src/HOL/Quotient.thy
 author paulson Tue Apr 25 16:39:54 2017 +0100 (2017-04-25) changeset 65578 e4997c181cce parent 63343 fb5d8a50c641 child 67091 1393c2340eec permissions -rw-r--r--
New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
1 (*  Title:      HOL/Quotient.thy
2     Author:     Cezary Kaliszyk and Christian Urban
3 *)
5 section \<open>Definition of Quotient Types\<close>
7 theory Quotient
8 imports Lifting
9 keywords
10   "print_quotmapsQ3" "print_quotientsQ3" "print_quotconsts" :: diag and
11   "quotient_type" :: thy_goal and "/" and
12   "quotient_definition" :: thy_goal
13 begin
15 text \<open>
16   Basic definition for equivalence relations
17   that are represented by predicates.
18 \<close>
20 text \<open>Composition of Relations\<close>
22 abbreviation
23   rel_conj :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool" (infixr "OOO" 75)
24 where
25   "r1 OOO r2 \<equiv> r1 OO r2 OO r1"
27 lemma eq_comp_r:
28   shows "((op =) OOO R) = R"
29   by (auto simp add: fun_eq_iff)
31 context includes lifting_syntax
32 begin
34 subsection \<open>Quotient Predicate\<close>
36 definition
37   "Quotient3 R Abs Rep \<longleftrightarrow>
38      (\<forall>a. Abs (Rep a) = a) \<and> (\<forall>a. R (Rep a) (Rep a)) \<and>
39      (\<forall>r s. R r s \<longleftrightarrow> R r r \<and> R s s \<and> Abs r = Abs s)"
41 lemma Quotient3I:
42   assumes "\<And>a. Abs (Rep a) = a"
43     and "\<And>a. R (Rep a) (Rep a)"
44     and "\<And>r s. R r s \<longleftrightarrow> R r r \<and> R s s \<and> Abs r = Abs s"
45   shows "Quotient3 R Abs Rep"
46   using assms unfolding Quotient3_def by blast
48 context
49   fixes R Abs Rep
50   assumes a: "Quotient3 R Abs Rep"
51 begin
53 lemma Quotient3_abs_rep:
54   "Abs (Rep a) = a"
55   using a
56   unfolding Quotient3_def
57   by simp
59 lemma Quotient3_rep_reflp:
60   "R (Rep a) (Rep a)"
61   using a
62   unfolding Quotient3_def
63   by blast
65 lemma Quotient3_rel:
66   "R r r \<and> R s s \<and> Abs r = Abs s \<longleftrightarrow> R r s" \<comment> \<open>orientation does not loop on rewriting\<close>
67   using a
68   unfolding Quotient3_def
69   by blast
71 lemma Quotient3_refl1:
72   "R r s \<Longrightarrow> R r r"
73   using a unfolding Quotient3_def
74   by fast
76 lemma Quotient3_refl2:
77   "R r s \<Longrightarrow> R s s"
78   using a unfolding Quotient3_def
79   by fast
81 lemma Quotient3_rel_rep:
82   "R (Rep a) (Rep b) \<longleftrightarrow> a = b"
83   using a
84   unfolding Quotient3_def
85   by metis
87 lemma Quotient3_rep_abs:
88   "R r r \<Longrightarrow> R (Rep (Abs r)) r"
89   using a unfolding Quotient3_def
90   by blast
92 lemma Quotient3_rel_abs:
93   "R r s \<Longrightarrow> Abs r = Abs s"
94   using a unfolding Quotient3_def
95   by blast
97 lemma Quotient3_symp:
98   "symp R"
99   using a unfolding Quotient3_def using sympI by metis
101 lemma Quotient3_transp:
102   "transp R"
103   using a unfolding Quotient3_def using transpI by (metis (full_types))
105 lemma Quotient3_part_equivp:
106   "part_equivp R"
107   by (metis Quotient3_rep_reflp Quotient3_symp Quotient3_transp part_equivpI)
109 lemma abs_o_rep:
110   "Abs o Rep = id"
111   unfolding fun_eq_iff
112   by (simp add: Quotient3_abs_rep)
114 lemma equals_rsp:
115   assumes b: "R xa xb" "R ya yb"
116   shows "R xa ya = R xb yb"
117   using b Quotient3_symp Quotient3_transp
118   by (blast elim: sympE transpE)
120 lemma rep_abs_rsp:
121   assumes b: "R x1 x2"
122   shows "R x1 (Rep (Abs x2))"
123   using b Quotient3_rel Quotient3_abs_rep Quotient3_rep_reflp
124   by metis
126 lemma rep_abs_rsp_left:
127   assumes b: "R x1 x2"
128   shows "R (Rep (Abs x1)) x2"
129   using b Quotient3_rel Quotient3_abs_rep Quotient3_rep_reflp
130   by metis
132 end
134 lemma identity_quotient3:
135   "Quotient3 (op =) id id"
136   unfolding Quotient3_def id_def
137   by blast
139 lemma fun_quotient3:
140   assumes q1: "Quotient3 R1 abs1 rep1"
141   and     q2: "Quotient3 R2 abs2 rep2"
142   shows "Quotient3 (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2)"
143 proof -
144   have "\<And>a.(rep1 ---> abs2) ((abs1 ---> rep2) a) = a"
145     using q1 q2 by (simp add: Quotient3_def fun_eq_iff)
146   moreover
147   have "\<And>a.(R1 ===> R2) ((abs1 ---> rep2) a) ((abs1 ---> rep2) a)"
148     by (rule rel_funI)
149       (insert q1 q2 Quotient3_rel_abs [of R1 abs1 rep1] Quotient3_rel_rep [of R2 abs2 rep2],
150         simp (no_asm) add: Quotient3_def, simp)
152   moreover
153   {
154   fix r s
155   have "(R1 ===> R2) r s = ((R1 ===> R2) r r \<and> (R1 ===> R2) s s \<and>
156         (rep1 ---> abs2) r  = (rep1 ---> abs2) s)"
157   proof -
159     have "(R1 ===> R2) r s \<Longrightarrow> (R1 ===> R2) r r" unfolding rel_fun_def
160       using Quotient3_part_equivp[OF q1] Quotient3_part_equivp[OF q2]
161       by (metis (full_types) part_equivp_def)
162     moreover have "(R1 ===> R2) r s \<Longrightarrow> (R1 ===> R2) s s" unfolding rel_fun_def
163       using Quotient3_part_equivp[OF q1] Quotient3_part_equivp[OF q2]
164       by (metis (full_types) part_equivp_def)
165     moreover have "(R1 ===> R2) r s \<Longrightarrow> (rep1 ---> abs2) r  = (rep1 ---> abs2) s"
166       apply(auto simp add: rel_fun_def fun_eq_iff) using q1 q2 unfolding Quotient3_def by metis
167     moreover have "((R1 ===> R2) r r \<and> (R1 ===> R2) s s \<and>
168         (rep1 ---> abs2) r  = (rep1 ---> abs2) s) \<Longrightarrow> (R1 ===> R2) r s"
169       apply(auto simp add: rel_fun_def fun_eq_iff) using q1 q2 unfolding Quotient3_def
170     by (metis map_fun_apply)
172     ultimately show ?thesis by blast
173  qed
174  }
175  ultimately show ?thesis by (intro Quotient3I) (assumption+)
176 qed
178 lemma lambda_prs:
179   assumes q1: "Quotient3 R1 Abs1 Rep1"
180   and     q2: "Quotient3 R2 Abs2 Rep2"
181   shows "(Rep1 ---> Abs2) (\<lambda>x. Rep2 (f (Abs1 x))) = (\<lambda>x. f x)"
182   unfolding fun_eq_iff
183   using Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2]
184   by simp
186 lemma lambda_prs1:
187   assumes q1: "Quotient3 R1 Abs1 Rep1"
188   and     q2: "Quotient3 R2 Abs2 Rep2"
189   shows "(Rep1 ---> Abs2) (\<lambda>x. (Abs1 ---> Rep2) f x) = (\<lambda>x. f x)"
190   unfolding fun_eq_iff
191   using Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2]
192   by simp
194 text\<open>
195   In the following theorem R1 can be instantiated with anything,
196   but we know some of the types of the Rep and Abs functions;
197   so by solving Quotient assumptions we can get a unique R1 that
198   will be provable; which is why we need to use \<open>apply_rsp\<close> and
199   not the primed version\<close>
201 lemma apply_rspQ3:
202   fixes f g::"'a \<Rightarrow> 'c"
203   assumes q: "Quotient3 R1 Abs1 Rep1"
204   and     a: "(R1 ===> R2) f g" "R1 x y"
205   shows "R2 (f x) (g y)"
206   using a by (auto elim: rel_funE)
208 lemma apply_rspQ3'':
209   assumes "Quotient3 R Abs Rep"
210   and "(R ===> S) f f"
211   shows "S (f (Rep x)) (f (Rep x))"
212 proof -
213   from assms(1) have "R (Rep x) (Rep x)" by (rule Quotient3_rep_reflp)
214   then show ?thesis using assms(2) by (auto intro: apply_rsp')
215 qed
217 subsection \<open>lemmas for regularisation of ball and bex\<close>
219 lemma ball_reg_eqv:
220   fixes P :: "'a \<Rightarrow> bool"
221   assumes a: "equivp R"
222   shows "Ball (Respects R) P = (All P)"
223   using a
224   unfolding equivp_def
225   by (auto simp add: in_respects)
227 lemma bex_reg_eqv:
228   fixes P :: "'a \<Rightarrow> bool"
229   assumes a: "equivp R"
230   shows "Bex (Respects R) P = (Ex P)"
231   using a
232   unfolding equivp_def
233   by (auto simp add: in_respects)
235 lemma ball_reg_right:
236   assumes a: "\<And>x. x \<in> R \<Longrightarrow> P x \<longrightarrow> Q x"
237   shows "All P \<longrightarrow> Ball R Q"
238   using a by fast
240 lemma bex_reg_left:
241   assumes a: "\<And>x. x \<in> R \<Longrightarrow> Q x \<longrightarrow> P x"
242   shows "Bex R Q \<longrightarrow> Ex P"
243   using a by fast
245 lemma ball_reg_left:
246   assumes a: "equivp R"
247   shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ball (Respects R) Q \<longrightarrow> All P"
248   using a by (metis equivp_reflp in_respects)
250 lemma bex_reg_right:
251   assumes a: "equivp R"
252   shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ex Q \<longrightarrow> Bex (Respects R) P"
253   using a by (metis equivp_reflp in_respects)
255 lemma ball_reg_eqv_range:
256   fixes P::"'a \<Rightarrow> bool"
257   and x::"'a"
258   assumes a: "equivp R2"
259   shows   "(Ball (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = All (\<lambda>f. P (f x)))"
260   apply(rule iffI)
261   apply(rule allI)
262   apply(drule_tac x="\<lambda>y. f x" in bspec)
263   apply(simp add: in_respects rel_fun_def)
264   apply(rule impI)
265   using a equivp_reflp_symp_transp[of "R2"]
266   apply (auto elim: equivpE reflpE)
267   done
269 lemma bex_reg_eqv_range:
270   assumes a: "equivp R2"
271   shows   "(Bex (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = Ex (\<lambda>f. P (f x)))"
272   apply(auto)
273   apply(rule_tac x="\<lambda>y. f x" in bexI)
274   apply(simp)
275   apply(simp add: Respects_def in_respects rel_fun_def)
276   apply(rule impI)
277   using a equivp_reflp_symp_transp[of "R2"]
278   apply (auto elim: equivpE reflpE)
279   done
281 (* Next four lemmas are unused *)
282 lemma all_reg:
283   assumes a: "!x :: 'a. (P x --> Q x)"
284   and     b: "All P"
285   shows "All Q"
286   using a b by fast
288 lemma ex_reg:
289   assumes a: "!x :: 'a. (P x --> Q x)"
290   and     b: "Ex P"
291   shows "Ex Q"
292   using a b by fast
294 lemma ball_reg:
295   assumes a: "!x :: 'a. (x \<in> R --> P x --> Q x)"
296   and     b: "Ball R P"
297   shows "Ball R Q"
298   using a b by fast
300 lemma bex_reg:
301   assumes a: "!x :: 'a. (x \<in> R --> P x --> Q x)"
302   and     b: "Bex R P"
303   shows "Bex R Q"
304   using a b by fast
307 lemma ball_all_comm:
308   assumes "\<And>y. (\<forall>x\<in>P. A x y) \<longrightarrow> (\<forall>x. B x y)"
309   shows "(\<forall>x\<in>P. \<forall>y. A x y) \<longrightarrow> (\<forall>x. \<forall>y. B x y)"
310   using assms by auto
312 lemma bex_ex_comm:
313   assumes "(\<exists>y. \<exists>x. A x y) \<longrightarrow> (\<exists>y. \<exists>x\<in>P. B x y)"
314   shows "(\<exists>x. \<exists>y. A x y) \<longrightarrow> (\<exists>x\<in>P. \<exists>y. B x y)"
315   using assms by auto
317 subsection \<open>Bounded abstraction\<close>
319 definition
320   Babs :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
321 where
322   "x \<in> p \<Longrightarrow> Babs p m x = m x"
324 lemma babs_rsp:
325   assumes q: "Quotient3 R1 Abs1 Rep1"
326   and     a: "(R1 ===> R2) f g"
327   shows      "(R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)"
328   apply (auto simp add: Babs_def in_respects rel_fun_def)
329   apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")
330   using a apply (simp add: Babs_def rel_fun_def)
331   apply (simp add: in_respects rel_fun_def)
332   using Quotient3_rel[OF q]
333   by metis
335 lemma babs_prs:
336   assumes q1: "Quotient3 R1 Abs1 Rep1"
337   and     q2: "Quotient3 R2 Abs2 Rep2"
338   shows "((Rep1 ---> Abs2) (Babs (Respects R1) ((Abs1 ---> Rep2) f))) = f"
339   apply (rule ext)
340   apply (simp add:)
341   apply (subgoal_tac "Rep1 x \<in> Respects R1")
342   apply (simp add: Babs_def Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2])
343   apply (simp add: in_respects Quotient3_rel_rep[OF q1])
344   done
346 lemma babs_simp:
347   assumes q: "Quotient3 R1 Abs Rep"
348   shows "((R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)) = ((R1 ===> R2) f g)"
349   apply(rule iffI)
350   apply(simp_all only: babs_rsp[OF q])
351   apply(auto simp add: Babs_def rel_fun_def)
352   apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")
353   apply(metis Babs_def)
354   apply (simp add: in_respects)
355   using Quotient3_rel[OF q]
356   by metis
358 (* If a user proves that a particular functional relation
359    is an equivalence this may be useful in regularising *)
360 lemma babs_reg_eqv:
361   shows "equivp R \<Longrightarrow> Babs (Respects R) P = P"
362   by (simp add: fun_eq_iff Babs_def in_respects equivp_reflp)
365 (* 3 lemmas needed for proving repabs_inj *)
366 lemma ball_rsp:
367   assumes a: "(R ===> (op =)) f g"
368   shows "Ball (Respects R) f = Ball (Respects R) g"
369   using a by (auto simp add: Ball_def in_respects elim: rel_funE)
371 lemma bex_rsp:
372   assumes a: "(R ===> (op =)) f g"
373   shows "(Bex (Respects R) f = Bex (Respects R) g)"
374   using a by (auto simp add: Bex_def in_respects elim: rel_funE)
376 lemma bex1_rsp:
377   assumes a: "(R ===> (op =)) f g"
378   shows "Ex1 (\<lambda>x. x \<in> Respects R \<and> f x) = Ex1 (\<lambda>x. x \<in> Respects R \<and> g x)"
379   using a by (auto elim: rel_funE simp add: Ex1_def in_respects)
381 (* 2 lemmas needed for cleaning of quantifiers *)
382 lemma all_prs:
383   assumes a: "Quotient3 R absf repf"
384   shows "Ball (Respects R) ((absf ---> id) f) = All f"
385   using a unfolding Quotient3_def Ball_def in_respects id_apply comp_def map_fun_def
386   by metis
388 lemma ex_prs:
389   assumes a: "Quotient3 R absf repf"
390   shows "Bex (Respects R) ((absf ---> id) f) = Ex f"
391   using a unfolding Quotient3_def Bex_def in_respects id_apply comp_def map_fun_def
392   by metis
394 subsection \<open>\<open>Bex1_rel\<close> quantifier\<close>
396 definition
397   Bex1_rel :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
398 where
399   "Bex1_rel R P \<longleftrightarrow> (\<exists>x \<in> Respects R. P x) \<and> (\<forall>x \<in> Respects R. \<forall>y \<in> Respects R. ((P x \<and> P y) \<longrightarrow> (R x y)))"
401 lemma bex1_rel_aux:
402   "\<lbrakk>\<forall>xa ya. R xa ya \<longrightarrow> x xa = y ya; Bex1_rel R x\<rbrakk> \<Longrightarrow> Bex1_rel R y"
403   unfolding Bex1_rel_def
404   apply (erule conjE)+
405   apply (erule bexE)
406   apply rule
407   apply (rule_tac x="xa" in bexI)
408   apply metis
409   apply metis
410   apply rule+
411   apply (erule_tac x="xaa" in ballE)
412   prefer 2
413   apply (metis)
414   apply (erule_tac x="ya" in ballE)
415   prefer 2
416   apply (metis)
417   apply (metis in_respects)
418   done
420 lemma bex1_rel_aux2:
421   "\<lbrakk>\<forall>xa ya. R xa ya \<longrightarrow> x xa = y ya; Bex1_rel R y\<rbrakk> \<Longrightarrow> Bex1_rel R x"
422   unfolding Bex1_rel_def
423   apply (erule conjE)+
424   apply (erule bexE)
425   apply rule
426   apply (rule_tac x="xa" in bexI)
427   apply metis
428   apply metis
429   apply rule+
430   apply (erule_tac x="xaa" in ballE)
431   prefer 2
432   apply (metis)
433   apply (erule_tac x="ya" in ballE)
434   prefer 2
435   apply (metis)
436   apply (metis in_respects)
437   done
439 lemma bex1_rel_rsp:
440   assumes a: "Quotient3 R absf repf"
441   shows "((R ===> op =) ===> op =) (Bex1_rel R) (Bex1_rel R)"
442   apply (simp add: rel_fun_def)
443   apply clarify
444   apply rule
445   apply (simp_all add: bex1_rel_aux bex1_rel_aux2)
446   apply (erule bex1_rel_aux2)
447   apply assumption
448   done
451 lemma ex1_prs:
452   assumes a: "Quotient3 R absf repf"
453   shows "((absf ---> id) ---> id) (Bex1_rel R) f = Ex1 f"
454 apply (simp add:)
455 apply (subst Bex1_rel_def)
456 apply (subst Bex_def)
457 apply (subst Ex1_def)
458 apply simp
459 apply rule
460  apply (erule conjE)+
461  apply (erule_tac exE)
462  apply (erule conjE)
463  apply (subgoal_tac "\<forall>y. R y y \<longrightarrow> f (absf y) \<longrightarrow> R x y")
464   apply (rule_tac x="absf x" in exI)
465   apply (simp)
466   apply rule+
467   using a unfolding Quotient3_def
468   apply metis
469  apply rule+
470  apply (erule_tac x="x" in ballE)
471   apply (erule_tac x="y" in ballE)
472    apply simp
473   apply (simp add: in_respects)
474  apply (simp add: in_respects)
475 apply (erule_tac exE)
476  apply rule
477  apply (rule_tac x="repf x" in exI)
478  apply (simp only: in_respects)
479   apply rule
480  apply (metis Quotient3_rel_rep[OF a])
481 using a unfolding Quotient3_def apply (simp)
482 apply rule+
483 using a unfolding Quotient3_def in_respects
484 apply metis
485 done
487 lemma bex1_bexeq_reg:
488   shows "(\<exists>!x\<in>Respects R. P x) \<longrightarrow> (Bex1_rel R (\<lambda>x. P x))"
489   by (auto simp add: Ex1_def Bex1_rel_def Bex_def Ball_def in_respects)
491 lemma bex1_bexeq_reg_eqv:
492   assumes a: "equivp R"
493   shows "(\<exists>!x. P x) \<longrightarrow> Bex1_rel R P"
494   using equivp_reflp[OF a]
495   apply (intro impI)
496   apply (elim ex1E)
497   apply (rule mp[OF bex1_bexeq_reg])
498   apply (rule_tac a="x" in ex1I)
499   apply (subst in_respects)
500   apply (rule conjI)
501   apply assumption
502   apply assumption
503   apply clarify
504   apply (erule_tac x="xa" in allE)
505   apply simp
506   done
508 subsection \<open>Various respects and preserve lemmas\<close>
510 lemma quot_rel_rsp:
511   assumes a: "Quotient3 R Abs Rep"
512   shows "(R ===> R ===> op =) R R"
513   apply(rule rel_funI)+
514   apply(rule equals_rsp[OF a])
515   apply(assumption)+
516   done
518 lemma o_prs:
519   assumes q1: "Quotient3 R1 Abs1 Rep1"
520   and     q2: "Quotient3 R2 Abs2 Rep2"
521   and     q3: "Quotient3 R3 Abs3 Rep3"
522   shows "((Abs2 ---> Rep3) ---> (Abs1 ---> Rep2) ---> (Rep1 ---> Abs3)) op \<circ> = op \<circ>"
523   and   "(id ---> (Abs1 ---> id) ---> Rep1 ---> id) op \<circ> = op \<circ>"
524   using Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2] Quotient3_abs_rep[OF q3]
525   by (simp_all add: fun_eq_iff)
527 lemma o_rsp:
528   "((R2 ===> R3) ===> (R1 ===> R2) ===> (R1 ===> R3)) op \<circ> op \<circ>"
529   "(op = ===> (R1 ===> op =) ===> R1 ===> op =) op \<circ> op \<circ>"
530   by (force elim: rel_funE)+
532 lemma cond_prs:
533   assumes a: "Quotient3 R absf repf"
534   shows "absf (if a then repf b else repf c) = (if a then b else c)"
535   using a unfolding Quotient3_def by auto
537 lemma if_prs:
538   assumes q: "Quotient3 R Abs Rep"
539   shows "(id ---> Rep ---> Rep ---> Abs) If = If"
540   using Quotient3_abs_rep[OF q]
541   by (auto simp add: fun_eq_iff)
543 lemma if_rsp:
544   assumes q: "Quotient3 R Abs Rep"
545   shows "(op = ===> R ===> R ===> R) If If"
546   by force
548 lemma let_prs:
549   assumes q1: "Quotient3 R1 Abs1 Rep1"
550   and     q2: "Quotient3 R2 Abs2 Rep2"
551   shows "(Rep2 ---> (Abs2 ---> Rep1) ---> Abs1) Let = Let"
552   using Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2]
553   by (auto simp add: fun_eq_iff)
555 lemma let_rsp:
556   shows "(R1 ===> (R1 ===> R2) ===> R2) Let Let"
557   by (force elim: rel_funE)
559 lemma id_rsp:
560   shows "(R ===> R) id id"
561   by auto
563 lemma id_prs:
564   assumes a: "Quotient3 R Abs Rep"
565   shows "(Rep ---> Abs) id = id"
566   by (simp add: fun_eq_iff Quotient3_abs_rep [OF a])
568 end
570 locale quot_type =
571   fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
572   and   Abs :: "'a set \<Rightarrow> 'b"
573   and   Rep :: "'b \<Rightarrow> 'a set"
574   assumes equivp: "part_equivp R"
575   and     rep_prop: "\<And>y. \<exists>x. R x x \<and> Rep y = Collect (R x)"
576   and     rep_inverse: "\<And>x. Abs (Rep x) = x"
577   and     abs_inverse: "\<And>c. (\<exists>x. ((R x x) \<and> (c = Collect (R x)))) \<Longrightarrow> (Rep (Abs c)) = c"
578   and     rep_inject: "\<And>x y. (Rep x = Rep y) = (x = y)"
579 begin
581 definition
582   abs :: "'a \<Rightarrow> 'b"
583 where
584   "abs x = Abs (Collect (R x))"
586 definition
587   rep :: "'b \<Rightarrow> 'a"
588 where
589   "rep a = (SOME x. x \<in> Rep a)"
591 lemma some_collect:
592   assumes "R r r"
593   shows "R (SOME x. x \<in> Collect (R r)) = R r"
594   apply simp
595   by (metis assms exE_some equivp[simplified part_equivp_def])
597 lemma Quotient:
598   shows "Quotient3 R abs rep"
599   unfolding Quotient3_def abs_def rep_def
600   proof (intro conjI allI)
601     fix a r s
602     show x: "R (SOME x. x \<in> Rep a) (SOME x. x \<in> Rep a)" proof -
603       obtain x where r: "R x x" and rep: "Rep a = Collect (R x)" using rep_prop[of a] by auto
604       have "R (SOME x. x \<in> Rep a) x"  using r rep some_collect by metis
605       then have "R x (SOME x. x \<in> Rep a)" using part_equivp_symp[OF equivp] by fast
606       then show "R (SOME x. x \<in> Rep a) (SOME x. x \<in> Rep a)"
607         using part_equivp_transp[OF equivp] by (metis \<open>R (SOME x. x \<in> Rep a) x\<close>)
608     qed
609     have "Collect (R (SOME x. x \<in> Rep a)) = (Rep a)" by (metis some_collect rep_prop)
610     then show "Abs (Collect (R (SOME x. x \<in> Rep a))) = a" using rep_inverse by auto
611     have "R r r \<Longrightarrow> R s s \<Longrightarrow> Abs (Collect (R r)) = Abs (Collect (R s)) \<longleftrightarrow> R r = R s"
612     proof -
613       assume "R r r" and "R s s"
614       then have "Abs (Collect (R r)) = Abs (Collect (R s)) \<longleftrightarrow> Collect (R r) = Collect (R s)"
615         by (metis abs_inverse)
616       also have "Collect (R r) = Collect (R s) \<longleftrightarrow> (\<lambda>A x. x \<in> A) (Collect (R r)) = (\<lambda>A x. x \<in> A) (Collect (R s))"
617         by rule simp_all
618       finally show "Abs (Collect (R r)) = Abs (Collect (R s)) \<longleftrightarrow> R r = R s" by simp
619     qed
620     then show "R r s \<longleftrightarrow> R r r \<and> R s s \<and> (Abs (Collect (R r)) = Abs (Collect (R s)))"
621       using equivp[simplified part_equivp_def] by metis
622     qed
624 end
626 subsection \<open>Quotient composition\<close>
628 lemma OOO_quotient3:
629   fixes R1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
630   fixes Abs1 :: "'a \<Rightarrow> 'b" and Rep1 :: "'b \<Rightarrow> 'a"
631   fixes Abs2 :: "'b \<Rightarrow> 'c" and Rep2 :: "'c \<Rightarrow> 'b"
632   fixes R2' :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
633   fixes R2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"
634   assumes R1: "Quotient3 R1 Abs1 Rep1"
635   assumes R2: "Quotient3 R2 Abs2 Rep2"
636   assumes Abs1: "\<And>x y. R2' x y \<Longrightarrow> R1 x x \<Longrightarrow> R1 y y \<Longrightarrow> R2 (Abs1 x) (Abs1 y)"
637   assumes Rep1: "\<And>x y. R2 x y \<Longrightarrow> R2' (Rep1 x) (Rep1 y)"
638   shows "Quotient3 (R1 OO R2' OO R1) (Abs2 \<circ> Abs1) (Rep1 \<circ> Rep2)"
639 apply (rule Quotient3I)
640    apply (simp add: o_def Quotient3_abs_rep [OF R2] Quotient3_abs_rep [OF R1])
641   apply simp
642   apply (rule_tac b="Rep1 (Rep2 a)" in relcomppI)
643    apply (rule Quotient3_rep_reflp [OF R1])
644   apply (rule_tac b="Rep1 (Rep2 a)" in relcomppI [rotated])
645    apply (rule Quotient3_rep_reflp [OF R1])
646   apply (rule Rep1)
647   apply (rule Quotient3_rep_reflp [OF R2])
648  apply safe
649     apply (rename_tac x y)
650     apply (drule Abs1)
651       apply (erule Quotient3_refl2 [OF R1])
652      apply (erule Quotient3_refl1 [OF R1])
653     apply (drule Quotient3_refl1 [OF R2], drule Rep1)
654     apply (subgoal_tac "R1 r (Rep1 (Abs1 x))")
655      apply (rule_tac b="Rep1 (Abs1 x)" in relcomppI, assumption)
656      apply (erule relcomppI)
657      apply (erule Quotient3_symp [OF R1, THEN sympD])
658     apply (rule Quotient3_rel[symmetric, OF R1, THEN iffD2])
659     apply (rule conjI, erule Quotient3_refl1 [OF R1])
660     apply (rule conjI, rule Quotient3_rep_reflp [OF R1])
661     apply (subst Quotient3_abs_rep [OF R1])
662     apply (erule Quotient3_rel_abs [OF R1])
663    apply (rename_tac x y)
664    apply (drule Abs1)
665      apply (erule Quotient3_refl2 [OF R1])
666     apply (erule Quotient3_refl1 [OF R1])
667    apply (drule Quotient3_refl2 [OF R2], drule Rep1)
668    apply (subgoal_tac "R1 s (Rep1 (Abs1 y))")
669     apply (rule_tac b="Rep1 (Abs1 y)" in relcomppI, assumption)
670     apply (erule relcomppI)
671     apply (erule Quotient3_symp [OF R1, THEN sympD])
672    apply (rule Quotient3_rel[symmetric, OF R1, THEN iffD2])
673    apply (rule conjI, erule Quotient3_refl2 [OF R1])
674    apply (rule conjI, rule Quotient3_rep_reflp [OF R1])
675    apply (subst Quotient3_abs_rep [OF R1])
676    apply (erule Quotient3_rel_abs [OF R1, THEN sym])
677   apply simp
678   apply (rule Quotient3_rel_abs [OF R2])
679   apply (rule Quotient3_rel_abs [OF R1, THEN ssubst], assumption)
680   apply (rule Quotient3_rel_abs [OF R1, THEN subst], assumption)
681   apply (erule Abs1)
682    apply (erule Quotient3_refl2 [OF R1])
683   apply (erule Quotient3_refl1 [OF R1])
684  apply (rename_tac a b c d)
685  apply simp
686  apply (rule_tac b="Rep1 (Abs1 r)" in relcomppI)
687   apply (rule Quotient3_rel[symmetric, OF R1, THEN iffD2])
688   apply (rule conjI, erule Quotient3_refl1 [OF R1])
689   apply (simp add: Quotient3_abs_rep [OF R1] Quotient3_rep_reflp [OF R1])
690  apply (rule_tac b="Rep1 (Abs1 s)" in relcomppI [rotated])
691   apply (rule Quotient3_rel[symmetric, OF R1, THEN iffD2])
692   apply (simp add: Quotient3_abs_rep [OF R1] Quotient3_rep_reflp [OF R1])
693   apply (erule Quotient3_refl2 [OF R1])
694  apply (rule Rep1)
695  apply (drule Abs1)
696    apply (erule Quotient3_refl2 [OF R1])
697   apply (erule Quotient3_refl1 [OF R1])
698  apply (drule Abs1)
699   apply (erule Quotient3_refl2 [OF R1])
700  apply (erule Quotient3_refl1 [OF R1])
701  apply (drule Quotient3_rel_abs [OF R1])
702  apply (drule Quotient3_rel_abs [OF R1])
703  apply (drule Quotient3_rel_abs [OF R1])
704  apply (drule Quotient3_rel_abs [OF R1])
705  apply simp
706  apply (rule Quotient3_rel[symmetric, OF R2, THEN iffD2])
707  apply simp
708 done
710 lemma OOO_eq_quotient3:
711   fixes R1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
712   fixes Abs1 :: "'a \<Rightarrow> 'b" and Rep1 :: "'b \<Rightarrow> 'a"
713   fixes Abs2 :: "'b \<Rightarrow> 'c" and Rep2 :: "'c \<Rightarrow> 'b"
714   assumes R1: "Quotient3 R1 Abs1 Rep1"
715   assumes R2: "Quotient3 op= Abs2 Rep2"
716   shows "Quotient3 (R1 OOO op=) (Abs2 \<circ> Abs1) (Rep1 \<circ> Rep2)"
717 using assms
718 by (rule OOO_quotient3) auto
720 subsection \<open>Quotient3 to Quotient\<close>
722 lemma Quotient3_to_Quotient:
723 assumes "Quotient3 R Abs Rep"
724 and "T \<equiv> \<lambda>x y. R x x \<and> Abs x = y"
725 shows "Quotient R Abs Rep T"
726 using assms unfolding Quotient3_def by (intro QuotientI) blast+
728 lemma Quotient3_to_Quotient_equivp:
729 assumes q: "Quotient3 R Abs Rep"
730 and T_def: "T \<equiv> \<lambda>x y. Abs x = y"
731 and eR: "equivp R"
732 shows "Quotient R Abs Rep T"
733 proof (intro QuotientI)
734   fix a
735   show "Abs (Rep a) = a" using q by(rule Quotient3_abs_rep)
736 next
737   fix a
738   show "R (Rep a) (Rep a)" using q by(rule Quotient3_rep_reflp)
739 next
740   fix r s
741   show "R r s = (R r r \<and> R s s \<and> Abs r = Abs s)" using q by(rule Quotient3_rel[symmetric])
742 next
743   show "T = (\<lambda>x y. R x x \<and> Abs x = y)" using T_def equivp_reflp[OF eR] by simp
744 qed
746 subsection \<open>ML setup\<close>
748 text \<open>Auxiliary data for the quotient package\<close>
750 named_theorems quot_equiv "equivalence relation theorems"
751   and quot_respect "respectfulness theorems"
752   and quot_preserve "preservation theorems"
753   and id_simps "identity simp rules for maps"
754   and quot_thm "quotient theorems"
755 ML_file "Tools/Quotient/quotient_info.ML"
757 declare [[mapQ3 "fun" = (rel_fun, fun_quotient3)]]
759 lemmas [quot_thm] = fun_quotient3
760 lemmas [quot_respect] = quot_rel_rsp if_rsp o_rsp let_rsp id_rsp
761 lemmas [quot_preserve] = if_prs o_prs let_prs id_prs
762 lemmas [quot_equiv] = identity_equivp
765 text \<open>Lemmas about simplifying id's.\<close>
766 lemmas [id_simps] =
767   id_def[symmetric]
768   map_fun_id
769   id_apply
770   id_o
771   o_id
772   eq_comp_r
773   vimage_id
775 text \<open>Translation functions for the lifting process.\<close>
776 ML_file "Tools/Quotient/quotient_term.ML"
779 text \<open>Definitions of the quotient types.\<close>
780 ML_file "Tools/Quotient/quotient_type.ML"
783 text \<open>Definitions for quotient constants.\<close>
784 ML_file "Tools/Quotient/quotient_def.ML"
787 text \<open>
788   An auxiliary constant for recording some information
789   about the lifted theorem in a tactic.
790 \<close>
791 definition
792   Quot_True :: "'a \<Rightarrow> bool"
793 where
794   "Quot_True x \<longleftrightarrow> True"
796 lemma
797   shows QT_all: "Quot_True (All P) \<Longrightarrow> Quot_True P"
798   and   QT_ex:  "Quot_True (Ex P) \<Longrightarrow> Quot_True P"
799   and   QT_ex1: "Quot_True (Ex1 P) \<Longrightarrow> Quot_True P"
800   and   QT_lam: "Quot_True (\<lambda>x. P x) \<Longrightarrow> (\<And>x. Quot_True (P x))"
801   and   QT_ext: "(\<And>x. Quot_True (a x) \<Longrightarrow> f x = g x) \<Longrightarrow> (Quot_True a \<Longrightarrow> f = g)"
802   by (simp_all add: Quot_True_def ext)
804 lemma QT_imp: "Quot_True a \<equiv> Quot_True b"
805   by (simp add: Quot_True_def)
807 context includes lifting_syntax
808 begin
810 text \<open>Tactics for proving the lifted theorems\<close>
811 ML_file "Tools/Quotient/quotient_tacs.ML"
813 end
815 subsection \<open>Methods / Interface\<close>
817 method_setup lifting =
818   \<open>Attrib.thms >> (fn thms => fn ctxt =>
819        SIMPLE_METHOD' (Quotient_Tacs.lift_tac ctxt [] thms))\<close>
820   \<open>lift theorems to quotient types\<close>
822 method_setup lifting_setup =
823   \<open>Attrib.thm >> (fn thm => fn ctxt =>
824        SIMPLE_METHOD' (Quotient_Tacs.lift_procedure_tac ctxt [] thm))\<close>
825   \<open>set up the three goals for the quotient lifting procedure\<close>
827 method_setup descending =
828   \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.descend_tac ctxt []))\<close>
829   \<open>decend theorems to the raw level\<close>
831 method_setup descending_setup =
832   \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.descend_procedure_tac ctxt []))\<close>
833   \<open>set up the three goals for the decending theorems\<close>
835 method_setup partiality_descending =
836   \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.partiality_descend_tac ctxt []))\<close>
837   \<open>decend theorems to the raw level\<close>
839 method_setup partiality_descending_setup =
840   \<open>Scan.succeed (fn ctxt =>
841        SIMPLE_METHOD' (Quotient_Tacs.partiality_descend_procedure_tac ctxt []))\<close>
842   \<open>set up the three goals for the decending theorems\<close>
844 method_setup regularize =
845   \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.regularize_tac ctxt))\<close>
846   \<open>prove the regularization goals from the quotient lifting procedure\<close>
848 method_setup injection =
849   \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.all_injection_tac ctxt))\<close>
850   \<open>prove the rep/abs injection goals from the quotient lifting procedure\<close>
852 method_setup cleaning =
853   \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.clean_tac ctxt))\<close>
854   \<open>prove the cleaning goals from the quotient lifting procedure\<close>
856 attribute_setup quot_lifted =
857   \<open>Scan.succeed Quotient_Tacs.lifted_attrib\<close>
858   \<open>lift theorems to quotient types\<close>
860 no_notation
861   rel_conj (infixr "OOO" 75)
863 end