(* Title: HOL/Codatatype/Tools/bnf_wrap.ML
Author: Jasmin Blanchette, TU Muenchen
Copyright 2012
Wrapping existing datatypes.
*)
signature BNF_WRAP =
sig
val no_binder: binding
val mk_half_pairss: 'a list -> ('a * 'a) list list
val wrap_datatype: ({prems: thm list, context: Proof.context} -> tactic) list list ->
(term list * term) * (binding list * binding list list) -> local_theory -> local_theory
end;
structure BNF_Wrap : BNF_WRAP =
struct
open BNF_Util
open BNF_Wrap_Tactics
val is_N = "is_";
val un_N = "un_";
fun mk_un_N 1 1 suf = un_N ^ suf
| mk_un_N _ l suf = un_N ^ suf ^ string_of_int l;
val case_congN = "case_cong";
val case_eqN = "case_eq";
val casesN = "cases";
val collapseN = "collapse";
val disc_excludeN = "disc_exclude";
val disc_exhaustN = "disc_exhaust";
val discsN = "discs";
val distinctN = "distinct";
val exhaustN = "exhaust";
val injectN = "inject";
val nchotomyN = "nchotomy";
val selsN = "sels";
val splitN = "split";
val split_asmN = "split_asm";
val weak_case_cong_thmsN = "weak_case_cong";
val no_binder = @{binding ""};
val fallback_binder = @{binding _};
fun pad_list x n xs = xs @ replicate (n - length xs) x;
fun mk_half_pairss' _ [] = []
| mk_half_pairss' indent (y :: ys) =
indent @ fold_rev (cons o single o pair y) ys (mk_half_pairss' ([] :: indent) ys);
fun mk_half_pairss ys = mk_half_pairss' [[]] ys;
(* TODO: provide a way to have a different default value, e.g. "tl Nil = Nil" *)
fun mk_undef T Ts = Const (@{const_name undefined}, Ts ---> T);
fun eta_expand_case_arg xs f_xs = fold_rev Term.lambda xs f_xs;
fun name_of_ctr t =
case head_of t of
Const (s, _) => s
| Free (s, _) => s
| _ => error "Cannot extract name of constructor";
fun prepare_wrap_datatype prep_term ((raw_ctrs, raw_case), (raw_disc_binders, raw_sel_binderss))
no_defs_lthy =
let
(* TODO: sanity checks on arguments *)
(* TODO: attributes (simp, case_names, etc.) *)
(* TODO: case syntax *)
(* TODO: integration with function package ("size") *)
val ctrs0 = map (prep_term no_defs_lthy) raw_ctrs;
val case0 = prep_term no_defs_lthy raw_case;
val n = length ctrs0;
val ks = 1 upto n;
val _ = if n > 0 then () else error "No constructors specified";
val Type (T_name, As0) = body_type (fastype_of (hd ctrs0));
val b = Binding.qualified_name T_name;
val (As, B) =
no_defs_lthy
|> mk_TFrees (length As0)
||> the_single o fst o mk_TFrees 1;
fun mk_ctr Ts ctr =
let val Type (_, Ts0) = body_type (fastype_of ctr) in
Term.subst_atomic_types (Ts0 ~~ Ts) ctr
end;
val T = Type (T_name, As);
val ctrs = map (mk_ctr As) ctrs0;
val ctr_Tss = map (binder_types o fastype_of) ctrs;
val ms = map length ctr_Tss;
val raw_disc_binders' = pad_list no_binder n raw_disc_binders;
fun can_really_rely_on_disc k =
not (Binding.eq_name (nth raw_disc_binders' (k - 1), no_binder)) orelse nth ms (k - 1) = 0;
fun can_rely_on_disc k =
can_really_rely_on_disc k orelse (k = 1 andalso not (can_really_rely_on_disc 2));
fun can_omit_disc_binder k m =
n = 1 orelse m = 0 orelse (n = 2 andalso can_rely_on_disc (3 - k));
val fallback_disc_binder = Binding.name o prefix is_N o Long_Name.base_name o name_of_ctr;
val disc_binders =
raw_disc_binders'
|> map4 (fn k => fn m => fn ctr => fn disc =>
if Binding.eq_name (disc, no_binder) then
if can_omit_disc_binder k m then NONE else SOME (fallback_disc_binder ctr)
else if Binding.eq_name (disc, fallback_binder) then
SOME (fallback_disc_binder ctr)
else
SOME disc) ks ms ctrs0;
val no_discs = map is_none disc_binders;
val no_discs_at_all = forall I no_discs;
fun fallback_sel_binder m l = Binding.name o mk_un_N m l o Long_Name.base_name o name_of_ctr;
val sel_binderss =
pad_list [] n raw_sel_binderss
|> map3 (fn ctr => fn m => map2 (fn l => fn sel =>
if Binding.eq_name (sel, no_binder) orelse Binding.eq_name (sel, fallback_binder) then
fallback_sel_binder m l ctr
else
sel) (1 upto m) o pad_list no_binder m) ctrs0 ms;
fun mk_case Ts T =
let
val (binders, body) = strip_type (fastype_of case0)
val Type (_, Ts0) = List.last binders
in Term.subst_atomic_types ((body, T) :: (Ts0 ~~ Ts)) case0 end;
val caseB = mk_case As B;
val caseB_Ts = map (fn Ts => Ts ---> B) ctr_Tss;
fun mk_caseB_term eta_fs = Term.list_comb (caseB, eta_fs);
val (((((((xss, yss), fs), gs), (v, v')), w), (p, p')), names_lthy) = no_defs_lthy |>
mk_Freess "x" ctr_Tss
||>> mk_Freess "y" ctr_Tss
||>> mk_Frees "f" caseB_Ts
||>> mk_Frees "g" caseB_Ts
||>> yield_singleton (apfst (op ~~) oo mk_Frees' "v") T
||>> yield_singleton (mk_Frees "w") T
||>> yield_singleton (apfst (op ~~) oo mk_Frees' "P") HOLogic.boolT;
val q = Free (fst p', B --> HOLogic.boolT);
fun mk_v_eq_v () = HOLogic.mk_eq (v, v);
val xctrs = map2 (curry Term.list_comb) ctrs xss;
val yctrs = map2 (curry Term.list_comb) ctrs yss;
val xfs = map2 (curry Term.list_comb) fs xss;
val xgs = map2 (curry Term.list_comb) gs xss;
val eta_fs = map2 eta_expand_case_arg xss xfs;
val eta_gs = map2 eta_expand_case_arg xss xgs;
val caseB_fs = Term.list_comb (caseB, eta_fs);
val exist_xs_v_eq_ctrs =
map2 (fn xctr => fn xs => list_exists_free xs (HOLogic.mk_eq (v, xctr))) xctrs xss;
fun mk_sel_case_args k xs x T =
map2 (fn Ts => fn i => if i = k then fold_rev Term.lambda xs x else mk_undef T Ts) ctr_Tss ks;
fun disc_free b = Free (Binding.name_of b, T --> HOLogic.boolT);
fun disc_spec b exist_xs_v_eq_ctr = mk_Trueprop_eq (disc_free b $ v, exist_xs_v_eq_ctr);
fun alternate_disc_lhs k =
HOLogic.mk_not
(case nth disc_binders (k - 1) of
NONE => nth exist_xs_v_eq_ctrs (k - 1)
| SOME b => disc_free b $ v);
fun alternate_disc k =
if n = 2 then Term.lambda v (alternate_disc_lhs (3 - k)) else error "Cannot use \"*\" here"
fun sel_spec b x xs k =
let val T' = fastype_of x in
mk_Trueprop_eq (Free (Binding.name_of b, T --> T') $ v,
Term.list_comb (mk_case As T', mk_sel_case_args k xs x T') $ v)
end;
val missing_unique_disc_def = TrueI; (*arbitrary marker*)
val missing_alternate_disc_def = FalseE; (*arbitrary marker*)
(* TODO: Allow use of same selector for several constructors *)
(* TODO: Allow use of same name for datatype and for constructor, e.g. "data L = L" *)
val (((raw_discs, raw_disc_defs), (raw_selss, raw_sel_defss)), (lthy', lthy)) =
no_defs_lthy
|> apfst split_list o fold_map4 (fn k => fn m => fn exist_xs_v_eq_ctr =>
fn NONE =>
if n = 1 then pair (Term.lambda v (mk_v_eq_v ()), missing_unique_disc_def)
else if m = 0 then pair (Term.lambda v exist_xs_v_eq_ctr, refl)
else pair (alternate_disc k, missing_alternate_disc_def)
| SOME b => Specification.definition (SOME (b, NONE, NoSyn),
((Thm.def_binding b, []), disc_spec b exist_xs_v_eq_ctr)) #>> apsnd snd)
ks ms exist_xs_v_eq_ctrs disc_binders
||>> apfst split_list o fold_map3 (fn bs => fn xs => fn k => apfst split_list o
fold_map2 (fn b => fn x => Specification.definition (SOME (b, NONE, NoSyn),
((Thm.def_binding b, []), sel_spec b x xs k)) #>> apsnd snd) bs xs) sel_binderss xss ks
||> `Local_Theory.restore;
(*transforms defined frees into consts (and more)*)
val phi = Proof_Context.export_morphism lthy lthy';
val disc_defs = map (Morphism.thm phi) raw_disc_defs;
val sel_defss = map (map (Morphism.thm phi)) raw_sel_defss;
val discs0 = map (Morphism.term phi) raw_discs;
val selss0 = map (map (Morphism.term phi)) raw_selss;
fun mk_disc_or_sel Ts t =
Term.subst_atomic_types (snd (Term.dest_Type (domain_type (fastype_of t))) ~~ Ts) t;
val discs = map (mk_disc_or_sel As) discs0;
val selss = map (map (mk_disc_or_sel As)) selss0;
fun mk_imp_p Qs = Logic.list_implies (Qs, HOLogic.mk_Trueprop p);
val goal_exhaust =
let fun mk_prem xctr xs = fold_rev Logic.all xs (mk_imp_p [mk_Trueprop_eq (v, xctr)]) in
fold_rev Logic.all [p, v] (mk_imp_p (map2 mk_prem xctrs xss))
end;
val goal_injectss =
let
fun mk_goal _ _ [] [] = []
| mk_goal xctr yctr xs ys =
[fold_rev Logic.all (xs @ ys) (mk_Trueprop_eq (HOLogic.mk_eq (xctr, yctr),
Library.foldr1 HOLogic.mk_conj (map2 (curry HOLogic.mk_eq) xs ys)))];
in
map4 mk_goal xctrs yctrs xss yss
end;
val goal_half_distinctss =
let
fun mk_goal ((xs, t), (xs', t')) =
fold_rev Logic.all (xs @ xs')
(HOLogic.mk_Trueprop (HOLogic.mk_not (HOLogic.mk_eq (t, t'))));
in
map (map mk_goal) (mk_half_pairss (xss ~~ xctrs))
end;
val goal_cases =
map3 (fn xs => fn xctr => fn xf =>
fold_rev Logic.all (fs @ xs) (mk_Trueprop_eq (caseB_fs $ xctr, xf))) xss xctrs xfs;
val goalss = [goal_exhaust] :: goal_injectss @ goal_half_distinctss @ [goal_cases];
fun after_qed thmss lthy =
let
val ([exhaust_thm], (inject_thmss, (half_distinct_thmss, [case_thms]))) =
(hd thmss, apsnd (chop (n * n)) (chop n (tl thmss)));
val exhaust_thm' =
let val Tinst = map (pairself (certifyT lthy)) (map Logic.varifyT_global As ~~ As) in
Drule.instantiate' [] [SOME (certify lthy v)]
(Thm.instantiate (Tinst, []) (Drule.zero_var_indexes exhaust_thm))
end;
val other_half_distinct_thmss = map (map (fn thm => thm RS not_sym)) half_distinct_thmss;
val (distinct_thmsss', distinct_thmsss) =
map2 (map2 append) (Library.chop_groups n half_distinct_thmss)
(transpose (Library.chop_groups n other_half_distinct_thmss))
|> `transpose;
val distinct_thms = interleave (flat half_distinct_thmss) (flat other_half_distinct_thmss);
val nchotomy_thm =
let
val goal =
HOLogic.mk_Trueprop (HOLogic.mk_all (fst v', snd v',
Library.foldr1 HOLogic.mk_disj exist_xs_v_eq_ctrs));
in
Skip_Proof.prove lthy [] [] goal (fn _ => mk_nchotomy_tac n exhaust_thm)
end;
val sel_thmss =
let
fun mk_thm k xs goal_case case_thm x sel_def =
let
val T = fastype_of x;
val cTs =
map ((fn T' => certifyT lthy (if T' = B then T else T')) o TFree)
(rev (Term.add_tfrees goal_case []));
val cxs = map (certify lthy) (mk_sel_case_args k xs x T);
in
Local_Defs.fold lthy [sel_def]
(Drule.instantiate' (map SOME cTs) (map SOME cxs) case_thm)
end;
fun mk_thms k xs goal_case case_thm sel_defs =
map2 (mk_thm k xs (strip_all_body goal_case) case_thm) xs sel_defs;
in
map5 mk_thms ks xss goal_cases case_thms sel_defss
end;
fun mk_unique_disc_def () =
let
val m = the_single ms;
val goal = mk_Trueprop_eq (mk_v_eq_v (), the_single exist_xs_v_eq_ctrs);
in
Skip_Proof.prove lthy [] [] goal (fn _ => mk_unique_disc_def_tac m exhaust_thm')
|> singleton (Proof_Context.export names_lthy lthy)
|> Thm.close_derivation
end;
fun mk_alternate_disc_def k =
let
val goal =
mk_Trueprop_eq (Morphism.term phi (alternate_disc_lhs (3 - k)),
nth exist_xs_v_eq_ctrs (k - 1));
in
Skip_Proof.prove lthy [] [] goal (fn {context = ctxt, ...} =>
mk_alternate_disc_def_tac ctxt k (nth disc_defs (2 - k)) (nth distinct_thms (2 - k))
exhaust_thm')
|> singleton (Proof_Context.export names_lthy lthy)
|> Thm.close_derivation
end;
val has_alternate_disc_def =
exists (fn def => Thm.eq_thm_prop (def, missing_alternate_disc_def)) disc_defs;
val disc_defs' =
map2 (fn k => fn def =>
if Thm.eq_thm_prop (def, missing_unique_disc_def) then mk_unique_disc_def ()
else if Thm.eq_thm_prop (def, missing_alternate_disc_def) then mk_alternate_disc_def k
else def)
ks disc_defs;
val discD_thms = map (fn def => def RS iffD1) disc_defs';
val discI_thms =
map2 (fn m => fn def => funpow m (fn thm => exI RS thm) (def RS iffD2)) ms disc_defs';
val not_discI_thms =
map2 (fn m => fn def => funpow m (fn thm => allI RS thm)
(Local_Defs.unfold lthy @{thms not_ex} (def RS @{thm ssubst[of _ _ Not]})))
ms disc_defs';
val (disc_thmss', disc_thmss) =
let
fun mk_thm discI _ [] = refl RS discI
| mk_thm _ not_discI [distinct] = distinct RS not_discI;
fun mk_thms discI not_discI distinctss = map (mk_thm discI not_discI) distinctss;
in
map3 mk_thms discI_thms not_discI_thms distinct_thmsss' |> `transpose
end;
val disc_thms = flat (map2 (fn true => K [] | false => I) no_discs disc_thmss);
val disc_exclude_thms =
if has_alternate_disc_def then
[]
else
let
fun mk_goal [] = []
| mk_goal [((_, true), (_, true))] = []
| mk_goal [(((_, disc), _), ((_, disc'), _))] =
[Logic.all v (Logic.mk_implies (HOLogic.mk_Trueprop (betapply (disc, v)),
HOLogic.mk_Trueprop (HOLogic.mk_not (betapply (disc', v)))))];
fun prove tac goal = Skip_Proof.prove lthy [] [] goal (K tac);
val bundles = ms ~~ discD_thms ~~ discs ~~ no_discs;
val half_pairss = mk_half_pairss bundles;
val goal_halvess = map mk_goal half_pairss;
val half_thmss =
map3 (fn [] => K (K []) | [goal] => fn [((((m, discD), _), _), _)] => fn disc_thm =>
[prove (mk_half_disc_exclude_tac m discD disc_thm) goal])
goal_halvess half_pairss (flat disc_thmss');
val goal_other_halvess = map (mk_goal o map swap) half_pairss;
val other_half_thmss =
map2 (map2 (prove o mk_other_half_disc_exclude_tac)) half_thmss goal_other_halvess;
in
interleave (flat half_thmss) (flat other_half_thmss)
end;
val disc_exhaust_thms =
if has_alternate_disc_def orelse no_discs_at_all then
[]
else
let
fun mk_prem disc = mk_imp_p [HOLogic.mk_Trueprop (betapply (disc, v))];
val goal = fold_rev Logic.all [p, v] (mk_imp_p (map mk_prem discs));
in
[Skip_Proof.prove lthy [] [] goal (fn _ =>
mk_disc_exhaust_tac n exhaust_thm discI_thms)]
end;
val collapse_thms =
let
fun mk_goal ctr disc sels =
let
val prem = HOLogic.mk_Trueprop (betapply (disc, v));
val concl =
mk_Trueprop_eq ((null sels ? swap)
(Term.list_comb (ctr, map (fn sel => sel $ v) sels), v));
in
if prem aconv concl then NONE
else SOME (Logic.all v (Logic.mk_implies (prem, concl)))
end;
val goals = map3 mk_goal ctrs discs selss;
in
map4 (fn m => fn discD => fn sel_thms => Option.map (fn goal =>
Skip_Proof.prove lthy [] [] goal (fn {context = ctxt, ...} =>
mk_collapse_tac ctxt m discD sel_thms)
|> perhaps (try (fn thm => refl RS thm)))) ms discD_thms sel_thmss goals
|> map_filter I
end;
val case_eq_thm =
let
fun mk_core f sels = Term.list_comb (f, map (fn sel => sel $ v) sels);
fun mk_rhs _ [f] [sels] = mk_core f sels
| mk_rhs (disc :: discs) (f :: fs) (sels :: selss) =
Const (@{const_name If}, HOLogic.boolT --> B --> B --> B) $
betapply (disc, v) $ mk_core f sels $ mk_rhs discs fs selss;
val goal = mk_Trueprop_eq (caseB_fs $ v, mk_rhs discs fs selss);
in
Skip_Proof.prove lthy [] [] goal (fn {context = ctxt, ...} =>
mk_case_eq_tac ctxt n exhaust_thm' case_thms disc_thmss' sel_thmss)
|> singleton (Proof_Context.export names_lthy lthy)
end;
val (case_cong_thm, weak_case_cong_thm) =
let
fun mk_prem xctr xs f g =
fold_rev Logic.all xs (Logic.mk_implies (mk_Trueprop_eq (w, xctr),
mk_Trueprop_eq (f, g)));
val v_eq_w = mk_Trueprop_eq (v, w);
val case_fs = mk_caseB_term eta_fs;
val case_gs = mk_caseB_term eta_gs;
val goal =
Logic.list_implies (v_eq_w :: map4 mk_prem xctrs xss fs gs,
mk_Trueprop_eq (case_fs $ v, case_gs $ w));
val goal_weak = Logic.mk_implies (v_eq_w, mk_Trueprop_eq (case_fs $ v, case_fs $ w));
in
(Skip_Proof.prove lthy [] [] goal (fn _ => mk_case_cong_tac exhaust_thm' case_thms),
Skip_Proof.prove lthy [] [] goal_weak (K (etac arg_cong 1)))
|> pairself (singleton (Proof_Context.export names_lthy lthy))
end;
val (split_thm, split_asm_thm) =
let
fun mk_conjunct xctr xs f_xs =
list_all_free xs (HOLogic.mk_imp (HOLogic.mk_eq (v, xctr), q $ f_xs));
fun mk_disjunct xctr xs f_xs =
list_exists_free xs (HOLogic.mk_conj (HOLogic.mk_eq (v, xctr),
HOLogic.mk_not (q $ f_xs)));
val lhs = q $ (mk_caseB_term eta_fs $ v);
val goal =
mk_Trueprop_eq (lhs, Library.foldr1 HOLogic.mk_conj (map3 mk_conjunct xctrs xss xfs));
val goal_asm =
mk_Trueprop_eq (lhs, HOLogic.mk_not (Library.foldr1 HOLogic.mk_disj
(map3 mk_disjunct xctrs xss xfs)));
val split_thm =
Skip_Proof.prove lthy [] [] goal
(fn _ => mk_split_tac exhaust_thm' case_thms inject_thmss distinct_thmsss)
|> singleton (Proof_Context.export names_lthy lthy)
val split_asm_thm =
Skip_Proof.prove lthy [] [] goal_asm (fn {context = ctxt, ...} =>
mk_split_asm_tac ctxt split_thm)
|> singleton (Proof_Context.export names_lthy lthy)
in
(split_thm, split_asm_thm)
end;
val notes =
[(case_congN, [case_cong_thm]),
(case_eqN, [case_eq_thm]),
(casesN, case_thms),
(collapseN, collapse_thms),
(discsN, disc_thms),
(disc_excludeN, disc_exclude_thms),
(disc_exhaustN, disc_exhaust_thms),
(distinctN, distinct_thms),
(exhaustN, [exhaust_thm]),
(injectN, flat inject_thmss),
(nchotomyN, [nchotomy_thm]),
(selsN, flat sel_thmss),
(splitN, [split_thm]),
(split_asmN, [split_asm_thm]),
(weak_case_cong_thmsN, [weak_case_cong_thm])]
|> filter_out (null o snd)
|> map (fn (thmN, thms) =>
((Binding.qualify true (Binding.name_of b) (Binding.name thmN), []), [(thms, [])]));
in
lthy |> Local_Theory.notes notes |> snd
end;
in
(goalss, after_qed, lthy')
end;
fun wrap_datatype tacss = (fn (goalss, after_qed, lthy) =>
map2 (map2 (Skip_Proof.prove lthy [] [])) goalss tacss
|> (fn thms => after_qed thms lthy)) oo
prepare_wrap_datatype (K I) (* FIXME? (singleton o Type_Infer_Context.infer_types) *)
val parse_bindings = Parse.$$$ "[" |-- Parse.list Parse.binding --| Parse.$$$ "]";
val parse_bindingss = Parse.$$$ "[" |-- Parse.list parse_bindings --| Parse.$$$ "]";
val wrap_datatype_cmd = (fn (goalss, after_qed, lthy) =>
Proof.theorem NONE after_qed (map (map (rpair [])) goalss) lthy) oo
prepare_wrap_datatype Syntax.read_term;
val _ =
Outer_Syntax.local_theory_to_proof @{command_spec "wrap_data"} "wraps an existing datatype"
(((Parse.$$$ "[" |-- Parse.list Parse.term --| Parse.$$$ "]") -- Parse.term --
Scan.optional (parse_bindings -- Scan.optional parse_bindingss []) ([], []))
>> wrap_datatype_cmd);
end;