(* Title: HOL/Tools/Datatype/datatype_prop.ML
Author: Stefan Berghofer, TU Muenchen
Datatype package: characteristic properties of datatypes.
*)
signature DATATYPE_PROP =
sig
include DATATYPE_COMMON
val indexify_names: string list -> string list
val make_tnames: typ list -> string list
val make_injs : descr list -> (string * sort) list -> term list list
val make_distincts : descr list ->
(string * sort) list -> (int * term list) list (*no symmetric inequalities*)
val make_ind : descr list -> (string * sort) list -> term
val make_casedists : descr list -> (string * sort) list -> term list
val make_primrec_Ts : descr list -> (string * sort) list ->
string list -> typ list * typ list
val make_primrecs : string list -> descr list ->
(string * sort) list -> theory -> term list
val make_cases : string list -> descr list ->
(string * sort) list -> theory -> term list list
val make_splits : string list -> descr list ->
(string * sort) list -> theory -> (term * term) list
val make_case_combs : string list -> descr list ->
(string * sort) list -> theory -> string -> term list
val make_weak_case_congs : string list -> descr list ->
(string * sort) list -> theory -> term list
val make_case_congs : string list -> descr list ->
(string * sort) list -> theory -> term list
val make_nchotomys : descr list ->
(string * sort) list -> term list
end;
structure Datatype_Prop : DATATYPE_PROP =
struct
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, _)) = unprefix "'" name
| type_name (Type (name, _)) =
let val name' = Long_Name.base_name name
in if Lexicon.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 (Datatype_Aux.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 @{const_name HOL.conj})
(map HOLogic.mk_eq (frees ~~ frees')))))
end;
in
map2 (fn d => fn T => fold_rev (make_inj T) (#3 (snd d)) [])
(hd descr) (take (length (hd descr)) (Datatype_Aux.get_rec_types descr' sorts))
end;
(************************* distinctness of constructors ***********************)
fun make_distincts descr sorts =
let
val descr' = flat descr;
val recTs = Datatype_Aux.get_rec_types descr' sorts;
val newTs = take (length (hd descr)) recTs;
fun prep_constr (cname, cargs) =
(cname, map (Datatype_Aux.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 (suffix "'") (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' = flat descr;
val recTs = Datatype_Aux.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 (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 (Datatype_Aux.body_index dt) U $
Datatype_Aux.app_bnds (Free (s, T)) (length Us)))
end;
val recs = filter Datatype_Aux.is_rec_type cargs;
val Ts = map (Datatype_Aux.typ_of_dtyp descr' sorts) cargs;
val recTs' = map (Datatype_Aux.typ_of_dtyp descr' sorts) recs;
val tnames = Name.variant_list pnames (make_tnames Ts);
val rec_tnames = map fst (filter (Datatype_Aux.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 =
maps (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 @{const_name HOL.conj})
(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' = flat descr;
fun make_casedist_prem T (cname, cargs) =
let
val Ts = map (Datatype_Aux.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
map2 make_casedist (hd descr)
(take (length (hd descr)) (Datatype_Aux.get_rec_types descr' sorts))
end;
(*************** characteristic equations for primrec combinator **************)
fun make_primrec_Ts descr sorts used =
let
val descr' = flat descr;
val rec_result_Ts =
map TFree
(Name.variant_list used (replicate (length descr') "'t") ~~
replicate (length descr') HOLogic.typeS);
val reccomb_fn_Ts = maps (fn (i, (_, _, constrs)) =>
map (fn (_, cargs) =>
let
val Ts = map (Datatype_Aux.typ_of_dtyp descr' sorts) cargs;
val recs = filter (Datatype_Aux.is_rec_type o fst) (cargs ~~ Ts);
fun mk_argT (dt, T) =
binder_types T ---> nth rec_result_Ts (Datatype_Aux.body_index dt);
val argTs = Ts @ map mk_argT recs
in argTs ---> 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' = flat descr;
val recTs = Datatype_Aux.get_rec_types descr' sorts;
val used = fold Term.add_tfree_namesT recTs [];
val (rec_result_Ts, reccomb_fn_Ts) = make_primrec_Ts descr sorts used;
val rec_fns =
map (uncurry (Datatype_Aux.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 (prefix (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 (cname, cargs) (ts, f :: fs) =
let
val recs = filter Datatype_Aux.is_rec_type cargs;
val Ts = map (Datatype_Aux.typ_of_dtyp descr' sorts) cargs;
val recTs' = map (Datatype_Aux.typ_of_dtyp descr' sorts) recs;
val tnames = make_tnames Ts;
val rec_tnames = map fst (filter (Datatype_Aux.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,
nth reccombs (Datatype_Aux.body_index dt) $ Datatype_Aux.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
fold (fn ((dt, T), comb_t) => fold (make_primrec T comb_t) (#3 (snd dt)))
(descr' ~~ recTs ~~ reccombs) ([], rec_fns)
|> fst
end;
(****************** make terms of form t_case f1 ... fn *********************)
fun make_case_combs new_type_names descr sorts thy fname =
let
val descr' = flat descr;
val recTs = Datatype_Aux.get_rec_types descr' sorts;
val used = fold Term.add_tfree_namesT recTs [];
val newTs = take (length (hd descr)) recTs;
val T' = TFree (singleton (Name.variant_list used) "'t", HOLogic.typeS);
val case_fn_Ts = map (fn (i, (_, _, constrs)) =>
map (fn (_, cargs) =>
let val Ts = map (Datatype_Aux.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 (Datatype_Aux.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' = flat descr;
val recTs = Datatype_Aux.get_rec_types descr' sorts;
val newTs = take (length (hd descr)) recTs;
fun make_case T comb_t ((cname, cargs), f) =
let
val Ts = map (Datatype_Aux.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' = flat descr;
val recTs = Datatype_Aux.get_rec_types descr' sorts;
val used' = fold Term.add_tfree_namesT recTs [];
val newTs = take (length (hd descr)) recTs;
val T' = TFree (singleton (Name.variant_list 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 (Datatype_Aux.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
(fold_rev (fn Free (s, T) => fn t => HOLogic.mk_all (s, T, t)) frees
(HOLogic.imp $ eqn $ P') :: t1s,
fold_rev (fn Free (s, T) => fn t => HOLogic.mk_exists (s, T, t)) frees
(HOLogic.conj $ eqn $ (HOLogic.Not $ P')) :: t2s)
end;
val (t1s, t2s) = fold_rev process_constr (constrs ~~ fs) ([], []);
val lhs = P $ (comb_t $ Free ("x", T));
in
(HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, Datatype_Aux.mk_conj t1s)),
HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, HOLogic.Not $ Datatype_Aux.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 = binder_types (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' = flat descr;
val recTs = Datatype_Aux.get_rec_types descr' sorts;
val newTs = take (length (hd descr)) recTs;
fun mk_eqn T (cname, cargs) =
let
val Ts = map (Datatype_Aux.typ_of_dtyp descr' sorts) cargs;
val tnames = Name.variant_list ["v"] (make_tnames Ts);
val frees = tnames ~~ Ts
in
fold_rev (fn (s, T') => fn t => HOLogic.mk_exists (s, T', t)) frees
(HOLogic.mk_eq (Free ("v", T),
list_comb (Const (cname, Ts ---> T), map Free frees)))
end;
in
map (fn ((_, (_, _, constrs)), T) =>
HOLogic.mk_Trueprop
(HOLogic.mk_all ("v", T, Datatype_Aux.mk_disj (map (mk_eqn T) constrs))))
(hd descr ~~ newTs)
end;
open Datatype_Aux;
end;