(*  Title:      HOL/Tools/datatype_abs_proofs.ML

    ID:         $Id$

    Author:     Stefan Berghofer, TU Muenchen

Proofs and defintions independent of concrete representation

of datatypes  (i.e. requiring only abstract properties such as

injectivity / distinctness of constructors and induction)

 - case distinction (exhaustion) theorems

 - characteristic equations for primrec combinators

 - characteristic equations for case combinators

 - equations for splitting "P (case ...)" expressions

 - datatype size function

 - "nchotomy" and "case_cong" theorems for TFL

*)

signature DATATYPE_ABS_PROOFS =

sig

  val prove_casedist_thms : string list ->

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

    attribute list -> theory -> thm list * theory

  val prove_primrec_thms : bool -> string list ->

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

      DatatypeAux.datatype_info Symtab.table -> thm list list -> thm list list ->

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

  val prove_case_thms : bool -> string list ->

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

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

  val prove_split_thms : string list ->

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

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

        (thm * thm) list * theory

  val prove_size_thms : bool -> string list ->

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

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

  val prove_nchotomys : string list -> DatatypeAux.descr list ->

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

  val prove_weak_case_congs : string list -> DatatypeAux.descr list ->

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

  val prove_case_congs : string list ->

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

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

end;

structure DatatypeAbsProofs: DATATYPE_ABS_PROOFS =

struct

open DatatypeAux;

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

fun prove_casedist_thms new_type_names descr sorts induct case_names_exhausts thy =

  let

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

    val descr' = List.concat descr;

    val recTs = get_rec_types descr' sorts;

    val newTs = Library.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 ("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 = Library.take (i, dummyPs) @ (P::(Library.drop (i + 1, dummyPs)));

        val cert = cterm_of thy;

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

        val induct' = refl RS ((List.nth

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

      in

        Goal.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 prove_casedist_thm ((0 upto (length newTs - 1)) ~~

      (DatatypeProp.make_casedists descr sorts) ~~ newTs)

  in

    thy

    |> store_thms_atts "exhaust" new_type_names (map single case_names_exhausts) casedist_thms

  end;

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

fun prove_primrec_thms flat_names new_type_names descr sorts

    (dt_info : datatype_info Symtab.table) constr_inject dist_rewrites dist_ss induct thy =

  let

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

    val big_name = space_implode "_" new_type_names;

    val thy0 = add_path flat_names big_name thy;

    val descr' = List.concat descr;

    val recTs = get_rec_types descr' sorts;

    val used = foldr add_typ_tfree_names [] recTs;

    val newTs = Library.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_name (Theory.sign_of thy0)) rec_set_names';

    val (rec_result_Ts, reccomb_fn_Ts) = DatatypeProp.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 (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 ((rec_intr_ts, l), (cname, cargs)) =

      let

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

          let val free1 = mk_Free "x" U j

          in (case (strip_dtyp dt, strip_type U) of

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

               let

                 val free2 = mk_Free "y" (Us ---> List.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, List.nth (rec_sets', m) $

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

                   free1::t1s, free2::t2s)

               end

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

          end;

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

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

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

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

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

      end;

    val (rec_intr_ts, _) = Library.foldl (fn (x, ((d, T), set_name)) =>

      Library.foldl (make_rec_intr T set_name) (x, #3 (snd d)))

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

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

      setmp InductivePackage.quiet_mode (!quiet_mode)

        (TheoryTarget.init NONE #>

         InductivePackage.add_inductive_i false big_rec_name' false false true

           (map (fn (s, T) => (s, SOME T, NoSyn)) (rec_set_names' ~~ rec_set_Ts))

           (map (apsnd SOME o dest_Free) rec_fns)

           (map (fn x => (("", []), x)) rec_intr_ts) [] #>

         apsnd (ProofContext.theory_of o LocalTheory.exit)) thy0;

    (* prove uniqueness and termination of primrec combinators *)

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

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

      let

        val distinct_tac =

          (if i < length newTs then

             full_simp_tac (HOL_ss addsimps (List.nth (dist_rewrites, i))) 1

           else full_simp_tac dist_ss 1);

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

          (if i < length newTs then List.nth (constr_inject, i)

            else #inject (the (Symtab.lookup dt_info tname)));

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

          let

            val k = length (List.filter 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', _) = Library.foldl (mk_unique_constr_tac (length constrs))

          ((tac, intrs, 0), constrs);

      in (tac', intrs') end;

    val rec_unique_thms =

      let

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

          Const ("Ex1", (T2 --> HOLogic.boolT) --> HOLogic.boolT) $

            absfree ("y", T2, set_t $ 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, _) = Library.foldl mk_unique_tac

          (((rtac induct' THEN_ALL_NEW ObjectLogic.atomize_tac) 1

              THEN rewtac (mk_meta_eq choice_eq), rec_intrs),

            descr' ~~ rec_elims ~~ recTs ~~ rec_result_Ts);

      in split_conj_thm (Goal.prove_global thy1 [] []

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

      end;

    val rec_total_thms = map (fn r => r RS theI') rec_unique_thms;

    (* define primrec combinators *)

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

    val reccomb_names = map (Sign.full_name (Theory.sign_of 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

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

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

          (reccomb_names ~~ recTs ~~ rec_result_Ts))

      |> (PureThy.add_defs_i false o map Thm.no_attributes) (map (fn ((((name, comb), set), T), T') =>

          ((Sign.base_name name) ^ "_def", Logic.mk_equals (comb, absfree ("x", T,

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

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

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

      ||> parent_path flat_names;

    (* prove characteristic equations for primrec combinators *)

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

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

      (fn _ => EVERY

        [rewrite_goals_tac reccomb_defs,

         rtac 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)]))

           (DatatypeProp.make_primrecs new_type_names descr sorts thy2)

  in

    thy2

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

    |> PureThy.add_thmss [(("recs", rec_thms), [])]

    ||> Theory.parent_path

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

  end;

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

fun prove_case_thms flat_names new_type_names descr sorts reccomb_names primrec_thms thy =

  let

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

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

    val descr' = List.concat descr;

    val recTs = get_rec_types descr' sorts;

    val used = foldr add_typ_tfree_names [] recTs;

    val newTs = Library.take (length (hd descr), recTs);

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

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

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

      let

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

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

      in Const ("arbitrary", Ts @ Ts' ---> T')

      end) constrs) descr';

    val case_names = map (fn s =>

      Sign.full_name (Theory.sign_of thy1) (s ^ "_case")) new_type_names;

    (* define case combinators via primrec combinators *)

    val (case_defs, thy2) = Library.foldl (fn ((defs, thy),

      ((((i, (_, _, constrs)), T), name), recname)) =>

        let

          val (fns1, fns2) = ListPair.unzip (map (fn ((_, cargs), j) =>

            let

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

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

              val frees' = map (uncurry (mk_Free "x")) (Ts' ~~ (1 upto length Ts'));

              val frees = Library.take (length cargs, frees');

              val free = 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 = (List.concat (Library.take (i, case_dummy_fns))) @

            fns2 @ (List.concat (Library.drop (i + 1, case_dummy_fns)));

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

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

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

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

              list_comb (reccomb, (List.concat (Library.take (i, case_dummy_fns))) @

                fns2 @ (List.concat (Library.drop (i + 1, case_dummy_fns))) )));

          val ([def_thm], thy') =

            thy

            |> Theory.add_consts_i [decl]

            |> (PureThy.add_defs_i false o map Thm.no_attributes) [def];

        in (defs @ [def_thm], thy')

        end) (([], thy1), (hd descr) ~~ newTs ~~ case_names ~~

          (Library.take (length newTs, reccomb_names)));

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

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

          (DatatypeProp.make_cases new_type_names descr sorts thy2)

  in

    thy2

    |> parent_path flat_names

    |> store_thmss "cases" new_type_names case_thms

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

  end;

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

fun prove_split_thms new_type_names descr sorts constr_inject dist_rewrites

    casedist_thms case_thms thy =

  let

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

    val descr' = List.concat descr;

    val recTs = get_rec_types descr' sorts;

    val newTs = Library.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

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

         Goal.prove_global thy [] [] t2 tacf)

      end;

    val split_thm_pairs = map prove_split_thms

      ((DatatypeProp.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

    |> store_thms "split" new_type_names split_thms

    ||>> store_thms "split_asm" new_type_names split_asm_thms

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

  end;

(******************************* size functions *******************************)

fun prove_size_thms flat_names new_type_names descr sorts reccomb_names primrec_thms thy =

  if exists (fn (_, (_, _, constrs)) => exists (fn (_, cargs) => exists (fn dt =>

    is_rec_type dt andalso not (null (fst (strip_dtyp dt)))) cargs) constrs)

      (List.concat descr)

  then

    ([], thy)

  else

  let

    val _ = message "Proving equations for size function ...";

    val big_name = space_implode "_" new_type_names;

    val thy1 = add_path flat_names big_name thy;

    val descr' = List.concat descr;

    val recTs = get_rec_types descr' sorts;

    val size_name = "Nat.size";

    val size_names = replicate (length (hd descr)) size_name @

      map (Sign.full_name (Theory.sign_of thy1)) (DatatypeProp.indexify_names

        (map (fn T => name_of_typ T ^ "_size") (Library.drop (length (hd descr), recTs))));

    val def_names = map (fn s => s ^ "_def") (DatatypeProp.indexify_names

      (map (fn T => name_of_typ T ^ "_size") recTs));

    fun plus (t1, t2) = Const ("HOL.plus", [HOLogic.natT, HOLogic.natT] ---> HOLogic.natT) $ t1 $ t2;

    fun make_sizefun (_, cargs) =

      let

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

        val k = length (List.filter is_rec_type cargs);

        val t = if k = 0 then HOLogic.zero else

          foldl1 plus (map Bound (k - 1 downto 0) @ [HOLogic.mk_nat 1])

      in

        foldr (fn (T, t') => Abs ("x", T, t')) t (Ts @ replicate k HOLogic.natT)

      end;

    val fs = List.concat (map (fn (_, (_, _, constrs)) => map make_sizefun constrs) descr');

    val fTs = map fastype_of fs;

    val (size_def_thms, thy') =

      thy1

      |> Theory.add_consts_i (map (fn (s, T) =>

           (Sign.base_name s, T --> HOLogic.natT, NoSyn))

           (Library.drop (length (hd descr), size_names ~~ recTs)))

      |> (PureThy.add_defs_i true o map Thm.no_attributes) (map (fn (((s, T), def_name), rec_name) =>

           (def_name, Logic.mk_equals (Const (s, T --> HOLogic.natT),

            list_comb (Const (rec_name, fTs @ [T] ---> HOLogic.natT), fs))))

            (size_names ~~ recTs ~~ def_names ~~ reccomb_names))

      ||> parent_path flat_names;

    val rewrites = size_def_thms @ map mk_meta_eq primrec_thms;

    val size_thms = map (fn t => Goal.prove_global thy' [] [] t

      (fn _ => EVERY [rewrite_goals_tac rewrites, rtac refl 1]))

        (DatatypeProp.make_size descr sorts thy')

  in

    thy'

    |> Theory.add_path big_name

    |> PureThy.add_thmss [(("size", size_thms), [])]

    ||> Theory.parent_path

    |-> (fn thmss => pair (Library.flat thmss))

  end;

fun prove_weak_case_congs new_type_names descr sorts thy =

  let

    fun prove_weak_case_cong t =

       Goal.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 (DatatypeProp.make_weak_case_congs

      new_type_names descr sorts thy)

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

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

fun prove_nchotomys new_type_names descr sorts casedist_thms thy =

  let

    val _ = message "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 

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

          EVERY [rtac allI 1,

           exh_tac (K exhaustion) 1,

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

      end;

    val nchotomys =

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

  in thy |> 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 ("Trueprop", _) $ (Const ("op =", _) $ _ $ Ma)) = tm;

        val cert = cterm_of (Theory.sign_of thy);

        val nchotomy' = nchotomy RS spec;

        val nchotomy'' = cterm_instantiate

          [(cert (hd (add_term_vars (concl_of nchotomy', []))), cert Ma)] nchotomy'

      in

        Goal.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 (DatatypeProp.make_case_congs

      new_type_names descr sorts thy ~~ nchotomys ~~ case_thms)

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

end;