(* Title: HOL/Tools/typedef_package.ML
Author: Markus Wenzel and Stefan Berghofer, TU Muenchen
Gordon/HOL-style type definitions: create a new syntactic type
represented by a non-empty subset.
*)
signature TYPEDEF_PACKAGE =
sig
type info =
{rep_type: typ, abs_type: typ, Rep_name: string, Abs_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 get_info: theory -> string -> info option
val add_typedef: bool -> binding option -> binding * string list * mixfix ->
term -> (binding * binding) option -> tactic -> theory -> (string * info) * theory
val typedef: (bool * binding) * (binding * string list * mixfix) * term
* (binding * binding) option -> theory -> Proof.state
val typedef_cmd: (bool * binding) * (binding * string list * mixfix) * string
* (binding * binding) option -> theory -> Proof.state
val interpretation: (string -> theory -> theory) -> theory -> theory
val setup: theory -> theory
end;
structure TypedefPackage: TYPEDEF_PACKAGE =
struct
(** type definitions **)
(* theory data *)
type info =
{rep_type: typ, abs_type: typ, Rep_name: string, Abs_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};
structure TypedefData = TheoryDataFun
(
type T = info Symtab.table;
val empty = Symtab.empty;
val copy = I;
val extend = I;
fun merge _ tabs : T = Symtab.merge (K true) tabs;
);
val get_info = Symtab.lookup o TypedefData.get;
fun put_info name info = TypedefData.map (Symtab.update (name, info));
(* prepare_typedef *)
fun declare_type_name a = Variable.declare_constraints (Logic.mk_type (TFree (a, dummyS)));
structure TypedefInterpretation = InterpretationFun(type T = string val eq = op =);
val interpretation = TypedefInterpretation.interpretation;
fun prepare_typedef prep_term def name (t, vs, mx) raw_set opt_morphs thy =
let
val _ = Theory.requires thy "Typedef" "typedefs";
val ctxt = ProofContext.init thy;
val full = Sign.full_name thy;
val full_name = full name;
val bname = Binding.name_of name;
(*rhs*)
val set = prep_term (ctxt |> fold declare_type_name vs) raw_set;
val setT = Term.fastype_of set;
val rhs_tfrees = Term.add_tfrees set [];
val rhs_tfreesT = Term.add_tfreesT setT [];
val oldT = HOLogic.dest_setT setT handle TYPE _ =>
error ("Not a set type: " ^ quote (Syntax.string_of_typ ctxt setT));
(*lhs*)
val defS = Sign.defaultS thy;
val lhs_tfrees = map (fn v => (v, the_default defS (AList.lookup (op =) rhs_tfrees v))) vs;
val args_setT = lhs_tfrees
|> filter (member (op =) rhs_tfrees andf (not o member (op =) rhs_tfreesT))
|> map TFree;
val tname = Binding.map_name (Syntax.type_name mx) t;
val full_tname = full tname;
val newT = Type (full_tname, map TFree lhs_tfrees);
val (Rep_name, Abs_name) =
(case opt_morphs of
NONE => (Binding.prefix_name "Rep_" name, Binding.prefix_name "Abs_" name)
| SOME morphs => morphs);
val setT' = map Term.itselfT args_setT ---> setT;
val setC = Term.list_comb (Const (full_name, setT'), map Logic.mk_type args_setT);
val RepC = Const (full Rep_name, newT --> oldT);
val AbsC = Const (full Abs_name, oldT --> newT);
(*inhabitance*)
fun mk_inhabited A =
HOLogic.mk_Trueprop (HOLogic.mk_exists ("x", oldT, HOLogic.mk_mem (Free ("x", oldT), A)));
val set' = if def then setC else set;
val goal' = mk_inhabited set';
val goal = mk_inhabited set;
val goal_pat = mk_inhabited (Var (the_default (bname, 0) (Syntax.read_variable bname), setT));
(*axiomatization*)
val typedef_name = Binding.prefix_name "type_definition_" name;
val typedefC =
Const (@{const_name type_definition},
(newT --> oldT) --> (oldT --> newT) --> setT --> HOLogic.boolT);
val typedef_prop = Logic.mk_implies (goal', HOLogic.mk_Trueprop (typedefC $ RepC $ AbsC $ set'));
val typedef_deps = Term.add_consts set' [];
(*set definition*)
fun add_def theory =
if def then
theory
|> Sign.add_consts_i [(name, setT', NoSyn)]
|> PureThy.add_defs false [Thm.no_attributes (apfst (Binding.name)
(PrimitiveDefs.mk_defpair (setC, set)))]
|-> (fn [th] => pair (SOME th))
else (NONE, theory);
fun contract_def NONE th = th
| contract_def (SOME def_eq) th =
let
val cert = Thm.cterm_of (Thm.theory_of_thm def_eq);
val goal_eq = MetaSimplifier.rewrite true [def_eq] (cert goal');
in Drule.standard (Drule.equal_elim_rule2 OF [goal_eq, th]) end;
fun typedef_result inhabited =
ObjectLogic.typedecl (t, vs, mx)
#> snd
#> Sign.add_consts_i
[(Rep_name, newT --> oldT, NoSyn),
(Abs_name, oldT --> newT, NoSyn)]
#> add_def
#-> (fn set_def =>
PureThy.add_axioms [((typedef_name, typedef_prop),
[Thm.rule_attribute (K (fn cond_axm => contract_def set_def inhabited RS cond_axm))])]
##>> pair set_def)
##> Theory.add_deps "" (dest_Const RepC) typedef_deps
##> Theory.add_deps "" (dest_Const AbsC) typedef_deps
#-> (fn ([type_definition], set_def) => fn thy1 =>
let
fun make th = Drule.standard (th OF [type_definition]);
val ([Rep, Rep_inverse, Abs_inverse, Rep_inject, Abs_inject,
Rep_cases, Abs_cases, Rep_induct, Abs_induct], thy2) =
thy1
|> Sign.add_path (Binding.name_of name)
|> PureThy.add_thms
[((Rep_name, make @{thm type_definition.Rep}), []),
((Binding.suffix_name "_inverse" Rep_name, make @{thm type_definition.Rep_inverse}), []),
((Binding.suffix_name "_inverse" Abs_name, make @{thm type_definition.Abs_inverse}), []),
((Binding.suffix_name "_inject" Rep_name, make @{thm type_definition.Rep_inject}), []),
((Binding.suffix_name "_inject" Abs_name, make @{thm type_definition.Abs_inject}), []),
((Binding.suffix_name "_cases" Rep_name, make @{thm type_definition.Rep_cases}),
[RuleCases.case_names [Binding.name_of Rep_name], Induct.cases_pred full_name]),
((Binding.suffix_name "_cases" Abs_name, make @{thm type_definition.Abs_cases}),
[RuleCases.case_names [Binding.name_of Abs_name], Induct.cases_type full_tname]),
((Binding.suffix_name "_induct" Rep_name, make @{thm type_definition.Rep_induct}),
[RuleCases.case_names [Binding.name_of Rep_name], Induct.induct_pred full_name]),
((Binding.suffix_name "_induct" Abs_name, make @{thm type_definition.Abs_induct}),
[RuleCases.case_names [Binding.name_of Abs_name], Induct.induct_type full_tname])]
||> Sign.parent_path;
val info = {rep_type = oldT, abs_type = newT,
Rep_name = full Rep_name, Abs_name = full Abs_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
thy2
|> put_info full_tname info
|> TypedefInterpretation.data full_tname
|> pair (full_tname, info)
end);
(* errors *)
fun show_names pairs = commas_quote (map fst pairs);
val illegal_vars =
if null (Term.add_vars set []) andalso null (Term.add_tvars set []) then []
else ["Illegal schematic variable(s) on rhs"];
val dup_lhs_tfrees =
(case duplicates (op =) lhs_tfrees of [] => []
| dups => ["Duplicate type variables on lhs: " ^ show_names dups]);
val extra_rhs_tfrees =
(case fold (remove (op =)) lhs_tfrees rhs_tfrees of [] => []
| extras => ["Extra type variables on rhs: " ^ show_names extras]);
val illegal_frees =
(case Term.add_frees set [] of [] => []
| xs => ["Illegal variables on rhs: " ^ show_names xs]);
val errs = illegal_vars @ dup_lhs_tfrees @ extra_rhs_tfrees @ illegal_frees;
val _ = if null errs then () else error (cat_lines errs);
(*test theory errors now!*)
val test_thy = Theory.copy thy;
val _ = test_thy
|> typedef_result (setmp quick_and_dirty true (SkipProof.make_thm test_thy) goal);
in (set, goal, goal_pat, typedef_result) end
handle ERROR msg =>
cat_error msg ("The error(s) above occurred in typedef " ^ quote (Binding.str_of name));
(* add_typedef: tactic interface *)
fun add_typedef def opt_name typ set opt_morphs tac thy =
let
val name = the_default (#1 typ) opt_name;
val (set, goal, _, typedef_result) =
prepare_typedef Syntax.check_term def name typ set opt_morphs thy;
val inhabited = Goal.prove_global thy [] [] goal (K tac)
handle ERROR msg => cat_error msg
("Failed to prove non-emptiness of " ^ quote (Syntax.string_of_term_global thy set));
in typedef_result inhabited thy end;
(* typedef: proof interface *)
local
fun gen_typedef prep_term ((def, name), typ, set, opt_morphs) thy =
let
val (_, goal, goal_pat, typedef_result) =
prepare_typedef prep_term def name typ set opt_morphs thy;
fun after_qed [[th]] = ProofContext.theory (snd o typedef_result th);
in Proof.theorem_i NONE after_qed [[(goal, [goal_pat])]] (ProofContext.init thy) end;
in
val typedef = gen_typedef Syntax.check_term;
val typedef_cmd = gen_typedef Syntax.read_term;
end;
(** outer syntax **)
local structure P = OuterParse in
val _ = OuterKeyword.keyword "morphisms";
val typedef_decl =
Scan.optional (P.$$$ "(" |--
((P.$$$ "open" >> K false) -- Scan.option P.binding || P.binding >> (fn s => (true, SOME s)))
--| P.$$$ ")") (true, NONE) --
(P.type_args -- P.binding) -- P.opt_infix -- (P.$$$ "=" |-- P.term) --
Scan.option (P.$$$ "morphisms" |-- P.!!! (P.binding -- P.binding));
fun mk_typedef ((((((def, opt_name), (vs, t)), mx), A), morphs)) =
typedef_cmd ((def, the_default (Binding.map_name (Syntax.type_name mx) t) opt_name),
(t, vs, mx), A, morphs);
val _ =
OuterSyntax.command "typedef" "HOL type definition (requires non-emptiness proof)"
OuterKeyword.thy_goal
(typedef_decl >> (Toplevel.print oo (Toplevel.theory_to_proof o mk_typedef)));
end;
val setup = TypedefInterpretation.init;
end;