src/HOL/Tools/Datatype/datatype_abs_proofs.ML
author wenzelm
Sun Mar 07 12:19:47 2010 +0100 (2010-03-07)
changeset 35625 9c818cab0dd0
parent 35410 1ea89d2a1bd4
child 38864 4abe644fcea5
permissions -rw-r--r--
modernized structure Object_Logic;
     1 (*  Title:      HOL/Tools/Datatype/datatype_abs_proofs.ML
     2     Author:     Stefan Berghofer, TU Muenchen
     3 
     4 Datatype package: proofs and defintions independent of concrete
     5 representation of datatypes  (i.e. requiring only abstract
     6 properties: injectivity / distinctness of constructors and induction).
     7 *)
     8 
     9 signature DATATYPE_ABS_PROOFS =
    10 sig
    11   include DATATYPE_COMMON
    12   val prove_casedist_thms : config -> string list ->
    13     descr list -> (string * sort) list -> thm ->
    14     attribute list -> theory -> thm list * theory
    15   val prove_primrec_thms : config -> string list ->
    16     descr list -> (string * sort) list ->
    17       (string -> thm list) -> thm list list -> thm list list * thm list list ->
    18         thm -> theory -> (string list * thm list) * theory
    19   val prove_case_thms : config -> string list ->
    20     descr list -> (string * sort) list ->
    21       string list -> thm list -> theory -> (thm list list * string list) * theory
    22   val prove_split_thms : config -> string list ->
    23     descr list -> (string * sort) list ->
    24       thm list list -> thm list list -> thm list -> thm list list -> theory ->
    25         (thm * thm) list * theory
    26   val prove_nchotomys : config -> string list -> descr list ->
    27     (string * sort) list -> thm list -> theory -> thm list * theory
    28   val prove_weak_case_congs : string list -> descr list ->
    29     (string * sort) list -> theory -> thm list * theory
    30   val prove_case_congs : string list ->
    31     descr list -> (string * sort) list ->
    32       thm list -> thm list list -> theory -> thm list * theory
    33 end;
    34 
    35 structure Datatype_Abs_Proofs: DATATYPE_ABS_PROOFS =
    36 struct
    37 
    38 open Datatype_Aux;
    39 
    40 (************************ case distinction theorems ***************************)
    41 
    42 fun prove_casedist_thms (config : config) new_type_names descr sorts induct case_names_exhausts thy =
    43   let
    44     val _ = message config "Proving case distinction theorems ...";
    45 
    46     val descr' = flat descr;
    47     val recTs = get_rec_types descr' sorts;
    48     val newTs = take (length (hd descr)) recTs;
    49 
    50     val {maxidx, ...} = rep_thm induct;
    51     val induct_Ps = map head_of (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct)));
    52 
    53     fun prove_casedist_thm (i, (T, t)) =
    54       let
    55         val dummyPs = map (fn (Var (_, Type (_, [T', T'']))) =>
    56           Abs ("z", T', Const (@{const_name True}, T''))) induct_Ps;
    57         val P = Abs ("z", T, HOLogic.imp $ HOLogic.mk_eq (Var (("a", maxidx+1), T), Bound 0) $
    58           Var (("P", 0), HOLogic.boolT))
    59         val insts = take i dummyPs @ (P::(drop (i + 1) dummyPs));
    60         val cert = cterm_of thy;
    61         val insts' = (map cert induct_Ps) ~~ (map cert insts);
    62         val induct' = refl RS ((nth
    63           (split_conj_thm (cterm_instantiate insts' induct)) i) RSN (2, rev_mp))
    64 
    65       in
    66         Skip_Proof.prove_global thy [] (Logic.strip_imp_prems t) (Logic.strip_imp_concl t)
    67           (fn {prems, ...} => EVERY
    68             [rtac induct' 1,
    69              REPEAT (rtac TrueI 1),
    70              REPEAT ((rtac impI 1) THEN (eresolve_tac prems 1)),
    71              REPEAT (rtac TrueI 1)])
    72       end;
    73 
    74     val casedist_thms = map_index prove_casedist_thm
    75       (newTs ~~ Datatype_Prop.make_casedists descr sorts)
    76   in
    77     thy
    78     |> store_thms_atts "exhaust" new_type_names (map single case_names_exhausts) casedist_thms
    79   end;
    80 
    81 
    82 (*************************** primrec combinators ******************************)
    83 
    84 fun prove_primrec_thms (config : config) new_type_names descr sorts
    85     injects_of constr_inject (dist_rewrites, other_dist_rewrites) induct thy =
    86   let
    87     val _ = message config "Constructing primrec combinators ...";
    88 
    89     val big_name = space_implode "_" new_type_names;
    90     val thy0 = Sign.add_path big_name thy;
    91 
    92     val descr' = flat descr;
    93     val recTs = get_rec_types descr' sorts;
    94     val used = List.foldr OldTerm.add_typ_tfree_names [] recTs;
    95     val newTs = take (length (hd descr)) recTs;
    96 
    97     val induct_Ps = map head_of (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct)));
    98 
    99     val big_rec_name' = big_name ^ "_rec_set";
   100     val rec_set_names' =
   101       if length descr' = 1 then [big_rec_name'] else
   102         map ((curry (op ^) (big_rec_name' ^ "_")) o string_of_int)
   103           (1 upto (length descr'));
   104     val rec_set_names = map (Sign.full_bname thy0) rec_set_names';
   105 
   106     val (rec_result_Ts, reccomb_fn_Ts) = Datatype_Prop.make_primrec_Ts descr sorts used;
   107 
   108     val rec_set_Ts = map (fn (T1, T2) =>
   109       reccomb_fn_Ts @ [T1, T2] ---> HOLogic.boolT) (recTs ~~ rec_result_Ts);
   110 
   111     val rec_fns = map (uncurry (mk_Free "f"))
   112       (reccomb_fn_Ts ~~ (1 upto (length reccomb_fn_Ts)));
   113     val rec_sets' = map (fn c => list_comb (Free c, rec_fns))
   114       (rec_set_names' ~~ rec_set_Ts);
   115     val rec_sets = map (fn c => list_comb (Const c, rec_fns))
   116       (rec_set_names ~~ rec_set_Ts);
   117 
   118     (* introduction rules for graph of primrec function *)
   119 
   120     fun make_rec_intr T rec_set (cname, cargs) (rec_intr_ts, l) =
   121       let
   122         fun mk_prem (dt, U) (j, k, prems, t1s, t2s) =
   123           let val free1 = mk_Free "x" U j
   124           in (case (strip_dtyp dt, strip_type U) of
   125              ((_, DtRec m), (Us, _)) =>
   126                let
   127                  val free2 = mk_Free "y" (Us ---> nth rec_result_Ts m) k;
   128                  val i = length Us
   129                in (j + 1, k + 1, HOLogic.mk_Trueprop (HOLogic.list_all
   130                      (map (pair "x") Us, nth rec_sets' m $
   131                        app_bnds free1 i $ app_bnds free2 i)) :: prems,
   132                    free1::t1s, free2::t2s)
   133                end
   134            | _ => (j + 1, k, prems, free1::t1s, t2s))
   135           end;
   136 
   137         val Ts = map (typ_of_dtyp descr' sorts) cargs;
   138         val (_, _, prems, t1s, t2s) = fold_rev mk_prem (cargs ~~ Ts) (1, 1, [], [], [])
   139 
   140       in (rec_intr_ts @ [Logic.list_implies (prems, HOLogic.mk_Trueprop
   141         (rec_set $ list_comb (Const (cname, Ts ---> T), t1s) $
   142           list_comb (nth rec_fns l, t1s @ t2s)))], l + 1)
   143       end;
   144 
   145     val (rec_intr_ts, _) = fold (fn ((d, T), set_name) =>
   146       fold (make_rec_intr T set_name) (#3 (snd d)))
   147         (descr' ~~ recTs ~~ rec_sets') ([], 0);
   148 
   149     val ({intrs = rec_intrs, elims = rec_elims, ...}, thy1) =
   150       thy0
   151       |> Sign.map_naming Name_Space.conceal
   152       |> Inductive.add_inductive_global
   153           {quiet_mode = #quiet config, verbose = false, alt_name = Binding.name big_rec_name',
   154             coind = false, no_elim = false, no_ind = true, skip_mono = true, fork_mono = false}
   155           (map (fn (s, T) => ((Binding.name s, T), NoSyn)) (rec_set_names' ~~ rec_set_Ts))
   156           (map dest_Free rec_fns)
   157           (map (fn x => (Attrib.empty_binding, x)) rec_intr_ts) []
   158       ||> Sign.restore_naming thy0
   159       ||> Theory.checkpoint;
   160 
   161     (* prove uniqueness and termination of primrec combinators *)
   162 
   163     val _ = message config "Proving termination and uniqueness of primrec functions ...";
   164 
   165     fun mk_unique_tac ((((i, (tname, _, constrs)), elim), T), T') (tac, intrs) =
   166       let
   167         val distinct_tac =
   168           (if i < length newTs then
   169              full_simp_tac (HOL_ss addsimps (nth dist_rewrites i)) 1
   170            else full_simp_tac (HOL_ss addsimps (flat other_dist_rewrites)) 1);
   171 
   172         val inject = map (fn r => r RS iffD1)
   173           (if i < length newTs then nth constr_inject i
   174             else injects_of tname);
   175 
   176         fun mk_unique_constr_tac n (cname, cargs) (tac, intr::intrs, j) =
   177           let
   178             val k = length (filter is_rec_type cargs)
   179 
   180           in (EVERY [DETERM tac,
   181                 REPEAT (etac ex1E 1), rtac ex1I 1,
   182                 DEPTH_SOLVE_1 (ares_tac [intr] 1),
   183                 REPEAT_DETERM_N k (etac thin_rl 1 THEN rotate_tac 1 1),
   184                 etac elim 1,
   185                 REPEAT_DETERM_N j distinct_tac,
   186                 TRY (dresolve_tac inject 1),
   187                 REPEAT (etac conjE 1), hyp_subst_tac 1,
   188                 REPEAT (EVERY [etac allE 1, dtac mp 1, atac 1]),
   189                 TRY (hyp_subst_tac 1),
   190                 rtac refl 1,
   191                 REPEAT_DETERM_N (n - j - 1) distinct_tac],
   192               intrs, j + 1)
   193           end;
   194 
   195         val (tac', intrs', _) = fold (mk_unique_constr_tac (length constrs))
   196           constrs (tac, intrs, 0);
   197 
   198       in (tac', intrs') end;
   199 
   200     val rec_unique_thms =
   201       let
   202         val rec_unique_ts = map (fn (((set_t, T1), T2), i) =>
   203           Const (@{const_name Ex1}, (T2 --> HOLogic.boolT) --> HOLogic.boolT) $
   204             absfree ("y", T2, set_t $ mk_Free "x" T1 i $ Free ("y", T2)))
   205               (rec_sets ~~ recTs ~~ rec_result_Ts ~~ (1 upto length recTs));
   206         val cert = cterm_of thy1
   207         val insts = map (fn ((i, T), t) => absfree ("x" ^ (string_of_int i), T, t))
   208           ((1 upto length recTs) ~~ recTs ~~ rec_unique_ts);
   209         val induct' = cterm_instantiate ((map cert induct_Ps) ~~
   210           (map cert insts)) induct;
   211         val (tac, _) = fold mk_unique_tac (descr' ~~ rec_elims ~~ recTs ~~ rec_result_Ts)
   212            (((rtac induct' THEN_ALL_NEW Object_Logic.atomize_prems_tac) 1
   213               THEN rewrite_goals_tac [mk_meta_eq @{thm choice_eq}], rec_intrs));
   214 
   215       in split_conj_thm (Skip_Proof.prove_global thy1 [] []
   216         (HOLogic.mk_Trueprop (mk_conj rec_unique_ts)) (K tac))
   217       end;
   218 
   219     val rec_total_thms = map (fn r => r RS @{thm theI'}) rec_unique_thms;
   220 
   221     (* define primrec combinators *)
   222 
   223     val big_reccomb_name = (space_implode "_" new_type_names) ^ "_rec";
   224     val reccomb_names = map (Sign.full_bname thy1)
   225       (if length descr' = 1 then [big_reccomb_name] else
   226         (map ((curry (op ^) (big_reccomb_name ^ "_")) o string_of_int)
   227           (1 upto (length descr'))));
   228     val reccombs = map (fn ((name, T), T') => list_comb
   229       (Const (name, reccomb_fn_Ts @ [T] ---> T'), rec_fns))
   230         (reccomb_names ~~ recTs ~~ rec_result_Ts);
   231 
   232     val (reccomb_defs, thy2) =
   233       thy1
   234       |> Sign.add_consts_i (map (fn ((name, T), T') =>
   235           (Binding.name (Long_Name.base_name name), reccomb_fn_Ts @ [T] ---> T', NoSyn))
   236           (reccomb_names ~~ recTs ~~ rec_result_Ts))
   237       |> (PureThy.add_defs false o map Thm.no_attributes) (map (fn ((((name, comb), set), T), T') =>
   238           (Binding.name (Long_Name.base_name name ^ "_def"), Logic.mk_equals (comb, absfree ("x", T,
   239            Const (@{const_name The}, (T' --> HOLogic.boolT) --> T') $ absfree ("y", T',
   240              set $ Free ("x", T) $ Free ("y", T'))))))
   241                (reccomb_names ~~ reccombs ~~ rec_sets ~~ recTs ~~ rec_result_Ts))
   242       ||> Sign.parent_path
   243       ||> Theory.checkpoint;
   244 
   245 
   246     (* prove characteristic equations for primrec combinators *)
   247 
   248     val _ = message config "Proving characteristic theorems for primrec combinators ..."
   249 
   250     val rec_thms = map (fn t => Skip_Proof.prove_global thy2 [] [] t
   251       (fn _ => EVERY
   252         [rewrite_goals_tac reccomb_defs,
   253          rtac @{thm the1_equality} 1,
   254          resolve_tac rec_unique_thms 1,
   255          resolve_tac rec_intrs 1,
   256          REPEAT (rtac allI 1 ORELSE resolve_tac rec_total_thms 1)]))
   257            (Datatype_Prop.make_primrecs new_type_names descr sorts thy2)
   258 
   259   in
   260     thy2
   261     |> Sign.add_path (space_implode "_" new_type_names)
   262     |> PureThy.add_thmss [((Binding.name "recs", rec_thms), [Nitpick_Simps.add])]
   263     ||> Sign.parent_path
   264     ||> Theory.checkpoint
   265     |-> (fn thms => pair (reccomb_names, flat thms))
   266   end;
   267 
   268 
   269 (***************************** case combinators *******************************)
   270 
   271 fun prove_case_thms (config : config) new_type_names descr sorts reccomb_names primrec_thms thy =
   272   let
   273     val _ = message config "Proving characteristic theorems for case combinators ...";
   274 
   275     val thy1 = Sign.add_path (space_implode "_" new_type_names) thy;
   276 
   277     val descr' = flat descr;
   278     val recTs = get_rec_types descr' sorts;
   279     val used = List.foldr OldTerm.add_typ_tfree_names [] recTs;
   280     val newTs = take (length (hd descr)) recTs;
   281     val T' = TFree (Name.variant used "'t", HOLogic.typeS);
   282 
   283     fun mk_dummyT dt = binder_types (typ_of_dtyp descr' sorts dt) ---> T';
   284 
   285     val case_dummy_fns = map (fn (_, (_, _, constrs)) => map (fn (_, cargs) =>
   286       let
   287         val Ts = map (typ_of_dtyp descr' sorts) cargs;
   288         val Ts' = map mk_dummyT (filter is_rec_type cargs)
   289       in Const (@{const_name undefined}, Ts @ Ts' ---> T')
   290       end) constrs) descr';
   291 
   292     val case_names = map (fn s => Sign.full_bname thy1 (s ^ "_case")) new_type_names;
   293 
   294     (* define case combinators via primrec combinators *)
   295 
   296     val (case_defs, thy2) = fold (fn ((((i, (_, _, constrs)), T), name), recname) => fn (defs, thy) =>
   297         let
   298           val (fns1, fns2) = split_list (map (fn ((_, cargs), j) =>
   299             let
   300               val Ts = map (typ_of_dtyp descr' sorts) cargs;
   301               val Ts' = Ts @ map mk_dummyT (filter is_rec_type cargs);
   302               val frees' = map2 (mk_Free "x") Ts' (1 upto length Ts');
   303               val frees = take (length cargs) frees';
   304               val free = mk_Free "f" (Ts ---> T') j
   305             in
   306              (free, list_abs_free (map dest_Free frees',
   307                list_comb (free, frees)))
   308             end) (constrs ~~ (1 upto length constrs)));
   309 
   310           val caseT = (map (snd o dest_Free) fns1) @ [T] ---> T';
   311           val fns = flat (take i case_dummy_fns) @
   312             fns2 @ flat (drop (i + 1) case_dummy_fns);
   313           val reccomb = Const (recname, (map fastype_of fns) @ [T] ---> T');
   314           val decl = ((Binding.name (Long_Name.base_name name), caseT), NoSyn);
   315           val def = (Binding.name (Long_Name.base_name name ^ "_def"),
   316             Logic.mk_equals (list_comb (Const (name, caseT), fns1),
   317               list_comb (reccomb, (flat (take i case_dummy_fns)) @
   318                 fns2 @ (flat (drop (i + 1) case_dummy_fns)))));
   319           val ([def_thm], thy') =
   320             thy
   321             |> Sign.declare_const decl |> snd
   322             |> (PureThy.add_defs false o map Thm.no_attributes) [def];
   323 
   324         in (defs @ [def_thm], thy')
   325         end) (hd descr ~~ newTs ~~ case_names ~~
   326           take (length newTs) reccomb_names) ([], thy1)
   327       ||> Theory.checkpoint;
   328 
   329     val case_thms = map (map (fn t => Skip_Proof.prove_global thy2 [] [] t
   330       (fn _ => EVERY [rewrite_goals_tac (case_defs @ map mk_meta_eq primrec_thms), rtac refl 1])))
   331           (Datatype_Prop.make_cases new_type_names descr sorts thy2)
   332   in
   333     thy2
   334     |> Context.theory_map ((fold o fold) Nitpick_Simps.add_thm case_thms)
   335     |> Sign.parent_path
   336     |> store_thmss "cases" new_type_names case_thms
   337     |-> (fn thmss => pair (thmss, case_names))
   338   end;
   339 
   340 
   341 (******************************* case splitting *******************************)
   342 
   343 fun prove_split_thms (config : config) new_type_names descr sorts constr_inject dist_rewrites
   344     casedist_thms case_thms thy =
   345   let
   346     val _ = message config "Proving equations for case splitting ...";
   347 
   348     val descr' = flat descr;
   349     val recTs = get_rec_types descr' sorts;
   350     val newTs = take (length (hd descr)) recTs;
   351 
   352     fun prove_split_thms ((((((t1, t2), inject), dist_rewrites'),
   353         exhaustion), case_thms'), T) =
   354       let
   355         val cert = cterm_of thy;
   356         val _ $ (_ $ lhs $ _) = hd (Logic.strip_assums_hyp (hd (prems_of exhaustion)));
   357         val exhaustion' = cterm_instantiate
   358           [(cert lhs, cert (Free ("x", T)))] exhaustion;
   359         val tacf = K (EVERY [rtac exhaustion' 1, ALLGOALS (asm_simp_tac
   360           (HOL_ss addsimps (dist_rewrites' @ inject @ case_thms')))])
   361       in
   362         (Skip_Proof.prove_global thy [] [] t1 tacf,
   363          Skip_Proof.prove_global thy [] [] t2 tacf)
   364       end;
   365 
   366     val split_thm_pairs = map prove_split_thms
   367       ((Datatype_Prop.make_splits new_type_names descr sorts thy) ~~ constr_inject ~~
   368         dist_rewrites ~~ casedist_thms ~~ case_thms ~~ newTs);
   369 
   370     val (split_thms, split_asm_thms) = ListPair.unzip split_thm_pairs
   371 
   372   in
   373     thy
   374     |> store_thms "split" new_type_names split_thms
   375     ||>> store_thms "split_asm" new_type_names split_asm_thms
   376     |-> (fn (thms1, thms2) => pair (thms1 ~~ thms2))
   377   end;
   378 
   379 fun prove_weak_case_congs new_type_names descr sorts thy =
   380   let
   381     fun prove_weak_case_cong t =
   382        Skip_Proof.prove_global thy [] (Logic.strip_imp_prems t) (Logic.strip_imp_concl t)
   383          (fn {prems, ...} => EVERY [rtac ((hd prems) RS arg_cong) 1])
   384 
   385     val weak_case_congs = map prove_weak_case_cong (Datatype_Prop.make_weak_case_congs
   386       new_type_names descr sorts thy)
   387 
   388   in thy |> store_thms "weak_case_cong" new_type_names weak_case_congs end;
   389 
   390 (************************* additional theorems for TFL ************************)
   391 
   392 fun prove_nchotomys (config : config) new_type_names descr sorts casedist_thms thy =
   393   let
   394     val _ = message config "Proving additional theorems for TFL ...";
   395 
   396     fun prove_nchotomy (t, exhaustion) =
   397       let
   398         (* For goal i, select the correct disjunct to attack, then prove it *)
   399         fun tac i 0 = EVERY [TRY (rtac disjI1 i),
   400               hyp_subst_tac i, REPEAT (rtac exI i), rtac refl i]
   401           | tac i n = rtac disjI2 i THEN tac i (n - 1)
   402       in 
   403         Skip_Proof.prove_global thy [] [] t (fn _ =>
   404           EVERY [rtac allI 1,
   405            exh_tac (K exhaustion) 1,
   406            ALLGOALS (fn i => tac i (i-1))])
   407       end;
   408 
   409     val nchotomys =
   410       map prove_nchotomy (Datatype_Prop.make_nchotomys descr sorts ~~ casedist_thms)
   411 
   412   in thy |> store_thms "nchotomy" new_type_names nchotomys end;
   413 
   414 fun prove_case_congs new_type_names descr sorts nchotomys case_thms thy =
   415   let
   416     fun prove_case_cong ((t, nchotomy), case_rewrites) =
   417       let
   418         val (Const ("==>", _) $ tm $ _) = t;
   419         val (Const (@{const_name Trueprop}, _) $ (Const (@{const_name "op ="}, _) $ _ $ Ma)) = tm;
   420         val cert = cterm_of thy;
   421         val nchotomy' = nchotomy RS spec;
   422         val [v] = Term.add_vars (concl_of nchotomy') [];
   423         val nchotomy'' = cterm_instantiate [(cert (Var v), cert Ma)] nchotomy'
   424       in
   425         Skip_Proof.prove_global thy [] (Logic.strip_imp_prems t) (Logic.strip_imp_concl t)
   426           (fn {prems, ...} => 
   427             let val simplify = asm_simp_tac (HOL_ss addsimps (prems @ case_rewrites))
   428             in EVERY [simp_tac (HOL_ss addsimps [hd prems]) 1,
   429                 cut_facts_tac [nchotomy''] 1,
   430                 REPEAT (etac disjE 1 THEN REPEAT (etac exE 1) THEN simplify 1),
   431                 REPEAT (etac exE 1) THEN simplify 1 (* Get last disjunct *)]
   432             end)
   433       end;
   434 
   435     val case_congs = map prove_case_cong (Datatype_Prop.make_case_congs
   436       new_type_names descr sorts thy ~~ nchotomys ~~ case_thms)
   437 
   438   in thy |> store_thms "case_cong" new_type_names case_congs end;
   439 
   440 end;