(*  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

   - "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_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

         SkipProof.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 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) =

         InductivePackage.add_inductive_global (serial_string ())

           {quiet_mode = ! quiet_mode, verbose = false, kind = Thm.internalK,

             alt_name = Name.binding big_rec_name', coind = false, no_elim = false, no_ind = true,

             skip_mono = true}

           (map (fn (s, T) => ((Name.binding s, T), NoSyn)) (rec_set_names' ~~ rec_set_Ts))

           (map dest_Free rec_fns)

           (map (fn x => ((Name.no_binding, []), x)) rec_intr_ts) [] 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_prems_tac) 1

               THEN rewtac (mk_meta_eq choice_eq), rec_intrs),

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

       in split_conj_thm (SkipProof.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 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') =>

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

           (reccomb_names ~~ recTs ~~ rec_result_Ts))

       |> (PureThy.add_defs 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 => SkipProof.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

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

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

     ||> Sign.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 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

             |> Sign.declare_const [] decl |> snd

             |> (PureThy.add_defs 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 => SkipProof.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

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

          SkipProof.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;

 fun prove_weak_case_congs new_type_names descr sorts thy =

   let

     fun prove_weak_case_cong t =

        SkipProof.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 

         SkipProof.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 thy;

         val nchotomy' = nchotomy RS spec;

         val nchotomy'' = cterm_instantiate

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

       in

         SkipProof.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;