(* Title: HOL/Tools/datatype_prop.ML
ID: $Id$
Author: Stefan Berghofer, TU Muenchen
License: GPL (GNU GENERAL PUBLIC LICENSE)
Characteristic properties of datatypes.
*)
signature DATATYPE_PROP =
sig
val dtK : int ref
val indexify_names: string list -> string list
val make_injs : (int * (string * DatatypeAux.dtyp list *
(string * DatatypeAux.dtyp list) list)) list list -> (string * sort) list ->
term list list
val make_ind : (int * (string * DatatypeAux.dtyp list *
(string * DatatypeAux.dtyp list) list)) list list -> (string * sort) list -> term
val make_casedists : (int * (string * DatatypeAux.dtyp list *
(string * DatatypeAux.dtyp list) list)) list list -> (string * sort) list -> term list
val make_primrecs : string list -> (int * (string * DatatypeAux.dtyp list *
(string * DatatypeAux.dtyp list) list)) list list -> (string * sort) list ->
theory -> term list
val make_cases : string list -> (int * (string * DatatypeAux.dtyp list *
(string * DatatypeAux.dtyp list) list)) list list -> (string * sort) list ->
theory -> term list list
val make_distincts : string list -> (int * (string * DatatypeAux.dtyp list *
(string * DatatypeAux.dtyp list) list)) list list -> (string * sort) list ->
theory -> term list list
val make_splits : string list -> (int * (string * DatatypeAux.dtyp list *
(string * DatatypeAux.dtyp list) list)) list list -> (string * sort) list ->
theory -> (term * term) list
val make_case_trrules : string list -> (int * (string * DatatypeAux.dtyp list *
(string * DatatypeAux.dtyp list) list)) list list -> ast Syntax.trrule list
val make_size : (int * (string * DatatypeAux.dtyp list *
(string * DatatypeAux.dtyp list) list)) list list -> (string * sort) list ->
theory -> term list
val make_weak_case_congs : string list -> (int * (string * DatatypeAux.dtyp list *
(string * DatatypeAux.dtyp list) list)) list list -> (string * sort) list ->
theory -> term list
val make_case_congs : string list -> (int * (string * DatatypeAux.dtyp list *
(string * DatatypeAux.dtyp list) list)) list list -> (string * sort) list ->
theory -> term list
val make_nchotomys : (int * (string * DatatypeAux.dtyp list *
(string * DatatypeAux.dtyp list) list)) list list -> (string * sort) list -> term list
end;
structure DatatypeProp : DATATYPE_PROP =
struct
open DatatypeAux;
(*the kind of distinctiveness axioms depends on number of constructors*)
val dtK = ref 7;
fun indexify_names names =
let
fun index (x :: xs) tab =
(case assoc (tab, x) of
None => if x mem xs then (x ^ "1") :: index xs ((x, 2) :: tab) else x :: index xs tab
| Some i => (x ^ Library.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' = Sign.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), injs) =
if null cargs then injs 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 (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')))))::injs
end;
in map (fn (d, T) => foldr (make_inj T) (#3 (snd d), []))
((hd descr) ~~ take (length (hd descr), get_rec_types descr' sorts))
end;
(********************************* induction **********************************)
fun make_ind descr sorts =
let
val descr' = flat 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 (nth_elem (i, pnames), T') end;
fun make_ind_prem k T (cname, cargs) =
let
fun mk_prem ((DtRec k, s), T) = HOLogic.mk_Trueprop
(make_pred k T $ Free (s, T))
| mk_prem ((DtType ("fun", [_, DtRec k]), s), T' as Type ("fun", [T, U])) =
(Const (InductivePackage.inductive_forall_name,
[T --> HOLogic.boolT] ---> HOLogic.boolT) $
Abs ("x", T, make_pred k U $ (Free (s, T') $ Bound 0))) |> HOLogic.mk_Trueprop;
val recs = 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 = variantlist (make_tnames Ts, pnames);
val rec_tnames = map fst (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 = flat (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' = flat descr;
fun make_casedist_prem T (cname, cargs) =
let
val Ts = map (typ_of_dtyp descr' sorts) cargs;
val frees = variantlist (make_tnames Ts, ["P", "y"]) ~~ 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) ~~ take (length (hd descr), get_rec_types descr' sorts))
end;
(*************** characteristic equations for primrec combinator **************)
fun make_primrecs new_type_names descr sorts thy =
let
val o_name = "Fun.op o";
val sign = Theory.sign_of thy;
val descr' = flat descr;
val recTs = get_rec_types descr' sorts;
val used = foldr add_typ_tfree_names (recTs, []);
val rec_result_Ts = map TFree (variantlist (replicate (length descr') "'t", used) ~~
replicate (length descr') HOLogic.termS);
val reccomb_fn_Ts = flat (map (fn (i, (_, _, constrs)) =>
map (fn (_, cargs) =>
let
val Ts = map (typ_of_dtyp descr' sorts) cargs;
val recs = filter (is_rec_type o fst) (cargs ~~ Ts);
fun mk_argT (DtRec k, _) = nth_elem (k, rec_result_Ts)
| mk_argT (DtType ("fun", [_, DtRec k]), Type ("fun", [T, _])) =
T --> nth_elem (k, rec_result_Ts);
val argTs = Ts @ map mk_argT recs
in argTs ---> nth_elem (i, rec_result_Ts)
end) constrs) descr');
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 sign)
(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 = 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 (filter (is_rec_type o snd) (tnames ~~ cargs));
val frees = map Free (tnames ~~ Ts);
val frees' = map Free (rec_tnames ~~ recTs');
fun mk_reccomb (DtRec i, _) = nth_elem (i, reccombs)
| mk_reccomb (DtType ("fun", [_, DtRec i]), Type ("fun", [T, U])) =
let val T' = nth_elem (i, rec_result_Ts)
in Const (o_name, [U --> T', T --> U, T] ---> T') $ nth_elem (i, reccombs)
end;
val reccombs' = map mk_reccomb (recs ~~ recTs')
in (ts @ [HOLogic.mk_Trueprop (HOLogic.mk_eq
(comb_t $ list_comb (Const (cname, Ts ---> T), frees),
list_comb (f, frees @ (map (op $) (reccombs' ~~ frees')))))], fs)
end
in fst (foldl (fn (x, ((dt, T), comb_t)) =>
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' = flat descr;
val recTs = get_rec_types descr' sorts;
val used = foldr add_typ_tfree_names (recTs, []);
val newTs = take (length (hd descr), recTs);
val T' = TFree (variant used "'t", HOLogic.termS);
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 (Theory.sign_of 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' = flat descr;
val recTs = 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 (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;
(************************* distinctness of constructors ***********************)
fun make_distincts new_type_names descr sorts thy =
let
val descr' = flat descr;
val recTs = get_rec_types descr' sorts;
val newTs = take (length (hd descr), recTs);
(**** number of constructors < dtK : C_i ... ~= C_j ... ****)
fun make_distincts_1 _ [] = []
| make_distincts_1 T ((cname, cargs)::constrs) =
let
val Ts = map (typ_of_dtyp descr' sorts) cargs;
val frees = map Free ((make_tnames Ts) ~~ Ts);
val t = list_comb (Const (cname, Ts ---> T), frees);
fun make_distincts' [] = []
| make_distincts' ((cname', cargs')::constrs') =
let
val Ts' = map (typ_of_dtyp descr' sorts) cargs';
val frees' = map Free ((map ((op ^) o (rpair "'"))
(make_tnames Ts')) ~~ Ts');
val t' = list_comb (Const (cname', Ts' ---> T), frees')
in
(HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.mk_eq (t, t')))::
(HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.mk_eq (t', t)))::
(make_distincts' constrs')
end
in (make_distincts' constrs) @ (make_distincts_1 T constrs)
end;
in map (fn (((_, (_, _, constrs)), T), tname) =>
if length constrs < !dtK then make_distincts_1 T constrs else [])
((hd descr) ~~ newTs ~~ new_type_names)
end;
(*************************** the "split" - equations **************************)
fun make_splits new_type_names descr sorts thy =
let
val descr' = flat descr;
val recTs = get_rec_types descr' sorts;
val used' = foldr add_typ_tfree_names (recTs, []);
val newTs = take (length (hd descr), recTs);
val T' = TFree (variant used' "'t", HOLogic.termS);
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 (variantlist (make_tnames Ts, used) ~~ Ts);
val eqn = HOLogic.mk_eq (Free ("x", T),
list_comb (Const (cname, Ts ---> T), frees));
val P' = P $ list_comb (f, frees)
in ((foldr (fn (Free (s, T), t) => HOLogic.mk_all (s, T, t))
(frees, HOLogic.imp $ eqn $ P'))::t1s,
(foldr (fn (Free (s, T), t) => HOLogic.mk_exists (s, T, t))
(frees, HOLogic.conj $ eqn $ (HOLogic.Not $ P')))::t2s)
end;
val (t1s, t2s) = 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;
(************************ translation rules for case **************************)
fun make_case_trrules new_type_names descr =
let
fun mk_asts i j ((cname, cargs)::constrs) =
let
val k = length cargs;
val xs = map (fn i => Variable ("x" ^ string_of_int i)) (i upto i + k - 1);
val t = Variable ("t" ^ string_of_int j);
val ast = Syntax.mk_appl (Constant "_case1")
[Syntax.mk_appl (Constant (Sign.base_name cname)) xs, t];
val ast' = foldr (fn (x, y) =>
Syntax.mk_appl (Constant "_abs") [x, y]) (xs, t)
in
(case constrs of
[] => (ast, [ast'])
| cs => let val (ast'', asts) = mk_asts (i + k) (j + 1) cs
in (Syntax.mk_appl (Constant "_case2") [ast, ast''],
ast'::asts)
end)
end;
fun mk_trrule ((_, (_, _, constrs)), tname) =
let val (ast, asts) = mk_asts 1 1 constrs
in Syntax.ParsePrintRule
(Syntax.mk_appl (Constant "_case_syntax") [Variable "t", ast],
Syntax.mk_appl (Constant (tname ^ "_case"))
(asts @ [Variable "t"]))
end
in
map mk_trrule (hd descr ~~ new_type_names)
end;
(******************************* size functions *******************************)
fun make_size descr sorts thy =
let
val descr' = flat descr;
val recTs = get_rec_types descr' sorts;
val size_name = "Nat.size";
val size_names = replicate (length (hd descr)) size_name @
map (Sign.intern_const (Theory.sign_of thy)) (indexify_names
(map (fn T => name_of_typ T ^ "_size") (drop (length (hd descr), recTs))));
val size_consts = map (fn (s, T) =>
Const (s, T --> HOLogic.natT)) (size_names ~~ recTs);
fun plus (t1, t2) = Const ("op +", [HOLogic.natT, HOLogic.natT] ---> HOLogic.natT) $ t1 $ t2;
fun make_size_eqn size_const T (cname, cargs) =
let
val recs = 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 (filter (is_rec_type o snd) (tnames ~~ cargs));
val ts = map (fn ((r, s), T) => nth_elem (dest_DtRec r, size_consts) $
Free (s, T)) (recs ~~ rec_tnames ~~ recTs);
val t = if ts = [] then HOLogic.zero else
foldl1 plus (ts @ [HOLogic.mk_nat 1])
in
HOLogic.mk_Trueprop (HOLogic.mk_eq (size_const $
list_comb (Const (cname, Ts ---> T), map Free (tnames ~~ Ts)), t))
end
in
flat (map (fn (((_, (_, _, constrs)), size_const), T) =>
map (make_size_eqn size_const T) constrs) (descr' ~~ size_consts ~~ recTs))
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 = variantlist (make_tnames Ts, used);
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 = get_rec_types descr' sorts;
val newTs = take (length (hd descr), recTs);
fun mk_eqn T (cname, cargs) =
let
val Ts = map (typ_of_dtyp descr' sorts) cargs;
val tnames = variantlist (make_tnames Ts, ["v"]);
val frees = tnames ~~ Ts
in
foldr (fn ((s, T'), 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, mk_disj (map (mk_eqn T) constrs))))
(hd descr ~~ newTs)
end;
end;