--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Tools/Old_Datatype/old_datatype_prop.ML Mon Sep 01 16:17:46 2014 +0200
@@ -0,0 +1,428 @@
+(* Title: HOL/Tools/Old_Datatype/old_datatype_prop.ML
+ Author: Stefan Berghofer, TU Muenchen
+
+Datatype package: characteristic properties of datatypes.
+*)
+
+signature OLD_DATATYPE_PROP =
+sig
+ type descr = Old_Datatype_Aux.descr
+ val indexify_names: string list -> string list
+ val make_tnames: typ list -> string list
+ val make_injs : descr list -> term list list
+ val make_distincts : descr list -> term list list (*no symmetric inequalities*)
+ val make_ind : descr list -> term
+ val make_casedists : descr list -> term list
+ val make_primrec_Ts : descr list -> string list -> typ list * typ list
+ val make_primrecs : string list -> descr list -> theory -> term list
+ val make_cases : string list -> descr list -> theory -> term list list
+ val make_splits : string list -> descr list -> theory -> (term * term) list
+ val make_case_combs : string list -> descr list -> theory -> string -> term list
+ val make_case_cong_weaks : string list -> descr list -> theory -> term list
+ val make_case_congs : string list -> descr list -> theory -> term list
+ val make_nchotomys : descr list -> term list
+end;
+
+structure Old_Datatype_Prop : OLD_DATATYPE_PROP =
+struct
+
+type descr = Old_Datatype_Aux.descr;
+
+
+val indexify_names = Case_Translation.indexify_names;
+val make_tnames = Case_Translation.make_tnames;
+
+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 Symbol_Pos.is_identifier name' then name' else "x" end;
+ in indexify_names (map type_name Ts) end;
+
+
+(************************* injectivity of constructors ************************)
+
+fun make_injs descr =
+ let
+ val descr' = flat descr;
+ fun make_inj T (cname, cargs) =
+ if null cargs then I
+ else
+ let
+ val Ts = map (Old_Datatype_Aux.typ_of_dtyp descr') cargs;
+ val constr_t = Const (cname, Ts ---> T);
+ val tnames = make_tnames Ts;
+ val frees = map Free (tnames ~~ Ts);
+ val frees' = map Free (map (suffix "'") 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)) (Old_Datatype_Aux.get_rec_types descr'))
+ end;
+
+
+(************************* distinctness of constructors ***********************)
+
+fun make_distincts descr =
+ let
+ val descr' = flat descr;
+ val recTs = Old_Datatype_Aux.get_rec_types descr';
+ val newTs = take (length (hd descr)) recTs;
+
+ fun prep_constr (cname, cargs) = (cname, map (Old_Datatype_Aux.typ_of_dtyp descr') 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 =>
+ make_distincts' T (map prep_constr constrs)) (hd descr) newTs
+ end;
+
+
+(********************************* induction **********************************)
+
+fun make_ind descr =
+ let
+ val descr' = flat descr;
+ val recTs = Old_Datatype_Aux.get_rec_types descr';
+ 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
+ Logic.list_all (map (pair "x") Us,
+ HOLogic.mk_Trueprop
+ (make_pred (Old_Datatype_Aux.body_index dt) U $
+ Old_Datatype_Aux.app_bnds (Free (s, T)) (length Us)))
+ end;
+
+ val recs = filter Old_Datatype_Aux.is_rec_type cargs;
+ val Ts = map (Old_Datatype_Aux.typ_of_dtyp descr') cargs;
+ val recTs' = map (Old_Datatype_Aux.typ_of_dtyp descr') recs;
+ val tnames = Name.variant_list pnames (make_tnames Ts);
+ val rec_tnames = map fst (filter (Old_Datatype_Aux.is_rec_type o snd) (tnames ~~ cargs));
+ val frees = tnames ~~ Ts;
+ val prems = map mk_prem (recs ~~ rec_tnames ~~ recTs');
+ in
+ fold_rev (Logic.all o 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 =
+ let
+ val descr' = flat descr;
+
+ fun make_casedist_prem T (cname, cargs) =
+ let
+ val Ts = map (Old_Datatype_Aux.typ_of_dtyp descr') cargs;
+ val frees = Name.variant_list ["P", "y"] (make_tnames Ts) ~~ Ts;
+ val free_ts = map Free frees;
+ in
+ fold_rev (Logic.all o 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)) (Old_Datatype_Aux.get_rec_types descr'))
+ end;
+
+(*************** characteristic equations for primrec combinator **************)
+
+fun make_primrec_Ts descr used =
+ let
+ val descr' = flat descr;
+
+ val rec_result_Ts =
+ map TFree
+ (Name.variant_list used (replicate (length descr') "'t") ~~
+ replicate (length descr') @{sort type});
+
+ val reccomb_fn_Ts = maps (fn (i, (_, _, constrs)) =>
+ map (fn (_, cargs) =>
+ let
+ val Ts = map (Old_Datatype_Aux.typ_of_dtyp descr') cargs;
+ val recs = filter (Old_Datatype_Aux.is_rec_type o fst) (cargs ~~ Ts);
+
+ fun mk_argT (dt, T) =
+ binder_types T ---> nth rec_result_Ts (Old_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 reccomb_names descr thy =
+ let
+ val descr' = flat descr;
+ val recTs = Old_Datatype_Aux.get_rec_types descr';
+ val used = fold Term.add_tfree_namesT recTs [];
+
+ val (rec_result_Ts, reccomb_fn_Ts) = make_primrec_Ts descr used;
+
+ val rec_fns =
+ map (uncurry (Old_Datatype_Aux.mk_Free "f"))
+ (reccomb_fn_Ts ~~ (1 upto (length reccomb_fn_Ts)));
+
+ 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 Old_Datatype_Aux.is_rec_type cargs;
+ val Ts = map (Old_Datatype_Aux.typ_of_dtyp descr') cargs;
+ val recTs' = map (Old_Datatype_Aux.typ_of_dtyp descr') recs;
+ val tnames = make_tnames Ts;
+ val rec_tnames = map fst (filter (Old_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
+ fold_rev (Term.abs o pair "x") Us
+ (nth reccombs (Old_Datatype_Aux.body_index dt) $
+ Old_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 case_names descr thy fname =
+ let
+ val descr' = flat descr;
+ val recTs = Old_Datatype_Aux.get_rec_types descr';
+ val used = fold Term.add_tfree_namesT recTs [];
+ val newTs = take (length (hd descr)) recTs;
+ val T' = TFree (singleton (Name.variant_list used) "'t", @{sort type});
+
+ val case_fn_Ts = map (fn (i, (_, _, constrs)) =>
+ map (fn (_, cargs) =>
+ let val Ts = map (Old_Datatype_Aux.typ_of_dtyp descr') cargs
+ in Ts ---> T' end) constrs) (hd descr);
+ in
+ map (fn ((name, Ts), T) => list_comb
+ (Const (name, Ts @ [T] ---> T'),
+ map (uncurry (Old_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 case_names descr thy =
+ let
+ val descr' = flat descr;
+ val recTs = Old_Datatype_Aux.get_rec_types descr';
+ val newTs = take (length (hd descr)) recTs;
+
+ fun make_case T comb_t ((cname, cargs), f) =
+ let
+ val Ts = map (Old_Datatype_Aux.typ_of_dtyp descr') 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 case_names descr thy "f")
+ end;
+
+
+(*************************** the "split" - equations **************************)
+
+fun make_splits case_names descr thy =
+ let
+ val descr' = flat descr;
+ val recTs = Old_Datatype_Aux.get_rec_types descr';
+ val used' = fold Term.add_tfree_namesT recTs [];
+ val newTs = take (length (hd descr)) recTs;
+ val T' = TFree (singleton (Name.variant_list used') "'t", @{sort type});
+ 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 (Old_Datatype_Aux.typ_of_dtyp descr') 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, Old_Datatype_Aux.mk_conj t1s)),
+ HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, HOLogic.Not $ Old_Datatype_Aux.mk_disj t2s)))
+ end
+
+ in
+ map make_split (hd descr ~~ newTs ~~ make_case_combs case_names descr thy "f")
+ end;
+
+(************************* additional rules for TFL ***************************)
+
+fun make_case_cong_weaks case_names descr thy =
+ let
+ val case_combs = make_case_combs case_names descr 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 case_names descr thy =
+ let
+ val case_combs = make_case_combs case_names descr thy "f";
+ val case_combs' = make_case_combs case_names descr 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
+ fold_rev Logic.all frees
+ (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 =
+ let
+ val descr' = flat descr;
+ val recTs = Old_Datatype_Aux.get_rec_types descr';
+ val newTs = take (length (hd descr)) recTs;
+
+ fun mk_eqn T (cname, cargs) =
+ let
+ val Ts = map (Old_Datatype_Aux.typ_of_dtyp descr') 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, Old_Datatype_Aux.mk_disj (map (mk_eqn T) constrs))))
+ (hd descr ~~ newTs)
+ end;
+
+end;