(* 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 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 list * mixfix) * string
* (binding * binding) option -> theory -> theory
end;
structure Repdef :> REPDEF =
struct
(** type definitions **)
type rep_info =
{ emb_def: thm, prj_def: thm, approx_def: thm, REP: thm };
(* building terms *)
fun adm_const T = Const (@{const_name adm}, (T --> HOLogic.boolT) --> HOLogic.boolT);
fun mk_adm (x, T, P) = adm_const T $ absfree (x, T, P);
fun below_const T = Const (@{const_name below}, T --> T --> HOLogic.boolT);
val natT = @{typ nat};
val udomT = @{typ udom};
fun alg_deflT T = Type (@{type_name alg_defl}, [T]);
fun cfunT (T, U) = Type (@{type_name "->"}, [T, U]);
fun emb_const T = Const (@{const_name emb}, cfunT (T, udomT));
fun prj_const T = Const (@{const_name prj}, cfunT (udomT, T));
fun approx_const T = Const (@{const_name approx}, natT --> cfunT (T, T));
fun LAM_const (T, U) = Const (@{const_name Abs_CFun}, (T --> U) --> cfunT (T, U));
fun APP_const (T, U) = Const (@{const_name Rep_CFun}, cfunT (T, U) --> (T --> U));
fun cast_const T = Const (@{const_name cast}, cfunT (alg_deflT T, cfunT (T, T)));
fun mk_cast (t, x) =
APP_const (udomT, udomT)
$ (APP_const (alg_deflT udomT, cfunT (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 (t, vs, mx) : binding * string 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";
val ctxt = ProofContext.init thy;
(*rhs*)
val defl = prep_term (ctxt |> fold declare_type_name vs) raw_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 ctxt deflT));
val rhs_tfrees = Term.add_tfrees defl [];
(*lhs*)
val defS = Sign.defaultS thy;
val lhs_tfrees = map (fn v => (v, the_default defS (AList.lookup (op =) rhs_tfrees v))) vs;
val lhs_sorts = map snd lhs_tfrees;
val tname = Binding.map_name (Syntax.type_name mx) t;
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), thy2) = 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, LAM_const (newT, udomT) $ Rep_const);
val prj_eqn = Logic.mk_equals (prj_const newT, LAM_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 --> cfunT (newT, newT));
val approx_eqn = Logic.mk_equals (approx_const newT,
repdef_approx_const $ Rep_const $ Abs_const $ defl);
(*instantiate class rep*)
val name_def = Binding.suffix_name "_def" name;
val ([emb_ldef, prj_ldef, approx_ldef], lthy3) = thy2
|> Theory_Target.instantiation ([full_tname], lhs_tfrees, @{sort rep})
|> fold_map Specification.definition
[ (NONE, ((Binding.prefix_name "emb_" name_def, []), emb_eqn))
, (NONE, ((Binding.prefix_name "prj_" name_def, []), prj_eqn))
, (NONE, ((Binding.prefix_name "approx_" name_def, []), approx_eqn)) ]
|>> map (snd o snd);
val ctxt_thy = ProofContext.init (ProofContext.theory_of lthy3);
val [emb_def, prj_def, approx_def] =
ProofContext.export lthy3 ctxt_thy [emb_ldef, prj_ldef, approx_ldef];
val typedef_thms =
[#type_definition info, #below_def cpo_info, emb_def, prj_def, approx_def];
val thy4 = lthy3
|> 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 thy4)
[#type_definition info, #below_def cpo_info, emb_def, prj_def];
val ([REP_thm], thy5) = thy4
|> Sign.add_path (Binding.name_of name)
|> PureThy.add_thms
[((Binding.prefix_name "REP_" name,
Drule.standard (@{thm typedef_REP} OF typedef_thms')), [])]
||> Sign.parent_path;
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), thy5)
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), typ, A, morphs) =
snd o gen_add_repdef Syntax.read_term def name typ A morphs;
(** 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.$$$ ")") (true, NONE) --
(P.type_args -- P.binding) -- P.opt_infix -- (P.$$$ "=" |-- P.term) --
Scan.option (P.$$$ "morphisms" |-- P.!!! (P.binding -- P.binding));
fun mk_repdef ((((((def, opt_name), (vs, t)), mx), A), morphs)) =
repdef_cmd
((def, the_default (Binding.map_name (Syntax.type_name mx) t) opt_name), (t, vs, 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;