src/HOL/Tools/SMT/smt_normalize.ML
author boehmes
Tue Oct 26 11:39:26 2010 +0200 (2010-10-26)
changeset 40161 539d07b00e5f
parent 39483 9f0e5684f04b
child 40162 7f58a9a843c2
permissions -rw-r--r--
keep track of theorems initially given to SMT (even if they are rewritten); provide interface to filter theorems necessary for SMT proofs
     1 (*  Title:      HOL/Tools/SMT/smt_normalize.ML
     2     Author:     Sascha Boehme, TU Muenchen
     3 
     4 Normalization steps on theorems required by SMT solvers:
     5   * simplify trivial distincts (those with less than three elements),
     6   * rewrite bool case expressions as if expressions,
     7   * normalize numerals (e.g. replace negative numerals by negated positive
     8     numerals),
     9   * embed natural numbers into integers,
    10   * add extra rules specifying types and constants which occur frequently,
    11   * fully translate into object logic, add universal closure,
    12   * monomorphize (create instances of schematic rules),
    13   * lift lambda terms,
    14   * make applications explicit for functions with varying number of arguments.
    15   * add (hypothetical definitions for) missing datatype selectors,
    16 *)
    17 
    18 signature SMT_NORMALIZE =
    19 sig
    20   type extra_norm = (int * thm) list -> Proof.context ->
    21     (int * thm) list * Proof.context
    22   val normalize: extra_norm -> bool -> (int * thm) list -> Proof.context ->
    23     (int * thm) list * Proof.context
    24   val atomize_conv: Proof.context -> conv
    25   val eta_expand_conv: (Proof.context -> conv) -> Proof.context -> conv
    26 end
    27 
    28 structure SMT_Normalize: SMT_NORMALIZE =
    29 struct
    30 
    31 infix 2 ??
    32 fun (test ?? f) x = if test x then f x else x
    33 
    34 fun if_conv c cv1 cv2 ct = (if c (Thm.term_of ct) then cv1 else cv2) ct
    35 fun if_true_conv c cv = if_conv c cv Conv.all_conv
    36 
    37 
    38 
    39 (* simplification of trivial distincts (distinct should have at least
    40    three elements in the argument list) *)
    41 
    42 local
    43   fun is_trivial_distinct (Const (@{const_name distinct}, _) $ t) =
    44         length (HOLogic.dest_list t) <= 2
    45     | is_trivial_distinct _ = false
    46 
    47   val thms = map mk_meta_eq @{lemma
    48     "distinct [] = True"
    49     "distinct [x] = True"
    50     "distinct [x, y] = (x ~= y)"
    51     by simp_all}
    52   fun distinct_conv _ =
    53     if_true_conv is_trivial_distinct (Conv.rewrs_conv thms)
    54 in
    55 fun trivial_distinct ctxt =
    56   map (apsnd ((Term.exists_subterm is_trivial_distinct o Thm.prop_of) ??
    57     Conv.fconv_rule (Conv.top_conv distinct_conv ctxt)))
    58 end
    59 
    60 
    61 
    62 (* rewrite bool case expressions as if expressions *)
    63 
    64 local
    65   val is_bool_case = (fn
    66       Const (@{const_name "bool.bool_case"}, _) $ _ $ _ $ _ => true
    67     | _ => false)
    68 
    69   val thms = map mk_meta_eq @{lemma
    70     "(case P of True => x | False => y) = (if P then x else y)"
    71     "(case P of False => y | True => x) = (if P then x else y)"
    72     by simp_all}
    73   val unfold_conv = if_true_conv is_bool_case (Conv.rewrs_conv thms)
    74 in
    75 fun rewrite_bool_cases ctxt =
    76   map (apsnd ((Term.exists_subterm is_bool_case o Thm.prop_of) ??
    77     Conv.fconv_rule (Conv.top_conv (K unfold_conv) ctxt)))
    78 end
    79 
    80 
    81 
    82 (* normalization of numerals: rewriting of negative integer numerals into
    83    positive numerals, Numeral0 into 0, Numeral1 into 1 *)
    84 
    85 local
    86   fun is_number_sort ctxt T =
    87     Sign.of_sort (ProofContext.theory_of ctxt) (T, @{sort number_ring})
    88 
    89   fun is_strange_number ctxt (t as Const (@{const_name number_of}, _) $ _) =
    90         (case try HOLogic.dest_number t of
    91           SOME (T, i) => is_number_sort ctxt T andalso i < 2
    92         | NONE => false)
    93     | is_strange_number _ _ = false
    94 
    95   val pos_numeral_ss = HOL_ss
    96     addsimps [@{thm Int.number_of_minus}, @{thm Int.number_of_Min}]
    97     addsimps [@{thm Int.number_of_Pls}, @{thm Int.numeral_1_eq_1}]
    98     addsimps @{thms Int.pred_bin_simps}
    99     addsimps @{thms Int.normalize_bin_simps}
   100     addsimps @{lemma
   101       "Int.Min = - Int.Bit1 Int.Pls"
   102       "Int.Bit0 (- Int.Pls) = - Int.Pls"
   103       "Int.Bit0 (- k) = - Int.Bit0 k"
   104       "Int.Bit1 (- k) = - Int.Bit1 (Int.pred k)"
   105       by simp_all (simp add: pred_def)}
   106 
   107   fun pos_conv ctxt = if_conv (is_strange_number ctxt)
   108     (Simplifier.rewrite (Simplifier.context ctxt pos_numeral_ss))
   109     Conv.no_conv
   110 in
   111 fun normalize_numerals ctxt =
   112   map (apsnd ((Term.exists_subterm (is_strange_number ctxt) o Thm.prop_of) ??
   113     Conv.fconv_rule (Conv.top_sweep_conv pos_conv ctxt)))
   114 end
   115 
   116 
   117 
   118 (* embedding of standard natural number operations into integer operations *)
   119 
   120 local
   121   val nat_embedding = map (pair ~1) @{lemma
   122     "nat (int n) = n"
   123     "i >= 0 --> int (nat i) = i"
   124     "i < 0 --> int (nat i) = 0"
   125     by simp_all}
   126 
   127   val nat_rewriting = @{lemma
   128     "0 = nat 0"
   129     "1 = nat 1"
   130     "number_of i = nat (number_of i)"
   131     "int (nat 0) = 0"
   132     "int (nat 1) = 1"
   133     "a < b = (int a < int b)"
   134     "a <= b = (int a <= int b)"
   135     "Suc a = nat (int a + 1)"
   136     "a + b = nat (int a + int b)"
   137     "a - b = nat (int a - int b)"
   138     "a * b = nat (int a * int b)"
   139     "a div b = nat (int a div int b)"
   140     "a mod b = nat (int a mod int b)"
   141     "min a b = nat (min (int a) (int b))"
   142     "max a b = nat (max (int a) (int b))"
   143     "int (nat (int a + int b)) = int a + int b"
   144     "int (nat (int a * int b)) = int a * int b"
   145     "int (nat (int a div int b)) = int a div int b"
   146     "int (nat (int a mod int b)) = int a mod int b"
   147     "int (nat (min (int a) (int b))) = min (int a) (int b)"
   148     "int (nat (max (int a) (int b))) = max (int a) (int b)"
   149     by (simp_all add: nat_mult_distrib nat_div_distrib nat_mod_distrib
   150       int_mult[symmetric] zdiv_int[symmetric] zmod_int[symmetric])}
   151 
   152   fun on_positive num f x = 
   153     (case try HOLogic.dest_number (Thm.term_of num) of
   154       SOME (_, i) => if i >= 0 then SOME (f x) else NONE
   155     | NONE => NONE)
   156 
   157   val cancel_int_nat_ss = HOL_ss
   158     addsimps [@{thm Nat_Numeral.nat_number_of}]
   159     addsimps [@{thm Nat_Numeral.int_nat_number_of}]
   160     addsimps @{thms neg_simps}
   161 
   162   fun cancel_int_nat_simproc _ ss ct = 
   163     let
   164       val num = Thm.dest_arg (Thm.dest_arg ct)
   165       val goal = Thm.mk_binop @{cterm "op == :: int => _"} ct num
   166       val simpset = Simplifier.inherit_context ss cancel_int_nat_ss
   167       fun tac _ = Simplifier.simp_tac simpset 1
   168     in on_positive num (Goal.prove_internal [] goal) tac end
   169 
   170   val nat_ss = HOL_ss
   171     addsimps nat_rewriting
   172     addsimprocs [Simplifier.make_simproc {
   173       name = "cancel_int_nat_num", lhss = [@{cpat "int (nat _)"}],
   174       proc = cancel_int_nat_simproc, identifier = [] }]
   175 
   176   fun conv ctxt = Simplifier.rewrite (Simplifier.context ctxt nat_ss)
   177 
   178   val uses_nat_type = Term.exists_type (Term.exists_subtype (equal @{typ nat}))
   179   val uses_nat_int =
   180     Term.exists_subterm (member (op aconv) [@{term int}, @{term nat}])
   181 in
   182 fun nat_as_int ctxt =
   183   map (apsnd ((uses_nat_type o Thm.prop_of) ?? Conv.fconv_rule (conv ctxt))) #>
   184   exists (uses_nat_int o Thm.prop_of o snd) ?? append nat_embedding
   185 end
   186 
   187 
   188 
   189 (* further normalizations: beta/eta, universal closure, atomize *)
   190 
   191 val eta_expand_eq = @{lemma "f == (%x. f x)" by (rule reflexive)}
   192 
   193 fun eta_expand_conv cv ctxt =
   194   Conv.rewr_conv eta_expand_eq then_conv Conv.abs_conv (cv o snd) ctxt
   195 
   196 local
   197   val eta_conv = eta_expand_conv
   198 
   199   fun keep_conv ctxt = Conv.binder_conv (norm_conv o snd) ctxt
   200   and eta_binder_conv ctxt = Conv.arg_conv (eta_conv norm_conv ctxt)
   201   and keep_let_conv ctxt = Conv.combination_conv
   202     (Conv.arg_conv (norm_conv ctxt)) (Conv.abs_conv (norm_conv o snd) ctxt)
   203   and unfold_let_conv ctxt = Conv.combination_conv
   204     (Conv.arg_conv (norm_conv ctxt)) (eta_conv norm_conv ctxt)
   205   and unfold_conv thm ctxt = Conv.rewr_conv thm then_conv keep_conv ctxt
   206   and unfold_ex1_conv ctxt = unfold_conv @{thm Ex1_def} ctxt
   207   and unfold_ball_conv ctxt = unfold_conv (mk_meta_eq @{thm Ball_def}) ctxt
   208   and unfold_bex_conv ctxt = unfold_conv (mk_meta_eq @{thm Bex_def}) ctxt
   209   and norm_conv ctxt ct =
   210     (case Thm.term_of ct of
   211       Const (@{const_name All}, _) $ Abs _ => keep_conv
   212     | Const (@{const_name All}, _) $ _ => eta_binder_conv
   213     | Const (@{const_name All}, _) => eta_conv eta_binder_conv
   214     | Const (@{const_name Ex}, _) $ Abs _ => keep_conv
   215     | Const (@{const_name Ex}, _) $ _ => eta_binder_conv
   216     | Const (@{const_name Ex}, _) => eta_conv eta_binder_conv
   217     | Const (@{const_name Let}, _) $ _ $ Abs _ => keep_let_conv
   218     | Const (@{const_name Let}, _) $ _ $ _ => unfold_let_conv
   219     | Const (@{const_name Let}, _) $ _ => eta_conv unfold_let_conv
   220     | Const (@{const_name Let}, _) => eta_conv (eta_conv unfold_let_conv)
   221     | Const (@{const_name Ex1}, _) $ _ => unfold_ex1_conv
   222     | Const (@{const_name Ex1}, _) => eta_conv unfold_ex1_conv 
   223     | Const (@{const_name Ball}, _) $ _ $ _ => unfold_ball_conv
   224     | Const (@{const_name Ball}, _) $ _ => eta_conv unfold_ball_conv
   225     | Const (@{const_name Ball}, _) => eta_conv (eta_conv unfold_ball_conv)
   226     | Const (@{const_name Bex}, _) $ _ $ _ => unfold_bex_conv
   227     | Const (@{const_name Bex}, _) $ _ => eta_conv unfold_bex_conv
   228     | Const (@{const_name Bex}, _) => eta_conv (eta_conv unfold_bex_conv)
   229     | Abs _ => Conv.abs_conv (norm_conv o snd)
   230     | _ $ _ => Conv.comb_conv o norm_conv
   231     | _ => K Conv.all_conv) ctxt ct
   232 
   233   fun is_normed t =
   234     (case t of
   235       Const (@{const_name All}, _) $ Abs (_, _, u) => is_normed u
   236     | Const (@{const_name All}, _) $ _ => false
   237     | Const (@{const_name All}, _) => false
   238     | Const (@{const_name Ex}, _) $ Abs (_, _, u) => is_normed u
   239     | Const (@{const_name Ex}, _) $ _ => false
   240     | Const (@{const_name Ex}, _) => false
   241     | Const (@{const_name Let}, _) $ u1 $ Abs (_, _, u2) =>
   242         is_normed u1 andalso is_normed u2
   243     | Const (@{const_name Let}, _) $ _ $ _ => false
   244     | Const (@{const_name Let}, _) $ _ => false
   245     | Const (@{const_name Let}, _) => false
   246     | Const (@{const_name Ex1}, _) => false
   247     | Const (@{const_name Ball}, _) => false
   248     | Const (@{const_name Bex}, _) => false
   249     | Abs (_, _, u) => is_normed u
   250     | u1 $ u2 => is_normed u1 andalso is_normed u2
   251     | _ => true)
   252 in
   253 fun norm_binder_conv ctxt = if_conv is_normed Conv.all_conv (norm_conv ctxt)
   254 end
   255 
   256 fun norm_def ctxt thm =
   257   (case Thm.prop_of thm of
   258     @{term Trueprop} $ (Const (@{const_name HOL.eq}, _) $ _ $ Abs _) =>
   259       norm_def ctxt (thm RS @{thm fun_cong})
   260   | Const (@{const_name "=="}, _) $ _ $ Abs _ =>
   261       norm_def ctxt (thm RS @{thm meta_eq_to_obj_eq})
   262   | _ => thm)
   263 
   264 fun atomize_conv ctxt ct =
   265   (case Thm.term_of ct of
   266     @{term "op ==>"} $ _ $ _ =>
   267       Conv.binop_conv (atomize_conv ctxt) then_conv
   268       Conv.rewr_conv @{thm atomize_imp}
   269   | Const (@{const_name "=="}, _) $ _ $ _ =>
   270       Conv.binop_conv (atomize_conv ctxt) then_conv
   271       Conv.rewr_conv @{thm atomize_eq}
   272   | Const (@{const_name all}, _) $ Abs _ =>
   273       Conv.binder_conv (atomize_conv o snd) ctxt then_conv
   274       Conv.rewr_conv @{thm atomize_all}
   275   | _ => Conv.all_conv) ct
   276 
   277 fun normalize_rule ctxt =
   278   Conv.fconv_rule (
   279     (* reduce lambda abstractions, except at known binders: *)
   280     Thm.beta_conversion true then_conv
   281     Thm.eta_conversion then_conv
   282     norm_binder_conv ctxt) #>
   283   norm_def ctxt #>
   284   Drule.forall_intr_vars #>
   285   Conv.fconv_rule (atomize_conv ctxt)
   286 
   287 
   288 
   289 (* lift lambda terms into additional rules *)
   290 
   291 local
   292   val meta_eq = @{cpat "op =="}
   293   val meta_eqT = hd (Thm.dest_ctyp (Thm.ctyp_of_term meta_eq))
   294   fun inst_meta cT = Thm.instantiate_cterm ([(meta_eqT, cT)], []) meta_eq
   295   fun mk_meta_eq ct cu = Thm.mk_binop (inst_meta (Thm.ctyp_of_term ct)) ct cu
   296 
   297   fun cert ctxt = Thm.cterm_of (ProofContext.theory_of ctxt)
   298 
   299   fun used_vars cvs ct =
   300     let
   301       val lookup = AList.lookup (op aconv) (map (` Thm.term_of) cvs)
   302       val add = (fn SOME ct => insert (op aconvc) ct | _ => I)
   303     in Term.fold_aterms (add o lookup) (Thm.term_of ct) [] end
   304 
   305   fun apply cv thm = 
   306     let val thm' = Thm.combination thm (Thm.reflexive cv)
   307     in Thm.transitive thm' (Thm.beta_conversion false (Thm.rhs_of thm')) end
   308   fun apply_def cvs eq = Thm.symmetric (fold apply cvs eq)
   309 
   310   fun replace_lambda cvs ct (cx as (ctxt, defs)) =
   311     let
   312       val cvs' = used_vars cvs ct
   313       val ct' = fold_rev Thm.cabs cvs' ct
   314     in
   315       (case Termtab.lookup defs (Thm.term_of ct') of
   316         SOME eq => (apply_def cvs' eq, cx)
   317       | NONE =>
   318           let
   319             val {T, ...} = Thm.rep_cterm ct' and n = Name.uu
   320             val (n', ctxt') = yield_singleton Variable.variant_fixes n ctxt
   321             val cu = mk_meta_eq (cert ctxt (Free (n', T))) ct'
   322             val (eq, ctxt'') = yield_singleton Assumption.add_assumes cu ctxt'
   323             val defs' = Termtab.update (Thm.term_of ct', eq) defs
   324           in (apply_def cvs' eq, (ctxt'', defs')) end)
   325     end
   326 
   327   fun none ct cx = (Thm.reflexive ct, cx)
   328   fun in_comb f g ct cx =
   329     let val (cu1, cu2) = Thm.dest_comb ct
   330     in cx |> f cu1 ||>> g cu2 |>> uncurry Thm.combination end
   331   fun in_arg f = in_comb none f
   332   fun in_abs f cvs ct (ctxt, defs) =
   333     let
   334       val (n, ctxt') = yield_singleton Variable.variant_fixes Name.uu ctxt
   335       val (cv, cu) = Thm.dest_abs (SOME n) ct
   336     in  (ctxt', defs) |> f (cv :: cvs) cu |>> Thm.abstract_rule n cv end
   337 
   338   fun traverse cvs ct =
   339     (case Thm.term_of ct of
   340       Const (@{const_name All}, _) $ Abs _ => in_arg (in_abs traverse cvs)
   341     | Const (@{const_name Ex}, _) $ Abs _ => in_arg (in_abs traverse cvs)
   342     | Const (@{const_name Let}, _) $ _ $ Abs _ =>
   343         in_comb (in_arg (traverse cvs)) (in_abs traverse cvs)
   344     | Abs _ => at_lambda cvs
   345     | _ $ _ => in_comb (traverse cvs) (traverse cvs)
   346     | _ => none) ct
   347 
   348   and at_lambda cvs ct =
   349     in_abs traverse cvs ct #-> (fn thm =>
   350     replace_lambda cvs (Thm.rhs_of thm) #>> Thm.transitive thm)
   351 
   352   fun has_free_lambdas t =
   353     (case t of
   354       Const (@{const_name All}, _) $ Abs (_, _, u) => has_free_lambdas u
   355     | Const (@{const_name Ex}, _) $ Abs (_, _, u) => has_free_lambdas u
   356     | Const (@{const_name Let}, _) $ u1 $ Abs (_, _, u2) =>
   357         has_free_lambdas u1 orelse has_free_lambdas u2
   358     | Abs _ => true
   359     | u1 $ u2 => has_free_lambdas u1 orelse has_free_lambdas u2
   360     | _ => false)
   361 
   362   fun lift_lm f thm cx =
   363     if not (has_free_lambdas (Thm.prop_of thm)) then (thm, cx)
   364     else cx |> f (Thm.cprop_of thm) |>> (fn thm' => Thm.equal_elim thm' thm)
   365 in
   366 fun lift_lambdas irules ctxt =
   367   let
   368     val cx = (ctxt, Termtab.empty)
   369     val (idxs, thms) = split_list irules
   370     val (thms', (ctxt', defs)) = fold_map (lift_lm (traverse [])) thms cx
   371     val eqs = Termtab.fold (cons o normalize_rule ctxt' o snd) defs []
   372   in (map (pair ~1) eqs @ (idxs ~~ thms'), ctxt') end
   373 end
   374 
   375 
   376 
   377 (* make application explicit for functions with varying number of arguments *)
   378 
   379 local
   380   val const = prefix "c" and free = prefix "f"
   381   fun min i (e as (_, j)) = if i <> j then (true, Int.min (i, j)) else e
   382   fun add t i = Symtab.map_default (t, (false, i)) (min i)
   383   fun traverse t =
   384     (case Term.strip_comb t of
   385       (Const (n, _), ts) => add (const n) (length ts) #> fold traverse ts 
   386     | (Free (n, _), ts) => add (free n) (length ts) #> fold traverse ts
   387     | (Abs (_, _, u), ts) => fold traverse (u :: ts)
   388     | (_, ts) => fold traverse ts)
   389   fun prune tab = Symtab.fold (fn (n, (true, i)) =>
   390     Symtab.update (n, i) | _ => I) tab Symtab.empty
   391 
   392   fun binop_conv cv1 cv2 = Conv.combination_conv (Conv.arg_conv cv1) cv2
   393   fun nary_conv conv1 conv2 ct =
   394     (Conv.combination_conv (nary_conv conv1 conv2) conv2 else_conv conv1) ct
   395   fun abs_conv conv tb = Conv.abs_conv (fn (cv, cx) =>
   396     let val n = fst (Term.dest_Free (Thm.term_of cv))
   397     in conv (Symtab.update (free n, 0) tb) cx end)
   398   val fun_app_rule = @{lemma "f x == fun_app f x" by (simp add: fun_app_def)}
   399 in
   400 fun explicit_application ctxt irules =
   401   let
   402     fun sub_conv tb ctxt ct =
   403       (case Term.strip_comb (Thm.term_of ct) of
   404         (Const (n, _), ts) => app_conv tb (const n) (length ts) ctxt
   405       | (Free (n, _), ts) => app_conv tb (free n) (length ts) ctxt
   406       | (Abs _, _) => nary_conv (abs_conv sub_conv tb ctxt) (sub_conv tb ctxt)
   407       | (_, _) => nary_conv Conv.all_conv (sub_conv tb ctxt)) ct
   408     and app_conv tb n i ctxt =
   409       (case Symtab.lookup tb n of
   410         NONE => nary_conv Conv.all_conv (sub_conv tb ctxt)
   411       | SOME j => fun_app_conv tb ctxt (i - j))
   412     and fun_app_conv tb ctxt i ct = (
   413       if i = 0 then nary_conv Conv.all_conv (sub_conv tb ctxt)
   414       else
   415         Conv.rewr_conv fun_app_rule then_conv
   416         binop_conv (fun_app_conv tb ctxt (i-1)) (sub_conv tb ctxt)) ct
   417 
   418     fun needs_exp_app tab = Term.exists_subterm (fn
   419         Bound _ $ _ => true
   420       | Const (n, _) => Symtab.defined tab (const n)
   421       | Free (n, _) => Symtab.defined tab (free n)
   422       | _ => false)
   423 
   424     fun rewrite tab ctxt thm =
   425       if not (needs_exp_app tab (Thm.prop_of thm)) then thm
   426       else Conv.fconv_rule (sub_conv tab ctxt) thm
   427 
   428     val tab = prune (fold (traverse o Thm.prop_of o snd) irules Symtab.empty)
   429   in map (apsnd (rewrite tab ctxt)) irules end
   430 end
   431 
   432 
   433 
   434 (* add missing datatype selectors via hypothetical definitions *)
   435 
   436 local
   437   val add = (fn Type (n, _) => Symtab.update (n, ()) | _ => I)
   438 
   439   fun collect t =
   440     (case Term.strip_comb t of
   441       (Abs (_, T, t), _) => add T #> collect t
   442     | (Const (_, T), ts) => collects T ts
   443     | (Free (_, T), ts) => collects T ts
   444     | _ => I)
   445   and collects T ts =
   446     let val ((Ts, Us), U) = Term.strip_type T |> apfst (chop (length ts))
   447     in fold add Ts #> add (Us ---> U) #> fold collect ts end
   448 
   449   fun add_constructors thy n =
   450     (case Datatype.get_info thy n of
   451       NONE => I
   452     | SOME {descr, ...} => fold (fn (_, (_, _, cs)) => fold (fn (n, ds) =>
   453         fold (insert (op =) o pair n) (1 upto length ds)) cs) descr)
   454 
   455   fun add_selector (c as (n, i)) ctxt =
   456     (case Datatype_Selectors.lookup_selector ctxt c of
   457       SOME _ => ctxt
   458     | NONE =>
   459         let
   460           val T = Sign.the_const_type (ProofContext.theory_of ctxt) n
   461           val U = Term.body_type T --> nth (Term.binder_types T) (i-1)
   462         in
   463           ctxt
   464           |> yield_singleton Variable.variant_fixes Name.uu
   465           |>> pair ((n, T), i) o rpair U
   466           |-> Context.proof_map o Datatype_Selectors.add_selector
   467         end)
   468 in
   469 
   470 fun datatype_selectors irules ctxt =
   471   let
   472     val ns = Symtab.keys (fold (collect o Thm.prop_of o snd) irules Symtab.empty)
   473     val cs = fold (add_constructors (ProofContext.theory_of ctxt)) ns []
   474   in (irules, fold add_selector cs ctxt) end
   475     (* FIXME: also generate hypothetical definitions for the selectors *)
   476 
   477 end
   478 
   479 
   480 
   481 (* combined normalization *)
   482 
   483 type extra_norm = (int * thm) list -> Proof.context ->
   484   (int * thm) list * Proof.context
   485 
   486 fun with_context f irules ctxt = (f ctxt irules, ctxt)
   487 
   488 fun normalize extra_norm with_datatypes irules ctxt =
   489   irules
   490   |> trivial_distinct ctxt
   491   |> rewrite_bool_cases ctxt
   492   |> normalize_numerals ctxt
   493   |> nat_as_int ctxt
   494   |> rpair ctxt
   495   |-> extra_norm
   496   |-> with_context (fn cx => map (apsnd (normalize_rule cx)))
   497   |-> SMT_Monomorph.monomorph
   498   |-> lift_lambdas
   499   |-> with_context explicit_application
   500   |-> (if with_datatypes then datatype_selectors else pair)
   501 
   502 end