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(* Title: HOL/Tools/datatype_rep_proofs.ML
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ID: $Id$
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Author: Stefan Berghofer
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Copyright 1998 TU Muenchen
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Definitional introduction of datatypes
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Proof of characteristic theorems:
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- injectivity of constructors
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- distinctness of constructors (internal version)
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- induction theorem
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*)
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signature DATATYPE_REP_PROOFS =
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sig
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val representation_proofs : DatatypeAux.datatype_info Symtab.table ->
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string list -> (int * (string * DatatypeAux.dtyp list *
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(string * DatatypeAux.dtyp list) list)) list list -> (string * sort) list ->
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(string * mixfix) list -> (string * mixfix) list list -> theory ->
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theory * thm list list * thm list list * thm
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end;
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structure DatatypeRepProofs : DATATYPE_REP_PROOFS =
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struct
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open DatatypeAux;
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val (_ $ (_ $ (_ $ (distinct_f $ _) $ _))) = hd (prems_of distinct_lemma);
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(* figure out internal names *)
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val image_name = Sign.intern_const (sign_of Set.thy) "op ``";
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val UNIV_name = Sign.intern_const (sign_of Set.thy) "UNIV";
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val inj_name = Sign.intern_const (sign_of Fun.thy) "inj";
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val inj_on_name = Sign.intern_const (sign_of Fun.thy) "inj_on";
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val inv_name = Sign.intern_const (sign_of Fun.thy) "inv";
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fun exh_thm_of (dt_info : datatype_info Symtab.table) tname =
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#exhaustion (the (Symtab.lookup (dt_info, tname)));
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(******************************************************************************)
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fun representation_proofs (dt_info : datatype_info Symtab.table)
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new_type_names descr sorts types_syntax constr_syntax thy =
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let
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val Univ_thy = the (get_thy "Univ" thy);
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val node_name = Sign.intern_tycon (sign_of Univ_thy) "node";
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val [In0_name, In1_name, Scons_name, Leaf_name, Numb_name] =
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map (Sign.intern_const (sign_of Univ_thy))
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["In0", "In1", "Scons", "Leaf", "Numb"];
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val [In0_inject, In1_inject, Scons_inject, Leaf_inject, In0_eq, In1_eq,
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In0_not_In1, In1_not_In0] = map (get_thm Univ_thy)
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["In0_inject", "In1_inject", "Scons_inject", "Leaf_inject", "In0_eq",
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"In1_eq", "In0_not_In1", "In1_not_In0"];
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val descr' = flat descr;
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val big_rec_name = (space_implode "_" new_type_names) ^ "_rep_set";
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val rep_set_names = map (Sign.full_name (sign_of thy))
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(if length descr' = 1 then [big_rec_name] else
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(map ((curry (op ^) (big_rec_name ^ "_")) o string_of_int)
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(1 upto (length descr'))));
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val leafTs = get_nonrec_types descr' sorts;
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val recTs = get_rec_types descr' sorts;
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val newTs = take (length (hd descr), recTs);
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val oldTs = drop (length (hd descr), recTs);
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val sumT = if null leafTs then HOLogic.unitT
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else fold_bal (fn (T, U) => Type ("+", [T, U])) leafTs;
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val Univ_elT = HOLogic.mk_setT (Type (node_name, [sumT]));
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val UnivT = HOLogic.mk_setT Univ_elT;
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val In0 = Const (In0_name, Univ_elT --> Univ_elT);
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val In1 = Const (In1_name, Univ_elT --> Univ_elT);
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val Leaf = Const (Leaf_name, sumT --> Univ_elT);
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(* make injections needed for embedding types in leaves *)
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fun mk_inj T' x =
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let
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fun mk_inj' T n i =
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if n = 1 then x else
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let val n2 = n div 2;
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val Type (_, [T1, T2]) = T
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in
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if i <= n2 then
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Const ("Inl", T1 --> T) $ (mk_inj' T1 n2 i)
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else
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Const ("Inr", T2 --> T) $ (mk_inj' T2 (n - n2) (i - n2))
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end
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in mk_inj' sumT (length leafTs) (1 + find_index_eq T' leafTs)
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end;
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(* make injections for constructors *)
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fun mk_univ_inj ts = access_bal (ap In0, ap In1, if ts = [] then
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Const ("arbitrary", Univ_elT)
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else
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foldr1 (HOLogic.mk_binop Scons_name) ts);
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(************** generate introduction rules for representing set **********)
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val _ = writeln "Constructing representing sets...";
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(* make introduction rule for a single constructor *)
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fun make_intr s n (i, (_, cargs)) =
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let
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fun mk_prem (DtRec k, (j, prems, ts)) =
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let val free_t = mk_Free "x" Univ_elT j
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in (j + 1, (HOLogic.mk_mem (free_t,
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Const (nth_elem (k, rep_set_names), UnivT)))::prems, free_t::ts)
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end
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| mk_prem (dt, (j, prems, ts)) =
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let val T = typ_of_dtyp descr' sorts dt
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in (j + 1, prems, (Leaf $ mk_inj T (mk_Free "x" T j))::ts)
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end;
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val (_, prems, ts) = foldr mk_prem (cargs, (1, [], []));
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val concl = HOLogic.mk_Trueprop (HOLogic.mk_mem
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(mk_univ_inj ts n i, Const (s, UnivT)))
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in Logic.list_implies (map HOLogic.mk_Trueprop prems, concl)
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end;
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val consts = map (fn s => Const (s, UnivT)) rep_set_names;
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val intr_ts = flat (map (fn ((_, (_, _, constrs)), rep_set_name) =>
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map (make_intr rep_set_name (length constrs))
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((1 upto (length constrs)) ~~ constrs)) (descr' ~~ rep_set_names));
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val (thy2, {raw_induct = rep_induct, intrs = rep_intrs, ...}) =
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InductivePackage.add_inductive_i false true big_rec_name false true false
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consts intr_ts [] [] thy;
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(********************************* typedef ********************************)
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val tyvars = map (fn (_, (_, Ts, _)) => map dest_DtTFree Ts) (hd descr);
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val thy3 = foldl (fn (thy, ((((name, mx), tvs), c), name')) =>
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TypedefPackage.add_typedef_i_no_def name' (name, tvs, mx) c [] []
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(Some (BREADTH_FIRST (has_fewer_prems 1) (resolve_tac rep_intrs 1))) thy)
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(thy2, types_syntax ~~ tyvars ~~ (take (length newTs, consts)) ~~
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new_type_names);
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(*********************** definition of constructors ***********************)
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val big_rep_name = (space_implode "_" new_type_names) ^ "_Rep_";
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val rep_names = map (curry op ^ "Rep_") new_type_names;
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val rep_names' = map (fn i => big_rep_name ^ (string_of_int i))
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(1 upto (length (flat (tl descr))));
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val all_rep_names = map (Sign.full_name (sign_of thy3)) (rep_names @ rep_names');
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(* isomorphism declarations *)
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val iso_decls = map (fn (T, s) => (s, T --> Univ_elT, NoSyn))
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(oldTs ~~ rep_names');
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(* constructor definitions *)
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fun make_constr_def tname T n ((thy, defs, eqns, i), ((cname, cargs), (cname', mx))) =
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let
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fun constr_arg (dt, (j, l_args, r_args)) =
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let val T = typ_of_dtyp descr' sorts dt;
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val free_t = mk_Free "x" T j
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in (case dt of
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DtRec m => (j + 1, free_t::l_args, (Const (nth_elem (m, all_rep_names),
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T --> Univ_elT) $ free_t)::r_args)
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| _ => (j + 1, free_t::l_args, (Leaf $ mk_inj T free_t)::r_args))
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end;
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val (_, l_args, r_args) = foldr constr_arg (cargs, (1, [], []));
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val constrT = (map (typ_of_dtyp descr' sorts) cargs) ---> T;
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val abs_name = Sign.intern_const (sign_of thy) ("Abs_" ^ tname);
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val rep_name = Sign.intern_const (sign_of thy) ("Rep_" ^ tname);
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val lhs = list_comb (Const (cname, constrT), l_args);
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val rhs = mk_univ_inj r_args n i;
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val def = equals T $ lhs $ (Const (abs_name, Univ_elT --> T) $ rhs);
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val def_name = (Sign.base_name cname) ^ "_def";
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val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq
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(Const (rep_name, T --> Univ_elT) $ lhs, rhs));
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val thy' = thy |>
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Theory.add_consts_i [(cname', constrT, mx)] |>
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Theory.add_defs_i [(def_name, def)];
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in (thy', defs @ [get_axiom thy' def_name], eqns @ [eqn], i + 1)
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end;
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(* constructor definitions for datatype *)
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fun dt_constr_defs ((thy, defs, eqns, rep_congs, dist_lemmas),
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((((_, (_, _, constrs)), tname), T), constr_syntax)) =
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let
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val _ $ (_ $ (cong_f $ _) $ _) = concl_of arg_cong;
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val sg = sign_of thy;
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val rep_const = cterm_of sg
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(Const (Sign.intern_const sg ("Rep_" ^ tname), T --> Univ_elT));
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val cong' = cterm_instantiate [(cterm_of sg cong_f, rep_const)] arg_cong;
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val dist = cterm_instantiate [(cterm_of sg distinct_f, rep_const)] distinct_lemma;
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val (thy', defs', eqns', _) = foldl ((make_constr_def tname T) (length constrs))
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((if length newTs = 1 then thy else Theory.add_path tname thy, defs, [], 1),
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constrs ~~ constr_syntax)
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in
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(if length newTs = 1 then thy' else Theory.parent_path thy', defs', eqns @ [eqns'],
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rep_congs @ [cong'], dist_lemmas @ [dist])
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end;
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val (thy4, constr_defs, constr_rep_eqns, rep_congs, dist_lemmas) = foldl dt_constr_defs
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((Theory.add_consts_i iso_decls thy3, [], [], [], []),
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hd descr ~~ new_type_names ~~ newTs ~~ constr_syntax);
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(*********** isomorphisms for new types (introduced by typedef) ***********)
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val _ = writeln "Proving isomorphism properties...";
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(* get axioms from theory *)
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val newT_iso_axms = map (fn s =>
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(get_axiom thy4 ("Abs_" ^ s ^ "_inverse"),
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get_axiom thy4 ("Rep_" ^ s ^ "_inverse"),
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get_axiom thy4 ("Rep_" ^ s))) new_type_names;
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(*------------------------------------------------*)
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(* prove additional theorems: *)
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(* inj_on dt_Abs_i rep_set_i and inj dt_Rep_i *)
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(*------------------------------------------------*)
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fun prove_newT_iso_inj_thm (((s, (thm1, thm2, _)), T), rep_set_name) =
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let
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val sg = sign_of thy4;
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val RepT = T --> Univ_elT;
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val Rep_name = Sign.intern_const sg ("Rep_" ^ s);
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val AbsT = Univ_elT --> T;
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val Abs_name = Sign.intern_const sg ("Abs_" ^ s);
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val inj_on_Abs_thm = prove_goalw_cterm [] (cterm_of sg
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(HOLogic.mk_Trueprop (Const (inj_on_name, [AbsT, UnivT] ---> HOLogic.boolT) $
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Const (Abs_name, AbsT) $ Const (rep_set_name, UnivT))))
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(fn _ => [rtac inj_on_inverseI 1, etac thm1 1]);
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val inj_Rep_thm = prove_goalw_cterm [] (cterm_of sg
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(HOLogic.mk_Trueprop (Const (inj_name, RepT --> HOLogic.boolT) $
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Const (Rep_name, RepT))))
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(fn _ => [rtac inj_inverseI 1, rtac thm2 1])
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in (inj_on_Abs_thm, inj_Rep_thm) end;
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val newT_iso_inj_thms = map prove_newT_iso_inj_thm
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(new_type_names ~~ newT_iso_axms ~~ newTs ~~
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take (length newTs, rep_set_names));
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(********* isomorphisms between existing types and "unfolded" types *******)
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(*---------------------------------------------------------------------*)
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(* isomorphisms are defined using primrec-combinators: *)
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(* generate appropriate functions for instantiating primrec-combinator *)
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(* *)
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(* e.g. dt_Rep_i = list_rec ... (%h t y. In1 ((Leaf h) $ y)) *)
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(* *)
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(* also generate characteristic equations for isomorphisms *)
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(* *)
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(* e.g. dt_Rep_i (cons h t) = In1 ((dt_Rep_j h) $ (dt_Rep_i t)) *)
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(*---------------------------------------------------------------------*)
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fun make_iso_def k ks n ((fs, eqns, i), (cname, cargs)) =
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let
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val argTs = map (typ_of_dtyp descr' sorts) cargs;
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val T = nth_elem (k, recTs);
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val rep_name = nth_elem (k, all_rep_names);
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val rep_const = Const (rep_name, T --> Univ_elT);
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val constr = Const (cname, argTs ---> T);
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fun process_arg ks' ((i2, i2', ts), dt) =
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let val T' = typ_of_dtyp descr' sorts dt
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in (case dt of
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DtRec j => if j mem ks' then
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(i2 + 1, i2' + 1, ts @ [mk_Free "y" Univ_elT i2'])
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else
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(i2 + 1, i2', ts @ [Const (nth_elem (j, all_rep_names),
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T' --> Univ_elT) $ mk_Free "x" T' i2])
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| _ => (i2 + 1, i2', ts @ [Leaf $ mk_inj T' (mk_Free "x" T' i2)]))
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end;
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val (i2, i2', ts) = foldl (process_arg ks) ((1, 1, []), cargs);
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val xs = map (uncurry (mk_Free "x")) (argTs ~~ (1 upto (i2 - 1)));
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val ys = map (mk_Free "y" Univ_elT) (1 upto (i2' - 1));
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val f = list_abs_free (map dest_Free (xs @ ys), mk_univ_inj ts n i);
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val (_, _, ts') = foldl (process_arg []) ((1, 1, []), cargs);
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val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq
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(rep_const $ list_comb (constr, xs), mk_univ_inj ts' n i))
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in (fs @ [f], eqns @ [eqn], i + 1) end;
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(* define isomorphisms for all mutually recursive datatypes in list ds *)
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fun make_iso_defs (ds, (thy, char_thms)) =
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let
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val ks = map fst ds;
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val (_, (tname, _, _)) = hd ds;
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val {rec_rewrites, rec_names, ...} = the (Symtab.lookup (dt_info, tname));
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fun process_dt ((fs, eqns, isos), (k, (tname, _, constrs))) =
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let
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val (fs', eqns', _) = foldl (make_iso_def k ks (length constrs))
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((fs, eqns, 1), constrs);
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val iso = (nth_elem (k, recTs), nth_elem (k, all_rep_names))
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in (fs', eqns', isos @ [iso]) end;
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val (fs, eqns, isos) = foldl process_dt (([], [], []), ds);
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val fTs = map fastype_of fs;
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val defs = map (fn (rec_name, (T, iso_name)) => ((Sign.base_name iso_name) ^ "_def",
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equals (T --> Univ_elT) $ Const (iso_name, T --> Univ_elT) $
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list_comb (Const (rec_name, fTs @ [T] ---> Univ_elT), fs))) (rec_names ~~ isos);
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val thy' = Theory.add_defs_i defs thy;
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val def_thms = map (get_axiom thy') (map fst defs);
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(* prove characteristic equations *)
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val rewrites = def_thms @ (map mk_meta_eq rec_rewrites);
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val char_thms' = map (fn eqn => prove_goalw_cterm rewrites
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(cterm_of (sign_of thy') eqn) (fn _ => [rtac refl 1])) eqns;
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in (thy', char_thms' @ char_thms) end;
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val (thy5, iso_char_thms) = foldr make_iso_defs (tl descr, (thy4, []));
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(* prove isomorphism properties *)
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(* prove x : dt_rep_set_i --> x : range dt_Rep_i *)
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331 |
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332 |
fun mk_iso_t (((set_name, iso_name), i), T) =
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333 |
let val isoT = T --> Univ_elT
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334 |
in HOLogic.imp $
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335 |
HOLogic.mk_mem (mk_Free "x" Univ_elT i, Const (set_name, UnivT)) $
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336 |
(if i < length newTs then Const ("True", HOLogic.boolT)
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337 |
else HOLogic.mk_mem (mk_Free "x" Univ_elT i,
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338 |
Const (image_name, [isoT, HOLogic.mk_setT T] ---> UnivT) $
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339 |
Const (iso_name, isoT) $ Const (UNIV_name, HOLogic.mk_setT T)))
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340 |
end;
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341 |
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342 |
val iso_t = HOLogic.mk_Trueprop (mk_conj (map mk_iso_t
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343 |
(rep_set_names ~~ all_rep_names ~~ (0 upto (length descr' - 1)) ~~ recTs)));
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344 |
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345 |
val newT_Abs_inverse_thms = map (fn (iso, _, _) => iso RS subst) newT_iso_axms;
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346 |
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347 |
(* all the theorems are proved by one single simultaneous induction *)
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348 |
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349 |
val iso_thms = if length descr = 1 then [] else
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350 |
drop (length newTs, split_conj_thm
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351 |
(prove_goalw_cterm [] (cterm_of (sign_of thy5) iso_t) (fn _ =>
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352 |
[indtac rep_induct 1,
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353 |
REPEAT (rtac TrueI 1),
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354 |
REPEAT (EVERY
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355 |
[REPEAT (etac rangeE 1),
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356 |
REPEAT (eresolve_tac newT_Abs_inverse_thms 1),
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357 |
TRY (hyp_subst_tac 1),
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358 |
rtac (sym RS range_eqI) 1,
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359 |
resolve_tac iso_char_thms 1])])));
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360 |
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361 |
val Abs_inverse_thms = newT_Abs_inverse_thms @ (map (fn r =>
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362 |
r RS mp RS f_inv_f RS subst) iso_thms);
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363 |
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364 |
(* prove inj dt_Rep_i and dt_Rep_i x : dt_rep_set_i *)
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365 |
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366 |
fun prove_iso_thms (ds, (inj_thms, elem_thms)) =
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367 |
let
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368 |
val (_, (tname, _, _)) = hd ds;
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369 |
val {induction, ...} = the (Symtab.lookup (dt_info, tname));
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370 |
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|
371 |
fun mk_ind_concl (i, _) =
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|
372 |
let
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|
373 |
val T = nth_elem (i, recTs);
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|
374 |
val Rep_t = Const (nth_elem (i, all_rep_names), T --> Univ_elT);
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|
375 |
val rep_set_name = nth_elem (i, rep_set_names)
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|
376 |
in (HOLogic.all_const T $ Abs ("y", T, HOLogic.imp $
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|
377 |
HOLogic.mk_eq (Rep_t $ mk_Free "x" T i, Rep_t $ Bound 0) $
|
|
378 |
HOLogic.mk_eq (mk_Free "x" T i, Bound 0)),
|
|
379 |
HOLogic.mk_mem (Rep_t $ mk_Free "x" T i, Const (rep_set_name, UnivT)))
|
|
380 |
end;
|
|
381 |
|
|
382 |
val (ind_concl1, ind_concl2) = ListPair.unzip (map mk_ind_concl ds);
|
|
383 |
|
|
384 |
val rewrites = map mk_meta_eq iso_char_thms;
|
|
385 |
val inj_thms' = map (fn r => r RS injD) inj_thms;
|
|
386 |
|
|
387 |
val inj_thm = prove_goalw_cterm [] (cterm_of (sign_of thy5)
|
|
388 |
(HOLogic.mk_Trueprop (mk_conj ind_concl1))) (fn _ =>
|
|
389 |
[indtac induction 1,
|
|
390 |
REPEAT (EVERY
|
|
391 |
[rtac allI 1, rtac impI 1,
|
|
392 |
exh_tac (exh_thm_of dt_info) 1,
|
|
393 |
REPEAT (EVERY
|
|
394 |
[hyp_subst_tac 1,
|
|
395 |
rewrite_goals_tac rewrites,
|
|
396 |
REPEAT (dresolve_tac [In0_inject, In1_inject] 1),
|
|
397 |
(eresolve_tac [In0_not_In1 RS notE, In1_not_In0 RS notE] 1)
|
|
398 |
ORELSE (EVERY
|
|
399 |
[REPEAT (etac Scons_inject 1),
|
|
400 |
REPEAT (dresolve_tac
|
|
401 |
(inj_thms' @ [Leaf_inject, Inl_inject, Inr_inject]) 1),
|
|
402 |
REPEAT (EVERY [etac allE 1, dtac mp 1, atac 1]),
|
|
403 |
TRY (hyp_subst_tac 1),
|
|
404 |
rtac refl 1])])])]);
|
|
405 |
|
|
406 |
val inj_thms'' = map (fn r =>
|
|
407 |
r RS (allI RS (inj_def RS meta_eq_to_obj_eq RS iffD2)))
|
|
408 |
(split_conj_thm inj_thm);
|
|
409 |
|
|
410 |
val elem_thm = prove_goalw_cterm [] (cterm_of (sign_of thy5)
|
|
411 |
(HOLogic.mk_Trueprop (mk_conj ind_concl2))) (fn _ =>
|
|
412 |
[indtac induction 1,
|
|
413 |
rewrite_goals_tac rewrites,
|
|
414 |
REPEAT (EVERY
|
|
415 |
[resolve_tac rep_intrs 1,
|
|
416 |
REPEAT ((atac 1) ORELSE (resolve_tac elem_thms 1))])]);
|
|
417 |
|
|
418 |
in (inj_thms @ inj_thms'', elem_thms @ (split_conj_thm elem_thm))
|
|
419 |
end;
|
|
420 |
|
|
421 |
val (iso_inj_thms, iso_elem_thms) = foldr prove_iso_thms
|
|
422 |
(tl descr, (map snd newT_iso_inj_thms, map #3 newT_iso_axms));
|
|
423 |
|
|
424 |
(******************* freeness theorems for constructors *******************)
|
|
425 |
|
|
426 |
val _ = writeln "Proving freeness of constructors...";
|
|
427 |
|
|
428 |
(* prove theorem Rep_i (Constr_j ...) = Inj_j ... *)
|
|
429 |
|
|
430 |
fun prove_constr_rep_thm eqn =
|
|
431 |
let
|
|
432 |
val inj_thms = map (fn (r, _) => r RS inj_onD) newT_iso_inj_thms;
|
|
433 |
val rewrites = constr_defs @ (map (mk_meta_eq o #2) newT_iso_axms)
|
|
434 |
in prove_goalw_cterm [] (cterm_of (sign_of thy5) eqn) (fn _ =>
|
|
435 |
[resolve_tac inj_thms 1,
|
|
436 |
rewrite_goals_tac rewrites,
|
|
437 |
rtac refl 1,
|
|
438 |
resolve_tac rep_intrs 2,
|
|
439 |
REPEAT (resolve_tac iso_elem_thms 1)])
|
|
440 |
end;
|
|
441 |
|
|
442 |
(*--------------------------------------------------------------*)
|
|
443 |
(* constr_rep_thms and rep_congs are used to prove distinctness *)
|
|
444 |
(* of constructors internally. *)
|
|
445 |
(* the external version uses dt_case which is not defined yet *)
|
|
446 |
(*--------------------------------------------------------------*)
|
|
447 |
|
|
448 |
val constr_rep_thms = map (map prove_constr_rep_thm) constr_rep_eqns;
|
|
449 |
|
|
450 |
val dist_rewrites = map (fn (rep_thms, dist_lemma) =>
|
|
451 |
dist_lemma::(rep_thms @ [In0_eq, In1_eq, In0_not_In1, In1_not_In0]))
|
|
452 |
(constr_rep_thms ~~ dist_lemmas);
|
|
453 |
|
|
454 |
(* prove injectivity of constructors *)
|
|
455 |
|
|
456 |
fun prove_constr_inj_thm rep_thms t =
|
|
457 |
let val inj_thms = Scons_inject::(map make_elim
|
|
458 |
((map (fn r => r RS injD) iso_inj_thms) @
|
|
459 |
[In0_inject, In1_inject, Leaf_inject, Inl_inject, Inr_inject]))
|
|
460 |
in prove_goalw_cterm [] (cterm_of (sign_of thy5) t) (fn _ =>
|
|
461 |
[rtac iffI 1,
|
|
462 |
REPEAT (etac conjE 2), hyp_subst_tac 2, rtac refl 2,
|
|
463 |
dresolve_tac rep_congs 1, dtac box_equals 1,
|
|
464 |
REPEAT (resolve_tac rep_thms 1),
|
|
465 |
REPEAT (eresolve_tac inj_thms 1),
|
|
466 |
hyp_subst_tac 1,
|
|
467 |
REPEAT (resolve_tac [conjI, refl] 1)])
|
|
468 |
end;
|
|
469 |
|
|
470 |
val constr_inject = map (fn (ts, thms) => map (prove_constr_inj_thm thms) ts)
|
|
471 |
((DatatypeProp.make_injs descr sorts) ~~ constr_rep_thms);
|
|
472 |
|
|
473 |
val thy6 = store_thmss "inject" new_type_names constr_inject thy5;
|
|
474 |
|
|
475 |
(*************************** induction theorem ****************************)
|
|
476 |
|
|
477 |
val _ = writeln "Proving induction rule for datatypes...";
|
|
478 |
|
|
479 |
val Rep_inverse_thms = (map (fn (_, iso, _) => iso RS subst) newT_iso_axms) @
|
|
480 |
(map (fn r => r RS inv_f_f RS subst) (drop (length newTs, iso_inj_thms)));
|
|
481 |
val Rep_inverse_thms' = map (fn r => r RS inv_f_f)
|
|
482 |
(drop (length newTs, iso_inj_thms));
|
|
483 |
|
|
484 |
fun mk_indrule_lemma ((prems, concls), ((i, _), T)) =
|
|
485 |
let
|
|
486 |
val Rep_t = Const (nth_elem (i, all_rep_names), T --> Univ_elT) $
|
|
487 |
mk_Free "x" T i;
|
|
488 |
|
|
489 |
val Abs_t = if i < length newTs then
|
|
490 |
Const (Sign.intern_const (sign_of thy6)
|
|
491 |
("Abs_" ^ (nth_elem (i, new_type_names))), Univ_elT --> T)
|
|
492 |
else Const (inv_name, [T --> Univ_elT, Univ_elT] ---> T) $
|
|
493 |
Const (nth_elem (i, all_rep_names), T --> Univ_elT)
|
|
494 |
|
|
495 |
in (prems @ [HOLogic.imp $ HOLogic.mk_mem (Rep_t,
|
|
496 |
Const (nth_elem (i, rep_set_names), UnivT)) $
|
|
497 |
(mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ (Abs_t $ Rep_t))],
|
|
498 |
concls @ [mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ mk_Free "x" T i])
|
|
499 |
end;
|
|
500 |
|
|
501 |
val (indrule_lemma_prems, indrule_lemma_concls) =
|
|
502 |
foldl mk_indrule_lemma (([], []), (descr' ~~ recTs));
|
|
503 |
|
|
504 |
val cert = cterm_of (sign_of thy6);
|
|
505 |
|
|
506 |
val indrule_lemma = prove_goalw_cterm [] (cert
|
|
507 |
(Logic.mk_implies
|
|
508 |
(HOLogic.mk_Trueprop (mk_conj indrule_lemma_prems),
|
|
509 |
HOLogic.mk_Trueprop (mk_conj indrule_lemma_concls)))) (fn prems =>
|
|
510 |
[cut_facts_tac prems 1, REPEAT (etac conjE 1),
|
|
511 |
REPEAT (EVERY
|
|
512 |
[TRY (rtac conjI 1), resolve_tac Rep_inverse_thms 1,
|
|
513 |
etac mp 1, resolve_tac iso_elem_thms 1])]);
|
|
514 |
|
|
515 |
val Ps = map head_of (dest_conj (HOLogic.dest_Trueprop (concl_of indrule_lemma)));
|
|
516 |
val frees = if length Ps = 1 then [Free ("P", snd (dest_Var (hd Ps)))] else
|
|
517 |
map (Free o apfst fst o dest_Var) Ps;
|
|
518 |
val indrule_lemma' = cterm_instantiate (map cert Ps ~~ map cert frees) indrule_lemma;
|
|
519 |
|
|
520 |
val dt_induct = prove_goalw_cterm [] (cert
|
|
521 |
(DatatypeProp.make_ind descr sorts)) (fn prems =>
|
|
522 |
[rtac indrule_lemma' 1, indtac rep_induct 1,
|
|
523 |
EVERY (map (fn (prem, r) => (EVERY
|
|
524 |
[REPEAT (eresolve_tac Abs_inverse_thms 1),
|
|
525 |
simp_tac (HOL_basic_ss addsimps ((symmetric r)::Rep_inverse_thms')) 1,
|
|
526 |
DEPTH_SOLVE_1 (ares_tac [prem] 1)]))
|
|
527 |
(prems ~~ (constr_defs @ (map mk_meta_eq iso_char_thms))))]);
|
|
528 |
|
|
529 |
val thy7 = PureThy.add_tthms [(("induct", Attribute.tthm_of dt_induct), [])] thy6;
|
|
530 |
|
|
531 |
in (thy7, constr_inject, dist_rewrites, dt_induct)
|
|
532 |
end;
|
|
533 |
|
|
534 |
end;
|