(* Title: HOLCF/Tools/repdef.ML
Author: Brian Huffman
Defining representable domains using algebraic deflations.
*)
signature REPDEF =
sig
type rep_info =
{ emb_def: thm, prj_def: thm, approx_def: thm, REP: thm }
val add_repdef: bool -> binding option -> binding * (string * sort) list * mixfix ->
term -> (binding * binding) option -> theory ->
(Typedef.info * Pcpodef.cpo_info * Pcpodef.pcpo_info * rep_info) * theory
val repdef_cmd: (bool * binding) * (binding * (string * string option) list * mixfix) * string
* (binding * binding) option -> theory -> theory
end;
structure Repdef :> REPDEF =
struct
open HOLCF_Library;
infixr 6 ->>;
infix -->>;
(** type definitions **)
type rep_info =
{ emb_def: thm, prj_def: thm, approx_def: thm, REP: thm };
(* building types and terms *)
val udomT = @{typ udom};
fun alg_deflT T = Type (@{type_name alg_defl}, [T]);
fun emb_const T = Const (@{const_name emb}, T ->> udomT);
fun prj_const T = Const (@{const_name prj}, udomT ->> T);
fun approx_const T = Const (@{const_name approx}, natT --> (T ->> T));
fun cast_const T = Const (@{const_name cast}, alg_deflT T ->> T ->> T);
fun mk_cast (t, x) =
capply_const (udomT, udomT)
$ (capply_const (alg_deflT udomT, udomT ->> udomT) $ cast_const udomT $ t)
$ x;
(* manipulating theorems *)
(* proving class instances *)
fun declare_type_name a =
Variable.declare_constraints (Logic.mk_type (TFree (a, dummyS)));
fun gen_add_repdef
(prep_term: Proof.context -> 'a -> term)
(def: bool)
(name: binding)
(typ as (tname, raw_args, mx) : binding * (string * sort) list * mixfix)
(raw_defl: 'a)
(opt_morphs: (binding * binding) option)
(thy: theory)
: (Typedef.info * Pcpodef.cpo_info * Pcpodef.pcpo_info * rep_info) * theory =
let
val _ = Theory.requires thy "Representable" "repdefs";
(*rhs*)
val (_, tmp_lthy) =
thy |> Theory.copy |> Theory_Target.init NONE
|> Typedecl.predeclare_constraints (tname, raw_args, mx);
val defl = prep_term tmp_lthy raw_defl;
val tmp_lthy = tmp_lthy |> Variable.declare_constraints defl;
val deflT = Term.fastype_of defl;
val _ = if deflT = @{typ "udom alg_defl"} then ()
else error ("Not type udom alg_defl: " ^ quote (Syntax.string_of_typ tmp_lthy deflT));
(*lhs*)
val lhs_tfrees = map (fn (a, _) => (a, ProofContext.default_sort tmp_lthy (a, ~1))) raw_args;
val lhs_sorts = map snd lhs_tfrees;
val full_tname = Sign.full_name thy tname;
val newT = Type (full_tname, map TFree lhs_tfrees);
(*morphisms*)
val morphs = opt_morphs
|> the_default (Binding.prefix_name "Rep_" name, Binding.prefix_name "Abs_" name);
(*set*)
val in_defl = @{term "in_deflation :: udom => udom alg_defl => bool"};
val set = HOLogic.Collect_const udomT $ Abs ("x", udomT, in_defl $ Bound 0 $ defl);
(*pcpodef*)
val tac1 = rtac @{thm CollectI} 1 THEN rtac @{thm bottom_in_deflation} 1;
val tac2 = rtac @{thm adm_mem_Collect_in_deflation} 1;
val ((info, cpo_info, pcpo_info), thy) = thy
|> Pcpodef.add_pcpodef def (SOME name) typ set (SOME morphs) (tac1, tac2);
(*definitions*)
val Rep_const = Const (#Rep_name info, newT --> udomT);
val Abs_const = Const (#Abs_name info, udomT --> newT);
val emb_eqn = Logic.mk_equals (emb_const newT, cabs_const (newT, udomT) $ Rep_const);
val prj_eqn = Logic.mk_equals (prj_const newT, cabs_const (udomT, newT) $
Abs ("x", udomT, Abs_const $ mk_cast (defl, Bound 0)));
val repdef_approx_const =
Const (@{const_name repdef_approx}, (newT --> udomT) --> (udomT --> newT)
--> alg_deflT udomT --> natT --> (newT ->> newT));
val approx_eqn = Logic.mk_equals (approx_const newT,
repdef_approx_const $ Rep_const $ Abs_const $ defl);
val name_def = Binding.suffix_name "_def" name;
val emb_bind = (Binding.prefix_name "emb_" name_def, []);
val prj_bind = (Binding.prefix_name "prj_" name_def, []);
val approx_bind = (Binding.prefix_name "approx_" name_def, []);
(*instantiate class rep*)
val lthy = thy
|> Theory_Target.instantiation ([full_tname], lhs_tfrees, @{sort rep});
val ((_, (_, emb_ldef)), lthy) =
Specification.definition (NONE, (emb_bind, emb_eqn)) lthy;
val ((_, (_, prj_ldef)), lthy) =
Specification.definition (NONE, (prj_bind, prj_eqn)) lthy;
val ((_, (_, approx_ldef)), lthy) =
Specification.definition (NONE, (approx_bind, approx_eqn)) lthy;
val ctxt_thy = ProofContext.init (ProofContext.theory_of lthy);
val emb_def = singleton (ProofContext.export lthy ctxt_thy) emb_ldef;
val prj_def = singleton (ProofContext.export lthy ctxt_thy) prj_ldef;
val approx_def = singleton (ProofContext.export lthy ctxt_thy) approx_ldef;
val type_definition_thm =
MetaSimplifier.rewrite_rule
(the_list (#set_def info))
(#type_definition info);
val typedef_thms =
[type_definition_thm, #below_def cpo_info, emb_def, prj_def, approx_def];
val thy = lthy
|> Class.prove_instantiation_instance
(K (Tactic.rtac (@{thm typedef_rep_class} OF typedef_thms) 1))
|> Local_Theory.exit_global;
(*other theorems*)
val typedef_thms' = map (Thm.transfer thy)
[type_definition_thm, #below_def cpo_info, emb_def, prj_def];
val (REP_thm, thy) = thy
|> Sign.add_path (Binding.name_of name)
|> PureThy.add_thm
((Binding.prefix_name "REP_" name,
Drule.export_without_context (@{thm typedef_REP} OF typedef_thms')), [])
||> Sign.restore_naming thy;
val rep_info =
{ emb_def = emb_def, prj_def = prj_def, approx_def = approx_def, REP = REP_thm };
in
((info, cpo_info, pcpo_info, rep_info), thy)
end
handle ERROR msg =>
cat_error msg ("The error(s) above occurred in repdef " ^ quote (Binding.str_of name));
fun add_repdef def opt_name typ defl opt_morphs thy =
let
val name = the_default (#1 typ) opt_name;
in
gen_add_repdef Syntax.check_term def name typ defl opt_morphs thy
end;
fun repdef_cmd ((def, name), (b, raw_args, mx), A, morphs) thy =
let
val ctxt = ProofContext.init thy;
val args = map (apsnd (Typedecl.read_constraint ctxt)) raw_args;
in snd (gen_add_repdef Syntax.read_term def name (b, args, mx) A morphs thy) end;
(** outer syntax **)
local structure P = OuterParse and K = OuterKeyword in
val repdef_decl =
Scan.optional (P.$$$ "(" |--
((P.$$$ "open" >> K false) -- Scan.option P.binding || P.binding >> (fn s => (true, SOME s)))
--| P.$$$ ")") (true, NONE) --
(P.type_args_constrained -- P.binding) -- P.opt_mixfix -- (P.$$$ "=" |-- P.term) --
Scan.option (P.$$$ "morphisms" |-- P.!!! (P.binding -- P.binding));
fun mk_repdef ((((((def, opt_name), (args, t)), mx), A), morphs)) =
repdef_cmd ((def, the_default t opt_name), (t, args, mx), A, morphs);
val _ =
OuterSyntax.command "repdef" "HOLCF definition of representable domains" K.thy_decl
(repdef_decl >>
(Toplevel.print oo (Toplevel.theory o mk_repdef)));
end;
end;