src/HOL/Tools/SMT/smt_normalize.ML
 author boehmes Wed Nov 24 13:31:12 2010 +0100 (2010-11-24) changeset 40685 dcb27631cb45 parent 40681 872b08416fb4 child 40686 4725ed462387 permissions -rw-r--r--
instantiate elimination rules (reduces number of quantified variables, and makes such theorems better amenable for SMT solvers)
```     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 = bool -> (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 structure U = SMT_Utils
```
```    32
```
```    33 infix 2 ??
```
```    34 fun (test ?? f) x = if test x then f x else x
```
```    35
```
```    36
```
```    37
```
```    38 (* instantiate elimination rules *)
```
```    39
```
```    40 local
```
```    41   val (cpfalse, cfalse) = `U.mk_cprop (Thm.cterm_of @{theory} @{const False})
```
```    42
```
```    43   fun inst f ct thm =
```
```    44     let val cv = f (Drule.strip_imp_concl (Thm.cprop_of thm))
```
```    45     in Thm.instantiate ([], [(cv, ct)]) thm end
```
```    46 in
```
```    47
```
```    48 fun instantiate_elim thm =
```
```    49   (case Thm.concl_of thm of
```
```    50     @{const Trueprop} \$ Var (_, @{typ bool}) => inst Thm.dest_arg cfalse thm
```
```    51   | Var _ => inst I cpfalse thm
```
```    52   | _ => thm)
```
```    53
```
```    54 end
```
```    55
```
```    56
```
```    57
```
```    58 (* simplification of trivial distincts (distinct should have at least
```
```    59    three elements in the argument list) *)
```
```    60
```
```    61 local
```
```    62   fun is_trivial_distinct (Const (@{const_name distinct}, _) \$ t) =
```
```    63         (case try HOLogic.dest_list t of
```
```    64           SOME [] => true
```
```    65         | SOME [_] => true
```
```    66         | _ => false)
```
```    67     | is_trivial_distinct _ = false
```
```    68
```
```    69   val thms = map mk_meta_eq @{lemma
```
```    70     "distinct [] = True"
```
```    71     "distinct [x] = True"
```
```    72     "distinct [x, y] = (x ~= y)"
```
```    73     by simp_all}
```
```    74   fun distinct_conv _ =
```
```    75     U.if_true_conv is_trivial_distinct (Conv.rewrs_conv thms)
```
```    76 in
```
```    77 fun trivial_distinct ctxt =
```
```    78   map (apsnd ((Term.exists_subterm is_trivial_distinct o Thm.prop_of) ??
```
```    79     Conv.fconv_rule (Conv.top_conv distinct_conv ctxt)))
```
```    80 end
```
```    81
```
```    82
```
```    83
```
```    84 (* rewrite bool case expressions as if expressions *)
```
```    85
```
```    86 local
```
```    87   val is_bool_case = (fn
```
```    88       Const (@{const_name "bool.bool_case"}, _) \$ _ \$ _ \$ _ => true
```
```    89     | _ => false)
```
```    90
```
```    91   val thm = mk_meta_eq @{lemma
```
```    92     "(case P of True => x | False => y) = (if P then x else y)" by simp}
```
```    93   val unfold_conv = U.if_true_conv is_bool_case (Conv.rewr_conv thm)
```
```    94 in
```
```    95 fun rewrite_bool_cases ctxt =
```
```    96   map (apsnd ((Term.exists_subterm is_bool_case o Thm.prop_of) ??
```
```    97     Conv.fconv_rule (Conv.top_conv (K unfold_conv) ctxt)))
```
```    98 end
```
```    99
```
```   100
```
```   101
```
```   102 (* normalization of numerals: rewriting of negative integer numerals into
```
```   103    positive numerals, Numeral0 into 0, Numeral1 into 1 *)
```
```   104
```
```   105 local
```
```   106   fun is_number_sort ctxt T =
```
```   107     Sign.of_sort (ProofContext.theory_of ctxt) (T, @{sort number_ring})
```
```   108
```
```   109   fun is_strange_number ctxt (t as Const (@{const_name number_of}, _) \$ _) =
```
```   110         (case try HOLogic.dest_number t of
```
```   111           SOME (T, i) => is_number_sort ctxt T andalso i < 2
```
```   112         | NONE => false)
```
```   113     | is_strange_number _ _ = false
```
```   114
```
```   115   val pos_numeral_ss = HOL_ss
```
```   116     addsimps [@{thm Int.number_of_minus}, @{thm Int.number_of_Min}]
```
```   117     addsimps [@{thm Int.number_of_Pls}, @{thm Int.numeral_1_eq_1}]
```
```   118     addsimps @{thms Int.pred_bin_simps}
```
```   119     addsimps @{thms Int.normalize_bin_simps}
```
```   120     addsimps @{lemma
```
```   121       "Int.Min = - Int.Bit1 Int.Pls"
```
```   122       "Int.Bit0 (- Int.Pls) = - Int.Pls"
```
```   123       "Int.Bit0 (- k) = - Int.Bit0 k"
```
```   124       "Int.Bit1 (- k) = - Int.Bit1 (Int.pred k)"
```
```   125       by simp_all (simp add: pred_def)}
```
```   126
```
```   127   fun pos_conv ctxt = U.if_conv (is_strange_number ctxt)
```
```   128     (Simplifier.rewrite (Simplifier.context ctxt pos_numeral_ss))
```
```   129     Conv.no_conv
```
```   130 in
```
```   131 fun normalize_numerals ctxt =
```
```   132   map (apsnd ((Term.exists_subterm (is_strange_number ctxt) o Thm.prop_of) ??
```
```   133     Conv.fconv_rule (Conv.top_sweep_conv pos_conv ctxt)))
```
```   134 end
```
```   135
```
```   136
```
```   137
```
```   138 (* embedding of standard natural number operations into integer operations *)
```
```   139
```
```   140 local
```
```   141   val nat_embedding = map (pair ~1) @{lemma
```
```   142     "nat (int n) = n"
```
```   143     "i >= 0 --> int (nat i) = i"
```
```   144     "i < 0 --> int (nat i) = 0"
```
```   145     by simp_all}
```
```   146
```
```   147   val nat_rewriting = @{lemma
```
```   148     "0 = nat 0"
```
```   149     "1 = nat 1"
```
```   150     "(number_of :: int => nat) = (%i. nat (number_of i))"
```
```   151     "int (nat 0) = 0"
```
```   152     "int (nat 1) = 1"
```
```   153     "op < = (%a b. int a < int b)"
```
```   154     "op <= = (%a b. int a <= int b)"
```
```   155     "Suc = (%a. nat (int a + 1))"
```
```   156     "op + = (%a b. nat (int a + int b))"
```
```   157     "op - = (%a b. nat (int a - int b))"
```
```   158     "op * = (%a b. nat (int a * int b))"
```
```   159     "op div = (%a b. nat (int a div int b))"
```
```   160     "op mod = (%a b. nat (int a mod int b))"
```
```   161     "min = (%a b. nat (min (int a) (int b)))"
```
```   162     "max = (%a b. nat (max (int a) (int b)))"
```
```   163     "int (nat (int a + int b)) = int a + int b"
```
```   164     "int (nat (int a + 1)) = int a + 1"  (* special rule due to Suc above *)
```
```   165     "int (nat (int a * int b)) = int a * int b"
```
```   166     "int (nat (int a div int b)) = int a div int b"
```
```   167     "int (nat (int a mod int b)) = int a mod int b"
```
```   168     "int (nat (min (int a) (int b))) = min (int a) (int b)"
```
```   169     "int (nat (max (int a) (int b))) = max (int a) (int b)"
```
```   170     by (auto intro!: ext simp add: nat_mult_distrib nat_div_distrib
```
```   171       nat_mod_distrib int_mult[symmetric] zdiv_int[symmetric]
```
```   172       zmod_int[symmetric])}
```
```   173
```
```   174   fun on_positive num f x =
```
```   175     (case try HOLogic.dest_number (Thm.term_of num) of
```
```   176       SOME (_, i) => if i >= 0 then SOME (f x) else NONE
```
```   177     | NONE => NONE)
```
```   178
```
```   179   val cancel_int_nat_ss = HOL_ss
```
```   180     addsimps [@{thm Nat_Numeral.nat_number_of}]
```
```   181     addsimps [@{thm Nat_Numeral.int_nat_number_of}]
```
```   182     addsimps @{thms neg_simps}
```
```   183
```
```   184   val int_eq = Thm.cterm_of @{theory} @{const "==" (int)}
```
```   185
```
```   186   fun cancel_int_nat_simproc _ ss ct =
```
```   187     let
```
```   188       val num = Thm.dest_arg (Thm.dest_arg ct)
```
```   189       val goal = Thm.mk_binop int_eq ct num
```
```   190       val simpset = Simplifier.inherit_context ss cancel_int_nat_ss
```
```   191       fun tac _ = Simplifier.simp_tac simpset 1
```
```   192     in on_positive num (Goal.prove_internal [] goal) tac end
```
```   193
```
```   194   val nat_ss = HOL_ss
```
```   195     addsimps nat_rewriting
```
```   196     addsimprocs [
```
```   197       Simplifier.make_simproc {
```
```   198         name = "cancel_int_nat_num", lhss = [@{cpat "int (nat _)"}],
```
```   199         proc = cancel_int_nat_simproc, identifier = [] }]
```
```   200
```
```   201   fun conv ctxt = Simplifier.rewrite (Simplifier.context ctxt nat_ss)
```
```   202
```
```   203   val uses_nat_type = Term.exists_type (Term.exists_subtype (equal @{typ nat}))
```
```   204   val uses_nat_int = Term.exists_subterm (member (op aconv)
```
```   205     [@{const of_nat (int)}, @{const nat}])
```
```   206 in
```
```   207 fun nat_as_int ctxt =
```
```   208   map (apsnd ((uses_nat_type o Thm.prop_of) ?? Conv.fconv_rule (conv ctxt))) #>
```
```   209   exists (uses_nat_int o Thm.prop_of o snd) ?? append nat_embedding
```
```   210 end
```
```   211
```
```   212
```
```   213
```
```   214 (* further normalizations: beta/eta, universal closure, atomize *)
```
```   215
```
```   216 val eta_expand_eq = @{lemma "f == (%x. f x)" by (rule reflexive)}
```
```   217
```
```   218 fun eta_expand_conv cv ctxt =
```
```   219   Conv.rewr_conv eta_expand_eq then_conv Conv.abs_conv (cv o snd) ctxt
```
```   220
```
```   221 local
```
```   222   val eta_conv = eta_expand_conv
```
```   223
```
```   224   fun args_conv cv ct =
```
```   225     (case Thm.term_of ct of
```
```   226       _ \$ _ => Conv.combination_conv (args_conv cv) cv
```
```   227     | _ => Conv.all_conv) ct
```
```   228
```
```   229   fun eta_args_conv cv 0 = args_conv o cv
```
```   230     | eta_args_conv cv i = eta_conv (eta_args_conv cv (i-1))
```
```   231
```
```   232   fun keep_conv ctxt = Conv.binder_conv (norm_conv o snd) ctxt
```
```   233   and eta_binder_conv ctxt = Conv.arg_conv (eta_conv norm_conv ctxt)
```
```   234   and keep_let_conv ctxt = Conv.combination_conv
```
```   235     (Conv.arg_conv (norm_conv ctxt)) (Conv.abs_conv (norm_conv o snd) ctxt)
```
```   236   and unfold_let_conv ctxt = Conv.combination_conv
```
```   237     (Conv.arg_conv (norm_conv ctxt)) (eta_conv norm_conv ctxt)
```
```   238   and unfold_conv thm ctxt = Conv.rewr_conv thm then_conv keep_conv ctxt
```
```   239   and unfold_ex1_conv ctxt = unfold_conv @{thm Ex1_def} ctxt
```
```   240   and unfold_ball_conv ctxt = unfold_conv (mk_meta_eq @{thm Ball_def}) ctxt
```
```   241   and unfold_bex_conv ctxt = unfold_conv (mk_meta_eq @{thm Bex_def}) ctxt
```
```   242   and norm_conv ctxt ct =
```
```   243     (case Thm.term_of ct of
```
```   244       Const (@{const_name All}, _) \$ Abs _ => keep_conv
```
```   245     | Const (@{const_name All}, _) \$ _ => eta_binder_conv
```
```   246     | Const (@{const_name All}, _) => eta_conv eta_binder_conv
```
```   247     | Const (@{const_name Ex}, _) \$ Abs _ => keep_conv
```
```   248     | Const (@{const_name Ex}, _) \$ _ => eta_binder_conv
```
```   249     | Const (@{const_name Ex}, _) => eta_conv eta_binder_conv
```
```   250     | Const (@{const_name Let}, _) \$ _ \$ Abs _ => keep_let_conv
```
```   251     | Const (@{const_name Let}, _) \$ _ \$ _ => unfold_let_conv
```
```   252     | Const (@{const_name Let}, _) \$ _ => eta_conv unfold_let_conv
```
```   253     | Const (@{const_name Let}, _) => eta_conv (eta_conv unfold_let_conv)
```
```   254     | Const (@{const_name Ex1}, _) \$ _ => unfold_ex1_conv
```
```   255     | Const (@{const_name Ex1}, _) => eta_conv unfold_ex1_conv
```
```   256     | Const (@{const_name Ball}, _) \$ _ \$ _ => unfold_ball_conv
```
```   257     | Const (@{const_name Ball}, _) \$ _ => eta_conv unfold_ball_conv
```
```   258     | Const (@{const_name Ball}, _) => eta_conv (eta_conv unfold_ball_conv)
```
```   259     | Const (@{const_name Bex}, _) \$ _ \$ _ => unfold_bex_conv
```
```   260     | Const (@{const_name Bex}, _) \$ _ => eta_conv unfold_bex_conv
```
```   261     | Const (@{const_name Bex}, _) => eta_conv (eta_conv unfold_bex_conv)
```
```   262     | Abs _ => Conv.abs_conv (norm_conv o snd)
```
```   263     | _ =>
```
```   264         (case Term.strip_comb (Thm.term_of ct) of
```
```   265           (Const (c as (_, T)), ts) =>
```
```   266             if SMT_Builtin.is_builtin ctxt c
```
```   267             then eta_args_conv norm_conv
```
```   268               (length (Term.binder_types T) - length ts)
```
```   269             else args_conv o norm_conv
```
```   270         | _ => args_conv o norm_conv)) ctxt ct
```
```   271
```
```   272   fun is_normed ctxt t =
```
```   273     (case t of
```
```   274       Const (@{const_name All}, _) \$ Abs (_, _, u) => is_normed ctxt u
```
```   275     | Const (@{const_name All}, _) \$ _ => false
```
```   276     | Const (@{const_name All}, _) => false
```
```   277     | Const (@{const_name Ex}, _) \$ Abs (_, _, u) => is_normed ctxt u
```
```   278     | Const (@{const_name Ex}, _) \$ _ => false
```
```   279     | Const (@{const_name Ex}, _) => false
```
```   280     | Const (@{const_name Let}, _) \$ u1 \$ Abs (_, _, u2) =>
```
```   281         is_normed ctxt u1 andalso is_normed ctxt u2
```
```   282     | Const (@{const_name Let}, _) \$ _ \$ _ => false
```
```   283     | Const (@{const_name Let}, _) \$ _ => false
```
```   284     | Const (@{const_name Let}, _) => false
```
```   285     | Const (@{const_name Ex1}, _) \$ _ => false
```
```   286     | Const (@{const_name Ex1}, _) => false
```
```   287     | Const (@{const_name Ball}, _) \$ _ \$ _ => false
```
```   288     | Const (@{const_name Ball}, _) \$ _ => false
```
```   289     | Const (@{const_name Ball}, _) => false
```
```   290     | Const (@{const_name Bex}, _) \$ _ \$ _ => false
```
```   291     | Const (@{const_name Bex}, _) \$ _ => false
```
```   292     | Const (@{const_name Bex}, _) => false
```
```   293     | Abs (_, _, u) => is_normed ctxt u
```
```   294     | _ =>
```
```   295         (case Term.strip_comb t of
```
```   296           (Const (c as (_, T)), ts) =>
```
```   297             if SMT_Builtin.is_builtin ctxt c
```
```   298             then length (Term.binder_types T) = length ts andalso
```
```   299               forall (is_normed ctxt) ts
```
```   300             else forall (is_normed ctxt) ts
```
```   301         | (_, ts) => forall (is_normed ctxt) ts))
```
```   302 in
```
```   303 fun norm_binder_conv ctxt =
```
```   304   U.if_conv (is_normed ctxt) Conv.all_conv (norm_conv ctxt)
```
```   305 end
```
```   306
```
```   307 fun norm_def ctxt thm =
```
```   308   (case Thm.prop_of thm of
```
```   309     @{const Trueprop} \$ (Const (@{const_name HOL.eq}, _) \$ _ \$ Abs _) =>
```
```   310       norm_def ctxt (thm RS @{thm fun_cong})
```
```   311   | Const (@{const_name "=="}, _) \$ _ \$ Abs _ =>
```
```   312       norm_def ctxt (thm RS @{thm meta_eq_to_obj_eq})
```
```   313   | _ => thm)
```
```   314
```
```   315 fun atomize_conv ctxt ct =
```
```   316   (case Thm.term_of ct of
```
```   317     @{const "==>"} \$ _ \$ _ =>
```
```   318       Conv.binop_conv (atomize_conv ctxt) then_conv
```
```   319       Conv.rewr_conv @{thm atomize_imp}
```
```   320   | Const (@{const_name "=="}, _) \$ _ \$ _ =>
```
```   321       Conv.binop_conv (atomize_conv ctxt) then_conv
```
```   322       Conv.rewr_conv @{thm atomize_eq}
```
```   323   | Const (@{const_name all}, _) \$ Abs _ =>
```
```   324       Conv.binder_conv (atomize_conv o snd) ctxt then_conv
```
```   325       Conv.rewr_conv @{thm atomize_all}
```
```   326   | _ => Conv.all_conv) ct
```
```   327
```
```   328 fun normalize_rule ctxt =
```
```   329   Conv.fconv_rule (
```
```   330     (* reduce lambda abstractions, except at known binders: *)
```
```   331     Thm.beta_conversion true then_conv
```
```   332     Thm.eta_conversion then_conv
```
```   333     norm_binder_conv ctxt) #>
```
```   334   norm_def ctxt #>
```
```   335   Drule.forall_intr_vars #>
```
```   336   Conv.fconv_rule (atomize_conv ctxt)
```
```   337
```
```   338
```
```   339
```
```   340 (* lift lambda terms into additional rules *)
```
```   341
```
```   342 local
```
```   343   fun used_vars cvs ct =
```
```   344     let
```
```   345       val lookup = AList.lookup (op aconv) (map (` Thm.term_of) cvs)
```
```   346       val add = (fn SOME ct => insert (op aconvc) ct | _ => I)
```
```   347     in Term.fold_aterms (add o lookup) (Thm.term_of ct) [] end
```
```   348
```
```   349   fun apply cv thm =
```
```   350     let val thm' = Thm.combination thm (Thm.reflexive cv)
```
```   351     in Thm.transitive thm' (Thm.beta_conversion false (Thm.rhs_of thm')) end
```
```   352   fun apply_def cvs eq = Thm.symmetric (fold apply cvs eq)
```
```   353
```
```   354   fun replace_lambda cvs ct (cx as (ctxt, defs)) =
```
```   355     let
```
```   356       val cvs' = used_vars cvs ct
```
```   357       val ct' = fold_rev Thm.cabs cvs' ct
```
```   358     in
```
```   359       (case Termtab.lookup defs (Thm.term_of ct') of
```
```   360         SOME eq => (apply_def cvs' eq, cx)
```
```   361       | NONE =>
```
```   362           let
```
```   363             val {T, ...} = Thm.rep_cterm ct' and n = Name.uu
```
```   364             val (n', ctxt') = yield_singleton Variable.variant_fixes n ctxt
```
```   365             val cu = U.mk_cequals (U.certify ctxt (Free (n', T))) ct'
```
```   366             val (eq, ctxt'') = yield_singleton Assumption.add_assumes cu ctxt'
```
```   367             val defs' = Termtab.update (Thm.term_of ct', eq) defs
```
```   368           in (apply_def cvs' eq, (ctxt'', defs')) end)
```
```   369     end
```
```   370
```
```   371   fun none ct cx = (Thm.reflexive ct, cx)
```
```   372   fun in_comb f g ct cx =
```
```   373     let val (cu1, cu2) = Thm.dest_comb ct
```
```   374     in cx |> f cu1 ||>> g cu2 |>> uncurry Thm.combination end
```
```   375   fun in_arg f = in_comb none f
```
```   376   fun in_abs f cvs ct (ctxt, defs) =
```
```   377     let
```
```   378       val (n, ctxt') = yield_singleton Variable.variant_fixes Name.uu ctxt
```
```   379       val (cv, cu) = Thm.dest_abs (SOME n) ct
```
```   380     in  (ctxt', defs) |> f (cv :: cvs) cu |>> Thm.abstract_rule n cv end
```
```   381
```
```   382   fun traverse cvs ct =
```
```   383     (case Thm.term_of ct of
```
```   384       Const (@{const_name All}, _) \$ Abs _ => in_arg (in_abs traverse cvs)
```
```   385     | Const (@{const_name Ex}, _) \$ Abs _ => in_arg (in_abs traverse cvs)
```
```   386     | Const (@{const_name Let}, _) \$ _ \$ Abs _ =>
```
```   387         in_comb (in_arg (traverse cvs)) (in_abs traverse cvs)
```
```   388     | Abs _ => at_lambda cvs
```
```   389     | _ \$ _ => in_comb (traverse cvs) (traverse cvs)
```
```   390     | _ => none) ct
```
```   391
```
```   392   and at_lambda cvs ct =
```
```   393     in_abs traverse cvs ct #-> (fn thm =>
```
```   394     replace_lambda cvs (Thm.rhs_of thm) #>> Thm.transitive thm)
```
```   395
```
```   396   fun has_free_lambdas t =
```
```   397     (case t of
```
```   398       Const (@{const_name All}, _) \$ Abs (_, _, u) => has_free_lambdas u
```
```   399     | Const (@{const_name Ex}, _) \$ Abs (_, _, u) => has_free_lambdas u
```
```   400     | Const (@{const_name Let}, _) \$ u1 \$ Abs (_, _, u2) =>
```
```   401         has_free_lambdas u1 orelse has_free_lambdas u2
```
```   402     | Abs _ => true
```
```   403     | u1 \$ u2 => has_free_lambdas u1 orelse has_free_lambdas u2
```
```   404     | _ => false)
```
```   405
```
```   406   fun lift_lm f thm cx =
```
```   407     if not (has_free_lambdas (Thm.prop_of thm)) then (thm, cx)
```
```   408     else cx |> f (Thm.cprop_of thm) |>> (fn thm' => Thm.equal_elim thm' thm)
```
```   409 in
```
```   410 fun lift_lambdas irules ctxt =
```
```   411   let
```
```   412     val cx = (ctxt, Termtab.empty)
```
```   413     val (idxs, thms) = split_list irules
```
```   414     val (thms', (ctxt', defs)) = fold_map (lift_lm (traverse [])) thms cx
```
```   415     val eqs = Termtab.fold (cons o normalize_rule ctxt' o snd) defs []
```
```   416   in (map (pair ~1) eqs @ (idxs ~~ thms'), ctxt') end
```
```   417 end
```
```   418
```
```   419
```
```   420
```
```   421 (* make application explicit for functions with varying number of arguments *)
```
```   422
```
```   423 local
```
```   424   val const = prefix "c" and free = prefix "f"
```
```   425   fun min i (e as (_, j)) = if i <> j then (true, Int.min (i, j)) else e
```
```   426   fun add t i = Symtab.map_default (t, (false, i)) (min i)
```
```   427   fun traverse t =
```
```   428     (case Term.strip_comb t of
```
```   429       (Const (n, _), ts) => add (const n) (length ts) #> fold traverse ts
```
```   430     | (Free (n, _), ts) => add (free n) (length ts) #> fold traverse ts
```
```   431     | (Abs (_, _, u), ts) => fold traverse (u :: ts)
```
```   432     | (_, ts) => fold traverse ts)
```
```   433   fun prune tab = Symtab.fold (fn (n, (true, i)) =>
```
```   434     Symtab.update (n, i) | _ => I) tab Symtab.empty
```
```   435
```
```   436   fun binop_conv cv1 cv2 = Conv.combination_conv (Conv.arg_conv cv1) cv2
```
```   437   fun nary_conv conv1 conv2 ct =
```
```   438     (Conv.combination_conv (nary_conv conv1 conv2) conv2 else_conv conv1) ct
```
```   439   fun abs_conv conv tb = Conv.abs_conv (fn (cv, cx) =>
```
```   440     let val n = fst (Term.dest_Free (Thm.term_of cv))
```
```   441     in conv (Symtab.update (free n, 0) tb) cx end)
```
```   442   val fun_app_rule = @{lemma "f x == fun_app f x" by (simp add: fun_app_def)}
```
```   443 in
```
```   444 fun explicit_application ctxt irules =
```
```   445   let
```
```   446     fun sub_conv tb ctxt ct =
```
```   447       (case Term.strip_comb (Thm.term_of ct) of
```
```   448         (Const (n, _), ts) => app_conv tb (const n) (length ts) ctxt
```
```   449       | (Free (n, _), ts) => app_conv tb (free n) (length ts) ctxt
```
```   450       | (Abs _, _) => nary_conv (abs_conv sub_conv tb ctxt) (sub_conv tb ctxt)
```
```   451       | (_, _) => nary_conv Conv.all_conv (sub_conv tb ctxt)) ct
```
```   452     and app_conv tb n i ctxt =
```
```   453       (case Symtab.lookup tb n of
```
```   454         NONE => nary_conv Conv.all_conv (sub_conv tb ctxt)
```
```   455       | SOME j => fun_app_conv tb ctxt (i - j))
```
```   456     and fun_app_conv tb ctxt i ct = (
```
```   457       if i = 0 then nary_conv Conv.all_conv (sub_conv tb ctxt)
```
```   458       else
```
```   459         Conv.rewr_conv fun_app_rule then_conv
```
```   460         binop_conv (fun_app_conv tb ctxt (i-1)) (sub_conv tb ctxt)) ct
```
```   461
```
```   462     fun needs_exp_app tab = Term.exists_subterm (fn
```
```   463         Bound _ \$ _ => true
```
```   464       | Const (n, _) => Symtab.defined tab (const n)
```
```   465       | Free (n, _) => Symtab.defined tab (free n)
```
```   466       | _ => false)
```
```   467
```
```   468     fun rewrite tab ctxt thm =
```
```   469       if not (needs_exp_app tab (Thm.prop_of thm)) then thm
```
```   470       else Conv.fconv_rule (sub_conv tab ctxt) thm
```
```   471
```
```   472     val tab = prune (fold (traverse o Thm.prop_of o snd) irules Symtab.empty)
```
```   473   in map (apsnd (rewrite tab ctxt)) irules end
```
```   474 end
```
```   475
```
```   476
```
```   477
```
```   478 (* add missing datatype selectors via hypothetical definitions *)
```
```   479
```
```   480 local
```
```   481   val add = (fn Type (n, _) => Symtab.update (n, ()) | _ => I)
```
```   482
```
```   483   fun collect t =
```
```   484     (case Term.strip_comb t of
```
```   485       (Abs (_, T, t), _) => add T #> collect t
```
```   486     | (Const (_, T), ts) => collects T ts
```
```   487     | (Free (_, T), ts) => collects T ts
```
```   488     | _ => I)
```
```   489   and collects T ts =
```
```   490     let val ((Ts, Us), U) = Term.strip_type T |> apfst (chop (length ts))
```
```   491     in fold add Ts #> add (Us ---> U) #> fold collect ts end
```
```   492
```
```   493   fun add_constructors thy n =
```
```   494     (case Datatype.get_info thy n of
```
```   495       NONE => I
```
```   496     | SOME {descr, ...} => fold (fn (_, (_, _, cs)) => fold (fn (n, ds) =>
```
```   497         fold (insert (op =) o pair n) (1 upto length ds)) cs) descr)
```
```   498
```
```   499   fun add_selector (c as (n, i)) ctxt =
```
```   500     (case Datatype_Selectors.lookup_selector ctxt c of
```
```   501       SOME _ => ctxt
```
```   502     | NONE =>
```
```   503         let
```
```   504           val T = Sign.the_const_type (ProofContext.theory_of ctxt) n
```
```   505           val U = Term.body_type T --> nth (Term.binder_types T) (i-1)
```
```   506         in
```
```   507           ctxt
```
```   508           |> yield_singleton Variable.variant_fixes Name.uu
```
```   509           |>> pair ((n, T), i) o rpair U
```
```   510           |-> Context.proof_map o Datatype_Selectors.add_selector
```
```   511         end)
```
```   512 in
```
```   513
```
```   514 fun datatype_selectors irules ctxt =
```
```   515   let
```
```   516     val ns = Symtab.keys (fold (collect o Thm.prop_of o snd) irules Symtab.empty)
```
```   517     val cs = fold (add_constructors (ProofContext.theory_of ctxt)) ns []
```
```   518   in (irules, fold add_selector cs ctxt) end
```
```   519     (* FIXME: also generate hypothetical definitions for the selectors *)
```
```   520
```
```   521 end
```
```   522
```
```   523
```
```   524
```
```   525 (* combined normalization *)
```
```   526
```
```   527 type extra_norm = bool -> (int * thm) list -> Proof.context ->
```
```   528   (int * thm) list * Proof.context
```
```   529
```
```   530 fun with_context f irules ctxt = (f ctxt irules, ctxt)
```
```   531
```
```   532 fun normalize extra_norm with_datatypes irules ctxt =
```
```   533   let
```
```   534     fun norm f ctxt' (i, thm) =
```
```   535       if Config.get ctxt' SMT_Config.drop_bad_facts then
```
```   536         (case try (f ctxt') thm of
```
```   537           SOME thm' => SOME (i, thm')
```
```   538         | NONE => (SMT_Config.verbose_msg ctxt' (prefix ("Warning: " ^
```
```   539             "dropping assumption: ") o Display.string_of_thm ctxt') thm; NONE))
```
```   540       else SOME (i, f ctxt' thm)
```
```   541   in
```
```   542     irules
```
```   543     |> map (apsnd instantiate_elim)
```
```   544     |> trivial_distinct ctxt
```
```   545     |> rewrite_bool_cases ctxt
```
```   546     |> normalize_numerals ctxt
```
```   547     |> nat_as_int ctxt
```
```   548     |> rpair ctxt
```
```   549     |-> extra_norm with_datatypes
```
```   550     |-> with_context (map_filter o norm normalize_rule)
```
```   551     |-> SMT_Monomorph.monomorph
```
```   552     |-> lift_lambdas
```
```   553     |-> with_context explicit_application
```
```   554     |-> (if with_datatypes then datatype_selectors else pair)
```
```   555   end
```
```   556
```
```   557 end
```