src/HOL/Tools/SMT/smt_normalize.ML
 author boehmes Wed May 12 23:54:04 2010 +0200 (2010-05-12) changeset 36899 bcd6fce5bf06 parent 36898 8e55aa1306c5 child 36936 c52d1c130898 permissions -rw-r--r--
1 (*  Title:      HOL/Tools/SMT/smt_normalize.ML
2     Author:     Sascha Boehme, TU Muenchen
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   * lift lambda terms,
13   * make applications explicit for functions with varying number of arguments.
14 *)
16 signature SMT_NORMALIZE =
17 sig
18   type extra_norm = thm list -> Proof.context -> thm list * Proof.context
19   val normalize: extra_norm -> thm list -> Proof.context ->
20     thm list * Proof.context
21   val atomize_conv: Proof.context -> conv
22   val eta_expand_conv: (Proof.context -> conv) -> Proof.context -> conv
23 end
25 structure SMT_Normalize: SMT_NORMALIZE =
26 struct
28 infix 2 ??
29 fun (test ?? f) x = if test x then f x else x
31 fun if_conv c cv1 cv2 ct = (if c (Thm.term_of ct) then cv1 else cv2) ct
32 fun if_true_conv c cv = if_conv c cv Conv.all_conv
36 (* simplification of trivial distincts (distinct should have at least
37    three elements in the argument list) *)
39 local
40   fun is_trivial_distinct (Const (@{const_name distinct}, _) \$ t) =
41         length (HOLogic.dest_list t) <= 2
42     | is_trivial_distinct _ = false
44   val thms = @{lemma
45     "distinct [] == True"
46     "distinct [x] == True"
47     "distinct [x, y] == (x ~= y)"
48     by simp_all}
49   fun distinct_conv _ =
50     if_true_conv is_trivial_distinct (More_Conv.rewrs_conv thms)
51 in
52 fun trivial_distinct ctxt =
53   map ((Term.exists_subterm is_trivial_distinct o Thm.prop_of) ??
54     Conv.fconv_rule (More_Conv.top_conv distinct_conv ctxt))
55 end
59 (* rewrite bool case expressions as if expressions *)
61 local
62   val is_bool_case = (fn
63       Const (@{const_name "bool.bool_case"}, _) \$ _ \$ _ \$ _ => true
64     | _ => false)
66   val thms = @{lemma
67     "(case P of True => x | False => y) == (if P then x else y)"
68     "(case P of False => y | True => x) == (if P then x else y)"
69     by (rule eq_reflection, simp)+}
70   val unfold_conv = if_true_conv is_bool_case (More_Conv.rewrs_conv thms)
71 in
72 fun rewrite_bool_cases ctxt =
73   map ((Term.exists_subterm is_bool_case o Thm.prop_of) ??
74     Conv.fconv_rule (More_Conv.top_conv (K unfold_conv) ctxt))
75 end
79 (* normalization of numerals: rewriting of negative integer numerals into
80    positive numerals, Numeral0 into 0, Numeral1 into 1 *)
82 local
83   fun is_number_sort ctxt T =
84     Sign.of_sort (ProofContext.theory_of ctxt) (T, @{sort number_ring})
86   fun is_strange_number ctxt (t as Const (@{const_name number_of}, _) \$ _) =
87         (case try HOLogic.dest_number t of
88           SOME (T, i) => is_number_sort ctxt T andalso i < 2
89         | NONE => false)
90     | is_strange_number _ _ = false
92   val pos_numeral_ss = HOL_ss
93     addsimps [@{thm Int.number_of_minus}, @{thm Int.number_of_Min}]
94     addsimps [@{thm Int.number_of_Pls}, @{thm Int.numeral_1_eq_1}]
98       "Int.Min = - Int.Bit1 Int.Pls"
99       "Int.Bit0 (- Int.Pls) = - Int.Pls"
100       "Int.Bit0 (- k) = - Int.Bit0 k"
101       "Int.Bit1 (- k) = - Int.Bit1 (Int.pred k)"
102       by simp_all (simp add: pred_def)}
104   fun pos_conv ctxt = if_conv (is_strange_number ctxt)
105     (Simplifier.rewrite (Simplifier.context ctxt pos_numeral_ss))
106     Conv.no_conv
107 in
108 fun normalize_numerals ctxt =
109   map ((Term.exists_subterm (is_strange_number ctxt) o Thm.prop_of) ??
110     Conv.fconv_rule (More_Conv.top_sweep_conv pos_conv ctxt))
111 end
115 (* embedding of standard natural number operations into integer operations *)
117 local
118   val nat_embedding = @{lemma
119     "nat (int n) = n"
120     "i >= 0 --> int (nat i) = i"
121     "i < 0 --> int (nat i) = 0"
122     by simp_all}
124   val nat_rewriting = @{lemma
125     "0 = nat 0"
126     "1 = nat 1"
127     "number_of i = nat (number_of i)"
128     "int (nat 0) = 0"
129     "int (nat 1) = 1"
130     "a < b = (int a < int b)"
131     "a <= b = (int a <= int b)"
132     "Suc a = nat (int a + 1)"
133     "a + b = nat (int a + int b)"
134     "a - b = nat (int a - int b)"
135     "a * b = nat (int a * int b)"
136     "a div b = nat (int a div int b)"
137     "a mod b = nat (int a mod int b)"
138     "min a b = nat (min (int a) (int b))"
139     "max a b = nat (max (int a) (int b))"
140     "int (nat (int a + int b)) = int a + int b"
141     "int (nat (int a * int b)) = int a * int b"
142     "int (nat (int a div int b)) = int a div int b"
143     "int (nat (int a mod int b)) = int a mod int b"
144     "int (nat (min (int a) (int b))) = min (int a) (int b)"
145     "int (nat (max (int a) (int b))) = max (int a) (int b)"
146     by (simp_all add: nat_mult_distrib nat_div_distrib nat_mod_distrib
147       int_mult[symmetric] zdiv_int[symmetric] zmod_int[symmetric])}
149   fun on_positive num f x =
150     (case try HOLogic.dest_number (Thm.term_of num) of
151       SOME (_, i) => if i >= 0 then SOME (f x) else NONE
152     | NONE => NONE)
154   val cancel_int_nat_ss = HOL_ss
159   fun cancel_int_nat_simproc _ ss ct =
160     let
161       val num = Thm.dest_arg (Thm.dest_arg ct)
162       val goal = Thm.mk_binop @{cterm "op == :: int => _"} ct num
163       val simpset = Simplifier.inherit_context ss cancel_int_nat_ss
164       fun tac _ = Simplifier.simp_tac simpset 1
165     in on_positive num (Goal.prove_internal [] goal) tac end
167   val nat_ss = HOL_ss
170       name = "cancel_int_nat_num", lhss = [@{cpat "int (nat _)"}],
171       proc = cancel_int_nat_simproc, identifier = [] }]
173   fun conv ctxt = Simplifier.rewrite (Simplifier.context ctxt nat_ss)
175   val uses_nat_type = Term.exists_type (Term.exists_subtype (equal @{typ nat}))
176   val uses_nat_int =
177     Term.exists_subterm (member (op aconv) [@{term int}, @{term nat}])
178 in
179 fun nat_as_int ctxt =
180   map ((uses_nat_type o Thm.prop_of) ?? Conv.fconv_rule (conv ctxt)) #>
181   exists (uses_nat_int o Thm.prop_of) ?? append nat_embedding
182 end
186 (* further normalizations: beta/eta, universal closure, atomize *)
188 val eta_expand_eq = @{lemma "f == (%x. f x)" by (rule reflexive)}
190 fun eta_expand_conv cv ctxt =
191   Conv.rewr_conv eta_expand_eq then_conv Conv.abs_conv (cv o snd) ctxt
193 local
194   val eta_conv = eta_expand_conv
196   fun keep_conv ctxt = More_Conv.binder_conv norm_conv ctxt
197   and eta_binder_conv ctxt = Conv.arg_conv (eta_conv norm_conv ctxt)
198   and keep_let_conv ctxt = Conv.combination_conv
199     (Conv.arg_conv (norm_conv ctxt)) (Conv.abs_conv (norm_conv o snd) ctxt)
200   and unfold_let_conv ctxt = Conv.combination_conv
201     (Conv.arg_conv (norm_conv ctxt)) (eta_conv norm_conv ctxt)
202   and unfold_conv thm ctxt = Conv.rewr_conv thm then_conv keep_conv ctxt
203   and unfold_ex1_conv ctxt = unfold_conv @{thm Ex1_def} ctxt
204   and unfold_ball_conv ctxt = unfold_conv @{thm Ball_def} ctxt
205   and unfold_bex_conv ctxt = unfold_conv @{thm Bex_def} ctxt
206   and norm_conv ctxt ct =
207     (case Thm.term_of ct of
208       Const (@{const_name All}, _) \$ Abs _ => keep_conv
209     | Const (@{const_name All}, _) \$ _ => eta_binder_conv
210     | Const (@{const_name All}, _) => eta_conv eta_binder_conv
211     | Const (@{const_name Ex}, _) \$ Abs _ => keep_conv
212     | Const (@{const_name Ex}, _) \$ _ => eta_binder_conv
213     | Const (@{const_name Ex}, _) => eta_conv eta_binder_conv
214     | Const (@{const_name Let}, _) \$ _ \$ Abs _ => keep_let_conv
215     | Const (@{const_name Let}, _) \$ _ \$ _ => unfold_let_conv
216     | Const (@{const_name Let}, _) \$ _ => eta_conv unfold_let_conv
217     | Const (@{const_name Let}, _) => eta_conv (eta_conv unfold_let_conv)
218     | Const (@{const_name Ex1}, _) \$ _ => unfold_ex1_conv
219     | Const (@{const_name Ex1}, _) => eta_conv unfold_ex1_conv
220     | Const (@{const_name Ball}, _) \$ _ \$ _ => unfold_ball_conv
221     | Const (@{const_name Ball}, _) \$ _ => eta_conv unfold_ball_conv
222     | Const (@{const_name Ball}, _) => eta_conv (eta_conv unfold_ball_conv)
223     | Const (@{const_name Bex}, _) \$ _ \$ _ => unfold_bex_conv
224     | Const (@{const_name Bex}, _) \$ _ => eta_conv unfold_bex_conv
225     | Const (@{const_name Bex}, _) => eta_conv (eta_conv unfold_bex_conv)
226     | Abs _ => Conv.abs_conv (norm_conv o snd)
227     | _ \$ _ => Conv.comb_conv o norm_conv
228     | _ => K Conv.all_conv) ctxt ct
230   fun is_normed t =
231     (case t of
232       Const (@{const_name All}, _) \$ Abs (_, _, u) => is_normed u
233     | Const (@{const_name All}, _) \$ _ => false
234     | Const (@{const_name All}, _) => false
235     | Const (@{const_name Ex}, _) \$ Abs (_, _, u) => is_normed u
236     | Const (@{const_name Ex}, _) \$ _ => false
237     | Const (@{const_name Ex}, _) => false
238     | Const (@{const_name Let}, _) \$ u1 \$ Abs (_, _, u2) =>
239         is_normed u1 andalso is_normed u2
240     | Const (@{const_name Let}, _) \$ _ \$ _ => false
241     | Const (@{const_name Let}, _) \$ _ => false
242     | Const (@{const_name Let}, _) => false
243     | Const (@{const_name Ex1}, _) => false
244     | Const (@{const_name Ball}, _) => false
245     | Const (@{const_name Bex}, _) => false
246     | Abs (_, _, u) => is_normed u
247     | u1 \$ u2 => is_normed u1 andalso is_normed u2
248     | _ => true)
249 in
250 fun norm_binder_conv ctxt = if_conv is_normed Conv.all_conv (norm_conv ctxt)
251 end
253 fun norm_def ctxt thm =
254   (case Thm.prop_of thm of
255     @{term Trueprop} \$ (Const (@{const_name "op ="}, _) \$ _ \$ Abs _) =>
256       norm_def ctxt (thm RS @{thm fun_cong})
257   | Const (@{const_name "=="}, _) \$ _ \$ Abs _ =>
258       norm_def ctxt (thm RS @{thm meta_eq_to_obj_eq})
259   | _ => thm)
261 fun atomize_conv ctxt ct =
262   (case Thm.term_of ct of
263     @{term "op ==>"} \$ _ \$ _ =>
264       Conv.binop_conv (atomize_conv ctxt) then_conv
265       Conv.rewr_conv @{thm atomize_imp}
266   | Const (@{const_name "=="}, _) \$ _ \$ _ =>
267       Conv.binop_conv (atomize_conv ctxt) then_conv
268       Conv.rewr_conv @{thm atomize_eq}
269   | Const (@{const_name all}, _) \$ Abs _ =>
270       More_Conv.binder_conv atomize_conv ctxt then_conv
271       Conv.rewr_conv @{thm atomize_all}
272   | _ => Conv.all_conv) ct
274 fun normalize_rule ctxt =
275   Conv.fconv_rule (
276     (* reduce lambda abstractions, except at known binders: *)
277     Thm.beta_conversion true then_conv
278     Thm.eta_conversion then_conv
279     norm_binder_conv ctxt) #>
280   norm_def ctxt #>
281   Drule.forall_intr_vars #>
282   Conv.fconv_rule (atomize_conv ctxt)
286 (* lift lambda terms into additional rules *)
288 local
289   val meta_eq = @{cpat "op =="}
290   val meta_eqT = hd (Thm.dest_ctyp (Thm.ctyp_of_term meta_eq))
291   fun inst_meta cT = Thm.instantiate_cterm ([(meta_eqT, cT)], []) meta_eq
292   fun mk_meta_eq ct cu = Thm.mk_binop (inst_meta (Thm.ctyp_of_term ct)) ct cu
294   fun cert ctxt = Thm.cterm_of (ProofContext.theory_of ctxt)
296   fun used_vars cvs ct =
297     let
298       val lookup = AList.lookup (op aconv) (map (` Thm.term_of) cvs)
299       val add = (fn SOME ct => insert (op aconvc) ct | _ => I)
300     in Term.fold_aterms (add o lookup) (Thm.term_of ct) [] end
302   fun apply cv thm =
303     let val thm' = Thm.combination thm (Thm.reflexive cv)
304     in Thm.transitive thm' (Thm.beta_conversion false (Thm.rhs_of thm')) end
305   fun apply_def cvs eq = Thm.symmetric (fold apply cvs eq)
307   fun replace_lambda cvs ct (cx as (ctxt, defs)) =
308     let
309       val cvs' = used_vars cvs ct
310       val ct' = fold_rev Thm.cabs cvs' ct
311     in
312       (case Termtab.lookup defs (Thm.term_of ct') of
313         SOME eq => (apply_def cvs' eq, cx)
314       | NONE =>
315           let
316             val {T, ...} = Thm.rep_cterm ct' and n = Name.uu
317             val (n', ctxt') = yield_singleton Variable.variant_fixes n ctxt
318             val cu = mk_meta_eq (cert ctxt (Free (n', T))) ct'
319             val (eq, ctxt'') = yield_singleton Assumption.add_assumes cu ctxt'
320             val defs' = Termtab.update (Thm.term_of ct', eq) defs
321           in (apply_def cvs' eq, (ctxt'', defs')) end)
322     end
324   fun none ct cx = (Thm.reflexive ct, cx)
325   fun in_comb f g ct cx =
326     let val (cu1, cu2) = Thm.dest_comb ct
327     in cx |> f cu1 ||>> g cu2 |>> uncurry Thm.combination end
328   fun in_arg f = in_comb none f
329   fun in_abs f cvs ct (ctxt, defs) =
330     let
331       val (n, ctxt') = yield_singleton Variable.variant_fixes Name.uu ctxt
332       val (cv, cu) = Thm.dest_abs (SOME n) ct
333     in  (ctxt', defs) |> f (cv :: cvs) cu |>> Thm.abstract_rule n cv end
335   fun traverse cvs ct =
336     (case Thm.term_of ct of
337       Const (@{const_name All}, _) \$ Abs _ => in_arg (in_abs traverse cvs)
338     | Const (@{const_name Ex}, _) \$ Abs _ => in_arg (in_abs traverse cvs)
339     | Const (@{const_name Let}, _) \$ _ \$ Abs _ =>
340         in_comb (in_arg (traverse cvs)) (in_abs traverse cvs)
341     | Abs _ => at_lambda cvs
342     | _ \$ _ => in_comb (traverse cvs) (traverse cvs)
343     | _ => none) ct
345   and at_lambda cvs ct =
346     in_abs traverse cvs ct #-> (fn thm =>
347     replace_lambda cvs (Thm.rhs_of thm) #>> Thm.transitive thm)
349   fun has_free_lambdas t =
350     (case t of
351       Const (@{const_name All}, _) \$ Abs (_, _, u) => has_free_lambdas u
352     | Const (@{const_name Ex}, _) \$ Abs (_, _, u) => has_free_lambdas u
353     | Const (@{const_name Let}, _) \$ u1 \$ Abs (_, _, u2) =>
354         has_free_lambdas u1 orelse has_free_lambdas u2
355     | Abs _ => true
356     | u1 \$ u2 => has_free_lambdas u1 orelse has_free_lambdas u2
357     | _ => false)
359   fun lift_lm f thm cx =
360     if not (has_free_lambdas (Thm.prop_of thm)) then (thm, cx)
361     else cx |> f (Thm.cprop_of thm) |>> (fn thm' => Thm.equal_elim thm' thm)
362 in
363 fun lift_lambdas thms ctxt =
364   let
365     val cx = (ctxt, Termtab.empty)
366     val (thms', (ctxt', defs)) = fold_map (lift_lm (traverse [])) thms cx
367     val eqs = Termtab.fold (cons o normalize_rule ctxt' o snd) defs []
368   in (eqs @ thms', ctxt') end
369 end
373 (* make application explicit for functions with varying number of arguments *)
375 local
376   val const = prefix "c" and free = prefix "f"
377   fun min i (e as (_, j)) = if i <> j then (true, Int.min (i, j)) else e
378   fun add t i = Symtab.map_default (t, (false, i)) (min i)
379   fun traverse t =
380     (case Term.strip_comb t of
381       (Const (n, _), ts) => add (const n) (length ts) #> fold traverse ts
382     | (Free (n, _), ts) => add (free n) (length ts) #> fold traverse ts
383     | (Abs (_, _, u), ts) => fold traverse (u :: ts)
384     | (_, ts) => fold traverse ts)
385   val prune = (fn (n, (true, i)) => Symtab.update (n, i) | _ => I)
386   fun prune_tab tab = Symtab.fold prune tab Symtab.empty
388   fun binop_conv cv1 cv2 = Conv.combination_conv (Conv.arg_conv cv1) cv2
389   fun nary_conv conv1 conv2 ct =
390     (Conv.combination_conv (nary_conv conv1 conv2) conv2 else_conv conv1) ct
391   fun abs_conv conv tb = Conv.abs_conv (fn (cv, cx) =>
392     let val n = fst (Term.dest_Free (Thm.term_of cv))
393     in conv (Symtab.update (free n, 0) tb) cx end)
394   val apply_rule = @{lemma "f x == apply f x" by (simp add: apply_def)}
395 in
396 fun explicit_application ctxt thms =
397   let
398     fun sub_conv tb ctxt ct =
399       (case Term.strip_comb (Thm.term_of ct) of
400         (Const (n, _), ts) => app_conv tb (const n) (length ts) ctxt
401       | (Free (n, _), ts) => app_conv tb (free n) (length ts) ctxt
402       | (Abs _, _) => nary_conv (abs_conv sub_conv tb ctxt) (sub_conv tb ctxt)
403       | (_, _) => nary_conv Conv.all_conv (sub_conv tb ctxt)) ct
404     and app_conv tb n i ctxt =
405       (case Symtab.lookup tb n of
406         NONE => nary_conv Conv.all_conv (sub_conv tb ctxt)
407       | SOME j => apply_conv tb ctxt (i - j))
408     and apply_conv tb ctxt i ct = (
409       if i = 0 then nary_conv Conv.all_conv (sub_conv tb ctxt)
410       else
411         Conv.rewr_conv apply_rule then_conv
412         binop_conv (apply_conv tb ctxt (i-1)) (sub_conv tb ctxt)) ct
414     fun needs_exp_app tab = Term.exists_subterm (fn
415         Bound _ \$ _ => true
416       | Const (n, _) => Symtab.defined tab (const n)
417       | Free (n, _) => Symtab.defined tab (free n)
418       | _ => false)
420     fun rewrite tab ctxt thm =
421       if not (needs_exp_app tab (Thm.prop_of thm)) then thm
422       else Conv.fconv_rule (sub_conv tab ctxt) thm
424     val tab = prune_tab (fold (traverse o Thm.prop_of) thms Symtab.empty)
425   in map (rewrite tab ctxt) thms end
426 end
430 (* combined normalization *)
432 type extra_norm = thm list -> Proof.context -> thm list * Proof.context
434 fun with_context f thms ctxt = (f ctxt thms, ctxt)
436 fun normalize extra_norm thms ctxt =
437   thms
438   |> trivial_distinct ctxt
439   |> rewrite_bool_cases ctxt
440   |> normalize_numerals ctxt
441   |> nat_as_int ctxt
442   |> rpair ctxt
443   |-> extra_norm
444   |-> with_context (fn cx => map (normalize_rule cx))
445   |-> SMT_Monomorph.monomorph
446   |-> lift_lambdas
447   |-> with_context explicit_application
449 end