src/HOL/Codatatype/Tools/bnf_sugar.ML
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rationalized data structure for distinctness theorems
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(*  Title:      HOL/Codatatype/Tools/bnf_sugar.ML
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    Author:     Jasmin Blanchette, TU Muenchen
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    Copyright   2012
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Sugar on top of a BNF.
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*)
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signature BNF_SUGAR =
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sig
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end;
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structure BNF_Sugar : BNF_SUGAR =
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struct
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open BNF_Util
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open BNF_FP_Util
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open BNF_Sugar_Tactics
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val is_N = "is_";
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val case_congN = "case_cong"
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val case_discsN = "case_discs"
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val casesN = "cases"
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val ctr_selsN = "ctr_sels"
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val disc_disjointN = "disc_disjoint"
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val disc_exhaustN = "disc_exhaust"
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val discsN = "discs"
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val distinctN = "distinct"
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val selsN = "sels"
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val splitN = "split"
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val split_asmN = "split_asm"
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val weak_case_cong_thmsN = "weak_case_cong"
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val default_name = @{binding _};
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fun mk_half_pairss' _ [] = []
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  | mk_half_pairss' pad (y :: ys) =
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    pad @ fold_rev (cons o single o pair y) ys (mk_half_pairss' ([] :: pad) ys);
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fun mk_half_pairss ys = mk_half_pairss' [[]] ys;
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val mk_Trueprop_eq = HOLogic.mk_Trueprop o HOLogic.mk_eq;
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fun eta_expand_caseof_arg xs f_xs = fold_rev Term.lambda xs f_xs;
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fun name_of_ctr t =
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  case head_of t of
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    Const (s, _) => s
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  | Free (s, _) => s
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  | _ => error "Cannot extract name of constructor";
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fun prepare_sugar prep_term (((raw_ctrs, raw_caseof), raw_disc_names), sel_namess) no_defs_lthy =
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  let
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    (* TODO: sanity checks on arguments *)
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    (* TODO: normalize types of constructors w.r.t. each other *)
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    val ctrs0 = map (prep_term no_defs_lthy) raw_ctrs;
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    val caseof0 = prep_term no_defs_lthy raw_caseof;
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    val disc_names =
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      map2 (fn ctr => fn disc =>
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        if Binding.eq_name (disc, default_name) then
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          Binding.name (prefix is_N (Long_Name.base_name (name_of_ctr ctr)))
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        else
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          disc) ctrs0 raw_disc_names;
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    val n = length ctrs0;
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    val ks = 1 upto n;
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    val (T_name, As0) = dest_Type (body_type (fastype_of (hd ctrs0)));
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    val b = Binding.qualified_name T_name;
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    val (As, B) =
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      no_defs_lthy
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      |> mk_TFrees (length As0)
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      ||> the_single o fst o mk_TFrees 1;
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    fun mk_undef T Ts = Const (@{const_name undefined}, Ts ---> T);
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    fun mk_ctr Ts ctr =
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      let val Ts0 = snd (dest_Type (body_type (fastype_of ctr))) in
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        Term.subst_atomic_types (Ts0 ~~ Ts) ctr
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      end;
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    fun mk_caseof Ts T =
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      let val (binders, body) = strip_type (fastype_of caseof0) in
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        Term.subst_atomic_types ((body, T) :: (snd (dest_Type (List.last binders)) ~~ Ts)) caseof0
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      end;
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    val T = Type (T_name, As);
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    val ctrs = map (mk_ctr As) ctrs0;
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    val ctr_Tss = map (binder_types o fastype_of) ctrs;
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    val ms = map length ctr_Tss;
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    val caseofB = mk_caseof As B;
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    val caseofB_Ts = map (fn Ts => Ts ---> B) ctr_Tss;
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    fun mk_caseofB_term eta_fs = Term.list_comb (caseofB, eta_fs);
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    val (((((((xss, yss), fs), gs), (v, v')), w), (p, p')), names_lthy) = no_defs_lthy |>
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      mk_Freess "x" ctr_Tss
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      ||>> mk_Freess "y" ctr_Tss
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      ||>> mk_Frees "f" caseofB_Ts
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      ||>> mk_Frees "g" caseofB_Ts
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      ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "v") T
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      ||>> yield_singleton (mk_Frees "w") T
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      ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "P") HOLogic.boolT;
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    val q = Free (fst p', B --> HOLogic.boolT);
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    val xctrs = map2 (curry Term.list_comb) ctrs xss;
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    val yctrs = map2 (curry Term.list_comb) ctrs yss;
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    val xfs = map2 (curry Term.list_comb) fs xss;
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    val xgs = map2 (curry Term.list_comb) gs xss;
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    val eta_fs = map2 eta_expand_caseof_arg xss xfs;
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    val eta_gs = map2 eta_expand_caseof_arg xss xgs;
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    val caseofB_fs = Term.list_comb (caseofB, eta_fs);
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    val exist_xs_v_eq_ctrs =
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      map2 (fn xctr => fn xs => list_exists_free xs (HOLogic.mk_eq (v, xctr))) xctrs xss;
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    fun mk_sel_caseof_args k xs x T =
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      map2 (fn Ts => fn i => if i = k then fold_rev Term.lambda xs x else mk_undef T Ts) ctr_Tss ks;
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    fun disc_spec b exist_xs_v_eq_ctr =
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      mk_Trueprop_eq (Free (Binding.name_of b, T --> HOLogic.boolT) $ v, exist_xs_v_eq_ctr);
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    fun sel_spec b x xs k =
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      let val T' = fastype_of x in
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        mk_Trueprop_eq (Free (Binding.name_of b, T --> T') $ v,
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          Term.list_comb (mk_caseof As T', mk_sel_caseof_args k xs x T') $ v)
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      end;
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    val (((raw_discs, (_, raw_disc_defs)), (raw_selss, (_, raw_sel_defss))), (lthy', lthy)) =
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      no_defs_lthy
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      |> apfst (apsnd split_list o split_list) o fold_map2 (fn b => fn exist_xs_v_eq_ctr =>
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        Specification.definition (SOME (b, NONE, NoSyn),
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          ((Thm.def_binding b, []), disc_spec b exist_xs_v_eq_ctr))) disc_names exist_xs_v_eq_ctrs
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      ||>> apfst (apsnd split_list o split_list) o fold_map3 (fn bs => fn xs => fn k =>
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        apfst (apsnd split_list o split_list) o fold_map2 (fn b => fn x =>
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          Specification.definition (SOME (b, NONE, NoSyn),
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            ((Thm.def_binding b, []), sel_spec b x xs k))) bs xs) sel_namess xss ks
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      ||> `Local_Theory.restore;
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    (*transforms defined frees into consts (and more)*)
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    val phi = Proof_Context.export_morphism lthy lthy';
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    val disc_defs = map (Morphism.thm phi) raw_disc_defs;
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    val sel_defss = map (map (Morphism.thm phi)) raw_sel_defss;
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    val discs0 = map (Morphism.term phi) raw_discs;
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    val selss0 = map (map (Morphism.term phi)) raw_selss;
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    fun mk_disc_or_sel Ts t =
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      Term.subst_atomic_types (snd (dest_Type (domain_type (fastype_of t))) ~~ Ts) t;
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    val discs = map (mk_disc_or_sel As) discs0;
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    val selss = map (map (mk_disc_or_sel As)) selss0;
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    fun mk_imp_p Qs = Logic.list_implies (Qs, HOLogic.mk_Trueprop p);
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    val goal_exhaust =
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      let fun mk_prem xctr xs = fold_rev Logic.all xs (mk_imp_p [mk_Trueprop_eq (v, xctr)]) in
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        mk_imp_p (map2 mk_prem xctrs xss)
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      end;
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    val goal_injectss =
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      let
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        fun mk_goal _ _ [] [] = []
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          | mk_goal xctr yctr xs ys =
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            [mk_Trueprop_eq (HOLogic.mk_eq (xctr, yctr),
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              Library.foldr1 HOLogic.mk_conj (map2 (curry HOLogic.mk_eq) xs ys))];
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      in
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        map4 mk_goal xctrs yctrs xss yss
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      end;
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    val goal_half_distinctss =
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      map (map (HOLogic.mk_Trueprop o HOLogic.mk_not o HOLogic.mk_eq)) (mk_half_pairss xctrs);
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    val goal_cases = map2 (fn xctr => fn xf => mk_Trueprop_eq (caseofB_fs $ xctr, xf)) xctrs xfs;
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    val goals = [goal_exhaust] :: goal_injectss @ goal_half_distinctss @ [goal_cases];
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    fun after_qed thmss lthy =
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      let
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        val ([exhaust_thm], (inject_thmss, (half_distinct_thmss, [case_thms]))) =
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          (hd thmss, apsnd (chop (n * n)) (chop n (tl thmss)));
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        val inject_thms = flat inject_thmss;
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        val exhaust_thm' =
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          let val Tinst = map (pairself (certifyT lthy)) (map Logic.varifyT_global As ~~ As) in
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            Drule.instantiate' [] [SOME (certify lthy v)]
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              (Thm.instantiate (Tinst, []) (Drule.zero_var_indexes exhaust_thm))
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          end;
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        val other_half_distinct_thmss = map (map (fn thm => thm RS not_sym)) half_distinct_thmss;
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        val distinct_thmsss =
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          map2 (map2 append) (Library.chop_groups n half_distinct_thmss)
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            (transpose (Library.chop_groups n other_half_distinct_thmss));
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        val distinct_thms = interleave (flat half_distinct_thmss) (flat other_half_distinct_thmss);
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        val nchotomy_thm =
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          let
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            val goal =
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              HOLogic.mk_Trueprop (HOLogic.mk_all (fst v', snd v',
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                Library.foldr1 HOLogic.mk_disj exist_xs_v_eq_ctrs));
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          in
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            Skip_Proof.prove lthy [] [] goal (fn _ => mk_nchotomy_tac n exhaust_thm)
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          end;
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        val sel_thmss =
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          let
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            fun mk_thm k xs goal_case case_thm x sel_def =
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              let
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                val T = fastype_of x;
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                val cTs =
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                  map ((fn T' => certifyT lthy (if T' = B then T else T')) o TFree)
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                    (rev (Term.add_tfrees goal_case []));
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                val cxs = map (certify lthy) (mk_sel_caseof_args k xs x T);
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              in
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                Local_Defs.fold lthy [sel_def]
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                  (Drule.instantiate' (map SOME cTs) (map SOME cxs) case_thm)
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              end;
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            fun mk_thms k xs goal_case case_thm sel_defs =
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              map2 (mk_thm k xs goal_case case_thm) xs sel_defs;
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          in
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            map5 mk_thms ks xss goal_cases case_thms sel_defss
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          end;
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        val discD_thms = map (fn def => def RS iffD1) disc_defs;
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        val discI_thms =
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          map2 (fn m => fn def => funpow m (fn thm => exI RS thm) (def RS iffD2)) ms disc_defs;
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        val not_disc_thms =
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          map2 (fn m => fn def => funpow m (fn thm => allI RS thm)
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                  (Local_Defs.unfold lthy @{thms not_ex} (def RS @{thm ssubst[of _ _ Not]})))
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            ms disc_defs;
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        val disc_thmss =
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          let
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            fun mk_thm discI _ [] = refl RS discI
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              | mk_thm _ not_disc [distinct] = distinct RS not_disc;
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            fun mk_thms discI not_disc distinctss = map (mk_thm discI not_disc) distinctss;
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          in
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            map3 mk_thms discI_thms not_disc_thms (transpose distinct_thmsss)
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          end;
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        val disc_disjoint_thms =
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          let
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            fun mk_goal ((_, disc), (_, disc')) =
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              Logic.all v (Logic.mk_implies (HOLogic.mk_Trueprop (disc $ v),
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                HOLogic.mk_Trueprop (HOLogic.mk_not (disc' $ v))));
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            fun prove tac goal = Skip_Proof.prove lthy [] [] goal (K tac);
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            val bundles = ms ~~ discD_thms ~~ discs;
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            val half_pairss = mk_half_pairss bundles;
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            val goal_halvess = map (map mk_goal) half_pairss;
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            val half_thmss =
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              map3 (fn [] => K (K [])
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                     | [(((m, discD), _), _)] => fn disc_thm => fn [goal] =>
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                [prove (mk_half_disc_disjoint_tac m discD disc_thm) goal])
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              half_pairss (flat (transpose disc_thmss)) goal_halvess;
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            val goal_other_halvess = map (map (mk_goal o swap)) half_pairss;
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            val other_half_thmss =
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              map2 (map2 (prove o mk_other_half_disc_disjoint_tac)) half_thmss goal_other_halvess;
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          in
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            interleave (flat half_thmss) (flat other_half_thmss)
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          end;
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        val disc_exhaust_thm =
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          let
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            fun mk_prem disc = mk_imp_p [HOLogic.mk_Trueprop (disc $ v)];
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            val goal = fold Logic.all [p, v] (mk_imp_p (map mk_prem discs));
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          in
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            Skip_Proof.prove lthy [] [] goal (fn _ => mk_disc_exhaust_tac n exhaust_thm discI_thms)
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          end;
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        val ctr_sel_thms =
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          let
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            fun mk_goal ctr disc sels =
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              Logic.all v (Logic.mk_implies (HOLogic.mk_Trueprop (disc $ v),
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                mk_Trueprop_eq ((null sels ? swap)
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                  (Term.list_comb (ctr, map (fn sel => sel $ v) sels), v))));
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            val goals = map3 mk_goal ctrs discs selss;
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          in
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            map4 (fn goal => fn m => fn discD => fn sel_thms =>
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              Skip_Proof.prove lthy [] [] goal (fn {context = ctxt, ...} =>
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                mk_ctr_sel_tac ctxt m discD sel_thms))
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              goals ms discD_thms sel_thmss
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          end;
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        val case_disc_thm =
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          let
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            fun mk_core f sels = Term.list_comb (f, map (fn sel => sel $ v) sels);
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            fun mk_rhs _ [f] [sels] = mk_core f sels
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              | mk_rhs (disc :: discs) (f :: fs) (sels :: selss) =
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                Const (@{const_name If}, HOLogic.boolT --> B --> B --> B) $
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                  (disc $ v) $ mk_core f sels $ mk_rhs discs fs selss;
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            val goal = mk_Trueprop_eq (caseofB_fs $ v, mk_rhs discs fs selss);
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          in
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            Skip_Proof.prove lthy [] [] goal (fn {context = ctxt, ...} =>
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              mk_case_disc_tac ctxt exhaust_thm' case_thms disc_thmss sel_thmss)
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            |> singleton (Proof_Context.export names_lthy lthy)
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          end;
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        val (case_cong_thm, weak_case_cong_thm) =
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          let
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            fun mk_prem xctr xs f g =
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              fold_rev Logic.all xs (Logic.mk_implies (mk_Trueprop_eq (w, xctr),
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                mk_Trueprop_eq (f, g)));
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            val v_eq_w = mk_Trueprop_eq (v, w);
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            val caseof_fs = mk_caseofB_term eta_fs;
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            val caseof_gs = mk_caseofB_term eta_gs;
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   323
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            val goal =
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              Logic.list_implies (v_eq_w :: map4 mk_prem xctrs xss fs gs,
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                 mk_Trueprop_eq (caseof_fs $ v, caseof_gs $ w));
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            val goal_weak =
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              Logic.mk_implies (v_eq_w, mk_Trueprop_eq (caseof_fs $ v, caseof_fs $ w));
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          in
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            (Skip_Proof.prove lthy [] [] goal (fn {context = ctxt, ...} =>
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               mk_case_cong_tac ctxt exhaust_thm' case_thms),
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             Skip_Proof.prove lthy [] [] goal_weak (K (etac arg_cong 1)))
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            |> pairself (singleton (Proof_Context.export names_lthy lthy))
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          end;
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        val (split_thm, split_asm_thm) =
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          let
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            fun mk_conjunct xctr xs f_xs =
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              list_all_free xs (HOLogic.mk_imp (HOLogic.mk_eq (v, xctr), q $ f_xs));
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            fun mk_disjunct xctr xs f_xs =
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              list_exists_free xs (HOLogic.mk_conj (HOLogic.mk_eq (v, xctr),
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                HOLogic.mk_not (q $ f_xs)));
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   343
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            val lhs = q $ (mk_caseofB_term eta_fs $ v);
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            val goal =
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              mk_Trueprop_eq (lhs, Library.foldr1 HOLogic.mk_conj (map3 mk_conjunct xctrs xss xfs));
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            val goal_asm =
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              mk_Trueprop_eq (lhs, HOLogic.mk_not (Library.foldr1 HOLogic.mk_disj
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                (map3 mk_disjunct xctrs xss xfs)));
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            val split_thm =
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              Skip_Proof.prove lthy [] [] goal (fn {context = ctxt, ...} =>
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                mk_split_tac ctxt exhaust_thm' case_thms inject_thms distinct_thms)
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              |> singleton (Proof_Context.export names_lthy lthy)
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            val split_asm_thm =
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              Skip_Proof.prove lthy [] [] goal_asm (fn {context = ctxt, ...} =>
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                mk_split_asm_tac ctxt split_thm)
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              |> singleton (Proof_Context.export names_lthy lthy)
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          in
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            (split_thm, split_asm_thm)
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          end;
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        (* TODO: case syntax *)
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        (* TODO: attributes (simp, case_names, etc.) *)
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        fun note thmN thms =
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          snd o Local_Theory.note
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            ((Binding.qualify true (Binding.name_of b) (Binding.name thmN), []), thms);
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      in
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        lthy
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        |> note case_congN [case_cong_thm]
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        |> note case_discsN [case_disc_thm]
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        |> note casesN case_thms
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        |> note ctr_selsN ctr_sel_thms
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        |> note discsN (flat disc_thmss)
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        |> note disc_disjointN disc_disjoint_thms
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        |> note disc_exhaustN [disc_exhaust_thm]
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        |> note distinctN distinct_thms
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        |> note exhaustN [exhaust_thm]
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        |> note injectN inject_thms
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        |> note nchotomyN [nchotomy_thm]
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        |> note selsN (flat sel_thmss)
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        |> note splitN [split_thm]
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        |> note split_asmN [split_asm_thm]
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        |> note weak_case_cong_thmsN [weak_case_cong_thm]
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      end;
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  in
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    (goals, after_qed, lthy')
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  end;
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val parse_binding_list = Parse.$$$ "[" |--  Parse.list Parse.binding --| Parse.$$$ "]";
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val bnf_sugar_cmd = (fn (goalss, after_qed, lthy) =>
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  Proof.theorem NONE after_qed (map (map (rpair [])) goalss) lthy) oo
49020
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   396
  prepare_sugar Syntax.read_term;
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66fc7fc2d49b started work on datatype sugar
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   398
val _ =
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   399
  Outer_Syntax.local_theory_to_proof @{command_spec "bnf_sugar"} "adds sugar on top of a BNF"
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    (((Parse.$$$ "[" |-- Parse.list Parse.term --| Parse.$$$ "]") -- Parse.term --
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      parse_binding_list -- (Parse.$$$ "[" |-- Parse.list parse_binding_list --| Parse.$$$ "]"))
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      >> bnf_sugar_cmd);
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66fc7fc2d49b started work on datatype sugar
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end;