(*  Title:      HOL/Tools/Datatype/datatype_abs_proofs.ML

    Author:     Stefan Berghofer, TU Muenchen

Datatype package: proofs and defintions independent of concrete

representation of datatypes  (i.e. requiring only abstract

properties: injectivity / distinctness of constructors and induction).

*)

signature DATATYPE_ABS_PROOFS =

sig

  include DATATYPE_COMMON

  val prove_casedist_thms : config -> string list ->

    descr list -> (string * sort) list -> thm ->

    attribute list -> theory -> thm list * theory

  val prove_primrec_thms : config -> string list ->

    descr list -> (string * sort) list ->

      (string -> thm list) -> thm list list -> thm list list * thm list list ->

        thm -> theory -> (string list * thm list) * theory

  val prove_case_thms : config -> string list ->

    descr list -> (string * sort) list ->

      string list -> thm list -> theory -> (thm list list * string list) * theory

  val prove_split_thms : config -> string list ->

    descr list -> (string * sort) list ->

      thm list list -> thm list list -> thm list -> thm list list -> theory ->

        (thm * thm) list * theory

  val prove_nchotomys : config -> string list -> descr list ->

    (string * sort) list -> thm list -> theory -> thm list * theory

  val prove_weak_case_congs : string list -> descr list ->

    (string * sort) list -> theory -> thm list * theory

  val prove_case_congs : string list ->

    descr list -> (string * sort) list ->

      thm list -> thm list list -> theory -> thm list * theory

end;

structure Datatype_Abs_Proofs: DATATYPE_ABS_PROOFS =

struct

(************************ case distinction theorems ***************************)

fun prove_casedist_thms (config : Datatype_Aux.config)

    new_type_names descr sorts induct case_names_exhausts thy =

  let

    val _ = Datatype_Aux.message config "Proving case distinction theorems ...";

    val descr' = flat descr;

    val recTs = Datatype_Aux.get_rec_types descr' sorts;

    val newTs = take (length (hd descr)) recTs;

    val {maxidx, ...} = rep_thm induct;

    val induct_Ps = map head_of (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct)));

    fun prove_casedist_thm (i, (T, t)) =

      let

        val dummyPs = map (fn (Var (_, Type (_, [T', T'']))) =>

          Abs ("z", T', Const (@{const_name True}, T''))) induct_Ps;

        val P = Abs ("z", T, HOLogic.imp $ HOLogic.mk_eq (Var (("a", maxidx+1), T), Bound 0) $

          Var (("P", 0), HOLogic.boolT))

        val insts = take i dummyPs @ (P::(drop (i + 1) dummyPs));

        val cert = cterm_of thy;

        val insts' = (map cert induct_Ps) ~~ (map cert insts);

        val induct' = refl RS ((nth

          (Datatype_Aux.split_conj_thm (cterm_instantiate insts' induct)) i) RSN (2, rev_mp))

      in

        Skip_Proof.prove_global thy [] (Logic.strip_imp_prems t) (Logic.strip_imp_concl t)

          (fn {prems, ...} => EVERY

            [rtac induct' 1,

             REPEAT (rtac TrueI 1),

             REPEAT ((rtac impI 1) THEN (eresolve_tac prems 1)),

             REPEAT (rtac TrueI 1)])

      end;

    val casedist_thms = map_index prove_casedist_thm

      (newTs ~~ Datatype_Prop.make_casedists descr sorts)

  in

    thy

    |> Datatype_Aux.store_thms_atts "exhaust" new_type_names

      (map single case_names_exhausts) casedist_thms

  end;

(*************************** primrec combinators ******************************)

fun prove_primrec_thms (config : Datatype_Aux.config) new_type_names descr sorts

    injects_of constr_inject (dist_rewrites, other_dist_rewrites) induct thy =

  let

    val _ = Datatype_Aux.message config "Constructing primrec combinators ...";

    val big_name = space_implode "_" new_type_names;

    val thy0 = Sign.add_path big_name thy;

    val descr' = flat descr;

    val recTs = Datatype_Aux.get_rec_types descr' sorts;

    val used = List.foldr OldTerm.add_typ_tfree_names [] recTs;

    val newTs = take (length (hd descr)) recTs;

    val induct_Ps = map head_of (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct)));

    val big_rec_name' = big_name ^ "_rec_set";

    val rec_set_names' =

      if length descr' = 1 then [big_rec_name'] else

        map ((curry (op ^) (big_rec_name' ^ "_")) o string_of_int)

          (1 upto (length descr'));

    val rec_set_names = map (Sign.full_bname thy0) rec_set_names';

    val (rec_result_Ts, reccomb_fn_Ts) = Datatype_Prop.make_primrec_Ts descr sorts used;

    val rec_set_Ts = map (fn (T1, T2) =>

      reccomb_fn_Ts @ [T1, T2] ---> HOLogic.boolT) (recTs ~~ rec_result_Ts);

    val rec_fns = map (uncurry (Datatype_Aux.mk_Free "f"))

      (reccomb_fn_Ts ~~ (1 upto (length reccomb_fn_Ts)));

    val rec_sets' = map (fn c => list_comb (Free c, rec_fns))

      (rec_set_names' ~~ rec_set_Ts);

    val rec_sets = map (fn c => list_comb (Const c, rec_fns))

      (rec_set_names ~~ rec_set_Ts);

    (* introduction rules for graph of primrec function *)

    fun make_rec_intr T rec_set (cname, cargs) (rec_intr_ts, l) =

      let

        fun mk_prem (dt, U) (j, k, prems, t1s, t2s) =

          let val free1 = Datatype_Aux.mk_Free "x" U j

          in (case (Datatype_Aux.strip_dtyp dt, strip_type U) of

             ((_, Datatype_Aux.DtRec m), (Us, _)) =>

               let

                 val free2 = Datatype_Aux.mk_Free "y" (Us ---> nth rec_result_Ts m) k;

                 val i = length Us

               in (j + 1, k + 1, HOLogic.mk_Trueprop (HOLogic.list_all

                     (map (pair "x") Us, nth rec_sets' m $

                       Datatype_Aux.app_bnds free1 i $ Datatype_Aux.app_bnds free2 i)) :: prems,

                   free1::t1s, free2::t2s)

               end

           | _ => (j + 1, k, prems, free1::t1s, t2s))

          end;

        val Ts = map (Datatype_Aux.typ_of_dtyp descr' sorts) cargs;

        val (_, _, prems, t1s, t2s) = fold_rev mk_prem (cargs ~~ Ts) (1, 1, [], [], [])

      in (rec_intr_ts @ [Logic.list_implies (prems, HOLogic.mk_Trueprop

        (rec_set $ list_comb (Const (cname, Ts ---> T), t1s) $

          list_comb (nth rec_fns l, t1s @ t2s)))], l + 1)

      end;

    val (rec_intr_ts, _) = fold (fn ((d, T), set_name) =>

      fold (make_rec_intr T set_name) (#3 (snd d)))

        (descr' ~~ recTs ~~ rec_sets') ([], 0);

    val ({intrs = rec_intrs, elims = rec_elims, ...}, thy1) =

      thy0

      |> Sign.map_naming Name_Space.conceal

      |> Inductive.add_inductive_global

          {quiet_mode = #quiet config, verbose = false, alt_name = Binding.name big_rec_name',

            coind = false, no_elim = false, no_ind = true, skip_mono = true, fork_mono = false}

          (map (fn (s, T) => ((Binding.name s, T), NoSyn)) (rec_set_names' ~~ rec_set_Ts))

          (map dest_Free rec_fns)

          (map (fn x => (Attrib.empty_binding, x)) rec_intr_ts) []

      ||> Sign.restore_naming thy0

      ||> Theory.checkpoint;

    (* prove uniqueness and termination of primrec combinators *)

    val _ = Datatype_Aux.message config "Proving termination and uniqueness of primrec functions ...";

    fun mk_unique_tac ((((i, (tname, _, constrs)), elim), T), T') (tac, intrs) =

      let

        val distinct_tac =

          (if i < length newTs then

             full_simp_tac (HOL_ss addsimps (nth dist_rewrites i)) 1

           else full_simp_tac (HOL_ss addsimps (flat other_dist_rewrites)) 1);

        val inject = map (fn r => r RS iffD1)

          (if i < length newTs then nth constr_inject i

            else injects_of tname);

        fun mk_unique_constr_tac n (cname, cargs) (tac, intr::intrs, j) =

          let

            val k = length (filter Datatype_Aux.is_rec_type cargs)

          in (EVERY [DETERM tac,

                REPEAT (etac ex1E 1), rtac ex1I 1,

                DEPTH_SOLVE_1 (ares_tac [intr] 1),

                REPEAT_DETERM_N k (etac thin_rl 1 THEN rotate_tac 1 1),

                etac elim 1,

                REPEAT_DETERM_N j distinct_tac,

                TRY (dresolve_tac inject 1),

                REPEAT (etac conjE 1), hyp_subst_tac 1,

                REPEAT (EVERY [etac allE 1, dtac mp 1, atac 1]),

                TRY (hyp_subst_tac 1),

                rtac refl 1,

                REPEAT_DETERM_N (n - j - 1) distinct_tac],

              intrs, j + 1)

          end;

        val (tac', intrs', _) = fold (mk_unique_constr_tac (length constrs))

          constrs (tac, intrs, 0);

      in (tac', intrs') end;

    val rec_unique_thms =

      let

        val rec_unique_ts = map (fn (((set_t, T1), T2), i) =>

          Const (@{const_name Ex1}, (T2 --> HOLogic.boolT) --> HOLogic.boolT) $

            absfree ("y", T2, set_t $ Datatype_Aux.mk_Free "x" T1 i $ Free ("y", T2)))

              (rec_sets ~~ recTs ~~ rec_result_Ts ~~ (1 upto length recTs));

        val cert = cterm_of thy1

        val insts = map (fn ((i, T), t) => absfree ("x" ^ (string_of_int i), T, t))

          ((1 upto length recTs) ~~ recTs ~~ rec_unique_ts);

        val induct' = cterm_instantiate ((map cert induct_Ps) ~~

          (map cert insts)) induct;

        val (tac, _) = fold mk_unique_tac (descr' ~~ rec_elims ~~ recTs ~~ rec_result_Ts)

           (((rtac induct' THEN_ALL_NEW Object_Logic.atomize_prems_tac) 1

              THEN rewrite_goals_tac [mk_meta_eq @{thm choice_eq}], rec_intrs));

      in

        Datatype_Aux.split_conj_thm (Skip_Proof.prove_global thy1 [] []

          (HOLogic.mk_Trueprop (Datatype_Aux.mk_conj rec_unique_ts)) (K tac))

      end;

    val rec_total_thms = map (fn r => r RS @{thm theI'}) rec_unique_thms;

    (* define primrec combinators *)

    val big_reccomb_name = (space_implode "_" new_type_names) ^ "_rec";

    val reccomb_names = map (Sign.full_bname thy1)

      (if length descr' = 1 then [big_reccomb_name] else

        (map ((curry (op ^) (big_reccomb_name ^ "_")) o string_of_int)

          (1 upto (length descr'))));

    val reccombs = map (fn ((name, T), T') => list_comb

      (Const (name, reccomb_fn_Ts @ [T] ---> T'), rec_fns))

        (reccomb_names ~~ recTs ~~ rec_result_Ts);

    val (reccomb_defs, thy2) =

      thy1

      |> Sign.add_consts_i (map (fn ((name, T), T') =>

          (Binding.name (Long_Name.base_name name), reccomb_fn_Ts @ [T] ---> T', NoSyn))

          (reccomb_names ~~ recTs ~~ rec_result_Ts))

      |> (Global_Theory.add_defs false o map Thm.no_attributes)

          (map (fn ((((name, comb), set), T), T') =>

            (Binding.name (Long_Name.base_name name ^ "_def"),

              Logic.mk_equals (comb, absfree ("x", T,

               Const (@{const_name The}, (T' --> HOLogic.boolT) --> T') $ absfree ("y", T',

                 set $ Free ("x", T) $ Free ("y", T'))))))

                   (reccomb_names ~~ reccombs ~~ rec_sets ~~ recTs ~~ rec_result_Ts))

      ||> Sign.parent_path

      ||> Theory.checkpoint;

    (* prove characteristic equations for primrec combinators *)

    val _ = Datatype_Aux.message config "Proving characteristic theorems for primrec combinators ...";

    val rec_thms = map (fn t => Skip_Proof.prove_global thy2 [] [] t

      (fn _ => EVERY

        [rewrite_goals_tac reccomb_defs,

         rtac @{thm the1_equality} 1,

         resolve_tac rec_unique_thms 1,

         resolve_tac rec_intrs 1,

         REPEAT (rtac allI 1 ORELSE resolve_tac rec_total_thms 1)]))

           (Datatype_Prop.make_primrecs new_type_names descr sorts thy2);

  in

    thy2

    |> Sign.add_path (space_implode "_" new_type_names)

    |> Global_Theory.add_thmss [((Binding.name "recs", rec_thms), [Nitpick_Simps.add])]

    ||> Sign.parent_path

    ||> Theory.checkpoint

    |-> (fn thms => pair (reccomb_names, flat thms))

  end;

(***************************** case combinators *******************************)

fun prove_case_thms (config : Datatype_Aux.config)

    new_type_names descr sorts reccomb_names primrec_thms thy =

  let

    val _ = Datatype_Aux.message config "Proving characteristic theorems for case combinators ...";

    val thy1 = Sign.add_path (space_implode "_" new_type_names) thy;

    val descr' = flat descr;

    val recTs = Datatype_Aux.get_rec_types descr' sorts;

    val used = List.foldr OldTerm.add_typ_tfree_names [] recTs;

    val newTs = take (length (hd descr)) recTs;

    val T' = TFree (singleton (Name.variant_list used) "'t", HOLogic.typeS);

    fun mk_dummyT dt = binder_types (Datatype_Aux.typ_of_dtyp descr' sorts dt) ---> T';

    val case_dummy_fns = map (fn (_, (_, _, constrs)) => map (fn (_, cargs) =>

      let

        val Ts = map (Datatype_Aux.typ_of_dtyp descr' sorts) cargs;

        val Ts' = map mk_dummyT (filter Datatype_Aux.is_rec_type cargs)

      in Const (@{const_name undefined}, Ts @ Ts' ---> T')

      end) constrs) descr';

    val case_names = map (fn s => Sign.full_bname thy1 (s ^ "_case")) new_type_names;

    (* define case combinators via primrec combinators *)

    val (case_defs, thy2) = fold (fn ((((i, (_, _, constrs)), T), name), recname) => fn (defs, thy) =>

        let

          val (fns1, fns2) = split_list (map (fn ((_, cargs), j) =>

            let

              val Ts = map (Datatype_Aux.typ_of_dtyp descr' sorts) cargs;

              val Ts' = Ts @ map mk_dummyT (filter Datatype_Aux.is_rec_type cargs);

              val frees' = map2 (Datatype_Aux.mk_Free "x") Ts' (1 upto length Ts');

              val frees = take (length cargs) frees';

              val free = Datatype_Aux.mk_Free "f" (Ts ---> T') j

            in

             (free, list_abs_free (map dest_Free frees',

               list_comb (free, frees)))

            end) (constrs ~~ (1 upto length constrs)));

          val caseT = (map (snd o dest_Free) fns1) @ [T] ---> T';

          val fns = flat (take i case_dummy_fns) @

            fns2 @ flat (drop (i + 1) case_dummy_fns);

          val reccomb = Const (recname, (map fastype_of fns) @ [T] ---> T');

          val decl = ((Binding.name (Long_Name.base_name name), caseT), NoSyn);

          val def = (Binding.name (Long_Name.base_name name ^ "_def"),

            Logic.mk_equals (list_comb (Const (name, caseT), fns1),

              list_comb (reccomb, (flat (take i case_dummy_fns)) @

                fns2 @ (flat (drop (i + 1) case_dummy_fns)))));

          val ([def_thm], thy') =

            thy

            |> Sign.declare_const_global decl |> snd

            |> (Global_Theory.add_defs false o map Thm.no_attributes) [def];

        in (defs @ [def_thm], thy')

        end) (hd descr ~~ newTs ~~ case_names ~~

          take (length newTs) reccomb_names) ([], thy1)

      ||> Theory.checkpoint;

    val case_thms = map (map (fn t => Skip_Proof.prove_global thy2 [] [] t

      (fn _ => EVERY [rewrite_goals_tac (case_defs @ map mk_meta_eq primrec_thms), rtac refl 1])))

          (Datatype_Prop.make_cases new_type_names descr sorts thy2)

  in

    thy2

    |> Context.theory_map ((fold o fold) Nitpick_Simps.add_thm case_thms)

    |> Sign.parent_path

    |> Datatype_Aux.store_thmss "cases" new_type_names case_thms

    |-> (fn thmss => pair (thmss, case_names))

  end;

(******************************* case splitting *******************************)

fun prove_split_thms (config : Datatype_Aux.config)

    new_type_names descr sorts constr_inject dist_rewrites casedist_thms case_thms thy =

  let

    val _ = Datatype_Aux.message config "Proving equations for case splitting ...";

    val descr' = flat descr;

    val recTs = Datatype_Aux.get_rec_types descr' sorts;

    val newTs = take (length (hd descr)) recTs;

    fun prove_split_thms ((((((t1, t2), inject), dist_rewrites'),

        exhaustion), case_thms'), T) =

      let

        val cert = cterm_of thy;

        val _ $ (_ $ lhs $ _) = hd (Logic.strip_assums_hyp (hd (prems_of exhaustion)));

        val exhaustion' = cterm_instantiate

          [(cert lhs, cert (Free ("x", T)))] exhaustion;

        val tacf = K (EVERY [rtac exhaustion' 1, ALLGOALS (asm_simp_tac

          (HOL_ss addsimps (dist_rewrites' @ inject @ case_thms')))])

      in

        (Skip_Proof.prove_global thy [] [] t1 tacf,

         Skip_Proof.prove_global thy [] [] t2 tacf)

      end;

    val split_thm_pairs = map prove_split_thms

      ((Datatype_Prop.make_splits new_type_names descr sorts thy) ~~ constr_inject ~~

        dist_rewrites ~~ casedist_thms ~~ case_thms ~~ newTs);

    val (split_thms, split_asm_thms) = ListPair.unzip split_thm_pairs

  in

    thy

    |> Datatype_Aux.store_thms "split" new_type_names split_thms

    ||>> Datatype_Aux.store_thms "split_asm" new_type_names split_asm_thms

    |-> (fn (thms1, thms2) => pair (thms1 ~~ thms2))

  end;

fun prove_weak_case_congs new_type_names descr sorts thy =

  let

    fun prove_weak_case_cong t =

       Skip_Proof.prove_global thy [] (Logic.strip_imp_prems t) (Logic.strip_imp_concl t)

         (fn {prems, ...} => EVERY [rtac ((hd prems) RS arg_cong) 1])

    val weak_case_congs = map prove_weak_case_cong (Datatype_Prop.make_weak_case_congs

      new_type_names descr sorts thy)

  in thy |> Datatype_Aux.store_thms "weak_case_cong" new_type_names weak_case_congs end;

(************************* additional theorems for TFL ************************)

fun prove_nchotomys (config : Datatype_Aux.config)

    new_type_names descr sorts casedist_thms thy =

  let

    val _ = Datatype_Aux.message config "Proving additional theorems for TFL ...";

    fun prove_nchotomy (t, exhaustion) =

      let

        (* For goal i, select the correct disjunct to attack, then prove it *)

        fun tac i 0 = EVERY [TRY (rtac disjI1 i),

              hyp_subst_tac i, REPEAT (rtac exI i), rtac refl i]

          | tac i n = rtac disjI2 i THEN tac i (n - 1)

      in 

        Skip_Proof.prove_global thy [] [] t (fn _ =>

          EVERY [rtac allI 1,

           Datatype_Aux.exh_tac (K exhaustion) 1,

           ALLGOALS (fn i => tac i (i-1))])

      end;

    val nchotomys =

      map prove_nchotomy (Datatype_Prop.make_nchotomys descr sorts ~~ casedist_thms)

  in thy |> Datatype_Aux.store_thms "nchotomy" new_type_names nchotomys end;

fun prove_case_congs new_type_names descr sorts nchotomys case_thms thy =

  let

    fun prove_case_cong ((t, nchotomy), case_rewrites) =

      let

        val (Const ("==>", _) $ tm $ _) = t;

        val (Const (@{const_name Trueprop}, _) $ (Const (@{const_name HOL.eq}, _) $ _ $ Ma)) = tm;

        val cert = cterm_of thy;

        val nchotomy' = nchotomy RS spec;

        val [v] = Term.add_vars (concl_of nchotomy') [];

        val nchotomy'' = cterm_instantiate [(cert (Var v), cert Ma)] nchotomy'

      in

        Skip_Proof.prove_global thy [] (Logic.strip_imp_prems t) (Logic.strip_imp_concl t)

          (fn {prems, ...} => 

            let val simplify = asm_simp_tac (HOL_ss addsimps (prems @ case_rewrites))

            in EVERY [simp_tac (HOL_ss addsimps [hd prems]) 1,

                cut_facts_tac [nchotomy''] 1,

                REPEAT (etac disjE 1 THEN REPEAT (etac exE 1) THEN simplify 1),

                REPEAT (etac exE 1) THEN simplify 1 (* Get last disjunct *)]

            end)

      end;

    val case_congs = map prove_case_cong (Datatype_Prop.make_case_congs

      new_type_names descr sorts thy ~~ nchotomys ~~ case_thms)

  in thy |> Datatype_Aux.store_thms "case_cong" new_type_names case_congs end;

open Datatype_Aux;

end;