(* 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, set_def: thm option, 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 -> theory -> theory) -> theory -> theory
val setup: theory -> theory
val add_typedef: bool -> binding option -> binding * (string * sort) list * mixfix ->
term -> (binding * binding) option -> tactic -> local_theory -> (string * info) * local_theory
val add_typedef_global: bool -> binding option -> binding * (string * sort) list * mixfix ->
term -> (binding * binding) option -> tactic -> theory -> (string * info) * theory
val typedef: (bool * binding) * (binding * (string * sort) list * mixfix) * term *
(binding * binding) option -> local_theory -> Proof.state
val typedef_cmd: (bool * binding) * (binding * (string * string option) list * mixfix) * string *
(binding * binding) 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, set_def: thm option, 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,
set_def, 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,
set_def = Option.map thm set_def, 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;
);
val get_info = Symtab.lookup_list o Data.get o Context.Proof;
val get_info_global = Symtab.lookup_list o Data.get o Context.Theory;
fun put_info name info = Data.map (Symtab.cons_list (name, info));
(* global interpretation *)
structure Typedef_Interpretation = Interpretation(type T = string val eq = op =);
val interpretation = Typedef_Interpretation.interpretation;
val setup = Typedef_Interpretation.init;
(* primitive typedef axiomatization -- for fresh typedecl *)
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 typedef_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 typedef_deps = Term.add_consts A [];
val ((axiom_name, axiom), axiom_lthy) = consts_lthy
|> Local_Theory.background_theory_result
(Thm.add_axiom consts_lthy (typedef_name, mk_typedef newT oldT RepC AbsC A) ##>
Theory.add_deps consts_lthy "" (dest_Const RepC) typedef_deps ##>
Theory.add_deps consts_lthy "" (dest_Const AbsC) typedef_deps);
in ((RepC, AbsC, axiom_name, axiom), axiom_lthy) end;
(* prepare_typedef *)
fun prepare_typedef prep_term def_set name (tname, raw_args, mx) raw_set opt_morphs lthy =
let
val full_name = Local_Theory.full_name lthy name;
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 (tname, args, mx)
||> Variable.declare_term set;
val Type (full_tname, type_args) = newT;
val lhs_tfrees = map Term.dest_TFree type_args;
(* set definition *)
val ((set', set_def), set_lthy) =
if def_set then
typedecl_lthy
|> Local_Theory.define ((name, NoSyn), ((Thm.def_binding name, []), set))
|>> (fn (set', (_, set_def)) => (set', SOME set_def))
else ((set, NONE), typedecl_lthy);
(* axiomatization *)
val (Rep_name, Abs_name) =
(case opt_morphs of
NONE => (Binding.prefix_name "Rep_" name, Binding.prefix_name "Abs_" name)
| SOME morphs => morphs);
val typedef_name = Binding.prefix_name "type_definition_" name;
val ((RepC, AbsC, axiom_name, typedef), typedef_lthy) =
let
val thy = Proof_Context.theory_of set_lthy;
val cert = Thm.cterm_of thy;
val ((defs, _), A) =
Local_Defs.export_cterm set_lthy (Proof_Context.init_global thy) (cert set')
||> Thm.term_of;
val ((RepC, AbsC, axiom_name, axiom), axiom_lthy) = set_lthy
|> primitive_typedef typedef_name newT oldT Rep_name Abs_name A;
val cert = Thm.cterm_of (Proof_Context.theory_of axiom_lthy);
val typedef =
Local_Defs.contract axiom_lthy defs (cert (mk_typedef newT oldT RepC AbsC set')) axiom;
in ((RepC, AbsC, axiom_name, typedef), axiom_lthy) end;
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 cert = Thm.cterm_of (Proof_Context.theory_of lthy1);
val inhabited' =
Local_Defs.contract lthy1 (the_list set_def) (cert (mk_inhabited set')) inhabited;
val typedef' = inhabited' RS typedef;
fun make th = Goal.norm_result (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 ((typedef_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_tname]),
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_tname]),
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, set_def = set_def,
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_tname (transform_info phi info))
|> Local_Theory.background_theory (Typedef_Interpretation.data full_tname)
|> pair (full_tname, 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 def opt_name typ set opt_morphs tac lthy =
let
val name = the_default (#1 typ) opt_name;
val ((goal, _, typedef_result), lthy') =
prepare_typedef Syntax.check_term def name typ set opt_morphs lthy;
val inhabited =
Goal.prove lthy' [] [] goal (K tac)
|> Goal.norm_result |> Thm.close_derivation;
in typedef_result inhabited lthy' end;
fun add_typedef_global def opt_name typ set opt_morphs tac =
Named_Target.theory_init
#> add_typedef def opt_name typ set opt_morphs tac
#> Local_Theory.exit_result_global (apsnd o transform_info);
(* typedef: proof interface *)
local
fun gen_typedef prep_term prep_constraint ((def, name), (b, raw_args, mx), set, opt_morphs) lthy =
let
val args = map (apsnd (prep_constraint lthy)) raw_args;
val ((goal, goal_pat, typedef_result), lthy') =
prepare_typedef prep_term def name (b, args, mx) set opt_morphs 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 "typedef" "HOL type definition (requires non-emptiness proof)"
Keyword.thy_goal
(Scan.optional (@{keyword "("} |--
((@{keyword "open"} >> K false) -- Scan.option Parse.binding ||
Parse.binding >> (fn s => (true, SOME s))) --| @{keyword ")"}) (true, NONE) --
(Parse.type_args_constrained -- Parse.binding) --
Parse.opt_mixfix -- (@{keyword "="} |-- Parse.term) --
Scan.option (@{keyword "morphisms"} |-- Parse.!!! (Parse.binding -- Parse.binding))
>> (fn ((((((def, opt_name), (args, t)), mx), A), morphs)) =>
typedef_cmd ((def, the_default t opt_name), (t, args, mx), A, morphs)));
end;