(* Title: HOL/Tools/datatype_rep_proofs.ML
ID: $Id$
Author: Stefan Berghofer, TU Muenchen
Definitional introduction of datatypes
Proof of characteristic theorems:
- injectivity of constructors
- distinctness of constructors
- induction theorem
*)
signature DATATYPE_REP_PROOFS =
sig
val representation_proofs : bool -> DatatypeAux.datatype_info Symtab.table ->
string list -> DatatypeAux.descr list -> (string * sort) list ->
(string * mixfix) list -> (string * mixfix) list list -> theory attribute
-> theory -> theory * thm list list * thm list list * thm list list *
DatatypeAux.simproc_dist list * thm
end;
structure DatatypeRepProofs : DATATYPE_REP_PROOFS =
struct
open DatatypeAux;
val (_ $ (_ $ (_ $ (distinct_f $ _) $ _))) = hd (prems_of distinct_lemma);
(** theory context references **)
val f_myinv_f = thm "f_myinv_f";
val myinv_f_f = thm "myinv_f_f";
fun exh_thm_of (dt_info : datatype_info Symtab.table) tname =
#exhaustion (the (Symtab.curried_lookup dt_info tname));
(******************************************************************************)
fun representation_proofs flat_names (dt_info : datatype_info Symtab.table)
new_type_names descr sorts types_syntax constr_syntax case_names_induct thy =
let
val Datatype_thy =
if Context.theory_name thy = "Datatype" then thy
else theory "Datatype";
val node_name = "Datatype_Universe.node";
val In0_name = "Datatype_Universe.In0";
val In1_name = "Datatype_Universe.In1";
val Scons_name = "Datatype_Universe.Scons";
val Leaf_name = "Datatype_Universe.Leaf";
val Numb_name = "Datatype_Universe.Numb";
val Lim_name = "Datatype_Universe.Lim";
val Suml_name = "Datatype.Suml";
val Sumr_name = "Datatype.Sumr";
val [In0_inject, In1_inject, Scons_inject, Leaf_inject,
In0_eq, In1_eq, In0_not_In1, In1_not_In0,
Lim_inject, Suml_inject, Sumr_inject] = map (get_thm Datatype_thy o Name)
["In0_inject", "In1_inject", "Scons_inject", "Leaf_inject",
"In0_eq", "In1_eq", "In0_not_In1", "In1_not_In0",
"Lim_inject", "Suml_inject", "Sumr_inject"];
val descr' = List.concat descr;
val big_name = space_implode "_" new_type_names;
val thy1 = add_path flat_names big_name thy;
val big_rec_name = big_name ^ "_rep_set";
val rep_set_names = map (Sign.full_name (Theory.sign_of thy1))
(if length descr' = 1 then [big_rec_name] else
(map ((curry (op ^) (big_rec_name ^ "_")) o string_of_int)
(1 upto (length descr'))));
val tyvars = map (fn (_, (_, Ts, _)) => map dest_DtTFree Ts) (hd descr);
val leafTs' = get_nonrec_types descr' sorts;
val branchTs = get_branching_types descr' sorts;
val branchT = if null branchTs then HOLogic.unitT
else fold_bal (fn (T, U) => Type ("+", [T, U])) branchTs;
val arities = get_arities descr' \ 0;
val unneeded_vars = hd tyvars \\ foldr add_typ_tfree_names [] (leafTs' @ branchTs);
val leafTs = leafTs' @ (map (fn n => TFree (n, valOf (assoc (sorts, n)))) unneeded_vars);
val recTs = get_rec_types descr' sorts;
val newTs = Library.take (length (hd descr), recTs);
val oldTs = Library.drop (length (hd descr), recTs);
val sumT = if null leafTs then HOLogic.unitT
else fold_bal (fn (T, U) => Type ("+", [T, U])) leafTs;
val Univ_elT = HOLogic.mk_setT (Type (node_name, [sumT, branchT]));
val UnivT = HOLogic.mk_setT Univ_elT;
val In0 = Const (In0_name, Univ_elT --> Univ_elT);
val In1 = Const (In1_name, Univ_elT --> Univ_elT);
val Leaf = Const (Leaf_name, sumT --> Univ_elT);
val Lim = Const (Lim_name, (branchT --> Univ_elT) --> Univ_elT);
(* make injections needed for embedding types in leaves *)
fun mk_inj T' x =
let
fun mk_inj' T n i =
if n = 1 then x else
let val n2 = n div 2;
val Type (_, [T1, T2]) = T
in
if i <= n2 then
Const ("Sum_Type.Inl", T1 --> T) $ (mk_inj' T1 n2 i)
else
Const ("Sum_Type.Inr", T2 --> T) $ (mk_inj' T2 (n - n2) (i - n2))
end
in mk_inj' sumT (length leafTs) (1 + find_index_eq T' leafTs)
end;
(* make injections for constructors *)
fun mk_univ_inj ts = access_bal (fn t => In0 $ t, fn t => In1 $ t, if ts = [] then
Const ("arbitrary", Univ_elT)
else
foldr1 (HOLogic.mk_binop Scons_name) ts);
(* function spaces *)
fun mk_fun_inj T' x =
let
fun mk_inj T n i =
if n = 1 then x else
let
val n2 = n div 2;
val Type (_, [T1, T2]) = T;
fun mkT U = (U --> Univ_elT) --> T --> Univ_elT
in
if i <= n2 then Const (Suml_name, mkT T1) $ mk_inj T1 n2 i
else Const (Sumr_name, mkT T2) $ mk_inj T2 (n - n2) (i - n2)
end
in mk_inj branchT (length branchTs) (1 + find_index_eq T' branchTs)
end;
val mk_lim = foldr (fn (T, t) => Lim $ mk_fun_inj T (Abs ("x", T, t)));
(************** generate introduction rules for representing set **********)
val _ = message "Constructing representing sets ...";
(* make introduction rule for a single constructor *)
fun make_intr s n (i, (_, cargs)) =
let
fun mk_prem (dt, (j, prems, ts)) = (case strip_dtyp dt of
(dts, DtRec k) =>
let
val Ts = map (typ_of_dtyp descr' sorts) dts;
val free_t =
app_bnds (mk_Free "x" (Ts ---> Univ_elT) j) (length Ts)
in (j + 1, list_all (map (pair "x") Ts,
HOLogic.mk_Trueprop (HOLogic.mk_mem (free_t,
Const (List.nth (rep_set_names, k), UnivT)))) :: prems,
mk_lim free_t Ts :: ts)
end
| _ =>
let val T = typ_of_dtyp descr' sorts dt
in (j + 1, prems, (Leaf $ mk_inj T (mk_Free "x" T j))::ts)
end);
val (_, prems, ts) = foldr mk_prem (1, [], []) cargs;
val concl = HOLogic.mk_Trueprop (HOLogic.mk_mem
(mk_univ_inj ts n i, Const (s, UnivT)))
in Logic.list_implies (prems, concl)
end;
val consts = map (fn s => Const (s, UnivT)) rep_set_names;
val intr_ts = List.concat (map (fn ((_, (_, _, constrs)), rep_set_name) =>
map (make_intr rep_set_name (length constrs))
((1 upto (length constrs)) ~~ constrs)) (descr' ~~ rep_set_names));
val (thy2, {raw_induct = rep_induct, intrs = rep_intrs, ...}) =
setmp InductivePackage.quiet_mode (!quiet_mode)
(InductivePackage.add_inductive_i false true big_rec_name false true false
consts (map (fn x => (("", x), [])) intr_ts) []) thy1;
(********************************* typedef ********************************)
val thy3 = add_path flat_names big_name (Library.foldl (fn (thy, ((((name, mx), tvs), c), name')) =>
setmp TypedefPackage.quiet_mode true
(TypedefPackage.add_typedef_i false (SOME name') (name, tvs, mx) c NONE
(rtac exI 1 THEN
QUIET_BREADTH_FIRST (has_fewer_prems 1)
(resolve_tac rep_intrs 1))) thy |> #1)
(parent_path flat_names thy2, types_syntax ~~ tyvars ~~
(Library.take (length newTs, consts)) ~~ new_type_names));
(*********************** definition of constructors ***********************)
val big_rep_name = (space_implode "_" new_type_names) ^ "_Rep_";
val rep_names = map (curry op ^ "Rep_") new_type_names;
val rep_names' = map (fn i => big_rep_name ^ (string_of_int i))
(1 upto (length (List.concat (tl descr))));
val all_rep_names = map (Sign.intern_const (Theory.sign_of thy3)) rep_names @
map (Sign.full_name (Theory.sign_of thy3)) rep_names';
(* isomorphism declarations *)
val iso_decls = map (fn (T, s) => (s, T --> Univ_elT, NoSyn))
(oldTs ~~ rep_names');
(* constructor definitions *)
fun make_constr_def tname T n ((thy, defs, eqns, i), ((cname, cargs), (cname', mx))) =
let
fun constr_arg (dt, (j, l_args, r_args)) =
let val T = typ_of_dtyp descr' sorts dt;
val free_t = mk_Free "x" T j
in (case (strip_dtyp dt, strip_type T) of
((_, DtRec m), (Us, U)) => (j + 1, free_t :: l_args, mk_lim
(Const (List.nth (all_rep_names, m), U --> Univ_elT) $
app_bnds free_t (length Us)) Us :: r_args)
| _ => (j + 1, free_t::l_args, (Leaf $ mk_inj T free_t)::r_args))
end;
val (_, l_args, r_args) = foldr constr_arg (1, [], []) cargs;
val constrT = (map (typ_of_dtyp descr' sorts) cargs) ---> T;
val abs_name = Sign.intern_const (Theory.sign_of thy) ("Abs_" ^ tname);
val rep_name = Sign.intern_const (Theory.sign_of thy) ("Rep_" ^ tname);
val lhs = list_comb (Const (cname, constrT), l_args);
val rhs = mk_univ_inj r_args n i;
val def = equals T $ lhs $ (Const (abs_name, Univ_elT --> T) $ rhs);
val def_name = (Sign.base_name cname) ^ "_def";
val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq
(Const (rep_name, T --> Univ_elT) $ lhs, rhs));
val (thy', [def_thm]) = thy |>
Theory.add_consts_i [(cname', constrT, mx)] |>
(PureThy.add_defs_i false o map Thm.no_attributes) [(def_name, def)];
in (thy', defs @ [def_thm], eqns @ [eqn], i + 1) end;
(* constructor definitions for datatype *)
fun dt_constr_defs ((thy, defs, eqns, rep_congs, dist_lemmas),
((((_, (_, _, constrs)), tname), T), constr_syntax)) =
let
val _ $ (_ $ (cong_f $ _) $ _) = concl_of arg_cong;
val sg = Theory.sign_of thy;
val rep_const = cterm_of sg
(Const (Sign.intern_const sg ("Rep_" ^ tname), T --> Univ_elT));
val cong' = standard (cterm_instantiate [(cterm_of sg cong_f, rep_const)] arg_cong);
val dist = standard (cterm_instantiate [(cterm_of sg distinct_f, rep_const)] distinct_lemma);
val (thy', defs', eqns', _) = Library.foldl ((make_constr_def tname T) (length constrs))
((add_path flat_names tname thy, defs, [], 1), constrs ~~ constr_syntax)
in
(parent_path flat_names thy', defs', eqns @ [eqns'],
rep_congs @ [cong'], dist_lemmas @ [dist])
end;
val (thy4, constr_defs, constr_rep_eqns, rep_congs, dist_lemmas) = Library.foldl dt_constr_defs
((thy3 |> Theory.add_consts_i iso_decls |> parent_path flat_names, [], [], [], []),
hd descr ~~ new_type_names ~~ newTs ~~ constr_syntax);
(*********** isomorphisms for new types (introduced by typedef) ***********)
val _ = message "Proving isomorphism properties ...";
(* get axioms from theory *)
val newT_iso_axms = map (fn s =>
(get_thm thy4 (Name ("Abs_" ^ s ^ "_inverse")),
get_thm thy4 (Name ("Rep_" ^ s ^ "_inverse")),
get_thm thy4 (Name ("Rep_" ^ s)))) new_type_names;
(*------------------------------------------------*)
(* prove additional theorems: *)
(* inj_on dt_Abs_i rep_set_i and inj dt_Rep_i *)
(*------------------------------------------------*)
fun prove_newT_iso_inj_thm (((s, (thm1, thm2, _)), T), rep_set_name) =
let
val sg = Theory.sign_of thy4;
val RepT = T --> Univ_elT;
val Rep_name = Sign.intern_const sg ("Rep_" ^ s);
val AbsT = Univ_elT --> T;
val Abs_name = Sign.intern_const sg ("Abs_" ^ s);
val inj_Abs_thm =
prove_goalw_cterm []
(cterm_of sg
(HOLogic.mk_Trueprop
(Const ("Fun.inj_on", [AbsT, UnivT] ---> HOLogic.boolT) $
Const (Abs_name, AbsT) $ Const (rep_set_name, UnivT))))
(fn _ => [rtac inj_on_inverseI 1, etac thm1 1]);
val setT = HOLogic.mk_setT T
val inj_Rep_thm =
prove_goalw_cterm []
(cterm_of sg
(HOLogic.mk_Trueprop
(Const ("Fun.inj_on", [RepT, setT] ---> HOLogic.boolT) $
Const (Rep_name, RepT) $ Const ("UNIV", setT))))
(fn _ => [rtac inj_inverseI 1, rtac thm2 1])
in (inj_Abs_thm, inj_Rep_thm) end;
val newT_iso_inj_thms = map prove_newT_iso_inj_thm
(new_type_names ~~ newT_iso_axms ~~ newTs ~~
Library.take (length newTs, rep_set_names));
(********* isomorphisms between existing types and "unfolded" types *******)
(*---------------------------------------------------------------------*)
(* isomorphisms are defined using primrec-combinators: *)
(* generate appropriate functions for instantiating primrec-combinator *)
(* *)
(* e.g. dt_Rep_i = list_rec ... (%h t y. In1 (Scons (Leaf h) y)) *)
(* *)
(* also generate characteristic equations for isomorphisms *)
(* *)
(* e.g. dt_Rep_i (cons h t) = In1 (Scons (dt_Rep_j h) (dt_Rep_i t)) *)
(*---------------------------------------------------------------------*)
fun make_iso_def k ks n ((fs, eqns, i), (cname, cargs)) =
let
val argTs = map (typ_of_dtyp descr' sorts) cargs;
val T = List.nth (recTs, k);
val rep_name = List.nth (all_rep_names, k);
val rep_const = Const (rep_name, T --> Univ_elT);
val constr = Const (cname, argTs ---> T);
fun process_arg ks' ((i2, i2', ts, Ts), dt) =
let
val T' = typ_of_dtyp descr' sorts dt;
val (Us, U) = strip_type T'
in (case strip_dtyp dt of
(_, DtRec j) => if j mem ks' then
(i2 + 1, i2' + 1, ts @ [mk_lim (app_bnds
(mk_Free "y" (Us ---> Univ_elT) i2') (length Us)) Us],
Ts @ [Us ---> Univ_elT])
else
(i2 + 1, i2', ts @ [mk_lim
(Const (List.nth (all_rep_names, j), U --> Univ_elT) $
app_bnds (mk_Free "x" T' i2) (length Us)) Us], Ts)
| _ => (i2 + 1, i2', ts @ [Leaf $ mk_inj T' (mk_Free "x" T' i2)], Ts))
end;
val (i2, i2', ts, Ts) = Library.foldl (process_arg ks) ((1, 1, [], []), cargs);
val xs = map (uncurry (mk_Free "x")) (argTs ~~ (1 upto (i2 - 1)));
val ys = map (uncurry (mk_Free "y")) (Ts ~~ (1 upto (i2' - 1)));
val f = list_abs_free (map dest_Free (xs @ ys), mk_univ_inj ts n i);
val (_, _, ts', _) = Library.foldl (process_arg []) ((1, 1, [], []), cargs);
val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq
(rep_const $ list_comb (constr, xs), mk_univ_inj ts' n i))
in (fs @ [f], eqns @ [eqn], i + 1) end;
(* define isomorphisms for all mutually recursive datatypes in list ds *)
fun make_iso_defs (ds, (thy, char_thms)) =
let
val ks = map fst ds;
val (_, (tname, _, _)) = hd ds;
val {rec_rewrites, rec_names, ...} = the (Symtab.curried_lookup dt_info tname);
fun process_dt ((fs, eqns, isos), (k, (tname, _, constrs))) =
let
val (fs', eqns', _) = Library.foldl (make_iso_def k ks (length constrs))
((fs, eqns, 1), constrs);
val iso = (List.nth (recTs, k), List.nth (all_rep_names, k))
in (fs', eqns', isos @ [iso]) end;
val (fs, eqns, isos) = Library.foldl process_dt (([], [], []), ds);
val fTs = map fastype_of fs;
val defs = map (fn (rec_name, (T, iso_name)) => ((Sign.base_name iso_name) ^ "_def",
equals (T --> Univ_elT) $ Const (iso_name, T --> Univ_elT) $
list_comb (Const (rec_name, fTs @ [T] ---> Univ_elT), fs))) (rec_names ~~ isos);
val (thy', def_thms) = (PureThy.add_defs_i false o map Thm.no_attributes) defs thy;
(* prove characteristic equations *)
val rewrites = def_thms @ (map mk_meta_eq rec_rewrites);
val char_thms' = map (fn eqn => prove_goalw_cterm rewrites
(cterm_of (Theory.sign_of thy') eqn) (fn _ => [rtac refl 1])) eqns;
in (thy', char_thms' @ char_thms) end;
val (thy5, iso_char_thms) = foldr make_iso_defs
(add_path flat_names big_name thy4, []) (tl descr);
(* prove isomorphism properties *)
fun mk_funs_inv thm =
let
val {sign, prop, ...} = rep_thm thm;
val _ $ (_ $ (Const (_, Type (_, [U, _])) $ _ $ S)) $
(_ $ (_ $ (r $ (a $ _)) $ _)) = Type.freeze prop;
val used = add_term_tfree_names (a, []);
fun mk_thm i =
let
val Ts = map (TFree o rpair HOLogic.typeS)
(variantlist (replicate i "'t", used));
val f = Free ("f", Ts ---> U)
in prove_goalw_cterm [] (cterm_of sign (Logic.mk_implies
(HOLogic.mk_Trueprop (HOLogic.list_all
(map (pair "x") Ts, HOLogic.mk_mem (app_bnds f i, S))),
HOLogic.mk_Trueprop (HOLogic.mk_eq (list_abs (map (pair "x") Ts,
r $ (a $ app_bnds f i)), f))))) (fn prems =>
[cut_facts_tac prems 1, REPEAT (rtac ext 1),
REPEAT (etac allE 1), rtac thm 1, atac 1])
end
in map (fn r => r RS subst) (thm :: map mk_thm arities) end;
(* prove inj dt_Rep_i and dt_Rep_i x : dt_rep_set_i *)
fun prove_iso_thms (ds, (inj_thms, elem_thms)) =
let
val (_, (tname, _, _)) = hd ds;
val {induction, ...} = the (Symtab.curried_lookup dt_info tname);
fun mk_ind_concl (i, _) =
let
val T = List.nth (recTs, i);
val Rep_t = Const (List.nth (all_rep_names, i), T --> Univ_elT);
val rep_set_name = List.nth (rep_set_names, i)
in (HOLogic.all_const T $ Abs ("y", T, HOLogic.imp $
HOLogic.mk_eq (Rep_t $ mk_Free "x" T i, Rep_t $ Bound 0) $
HOLogic.mk_eq (mk_Free "x" T i, Bound 0)),
HOLogic.mk_mem (Rep_t $ mk_Free "x" T i, Const (rep_set_name, UnivT)))
end;
val (ind_concl1, ind_concl2) = ListPair.unzip (map mk_ind_concl ds);
val rewrites = map mk_meta_eq iso_char_thms;
val inj_thms' = map (fn r => r RS injD)
(map snd newT_iso_inj_thms @ inj_thms);
val inj_thm = prove_goalw_cterm [] (cterm_of (Theory.sign_of thy5)
(HOLogic.mk_Trueprop (mk_conj ind_concl1))) (fn _ =>
[(indtac induction THEN_ALL_NEW ObjectLogic.atomize_tac) 1,
REPEAT (EVERY
[rtac allI 1, rtac impI 1,
exh_tac (exh_thm_of dt_info) 1,
REPEAT (EVERY
[hyp_subst_tac 1,
rewrite_goals_tac rewrites,
REPEAT (dresolve_tac [In0_inject, In1_inject] 1),
(eresolve_tac [In0_not_In1 RS notE, In1_not_In0 RS notE] 1)
ORELSE (EVERY
[REPEAT (eresolve_tac (Scons_inject ::
map make_elim [Leaf_inject, Inl_inject, Inr_inject]) 1),
REPEAT (cong_tac 1), rtac refl 1,
REPEAT (atac 1 ORELSE (EVERY
[REPEAT (rtac ext 1),
REPEAT (eresolve_tac (mp :: allE ::
map make_elim (Suml_inject :: Sumr_inject ::
Lim_inject :: fun_cong :: inj_thms')) 1),
atac 1]))])])])]);
val inj_thms'' = map (fn r => r RS datatype_injI)
(split_conj_thm inj_thm);
val elem_thm =
prove_goalw_cterm []
(cterm_of (Theory.sign_of thy5)
(HOLogic.mk_Trueprop (mk_conj ind_concl2)))
(fn _ =>
[(indtac induction THEN_ALL_NEW ObjectLogic.atomize_tac) 1,
rewrite_goals_tac rewrites,
REPEAT ((resolve_tac rep_intrs THEN_ALL_NEW
((REPEAT o etac allE) THEN' ares_tac elem_thms)) 1)]);
in (inj_thms'' @ inj_thms, elem_thms @ (split_conj_thm elem_thm))
end;
val (iso_inj_thms_unfolded, iso_elem_thms) = foldr prove_iso_thms
([], map #3 newT_iso_axms) (tl descr);
val iso_inj_thms = map snd newT_iso_inj_thms @ iso_inj_thms_unfolded;
(* prove x : dt_rep_set_i --> x : range dt_Rep_i *)
fun mk_iso_t (((set_name, iso_name), i), T) =
let val isoT = T --> Univ_elT
in HOLogic.imp $
HOLogic.mk_mem (mk_Free "x" Univ_elT i, Const (set_name, UnivT)) $
(if i < length newTs then Const ("True", HOLogic.boolT)
else HOLogic.mk_mem (mk_Free "x" Univ_elT i,
Const ("image", [isoT, HOLogic.mk_setT T] ---> UnivT) $
Const (iso_name, isoT) $ Const ("UNIV", HOLogic.mk_setT T)))
end;
val iso_t = HOLogic.mk_Trueprop (mk_conj (map mk_iso_t
(rep_set_names ~~ all_rep_names ~~ (0 upto (length descr' - 1)) ~~ recTs)));
(* all the theorems are proved by one single simultaneous induction *)
val range_eqs = map (fn r => mk_meta_eq (r RS range_ex1_eq))
iso_inj_thms_unfolded;
val iso_thms = if length descr = 1 then [] else
Library.drop (length newTs, split_conj_thm
(prove_goalw_cterm [] (cterm_of (Theory.sign_of thy5) iso_t) (fn _ =>
[(indtac rep_induct THEN_ALL_NEW ObjectLogic.atomize_tac) 1,
REPEAT (rtac TrueI 1),
rewrite_goals_tac (mk_meta_eq choice_eq ::
symmetric (mk_meta_eq expand_fun_eq) :: range_eqs),
rewrite_goals_tac (map symmetric range_eqs),
REPEAT (EVERY
[REPEAT (eresolve_tac ([rangeE, ex1_implies_ex RS exE] @
List.concat (map (mk_funs_inv o #1) newT_iso_axms)) 1),
TRY (hyp_subst_tac 1),
rtac (sym RS range_eqI) 1,
resolve_tac iso_char_thms 1])])));
val Abs_inverse_thms' =
map #1 newT_iso_axms @
map2 (fn (r_inj, r) => f_myinv_f OF [r_inj, r RS mp])
(iso_inj_thms_unfolded, iso_thms);
val Abs_inverse_thms = List.concat (map mk_funs_inv Abs_inverse_thms');
(******************* freeness theorems for constructors *******************)
val _ = message "Proving freeness of constructors ...";
(* prove theorem Rep_i (Constr_j ...) = Inj_j ... *)
fun prove_constr_rep_thm eqn =
let
val inj_thms = map (fn (r, _) => r RS inj_onD) newT_iso_inj_thms;
val rewrites = o_def :: constr_defs @ (map (mk_meta_eq o #2) newT_iso_axms)
in prove_goalw_cterm [] (cterm_of (Theory.sign_of thy5) eqn) (fn _ =>
[resolve_tac inj_thms 1,
rewrite_goals_tac rewrites,
rtac refl 1,
resolve_tac rep_intrs 2,
REPEAT (resolve_tac iso_elem_thms 1)])
end;
(*--------------------------------------------------------------*)
(* constr_rep_thms and rep_congs are used to prove distinctness *)
(* of constructors. *)
(*--------------------------------------------------------------*)
val constr_rep_thms = map (map prove_constr_rep_thm) constr_rep_eqns;
val dist_rewrites = map (fn (rep_thms, dist_lemma) =>
dist_lemma::(rep_thms @ [In0_eq, In1_eq, In0_not_In1, In1_not_In0]))
(constr_rep_thms ~~ dist_lemmas);
fun prove_distinct_thms (_, []) = []
| prove_distinct_thms (dist_rewrites', t::_::ts) =
let
val dist_thm = prove_goalw_cterm [] (cterm_of (Theory.sign_of thy5) t) (fn _ =>
[simp_tac (HOL_ss addsimps dist_rewrites') 1])
in dist_thm::(standard (dist_thm RS not_sym))::
(prove_distinct_thms (dist_rewrites', ts))
end;
val distinct_thms = map prove_distinct_thms (dist_rewrites ~~
DatatypeProp.make_distincts new_type_names descr sorts thy5);
val simproc_dists = map (fn ((((_, (_, _, constrs)), rep_thms), congr), dists) =>
if length constrs < !DatatypeProp.dtK then FewConstrs dists
else ManyConstrs (congr, HOL_basic_ss addsimps rep_thms)) (hd descr ~~
constr_rep_thms ~~ rep_congs ~~ distinct_thms);
(* prove injectivity of constructors *)
fun prove_constr_inj_thm rep_thms t =
let val inj_thms = Scons_inject :: (map make_elim
((map (fn r => r RS injD) iso_inj_thms) @
[In0_inject, In1_inject, Leaf_inject, Inl_inject, Inr_inject,
Lim_inject, Suml_inject, Sumr_inject]))
in prove_goalw_cterm [] (cterm_of (Theory.sign_of thy5) t) (fn _ =>
[rtac iffI 1,
REPEAT (etac conjE 2), hyp_subst_tac 2, rtac refl 2,
dresolve_tac rep_congs 1, dtac box_equals 1,
REPEAT (resolve_tac rep_thms 1),
REPEAT (eresolve_tac inj_thms 1),
REPEAT (ares_tac [conjI] 1 ORELSE (EVERY [REPEAT (rtac ext 1),
REPEAT (eresolve_tac (make_elim fun_cong :: inj_thms) 1),
atac 1]))])
end;
val constr_inject = map (fn (ts, thms) => map (prove_constr_inj_thm thms) ts)
((DatatypeProp.make_injs descr sorts) ~~ constr_rep_thms);
val (thy6, (constr_inject', distinct_thms'))= thy5 |> parent_path flat_names |>
store_thmss "inject" new_type_names constr_inject |>>>
store_thmss "distinct" new_type_names distinct_thms;
(*************************** induction theorem ****************************)
val _ = message "Proving induction rule for datatypes ...";
val Rep_inverse_thms = (map (fn (_, iso, _) => iso RS subst) newT_iso_axms) @
(map (fn r => r RS myinv_f_f RS subst) iso_inj_thms_unfolded);
val Rep_inverse_thms' = map (fn r => r RS myinv_f_f) iso_inj_thms_unfolded;
fun mk_indrule_lemma ((prems, concls), ((i, _), T)) =
let
val Rep_t = Const (List.nth (all_rep_names, i), T --> Univ_elT) $
mk_Free "x" T i;
val Abs_t = if i < length newTs then
Const (Sign.intern_const (Theory.sign_of thy6)
("Abs_" ^ (List.nth (new_type_names, i))), Univ_elT --> T)
else Const ("Inductive.myinv", [T --> Univ_elT, Univ_elT] ---> T) $
Const (List.nth (all_rep_names, i), T --> Univ_elT)
in (prems @ [HOLogic.imp $ HOLogic.mk_mem (Rep_t,
Const (List.nth (rep_set_names, i), UnivT)) $
(mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ (Abs_t $ Rep_t))],
concls @ [mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ mk_Free "x" T i])
end;
val (indrule_lemma_prems, indrule_lemma_concls) =
Library.foldl mk_indrule_lemma (([], []), (descr' ~~ recTs));
val cert = cterm_of (Theory.sign_of thy6);
val indrule_lemma = prove_goalw_cterm [] (cert
(Logic.mk_implies
(HOLogic.mk_Trueprop (mk_conj indrule_lemma_prems),
HOLogic.mk_Trueprop (mk_conj indrule_lemma_concls)))) (fn prems =>
[cut_facts_tac prems 1, REPEAT (etac conjE 1),
REPEAT (EVERY
[TRY (rtac conjI 1), resolve_tac Rep_inverse_thms 1,
etac mp 1, resolve_tac iso_elem_thms 1])]);
val Ps = map head_of (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of indrule_lemma)));
val frees = if length Ps = 1 then [Free ("P", snd (dest_Var (hd Ps)))] else
map (Free o apfst fst o dest_Var) Ps;
val indrule_lemma' = cterm_instantiate (map cert Ps ~~ map cert frees) indrule_lemma;
val dt_induct = prove_goalw_cterm [] (cert
(DatatypeProp.make_ind descr sorts)) (fn prems =>
[rtac indrule_lemma' 1,
(indtac rep_induct THEN_ALL_NEW ObjectLogic.atomize_tac) 1,
EVERY (map (fn (prem, r) => (EVERY
[REPEAT (eresolve_tac Abs_inverse_thms 1),
simp_tac (HOL_basic_ss addsimps ((symmetric r)::Rep_inverse_thms')) 1,
DEPTH_SOLVE_1 (ares_tac [prem] 1 ORELSE etac allE 1)]))
(prems ~~ (constr_defs @ (map mk_meta_eq iso_char_thms))))]);
val (thy7, [dt_induct']) = thy6 |>
Theory.add_path big_name |>
PureThy.add_thms [(("induct", dt_induct), [case_names_induct])] |>>
Theory.parent_path;
in (thy7, constr_inject', distinct_thms', dist_rewrites, simproc_dists, dt_induct')
end;
end;