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 *)
```
```     4
```
```     5 section \<open>Definition of Quotient Types\<close>
```
```     6
```
```     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
```
```    14
```
```    15 text \<open>
```
```    16   Basic definition for equivalence relations
```
```    17   that are represented by predicates.
```
```    18 \<close>
```
```    19
```
```    20 text \<open>Composition of Relations\<close>
```
```    21
```
```    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"
```
```    26
```
```    27 lemma eq_comp_r:
```
```    28   shows "((op =) OOO R) = R"
```
```    29   by (auto simp add: fun_eq_iff)
```
```    30
```
```    31 context includes lifting_syntax
```
```    32 begin
```
```    33
```
```    34 subsection \<open>Quotient Predicate\<close>
```
```    35
```
```    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)"
```
```    40
```
```    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
```
```    47
```
```    48 context
```
```    49   fixes R Abs Rep
```
```    50   assumes a: "Quotient3 R Abs Rep"
```
```    51 begin
```
```    52
```
```    53 lemma Quotient3_abs_rep:
```
```    54   "Abs (Rep a) = a"
```
```    55   using a
```
```    56   unfolding Quotient3_def
```
```    57   by simp
```
```    58
```
```    59 lemma Quotient3_rep_reflp:
```
```    60   "R (Rep a) (Rep a)"
```
```    61   using a
```
```    62   unfolding Quotient3_def
```
```    63   by blast
```
```    64
```
```    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
```
```    70
```
```    71 lemma Quotient3_refl1:
```
```    72   "R r s \<Longrightarrow> R r r"
```
```    73   using a unfolding Quotient3_def
```
```    74   by fast
```
```    75
```
```    76 lemma Quotient3_refl2:
```
```    77   "R r s \<Longrightarrow> R s s"
```
```    78   using a unfolding Quotient3_def
```
```    79   by fast
```
```    80
```
```    81 lemma Quotient3_rel_rep:
```
```    82   "R (Rep a) (Rep b) \<longleftrightarrow> a = b"
```
```    83   using a
```
```    84   unfolding Quotient3_def
```
```    85   by metis
```
```    86
```
```    87 lemma Quotient3_rep_abs:
```
```    88   "R r r \<Longrightarrow> R (Rep (Abs r)) r"
```
```    89   using a unfolding Quotient3_def
```
```    90   by blast
```
```    91
```
```    92 lemma Quotient3_rel_abs:
```
```    93   "R r s \<Longrightarrow> Abs r = Abs s"
```
```    94   using a unfolding Quotient3_def
```
```    95   by blast
```
```    96
```
```    97 lemma Quotient3_symp:
```
```    98   "symp R"
```
```    99   using a unfolding Quotient3_def using sympI by metis
```
```   100
```
```   101 lemma Quotient3_transp:
```
```   102   "transp R"
```
```   103   using a unfolding Quotient3_def using transpI by (metis (full_types))
```
```   104
```
```   105 lemma Quotient3_part_equivp:
```
```   106   "part_equivp R"
```
```   107   by (metis Quotient3_rep_reflp Quotient3_symp Quotient3_transp part_equivpI)
```
```   108
```
```   109 lemma abs_o_rep:
```
```   110   "Abs o Rep = id"
```
```   111   unfolding fun_eq_iff
```
```   112   by (simp add: Quotient3_abs_rep)
```
```   113
```
```   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)
```
```   119
```
```   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
```
```   125
```
```   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
```
```   131
```
```   132 end
```
```   133
```
```   134 lemma identity_quotient3:
```
```   135   "Quotient3 (op =) id id"
```
```   136   unfolding Quotient3_def id_def
```
```   137   by blast
```
```   138
```
```   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)
```
```   151
```
```   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 -
```
```   158
```
```   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)
```
```   171
```
```   172     ultimately show ?thesis by blast
```
```   173  qed
```
```   174  }
```
```   175  ultimately show ?thesis by (intro Quotient3I) (assumption+)
```
```   176 qed
```
```   177
```
```   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
```
```   185
```
```   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
```
```   193
```
```   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>
```
```   200
```
```   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)
```
```   207
```
```   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
```
```   216
```
```   217 subsection \<open>lemmas for regularisation of ball and bex\<close>
```
```   218
```
```   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)
```
```   226
```
```   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)
```
```   234
```
```   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
```
```   239
```
```   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
```
```   244
```
```   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)
```
```   249
```
```   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)
```
```   254
```
```   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
```
```   268
```
```   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
```
```   280
```
```   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
```
```   287
```
```   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
```
```   293
```
```   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
```
```   299
```
```   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
```
```   305
```
```   306
```
```   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
```
```   311
```
```   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
```
```   316
```
```   317 subsection \<open>Bounded abstraction\<close>
```
```   318
```
```   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"
```
```   323
```
```   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
```
```   334
```
```   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
```
```   345
```
```   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
```
```   357
```
```   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)
```
```   363
```
```   364
```
```   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)
```
```   370
```
```   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)
```
```   375
```
```   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)
```
```   380
```
```   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
```
```   387
```
```   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
```
```   393
```
```   394 subsection \<open>\<open>Bex1_rel\<close> quantifier\<close>
```
```   395
```
```   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)))"
```
```   400
```
```   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
```
```   419
```
```   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
```
```   438
```
```   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
```
```   449
```
```   450
```
```   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
```
```   486
```
```   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)
```
```   490
```
```   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
```
```   507
```
```   508 subsection \<open>Various respects and preserve lemmas\<close>
```
```   509
```
```   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
```
```   517
```
```   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)
```
```   526
```
```   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)+
```
```   531
```
```   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
```
```   536
```
```   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)
```
```   542
```
```   543 lemma if_rsp:
```
```   544   assumes q: "Quotient3 R Abs Rep"
```
```   545   shows "(op = ===> R ===> R ===> R) If If"
```
```   546   by force
```
```   547
```
```   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)
```
```   554
```
```   555 lemma let_rsp:
```
```   556   shows "(R1 ===> (R1 ===> R2) ===> R2) Let Let"
```
```   557   by (force elim: rel_funE)
```
```   558
```
```   559 lemma id_rsp:
```
```   560   shows "(R ===> R) id id"
```
```   561   by auto
```
```   562
```
```   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])
```
```   567
```
```   568 end
```
```   569
```
```   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
```
```   580
```
```   581 definition
```
```   582   abs :: "'a \<Rightarrow> 'b"
```
```   583 where
```
```   584   "abs x = Abs (Collect (R x))"
```
```   585
```
```   586 definition
```
```   587   rep :: "'b \<Rightarrow> 'a"
```
```   588 where
```
```   589   "rep a = (SOME x. x \<in> Rep a)"
```
```   590
```
```   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])
```
```   596
```
```   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
```
```   623
```
```   624 end
```
```   625
```
```   626 subsection \<open>Quotient composition\<close>
```
```   627
```
```   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
```
```   709
```
```   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
```
```   719
```
```   720 subsection \<open>Quotient3 to Quotient\<close>
```
```   721
```
```   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+
```
```   727
```
```   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
```
```   745
```
```   746 subsection \<open>ML setup\<close>
```
```   747
```
```   748 text \<open>Auxiliary data for the quotient package\<close>
```
```   749
```
```   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"
```
```   756
```
```   757 declare [[mapQ3 "fun" = (rel_fun, fun_quotient3)]]
```
```   758
```
```   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
```
```   763
```
```   764
```
```   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
```
```   774
```
```   775 text \<open>Translation functions for the lifting process.\<close>
```
```   776 ML_file "Tools/Quotient/quotient_term.ML"
```
```   777
```
```   778
```
```   779 text \<open>Definitions of the quotient types.\<close>
```
```   780 ML_file "Tools/Quotient/quotient_type.ML"
```
```   781
```
```   782
```
```   783 text \<open>Definitions for quotient constants.\<close>
```
```   784 ML_file "Tools/Quotient/quotient_def.ML"
```
```   785
```
```   786
```
```   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"
```
```   795
```
```   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)
```
```   803
```
```   804 lemma QT_imp: "Quot_True a \<equiv> Quot_True b"
```
```   805   by (simp add: Quot_True_def)
```
```   806
```
```   807 context includes lifting_syntax
```
```   808 begin
```
```   809
```
```   810 text \<open>Tactics for proving the lifted theorems\<close>
```
```   811 ML_file "Tools/Quotient/quotient_tacs.ML"
```
```   812
```
```   813 end
```
```   814
```
```   815 subsection \<open>Methods / Interface\<close>
```
```   816
```
```   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>
```
```   821
```
```   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>
```
```   826
```
```   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>
```
```   830
```
```   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>
```
```   834
```
```   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>
```
```   838
```
```   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>
```
```   843
```
```   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>
```
```   847
```
```   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>
```
```   851
```
```   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>
```
```   855
```
```   856 attribute_setup quot_lifted =
```
```   857   \<open>Scan.succeed Quotient_Tacs.lifted_attrib\<close>
```
```   858   \<open>lift theorems to quotient types\<close>
```
```   859
```
```   860 no_notation
```
```   861   rel_conj (infixr "OOO" 75)
```
```   862
```
```   863 end
```
```   864
```