(* Title: HOLCF/Tools/domain/domain_isomorphism.ML
Author: Brian Huffman
Defines new types satisfying the given domain equations.
*)
signature DOMAIN_ISOMORPHISM =
sig
val domain_isomorphism:
(string list * binding * mixfix * typ) list -> theory -> theory
val domain_isomorphism_cmd:
(string list * binding * mixfix * string) list -> theory -> theory
end;
structure Domain_Isomorphism :> DOMAIN_ISOMORPHISM =
struct
val beta_ss =
HOL_basic_ss
addsimps simp_thms
addsimps [@{thm beta_cfun}]
addsimprocs [@{simproc cont_proc}];
val beta_tac = simp_tac beta_ss;
(******************************************************************************)
(******************************* building types *******************************)
(******************************************************************************)
(* ->> is taken from holcf_logic.ML *)
fun cfunT (T, U) = Type(@{type_name "->"}, [T, U]);
infixr 6 ->>; val (op ->>) = cfunT;
fun dest_cfunT (Type(@{type_name "->"}, [T, U])) = (T, U)
| dest_cfunT T = raise TYPE ("dest_cfunT", [T], []);
fun tupleT [] = HOLogic.unitT
| tupleT [T] = T
| tupleT (T :: Ts) = HOLogic.mk_prodT (T, tupleT Ts);
val deflT = @{typ "udom alg_defl"};
(******************************************************************************)
(******************************* building terms *******************************)
(******************************************************************************)
(* builds the expression (v1,v2,..,vn) *)
fun mk_tuple [] = HOLogic.unit
| mk_tuple (t::[]) = t
| mk_tuple (t::ts) = HOLogic.mk_prod (t, mk_tuple ts);
(* builds the expression (%(v1,v2,..,vn). rhs) *)
fun lambda_tuple [] rhs = Term.lambda (Free("unit", HOLogic.unitT)) rhs
| lambda_tuple (v::[]) rhs = Term.lambda v rhs
| lambda_tuple (v::vs) rhs =
HOLogic.mk_split (Term.lambda v (lambda_tuple vs rhs));
(* continuous application and abstraction *)
fun capply_const (S, T) =
Const(@{const_name Rep_CFun}, (S ->> T) --> (S --> T));
fun cabs_const (S, T) =
Const(@{const_name Abs_CFun}, (S --> T) --> (S ->> T));
fun mk_cabs t =
let val T = Term.fastype_of t
in cabs_const (Term.domain_type T, Term.range_type T) $ t end
(* builds the expression (LAM v. rhs) *)
fun big_lambda v rhs =
cabs_const (Term.fastype_of v, Term.fastype_of rhs) $ Term.lambda v rhs;
(* builds the expression (LAM v1 v2 .. vn. rhs) *)
fun big_lambdas [] rhs = rhs
| big_lambdas (v::vs) rhs = big_lambda v (big_lambdas vs rhs);
fun mk_capply (t, u) =
let val (S, T) =
case Term.fastype_of t of
Type(@{type_name "->"}, [S, T]) => (S, T)
| _ => raise TERM ("mk_capply " ^ ML_Syntax.print_list ML_Syntax.print_term [t, u], [t, u]);
in capply_const (S, T) $ t $ u end;
(* miscellaneous term constructions *)
val mk_trp = HOLogic.mk_Trueprop;
val mk_fst = HOLogic.mk_fst;
val mk_snd = HOLogic.mk_snd;
fun mk_cont t =
let val T = Term.fastype_of t
in Const(@{const_name cont}, T --> HOLogic.boolT) $ t end;
fun mk_fix t =
let val (T, _) = dest_cfunT (Term.fastype_of t)
in mk_capply (Const(@{const_name fix}, (T ->> T) ->> T), t) end;
fun mk_Rep_of T =
Const (@{const_name Rep_of}, Term.itselfT T --> deflT) $ Logic.mk_type T;
(* splits a cterm into the right and lefthand sides of equality *)
fun dest_eqs t = HOLogic.dest_eq (HOLogic.dest_Trueprop t);
fun mk_eqs (t, u) = HOLogic.mk_Trueprop (HOLogic.mk_eq (t, u));
(******************************************************************************)
(*************** fixed-point definitions and unfolding theorems ***************)
(******************************************************************************)
fun add_fixdefs
(spec : (binding * term) list)
(thy : theory) : thm list * theory =
let
val binds = map fst spec;
val (lhss, rhss) = ListPair.unzip (map (dest_eqs o snd) spec);
val functional = lambda_tuple lhss (mk_tuple rhss);
val fixpoint = mk_fix (mk_cabs functional);
(* project components of fixpoint *)
fun mk_projs (x::[]) t = [(x, t)]
| mk_projs (x::xs) t = (x, mk_fst t) :: mk_projs xs (mk_snd t);
val projs = mk_projs lhss fixpoint;
(* convert parameters to lambda abstractions *)
fun mk_eqn (lhs, rhs) =
case lhs of
Const (@{const_name Rep_CFun}, _) $ f $ (x as Free _) =>
mk_eqn (f, big_lambda x rhs)
| Const _ => Logic.mk_equals (lhs, rhs)
| _ => raise TERM ("lhs not of correct form", [lhs, rhs]);
val eqns = map mk_eqn projs;
(* register constant definitions *)
val (fixdef_thms, thy) =
(PureThy.add_defs false o map Thm.no_attributes)
(map (Binding.suffix_name "_def") binds ~~ eqns) thy;
(* prove applied version of definitions *)
fun prove_proj (lhs, rhs) =
let
val tac = rewrite_goals_tac fixdef_thms THEN beta_tac 1;
val goal = Logic.mk_equals (lhs, rhs);
in Goal.prove_global thy [] [] goal (K tac) end;
val proj_thms = map prove_proj projs;
(* mk_tuple lhss == fixpoint *)
fun pair_equalI (thm1, thm2) = @{thm Pair_equalI} OF [thm1, thm2];
val tuple_fixdef_thm = foldr1 pair_equalI proj_thms;
val cont_thm =
Goal.prove_global thy [] [] (mk_trp (mk_cont functional))
(K (beta_tac 1));
val tuple_unfold_thm =
(@{thm def_cont_fix_eq} OF [tuple_fixdef_thm, cont_thm])
|> LocalDefs.unfold (ProofContext.init thy) @{thms split_conv};
fun mk_unfold_thms [] thm = []
| mk_unfold_thms (n::[]) thm = [(n, thm)]
| mk_unfold_thms (n::ns) thm = let
val thmL = thm RS @{thm Pair_eqD1};
val thmR = thm RS @{thm Pair_eqD2};
in (n, thmL) :: mk_unfold_thms ns thmR end;
val unfold_binds = map (Binding.suffix_name "_unfold") binds;
(* register unfold theorems *)
val (unfold_thms, thy) =
(PureThy.add_thms o map (Thm.no_attributes o apsnd Drule.standard))
(mk_unfold_thms unfold_binds tuple_unfold_thm) thy;
in
(unfold_thms, thy)
end;
(******************************************************************************)
fun typ_of_dtyp
(descr : (string * string list) list)
(sorts : (string * sort) list)
: DatatypeAux.dtyp -> typ =
let
fun tfree a = TFree (a, the (AList.lookup (op =) sorts a))
fun typ_of (DatatypeAux.DtTFree a) = tfree a
| typ_of (DatatypeAux.DtType (s, ds)) = Type (s, map typ_of ds)
| typ_of (DatatypeAux.DtRec i) =
let val (s, vs) = nth descr i
in Type (s, map tfree vs) end
in typ_of end;
fun is_closed_dtyp (DatatypeAux.DtTFree a) = false
| is_closed_dtyp (DatatypeAux.DtRec i) = false
| is_closed_dtyp (DatatypeAux.DtType (s, ds)) = forall is_closed_dtyp ds;
(* FIXME: use theory data for this *)
val defl_tab : term Symtab.table =
Symtab.make [(@{type_name "->"}, @{term "cfun_typ"}),
(@{type_name "++"}, @{term "ssum_typ"}),
(@{type_name "**"}, @{term "sprod_typ"}),
(@{type_name "*"}, @{term "cprod_typ"}),
(@{type_name "u"}, @{term "u_typ"}),
(@{type_name "upper_pd"}, @{term "upper_typ"}),
(@{type_name "lower_pd"}, @{term "lower_typ"}),
(@{type_name "convex_pd"}, @{term "convex_typ"})];
fun defl_of_typ
(tab : term Symtab.table)
(free : string -> term)
(T : typ) : term =
let
fun is_closed_typ (Type (_, Ts)) = forall is_closed_typ Ts
| is_closed_typ _ = false;
fun defl_of (TFree (a, _)) = free a
| defl_of (TVar _) = error ("defl_of_typ: TVar")
| defl_of (T as Type (c, Ts)) =
case Symtab.lookup tab c of
SOME t => Library.foldl mk_capply (t, map defl_of Ts)
| NONE => if is_closed_typ T
then mk_Rep_of T
else error ("defl_of_typ: type variable under unsupported type constructor " ^ c);
in defl_of T end;
fun defl_of_dtyp
(descr : (string * string list) list)
(sorts : (string * sort) list)
(f : string -> term)
(r : int -> term)
(dt : DatatypeAux.dtyp) : term =
let
fun tfree a = TFree (a, the (AList.lookup (op =) sorts a))
fun defl_of (DatatypeAux.DtTFree a) = f a
| defl_of (DatatypeAux.DtRec i) = r i
| defl_of (dt as DatatypeAux.DtType (s, ds)) =
case Symtab.lookup defl_tab s of
SOME t => Library.foldl mk_capply (t, map defl_of ds)
| NONE => if DatatypeAux.is_rec_type dt
then error ("defl_of_dtyp: recursion under unsupported type constructor " ^ s)
else if is_closed_dtyp dt
then mk_Rep_of (typ_of_dtyp descr sorts dt)
else error ("defl_of_dtyp: type variable under unsupported type constructor " ^ s);
in defl_of dt end;
(******************************************************************************)
(* prepare datatype specifications *)
fun read_typ thy str sorts =
let
val ctxt = ProofContext.init thy
|> fold (Variable.declare_typ o TFree) sorts;
val T = Syntax.read_typ ctxt str;
in (T, Term.add_tfreesT T sorts) end;
fun cert_typ sign raw_T sorts =
let
val T = Type.no_tvars (Sign.certify_typ sign raw_T)
handle TYPE (msg, _, _) => error msg;
val sorts' = Term.add_tfreesT T sorts;
val _ =
case duplicates (op =) (map fst sorts') of
[] => ()
| dups => error ("Inconsistent sort constraints for " ^ commas dups)
in (T, sorts') end;
fun gen_domain_isomorphism
(prep_typ: theory -> 'a -> (string * sort) list -> typ * (string * sort) list)
(doms_raw: (string list * binding * mixfix * 'a) list)
(thy: theory)
: theory =
let
val _ = Theory.requires thy "Domain" "domain definitions";
(* this theory is used just for parsing *)
val tmp_thy = thy |>
Theory.copy |>
Sign.add_types (map (fn (tvs, tname, mx, _) =>
(tname, length tvs, mx)) doms_raw);
fun prep_dom thy (vs, t, mx, typ_raw) sorts =
let val (typ, sorts') = prep_typ thy typ_raw sorts
in ((vs, t, mx, typ), sorts') end;
val (doms : (string list * binding * mixfix * typ) list,
sorts : (string * sort) list) =
fold_map (prep_dom tmp_thy) doms_raw [];
(* domain equations *)
fun mk_dom_eqn (vs, tbind, mx, rhs) =
let fun arg v = TFree (v, the (AList.lookup (op =) sorts v));
in (Type (Sign.full_name tmp_thy tbind, map arg vs), rhs) end;
val dom_eqns = map mk_dom_eqn doms;
(* check for valid type parameters *)
val (tyvars, _, _, _)::_ = doms;
val new_doms = map (fn (tvs, tname, mx, _) =>
let val full_tname = Sign.full_name tmp_thy tname
in
(case duplicates (op =) tvs of
[] =>
if eq_set (op =) (tyvars, tvs) then (full_tname, tvs)
else error ("Mutually recursive domains must have same type parameters")
| dups => error ("Duplicate parameter(s) for domain " ^ quote (Binding.str_of tname) ^
" : " ^ commas dups))
end) doms;
val dom_names = map fst new_doms;
(* declare type combinator constants *)
fun declare_typ_const (vs, tbind, mx, rhs) thy =
let
val typ_type = Library.foldr cfunT (map (K deflT) vs, deflT);
val typ_bind = Binding.suffix_name "_typ" tbind;
in
Sign.declare_const ((typ_bind, typ_type), NoSyn) thy
end;
val (typ_consts, thy) = fold_map declare_typ_const doms thy;
(* defining equations for type combinators *)
val defl_tab1 = defl_tab; (* FIXME: use theory data *)
val defl_tab2 =
Symtab.make (map (fst o dest_Type o fst) dom_eqns ~~ typ_consts);
val defl_tab' = Symtab.merge (K true) (defl_tab1, defl_tab2);
fun free a = Free (Library.unprefix "'" a, deflT);
fun mk_defl_spec (lhsT, rhsT) =
mk_eqs (defl_of_typ defl_tab' free lhsT,
defl_of_typ defl_tab' free rhsT);
val defl_specs = map mk_defl_spec dom_eqns;
(* register recursive definition of type combinators *)
val typ_binds = map (Binding.suffix_name "_typ" o #2) doms;
val (typ_unfold_thms, thy) = add_fixdefs (typ_binds ~~ defl_specs) thy;
(* define types using deflation combinators *)
fun make_repdef ((vs, tbind, mx, _), typ_const) thy =
let
fun tfree a = TFree (a, the (AList.lookup (op =) sorts a))
val reps = map (mk_Rep_of o tfree) vs;
val defl = Library.foldl mk_capply (typ_const, reps);
val ((_, _, _, {REP, ...}), thy) =
Repdef.add_repdef false NONE (tbind, vs, mx) defl NONE thy;
in
(REP, thy)
end;
val (REP_thms, thy) =
fold_map make_repdef (doms ~~ typ_consts) thy;
(* FIXME: use theory data for this *)
val REP_simps = REP_thms @
@{thms REP_cfun REP_ssum REP_sprod REP_cprod REP_up
REP_upper REP_lower REP_convex};
(* prove REP equations *)
fun mk_REP_eqn_thm (lhsT, rhsT) =
let
val goal = mk_eqs (mk_Rep_of lhsT, mk_Rep_of rhsT);
val tac =
simp_tac (HOL_basic_ss addsimps REP_simps) 1
THEN resolve_tac typ_unfold_thms 1;
in
Goal.prove_global thy [] [] goal (K tac)
end;
val REP_eqn_thms = map mk_REP_eqn_thm dom_eqns;
in
thy
end;
val domain_isomorphism = gen_domain_isomorphism cert_typ;
val domain_isomorphism_cmd = gen_domain_isomorphism read_typ;
(******************************************************************************)
(******************************** outer syntax ********************************)
(******************************************************************************)
local
structure P = OuterParse and K = OuterKeyword
val parse_domain_iso : (string list * binding * mixfix * string) parser =
(P.type_args -- P.binding -- P.opt_infix -- (P.$$$ "=" |-- P.typ))
>> (fn (((vs, t), mx), rhs) => (vs, t, mx, rhs));
val parse_domain_isos = P.and_list1 parse_domain_iso;
in
val _ =
OuterSyntax.command "domain_isomorphism" "define domain isomorphisms (HOLCF)" K.thy_decl
(parse_domain_isos >> (Toplevel.theory o domain_isomorphism_cmd));
end;
end;