src/HOL/typedef.ML

+(*  Title:      HOL/subtype.ML
+    ID:         $Id$
+    Author:     Markus Wenzel, TU Muenchen
+
+Internal interface for subtype definitions.
+*)
+
+signature SUBTYPE =
+sig
+  val prove_nonempty: cterm -> thm list -> tactic option -> thm
+  val add_subtype: string -> string * string list * mixfix ->
+    string -> string list -> thm list -> tactic option -> theory -> theory
+  val add_subtype_i: string -> string * string list * mixfix ->
+    term -> string list -> thm list -> tactic option -> theory -> theory
+end;
+
+structure Subtype: SUBTYPE =
+struct
+
+open Syntax Logic HOLogic;
+
+
+(* prove non-emptyness of a set *)   (*exception ERROR*)
+
+val is_def = is_equals o #prop o rep_thm;
+
+fun prove_nonempty cset thms usr_tac =
+  let
+    val {T = setT, t = set, maxidx, sign} = rep_cterm cset;
+    val T = dest_setT setT;
+    val goal =
+      cterm_of sign (mk_Trueprop (mk_mem (Var (("x", maxidx + 1), T), set)));
+    val tac =
+      TRY (rewrite_goals_tac (filter is_def thms)) THEN
+      TRY (REPEAT_FIRST (resolve_tac (filter_out is_def thms))) THEN
+      if_none usr_tac (TRY (ALLGOALS (fast_tac set_cs)));
+  in
+    prove_goalw_cterm [] goal (K [tac])
+  end
+  handle ERROR =>
+    error ("Failed to prove non-emptyness of " ^ quote (string_of_cterm cset));
+
+
+(* ext_subtype *)
+
+fun ext_subtype prep_term name (t, vs, mx) raw_set axms thms usr_tac thy =
+  let
+    val _ = require_thy thy "Set" "subtype definitions";
+    val sign = sign_of thy;
+
+    (*rhs*)
+    val cset = prep_term sign raw_set;
+    val {T = setT, t = set, ...} = rep_cterm cset;
+    val rhs_tfrees = term_tfrees set;
+    val oldT = dest_setT setT handle TYPE _ =>
+      error ("Not a set type: " ^ quote (Sign.string_of_typ sign setT));
+
+    (*lhs*)
+    val lhs_tfrees =
+      map (fn v => (v, if_none (assoc (rhs_tfrees, v)) termS)) vs;
+
+    val tname = type_name t mx;
+    val tlen = length vs;
+    val newT = Type (tname, map TFree lhs_tfrees);
+
+    val Rep_name = "Rep_" ^ name;
+    val Abs_name = "Abs_" ^ name;
+    val setC = Const (name, setT);
+    val RepC = Const (Rep_name, newT --> oldT);
+    val AbsC = Const (Abs_name, oldT --> newT);
+    val x_new = Free ("x", newT);
+    val y_old = Free ("y", oldT);
+
+    (*axioms*)
+    val rep_type = mk_Trueprop (mk_mem (RepC $ x_new, setC));
+    val rep_type_inv = mk_Trueprop (mk_eq (AbsC $ (RepC $ x_new), x_new));
+    val abs_type_inv = mk_implies (mk_Trueprop (mk_mem (y_old, setC)),
+      mk_Trueprop (mk_eq (RepC $ (AbsC $ y_old), y_old)));
+
+
+    (* errors *)
+
+    val show_names = commas_quote o map fst;
+
+    val illegal_vars =
+      if null (term_vars set) andalso null (term_tvars set) then []
+      else ["Illegal schematic variable(s) on rhs"];
+
+    val dup_lhs_tfrees =
+      (case duplicates lhs_tfrees of [] => []
+      | dups => ["Duplicate type variables on lhs: " ^ show_names dups]);
+
+    val extra_rhs_tfrees =
+      (case gen_rems (op =) (rhs_tfrees, lhs_tfrees) of [] => []
+      | extras => ["Extra type variables on rhs: " ^ show_names extras]);
+
+    val illegal_frees =
+      (case term_frees set of [] => []
+      | xs => ["Illegal variables on rhs: " ^ show_names (map dest_Free xs)]);
+
+    val errs = illegal_vars @ dup_lhs_tfrees @ extra_rhs_tfrees @ illegal_frees;
+  in
+    if null errs then ()
+    else error (cat_lines errs);
+
+    prove_nonempty cset (map (get_axiom thy) axms @ thms) usr_tac;
+
+    thy
+    |> add_types [(t, tlen, mx)]
+    |> add_arities
+     [(tname, replicate tlen logicS, logicS),
+      (tname, replicate tlen termS, termS)]
+    |> add_consts_i
+     [(name, setT, NoSyn),
+      (Rep_name, newT --> oldT, NoSyn),
+      (Abs_name, oldT --> newT, NoSyn)]
+    |> add_defs_i
+     [(name ^ "_def", mk_equals (setC, set))]
+    |> add_axioms_i
+     [(Rep_name, rep_type),
+      (Rep_name ^ "_inverse", rep_type_inv),
+      (Abs_name ^ "_inverse", abs_type_inv)]
+  end
+  handle ERROR =>
+    error ("The error(s) above occurred in subtype definition " ^ quote name);
+
+
+(* external interfaces *)
+
+fun cert_term sg tm =
+  cterm_of sg tm handle TERM (msg, _) => error msg;
+
+fun read_term sg str =
+  read_cterm sg (str, termTVar);
+
+val add_subtype = ext_subtype read_term;
+val add_subtype_i = ext_subtype cert_term;
+
+
+end;
+