(*  Title:      HOL/Codatatype/Tools/bnf_fp_sugar_tactics.ML

     Author:     Jasmin Blanchette, TU Muenchen

     Copyright   2012

 Tactics for datatype and codatatype sugar.

 *)

 signature BNF_FP_SUGAR_TACTICS =

 sig

   val mk_case_tac: Proof.context -> int -> int -> int -> thm -> thm -> thm -> tactic

   val mk_coiter_like_tac: thm list -> thm list -> thm -> thm -> thm -> Proof.context -> tactic

   val mk_exhaust_tac: Proof.context -> int -> thm list -> thm -> thm -> tactic

   val mk_fld_iff_unf_tac: Proof.context -> ctyp option list -> cterm -> cterm -> thm -> thm ->

     tactic

   val mk_half_distinct_tac: Proof.context -> thm -> thm list -> tactic

   val mk_induct_tac: Proof.context -> int list -> int list list ->

     ((int * int) * (int * int)) list list list -> thm list -> thm -> thm list -> thm list list ->

     tactic

   val mk_inject_tac: Proof.context -> thm -> thm -> tactic

   val mk_iter_like_tac: thm list -> thm list -> thm list -> thm -> thm -> Proof.context -> tactic

 end;

 structure BNF_FP_Sugar_Tactics : BNF_FP_SUGAR_TACTICS =

 struct

 open BNF_Tactics

 open BNF_Util

 open BNF_FP_Util

 val meta_mp = @{thm meta_mp};

 val meta_spec = @{thm meta_spec};

 fun inst_spurious_fs lthy thm =

   let

     val fs =

       Term.add_vars (prop_of thm) []

       |> filter (fn (_, Type (@{type_name fun}, [_, T'])) => T' <> HOLogic.boolT | _ => false);

     val cfs =

       map (fn f as (_, T) => (certify lthy (Var f), certify lthy (id_abs (domain_type T)))) fs;

   in

     Drule.cterm_instantiate cfs thm

   end;

 val inst_spurious_fs_tac = PRIMITIVE o inst_spurious_fs;

 fun mk_set_rhs def T =

   let

     val lhs = snd (Logic.dest_equals (prop_of def));

     val Type (_, Ts0) = domain_type (fastype_of lhs);

     val Type (_, Ts) = domain_type T;

   in

     Term.subst_atomic_types (Ts0 ~~ Ts) lhs

   end;

 val mk_fsts_rhs = mk_set_rhs @{thm fsts_def[abs_def]};

 val mk_snds_rhs = mk_set_rhs @{thm snds_def[abs_def]};

 val mk_setl_rhs = mk_set_rhs @{thm sum_setl_def[abs_def]};

 val mk_setr_rhs = mk_set_rhs @{thm sum_setr_def[abs_def]};

 (* TODO: Put this in "Balanced_Tree" *)

 fun balanced_tree_middle n = n div 2;

 val sum_prod_sel_defs =

   @{thms fsts_def[abs_def] snds_def[abs_def] sum_setl_def[abs_def] sum_setr_def[abs_def]};

 fun unfold_sum_prod_sets ctxt ms thm =

   let

     fun unf_prod 0 f = f

       | unf_prod 1 f = f

       | unf_prod m (t1 $ (t2 $ (t3 $ (t4 $ Const (@{const_name fsts}, T1) $

           (t5 $ Const (@{const_name snds}, T2) $ t6)))) $ (t7 $ f $ g)) =

         t1 $ (t2 $ (t3 $ (t4 $ mk_fsts_rhs T1 $ (t5 $ mk_snds_rhs T2 $ t6))))

           $ (t7 $ f $ unf_prod (m - 1) g)

       | unf_prod _ f = f;

     fun unf_sum [m] f = unf_prod m f

       | unf_sum ms (t1 $ (t2 $ (t3 $ (t4 $ Const (@{const_name sum_setl}, T1) $

           (t5 $ Const (@{const_name sum_setr}, T2) $ t6)))) $ (t7 $ f $ g)) =

         let val (ms1, ms2) = chop (balanced_tree_middle (length ms)) ms in

           t1 $ (t2 $ (t3 $ (t4 $ mk_setl_rhs T1 $ (t5 $ mk_setr_rhs T2 $ t6))))

             $ (t7 $ unf_sum ms1 f $ unf_sum ms2 g)

         end

       | unf_sum _ f = f;

     val P = prop_of thm;

     val P' = Logic.dest_equals P ||> unf_sum ms;

     val goal = Logic.mk_implies (P, Logic.mk_equals P');

   in

     Skip_Proof.prove ctxt [] [] goal (fn {context = ctxt, ...} =>

       Local_Defs.unfold_tac ctxt sum_prod_sel_defs THEN atac 1)

     OF [thm]

   end;

 fun mk_case_tac ctxt n k m case_def ctr_def unf_fld =

   Local_Defs.unfold_tac ctxt [case_def, ctr_def, unf_fld] THEN

   (rtac (mk_sum_casesN_balanced n k RS ssubst) THEN'

    REPEAT_DETERM_N (Int.max (0, m - 1)) o rtac (@{thm split} RS ssubst) THEN'

    rtac refl) 1;

 fun mk_exhaust_tac ctxt n ctr_defs fld_iff_unf sumEN' =

   Local_Defs.unfold_tac ctxt (fld_iff_unf :: ctr_defs) THEN rtac sumEN' 1 THEN

   Local_Defs.unfold_tac ctxt @{thms all_prod_eq} THEN

   EVERY' (maps (fn k => [select_prem_tac n (rotate_tac 1) k, REPEAT_DETERM o dtac meta_spec,

     etac meta_mp, atac]) (1 upto n)) 1;

 fun mk_fld_iff_unf_tac ctxt cTs cfld cunf fld_unf unf_fld =

   (rtac iffI THEN'

    EVERY' (map3 (fn cTs => fn cx => fn th =>

      dtac (Drule.instantiate' cTs [NONE, NONE, SOME cx] arg_cong) THEN'

      SELECT_GOAL (Local_Defs.unfold_tac ctxt [th]) THEN'

      atac) [rev cTs, cTs] [cunf, cfld] [unf_fld, fld_unf])) 1;

 fun mk_half_distinct_tac ctxt fld_inject ctr_defs =

   Local_Defs.unfold_tac ctxt (fld_inject :: @{thms sum.inject} @ ctr_defs) THEN

   rtac @{thm sum.distinct(1)} 1;

 fun mk_inject_tac ctxt ctr_def fld_inject =

   Local_Defs.unfold_tac ctxt [ctr_def] THEN rtac (fld_inject RS ssubst) 1 THEN

   Local_Defs.unfold_tac ctxt @{thms sum.inject Pair_eq conj_assoc} THEN rtac refl 1;

 val iter_like_thms =

   @{thms case_unit comp_def convol_def id_apply map_pair_def sum.simps(5,6) sum_map.simps

       split_conv};

 fun mk_iter_like_tac pre_map_defs map_ids iter_like_defs fld_iter_like ctr_def ctxt =

   Local_Defs.unfold_tac ctxt (ctr_def :: fld_iter_like :: iter_like_defs @ pre_map_defs @ map_ids @

     iter_like_thms) THEN Local_Defs.unfold_tac ctxt @{thms id_def} THEN rtac refl 1;

 val coiter_like_ss = ss_only @{thms if_True if_False};

 val coiter_like_thms = @{thms id_apply map_pair_def sum_map.simps prod.cases};

 fun mk_coiter_like_tac coiter_like_defs map_ids fld_unf_coiter_like pre_map_def ctr_def ctxt =

   Local_Defs.unfold_tac ctxt (ctr_def :: coiter_like_defs) THEN

   subst_tac ctxt [fld_unf_coiter_like] 1 THEN asm_simp_tac coiter_like_ss 1 THEN

   Local_Defs.unfold_tac ctxt (pre_map_def :: coiter_like_thms @ map_ids) THEN

   Local_Defs.unfold_tac ctxt @{thms id_def} THEN

   TRY ((rtac refl ORELSE' subst_tac ctxt @{thms unit_eq} THEN' rtac refl) 1);

 val maybe_singletonI_tac = atac ORELSE' rtac @{thm singletonI};

 fun solve_prem_prem_tac ctxt =

   SELECT_GOAL (Local_Defs.unfold_tac ctxt

     @{thms Un_iff eq_UN_compreh_Un mem_Collect_eq mem_UN_compreh_eq}) THEN'

   REPEAT o (etac @{thm rev_bexI} ORELSE' resolve_tac @{thms disjI1 disjI2}) THEN'

   (atac ORELSE' rtac refl ORELSE' rtac @{thm singletonI});

 val induct_prem_prem_thms =

   @{thms SUP_empty Sup_empty Sup_insert UN_compreh_bex_eq_empty UN_compreh_bex_eq_singleton

       UN_insert Un_assoc Un_empty_left Un_empty_right Union_Un_distrib collect_def[abs_def] fst_conv

       image_def o_apply snd_conv snd_prod_fun sum.cases sup_bot_right fst_map_pair map_pair_simp

       sum_map.simps};

 fun mk_induct_leverage_prem_prems_tac ctxt nn ppjjqqkks set_natural's pre_set_defs =

   EVERY' (maps (fn ((pp, jj), (qq, kk)) =>

     [select_prem_tac nn (dtac meta_spec) kk, etac meta_mp,

      SELECT_GOAL (Local_Defs.unfold_tac ctxt

        (pre_set_defs @ set_natural's @ induct_prem_prem_thms)),

      solve_prem_prem_tac ctxt]) (rev ppjjqqkks)) 1;

 fun mk_induct_discharge_prem_tac ctxt nn n set_natural's pre_set_defs m k ppjjqqkks =

   let val r = length ppjjqqkks in

     EVERY' [select_prem_tac n (rotate_tac 1) k, rotate_tac ~1, hyp_subst_tac,

       REPEAT_DETERM_N m o (dtac meta_spec THEN' rotate_tac ~1)] 1 THEN

     EVERY [REPEAT_DETERM_N r

         (rotate_tac ~1 1 THEN dtac meta_mp 1 THEN rotate_tac 1 1 THEN prefer_tac 2),

       if r > 0 then PRIMITIVE Raw_Simplifier.norm_hhf else all_tac, atac 1,

       mk_induct_leverage_prem_prems_tac ctxt nn ppjjqqkks set_natural's pre_set_defs]

   end;

 fun mk_induct_tac ctxt ns mss ppjjqqkksss ctr_defs fld_induct' set_natural's pre_set_defss =

   let

     val nn = length ns;

     val n = Integer.sum ns;

     val pre_set_defss' = map2 (map o unfold_sum_prod_sets ctxt) mss pre_set_defss;

   in

     Local_Defs.unfold_tac ctxt ctr_defs THEN rtac fld_induct' 1 THEN inst_spurious_fs_tac ctxt THEN

     EVERY (map4 (EVERY oooo map3 o mk_induct_discharge_prem_tac ctxt nn n set_natural's)

       pre_set_defss' mss (unflat mss (1 upto n)) ppjjqqkksss)

   end;

 end;