(* Title: HOL/Tools/Datatype/datatype.ML
Author: Stefan Berghofer, TU Muenchen
Datatype package: definitional introduction of datatypes
with proof of characteristic theorems: injectivity / distinctness
of constructors and induction. Main interface to datatypes
after full bootstrap of datatype package.
*)
signature DATATYPE =
sig
include DATATYPE_DATA
val add_datatype: config ->
(string list * binding * mixfix * (binding * typ list * mixfix) list) list ->
theory -> string list * theory
val add_datatype_cmd:
(string list * binding * mixfix * (binding * string list * mixfix) list) list ->
theory -> theory
val parse_decl: (string list * binding * mixfix * (binding * string list * mixfix) list) parser
end;
structure Datatype : DATATYPE =
struct
(** auxiliary **)
val (_ $ (_ $ (_ $ (distinct_f $ _) $ _))) = hd (prems_of distinct_lemma);
val collect_simp = rewrite_rule [mk_meta_eq mem_Collect_eq];
fun exh_thm_of (dt_info : Datatype_Aux.info Symtab.table) tname =
#exhaust (the (Symtab.lookup dt_info tname));
val node_name = @{type_name "Datatype.node"};
val In0_name = @{const_name "Datatype.In0"};
val In1_name = @{const_name "Datatype.In1"};
val Scons_name = @{const_name "Datatype.Scons"};
val Leaf_name = @{const_name "Datatype.Leaf"};
val Lim_name = @{const_name "Datatype.Lim"};
val Suml_name = @{const_name "Sum_Type.Suml"};
val Sumr_name = @{const_name "Sum_Type.Sumr"};
val In0_inject = @{thm In0_inject};
val In1_inject = @{thm In1_inject};
val Scons_inject = @{thm Scons_inject};
val Leaf_inject = @{thm Leaf_inject};
val In0_eq = @{thm In0_eq};
val In1_eq = @{thm In1_eq};
val In0_not_In1 = @{thm In0_not_In1};
val In1_not_In0 = @{thm In1_not_In0};
val Lim_inject = @{thm Lim_inject};
val Inl_inject = @{thm Inl_inject};
val Inr_inject = @{thm Inr_inject};
val Suml_inject = @{thm Suml_inject};
val Sumr_inject = @{thm Sumr_inject};
val datatype_injI =
@{lemma "(!!x. ALL y. f x = f y --> x = y) ==> inj f" by (simp add: inj_on_def)};
(** proof of characteristic theorems **)
fun representation_proofs (config : Datatype_Aux.config) (dt_info : Datatype_Aux.info Symtab.table)
descr types_syntax constr_syntax case_names_induct thy =
let
val descr' = flat descr;
val new_type_names = map (Binding.name_of o fst) types_syntax;
val big_name = space_implode "_" new_type_names;
val thy1 = Sign.add_path big_name thy;
val big_rec_name = big_name ^ "_rep_set";
val rep_set_names' =
if length descr' = 1 then [big_rec_name]
else map (prefix (big_rec_name ^ "_") o string_of_int) (1 upto length descr');
val rep_set_names = map (Sign.full_bname thy1) rep_set_names';
val tyvars = map (fn (_, (_, Ts, _)) => map Datatype_Aux.dest_DtTFree Ts) (hd descr);
val leafTs' = Datatype_Aux.get_nonrec_types descr';
val branchTs = Datatype_Aux.get_branching_types descr';
val branchT =
if null branchTs then HOLogic.unitT
else Balanced_Tree.make (fn (T, U) => Type (@{type_name Sum_Type.sum}, [T, U])) branchTs;
val arities = remove (op =) 0 (Datatype_Aux.get_arities descr');
val unneeded_vars =
subtract (op =) (fold Term.add_tfreesT (leafTs' @ branchTs) []) (hd tyvars);
val leafTs = leafTs' @ map TFree unneeded_vars;
val recTs = Datatype_Aux.get_rec_types descr';
val (newTs, oldTs) = chop (length (hd descr)) recTs;
val sumT =
if null leafTs then HOLogic.unitT
else Balanced_Tree.make (fn (T, U) => Type (@{type_name Sum_Type.sum}, [T, U])) leafTs;
val Univ_elT = HOLogic.mk_setT (Type (node_name, [sumT, branchT]));
val UnivT = HOLogic.mk_setT Univ_elT;
val UnivT' = Univ_elT --> HOLogic.boolT;
val Collect = Const (@{const_name Collect}, UnivT' --> UnivT);
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 (@{const_name Inl}, T1 --> T) $ mk_inj' T1 n2 i
else Const (@{const_name Inr}, T2 --> T) $ mk_inj' T2 (n - n2) (i - n2)
end;
in mk_inj' sumT (length leafTs) (1 + find_index (fn T'' => T'' = T') leafTs) end;
(* make injections for constructors *)
fun mk_univ_inj ts = Balanced_Tree.access
{left = fn t => In0 $ t,
right = fn t => In1 $ t,
init =
if ts = [] then Const (@{const_name undefined}, 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 (fn T'' => T'' = T') branchTs) end;
fun mk_lim t Ts = fold_rev (fn T => fn t => Lim $ mk_fun_inj T (Abs ("x", T, t))) Ts t;
(************** generate introduction rules for representing set **********)
val _ = Datatype_Aux.message config "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 Datatype_Aux.strip_dtyp dt of
(dts, Datatype_Aux.DtRec k) =>
let
val Ts = map (Datatype_Aux.typ_of_dtyp descr') dts;
val free_t =
Datatype_Aux.app_bnds (Datatype_Aux.mk_Free "x" (Ts ---> Univ_elT) j) (length Ts)
in
(j + 1, list_all (map (pair "x") Ts,
HOLogic.mk_Trueprop
(Free (nth rep_set_names' k, UnivT') $ free_t)) :: prems,
mk_lim free_t Ts :: ts)
end
| _ =>
let val T = Datatype_Aux.typ_of_dtyp descr' dt
in (j + 1, prems, (Leaf $ mk_inj T (Datatype_Aux.mk_Free "x" T j)) :: ts) end);
val (_, prems, ts) = fold_rev mk_prem cargs (1, [], []);
val concl = HOLogic.mk_Trueprop (Free (s, UnivT') $ mk_univ_inj ts n i);
in Logic.list_implies (prems, concl) end;
val intr_ts = maps (fn ((_, (_, _, constrs)), rep_set_name) =>
map (make_intr rep_set_name (length constrs))
((1 upto length constrs) ~~ constrs)) (descr' ~~ rep_set_names');
val ({raw_induct = rep_induct, intrs = rep_intrs, ...}, thy2) =
thy1
|> Sign.map_naming Name_Space.conceal
|> Inductive.add_inductive_global
{quiet_mode = #quiet config, verbose = false, alt_name = Binding.name big_rec_name,
coind = false, no_elim = true, no_ind = false, skip_mono = true, fork_mono = false}
(map (fn s => ((Binding.name s, UnivT'), NoSyn)) rep_set_names') []
(map (fn x => (Attrib.empty_binding, x)) intr_ts) []
||> Sign.restore_naming thy1
||> Theory.checkpoint;
(********************************* typedef ********************************)
val (typedefs, thy3) = thy2
|> Sign.parent_path
|> fold_map
(fn (((name, mx), tvs), c) =>
Typedef.add_typedef_global false NONE (name, tvs, mx)
(Collect $ Const (c, UnivT')) NONE
(rtac exI 1 THEN rtac CollectI 1 THEN
QUIET_BREADTH_FIRST (has_fewer_prems 1)
(resolve_tac rep_intrs 1)))
(types_syntax ~~ tyvars ~~ (take (length newTs) rep_set_names))
||> Sign.add_path big_name;
(*********************** definition of constructors ***********************)
val big_rep_name = space_implode "_" new_type_names ^ "_Rep_";
val rep_names = map (prefix "Rep_") new_type_names;
val rep_names' = map (fn i => big_rep_name ^ string_of_int i) (1 upto length (flat (tl descr)));
val all_rep_names =
map (Sign.intern_const thy3) rep_names @
map (Sign.full_bname thy3) rep_names';
(* isomorphism declarations *)
val iso_decls = map (fn (T, s) => (Binding.name s, T --> Univ_elT, NoSyn))
(oldTs ~~ rep_names');
(* constructor definitions *)
fun make_constr_def tname T n ((cname, cargs), (cname', mx)) (thy, defs, eqns, i) =
let
fun constr_arg dt (j, l_args, r_args) =
let
val T = Datatype_Aux.typ_of_dtyp descr' dt;
val free_t = Datatype_Aux.mk_Free "x" T j;
in
(case (Datatype_Aux.strip_dtyp dt, strip_type T) of
((_, Datatype_Aux.DtRec m), (Us, U)) =>
(j + 1, free_t :: l_args, mk_lim
(Const (nth all_rep_names m, U --> Univ_elT) $
Datatype_Aux.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) = fold_rev constr_arg cargs (1, [], []);
val constrT = map (Datatype_Aux.typ_of_dtyp descr') cargs ---> T;
val abs_name = Sign.intern_const thy ("Abs_" ^ tname);
val rep_name = Sign.intern_const thy ("Rep_" ^ tname);
val lhs = list_comb (Const (cname, constrT), l_args);
val rhs = mk_univ_inj r_args n i;
val def = Logic.mk_equals (lhs, Const (abs_name, Univ_elT --> T) $ rhs);
val def_name = Long_Name.base_name cname ^ "_def";
val eqn =
HOLogic.mk_Trueprop (HOLogic.mk_eq (Const (rep_name, T --> Univ_elT) $ lhs, rhs));
val ([def_thm], thy') =
thy
|> Sign.add_consts_i [(cname', constrT, mx)]
|> (Global_Theory.add_defs false o map Thm.no_attributes) [(Binding.name def_name, def)];
in (thy', defs @ [def_thm], eqns @ [eqn], i + 1) end;
(* constructor definitions for datatype *)
fun dt_constr_defs ((((_, (_, _, constrs)), tname), T), constr_syntax)
(thy, defs, eqns, rep_congs, dist_lemmas) =
let
val _ $ (_ $ (cong_f $ _) $ _) = concl_of arg_cong;
val rep_const =
cterm_of thy (Const (Sign.intern_const thy ("Rep_" ^ tname), T --> Univ_elT));
val cong' =
Drule.export_without_context
(cterm_instantiate [(cterm_of thy cong_f, rep_const)] arg_cong);
val dist =
Drule.export_without_context
(cterm_instantiate [(cterm_of thy distinct_f, rep_const)] distinct_lemma);
val (thy', defs', eqns', _) =
fold ((make_constr_def tname T) (length constrs))
(constrs ~~ constr_syntax) (Sign.add_path tname thy, defs, [], 1);
in
(Sign.parent_path thy', defs', eqns @ [eqns'],
rep_congs @ [cong'], dist_lemmas @ [dist])
end;
val (thy4, constr_defs, constr_rep_eqns, rep_congs, dist_lemmas) =
fold dt_constr_defs
(hd descr ~~ new_type_names ~~ newTs ~~ constr_syntax)
(thy3 |> Sign.add_consts_i iso_decls |> Sign.parent_path, [], [], [], []);
(*********** isomorphisms for new types (introduced by typedef) ***********)
val _ = Datatype_Aux.message config "Proving isomorphism properties ...";
val newT_iso_axms = map (fn (_, (_, td)) =>
(collect_simp (#Abs_inverse td), #Rep_inverse td,
collect_simp (#Rep td))) typedefs;
val newT_iso_inj_thms = map (fn (_, (_, td)) =>
(collect_simp (#Abs_inject td) RS iffD1, #Rep_inject td RS iffD1)) typedefs;
(********* 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 (cname, cargs) (fs, eqns, i) =
let
val argTs = map (Datatype_Aux.typ_of_dtyp descr') cargs;
val T = nth recTs k;
val rep_name = nth all_rep_names k;
val rep_const = Const (rep_name, T --> Univ_elT);
val constr = Const (cname, argTs ---> T);
fun process_arg ks' dt (i2, i2', ts, Ts) =
let
val T' = Datatype_Aux.typ_of_dtyp descr' dt;
val (Us, U) = strip_type T'
in
(case Datatype_Aux.strip_dtyp dt of
(_, Datatype_Aux.DtRec j) =>
if member (op =) ks' j then
(i2 + 1, i2' + 1, ts @ [mk_lim (Datatype_Aux.app_bnds
(Datatype_Aux.mk_Free "y" (Us ---> Univ_elT) i2') (length Us)) Us],
Ts @ [Us ---> Univ_elT])
else
(i2 + 1, i2', ts @ [mk_lim
(Const (nth all_rep_names j, U --> Univ_elT) $
Datatype_Aux.app_bnds (Datatype_Aux.mk_Free "x" T' i2) (length Us)) Us], Ts)
| _ => (i2 + 1, i2', ts @ [Leaf $ mk_inj T' (Datatype_Aux.mk_Free "x" T' i2)], Ts))
end;
val (i2, i2', ts, Ts) = fold (process_arg ks) cargs (1, 1, [], []);
val xs = map (uncurry (Datatype_Aux.mk_Free "x")) (argTs ~~ (1 upto (i2 - 1)));
val ys = map (uncurry (Datatype_Aux.mk_Free "y")) (Ts ~~ (1 upto (i2' - 1)));
val f = fold_rev (absfree o dest_Free) (xs @ ys) (mk_univ_inj ts n i);
val (_, _, ts', _) = fold (process_arg []) cargs (1, 1, [], []);
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.lookup dt_info tname);
fun process_dt (k, (tname, _, constrs)) (fs, eqns, isos) =
let
val (fs', eqns', _) = fold (make_iso_def k ks (length constrs)) constrs (fs, eqns, 1);
val iso = (nth recTs k, nth all_rep_names k);
in (fs', eqns', isos @ [iso]) end;
val (fs, eqns, isos) = fold process_dt ds ([], [], []);
val fTs = map fastype_of fs;
val defs =
map (fn (rec_name, (T, iso_name)) =>
(Binding.name (Long_Name.base_name iso_name ^ "_def"),
Logic.mk_equals (Const (iso_name, T --> Univ_elT),
list_comb (Const (rec_name, fTs @ [T] ---> Univ_elT), fs)))) (rec_names ~~ isos);
val (def_thms, thy') =
apsnd Theory.checkpoint ((Global_Theory.add_defs 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 => Skip_Proof.prove_global thy' [] [] eqn
(fn _ => EVERY [rewrite_goals_tac rewrites, rtac refl 1])) eqns;
in (thy', char_thms' @ char_thms) end;
val (thy5, iso_char_thms) =
apfst Theory.checkpoint (fold_rev make_iso_defs (tl descr) (Sign.add_path big_name thy4, []));
(* prove isomorphism properties *)
fun mk_funs_inv thy thm =
let
val prop = Thm.prop_of thm;
val _ $ (_ $ ((S as Const (_, Type (_, [U, _]))) $ _ )) $
(_ $ (_ $ (r $ (a $ _)) $ _)) = Type.legacy_freeze prop;
val used = Term.add_tfree_names a [];
fun mk_thm i =
let
val Ts = map (TFree o rpair HOLogic.typeS) (Name.variant_list used (replicate i "'t"));
val f = Free ("f", Ts ---> U);
in
Skip_Proof.prove_global thy [] []
(Logic.mk_implies
(HOLogic.mk_Trueprop (HOLogic.list_all
(map (pair "x") Ts, S $ Datatype_Aux.app_bnds f i)),
HOLogic.mk_Trueprop (HOLogic.mk_eq (list_abs (map (pair "x") Ts,
r $ (a $ Datatype_Aux.app_bnds f i)), f))))
(fn _ => EVERY [REPEAT_DETERM_N i (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 *)
val fun_congs =
map (fn T => make_elim (Drule.instantiate' [SOME (ctyp_of thy5 T)] [] fun_cong)) branchTs;
fun prove_iso_thms ds (inj_thms, elem_thms) =
let
val (_, (tname, _, _)) = hd ds;
val induct = #induct (the (Symtab.lookup dt_info tname));
fun mk_ind_concl (i, _) =
let
val T = nth recTs i;
val Rep_t = Const (nth all_rep_names i, T --> Univ_elT);
val rep_set_name = nth rep_set_names i;
in
(HOLogic.all_const T $ Abs ("y", T, HOLogic.imp $
HOLogic.mk_eq (Rep_t $ Datatype_Aux.mk_Free "x" T i, Rep_t $ Bound 0) $
HOLogic.mk_eq (Datatype_Aux.mk_Free "x" T i, Bound 0)),
Const (rep_set_name, UnivT') $ (Rep_t $ Datatype_Aux.mk_Free "x" T i))
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 snd newT_iso_inj_thms @ map (fn r => r RS @{thm injD}) inj_thms;
val inj_thm =
Skip_Proof.prove_global thy5 [] []
(HOLogic.mk_Trueprop (Datatype_Aux.mk_conj ind_concl1))
(fn _ => EVERY
[(Datatype_Aux.ind_tac induct [] THEN_ALL_NEW Object_Logic.atomize_prems_tac) 1,
REPEAT (EVERY
[rtac allI 1, rtac impI 1,
Datatype_Aux.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 :: inj_thms') @ fun_congs) 1),
atac 1]))])])])]);
val inj_thms'' = map (fn r => r RS datatype_injI) (Datatype_Aux.split_conj_thm inj_thm);
val elem_thm =
Skip_Proof.prove_global thy5 [] []
(HOLogic.mk_Trueprop (Datatype_Aux.mk_conj ind_concl2))
(fn _ =>
EVERY [(Datatype_Aux.ind_tac induct [] THEN_ALL_NEW Object_Logic.atomize_prems_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 @ (Datatype_Aux.split_conj_thm elem_thm)) end;
val (iso_inj_thms_unfolded, iso_elem_thms) =
fold_rev prove_iso_thms (tl descr) ([], map #3 newT_iso_axms);
val iso_inj_thms =
map snd newT_iso_inj_thms @ map (fn r => r RS @{thm injD}) iso_inj_thms_unfolded;
(* prove dt_rep_set_i x --> x : range dt_Rep_i *)
fun mk_iso_t (((set_name, iso_name), i), T) =
let val isoT = T --> Univ_elT in
HOLogic.imp $
(Const (set_name, UnivT') $ Datatype_Aux.mk_Free "x" Univ_elT i) $
(if i < length newTs then @{term True}
else HOLogic.mk_mem (Datatype_Aux.mk_Free "x" Univ_elT i,
Const (@{const_name image}, isoT --> HOLogic.mk_setT T --> UnivT) $
Const (iso_name, isoT) $ Const (@{const_abbrev UNIV}, HOLogic.mk_setT T)))
end;
val iso_t = HOLogic.mk_Trueprop (Datatype_Aux.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 @{thm range_ex1_eq})) iso_inj_thms_unfolded;
val iso_thms =
if length descr = 1 then []
else
drop (length newTs) (Datatype_Aux.split_conj_thm
(Skip_Proof.prove_global thy5 [] [] iso_t (fn _ => EVERY
[(Datatype_Aux.ind_tac rep_induct [] THEN_ALL_NEW Object_Logic.atomize_prems_tac) 1,
REPEAT (rtac TrueI 1),
rewrite_goals_tac (mk_meta_eq @{thm choice_eq} ::
Thm.symmetric (mk_meta_eq @{thm fun_eq_iff}) :: range_eqs),
rewrite_goals_tac (map Thm.symmetric range_eqs),
REPEAT (EVERY
[REPEAT (eresolve_tac ([rangeE, ex1_implies_ex RS exE] @
maps (mk_funs_inv thy5 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 => fn r => @{thm f_the_inv_into_f} OF [r_inj, r RS mp])
iso_inj_thms_unfolded iso_thms;
val Abs_inverse_thms = maps (mk_funs_inv thy5) Abs_inverse_thms';
(******************* freeness theorems for constructors *******************)
val _ = Datatype_Aux.message config "Proving freeness of constructors ...";
(* prove theorem Rep_i (Constr_j ...) = Inj_j ... *)
fun prove_constr_rep_thm eqn =
let
val inj_thms = map fst newT_iso_inj_thms;
val rewrites = @{thm o_def} :: constr_defs @ (map (mk_meta_eq o #2) newT_iso_axms);
in
Skip_Proof.prove_global thy5 [] [] eqn
(fn _ => EVERY
[resolve_tac inj_thms 1,
rewrite_goals_tac rewrites,
rtac refl 3,
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 dist_rewrites' (k, ts) =
let
fun prove [] = []
| prove (t :: ts) =
let
val dist_thm = Skip_Proof.prove_global thy5 [] [] t (fn _ =>
EVERY [simp_tac (HOL_ss addsimps dist_rewrites') 1])
in dist_thm :: Drule.export_without_context (dist_thm RS not_sym) :: prove ts end;
in prove ts end;
val distinct_thms =
map2 (prove_distinct_thms) dist_rewrites (Datatype_Prop.make_distincts descr);
(* prove injectivity of constructors *)
fun prove_constr_inj_thm rep_thms t =
let
val inj_thms = Scons_inject ::
map make_elim
(iso_inj_thms @
[In0_inject, In1_inject, Leaf_inject, Inl_inject, Inr_inject,
Lim_inject, Suml_inject, Sumr_inject])
in
Skip_Proof.prove_global thy5 [] [] t
(fn _ => EVERY
[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)
(Datatype_Prop.make_injs descr ~~ constr_rep_thms);
val ((constr_inject', distinct_thms'), thy6) =
thy5
|> Sign.parent_path
|> Datatype_Aux.store_thmss "inject" new_type_names constr_inject
||>> Datatype_Aux.store_thmss "distinct" new_type_names distinct_thms;
(*************************** induction theorem ****************************)
val _ = Datatype_Aux.message config "Proving induction rule for datatypes ...";
val Rep_inverse_thms =
map (fn (_, iso, _) => iso RS subst) newT_iso_axms @
map (fn r => r RS @{thm the_inv_f_f} RS subst) iso_inj_thms_unfolded;
val Rep_inverse_thms' = map (fn r => r RS @{thm the_inv_f_f}) iso_inj_thms_unfolded;
fun mk_indrule_lemma ((i, _), T) (prems, concls) =
let
val Rep_t =
Const (nth all_rep_names i, T --> Univ_elT) $ Datatype_Aux.mk_Free "x" T i;
val Abs_t =
if i < length newTs then
Const (Sign.intern_const thy6 ("Abs_" ^ nth new_type_names i), Univ_elT --> T)
else Const (@{const_name the_inv_into},
[HOLogic.mk_setT T, T --> Univ_elT, Univ_elT] ---> T) $
HOLogic.mk_UNIV T $ Const (nth all_rep_names i, T --> Univ_elT);
in
(prems @
[HOLogic.imp $
(Const (nth rep_set_names i, UnivT') $ Rep_t) $
(Datatype_Aux.mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ (Abs_t $ Rep_t))],
concls @
[Datatype_Aux.mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ Datatype_Aux.mk_Free "x" T i])
end;
val (indrule_lemma_prems, indrule_lemma_concls) =
fold mk_indrule_lemma (descr' ~~ recTs) ([], []);
val cert = cterm_of thy6;
val indrule_lemma =
Skip_Proof.prove_global thy6 [] []
(Logic.mk_implies
(HOLogic.mk_Trueprop (Datatype_Aux.mk_conj indrule_lemma_prems),
HOLogic.mk_Trueprop (Datatype_Aux.mk_conj indrule_lemma_concls)))
(fn _ =>
EVERY
[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_prop = Datatype_Prop.make_ind descr;
val dt_induct =
Skip_Proof.prove_global thy6 []
(Logic.strip_imp_prems dt_induct_prop)
(Logic.strip_imp_concl dt_induct_prop)
(fn {prems, ...} =>
EVERY
[rtac indrule_lemma' 1,
(Datatype_Aux.ind_tac rep_induct [] THEN_ALL_NEW Object_Logic.atomize_prems_tac) 1,
EVERY (map (fn (prem, r) => (EVERY
[REPEAT (eresolve_tac Abs_inverse_thms 1),
simp_tac (HOL_basic_ss addsimps (Thm.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 ([dt_induct'], thy7) =
thy6
|> Global_Theory.add_thms
[((Binding.qualify true big_name (Binding.name "induct"), dt_induct),
[case_names_induct])]
||> Theory.checkpoint;
in
((constr_inject', distinct_thms', dt_induct'), thy7)
end;
(** datatype definition **)
fun gen_add_datatype prep_typ config dts thy =
let
val _ = Theory.requires thy "Datatype" "datatype definitions";
(* this theory is used just for parsing *)
val tmp_thy = thy
|> Theory.copy
|> Sign.add_types_global (map (fn (tvs, tname, mx, _) => (tname, length tvs, mx)) dts);
val tmp_ctxt = Proof_Context.init_global tmp_thy;
val (tyvars, _, _, _) ::_ = dts;
val (new_dts, types_syntax) = ListPair.unzip (map (fn (tvs, tname, mx, _) =>
let val full_tname = Sign.full_name tmp_thy tname in
(case duplicates (op =) tvs of
[] =>
if eq_set (op =) (tyvars, tvs) then ((full_tname, tvs), (tname, mx))
else error ("Mutually recursive datatypes must have same type parameters")
| dups =>
error ("Duplicate parameter(s) for datatype " ^ Binding.print tname ^
" : " ^ commas dups))
end) dts);
val dt_names = map fst new_dts;
val _ =
(case duplicates (op =) (map fst new_dts) of
[] => ()
| dups => error ("Duplicate datatypes: " ^ commas_quote dups));
fun prep_dt_spec (tvs, tname, mx, constrs) (dts', constr_syntax, sorts, i) =
let
fun prep_constr (cname, cargs, mx') (constrs, constr_syntax', sorts') =
let
val (cargs', sorts'') = fold_map (prep_typ tmp_thy) cargs sorts';
val _ =
(case subtract (op =) tvs (fold Term.add_tfree_namesT cargs' []) of
[] => ()
| vs => error ("Extra type variables on rhs: " ^ commas vs));
val c = Sign.full_name_path tmp_thy (Binding.name_of tname) cname;
in
(constrs @ [(c, map (Datatype_Aux.dtyp_of_typ new_dts) cargs')],
constr_syntax' @ [(cname, mx')], sorts'')
end handle ERROR msg =>
cat_error msg ("The error above occurred in constructor " ^ Binding.print cname ^
" of datatype " ^ Binding.print tname);
val (constrs', constr_syntax', sorts') = fold prep_constr constrs ([], [], sorts);
in
(case duplicates (op =) (map fst constrs') of
[] =>
(dts' @ [(i, (Sign.full_name tmp_thy tname, tvs, constrs'))],
constr_syntax @ [constr_syntax'], sorts', i + 1)
| dups =>
error ("Duplicate constructors " ^ commas_quote dups ^
" in datatype " ^ Binding.print tname))
end;
val (dts0, constr_syntax, sorts', i) = fold prep_dt_spec dts ([], [], [], 0);
val tmp_ctxt' = tmp_ctxt |> fold (Variable.declare_typ o TFree) sorts';
val dts' = dts0 |> map (fn (i, (name, tvs, cs)) =>
let
val args = tvs |>
map (fn a => Datatype_Aux.DtTFree (a, Proof_Context.default_sort tmp_ctxt' (a, ~1)));
in (i, (name, args, cs)) end);
val dt_info = Datatype_Data.get_all thy;
val (descr, _) = Datatype_Aux.unfold_datatypes tmp_ctxt dts' dt_info dts' i;
val _ =
Datatype_Aux.check_nonempty descr
handle (exn as Datatype_Aux.Datatype_Empty s) =>
if #strict config then error ("Nonemptiness check failed for datatype " ^ quote s)
else reraise exn;
val _ =
Datatype_Aux.message config
("Constructing datatype(s) " ^ commas_quote (map (Binding.name_of o #2) dts));
in
thy
|> representation_proofs config dt_info descr types_syntax constr_syntax
(Datatype_Data.mk_case_names_induct (flat descr))
|-> (fn (inject, distinct, induct) =>
Datatype_Data.derive_datatype_props config dt_names descr induct inject distinct)
end;
val add_datatype = gen_add_datatype Datatype_Data.cert_typ;
val add_datatype_cmd = snd oo gen_add_datatype Datatype_Data.read_typ Datatype_Aux.default_config;
(* concrete syntax *)
val parse_decl =
Parse.type_args -- Parse.binding -- Parse.opt_mixfix --
(Parse.$$$ "=" |-- Parse.enum1 "|" (Parse.binding -- Scan.repeat Parse.typ -- Parse.opt_mixfix))
>> (fn (((vs, t), mx), cons) => (vs, t, mx, map Parse.triple1 cons));
val _ =
Outer_Syntax.command "datatype" "define inductive datatypes" Keyword.thy_decl
(Parse.and_list1 parse_decl >> (Toplevel.theory o add_datatype_cmd));
open Datatype_Data;
end;