src/HOL/BNF/Tools/bnf_fp_def_sugar_tactics.ML

fixed simplification of prod and sum relators to avoid issues with e.g. codata ('a, 'b) k = K "'a + 'b"

(*  Title:      HOL/BNF/Tools/bnf_fp_def_sugar_tactics.ML
    Author:     Jasmin Blanchette, TU Muenchen
    Copyright   2012

Tactics for datatype and codatatype sugar.
*)

signature BNF_FP_DEF_SUGAR_TACTICS =
sig
  val sum_prod_thms_map: thm list
  val sum_prod_thms_set: thm list
  val sum_prod_thms_rel: thm list

  val mk_case_tac: Proof.context -> int -> int -> int -> thm -> thm -> thm -> tactic
  val mk_coinduct_tac: Proof.context -> thm list -> int -> int list -> thm -> thm list ->
    thm list -> thm list -> thm list list -> thm list list list -> thm list list list -> tactic
  val mk_corec_like_tac: thm list -> thm list -> thm -> thm -> thm -> Proof.context -> tactic
  val mk_ctor_iff_dtor_tac: Proof.context -> ctyp option list -> cterm -> cterm -> thm -> thm ->
    tactic
  val mk_disc_corec_like_iff_tac: thm list -> thm list -> thm list -> Proof.context -> tactic
  val mk_exhaust_tac: Proof.context -> int -> thm list -> thm -> thm -> tactic
  val mk_half_distinct_tac: Proof.context -> thm -> thm list -> tactic
  val mk_induct_tac: Proof.context -> int -> int list -> int list list -> int list list list ->
    thm list -> thm -> thm list -> thm list list -> tactic
  val mk_inject_tac: Proof.context -> thm -> thm -> tactic
  val mk_rec_like_tac: thm list -> thm list -> thm list -> thm -> thm -> Proof.context -> tactic
end;

structure BNF_FP_Def_Sugar_Tactics : BNF_FP_DEF_SUGAR_TACTICS =
struct

open BNF_Tactics
open BNF_Util
open BNF_FP

val basic_simp_thms = @{thms simp_thms(7,8,12,14,22,24)};
val more_simp_thms = basic_simp_thms @ @{thms simp_thms(11,15,16,21)};

val sum_prod_thms_map = @{thms id_apply map_pair_simp sum_map.simps prod.cases};
val sum_prod_thms_set0 =
  @{thms SUP_empty Sup_empty Sup_insert UN_insert Un_empty_left Un_empty_right Un_iff
      Union_Un_distrib collect_def[abs_def] image_def o_apply map_pair_simp
      mem_Collect_eq mem_UN_compreh_eq prod_set_simps sum_map.simps sum_set_simps};
val sum_prod_thms_set = @{thms UN_compreh_eq_eq} @ sum_prod_thms_set0;
(* FIXME: Need "prod.cases" and "sum.cases"? And why "sum.inject" but no "prod.inject"? *)
val sum_prod_thms_rel =
  @{thms prod.cases prod_rel_simp sum.cases sum_rel_simps
      sum.inject sum.distinct[THEN eq_False[THEN iffD2]]};

val ss_if_True_False = ss_only @{thms if_True if_False};

fun mk_proj T k =
  let val binders = binder_types T in
    fold_rev (fn T => fn t => Abs (Name.uu, T, t)) binders (Bound (length binders - k))
  end;

fun inst_as_projs ctxt k 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 ctxt (Var f), certify ctxt (mk_proj T k))) fs;
  in
    Drule.cterm_instantiate cfs thm
  end;

val inst_as_projs_tac = PRIMITIVE oo inst_as_projs;

fun mk_case_tac ctxt n k m case_def ctr_def dtor_ctor =
  unfold_thms_tac ctxt [case_def, ctr_def, dtor_ctor] 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 ctor_iff_dtor sumEN' =
  unfold_thms_tac ctxt (ctor_iff_dtor :: ctr_defs) THEN rtac sumEN' 1 THEN
  unfold_thms_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_ctor_iff_dtor_tac ctxt cTs cctor cdtor ctor_dtor dtor_ctor =
  (rtac iffI THEN'
   EVERY' (map3 (fn cTs => fn cx => fn th =>
     dtac (Drule.instantiate' cTs [NONE, NONE, SOME cx] arg_cong) THEN'
     SELECT_GOAL (unfold_thms_tac ctxt [th]) THEN'
     atac) [rev cTs, cTs] [cdtor, cctor] [dtor_ctor, ctor_dtor])) 1;

fun mk_half_distinct_tac ctxt ctor_inject ctr_defs =
  unfold_thms_tac ctxt (ctor_inject :: @{thms sum.inject} @ ctr_defs) THEN
  rtac @{thm sum.distinct(1)} 1;

fun mk_inject_tac ctxt ctr_def ctor_inject =
  unfold_thms_tac ctxt [ctr_def] THEN rtac (ctor_inject RS ssubst) 1 THEN
  unfold_thms_tac ctxt @{thms sum.inject Pair_eq conj_assoc} THEN rtac refl 1;

(*TODO: Try "sum_prod_thms_map" here, enriched with a few theorems*)
val rec_like_unfold_thms =
  @{thms comp_def convol_def id_apply map_pair_def prod_case_Pair_iden sum.simps(5,6) sum_map.simps
      split_conv unit_case_Unity};

fun mk_rec_like_tac pre_map_defs map_ids rec_like_defs ctor_rec_like ctr_def ctxt =
  unfold_thms_tac ctxt (ctr_def :: ctor_rec_like :: rec_like_defs @ pre_map_defs @ map_ids @
    rec_like_unfold_thms) THEN unfold_thms_tac ctxt @{thms id_def} THEN rtac refl 1;

fun mk_corec_like_tac corec_like_defs map_ids ctor_dtor_corec_like pre_map_def ctr_def ctxt =
  unfold_thms_tac ctxt (ctr_def :: corec_like_defs) THEN
  (rtac (ctor_dtor_corec_like RS trans) THEN' asm_simp_tac ss_if_True_False) 1 THEN
  unfold_thms_tac ctxt (pre_map_def :: sum_prod_thms_map @ map_ids) THEN
  unfold_thms_tac ctxt @{thms id_def} THEN
  (rtac refl ORELSE' rtac (@{thm unit_eq} RS arg_cong)) 1;

fun mk_disc_corec_like_iff_tac case_splits' corec_likes discs ctxt =
  EVERY (map3 (fn case_split_tac => fn corec_like_thm => fn disc =>
      case_split_tac 1 THEN unfold_thms_tac ctxt [corec_like_thm] THEN
      asm_simp_tac (ss_only basic_simp_thms) 1 THEN
      (if is_refl disc then all_tac else rtac disc 1))
    (map rtac case_splits' @ [K all_tac]) corec_likes discs);

val solve_prem_prem_tac =
  REPEAT o (eresolve_tac @{thms bexE rev_bexI} ORELSE' rtac @{thm rev_bexI[OF UNIV_I]} ORELSE'
    hyp_subst_tac ORELSE' resolve_tac @{thms disjI1 disjI2}) THEN'
  (rtac refl ORELSE' atac ORELSE' rtac @{thm singletonI});

fun mk_induct_leverage_prem_prems_tac ctxt nn kks set_natural's pre_set_defs =
  EVERY' (maps (fn kk => [select_prem_tac nn (dtac meta_spec) kk, etac meta_mp,
     SELECT_GOAL (unfold_thms_tac ctxt (pre_set_defs @ set_natural's @ sum_prod_thms_set0)),
     solve_prem_prem_tac]) (rev kks)) 1;

fun mk_induct_discharge_prem_tac ctxt nn n set_natural's pre_set_defs m k kks =
  let val r = length kks 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 kks set_natural's pre_set_defs]
  end;

fun mk_induct_tac ctxt nn ns mss kkss ctr_defs ctor_induct' set_natural's pre_set_defss =
  let val n = Integer.sum ns in
    unfold_thms_tac ctxt ctr_defs THEN rtac ctor_induct' 1 THEN inst_as_projs_tac ctxt 1 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)) kkss)
  end;

fun mk_coinduct_same_ctr ctxt rel_eqs pre_rel_def dtor_ctor ctr_def discs sels =
  hyp_subst_tac THEN'
  subst_tac ctxt (SOME [1, 2]) [ctr_def] THEN'
  SELECT_GOAL (unfold_thms_tac ctxt (pre_rel_def :: dtor_ctor :: sels)) THEN'
  SELECT_GOAL (unfold_thms_tac ctxt (pre_rel_def :: dtor_ctor :: sels @ sum_prod_thms_rel)) THEN'
  (atac ORELSE' REPEAT o etac conjE THEN'
     full_simp_tac
       (ss_only (@{thm prod.inject} :: no_refl discs @ rel_eqs @ more_simp_thms)) THEN_MAYBE'
     REPEAT o hyp_subst_tac THEN' REPEAT o rtac conjI THEN' REPEAT o rtac refl);

fun mk_coinduct_distinct_ctrs discs discs' =
  hyp_subst_tac THEN' REPEAT o etac conjE THEN'
  full_simp_tac (ss_only (refl :: no_refl (discs @ discs') @ basic_simp_thms));

fun mk_coinduct_discharge_prem_tac ctxt rel_eqs' nn kk n pre_rel_def dtor_ctor exhaust ctr_defs
    discss selss =
  let val ks = 1 upto n in
    EVERY' ([rtac allI, rtac allI, rtac impI, select_prem_tac nn (dtac meta_spec) kk, dtac
        meta_spec, dtac meta_mp, atac, rtac exhaust, K (inst_as_projs_tac ctxt 1), hyp_subst_tac] @
      map4 (fn k => fn ctr_def => fn discs => fn sels =>
        EVERY' ([rtac exhaust, K (inst_as_projs_tac ctxt 2)] @
          map2 (fn k' => fn discs' =>
            if k' = k then
              mk_coinduct_same_ctr ctxt rel_eqs' pre_rel_def dtor_ctor ctr_def discs sels
            else
              mk_coinduct_distinct_ctrs discs discs') ks discss)) ks ctr_defs discss selss)
  end;

fun mk_coinduct_tac ctxt rel_eqs' nn ns dtor_coinduct' pre_rel_defs dtor_ctors exhausts ctr_defss
    discsss selsss =
  (rtac dtor_coinduct' THEN'
   EVERY' (map8 (mk_coinduct_discharge_prem_tac ctxt rel_eqs' nn)
     (1 upto nn) ns pre_rel_defs dtor_ctors exhausts ctr_defss discsss selsss)) 1;

end;