rep_inverse})
||>> note_qualify ((Binding.suffix_name "_inverse" Abs_name, []),
make @{thm type_definition.Abs_inverse})
||>> note_qualify ((Binding.suffix_name "_inject" Rep_name, []),
make @{thm type_definition.Rep_inject})
||>> note_qualify ((Binding.suffix_name "_inject" Abs_name, []),
make @{thm type_definition.Abs_inject})
||>> note_qualify ((Binding.suffix_name "_cases" Rep_name,
[Rule_Cases.case_names [Binding.name_of Rep_name], Induct.cases_pred full_name]),
make @{thm type_definition.Rep_cases})
||>> note_qualify ((Binding.suffix_name "_cases" Abs_name,
[Rule_Cases.case_names [Binding.name_of Abs_name], Induct.cases_type full_name]),
make @{thm type_definition.Abs_cases})
||>> note_qualify ((Binding.suffix_name "_induct" Rep_name,
[Rule_Cases.case_names [Binding.name_of Rep_name], Induct.induct_pred full_name]),
make @{thm type_definition.Rep_induct})
||>> note_qualify ((Binding.suffix_name "_induct" Abs_name,
[Rule_Cases.case_names [Binding.name_of Abs_name], Induct.induct_type full_name]),
make @{thm type_definition.Abs_induct});
val info =
({rep_type = oldT, abs_type = newT, Rep_name = #1 (Term.dest_Const RepC),
Abs_name = #1 (Term.dest_Const AbsC), axiom_name = axiom_name},
{inhabited = inhabited, type_definition = type_definition,
Rep = Rep, Rep_inverse = Rep_inverse, Abs_inverse = Abs_inverse,
Rep_inject = Rep_inject, Abs_inject = Abs_inject, Rep_cases = Rep_cases,
Abs_cases = Abs_cases, Rep_induct = Rep_induct, Abs_induct = Abs_induct});
in
lthy2
|> Local_Theory.declaration {syntax = false, pervasive = true}
(fn phi => put_info full_name (transform_info phi info))
|> Typedef_Plugin.data Plugin_Name.default_filter full_name
|> pair (full_name, info)
end;
in ((goal, goal_pat, typedef_result), alias_lthy) end
handle ERROR msg =>
cat_error msg ("The error(s) above occurred in typedef " ^ Binding.print name);
(* add_typedef: tactic interface *)
fun add_typedef overloaded typ set opt_bindings tac lthy =
let
val ((goal, _, typedef_result), lthy') =
prepare_typedef Syntax.check_term overloaded typ set opt_bindings lthy;
val inhabited =
Goal.prove lthy' [] [] goal (tac o #context)
|> Goal.norm_result lthy' |> Thm.close_derivation;
in typedef_result inhabited lthy' end;
fun add_typedef_global overloaded typ set opt_bindings tac =
Named_Target.theory_init
#> add_typedef overloaded typ set opt_bindings tac
#> Local_Theory.exit_result_global (apsnd o transform_info);
(* typedef: proof interface *)
local
fun gen_typedef prep_term prep_constraint overloaded (b, raw_args, mx) set opt_bindings lthy =
let
val args = map (apsnd (prep_constraint lthy)) raw_args;
val ((goal, goal_pat, typedef_result), lthy') =
prepare_typedef prep_term overloaded (b, args, mx) set opt_bindings lthy;
fun after_qed [[th]] = snd o typedef_result th;
in Proof.theorem NONE after_qed [[(goal, [goal_pat])]] lthy' end;
in
val typedef = gen_typedef Syntax.check_term (K I);
val typedef_cmd = gen_typedef Syntax.read_term Typedecl.read_constraint;
end;
(** outer syntax **)
val _ =
Outer_Syntax.local_theory_to_proof @{command_keyword typedef}
"HOL type definition (requires non-emptiness proof)"
(Parse_Spec.overloaded -- Parse.type_args_constrained -- Parse.binding -- Parse.opt_mixfix --
(@{keyword "="} |-- Parse.term) --
Scan.option (@{keyword "morphisms"} |-- Parse.!!! (Parse.binding -- Parse.binding))
>> (fn (((((overloaded, vs), t), mx), A), opt_morphs) => fn lthy =>
typedef_cmd {overloaded = overloaded} (t, vs, mx) A
(SOME (make_morphisms t opt_morphs)) lthy));
val overloaded = typedef_overloaded;
end;