src/HOL/Tools/datatype_prop.ML
changeset 5177 0d3a168e4d44
child 5578 7de426cf179c
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Tools/datatype_prop.ML	Fri Jul 24 12:50:06 1998 +0200
@@ -0,0 +1,513 @@
+(*  Title:      HOL/Tools/datatype_prop.ML
+    ID:         $Id$
+    Author:     Stefan Berghofer
+    Copyright   1998  TU Muenchen
+
+Characteristic properties of datatypes
+*)
+
+signature DATATYPE_PROP =
+sig
+  val dtK : int
+  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 : 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 = 7;
+
+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;
+
+    fun index_vnames (vn::vns) tab =
+          (case assoc (tab, vn) of
+             None => if vn mem vns
+                     then (vn ^ "1") :: index_vnames vns ((vn, 2)::tab)
+                     else vn :: index_vnames vns tab
+           | Some i => (vn ^ (string_of_int i))::
+                     index_vnames vns ((vn, i + 1)::tab))
+      | index_vnames [] _ = []
+
+  in index_vnames (map type_name Ts) []
+  end;
+
+(** FIXME: move to hologic.ML ? **)
+val Not = Const ("Not", HOLogic.boolT --> HOLogic.boolT);
+
+(************************* 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
+        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 (fn ((r, s), T) => HOLogic.mk_Trueprop
+          (make_pred (dest_DtRec r) T $ Free (s, T))) (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 sign = sign_of thy;
+
+    val descr' = flat descr;
+    val recTs = get_rec_types descr' sorts;
+
+    val rec_result_Ts = map (fn (i, _) =>
+      TFree ("'t" ^ (string_of_int (i + 1)), HOLogic.termS)) descr';
+
+    val reccomb_fn_Ts = flat (map (fn (i, (_, _, constrs)) =>
+      map (fn (_, cargs) =>
+        let
+          val recs = filter is_rec_type cargs;
+          val argTs = (map (typ_of_dtyp descr' sorts) cargs) @
+            (map (fn r => nth_elem (dest_DtRec r, rec_result_Ts)) 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');
+        val reccombs' = map (fn (DtRec i) => nth_elem (i, reccombs)) recs
+
+      in (ts @ [HOLogic.mk_Trueprop (HOLogic.mk_eq
+        (comb_t $ list_comb (Const (cname, Ts ---> T), frees),
+         list_comb (f, frees @ (map (uncurry ap) (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 newTs = take (length (hd descr), recTs);
+    val T' = TFree ("'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 (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 (Not $ HOLogic.mk_eq (t, t')))::
+                    (HOLogic.mk_Trueprop (Not $ HOLogic.mk_eq (t', t)))::
+                      (make_distincts' constrs')
+                  end
+
+          in (make_distincts' constrs) @ (make_distincts_1 T constrs)
+          end;
+
+    (**** number of constructors >= dtK : t_ord C_i ... = i ****)
+
+    fun make_distincts_2 T tname i constrs =
+      let
+        val ord_name = Sign.intern_const (sign_of thy) (tname ^ "_ord");
+        val ord_t = Const (ord_name, T --> HOLogic.natT)
+
+      in (case constrs of
+          [] => [Logic.mk_implies (HOLogic.mk_Trueprop (Not $ HOLogic.mk_eq
+             (ord_t $ Free ("x", T), ord_t $ Free ("y", T))),
+               HOLogic.mk_Trueprop (Not $ HOLogic.mk_eq
+                 (Free ("x", T), Free ("y", T))))]
+        | ((cname, cargs)::constrs) =>
+            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 (ord_t $
+                list_comb (Const (cname, Ts ---> T), frees), HOLogic.mk_nat i)))::
+                  (make_distincts_2 T tname (i + 1) constrs)
+            end)
+      end;
+
+  in map (fn (((_, (_, _, constrs)), T), tname) =>
+      if length constrs < dtK then make_distincts_1 T constrs
+      else make_distincts_2 T tname 0 constrs)
+        ((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 newTs = take (length (hd descr), recTs);
+    val T' = TFree ("'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 $ (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, 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 = Ast.mk_appl (Constant "@case1")
+          [Ast.mk_appl (Constant (Sign.base_name cname)) xs, t];
+        val ast' = foldr (fn (x, y) =>
+          Ast.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 (Ast.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
+        (Ast.mk_appl (Constant "@case") [Variable "t", ast],
+         Ast.mk_appl (Constant (tname ^ "_case"))
+           (asts @ [Variable "t"]))
+      end
+
+  in
+    map mk_trrule (hd descr ~~ new_type_names)
+  end;
+
+(******************************* size functions *******************************)
+
+fun make_size new_type_names descr sorts thy =
+  let
+    val descr' = flat descr;
+    val recTs = get_rec_types descr' sorts;
+
+    val big_size_name = space_implode "_" new_type_names ^ "_size";
+    val size_name = Sign.intern_const (sign_of (the (get_thy "Arith" thy))) "size";
+    val size_names = replicate (length (hd descr)) size_name @
+      map (Sign.intern_const (sign_of thy))
+        (if length (flat (tl descr)) = 1 then [big_size_name] else
+          map (fn i => big_size_name ^ "_" ^ string_of_int i)
+            (1 upto length (flat (tl descr))));
+    val size_consts = map (fn (s, T) =>
+      Const (s, T --> HOLogic.natT)) (size_names ~~ recTs);
+
+    val plus_t = Const ("op +", [HOLogic.natT, HOLogic.natT] ---> HOLogic.natT);
+
+    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 (app plus_t) (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 ***************************)
+
+(*---------------------------------------------------------------------------
+ * 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;