--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Tools/datatype_package/datatype_prop.ML Wed Jun 10 15:04:33 2009 +0200
@@ -0,0 +1,435 @@
+(* Title: HOL/Tools/datatype_prop.ML
+ Author: Stefan Berghofer, TU Muenchen
+
+Characteristic properties of datatypes.
+*)
+
+signature DATATYPE_PROP =
+sig
+ val indexify_names: string list -> string list
+ val make_tnames: typ list -> string list
+ val make_injs : DatatypeAux.descr list -> (string * sort) list -> term list list
+ val make_distincts : DatatypeAux.descr list ->
+ (string * sort) list -> (int * term list) list (*no symmetric inequalities*)
+ val make_ind : DatatypeAux.descr list -> (string * sort) list -> term
+ val make_casedists : DatatypeAux.descr list -> (string * sort) list -> term list
+ val make_primrec_Ts : DatatypeAux.descr list -> (string * sort) list ->
+ string list -> typ list * typ list
+ val make_primrecs : string list -> DatatypeAux.descr list ->
+ (string * sort) list -> theory -> term list
+ val make_cases : string list -> DatatypeAux.descr list ->
+ (string * sort) list -> theory -> term list list
+ val make_splits : string list -> DatatypeAux.descr list ->
+ (string * sort) list -> theory -> (term * term) list
+ val make_weak_case_congs : string list -> DatatypeAux.descr list ->
+ (string * sort) list -> theory -> term list
+ val make_case_congs : string list -> DatatypeAux.descr list ->
+ (string * sort) list -> theory -> term list
+ val make_nchotomys : DatatypeAux.descr list ->
+ (string * sort) list -> term list
+end;
+
+structure DatatypeProp : DATATYPE_PROP =
+struct
+
+open DatatypeAux;
+
+fun indexify_names names =
+ let
+ fun index (x :: xs) tab =
+ (case AList.lookup (op =) tab x of
+ NONE => if member (op =) xs x then (x ^ "1") :: index xs ((x, 2) :: tab) else x :: index xs tab
+ | SOME i => (x ^ string_of_int i) :: index xs ((x, i + 1) :: tab))
+ | index [] _ = [];
+ in index names [] end;
+
+fun make_tnames Ts =
+ let
+ fun type_name (TFree (name, _)) = implode (tl (explode name))
+ | type_name (Type (name, _)) =
+ let val name' = Long_Name.base_name name
+ in if Syntax.is_identifier name' then name' else "x" end;
+ in indexify_names (map type_name Ts) end;
+
+
+(************************* injectivity of constructors ************************)
+
+fun make_injs descr sorts =
+ let
+ val descr' = flat descr;
+ fun make_inj T (cname, cargs) =
+ if null cargs then I else
+ let
+ val Ts = map (typ_of_dtyp descr' sorts) cargs;
+ val constr_t = Const (cname, Ts ---> T);
+ val tnames = make_tnames Ts;
+ val frees = map Free (tnames ~~ Ts);
+ val frees' = map Free ((map ((op ^) o (rpair "'")) tnames) ~~ Ts);
+ in cons (HOLogic.mk_Trueprop (HOLogic.mk_eq
+ (HOLogic.mk_eq (list_comb (constr_t, frees), list_comb (constr_t, frees')),
+ foldr1 (HOLogic.mk_binop "op &")
+ (map HOLogic.mk_eq (frees ~~ frees')))))
+ end;
+ in
+ map2 (fn d => fn T => fold_rev (make_inj T) (#3 (snd d)) [])
+ (hd descr) (Library.take (length (hd descr), get_rec_types descr' sorts))
+ end;
+
+
+(************************* distinctness of constructors ***********************)
+
+fun make_distincts descr sorts =
+ let
+ val descr' = flat descr;
+ val recTs = get_rec_types descr' sorts;
+ val newTs = Library.take (length (hd descr), recTs);
+
+ fun prep_constr (cname, cargs) = (cname, map (typ_of_dtyp descr' sorts) cargs);
+
+ fun make_distincts' _ [] = []
+ | make_distincts' T ((cname, cargs)::constrs) =
+ let
+ val frees = map Free ((make_tnames cargs) ~~ cargs);
+ val t = list_comb (Const (cname, cargs ---> T), frees);
+
+ fun make_distincts'' (cname', cargs') =
+ let
+ val frees' = map Free ((map ((op ^) o (rpair "'"))
+ (make_tnames cargs')) ~~ cargs');
+ val t' = list_comb (Const (cname', cargs' ---> T), frees')
+ in
+ HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.mk_eq (t, t'))
+ end
+
+ in map make_distincts'' constrs @ make_distincts' T constrs end;
+
+ in
+ map2 (fn ((_, (_, _, constrs))) => fn T =>
+ (length constrs, make_distincts' T (map prep_constr constrs))) (hd descr) newTs
+ end;
+
+
+(********************************* induction **********************************)
+
+fun make_ind descr sorts =
+ let
+ val descr' = List.concat descr;
+ val recTs = get_rec_types descr' sorts;
+ val pnames = if length descr' = 1 then ["P"]
+ else map (fn i => "P" ^ string_of_int i) (1 upto length descr');
+
+ fun make_pred i T =
+ let val T' = T --> HOLogic.boolT
+ in Free (List.nth (pnames, i), T') end;
+
+ fun make_ind_prem k T (cname, cargs) =
+ let
+ fun mk_prem ((dt, s), T) =
+ let val (Us, U) = strip_type T
+ in list_all (map (pair "x") Us, HOLogic.mk_Trueprop
+ (make_pred (body_index dt) U $ app_bnds (Free (s, T)) (length Us)))
+ end;
+
+ val recs = List.filter is_rec_type cargs;
+ val Ts = map (typ_of_dtyp descr' sorts) cargs;
+ val recTs' = map (typ_of_dtyp descr' sorts) recs;
+ val tnames = Name.variant_list pnames (make_tnames Ts);
+ val rec_tnames = map fst (List.filter (is_rec_type o snd) (tnames ~~ cargs));
+ val frees = tnames ~~ Ts;
+ val prems = map mk_prem (recs ~~ rec_tnames ~~ recTs');
+
+ in list_all_free (frees, Logic.list_implies (prems,
+ HOLogic.mk_Trueprop (make_pred k T $
+ list_comb (Const (cname, Ts ---> T), map Free frees))))
+ end;
+
+ val prems = List.concat (map (fn ((i, (_, _, constrs)), T) =>
+ map (make_ind_prem i T) constrs) (descr' ~~ recTs));
+ val tnames = make_tnames recTs;
+ val concl = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop "op &")
+ (map (fn (((i, _), T), tname) => make_pred i T $ Free (tname, T))
+ (descr' ~~ recTs ~~ tnames)))
+
+ in Logic.list_implies (prems, concl) end;
+
+(******************************* case distinction *****************************)
+
+fun make_casedists descr sorts =
+ let
+ val descr' = List.concat descr;
+
+ fun make_casedist_prem T (cname, cargs) =
+ let
+ val Ts = map (typ_of_dtyp descr' sorts) cargs;
+ val frees = Name.variant_list ["P", "y"] (make_tnames Ts) ~~ Ts;
+ val free_ts = map Free frees
+ in list_all_free (frees, Logic.mk_implies (HOLogic.mk_Trueprop
+ (HOLogic.mk_eq (Free ("y", T), list_comb (Const (cname, Ts ---> T), free_ts))),
+ HOLogic.mk_Trueprop (Free ("P", HOLogic.boolT))))
+ end;
+
+ fun make_casedist ((_, (_, _, constrs)), T) =
+ let val prems = map (make_casedist_prem T) constrs
+ in Logic.list_implies (prems, HOLogic.mk_Trueprop (Free ("P", HOLogic.boolT)))
+ end
+
+ in map make_casedist
+ ((hd descr) ~~ Library.take (length (hd descr), get_rec_types descr' sorts))
+ end;
+
+(*************** characteristic equations for primrec combinator **************)
+
+fun make_primrec_Ts descr sorts used =
+ let
+ val descr' = List.concat descr;
+
+ val rec_result_Ts = map TFree (Name.variant_list used (replicate (length descr') "'t") ~~
+ replicate (length descr') HOLogic.typeS);
+
+ val reccomb_fn_Ts = List.concat (map (fn (i, (_, _, constrs)) =>
+ map (fn (_, cargs) =>
+ let
+ val Ts = map (typ_of_dtyp descr' sorts) cargs;
+ val recs = List.filter (is_rec_type o fst) (cargs ~~ Ts);
+
+ fun mk_argT (dt, T) =
+ binder_types T ---> List.nth (rec_result_Ts, body_index dt);
+
+ val argTs = Ts @ map mk_argT recs
+ in argTs ---> List.nth (rec_result_Ts, i)
+ end) constrs) descr');
+
+ in (rec_result_Ts, reccomb_fn_Ts) end;
+
+fun make_primrecs new_type_names descr sorts thy =
+ let
+ val descr' = List.concat descr;
+ val recTs = get_rec_types descr' sorts;
+ val used = List.foldr OldTerm.add_typ_tfree_names [] recTs;
+
+ val (rec_result_Ts, reccomb_fn_Ts) = make_primrec_Ts descr sorts used;
+
+ val rec_fns = map (uncurry (mk_Free "f"))
+ (reccomb_fn_Ts ~~ (1 upto (length reccomb_fn_Ts)));
+
+ val big_reccomb_name = (space_implode "_" new_type_names) ^ "_rec";
+ val reccomb_names = map (Sign.intern_const thy)
+ (if length descr' = 1 then [big_reccomb_name] else
+ (map ((curry (op ^) (big_reccomb_name ^ "_")) o string_of_int)
+ (1 upto (length descr'))));
+ val reccombs = map (fn ((name, T), T') => list_comb
+ (Const (name, reccomb_fn_Ts @ [T] ---> T'), rec_fns))
+ (reccomb_names ~~ recTs ~~ rec_result_Ts);
+
+ fun make_primrec T comb_t ((ts, f::fs), (cname, cargs)) =
+ let
+ val recs = List.filter is_rec_type cargs;
+ val Ts = map (typ_of_dtyp descr' sorts) cargs;
+ val recTs' = map (typ_of_dtyp descr' sorts) recs;
+ val tnames = make_tnames Ts;
+ val rec_tnames = map fst (List.filter (is_rec_type o snd) (tnames ~~ cargs));
+ val frees = map Free (tnames ~~ Ts);
+ val frees' = map Free (rec_tnames ~~ recTs');
+
+ fun mk_reccomb ((dt, T), t) =
+ let val (Us, U) = strip_type T
+ in list_abs (map (pair "x") Us,
+ List.nth (reccombs, body_index dt) $ app_bnds t (length Us))
+ end;
+
+ val reccombs' = map mk_reccomb (recs ~~ recTs' ~~ frees')
+
+ in (ts @ [HOLogic.mk_Trueprop (HOLogic.mk_eq
+ (comb_t $ list_comb (Const (cname, Ts ---> T), frees),
+ list_comb (f, frees @ reccombs')))], fs)
+ end
+
+ in fst (Library.foldl (fn (x, ((dt, T), comb_t)) =>
+ Library.foldl (make_primrec T comb_t) (x, #3 (snd dt)))
+ (([], rec_fns), descr' ~~ recTs ~~ reccombs))
+ end;
+
+(****************** make terms of form t_case f1 ... fn *********************)
+
+fun make_case_combs new_type_names descr sorts thy fname =
+ let
+ val descr' = List.concat descr;
+ val recTs = get_rec_types descr' sorts;
+ val used = List.foldr OldTerm.add_typ_tfree_names [] recTs;
+ val newTs = Library.take (length (hd descr), recTs);
+ val T' = TFree (Name.variant used "'t", HOLogic.typeS);
+
+ val case_fn_Ts = map (fn (i, (_, _, constrs)) =>
+ map (fn (_, cargs) =>
+ let val Ts = map (typ_of_dtyp descr' sorts) cargs
+ in Ts ---> T' end) constrs) (hd descr);
+
+ val case_names = map (fn s =>
+ Sign.intern_const thy (s ^ "_case")) new_type_names
+ in
+ map (fn ((name, Ts), T) => list_comb
+ (Const (name, Ts @ [T] ---> T'),
+ map (uncurry (mk_Free fname)) (Ts ~~ (1 upto length Ts))))
+ (case_names ~~ case_fn_Ts ~~ newTs)
+ end;
+
+(**************** characteristic equations for case combinator ****************)
+
+fun make_cases new_type_names descr sorts thy =
+ let
+ val descr' = List.concat descr;
+ val recTs = get_rec_types descr' sorts;
+ val newTs = Library.take (length (hd descr), recTs);
+
+ fun make_case T comb_t ((cname, cargs), f) =
+ let
+ val Ts = map (typ_of_dtyp descr' sorts) cargs;
+ val frees = map Free ((make_tnames Ts) ~~ Ts)
+ in HOLogic.mk_Trueprop (HOLogic.mk_eq
+ (comb_t $ list_comb (Const (cname, Ts ---> T), frees),
+ list_comb (f, frees)))
+ end
+
+ in map (fn (((_, (_, _, constrs)), T), comb_t) =>
+ map (make_case T comb_t) (constrs ~~ (snd (strip_comb comb_t))))
+ ((hd descr) ~~ newTs ~~ (make_case_combs new_type_names descr sorts thy "f"))
+ end;
+
+
+(*************************** the "split" - equations **************************)
+
+fun make_splits new_type_names descr sorts thy =
+ let
+ val descr' = List.concat descr;
+ val recTs = get_rec_types descr' sorts;
+ val used' = List.foldr OldTerm.add_typ_tfree_names [] recTs;
+ val newTs = Library.take (length (hd descr), recTs);
+ val T' = TFree (Name.variant used' "'t", HOLogic.typeS);
+ val P = Free ("P", T' --> HOLogic.boolT);
+
+ fun make_split (((_, (_, _, constrs)), T), comb_t) =
+ let
+ val (_, fs) = strip_comb comb_t;
+ val used = ["P", "x"] @ (map (fst o dest_Free) fs);
+
+ fun process_constr (((cname, cargs), f), (t1s, t2s)) =
+ let
+ val Ts = map (typ_of_dtyp descr' sorts) cargs;
+ val frees = map Free (Name.variant_list used (make_tnames Ts) ~~ Ts);
+ val eqn = HOLogic.mk_eq (Free ("x", T),
+ list_comb (Const (cname, Ts ---> T), frees));
+ val P' = P $ list_comb (f, frees)
+ in ((List.foldr (fn (Free (s, T), t) => HOLogic.mk_all (s, T, t))
+ (HOLogic.imp $ eqn $ P') frees)::t1s,
+ (List.foldr (fn (Free (s, T), t) => HOLogic.mk_exists (s, T, t))
+ (HOLogic.conj $ eqn $ (HOLogic.Not $ P')) frees)::t2s)
+ end;
+
+ val (t1s, t2s) = List.foldr process_constr ([], []) (constrs ~~ fs);
+ val lhs = P $ (comb_t $ Free ("x", T))
+ in
+ (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, mk_conj t1s)),
+ HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, HOLogic.Not $ mk_disj t2s)))
+ end
+
+ in map make_split ((hd descr) ~~ newTs ~~
+ (make_case_combs new_type_names descr sorts thy "f"))
+ end;
+
+(************************* additional rules for TFL ***************************)
+
+fun make_weak_case_congs new_type_names descr sorts thy =
+ let
+ val case_combs = make_case_combs new_type_names descr sorts thy "f";
+
+ fun mk_case_cong comb =
+ let
+ val Type ("fun", [T, _]) = fastype_of comb;
+ val M = Free ("M", T);
+ val M' = Free ("M'", T);
+ in
+ Logic.mk_implies (HOLogic.mk_Trueprop (HOLogic.mk_eq (M, M')),
+ HOLogic.mk_Trueprop (HOLogic.mk_eq (comb $ M, comb $ M')))
+ end
+ in
+ map mk_case_cong case_combs
+ end;
+
+
+(*---------------------------------------------------------------------------
+ * Structure of case congruence theorem looks like this:
+ *
+ * (M = M')
+ * ==> (!!x1,...,xk. (M' = C1 x1..xk) ==> (f1 x1..xk = g1 x1..xk))
+ * ==> ...
+ * ==> (!!x1,...,xj. (M' = Cn x1..xj) ==> (fn x1..xj = gn x1..xj))
+ * ==>
+ * (ty_case f1..fn M = ty_case g1..gn M')
+ *---------------------------------------------------------------------------*)
+
+fun make_case_congs new_type_names descr sorts thy =
+ let
+ val case_combs = make_case_combs new_type_names descr sorts thy "f";
+ val case_combs' = make_case_combs new_type_names descr sorts thy "g";
+
+ fun mk_case_cong ((comb, comb'), (_, (_, _, constrs))) =
+ let
+ val Type ("fun", [T, _]) = fastype_of comb;
+ val (_, fs) = strip_comb comb;
+ val (_, gs) = strip_comb comb';
+ val used = ["M", "M'"] @ map (fst o dest_Free) (fs @ gs);
+ val M = Free ("M", T);
+ val M' = Free ("M'", T);
+
+ fun mk_clause ((f, g), (cname, _)) =
+ let
+ val (Ts, _) = strip_type (fastype_of f);
+ val tnames = Name.variant_list used (make_tnames Ts);
+ val frees = map Free (tnames ~~ Ts)
+ in
+ list_all_free (tnames ~~ Ts, Logic.mk_implies
+ (HOLogic.mk_Trueprop
+ (HOLogic.mk_eq (M', list_comb (Const (cname, Ts ---> T), frees))),
+ HOLogic.mk_Trueprop
+ (HOLogic.mk_eq (list_comb (f, frees), list_comb (g, frees)))))
+ end
+
+ in
+ Logic.list_implies (HOLogic.mk_Trueprop (HOLogic.mk_eq (M, M')) ::
+ map mk_clause (fs ~~ gs ~~ constrs),
+ HOLogic.mk_Trueprop (HOLogic.mk_eq (comb $ M, comb' $ M')))
+ end
+
+ in
+ map mk_case_cong (case_combs ~~ case_combs' ~~ hd descr)
+ end;
+
+(*---------------------------------------------------------------------------
+ * Structure of exhaustion theorem looks like this:
+ *
+ * !v. (? y1..yi. v = C1 y1..yi) | ... | (? y1..yj. v = Cn y1..yj)
+ *---------------------------------------------------------------------------*)
+
+fun make_nchotomys descr sorts =
+ let
+ val descr' = List.concat descr;
+ val recTs = get_rec_types descr' sorts;
+ val newTs = Library.take (length (hd descr), recTs);
+
+ fun mk_eqn T (cname, cargs) =
+ let
+ val Ts = map (typ_of_dtyp descr' sorts) cargs;
+ val tnames = Name.variant_list ["v"] (make_tnames Ts);
+ val frees = tnames ~~ Ts
+ in
+ List.foldr (fn ((s, T'), t) => HOLogic.mk_exists (s, T', t))
+ (HOLogic.mk_eq (Free ("v", T),
+ list_comb (Const (cname, Ts ---> T), map Free frees))) frees
+ end
+
+ in map (fn ((_, (_, _, constrs)), T) =>
+ HOLogic.mk_Trueprop (HOLogic.mk_all ("v", T, mk_disj (map (mk_eqn T) constrs))))
+ (hd descr ~~ newTs)
+ end;
+
+end;