src/HOL/MicroJava/Comp/LemmasComp.thy
 author wenzelm Mon Dec 29 21:02:49 2014 +0100 (2014-12-29) changeset 59199 cb8e5f7a5e4a parent 58184 db1381d811ab child 60304 3f429b7d8eb5 permissions -rw-r--r--
tuned;
1 (*  Title:      HOL/MicroJava/Comp/LemmasComp.thy
2     Author:     Martin Strecker
3 *)
5 (* Lemmas for compiler correctness proof *)
7 theory LemmasComp
8 imports TranslComp
9 begin
12 declare split_paired_All [simp del]
13 declare split_paired_Ex [simp del]
16 (**********************************************************************)
17 (* misc lemmas *)
21 lemma split_pairs: "(\<lambda>(a,b). (F a b)) (ab) = F (fst ab) (snd ab)"
22 proof -
23   have "(\<lambda>(a,b). (F a b)) (fst ab,snd ab) = F (fst ab) (snd ab)"
24     by (simp add: split_def)
25   then show ?thesis by simp
26 qed
29 lemma c_hupd_conv:
30   "c_hupd h' (xo, (h,l)) = (xo, (if xo = None then h' else h),l)"
31 by (simp add: c_hupd_def)
33 lemma gl_c_hupd [simp]: "(gl (c_hupd h xs)) = (gl xs)"
34 by (simp add: gl_def c_hupd_def split_pairs)
36 lemma c_hupd_xcpt_invariant [simp]: "gx (c_hupd h' (xo, st)) = xo"
37 by (cases st) (simp only: c_hupd_conv gx_conv)
39 (* not added to simpset because of interference with c_hupd_conv *)
40 lemma c_hupd_hp_invariant: "gh (c_hupd hp (None, st)) = hp"
41 by (cases st) (simp add: c_hupd_conv gh_def)
44 lemma unique_map_fst [rule_format]: "(\<forall> x \<in> set xs. (fst x = fst (f x) )) \<longrightarrow>
45   unique (map f xs) = unique xs"
46 proof (induct xs)
47   case Nil show ?case by simp
48 next
49   case (Cons a list)
50   show ?case
51   proof
52     assume fst_eq: "\<forall>x\<in>set (a # list). fst x = fst (f x)"
54     have fst_set: "(fst a \<in> fst ` set list) = (fst (f a) \<in> fst ` f ` set list)"
55     proof
56       assume as: "fst a \<in> fst ` set list"
57       then obtain x where fst_a_x: "x\<in>set list \<and> fst a = fst x"
58         by (auto simp add: image_iff)
59       then have "fst (f a) = fst (f x)" by (simp add: fst_eq)
60       with as show "(fst (f a) \<in> fst ` f ` set list)" by (simp add: fst_a_x)
61     next
62       assume as: "fst (f a) \<in> fst ` f ` set list"
63       then obtain x where fst_a_x: "x\<in>set list \<and> fst (f a) = fst (f x)"
64         by (auto simp add: image_iff)
65       then have "fst a = fst x" by (simp add: fst_eq)
66       with as show "fst a \<in> fst ` set list" by (simp add: fst_a_x)
67     qed
69     with fst_eq Cons
70     show "unique (map f (a # list)) = unique (a # list)"
71       by (simp add: unique_def fst_set image_comp)
72   qed
73 qed
75 lemma comp_unique: "unique (comp G) = unique G"
76 apply (simp add: comp_def)
77 apply (rule unique_map_fst)
78 apply (simp add: compClass_def split_beta)
79 done
82 (**********************************************************************)
83 (* invariance of properties under compilation *)
85 lemma comp_class_imp:
86   "(class G C = Some(D, fs, ms)) \<Longrightarrow>
87   (class (comp G) C = Some(D, fs, map (compMethod G C) ms))"
88 apply (simp add: class_def comp_def compClass_def)
89 apply (rule HOL.trans)
90 apply (rule map_of_map2)
91 apply auto
92 done
94 lemma comp_class_None:
95 "(class G C = None) = (class (comp G) C = None)"
96 apply (simp add: class_def comp_def compClass_def)
97 apply (simp add: map_of_in_set)
98 apply (simp add: image_comp [symmetric] o_def split_beta)
99 done
101 lemma comp_is_class: "is_class (comp G) C = is_class G C"
102 by (cases "class G C") (auto simp: is_class_def comp_class_None dest: comp_class_imp)
105 lemma comp_is_type: "is_type (comp G) T = is_type G T"
106   apply (cases T)
107   apply simp
108   apply (induct G)
109   apply simp
110   apply (simp only: comp_is_class)
111   apply (simp add: comp_is_class)
112   apply (simp only: comp_is_class)
113   done
115 lemma comp_classname: "is_class G C
116   \<Longrightarrow> fst (the (class G C)) = fst (the (class (comp G) C))"
117 by (cases "class G C") (auto simp: is_class_def dest: comp_class_imp)
119 lemma comp_subcls1: "subcls1 (comp G) = subcls1 G"
120 by (auto simp add: subcls1_def2 comp_classname comp_is_class)
122 lemma comp_widen: "widen (comp G) = widen G"
123   apply (simp add: fun_eq_iff)
124   apply (intro allI iffI)
125   apply (erule widen.cases)
126   apply (simp_all add: comp_subcls1 widen.null)
127   apply (erule widen.cases)
128   apply (simp_all add: comp_subcls1 widen.null)
129   done
131 lemma comp_cast: "cast (comp G) = cast G"
132   apply (simp add: fun_eq_iff)
133   apply (intro allI iffI)
134   apply (erule cast.cases)
135   apply (simp_all add: comp_subcls1 cast.widen cast.subcls)
136   apply (rule cast.widen) apply (simp add: comp_widen)
137   apply (erule cast.cases)
138   apply (simp_all add: comp_subcls1 cast.widen cast.subcls)
139   apply (rule cast.widen) apply (simp add: comp_widen)
140   done
142 lemma comp_cast_ok: "cast_ok (comp G) = cast_ok G"
143   by (simp add: fun_eq_iff cast_ok_def comp_widen)
146 lemma compClass_fst [simp]: "(fst (compClass G C)) = (fst C)"
147 by (simp add: compClass_def split_beta)
149 lemma compClass_fst_snd [simp]: "(fst (snd (compClass G C))) = (fst (snd C))"
150 by (simp add: compClass_def split_beta)
152 lemma compClass_fst_snd_snd [simp]: "(fst (snd (snd (compClass G C)))) = (fst (snd (snd C)))"
153 by (simp add: compClass_def split_beta)
155 lemma comp_wf_fdecl [simp]: "wf_fdecl (comp G) fd = wf_fdecl G fd"
156 by (cases fd) (simp add: wf_fdecl_def comp_is_type)
159 lemma compClass_forall [simp]: "
160   (\<forall>x\<in>set (snd (snd (snd (compClass G C)))). P (fst x) (fst (snd x))) =
161   (\<forall>x\<in>set (snd (snd (snd C))). P (fst x) (fst (snd x)))"
162 by (simp add: compClass_def compMethod_def split_beta)
164 lemma comp_wf_mhead: "wf_mhead (comp G) S rT =  wf_mhead G S rT"
165 by (simp add: wf_mhead_def split_beta comp_is_type)
167 lemma comp_ws_cdecl: "
168   ws_cdecl (TranslComp.comp G) (compClass G C) = ws_cdecl G C"
169 apply (simp add: ws_cdecl_def split_beta comp_is_class comp_subcls1)
171 apply (simp add: compClass_def compMethod_def split_beta unique_map_fst)
172 done
175 lemma comp_wf_syscls: "wf_syscls (comp G) = wf_syscls G"
176 apply (simp add: wf_syscls_def comp_def compClass_def split_beta)
177 apply (simp add: image_comp)
178 apply (subgoal_tac "(Fun.comp fst (\<lambda>(C, cno::cname, fdls::fdecl list, jmdls).
179   (C, cno, fdls, map (compMethod G C) jmdls))) = fst")
180 apply simp
181 apply (simp add: fun_eq_iff split_beta)
182 done
185 lemma comp_ws_prog: "ws_prog (comp G) = ws_prog G"
186 apply (rule sym)
187 apply (simp add: ws_prog_def comp_ws_cdecl comp_unique)
188 apply (simp add: comp_wf_syscls)
189 apply (auto simp add: comp_ws_cdecl [symmetric] TranslComp.comp_def)
190 done
193 lemma comp_class_rec: " wf ((subcls1 G)^-1) \<Longrightarrow>
194 class_rec (comp G) C t f =
195   class_rec G C t (\<lambda> C' fs' ms' r'. f C' fs' (map (compMethod G C') ms') r')"
196 apply (rule_tac a = C in  wf_induct) apply assumption
197 apply (subgoal_tac "wf ((subcls1 (comp G))^-1)")
198 apply (subgoal_tac "(class G x = None) \<or> (\<exists> D fs ms. (class G x = Some (D, fs, ms)))")
199 apply (erule disjE)
201   (* case class G x = None *)
202 apply (simp (no_asm_simp) add: class_rec_def comp_subcls1
203   wfrec [where R="(subcls1 G)^-1", simplified])
204 apply (simp add: comp_class_None)
206   (* case \<exists> D fs ms. (class G x = Some (D, fs, ms)) *)
207 apply (erule exE)+
208 apply (frule comp_class_imp)
209 apply (frule_tac G="comp G" and C=x and t=t and f=f in class_rec_lemma)
210   apply assumption
211 apply (frule_tac G=G and C=x and t=t
212    and f="(\<lambda>C' fs' ms'. f C' fs' (map (compMethod G C') ms'))" in class_rec_lemma)
213   apply assumption
214 apply (simp only:)
216 apply (case_tac "x = Object")
217   apply simp
218   apply (frule subcls1I, assumption)
219     apply (drule_tac x=D in spec, drule mp, simp)
220     apply simp
222   (* subgoals *)
223 apply (case_tac "class G x")
224 apply auto
225 apply (simp add: comp_subcls1)
226 done
228 lemma comp_fields: "wf ((subcls1 G)^-1) \<Longrightarrow>
229   fields (comp G,C) = fields (G,C)"
230 by (simp add: fields_def comp_class_rec)
232 lemma comp_field: "wf ((subcls1 G)^-1) \<Longrightarrow>
233   field (comp G,C) = field (G,C)"
234 by (simp add: TypeRel.field_def comp_fields)
238 lemma class_rec_relation [rule_format (no_asm)]: "\<lbrakk>  ws_prog G;
239   \<forall> fs ms. R (f1 Object fs ms t1) (f2 Object fs ms t2);
240    \<forall> C fs ms r1 r2. (R r1 r2) \<longrightarrow> (R (f1 C fs ms r1) (f2 C fs ms r2)) \<rbrakk>
241   \<Longrightarrow> ((class G C) \<noteq> None) \<longrightarrow>
242   R (class_rec G C t1 f1) (class_rec G C t2 f2)"
243 apply (frule wf_subcls1) (* establish wf ((subcls1 G)^-1) *)
244 apply (rule_tac a = C in  wf_induct) apply assumption
245 apply (intro strip)
246 apply (subgoal_tac "(\<exists>D rT mb. class G x = Some (D, rT, mb))")
247   apply (erule exE)+
248 apply (frule_tac C=x and t=t1 and f=f1 in class_rec_lemma)
249   apply assumption
250 apply (frule_tac C=x and t=t2 and f=f2 in class_rec_lemma)
251   apply assumption
252 apply (simp only:)
254 apply (case_tac "x = Object")
255   apply simp
256   apply (frule subcls1I, assumption)
257     apply (drule_tac x=D in spec, drule mp, simp)
258     apply simp
259     apply (subgoal_tac "(\<exists>D' rT' mb'. class G D = Some (D', rT', mb'))")
260     apply blast
262   (* subgoals *)
264 apply (frule class_wf_struct) apply assumption
265 apply (simp add: ws_cdecl_def is_class_def)
267 apply (simp add: subcls1_def2 is_class_def)
268 apply auto
269 done
272 abbreviation (input)
273   "mtd_mb == snd o snd"
275 lemma map_of_map:
276   "map_of (map (\<lambda>(k, v). (k, f v)) xs) k = map_option f (map_of xs k)"
277   by (simp add: map_of_map)
279 lemma map_of_map_fst: "\<lbrakk> inj f;
280   \<forall>x\<in>set xs. fst (f x) = fst x; \<forall>x\<in>set xs. fst (g x) = fst x \<rbrakk>
281   \<Longrightarrow>  map_of (map g xs) k
282   = map_option (\<lambda> e. (snd (g ((inv f) (k, e))))) (map_of (map f xs) k)"
283 apply (induct xs)
284 apply simp
285 apply simp
286 apply (case_tac "k = fst a")
287 apply simp
288 apply (subgoal_tac "(inv f (fst a, snd (f a))) = a", simp)
289 apply (subgoal_tac "(fst a, snd (f a)) = f a", simp)
290 apply (erule conjE)+
291 apply (drule_tac s ="fst (f a)" and t="fst a" in sym)
292 apply simp
293 apply (simp add: surjective_pairing)
294 done
296 lemma comp_method [rule_format (no_asm)]: "\<lbrakk> ws_prog G; is_class G C\<rbrakk> \<Longrightarrow>
297   ((method (comp G, C) S) =
298   map_option (\<lambda> (D,rT,b).  (D, rT, mtd_mb (compMethod G D (S, rT, b))))
299              (method (G, C) S))"
300 apply (simp add: method_def)
301 apply (frule wf_subcls1)
302 apply (simp add: comp_class_rec)
303 apply (simp add: split_iter split_compose map_map [symmetric] del: map_map)
304 apply (rule_tac R="%x y. ((x S) = (map_option (\<lambda>(D, rT, b).
305   (D, rT, snd (snd (compMethod G D (S, rT, b))))) (y S)))"
306   in class_rec_relation)
307 apply assumption
309 apply (intro strip)
310 apply simp
312 apply (rule trans)
314 apply (rule_tac f="(\<lambda>(s, m). (s, Object, m))" and
315   g="(Fun.comp (\<lambda>(s, m). (s, Object, m)) (compMethod G Object))"
316   in map_of_map_fst)
317 defer                           (* inj \<dots> *)
318 apply (simp add: inj_on_def split_beta)
319 apply (simp add: split_beta compMethod_def)
320 apply (simp add: map_of_map [symmetric])
321 apply (simp add: split_beta)
322 apply (simp add: Fun.comp_def split_beta)
323 apply (subgoal_tac "(\<lambda>x\<Colon>(vname list \<times> fdecl list \<times> stmt \<times> expr) mdecl.
324                     (fst x, Object,
325                      snd (compMethod G Object
326                            (inv (\<lambda>(s\<Colon>sig, m\<Colon>ty \<times> vname list \<times> fdecl list \<times> stmt \<times> expr).
327                                     (s, Object, m))
328                              (S, Object, snd x)))))
329   = (\<lambda>x. (fst x, Object, fst (snd x),
330                         snd (snd (compMethod G Object (S, snd x)))))")
331 apply (simp only:)
332 apply (simp add: fun_eq_iff)
333 apply (intro strip)
334 apply (subgoal_tac "(inv (\<lambda>(s, m). (s, Object, m)) (S, Object, snd x)) = (S, snd x)")
335 apply (simp only:)
336 apply (simp add: compMethod_def split_beta)
337 apply (rule inv_f_eq)
338 defer
339 defer
341 apply (intro strip)
344 apply (subgoal_tac "inj (\<lambda>(s, m). (s, Ca, m))")
345 apply (drule_tac g="(Fun.comp (\<lambda>(s, m). (s, Ca, m)) (compMethod G Ca))" and xs=ms
346   and k=S in map_of_map_fst)
347 apply (simp add: split_beta)
348 apply (simp add: compMethod_def split_beta)
349 apply (case_tac "(map_of (map (\<lambda>(s, m). (s, Ca, m)) ms) S)")
350 apply simp
351 apply simp apply (simp add: split_beta map_of_map) apply (erule exE) apply (erule conjE)+
352 apply (drule_tac t=a in sym)
353 apply (subgoal_tac "(inv (\<lambda>(s, m). (s, Ca, m)) (S, a)) = (S, snd a)")
354 apply simp
355 apply (subgoal_tac "\<forall>x\<in>set ms. fst ((Fun.comp (\<lambda>(s, m). (s, Ca, m)) (compMethod G Ca)) x) = fst x")
356    prefer 2 apply (simp (no_asm_simp) add: compMethod_def split_beta)
357 apply (simp add: map_of_map2)
358 apply (simp (no_asm_simp) add: compMethod_def split_beta)
360   -- "remaining subgoals"
361 apply (auto intro: inv_f_eq simp add: inj_on_def is_class_def)
362 done
366 lemma comp_wf_mrT: "\<lbrakk> ws_prog G; is_class G D\<rbrakk> \<Longrightarrow>
367   wf_mrT (TranslComp.comp G) (C, D, fs, map (compMethod G a) ms) =
368   wf_mrT G (C, D, fs, ms)"
369 apply (simp add: wf_mrT_def compMethod_def split_beta)
370 apply (simp add: comp_widen)
371 apply (rule iffI)
372 apply (intro strip)
373 apply simp
374 apply (drule bspec) apply assumption
375 apply (drule_tac x=D' in spec) apply (drule_tac x=rT' in spec) apply (drule mp)
376 prefer 2 apply assumption
377 apply (simp add: comp_method [of G D])
378 apply (erule exE)+
379 apply (simp add: split_paired_all)
380 apply (intro strip)
381 apply (simp add: comp_method)
382 apply auto
383 done
386 (**********************************************************************)
387   (* DIVERSE OTHER LEMMAS *)
388 (**********************************************************************)
390 lemma max_spec_preserves_length:
391   "max_spec G C (mn, pTs) = {((md,rT),pTs')}
392   \<Longrightarrow> length pTs = length pTs'"
393 apply (frule max_spec2mheads)
394 apply (erule exE)+
395 apply (simp add: list_all2_iff)
396 done
399 lemma ty_exprs_length [simp]: "(E\<turnstile>es[::]Ts \<longrightarrow> length es = length Ts)"
400 apply (subgoal_tac "(E\<turnstile>e::T \<longrightarrow> True) \<and> (E\<turnstile>es[::]Ts \<longrightarrow> length es = length Ts) \<and> (E\<turnstile>s\<surd> \<longrightarrow> True)")
401 apply blast
402 apply (rule ty_expr_ty_exprs_wt_stmt.induct)
403 apply auto
404 done
407 lemma max_spec_preserves_method_rT [simp]:
408   "max_spec G C (mn, pTs) = {((md,rT),pTs')}
409   \<Longrightarrow> method_rT (the (method (G, C) (mn, pTs'))) = rT"
410 apply (frule max_spec2mheads)
411 apply (erule exE)+
412 apply (simp add: method_rT_def)
413 done
415   (**********************************************************************************)
417 declare compClass_fst [simp del]
418 declare compClass_fst_snd [simp del]
419 declare compClass_fst_snd_snd [simp del]
421 declare split_paired_All [simp add]
422 declare split_paired_Ex [simp add]
424 end