(* Title: HOL/Codatatype/Tools/bnf_sugar.ML
Author: Jasmin Blanchette, TU Muenchen
Copyright 2012
Sugar on top of a BNF.
*)
signature BNF_SUGAR =
sig
end;
structure BNF_Sugar : BNF_SUGAR =
struct
open BNF_Util
open BNF_FP_Util
open BNF_Sugar_Tactics
val case_congN = "case_cong"
val case_discsN = "case_discs"
val casesN = "cases"
val ctr_selsN = "ctr_sels"
val disc_disjointN = "disc_disjoint"
val disc_exhaustN = "disc_exhaust"
val discsN = "discs"
val distinctN = "distinct"
val selsN = "sels"
val splitN = "split"
val split_asmN = "split_asm"
val weak_case_cong_thmsN = "weak_case_cong"
fun mk_half_pairs [] = []
| mk_half_pairs (x :: xs) = fold_rev (cons o pair x) xs (mk_half_pairs xs);
fun index_of_half_row _ 0 = 0
| index_of_half_row n j = index_of_half_row n (j - 1) + n - j;
fun index_of_half_cell n j k = index_of_half_row n j + k - (j + 1);
fun prepare_sugar prep_term (((raw_ctrs, raw_caseof), disc_names), sel_namess) no_defs_lthy =
let
(* TODO: sanity checks on arguments *)
(* TODO: normalize types of constructors w.r.t. each other *)
val ctrs0 = map (prep_term no_defs_lthy) raw_ctrs;
val caseof0 = prep_term no_defs_lthy raw_caseof;
val n = length ctrs0;
val ks = 1 upto n;
val (T_name, As0) = dest_Type (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_undef T Ts = Const (@{const_name undefined}, Ts ---> T);
fun mk_ctr Ts ctr =
let val Ts0 = snd (dest_Type (body_type (fastype_of ctr))) in
Term.subst_atomic_types (Ts0 ~~ Ts) ctr
end;
fun mk_caseof Ts T =
let val (binders, body) = strip_type (fastype_of caseof0) in
Term.subst_atomic_types ((body, T) :: (snd (dest_Type (List.last binders)) ~~ Ts)) caseof0
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 caseofB = mk_caseof As B;
val caseofB_Ts = map (fn Ts => Ts ---> B) ctr_Tss;
val (((((xss, yss), fs), (v, v')), p), names_lthy) = no_defs_lthy |>
mk_Freess "x" ctr_Tss
||>> mk_Freess "y" ctr_Tss
||>> mk_Frees "f" caseofB_Ts
||>> yield_singleton (apfst (op ~~) oo mk_Frees' "v") T
||>> yield_singleton (mk_Frees "P") HOLogic.boolT;
val xctrs = map2 (curry Term.list_comb) ctrs xss;
val yctrs = map2 (curry Term.list_comb) ctrs yss;
val eta_fs = map2 (fn f => fn xs => fold_rev Term.lambda xs (Term.list_comb (f, xs))) fs xss;
val exist_xs_v_eq_ctrs =
map2 (fn xctr => fn xs => list_exists_free xs (HOLogic.mk_eq (v, xctr))) xctrs xss;
fun mk_caseof_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_spec b exist_xs_v_eq_ctr =
HOLogic.mk_Trueprop (HOLogic.mk_eq (Free (Binding.name_of b, T --> HOLogic.boolT) $ v,
exist_xs_v_eq_ctr));
fun sel_spec b x xs k =
let val T' = fastype_of x in
HOLogic.mk_Trueprop (HOLogic.mk_eq (Free (Binding.name_of b, T --> T') $ v,
Term.list_comb (mk_caseof As T', mk_caseof_args k xs x T') $ v))
end;
val (((raw_discs, (_, raw_disc_defs)), (raw_selss, (_, raw_sel_defss))), (lthy', lthy)) =
no_defs_lthy
|> apfst (apsnd split_list o split_list) o fold_map2 (fn b => fn exist_xs_v_eq_ctr =>
Specification.definition (SOME (b, NONE, NoSyn),
((Thm.def_binding b, []), disc_spec b exist_xs_v_eq_ctr))) disc_names exist_xs_v_eq_ctrs
||>> apfst (apsnd split_list o split_list) o fold_map3 (fn bs => fn xs => fn k =>
apfst (apsnd split_list o 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))) bs xs) sel_namess 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 (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 Q = Logic.list_implies (Q, HOLogic.mk_Trueprop p);
val goal_exhaust =
let
fun mk_prem xctr xs =
fold_rev Logic.all xs (mk_imp_p [HOLogic.mk_Trueprop (HOLogic.mk_eq (v, xctr))]);
in
mk_imp_p (map2 mk_prem xctrs xss)
end;
val goal_injects =
let
fun mk_goal _ _ [] [] = NONE
| mk_goal xctr yctr xs ys =
SOME (HOLogic.mk_Trueprop (HOLogic.mk_eq
(HOLogic.mk_eq (xctr, yctr),
Library.foldr1 HOLogic.mk_conj (map2 (curry HOLogic.mk_eq) xs ys))));
in
map_filter I (map4 mk_goal xctrs yctrs xss yss)
end;
val goal_half_distincts =
map (HOLogic.mk_Trueprop o HOLogic.mk_not o HOLogic.mk_eq) (mk_half_pairs xctrs);
val goal_cases =
let
val lhs0 = Term.list_comb (caseofB, eta_fs);
fun mk_goal xctr xs f =
HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs0 $ xctr, Term.list_comb (f, xs)));
in
map3 mk_goal xctrs xss fs
end;
val goals = [[goal_exhaust], goal_injects, goal_half_distincts, goal_cases];
fun after_qed thmss lthy =
let
val [[exhaust_thm], inject_thms, half_distinct_thms, case_thms] = thmss;
val other_half_distinct_thms = map (fn thm => thm RS not_sym) half_distinct_thms;
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_caseof_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 goal_case case_thm) xs sel_defs;
in
map5 mk_thms ks xss goal_cases case_thms sel_defss
end;
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_disc_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_thms =
let
fun get_distinct_thm k k' =
if k > k' then nth half_distinct_thms (index_of_half_cell n (k' - 1) (k - 1))
else nth other_half_distinct_thms (index_of_half_cell n (k' - 1) (k' - 1))
fun mk_thm ((k, discI), not_disc) k' =
if k = k' then refl RS discI else get_distinct_thm k k' RS not_disc;
in
map_product mk_thm (ks ~~ discI_thms ~~ not_disc_thms) ks
end;
val disc_disjoint_thms =
let
fun get_disc_thm k k' = nth disc_thms ((k' - 1) * n + (k - 1));
fun mk_goal ((_, disc), (_, disc')) =
Logic.all v (Logic.mk_implies (HOLogic.mk_Trueprop (disc $ v),
HOLogic.mk_Trueprop (HOLogic.mk_not (disc' $ v))));
fun prove tac goal = Skip_Proof.prove lthy [] [] goal (K tac);
val bundles = ks ~~ ms ~~ discD_thms ~~ discs;
val half_pairs = mk_half_pairs bundles;
val goal_halves = map mk_goal half_pairs;
val half_thms =
map2 (fn ((((k, m), discD), _), (((k', _), _), _)) =>
prove (mk_half_disc_disjoint_tac m discD (get_disc_thm k k')))
half_pairs goal_halves;
val goal_other_halves = map (mk_goal o swap) half_pairs;
val other_half_thms =
map2 (prove o mk_other_half_disc_disjoint_tac) half_thms goal_other_halves;
in
half_thms @ other_half_thms
end;
val disc_exhaust_thm =
let
fun mk_prem disc = mk_imp_p [HOLogic.mk_Trueprop (disc $ v)];
val goal = fold 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 ctr_sel_thms =
let
fun mk_goal ctr disc sels =
Logic.all v (Logic.mk_implies (HOLogic.mk_Trueprop (disc $ v),
HOLogic.mk_Trueprop (HOLogic.mk_eq ((null sels ? swap)
(Term.list_comb (ctr, map (fn sel => sel $ v) sels), v)))));
val goals = map3 mk_goal ctrs discs selss;
in
map4 (fn goal => fn m => fn discD => fn sel_thms =>
Skip_Proof.prove lthy [] [] goal (fn {context = ctxt, ...} =>
mk_ctr_sel_tac ctxt m discD sel_thms))
goals ms discD_thms sel_thmss
end;
val case_disc_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) $
(disc $ v) $ mk_core f sels $ mk_rhs discs fs selss;
val lhs = Term.list_comb (caseofB, eta_fs) $ v;
val rhs = mk_rhs discs fs selss;
val goal = HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs));
val Tinst = map (pairself (certifyT lthy)) (map Logic.varifyT_global As ~~ As);
val exhaust_thm' =
Drule.instantiate' [] [SOME (certify lthy v)]
(Thm.instantiate (Tinst, []) (Drule.zero_var_indexes exhaust_thm));
in
Skip_Proof.prove lthy [] [] goal (fn {context = ctxt, ...} =>
mk_case_disc_tac ctxt exhaust_thm' case_thms disc_thms sel_thmss)
|> singleton (Proof_Context.export names_lthy lthy)
end;
val case_cong_thm = TrueI;
val weak_case_cong_thms = TrueI;
val split_thms = [];
val split_asm_thms = [];
(* case syntax *)
fun note thmN thms =
snd o Local_Theory.note
((Binding.qualify true (Binding.name_of b) (Binding.name thmN), []), thms);
in
lthy
|> note case_congN [case_cong_thm]
|> note case_discsN [case_disc_thm]
|> note casesN case_thms
|> note ctr_selsN ctr_sel_thms
|> note discsN disc_thms
|> note disc_disjointN disc_disjoint_thms
|> note disc_exhaustN [disc_exhaust_thm]
|> note distinctN (half_distinct_thms @ other_half_distinct_thms)
|> note exhaustN [exhaust_thm]
|> note injectN inject_thms
|> note nchotomyN [nchotomy_thm]
|> note selsN (flat sel_thmss)
|> note splitN split_thms
|> note split_asmN split_asm_thms
|> note weak_case_cong_thmsN [weak_case_cong_thms]
end;
in
(goals, after_qed, lthy')
end;
val parse_binding_list = Parse.$$$ "[" |-- Parse.list Parse.binding --| Parse.$$$ "]";
val bnf_sugar_cmd = (fn (goalss, after_qed, lthy) =>
Proof.theorem NONE after_qed (map (map (rpair [])) goalss) lthy) oo
prepare_sugar Syntax.read_term;
val _ =
Outer_Syntax.local_theory_to_proof @{command_spec "bnf_sugar"} "adds sugar on top of a BNF"
(((Parse.$$$ "[" |-- Parse.list Parse.term --| Parse.$$$ "]") -- Parse.term --
parse_binding_list -- (Parse.$$$ "[" |-- Parse.list parse_binding_list --| Parse.$$$ "]"))
>> bnf_sugar_cmd);
end;