(* Title: HOL/Tools/datatype_aux.ML
Author: Stefan Berghofer, TU Muenchen
Auxiliary functions for defining datatypes.
*)
signature DATATYPE_AUX =
sig
val quiet_mode : bool ref
val message : string -> unit
val add_path : bool -> string -> theory -> theory
val parent_path : bool -> theory -> theory
val store_thmss_atts : string -> string list -> attribute list list -> thm list list
-> theory -> thm list list * theory
val store_thmss : string -> string list -> thm list list -> theory -> thm list list * theory
val store_thms_atts : string -> string list -> attribute list list -> thm list
-> theory -> thm list * theory
val store_thms : string -> string list -> thm list -> theory -> thm list * theory
val split_conj_thm : thm -> thm list
val mk_conj : term list -> term
val mk_disj : term list -> term
val app_bnds : term -> int -> term
val cong_tac : int -> tactic
val indtac : thm -> string list -> int -> tactic
val exh_tac : (string -> thm) -> int -> tactic
datatype simproc_dist = FewConstrs of thm list
| ManyConstrs of thm * simpset;
datatype dtyp =
DtTFree of string
| DtType of string * (dtyp list)
| DtRec of int;
type descr
type datatype_info
exception Datatype
exception Datatype_Empty of string
val name_of_typ : typ -> string
val dtyp_of_typ : (string * string list) list -> typ -> dtyp
val mk_Free : string -> typ -> int -> term
val is_rec_type : dtyp -> bool
val typ_of_dtyp : descr -> (string * sort) list -> dtyp -> typ
val dest_DtTFree : dtyp -> string
val dest_DtRec : dtyp -> int
val strip_dtyp : dtyp -> dtyp list * dtyp
val body_index : dtyp -> int
val mk_fun_dtyp : dtyp list -> dtyp -> dtyp
val get_nonrec_types : descr -> (string * sort) list -> typ list
val get_branching_types : descr -> (string * sort) list -> typ list
val get_arities : descr -> int list
val get_rec_types : descr -> (string * sort) list -> typ list
val check_nonempty : descr list -> unit
val unfold_datatypes :
theory -> descr -> (string * sort) list -> datatype_info Symtab.table ->
descr -> int -> descr list * int
end;
structure DatatypeAux : DATATYPE_AUX =
struct
val quiet_mode = ref false;
fun message s = if !quiet_mode then () else writeln s;
fun add_path flat_names s = if flat_names then I else Sign.add_path s;
fun parent_path flat_names = if flat_names then I else Sign.parent_path;
(* store theorems in theory *)
fun store_thmss_atts label tnames attss thmss =
fold_map (fn ((tname, atts), thms) =>
Sign.add_path tname
#> PureThy.add_thmss [((Binding.name label, thms), atts)]
#-> (fn thm::_ => Sign.parent_path #> pair thm)) (tnames ~~ attss ~~ thmss)
##> Theory.checkpoint;
fun store_thmss label tnames = store_thmss_atts label tnames (replicate (length tnames) []);
fun store_thms_atts label tnames attss thmss =
fold_map (fn ((tname, atts), thms) =>
Sign.add_path tname
#> PureThy.add_thms [((Binding.name label, thms), atts)]
#-> (fn thm::_ => Sign.parent_path #> pair thm)) (tnames ~~ attss ~~ thmss)
##> Theory.checkpoint;
fun store_thms label tnames = store_thms_atts label tnames (replicate (length tnames) []);
(* split theorem thm_1 & ... & thm_n into n theorems *)
fun split_conj_thm th =
((th RS conjunct1)::(split_conj_thm (th RS conjunct2))) handle THM _ => [th];
val mk_conj = foldr1 (HOLogic.mk_binop "op &");
val mk_disj = foldr1 (HOLogic.mk_binop "op |");
fun app_bnds t i = list_comb (t, map Bound (i - 1 downto 0));
fun cong_tac i st = (case Logic.strip_assums_concl
(List.nth (prems_of st, i - 1)) of
_ $ (_ $ (f $ x) $ (g $ y)) =>
let
val cong' = Thm.lift_rule (Thm.cprem_of st i) cong;
val _ $ (_ $ (f' $ x') $ (g' $ y')) =
Logic.strip_assums_concl (prop_of cong');
val insts = map (pairself (cterm_of (Thm.theory_of_thm st)) o
apsnd (curry list_abs (Logic.strip_params (concl_of cong'))) o
apfst head_of) [(f', f), (g', g), (x', x), (y', y)]
in compose_tac (false, cterm_instantiate insts cong', 2) i st
handle THM _ => no_tac st
end
| _ => no_tac st);
(* instantiate induction rule *)
fun indtac indrule indnames i st =
let
val ts = HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of indrule));
val ts' = HOLogic.dest_conj (HOLogic.dest_Trueprop
(Logic.strip_imp_concl (List.nth (prems_of st, i - 1))));
val getP = if can HOLogic.dest_imp (hd ts) then
(apfst SOME) o HOLogic.dest_imp else pair NONE;
val flt = if null indnames then I else
filter (fn Free (s, _) => s mem indnames | _ => false);
fun abstr (t1, t2) = (case t1 of
NONE => (case flt (OldTerm.term_frees t2) of
[Free (s, T)] => SOME (absfree (s, T, t2))
| _ => NONE)
| SOME (_ $ t') => SOME (Abs ("x", fastype_of t', abstract_over (t', t2))))
val cert = cterm_of (Thm.theory_of_thm st);
val insts = List.mapPartial (fn (t, u) => case abstr (getP u) of
NONE => NONE
| SOME u' => SOME (t |> getP |> snd |> head_of |> cert, cert u')) (ts ~~ ts');
val indrule' = cterm_instantiate insts indrule
in
rtac indrule' i st
end;
(* perform exhaustive case analysis on last parameter of subgoal i *)
fun exh_tac exh_thm_of i state =
let
val thy = Thm.theory_of_thm state;
val prem = nth (prems_of state) (i - 1);
val params = Logic.strip_params prem;
val (_, Type (tname, _)) = hd (rev params);
val exhaustion = Thm.lift_rule (Thm.cprem_of state i) (exh_thm_of tname);
val prem' = hd (prems_of exhaustion);
val _ $ (_ $ lhs $ _) = hd (rev (Logic.strip_assums_hyp prem'));
val exhaustion' = cterm_instantiate [(cterm_of thy (head_of lhs),
cterm_of thy (List.foldr (fn ((_, T), t) => Abs ("z", T, t))
(Bound 0) params))] exhaustion
in compose_tac (false, exhaustion', nprems_of exhaustion) i state
end;
(* handling of distinctness theorems *)
datatype simproc_dist = FewConstrs of thm list
| ManyConstrs of thm * simpset;
(********************** Internal description of datatypes *********************)
datatype dtyp =
DtTFree of string
| DtType of string * (dtyp list)
| DtRec of int;
(* information about datatypes *)
(* index, datatype name, type arguments, constructor name, types of constructor's arguments *)
type descr = (int * (string * dtyp list * (string * dtyp list) list)) list;
type datatype_info =
{index : int,
alt_names : string list option,
descr : descr,
sorts : (string * sort) list,
rec_names : string list,
rec_rewrites : thm list,
case_name : string,
case_rewrites : thm list,
induction : thm,
exhaustion : thm,
distinct : simproc_dist,
inject : thm list,
nchotomy : thm,
case_cong : thm,
weak_case_cong : thm};
fun mk_Free s T i = Free (s ^ (string_of_int i), T);
fun subst_DtTFree _ substs (T as (DtTFree name)) =
AList.lookup (op =) substs name |> the_default T
| subst_DtTFree i substs (DtType (name, ts)) =
DtType (name, map (subst_DtTFree i substs) ts)
| subst_DtTFree i _ (DtRec j) = DtRec (i + j);
exception Datatype;
exception Datatype_Empty of string;
fun dest_DtTFree (DtTFree a) = a
| dest_DtTFree _ = raise Datatype;
fun dest_DtRec (DtRec i) = i
| dest_DtRec _ = raise Datatype;
fun is_rec_type (DtType (_, dts)) = exists is_rec_type dts
| is_rec_type (DtRec _) = true
| is_rec_type _ = false;
fun strip_dtyp (DtType ("fun", [T, U])) = apfst (cons T) (strip_dtyp U)
| strip_dtyp T = ([], T);
val body_index = dest_DtRec o snd o strip_dtyp;
fun mk_fun_dtyp [] U = U
| mk_fun_dtyp (T :: Ts) U = DtType ("fun", [T, mk_fun_dtyp Ts U]);
fun name_of_typ (Type (s, Ts)) =
let val s' = Long_Name.base_name s
in space_implode "_" (List.filter (not o equal "") (map name_of_typ Ts) @
[if Syntax.is_identifier s' then s' else "x"])
end
| name_of_typ _ = "";
fun dtyp_of_typ _ (TFree (n, _)) = DtTFree n
| dtyp_of_typ _ (TVar _) = error "Illegal schematic type variable(s)"
| dtyp_of_typ new_dts (Type (tname, Ts)) =
(case AList.lookup (op =) new_dts tname of
NONE => DtType (tname, map (dtyp_of_typ new_dts) Ts)
| SOME vs => if map (try (fst o dest_TFree)) Ts = map SOME vs then
DtRec (find_index (curry op = tname o fst) new_dts)
else error ("Illegal occurrence of recursive type " ^ tname));
fun typ_of_dtyp descr sorts (DtTFree a) = TFree (a, (the o AList.lookup (op =) sorts) a)
| typ_of_dtyp descr sorts (DtRec i) =
let val (s, ds, _) = (the o AList.lookup (op =) descr) i
in Type (s, map (typ_of_dtyp descr sorts) ds) end
| typ_of_dtyp descr sorts (DtType (s, ds)) =
Type (s, map (typ_of_dtyp descr sorts) ds);
(* find all non-recursive types in datatype description *)
fun get_nonrec_types descr sorts =
map (typ_of_dtyp descr sorts) (Library.foldl (fn (Ts, (_, (_, _, constrs))) =>
Library.foldl (fn (Ts', (_, cargs)) =>
filter_out is_rec_type cargs union Ts') (Ts, constrs)) ([], descr));
(* get all recursive types in datatype description *)
fun get_rec_types descr sorts = map (fn (_ , (s, ds, _)) =>
Type (s, map (typ_of_dtyp descr sorts) ds)) descr;
(* get all branching types *)
fun get_branching_types descr sorts =
map (typ_of_dtyp descr sorts) (fold (fn (_, (_, _, constrs)) =>
fold (fn (_, cargs) => fold (strip_dtyp #> fst #> fold (insert op =)) cargs)
constrs) descr []);
fun get_arities descr = fold (fn (_, (_, _, constrs)) =>
fold (fn (_, cargs) => fold (insert op =) (map (length o fst o strip_dtyp)
(List.filter is_rec_type cargs))) constrs) descr [];
(* nonemptiness check for datatypes *)
fun check_nonempty descr =
let
val descr' = List.concat descr;
fun is_nonempty_dt is i =
let
val (_, _, constrs) = (the o AList.lookup (op =) descr') i;
fun arg_nonempty (_, DtRec i) = if i mem is then false
else is_nonempty_dt (i::is) i
| arg_nonempty _ = true;
in exists ((forall (arg_nonempty o strip_dtyp)) o snd) constrs
end
in assert_all (fn (i, _) => is_nonempty_dt [i] i) (hd descr)
(fn (_, (s, _, _)) => raise Datatype_Empty s)
end;
(* unfold a list of mutually recursive datatype specifications *)
(* all types of the form DtType (dt_name, [..., DtRec _, ...]) *)
(* need to be unfolded *)
fun unfold_datatypes sign orig_descr sorts (dt_info : datatype_info Symtab.table) descr i =
let
fun typ_error T msg = error ("Non-admissible type expression\n" ^
Syntax.string_of_typ_global sign (typ_of_dtyp (orig_descr @ descr) sorts T) ^ "\n" ^ msg);
fun get_dt_descr T i tname dts =
(case Symtab.lookup dt_info tname of
NONE => typ_error T (tname ^ " is not a datatype - can't use it in\
\ nested recursion")
| (SOME {index, descr, ...}) =>
let val (_, vars, _) = (the o AList.lookup (op =) descr) index;
val subst = ((map dest_DtTFree vars) ~~ dts) handle Library.UnequalLengths =>
typ_error T ("Type constructor " ^ tname ^ " used with wrong\
\ number of arguments")
in (i + index, map (fn (j, (tn, args, cs)) => (i + j,
(tn, map (subst_DtTFree i subst) args,
map (apsnd (map (subst_DtTFree i subst))) cs))) descr)
end);
(* unfold a single constructor argument *)
fun unfold_arg ((i, Ts, descrs), T) =
if is_rec_type T then
let val (Us, U) = strip_dtyp T
in if exists is_rec_type Us then
typ_error T "Non-strictly positive recursive occurrence of type"
else (case U of
DtType (tname, dts) =>
let
val (index, descr) = get_dt_descr T i tname dts;
val (descr', i') = unfold_datatypes sign orig_descr sorts
dt_info descr (i + length descr)
in (i', Ts @ [mk_fun_dtyp Us (DtRec index)], descrs @ descr') end
| _ => (i, Ts @ [T], descrs))
end
else (i, Ts @ [T], descrs);
(* unfold a constructor *)
fun unfold_constr ((i, constrs, descrs), (cname, cargs)) =
let val (i', cargs', descrs') = Library.foldl unfold_arg ((i, [], descrs), cargs)
in (i', constrs @ [(cname, cargs')], descrs') end;
(* unfold a single datatype *)
fun unfold_datatype ((i, dtypes, descrs), (j, (tname, tvars, constrs))) =
let val (i', constrs', descrs') =
Library.foldl unfold_constr ((i, [], descrs), constrs)
in (i', dtypes @ [(j, (tname, tvars, constrs'))], descrs')
end;
val (i', descr', descrs) = Library.foldl unfold_datatype ((i, [],[]), descr);
in (descr' :: descrs, i') end;
end;