(* 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;